Dynamic Substructures, Volume 4: Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics 2020 [1st ed.] 9783030476298, 9783030476304

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Conference Proceedings of the Society for Experimental Mechanics Series

Andreas Linderholt · Matt Allen Walter D’Ambrogio  Editors

Dynamic Substructures, Volume 4 Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics 2020

Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics, Inc., Bethel, CT, USA

The Conference Proceedings of the Society for Experimental Mechanics Series presents early findings and case studies from a wide range of fundamental and applied work across the broad range of fields that comprise Experimental Mechanics. Series volumes follow the principle tracks or focus topics featured in each of the Society’s two annual conferences: IMAC, A Conference and Exposition on Structural Dynamics, and the Society’s Annual Conference & Exposition and will address critical areas of interest to researchers and design engineers working in all areas of Structural Dynamics, Solid Mechanics and Materials Research.

More information about this series at http://www.springer.com/series/8922

Andreas Linderholt • Matt Allen • Walter D’Ambrogio Editors

Dynamic Substructures, Volume 4 Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics 2020

Editors Andreas Linderholt Department of Mechanical Engineering Linnaeus University V¨axj¨o, Sweden

Matt Allen University of Wisconsin – Madison Madison, Wisconsin, USA

Walter D’Ambrogio University of L’Aquila L’Aquila, Italy

ISSN 2191-5644 ISSN 2191-5652 (electronic) Conference Proceedings of the Society for Experimental Mechanics Series ISBN 978-3-030-47629-8 ISBN 978-3-030-47630-4 (eBook) https://doi.org/10.1007/978-3-030-47630-4 © The Society for Experimental Mechanics, Inc. 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Dynamic Substructures represents one of eight volumes of technical papers presented at the 38th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics, and held in Houston, Texas, February 10–13, 2020. The full proceedings also include volumes on Nonlinear Structures and Systems; Dynamics of Civil Structures; Model Validation and Uncertainty Quantification; Special Topics in Structural Dynamics & Experimental Techniques; Rotating Machinery, Optical Methods & Scanning LDV Methods; Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing; and Topics in Modal Analysis & Testing. Each collection presents early findings from experimental and computational investigations on an important area within Structural Dynamics. Coupled structures, or substructuring, is one of these areas. Substructuring is a general paradigm in engineering dynamics where a complicated system is analyzed by considering the dynamic interactions between subcomponents. In numerical simulations, substructuring allows one to reduce the complexity of parts of the system in order to construct a computationally efficient model of the assembled system. A subcomponent model can also be derived experimentally, allowing one to predict the dynamic behavior of an assembly by combining experimentally and/or analytically derived models. This can be advantageous for subcomponents that are expensive or difficult to model analytically. Substructuring can also be used to couple numerical simulation with real-time testing of components. Such approaches are known as hardware-in-the-loop or hybrid testing. Whether experimental or numerical, all substructuring approaches have a common basis, namely the equilibrium of the substructures under the action of the applied and interface forces and the compatibility of displacements at the interfaces of the subcomponents. Experimental substructuring requires special care in the way the measurements are obtained and processed in order to assure that measurement inaccuracies and noise do not invalidate the results. In numerical approaches, the fundamental quest is the efficient computation of reduced order models describing the substructure’s dynamic motion. For hardware-in-the-loop applications, difficulties include the fast computation of the numerical components and the proper sensing and actuation of the hardware component. Recent advances in experimental techniques, sensor/actuator technologies, novel numerical methods, and parallel computing have rekindled interest in substructuring in recent years leading to new insights and improved experimental and analytical techniques. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Växjö, Sweden Madison, WI, USA L’Aquila, Italy

Andreas Linderholt Matt Allen Walter D’Ambrogio

v

Contents

1

Comparison of Feedforward Control Schemes for Real-Time Hybrid Substructuring (RTHS) . . . . . . . . . . . . . . . Christina Insam, Mert Göldeli, Tobias Klotz, and Daniel J. Rixen

1

2

Proposed 12-DOF Shaker Control of BARC Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kevin Napolitano and Melissa Schmidt-Landin

15

3

Mechanical Environment Test Specifications Derived from Equivalent Energy in Fixed Base Modes . . . . . . . . Troy J. Skousen and Randall L. Mayes

27

4

Implementing Experimental Substructuring in Abaqus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Benjamin Moldenhauer, Matt Allen, Daniel Roettgen, and Brian Owens

43

5

Vibration Test Design with Integrated Shaker Electro-Mechanical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ryan Schultz

63

6

Reproducing a Component Field Environment on a Six Degree-of-Freedom Shaker . . . . . . . . . . . . . . . . . . . . . . . . . . . Jelena Paripovic and Randall L. Mayes

73

7

In-Situ Source Characterization for NVH Analysis of the Engine-Transmission Unit . . . . . . . . . . . . . . . . . . . . . . . . . . Ahmed El Mahmoudi, Francesco Trainotti, Keychun Park, and Daniel J. Rixen

79

8

Using Modal Projection Error to Predict Success of a Six Degree of Freedom Shaker Test . . . . . . . . . . . . . . . . . . . . Tyler F. Schoenherr, Janelle K. Lee, and Justin Porter

93

9

On Dynamic Substructuring of Systems with Localised Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Thomas Simpson, Dimitrios Giagopoulos, Vasilis Dertimanis, and Eleni Chatzi

10

Source Characterization for Automotive Applications Using Innovative Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 J. Harvie and D. de Klerk

11

Impact of Junction Properties on the Modal Behavior of Assembled Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Jean-Baptiste Chassang, Adrien Pelat, Frédéric Ablitzer, Laurent Polac, and Charles Pezerat

12

Quantifying Joint Uncertainties for Hybrid System Vibration Testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Nadim A. Bari, Manuel Serrano, Safwat M. Shenouda, Stuart Taylor, John Schultze, and Garrison Flynn

13

Damping Identification and Model Updating of Boundary Conditions for a Cantilever Beam . . . . . . . . . . . . . . . . 139 Nimai Domenico Bibbo and Vikas Arora

14

An Experimental Substructure Test Object: Components Cut Out From a Steel Structure . . . . . . . . . . . . . . . . . . . 149 Andreas Linderholt

15

Frequency Based Model Mixing for Machine Condition Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Samuel Krügel and Daniel J. Rixen

16

Using a Machine Learning Approach for Computational Substructure in Real-Time Hybrid Simulation . . . 163 Elif Ecem Bas, Mohamed A. Moustafa, David Feil-Seifer, and Janelle Blankenburg

vii

viii

Contents

17

On the Stability of a Discrete Convolution with Measured Impulse Response Functions of Mechanical Components in Numerical Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Wolfgang Witteveen, Lukas Koller, and Florian Pichler

18

Development of an Electrodynamic Actuator for an Automatic Modal Impulse Hammer . . . . . . . . . . . . . . . . . . . . . 189 Johannes Maierhofer and Daniel J. Rixen

Chapter 1

Comparison of Feedforward Control Schemes for Real-Time Hybrid Substructuring (RTHS) Christina Insam, Mert Göldeli, Tobias Klotz, and Daniel J. Rixen

Abstract In order to meet the high demands in testing, actuators must be able to follow their desired displacement with high precision. Feedforward control enables high tracking performance of actuators. In combination with feedback controllers, an actuator can follow a prescribed trajectory quickly, stably and robustly under varying conditions. In RealTime Hybrid Substructuring (RTHS), a method where parts can be tested under realistic boundary conditions, high tracking performance of the actuator is vital. It not only increases fidelity of the RTHS test outcome—meaning that the test replicates the environment and boundary conditions of the test specimen well—but it also prevents the RTHS loop from becoming unstable. Hence, research is carried out in the field of control schemes being applied to RTHS systems. In this work, the existing cascaded feedback control of the position controlled Stewart Platform is expanded by three different feedforward control schemes: model-based dynamic feedforward, modeling-free iterative learning control and velocity feedforward. The tracking performances are compared and discussed using a commanded sine trajectory. Results reveal that modeling-free iterative learning control and velocity feedforward outperform model-based dynamic feedforward and follow the desired trajectory with high amplitude and phase accuracy. Velocity feedforward is simple and requires almost no implementation effort. Thus it is recommended for applications with stiff actuators. In contrast, modeling-free iterative learning control is recommended for tasks where the actuator is not stiff compared to the test specimen. As all these feedforward control schemes improve the tracking performance compared to feedback control, the fidelity of the RTHS test will improve using them. Keywords Feedforward control for RTHS · Parallel manipulators · Model-based dynamic feedforward · Modeling-free iterative learning control · Velocity feedforward

1.1 Introduction In Real-Time Hybrid Substructuring (RTHS), mechanical components can be tested with realistic boundary conditions. By realistic boundary conditions we mean that the mechanical component is excited by the same forces and displacements that it will be subject to in future applications. Using RTHS, we investigate whether the test specimen will withstand the loads in operation, dynamically influence the movement of the whole structure as intended and functions correctly. This is achieved by running a co-simulation of the structure surrounding the mechanical component (test specimen) in the future application while testing the test specimen on a test bench. The test specimen is referred to as the experimental part (EXP ) and the co-simulated surrounding structure is referred to as the numerical part (N U M).1 The co-simulation and the test specimen are coupled at their interface points in real-time by a so-called transfer system (T S). It consists of an actuator, a force-torque sensor and a digital signal processor [1]. Current applications of RTHS can be found in civil and mechanical engineering. Applications in civil engineering include testing of buildings under earthquake loads, testing of tall buildings under wind loads and testing of road/rail bridges under wind and wave loads. In mechanical engineering, the method has been applied to common problems in the aerospace and

1 Note,

that instead of naming it parts, one can also find the naming components or substructures in literature.

C. Insam () · M. Göldeli · T. Klotz · D. J. Rixen Chair of Applied Mechanics, Faculty of Mechanical Engineering, Technical University of Munich, Garching, Germany e-mail: [email protected]; [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2021 A. Linderholt et al. (eds.), Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47630-4_1

1

2

C. Insam et al.

Fext

NUM GN

EXP

TS z

−

z

GE−1

Fint

GT S Fint

Fig. 1.1 Coupling in RTHS: The numerical part (N U M, in blue) outputs displacements z that are realized by the transfer system (T S, in orange). The experimental part (N U M, in green) reacts to these displacements by forces Fint

automotive/transportation sector as well as in robotics and manufacturing systems. Olma et al. [2] In the automotive sector, e.g., semiactive Macpherson suspension systems [3] or vehicle axles [2] have been tested using RTHS. Biomechanical analysis of endoprosthesis have been performed by [4] and preliminary studies for testing of prosthetic feet have been carried out by [5]. A commonly used way to couple the RTHS loop is visualized in Fig. 1.1. The numerical part (N U M, transfer function GN , in blue) is simulated for one numerical time integration step. The solution of this time integration step yields the displacement at the interface points z. The displacements are applied to the experimental part (EXP , transfer function GE , in green)2 by the position controlled actuator (the upper part of T S, in orange). The experimental part reacts to the real displacements z3 by restoring forces Fint that are measurable by the force-torque sensor (lower part of T S, in orange) at the  are transferred to the numerical part by the negative sign (action and reaction) and act together interface. These forces Fint with external forces Fext on the numerical part. The next numerical time integration step is performed. The dynamics of the digital signal processor are neglected here.  = F , i.e., G In the case of ideal coupling, the actuator outputs z = z and the force-torque sensor outputs Fint int T S = 1. If both conditions are satisfied, both parts—the numerical and the experimental—are in equilibrium and compatibility is fulfilled. In case compatibility and equilibrium are fulfilled, the RTHS test emulates the dynamic behavior of the overall structure perfectly. By overall structure we denote the virtual and experimental part combined in one mechanical system. If the true dynamic behavior is replicated, it is stated that the RTHS test has high fidelity.  = F can be assumed, the actuator introduces While the dynamics of the force-torque sensor are often negligible and Fint int its own dynamics that are unwanted in RTHS. This is an important aspect that deteriorates the fidelity of the RTHS test. The test can also become unstable if the introduced dynamics and time lag are too large. Over the last two centuries, control schemes that aim at achieving better tracking, i.e., trying to achieve z = z, have been developed. One of the first ideas was called polynomial extrapolation and was developed by [6]. This control scheme alters the input z that is sent to the actuator but does not improve the position control of the actuator itself. Improved control schemes include adaptive controllers [7], sliding mode controllers with adaption layers [8] or model-based controllers [9]. In this contribution, we investigate three different types of feedforward controllers that extend an existing feedback control scheme. In Sect. 1.2, we introduce three different feedforward control schemes. The test bench and the experimental setup are described in Sect. 1.3. In Sect. 1.4, results are displayed and discussed. Lastly, in Sect. 1.5 we summarize the results and give recommendations.

1.2 Feedforward Control Schemes Control can be divided into feedforward and feedback control. Feedforward control calculates the input to the actuator based on the desired trajectory z. The goal is that the actuator follows the desired trajectory as fast and as accurately as possible, while maintaining stability and robustness. If the dynamic behavior of the actuator is perfectly known, one can find a feedforward controller that uses the inverted dynamics and perfect reference tracking can be achieved. Thus, the feedforward controller determines the command input response, i.e., the dynamics with which the actuator follows the desired trajectory. In reality however, there are modeling errors and disturbances. Hence an additional controller, called feedback controller, 2 The

transfer function is written as G−1 E in the figure, as in our definition of the transfer function it receives forces and outputs displacements. achieved values are denoted with a • throughout the whole paper.

3 Real

1 Comparison of Feedforward Control Schemes for Real-Time Hybrid Substructuring (RTHS)

3

GT S

u zF F

FF

GS

Memory ez

z

z

Pz

z

P Iv

z

Fint

itot

id

P Ii

ud

utot

z

i

Fint

Fig. 1.2 The transfer system for RTHS experiments with actuator control and force feedback. The current P-PI-PI cascade to control the Stewart platform with transfer function GS is extended by feedforward control (FF). The signals that are needed for feedforward control are shown for model-based dynamic feedforward control in blue, model-free inversion-based iterative learning control in green and velocity feedforward in orange

needs to be added to deal with such uncertainties. The parameters of the feedback controller determine the dynamic response to disturbances [10]. In this section, we explain how an existing cascaded control scheme (feedback control) can be expanded by three different types of feedforward controllers. Because it is intended to use the actuator within RTHS, Fig. 1.2 visualizes the entire transfer system with transfer function GT S . It comprises the actuator with the existing control loop and the signals that are used for feedforward control as well as the force feedback signal. The dashed line pointing from Fint to the robot representation indicates the actuator’s sensitivity with respect to external forces. This means that the actuator is not infinitely stiff, and depending on external forces, its dynamics change. The phenomenon is called control-structure interaction (CSI) and is further described in [11] for hydraulic actuators. The existing control loop of the Stewart Platform, which is our actuator and is explained in Sect. 1.3, can be seen in black in Fig. 1.2. This Stewart Platform is driven by electric motors. To control electric motors, cascaded control has proved to be successful [12]. The cascaded control consists of three loops, where the inner loop is the current control loop, the middle loop is the velocity control loop and the outer loop is the position control loop. The dynamics of the inner loop are the fastest, while the dynamics of the outer loop are the slowest. The position loop is controlled by a proportional controller Pz and outputs the desired velocity z˙ d . The velocity loop with velocity error z˙ d − z˙ is controlled by a proportional-integral controller P Iv and outputs the desired current. The current loop with current error id − i  is again controlled by a proportional-integral controller P Ii and sends a voltage to the electric motors.

1.2.1 Model-Based Dynamic Feedforward In model-based dynamic feedforward control (MBDC), a model of the plant dynamics is used to generate an appropriate feedforward signal. The block diagram for MBDC is shown in Fig. 1.3. This feedforward controller is an independent control loop, where the controller C˜ in state space controls the model of the plant G˜S . Note that the modeled transfer function for the plant GS is denoted by G˜S . If the model of the actuator is perfect, i.e., G˜S = GS and linear behavior can be assumed, the real actuator response is the same as the actuator response of the plant model in the feedforward controller [10, 13, 14]. In order to set up a state controller that controls the states x, the feedforward control matrices M˜ x and M˜ u are necessary, where the matrix M˜ x outputs the desired states xd . This control loop is like a simulated control loop, where no disturbances occur. The control voltage u˜ that is calculated using the state controller C˜ is used as feedforward command signal and added to the output voltage ud of the P-PI-PI cascade (see blue arrows in Fig. 1.2). This scheme assumes linear behavior of the plant, which is not true for many actuators. It has the advantage that the state controller C˜ in the feedforward controller can be

4

C. Insam et al.

z

∼u

∼ Mu

∼ GS

∼ C ∼ Mx

xd

x

Fig. 1.3 In model-based dynamic feedforward control, a state controller C˜ is designed such that the command input response of the actuator is highly dynamic

chosen with speed and agility because there are no disturbances, such that the command input response is quick [10, 13, 14]. The choice of the state controller C˜ is arbitrary and possible methods are pole placement or a linear quadratic (LQ) control law. For our experiments, we chose the LQ control law, which is also known as the Riccati controller and is referred to as optimal control. It considers the time and oscillation that arises to go from one state to another state and also the effort (output voltage) that it takes to perform the state transition. A cost function containing both properties is set up, which is optimally fulfilled by the LQ control law. The two properties can be weighted depending on the desired control characteristics [15]. Note that the model of the actuator transfer function G˜S needs to be identified in the experimental setup where it will be used. The reason is that its dynamics change depending on the external forces exerted on the actuator (CSI), which are the forces Fint coming from the experimental part in the case of RTHS. Because the goal is to cancel the dynamics of the actuator during operation, the dynamics need to be identified under operating conditions. A related approach, where the inverse of G˜S is used to generate the feedforward signal, has been successfully applied to RTHS by [16–19]. Nevertheless, when inverting the actuator transfer function, additional dynamics need to be added in −1 order to get a realizable inverse transfer function G˜S . MBDC circumvents this problem due to the independent state space control loop.

1.2.2 Model-Free Inversion-Based Iterative Feedforward Control Iterative learning control can be used for systems that perform a task repeatedly, such as scanning mechanisms of micromirrors, rotating discs or printers. From one iteration to the next, this control scheme learns from the error made and tries to reduce it by the iterative learning control law. In RTHS, applications also exist where the experiment can be done several times in a row and iterative learning control can be applicable. Hochrainer and Puhwein [20] applied iterative learning control to RTHS and showed its applicability and usefulness. The idea of iterative learning control was originally proposed by [21] and developed by [22–24] as early as in the 1980s. Over the past three decades, different methods have developed, some using model knowledge about the actuator from system identification, others trying to learn the transfer function through the cycles as well. In this work, we propose an implementation that is based on a model-free method called model-free inversion-based iterative feedforward control (MFIIC). This method was proposed by [25] and further improved by [26]. In MFIIC, the following data are recorded for the j th iteration: the position error signal ez,j = z − zj , the achieved displacement zj and the desired velocity z˙ j , which is the output of the position controller Pz (see the green arrows in Fig. 1.2). As the reference trajectory z is the same in each iteration, we do not need to add the index j . For each time step of the entire cycle j + 1, the feedforward signal is calculated by: z˙ F F,j +1 = z˙ F F,j +

z˙ j · ez,j , zj

(1.1)

where the feedforward command from the previous iteration j is denoted by z˙ F F,j . Thus, the feedforward signal that is added in iteration j + 1 is based on signals from cycle j . The feedforward signal reduces the error, i.e., ez,j +1 < ez,j if

1 Comparison of Feedforward Control Schemes for Real-Time Hybrid Substructuring (RTHS)

5

the algorithm converges. The main idea is to approximate the inverse transfer function of the plant for each time step of the z˙ cycle by a static factor, which is zj in our case. Here, the transfer function of the plant includes the PI-controllers for velocity j

and current (P Ii and P Iv ) and the Stewart Platform (GS ). This is because the feedforward signal is introduced on velocity level, which means that the transfer path that the feedforward signal is being put through includes the PI-controllers and the Stewart Platform. The original idea proposed by [25, 26] is slightly different: they use the feedforward signal z˙ F F to approximate the transfer function. Furthermore, they perform the calculation (1.1) in the frequency domain, meaning that all signals are transferred to the frequency domain by the Fourier transform, the calculation is performed and then the feedforward signal is transferred back to the time domain by the inverse Fourier transform. However, in our application the approach described in (1.1), which is in the time domain, performs better. Note that we could also inject the feedforward signal on the position, current or voltage level. In these respective cases, we would need to adapt (1.1) and use other signals to calculate the static factor that approximates the transfer function of the plant. In MFIIC no model knowledge of the actuator and external forces is necessary. Hence, it could be of great benefit in RTHS. The transfer function that is learned over the cycles represents the transfer function under operating conditions and possible CSI does not need to be considered.

1.2.3 Velocity Feedforward A very simple approach is velocity feedforward (VFF). Instead of only prescribing a position trajectory, its derivative, the velocity, is also prescribed [12]. Therefore, the feedforward block only contains a derivative block. The calculated feedforward signal must be injected on the velocity level, see orange errors in Fig. 1.2. This approach is only applicable if the existing control scheme contains a velocity controller, such as is the case in a P-PI-PI cascaded controller. The other two approaches can also be applied to standard PID-controllers or black box systems, as is the case in industrial robots. The approach of VFF is simple to implement. However, it does not consider CSI and thus is only successfully applicable if the actuator is stiff compared to the environment it is interacting with.

1.3 Experimental Setup As previously mentioned, we use a Stewart Platform as an actuator for our RTHS experiments. In order to make RTHS tests with high fidelity, good tracking performance of the actuator is vital. Hence, we want to improve the tracking performance of this Stewart Platform and extend the existing P-PI-PI cascade for position control by feedforward controllers.

1.3.1 Stewart Platform Stewart Platforms, as visualized in Fig. 1.4, consist of six legs that are variable in length. They belong to the group of parallel manipulators, which possess the property that they are stiff actuators with high dynamics. Our Stewart Platform is driven by six electric motors that are controlled by a cascaded controller. The upper platform has six degrees of freedom, i.e., translations in X, Y and Z and rotations about the respective axes, namely φ, θ , ψ. The existing P-PI-PI cascade for position control is implemented decentralized, i.e., each leg is controlled individually. Coupling between legs is assumed to be negligible and controlled by the feedback controller.

1.3.2 System Identification In order to implement MBDC, which was introduced in Sect. 1.2, system identification of the Stewart Platform needs to be performed, meaning that G˜S needs to be found. As in the existing Stewart Platform, each leg is controlled independently and system identification of each leg is carried out individually as well. In literature, different ways for system identification are

6

C. Insam et al.

Fig. 1.4 Stewart Platform used in the experiments

stated, see e.g., [27]. The main idea is that all frequencies of interest are excited by the input variable and the output response is measured. In our Stewart Platform, the current control loop is implemented on the servo controllers (hardware) and the current signals cannot be accessed. We assume that the dynamics of the current control loop can be neglected as this loop is supposed to be very fast. Hence, the input signal for system identification of the Stewart Platform is itot and the output is z . The transfer function GS,i for each leg i = 1 . . . 6 writes GS,i = GS,i (s) =

Zi (s) Zi (s) ≈ . Utot,i (s) Itot,i (s)

(1.2)

The variables are written in capital letters to clarify that they are in the frequency domain with s = iω. The Laplace variable is denoted by s, but omitted throughout the rest of this paper. Because the Stewart Platform can only operate safely if it is controlled by a feedback controller, we decided to use closed-loop system identification. In closed-loop system identification it is possible to identify the controlled plant without the input of the controller, even though the feedback controller is active. For this, the requirement is that the correlation between the excitation signal and the plant output (zi ) is high, i.e., coherence must be guaranteed (as a rule of thumb larger than 75%). There are different types of excitation signals that can be used, e.g., white noise, pseudo-random binary signal or multi-sine signal. Based on experience, we chose the multi-sine signal for excitation, which is a sum of multiple sine signals. The phases of these sine signals during summation need to be chosen according to the so-called Schroeder phases. The Schroeder phases distribute the sine signals in the time domain and thus spread the input power [27]. The system identification was performed in the frequency range from 0.5 to 100 Hz for each leg. We selected a frequency resolution of 0.5 Hz. In order to inject enough energy for each individual frequency during the experiments, we split the whole frequency range into 10 smaller frequency ranges, namely [0.5, 10], [10.5, 20], . . . , [90.5, 100] Hz. We performed 10 measurements for each leg, where the multi-sine signal in each of the experiments comprised 20 different frequencies. Each measurement was performed for a time span of 200 s. In the post-processing of the measured data, a phase correction needs to be performed. The sampling frequency during measurement was 1000 Hz, i.e., the sampling time was 1 ms. The discrete measuring introduces frequency-dependent phase delay, as during 1 ms the oscillation goes on. For example, for the measurement at 100 Hz, a sampling frequency of 1000 Hz means that a phase of −36◦ is artificially introduced. Hence, the respective phase needs to be subtracted from the measured phase values of GS,i to obtain the real phase values of GS,i . Based on considerations in [28], the dynamic behavior of the Stewart Platform can be represented with PT1-dynamics and an integrator in this frequency range. In the considered frequency range, only dynamics coming from the mechanical system are relevant. The dynamics from the electric system and the controllers are relevant in a higher frequency regime and do not

1 Comparison of Feedforward Control Schemes for Real-Time Hybrid Substructuring (RTHS) Table 1.1 Summary of the transfer functions of the Stewart Platform’s legs i = 1 . . . 6

i=1

7 i=1

i=3

i=4

i=5

i=6

1011 1053.1 1123.5 780.9 988.1 1070.9 ˜ S,i G s·(s+122.9) s·(s+55.5) s·(s+54.7) s·(s+55.3) s·(s+66.2) s·(s+59)

have an influence up to 100 Hz. Therefore, we fitted a transfer function G˜S of the following form to the measured data of the real plant GS : ˜ S,i (s) = G

k · ωc k 1 = , with ωc = . s · (T s + 1) s · (s + ωc ) T

(1.3)

This dynamic behavior can also be interpreted as a mass with damping and no stiffness. This is quite illustrative, as the torques coming from the electric motors need to push the mass of the Stewart Platform’s legs and upper platform, which possess damping (friction, compliance, . . . ) but almost no stiffness. We identified the transfer functions listed in Table 1.1 for each leg. The cutoff frequencies, i.e., the frequency above which the amplitudes fall with a slope of −40 dB/decade is at frequencies of approximately 8–10 Hz. The amplitude falls below zero above 2–3 Hz, which implies that the input signal is mainly attenuated above and the dynamic limits of the Stewart Platform are in this range. The task of the feedforward controllers is to optimally use and even increase the dynamic range of the actuator to its mechanical limits such as velocity limit and maximum motor voltage.

1.3.3 Benchmark Problem To compare the tracking performance of the three different feedforward control schemes in combination with the existing P-PI-PI cascade, the following benchmark problem is used: A sine trajectory with frequency f and amplitude A around the initial position t0 should be followed. Hence, the desired position trajectory td is of the form ⎞ ⎞ ⎛ td,X 0 ⎟ ⎜t ⎟ ⎜ 0 ⎟ ⎜ d,Z ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ ⎜ td,Z ⎟ ⎜ A · sin(2πf t) ⎟ td (t) = ⎜ ⎟ , with ⎟ = t0 + ⎜ ⎟ ⎜ td,φ ⎟ ⎜ 0 ⎟ ⎟ ⎜ ⎜ ⎠ ⎝ td,θ ⎠ ⎝ 0 0 td,ψ ⎛

 1 t ∈ 0, f

(1.4)

and the Stewart Platform is at its initial position t0 for t < 0 and t > f1 . This trajectory command only prescribes the trajectory in Stewart Platform’s Z-direction, which is the vertical direction, and the other translations (X and Y ) and rotations (φ, θ , ψ) are set to 0. This trajectory command td is transformed by inverse kinematics, which gives the kinematic relationship between the upper platform and the leg lengths, to the desired leg lengths zi (compare to Fig. 1.2), for each leg i = 1 . . . 6. Between two successive sine trajectories there was a pause of 1 s at position t0 .

1.3.4 Parameters Setting for the Experiments To keep the results comparable, all experiments have been conducted several times with the same conditions, such as the same maximum actuator velocity of 40 mm s and the same constants in the P-PI-PI cascaded feedback controller. For MBDC, a scaling factor αMBDC was introduced to scale the feedforward signal. This is necessary because of the nonlinear transfer behavior of the Stewart Platform, i.e., the amplification from input voltage to leg displacement depends on the amplitude of the input voltage. In the tuning process, a continuous sine command with f = 2 Hz was used and the factor αMBDC was tuned for each leg to its optimum. The obtained values lie in the range of (1.8, 3.8). Likewise, a scaling factor αMF I I C was used for MFIIC. Here, the scaling factor determines the convergence rate and the achievable minimum error with this scheme. If the scaling factor is chosen too small, the convergence is slow and the remaining error is large compared to the

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optimum scaling factor. However, if the scaling factor is too high, the algorithm diverges. In the tuning process for MFIIC, experience on different trajectories showed that a scaling value of αMF I I C = 50 yields good results. The MFIIC algorithm was trained for 26 iterations. The identified transfer functions stated in Sect. 1.3.2 were used for MBDC and the state controller C˜ was set up using the LQ control law. The state controller contains only two states x for each leg, namely the leg length and the leg velocities ˜ S,i . The weighting in the cost function of the LQ control law was chosen such that the coming from the modeled plant G control effort was almost neglected: The costs for slow and oscillating behavior were weighted 1012 (106 ) times higher than the cost for the voltage output for the leg length (leg velocity). For MBDC we use a low-pass filter with cutoff frequency at 100 Hz in the implementation. This dampens unwanted highly dynamic effects. This is necessary because system identification was carried out up to 100 Hz and therefore the dynamics above this frequency cannot be predicted correctly.

1.4 Results and Discussion This section presents the results of the experiments described in Sect. 1.3. The desired position trajectory that was carried out by the Stewart Platform varied in amplitude A and frequency f . These were applied with each of the three feedforward control schemes (see Sect. 1.2) to investigate their tracking performance. Experiments were conducted with amplitudes of A ∈ {1, 3, 5} mm and frequencies of f ∈ {0.25, 0.5, 1, 2} Hz. We also performed experiments with higher frequencies, but all methods could only achieve maximum amplitudes of 0.5 mm, therefore we do not show the results. This implies that mechanical limits like maximum velocity and motor voltages were achieved.

1.4.1 Convergence of MFIIC

10 –1

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First, we needed to check the convergence of our implemented MFIIC algorithm. The results are shown in Fig. 1.5. In Fig. 1.5a, the desired trajectory and the achieved trajectory over the time interval f1 are shown for the first, the 12th and the 26th (which was the last) iteration. As the iterations increase, the achieved displacement comes closer to the desired displacement, especially during the dynamic scenarios. However, when the direction of motion changes and the velocity approaches zero, it takes some time for the Stewart Platform to overcome the friction, thus the error is large at these points. The plot on the right, Fig. 1.5b, shows the root-mean-square (RMS) error between the desired reference signal td and the true

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Fig. 1.5 The error between the desired signal td,Z and the achieved displacement tZ in Z-direction using MFIIC as the feedforward control scheme is shown for f = 1 Hz and A = 1 mm. (a) Desired trajectory (black, dashed line) and achieved displacement over time in the 1st (blue), the 12th (green) and the 26th (orange) iteration. (b) RMS error over the number of iterations

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achieved displacement t in Z-direction (td,Z and tZ ). The RMS error was calculated by 1000

f

 2 f · td,Z − tZ , rms = 1000

(1.5)

i=1

because the sampling frequency was 1000 Hz and thus the number of measured samples over the time period f1 was 1000 f . The error in the first iteration corresponds to the error of the P-PI-PI cascade, as learning begins in the second iteration based on the error signal from the first iteration. We see that the algorithm converges and the error decreases over time. Our maximum number of iterations was 26 and convergence was in the order of ≈ 12 , where j counts the iterations. This means j

that doubling the number of iterations reduces the error to 14 in the considered range. However, there will be a convergence limit due to mechanical limits and maximum motor voltages. Nevertheless, the potential of MFIIC and its convergence looks promising. Using MFIIC, the learned feedforward signal can be saved and used in future experiments. Hence, the results shown in the following sections take the feedforward signal of MFIIC in its converged state, i.e., from the 26th iteration.

1.4.2 Comparison of the Feedforward Control Schemes We first show the representative course over time for A = 1 mm and f = 0.5 Hz for the existing P-PI-PI cascade and the three different feedforward control schemes in Fig. 1.6. It can be seen that the achieved displacement tZ comes closer to the desired displacement td,Z for all feedforward control schemes. In Fig. 1.6b, the error between desired and achieved displacement, i.e., tdZ − tZ , is visualized. Here, it is also obvious that all feedforward control schemes reduce the error. The MFIIC and VFF especially reduce the magnitude of the error to approximately 10–20%. The largest magnitude of the error occurs at the turning points of the trajectory, where the velocity is zero and static friction has large influence. In order to make general statements about the tracking performance of all methods for different trajectories, we investigate the results for all amplitudes A ∈ {1, 3, 5} mm and frequencies f ∈ {0.25, 0.5, 1, 2} Hz in a synchronization subspace plot (SSP). The SSP plot can be used to analyze the tracking performance for sine trajectories. The desired displacement td,Z is plotted over the achieved displacement tZ . In the case of ideal tracking, a slope with an incline of 45◦ results. If the incline is lower it means that amplitude overshoot has occurred, if the incline is higher, it means that the Stewart Platform undershot. An ellipse forms in the case of phase errors: In the case of a phase lag, the ellipse is passed through clockwise, and in the case of phase lead the ellipse is passed through counterclockwise. Figure 1.7 visualizes the SSP plots for the measured amplitudes and frequencies. All amplitudes A ∈ {1, 3, 5} mm were only investigated for frequencies f ∈ {0.25, 0.5} Hz. For frequency f = 1 Hz we investigated amplitudes A ∈ {1, 3} mm 0.2 Desired P-PI-PI MBDC MFIIC VFF

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Fig. 1.6 The plots show the results for the P-PI-PI cascade (in black, dashed line), the MBDC (in blue, solid line), the MFIIC (in green, solid line) and the VFF (in orange, solid line) for the desired trajectory with f = 0.5 Hz and A = 1 mm. (a) The desired (td,Z ) and achieved displacement tZ in Z-direction. (b) The error signal between desired and achieved displacement over time

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A = 1 mm

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tZ in mm Fig. 1.7 Synchronization subspace plots are shown for amplitudes A ∈ {1, 3, 5} mm and frequencies f ∈ {0.25, 0.5, 1, 2} Hz. The results of the P-PI-PI cascade (in black, dashed), the MBDC (in blue, solid), the MFIIC (in green, solid) and the VFF (in orange, solid) are shown

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and for f = 2 Hz we only investigated A = 1 mm. The reason is that our velocity limit was set to 60 mm s throughout the experiments to prevent the Stewart Platform from damage and execution of these experiments included such high velocities. From the SSP plots, it is clear that all feedforward schemes (MBDC, MFIIC, VFF) significantly improve the phase error compared to the P-PI-PI cascade alone. The P-PI-PI cascade suffers from phase lag (the ellipse in the SSP plot are passed through clockwise) and for higher frequencies (f ∈ {1, 2} Hz) from amplitude errors (undershoot) as well. For f = 0.25 Hz, the MBDC improves the tracking performance only slightly. For f = 1 Hz, the amplitude error is larger than for the P-PI-PI cascade, though the phase error decreases. For f = 2 Hz, MBDC outperforms the P-PI-PI cascade, because the optimum value αMBDC for MBDC was tuned for each leg at the excitation frequency of f = 2 Hz. MBDC suffers from amplitude undershoot, especially for higher frequencies, which can be observed from its incline >45◦ in the SSP plot. MFIIC and VFF perform comparably. They reduce the phase and amplitude error the most for all frequencies and amplitudes. The tracking performance achieved is almost ideal (straight line, 45◦ incline) in the investigated frequency range.

1.4.3 Coupling Between Directions The Stewart Platform consists of six legs and each of the legs is actuated and controlled independently and coupling effects are neglected. Thus, we need to investigate how large the effect of coupling is and how much the other directions are influenced by a desired movement in Stewart Platform’s Z-direction. We representatively show the movement during one sine trajectory with A = 1 mm and f = 0.5 Hz in Fig. 1.8 for the X-direction, i.e., tX . The results are similar for the Y direction as well as the rotations φ, θ and ψ. We can see that for all methods, the movement induced in X-direction is of the same order of magnitude and the maximum value is 0.04 mm. As the desired amplitude is A = 1 mm, i.e., the entire movement in Z-direction is 2 mm, this induced movement is 0 − λk +  λk ≤ 0

(4.10)

This is then put into diagonal matrix, as in Eq. (4.11), where the only nonzero entries will be negative eigenvalues with a flipped sign and a numerically small amount added. ˆ = Λ

\

λ\



(4.11)

The modeshapes corresponding to the negative eigenvalues, φk , are then collected into a matrix φMD , as in Eq. (4.12).   φ MD = φ k · · · φ n

(4.12)

The mass matrix is then made positive definite in Eq. (4.13), which essentially adds to the negative eigenvalues to make them equal to . ˆD=M ˆ D + φT Λ ˆ φM M MD D

(4.13)

Ideally, very little mass would need to be added to make the matrix positive definite. This amount can be quantified by computing the ratio of the matrix norms as given in Eq. (4.14). nratio =

! ! T !φM

! ˆ φM ! Λ ! D ! !ˆ ! !MD !

D !

(4.14)

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The modal properties of the decoupled system, D and ωn, D , are then found by transforming the eigensolution of the corrected mass and stiffness matrices as in Eq. (4.15), where EX, F and TS, F are matrices of the subsystem basis modeshapes at all DOF. If this continues to yield negative eigenvalues, they must simply be removed from the model.   ˆ D − λCON,D M ˆ D CON,D = [0] K

 ; D =

 " EX,F 0 LD CON,D ; ωn,D = λCON,D 0 TS,F

(4.15)

The decoupled model may then be imported into Abaqus as a set of modal DOF with unit mass and a stiffness equal to the square of the natural frequency. To constrain these to the analytical subcomponent in Abaqus, the compatibility condition is formed as shown in Eq. (4.16), where xFE corresponds to a set of physical DOF in the Abaqus FEM to constrain to.  †TS D

I





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 = [0]

(4.16)

The constraint equations are then as given in Eq. (4.17), which can be implemented in Abaqus as a set of linear equation multipoint constraints.   BABQ = †TS D †TS

(4.17)

4.3 Implementation The general process for implementing the TS method in Abaqus as presented in the previous section is detailed in the following steps. This procedure will require the use of MATLAB and Abaqus and assumes that work has already been completed in designing the subcomponents, measuring experimental data, and creating the necessary FEMs. This data is then imported into MATLAB where constraint DOF are identified and the TS is decoupled from the experimental subsystem. The resultant decoupled model is then imported into Abaqus and constrained to the analytical subsystem.

4.3.1 Gather Subsystem Data and Import into MATLAB The substructuring process begins with designing the experimental subsystem, the TS, and the analytical subsystem; several guidelines for creating a suitable TS are given in [5]. From a modal test of the experimental component and a FEM of the TS, the linear natural frequencies and modeshapes of both may be determined; damping is optional. This data is then imported into MATLAB, along with node locations and DOF directions for every subsystem, including the analytical component.

4.3.2 Identify Constraint DOF In order to define the substructuring constraint equations, the DOF to be constrained between each subsystem must be determined. After ensuring that all components are defined within the same global coordinate system, or performing any transformations required to make them so, this can be done by first locating node pairs between the components and then matching any common DOF directions at each pairing. Typically, the constraints are defined in terms of the experimental subsystem DOF, as in pairings are only found between it and the other subsystems. However, since the TS and analytical component are FEMs, it is possible to define the constraints at every viable node pair between them. While this may yield improved results, it can also drastically increase the number of terms in each constraint equation. The Abaqus documentation advises against this, as implementing long equations can severely degrade solver performance [6].

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4.3.3 Decouple the TS from the Experimental Subsystem To implement Eqs. (4.8)–(4.15) in MATLAB, one must first determine a suitable modal basis for the experimental subsystem and the TS. This is arguably the most critical step in the substructuring process as the resulting model can be incredibly sensitive to what modes are used in its creation. In general, a suitable modal basis is one that adequately spans the system dynamics in the frequency range of interest and, for the TS, it is critical that it does not contain linearly dependent modeshapes when partitioned to the test DOF in order to avoid numerical ill-conditioning in its pseudo inverse. Several techniques and metrics that assist in selecting appropriate modes are given in [4, 8].

4.3.4 Form Constraint Equations for Use in Abaqus The decoupled model computed in the previous step is then coupled to the Abaqus analytical model as in Eq. (4.16). The constraint equations, shown in matrix form in Eq. (4.17), are computed with the pseudo inverse of the TS modal basis and the modeshapes of the decoupled model. These can be implemented in Abaqus as linear equation multipoint constraints as each column corresponds to a constraint DOF, each row an equation, and each entry a term in that equation. However, Abaqus enforces linear equations by eliminating the first term DOF from the model, meaning that it cannot appear in any other equation [6]. Since Eq. (4.17) likely produces a fully populated matrix, these constraint equations must be altered, such as putting them into reduced row echelon form (RREF), which removes the pivot, or leading nonzero term in that row, from all other rows. If the built-in MATLAB function for RREF is used, it may use a numerically zero term as a pivot and produce arbitrarily large terms in the other rows. This destroys the conditioning of those equations, which can cause errors in Abaqus later. To avoid this, the columns of the constraint matrix should be reordered such that the leading diagonal does not contain any arbitrarily small terms.

4.3.5 Write Auxiliary Abaqus Input File To complete the substructuring procedure, the decoupled model and constraint equations are imported into Abaqus by writing a text file to be processed with the input file for the analytical subsystem FEM. This auxiliary input file defines the decoupled model in terms of its modal DOF by generating an appropriate quantity of nodes, assigning each a point mass element with unit magnitude, attaching a grounded spring element with stiffness equal to the square of each natural frequency, and constraining out the other five DOF to make them single DOF oscillators. The constraint equations are then written as linear equations relating the motion of these decoupled model DOF to the physical FEM DOF. If the constraint equation matrix was reordered in the previous step, care must be taken to ensure that each equation coefficient is associated with the correct DOF. A command must then be placed in the analytical subsystem input file instructing Abaqus to include the auxiliary input file when processing that analysis. To enable a painless application of this process, Appendix A contains a MATLAB script that, given the necessary data, writes the auxiliary input file in the correct Abaqus syntax.

4.4 Numerical Case Study To evaluate the basic usability and practicality of implementing the proposed substructuring method, a numerical case study was conducted. To facilitate ease of application and interpretation, a system of simple 2D beam FEMs were chosen for the subsystems as presented in Fig. 4.2. The experimental subsystem is a 1 [m] long cantilever, the TS is a 0.2 [m] long free-free beam, and the analytical subsystem is a 1 [m] long free-free beam. A FEM of each was constructed in Abaqus from 500 B21 two node linear beam elements with a 0.01 [m] square cross section, 70 [GPa] elastic modulus, Poisson’s Ratio of 0.33, and 2700 [kg/m3 ] density. The experimental subsystem modeshapes were truncated to only two translational DOF every 0.1 [m] along the beam to simulate a sparse sensor placement. The subsystems were assembled as shown in Fig. 4.2, where the vertical and horizontal DOF of the last three nodes at the tip of the cantilever were constrained to the six matching DOF on the TS and analytical subsystem. In this scenario, the tip of the cantilever is effectively removed and replaced by a longer

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Fig. 4.2 Subsystem layout for the simple beam numerical case study

Fig. 4.3 Self MAC of TS and EX modes at the constraint DOF showing where each begins to alias

beam section, yielding in a longer cantilever. Thus, the truth data for this setup is a 1.8 [m] cantilever beam composed of 900 elements with the properties stated above. After importing the subsystem data into MATLAB and locating the indices of the DOF to constrain in each, the modal bases for the TS and experimental subsystem are formed. In this case study, the first step was to select TS modes based on examining their modal assurance criterion (MAC) at the constraint DOF. A MAC is essentially a measure of how similar a pair of modes are and was used as an indication of linear independence; a low MAC value signifies linear independence and a high value represents linear dependence. The left side of Fig. 4.3 depicts a plot of the self MAC of the TS modes, where large off-diagonal terms indicate dependent modes. This shows that modes 5, 6, 7, 9, 10, 12 and beyond are aliased versions of modes 1, 2, and 4, which are the rigid body modes (RBM) and first flexural mode in the transverse direction. Modes 3, 8, and 11 are the RBM and first two flexural modes in the axial direction; the aliased copies of these are beyond the 12 modes shown in the plot. As there are three constraint DOF in each of these directions, transverse and axial, only three modes in each direction can be clearly resolved. Thus, the MAC indicates that modes 1, 2, 3, 4, 8, and 11 of the TS are suitable candidates for its modal basis. For the experimental subsystem, the standard substructuring procedure would be to use all available modes in its modal basis. However, in this case study, it was observed that as more modes were included in the experimental subsystem, a larger

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correction was required to eliminate negative eigenvalues from the decoupled model, as computed in Eqs. (4.8)–(4.14). This negatively impacted the accuracy of the subsequent assembly of the corrected decoupled model and the analytical subsystem, with larger corrections yielding a worse final substructuring result. Therefore, in this work modes were removed from both the TS and experimental subsystem modal bases in order to minimize the amount of correction needed for the decoupled model. This resulted in the experimental subsystem containing modes 1–11, 15, 20, and 24, and mode 11 being removed from the TS, which reduced the mass matrix correction ratio, Eq. (4.14), from 0.96 to 0.03. A possible explanation for this is given on the right side of Fig. 4.3, which illustrates the self MAC of the experimental subsystem modes at the constraint DOF. This shows that the transverse bending modes begin to significantly alias past mode 11; modes 8, 15, 20, and 24 are axial modes and show a similar trend beyond what is shown in the plot. Thus, contrary to standard substructuring practice, it seems that aliased experimental subsystem modes decrease substructuring accuracy in the separated approach proposed in this work. With the TS and experimental subsystem modal bases determined above, the resultant decoupled model is constrained to the analytical subsystem as in Eqs. (4.16) and (4.17). This is completed in Abaqus by generating the text file to be included with the input file for the analytical subsystem FEM. Appendix B contains a truncated version of this auxiliary input file as written by the script in Appendix A as an example of what should be expected when this proposed procedure is properly applied. In the following results, the truth data is compared to three different methods for completing the substructuring computations. ‘Control’ is the result if the standard substructuring procedure presented in Eqs. (4.1)–(4.7) is implemented in MATLAB, ‘MATLAB’ is the result if the analytical subsystem is imported into MATLAB and constrained to the decoupled model in modal coordinates, and ‘Abaqus’ is the result of implementing the method proposed in this work where the decoupled model is imported into Abaqus and constrained to the FEM there. A MAC between the modeshapes from the MATLAB and Abaqus results, displayed in Fig. 4.4, shows that the two methods yield essentially identical models, as the MAC matrix is nearly symmetric. Natural frequencies are also effectively equivalent, as there is less than 0.003% difference in the prediction from each. A similar result is found when comparing the Abaqus and Control results, where the modeshapes are indistinguishable and natural frequencies estimates are within 0.4%. To easily compare the substructuring predictions to the truth model, frequency response functions (FRFs) at the beam tip were generated from the modeshapes, natural frequencies, and an assumed constant damping ratio of 0.5%. The beam motion in the transverse direction is given in Fig. 4.5, and in the axial direction in Fig. 4.6. These show that, while the results from each of the three substructuring methods are functionally identical, they only agree with the truth data up to 2000 [Hz] in the transverse direction and 9000 [Hz] in the axial direction. This is due to the experimental subsystem modal basis containing transverse modes up to 2050 [Hz] and axial modes up to 8910 [Hz]; there is no dynamic information past these from which the substructuring methods can accurately predict resultant behavior.

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As a final check, the MAC between the truth model and Abaqus substructuring results is shown in Fig. 4.7. The diagonal up to mode 18 shows perfect agreement, with small off diagonal terms simply due to subsequent beam modes being inherently somewhat similar. While the results seem to diverge past mode 18, this is what was previously observed in the beam tip FRFs; the transverse modes are accurate up to 2000 [Hz], which consists of the first 18 modes. Past this point in the MAC, several mode pairs corresponding to the axial modes show high correlation which is also in agreement with the axial FRF. This numerical case study demonstrates that the TS method can be implemented in Abaqus to very great effect; the excellent level of agreement between all substructuring methods and the truth model is very encouraging. Assuming the standard substructuring method yields the optimal model from the selected modal bases, the proposed separated method

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approaches the same result by minimizing the amount of correction needed in the decoupled model. Although the assembled model is accurate throughout the entire frequency range of the experimental modal basis, this is largely limited by the selected constraint DOF. If more DOF were included in the interface region of the experimental subsystem, higher order modes could be included in the TS modal basis without aliasing, allowing the TS to be more flexible and accommodate higher frequency experimental subsystem dynamics at the interface. This shows that, while the strengths of the TS method arise from satisfying the subsystem constraints via the TS, this also limits its applicable dynamic range to what can be projected through the TS modal basis.

4.5 Experimental Test Case The proposed Abaqus substructuring procedure was then applied to an experimental test case where it may be compared to the other methods in a more realistic setting. The subsystems used in this test case are shown in Fig. 4.8, where the experimental subsystem is an electrodynamic shaker with a three sided half cube adapter mounted to the armature, the TS is the half cube adapter, and the analytical subsystem is the half cube with an attached cantilever beam. See [9] for a more detailed description of these subsystems. The resulting substructured model is a prediction of how the dynamics of the shaker and adapter will change due to the addition of the beam. In general, the analytical subsystem could be a FEM of any proposed test setup and the substructured model can be used to identify any problematic dynamics of the system prior to a test taking place. This was previously implemented in [9] with the standard TS method, where all subcomponent data was imported into MATLAB and the constraints were applied simultaneously. As in the numerical test case, that method, deemed the ‘Control’ result, will here be compared against coupling the decoupled model to the analytical subsystem in either MATLAB or Abaqus, again referred to as the ‘MATLAB’ and ‘Abaqus’ results. The subsystems are constrained at every measurement point utilized in characterizing the experimental subsystem. The location and direction of these 51 DOF, shown in Fig. 4.9, were chosen with Effective Independence [10] to maximize the linear independence of the first 35 modes of the TS. This optimizes the test setup while providing the best possible array of modes from which the TS modal basis can be selected. Thus, by minimizing the amount of correction needed in decoupling the TS from the experimental subsystem, the first 22 modes of the TS, up to 3400 [Hz], and the first 26 modes of the experimental subsystem, up to 3875 [Hz], were selected for the modal bases of each. For the decoupled model, these require a correction of 0.22% to the stiffness matrix and no correction to the mass matrix.

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Fig. 4.8 Experimental test case subsystems

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Fig. 4.9 Location and direction of experimental test case constraint DOF

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The results of the different substructuring methods are presented as FRF and MAC plots in Fig. 4.10. As damping was not computed in any of the substructuring computations, a damping ratio of 0.1% was assumed when generating the FRFs at each measurement point, which are then averaged into the single curve shown. Each row of plots corresponds to an implementation of substructuring, with the top showing the ‘Control’ result, the middle the ‘MATLAB’ result, and the bottom displays the ‘Abaqus’ model. In each of these FRF plots, the substructured model is shown relative to a set of truth data measured from the physical system with a cantilever beam mounted on the half cube and shaker. To the right of the FRFs, MAC plots between the substructured and truth model modeshapes show how closely the two correspond. In the top FRF, which compares the standard substructuring method to the truth model, one can see that the substructured prediction closely follows the truth data throughout the frequency range shown; up to 2000 [Hz]. The additional peak in the substructured model at 1500 [Hz] is caused by a torsional mode of the cantilever beam; this was not observable in the truth data due to the placement of sensors on the beam. The MAC plot generally agrees with this, as there is strong correlation for most of the truth modes. The seemingly duplicated truth modes, such as 4, 5, and 6, and 7 and 8, are the result of internal shaker modes producing identical motion when viewed from the measurement points on the half cube. The substructure modes under 100 [Hz] do not correspond to any truth modes in the MAC plot due to these not having been significant enough to curve fit in the measured truth data. When moving down to the middle row of plots, which show the ‘MATLAB’ results where the decoupling and coupling steps are separated, it can be seen that the resultant model is essentially the same as the ‘Control’ model. The only significant difference is observed at low frequency, where the ‘MATLAB’ result has increased the frequency of the modes. This is likely a consequence of the correction that was applied to the stiffness matrix of the decoupled model; a small amount of stiffness was added to correct for negative eigenvalues, which in turn increased the frequency of the modes. Comparing these two to the bottom row of plots, which present the model from Abaqus, there is good agreement under 800 [Hz], with the same increase in frequency with the modes under 100 [Hz] that was seen in the ‘MATLAB’ model. Between 800 [Hz] and 1500 [Hz], the Abaqus result seems to be shifted to a lower frequency, or simply not well predicted at all. From 1500 [Hz] to 1800 [Hz], the Abaqus model seems to have generated a set of modes that are not present in either of the other models, or in the truth data. However, the Abaqus result does then correctly predict the large peak near 1900 [Hz]. The MAC plot agrees with these results, in that the modes between 100 [Hz] and 800 [Hz] are correct relative to the truth model, and while those between 800 [Hz] and 1800 [Hz] vaguely resemble the trend shown by the other models, there is worse correlation to the truth model. As an additional, more focused, evaluation of the substructuring methods, each was used to generate a drive point FRF at the tip of the cantilever beam in its primary bending direction. As before, a damping ratio of 0.1% was assumed while computing the FRF. The results from the truth test and all three substructured models are given below in Fig. 4.11. Unlike the results previously discussed above, the beam tip FRFs from all three substructuring models agree very well with each other and to the truth data, up through 2500 [Hz]; the component of interest that is being substructured to the experimental subsystem, the cantilever, is accurately modeled by Abaqus. The key strength of performing the substructuring computations in Abaqus is the ability to utilize its extensive capabilities for post-processing FEM results. In Abaqus, one can easily visualize the resultant substructured model in the form of the entire analytical subsystem finite element mesh, whereas attempting to do the same in MATLAB would require complex custom scripts and be much more computationally expensive. As a demonstration of the how these results can be viewed in Abaqus, Fig. 4.12 shows an image from Abaqus/CAE of the 700 [Hz] mode of the substructured model. Instead of inferring how the structure is responding based on reconstructed FRFs or unwieldy plots in MATLAB, one can easily observe the dynamic motion of the model directly via the deformed FEM mesh. For this mode, the free corners of the two half cube side walls are flexing while the base remains nearly fixed, as it is now modeling a bolted connection to the shaker armature. Also, since the second cantilever beam mode immediately precedes this at 650 [Hz], the beam is still approximately responding in that shape. Gleaning the same information from the MATLAB model would likely be significantly more cumbersome. This experimental test case presents a difficult challenge; the TS and experimental subsystem are both quite dynamically complex. When combined in MATLAB in the standard way, the resulting model agrees quite well with the associated truth data. The same can be said about the results when the substructuring computations are separated into two distinct steps in MATLAB. However, when the decoupled model is imported into Abaqus and assembled to the analytical subsystem there, the accuracy of the results for the half cube seem to decrease, while at the same time, the dynamics of the cantilever beam are perfectly represented. The degraded results may, in some way, be due to the analytical subsystem FEM in Abaqus implicitly containing every mode of the model. To assemble the decoupled model and the analytical subsystem in MATLAB, only the first 100 modes of the latter are exported from Abaqus to represent it. If the mathematical computations to assemble the components are effectively the same whether done in MATLAB or Abaqus, as was the situation in the numerical case study, the only difference between the two substructuring methods is how many modes are contained in the analytical subsystem.

4 Implementing Experimental Substructuring in Abaqus

Fig. 4.10 Comparison of half cube substructuring methods; reconstructed FRFs and MACs

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Magnitude [m/s2 / N]

10 4

Truth Control MATLAB Abaqus

10 3

10 2

10 1

10 0 0

500

1000

1500

2000

2500

Frequency [Hz] Fig. 4.11 Reconstructed FRFs at the beam tip

Fig. 4.12 Substructuring result at 700 Hz visualized in Abaqus/CAE

This conclusion seems contrary to the generally accepted substructuring guideline that the resultant model should become more accurate as modes are added to the analytical subsystem.

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4.6 Conclusions and Future Work A standard approach for implementing the TS method of experimental substructuring is to import and assemble all the subsystems in MATLAB. The preceding sections presented a procedure for performing this in Abaqus, producing a model that can leverage the full post-processing capabilities of a FEM. This was accomplished by separating the usually simultaneously computed substructuring constraints into two separate actions; decouple the TS from the experimental subsystem in MATLAB and then couple the result to the analytical subsystem FEM in Abaqus. This approach introduces some intricacies, as it is likely that a correction must be applied to the decoupled model before it can be imported into Abaqus. When applied to a numerical case study with simple beam models, the proposed method was found to yield excellent results that agreed with those from the standard substructuring method as well as a set of truth data. However, contrary to standard TS method substructuring practices, this was achieved through limiting what modes are included in the TS and experimental subsystem modal bases, such that the correction to the decoupled model was minimized. While this resulted in near perfect agreement between all models, it likely limited their applicable frequency range by restricting the modal bases. The process was then implemented on a complex experimental test case in which a FEM of a shaker adapter and potential test setup, a half cube with an attached cantilever beam, was attached to an experimental model of the half cube adapter mounted to an electrodynamic shaker. In this case, while the standard and separated approaches in MATLAB agree very well, the results from Abaqus are noticeably less accurate for the half cube, but still very accurate for the cantilever beam. The cause of this discrepancy is possibly due to the Abaqus FEM implicitly containing every mode of the half cube and beam system, as only the first 100 modes were imported into MATLAB for assembly there. This is an additional instance of observed results seemingly contradicting common substructuring practices. This work shows that the TS method of experimental substructuring can be implemented in Abaqus by importing an experimental model with decoupled TS from MATLAB and constraining it to the native Abaqus FEM. This process is limited by the need to correct the decoupled model before importing it into Abaqus, reducing the TS and experimental subsystem modal bases to minimize this correction, and the possibility that several normal substructuring practices seem to be detrimental to the results. Thus, while some accuracy penalty is likely to be observed, experimental substructuring can be completed within Abaqus, allowing for expanded accessibility and much more practical results. Future work includes establishing metrics for assessing the quality of the decoupled experimental subsystem and exploring other options for importing that model into Abaqus, such as generating it as a Hurty/Craig-Bampton model that may be imported as a standard super-element into Abaqus. Sandia Acknowledgement Notice: This manuscript has been authored by National Technology and Engineering Solutions of Sandia, LLC. under Contract No. DE-NA0003525 with the U.S. Department of Energy/National Nuclear Security Administration. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.

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Appendix A: MATLAB Function to Generate Auxillary Abaqus Input File function ABQ_Constr(filename,constr,fn,FEM_DOF,Num_FEM_Nodes,Num_FEM_Eles) % Writes a text file for performing experimental substructuring in Abaqus. % Generates nodes, constrains DOF 2-6, assigns unit mass, attaches a % grounded spring with stiffness equal to the square of the decoupled model % natural frequencies, then writes out constraint equations as Linear % Equation Multipoint Constraints. % % INPUTS % filename - name of written text file % constr - Constraint Equations in RREF % fn - Natural Frequencies of decoupled model [Hz] % FEM_DOF - DOF in Abaqus FEM to constrain to % Num_FEM_Nodes - Number of nodes in the Abaqus FEM % Num_FEM_Eles - Number of elements in the Abaqus FEM % % OUTPUTS % Writes a text file to the current directory % % In the .inp file for the Abaqus FEM, include the following line: % *INCLUDE,INPUT=filename.inp % Both .inp files must be in the same directory. % % EXAMPLE OF PROPER INPUTS % filename = 'example.inp'; % constr = [1 0 .5 .5 ; 0 1 .5 .5]; % fn = [1 2]; % FEM_DOF = [1.2 2.2]; % 2nd DOF in nodes 1 and 2 % Num_FEM_Nodes = 2; % Num_FEM_Eles = 1;

% Open Text file to write Abaqus Auxiliary Input File fid = fopen(filename,'wt');

% Create the Modal Nodes % Number of Modal Nodes Num_Q_DOF = size(fn(:),1); % Modal Nodes Q_nodes = [((Num_FEM_Nodes+1):(Num_FEM_Nodes+Num_Q_DOF)).',zeros(Num_Q_DOF,3)]; % Reset local coord system to global fprintf(fid,'*SYSTEM\n'); % Initialize node creation fprintf(fid,'*NODE\n'); % Write nodes fprintf(fid,'%i,\t%.8g,\t%.8g,\t%.8g\n',Q_nodes');

% Assign Boundary Conditions % Initialize Node Set fprintf(fid,'*NSET, NSET=Modal_DOF, GENERATE\n'); % Create Node Set fprintf(fid,'%i, %i, 1\n',Q_nodes(1,1),Q_nodes(end,1)); % Initialize Boundary definitions

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fprintf(fid,'*BOUNDARY\n'); % Constrain DOF 2-6, DOF 1 will be used for spring stiffness fprintf(fid,'Modal_DOF, 2, 6\n');

% Assign Point Mass % Modal Mass Elements Q_m_eles = [((Num_FEM_Eles+1):(Num_FEM_Eles+Num_Q_DOF)).' Q_nodes(1:end,1)]; % Initialize Point Mass Element Set fprintf(fid,'*Element, type=MASS, elset=Modal_Masses\n'); % Write Mass Elements fprintf(fid,'%i,\t%.8g\n',Q_m_eles'); % Initialize Point Mass Element Set fprintf(fid,'*Mass, elset=Modal_Masses\n1,\n');

% Assign Grounded Spring % Modal Stiffness Elements Q_k_eles = [((Q_m_eles(end,1)+1):(Q_m_eles(end,1)+Num_Q_DOF)).',Q_nodes(1:end,1)]; for ii=1:Num_Q_DOF fprintf(fid,'*Spring, elset=K%i\n',ii); % Initialize Spring Element fprintf(fid,'1\n'); % Spring DOF fprintf(fid,'%.19G\n',(fn(ii)*2*pi)^2); % Spring Stiffness (Maximum 20 characters for a data line) fprintf(fid,'*Element, type=Spring1, elset=K%i\n',ii); % Grounded Type Spring fprintf(fid,'%i, %i\n',Q_k_eles(ii,1),Q_k_eles(ii,2)); % Spring Element and Node end

% Write Linear Equation Constraints % DOF to constrain together constr_DOF = [Q_nodes(1:end,1)+.1 ; FEM_DOF(:)]; % Seperating into Node and Direction constr_inds = [floor(constr_DOF) round(10*mod(constr_DOF,1))]; for jj=1:size(constr,1) % Indices of Nonzero Equation Terms term_ind = find(constr(jj,:)); % Initialize Equation Field fprintf(fid,'*EQUATION\n'); % Number of Terms in the Equation fprintf(fid,'%i\n',length(term_ind)); % Write Equation Terms fprintf(fid,'%i,\t%i,\t%.19G,\n',[constr_inds(term_ind(1:end),1),... constr_inds(term_ind(1:end),2),constr(jj,term_ind(1:end))']'); end

% Close text file fclose(fid); end

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Truncated Auxillary Abaqus Input File for Beam Case Study * SYSTEM * NODE 5001, 0, 0, 0 ... 5014, 0, 0, 0 * NSET, NSET=Modal_DOF, GENERATE 5001, 5014, 1 * BOUNDARY Modal_DOF, 2, 6 * Element, type=MASS, elset=Modal_Masses 5001, 5001 ... 5014, 5014 * Mass, elset=Modal_Masses 1, * Spring, elset=K1 1 242212591293406.625 *Element, type=Spring1, elset=K1 5015, 5001 ... * Spring, elset=K14 1 99889894.06944584846 * Element, type=Spring1, elset=K14 5028, 5014 * EQUATION 11 5001, 1, 1, 5006, 1, -0.000264860351395888741, 5007, 1, -0.0001025417654326699714, 5008, 1, -0.0001571114822033104104, 5009, 1, -0.0002081731213936832193, 5010, 1, 0.0002870250227079269531, 5011, 1, 0.0003337867566874962231, 5012, 1, -0.0003829544046016372617, 1, 2, -0.0001271774056828475232, 51, 2, 4.163977166513179649E-05, 101, 2, -8.145943855598651294E-06, ... * EQUATION 6 5005, 1, 1, 5013, 1, -0.9985942660576510033, 5014, 1, 0.9774124704222225679, 1, 1, -0.389357596507627024, 51, 1, -0.106769751316806466, 101, 1, 0.1758183602094033071,

References 1. de Klerk, D., et al.: General framework for dynamic substructuring: history, review, and classification of techniques. AIAA J. 46, 1169–1181 (2008) 2. Allen, M.S., Mayes, R.L., Bergman, E.J.: Experimental modal substructuring to couple and uncouple substructures with flexible fixtures and multi-point connections. J. Sound Vib. 329, 4891–4906 (2010) 3. Dynamic Substructuring Wiki, [Online]. Available: http://substructure.engr.wisc.edu 4. Allen, M.S., Mayes, R.L.: Comparison of FRF and Modal methods for combining experimental and analytical substructures. In Proceedings of the 25th IMAC, Orlando, FL, 2007 5. Mayes, R.L., Arviso, M.: Design studies for the transmission simulator method of experimental dynamic substructuring. In International Seminar on Modal Analysis, Lueven, Belgium, 2010

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6. Simulia, D.S.: Abaqus Analysis User’s Guide, 2016. [Online]. Available: http://130.149.89.49:2080/v2016/books/usb/default.htm. [Accessed 30 Setember 2019] 7. Mayes, R.L., Allen, M.S., Kammer, D.C.: Correcting indefinite mass matrices due to substructure uncoupling. J. Sound Vib. 5856–5866 (2013) 8. Allen, M.S., Kammer, D.C., Mayes, R.L.: Metrics for diagnosing negative mass and stiffness when uncoupling experimental and analytical substructures. J. Sound Vib. 331, 5435–5448 (2012) 9. Moldenhauer, B., Allen, M.S., DeLima, W.J., Dodgen, E.: Using hybrid modal substructuring with a complex transmission simulator to model an electrodynamic shaker. In Proceedings of the 37th IMAC, Orlando, FL, 2019 10. Kammer, D.C.: Sensor placement for on-orbit modal identification and correlation of large space structures. J. Guid. Control. Dyn. 14(2), 251–259 (1991)

Chapter 5

Vibration Test Design with Integrated Shaker Electro-Mechanical Models Ryan Schultz

Abstract Design of multi-shaker tests relies on locating shakers on the structure such that the desired vibration response is obtained within the shaker force, acceleration, voltage, and current requirements. While shaker electro-mechanical models can be used to relate the shaker force and acceleration to voltage and current requirements, they need to be integrated with a structural dynamics model of the device under test. This connection of a shaker to a structure is a substructuring problem, with the structure representing one component and the shaker representing a second component. Here, frequency based substructuring is used to connect a shaker electro-mechanical model to a model of device under test. This provides a straightforward methodology for predicting shaker requirements given a target vibration response in a multi-shaker test. Predictions of the coupled shaker-structure model yield the shaker force, acceleration, voltage and current requirements which can be compared with the shaker capabilities to choose optimal shaker locations. Keywords Electro-mechanical model · Electrodynamic shaker · Substructuring · Frequency based substructuring

5.1 Introduction Utilizing models for test design is particularly important for multi-shaker vibration tests because the location of shakers can have a large effect on test performance and shaker force, voltage, and current requirements. Additionally, determining good shaker locations before starting test setup can save time and rework. A model which integrates the dynamics of the device under test (DUT) with the coupled electro-mechanical behavior of the shakers is therefore very useful. The connection of two or more dynamic components, in this case a DUT and several shakers, represents a substructuring problem. As shaker electro-mechanical models can conveniently be modeled in terms of impedance of a coupled system, it is natural to use frequency based substructuring (FBS). In this work, a lumped parameter shaker electro-mechanical model is presented which represents the main dynamics of interest for most modal shakers. This lumped parameter model is used to generate an impedance model of an electrodynamic shaker which can then be integrated with a model or measurements of an example DUT. Theory on FBS is presented and then utilized to demonstrate how an impedance model of a shaker can be integrated with an impedance model of a DUT to predict the effects of the shaker on the DUT and the electrical response of the shaker.

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. R. Schultz () Structural Dynamics Department, Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2021 A. Linderholt et al. (eds.), Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47630-4_5

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5.2 Theory This section describes some of the frequency based substructuring and shaker electro-mechanical modeling which was used in this work. This is not intended to be an in-depth examination into either topic. Instead, the following sections provide some of the concepts, terminology, and techniques used to model shakers and connect shaker models to DUT models in this work.

5.2.1 Frequency Based Substructuring This section describes the basic theory behind the FBS used in this work. Far more depth can be found in many papers, including Allen et al. [1] and the course notes by Avitabile [2]. Consider a two-component system consisting of Component A and Component B, as shown in Fig. 5.1. Component B has an input applied at the XB,i degree of freedom (DOF), the outputs are at the o DOF and there is a single connection at the c DOF. The frequency response functions (FRFs) needed for FBS are the between the connections, inputs, and outputs as shown in Eq. (5.1) where the FRF matrices are defined to have outputs on the row dimension, inputs on the column dimension, and frequency lines on the page (third) dimension. The drive point, or connection-connection, FRFs are also needed. With these various FRFs, the FRFs of the coupled system comprised

x6 M6 K56

C

x5 M5

xA,o

f5 K45

A xA,c xB,c

K34

B

xB,o

x3

M3

B fin

x4

M4

f3 xB,i

K23 M2 A f1

K12

x1 M1

K01 Fig. 5.1 Example two component (left) and three component (right) systems

x2

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of A and B together can be formed using Eq. (5.2). Thus, the response at the output on component A due to an input on component B can be determined. HA,oc = HA (o, c, :) HA,cc = HA (c, c, :) HB,oi = HB (o, i, :) HB,oc = HB (o, c, :) HB,ci = HB (c, i, :) HB,cc = HB (c, c, :)

(5.1)

 −1 Hsys,A = HA,oc HA,cc + HB,cc HB,ci  −1 Hsys,B = HB,oi − HB,oc HA,cc + HB,cc HB,ci

(5.2)

The FBS equations can be written more generally using matrix forms. Take for example a more complicated, three component system as shown in Fig. 5.1, with Components A, B, and C are connected together. First, the component matrices, HA , HB , and HC need to be formed. These can have input and output DOFs at any DOFs of interest but must at least have outputs and inputs at the connection DOFs for the drive point FRFs needed in the FBS procedure. The coupled system FRF matrix is given by Eq. (5.3), where Hu is the uncoupled matrix shown in Eq. (5.4) [1].

−1 Bo Hu Hsys = Hu − Hu Bi T Bo Hu Bi T ⎤ HA 0 0 Hu = ⎣ 0 HB 0 ⎦ 0 0 HC

(5.3)

⎡

(5.4)

Boolean constraint matrices, Bi, and Bo, connect the components at specified input and output DOFs, respectively. For example, in the three-component example in Fig. 5.1 the connection between A and B is at DOF 3 and the connection between B and C is at DOF 5. Say there are inputs at DOF 1, 3 and 5 and outputs at all six DOF, 1 through 6. In that case, the Boolean matrix at the input DOF is as shown in Eq. (5.5), with five columns, one for each of the input DOFs of the uncoupled system. Inputs 3 and 5 have two columns each in the Boolean matrix because they belong to two components. The Boolean matrix at the output DOF is similarly formed, Eq. (5.6). There are eight columns, with the two connection DOF, 3 and 5 each having two columns because they belong to two components. The two rows indicate the two connections between the components. 

0 −1 1 0 0 Bi = 0 0 0 −1 1 



0 0 −1 1 0 0 0 0 Bo = 0 0 0 0 0 −1 1 0

(5.5)  (5.6)

5.3 Shaker Electro-Mechanical Model An electrodynamic shaker consists of a coil-wrapped armature which translates inside a shaker body when current is applied, creating an electric field which interacts with a set of magnets in the shaker body. A simplified diagram in Fig. 5.2 shows the main components in a typical modal shaker. Leaf springs or flexures connect the armature to the shaker body and constrain the armature motion to be primarily along the axis of the shaker body. A stinger is mounted to one end of the armature and connects the shaker to the DUT. A lumped parameter model can be used to represent the shaker mechanical and electrical components, as shown in Fig. 5.3 [3, 4]. The interaction between the electrical current in the coil of wire and the magnets in the shaker body couple the mechanical and electrical components. Here, three DOF are used for the mechanical component, with M1 representing the armature mass, M2 representing the body mass, and M3 representing the

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Stinger

Body

Armature

Magnet

Coil

Flexure

Fig. 5.2 Diagram of the internal components of a typical electro-dynamic shaker

Fig. 5.3 Lumped parameter models of the mechanical (left) and electrical (right) components of the shaker model

mass of the stinger and attached components such as load cells. The spring K12 represents the flexures and K13 represents the stinger stiffness. For the electrical component, a resistor (Re ) and an inductor (Le ) in series represent the coil of wire. The coupling is indicated by the coil force on M1 in the mechanical component and the back electro-motive force (EMF) in the electrical component. This coupling is determined by the value of the force factor (BL) in this lumped parameter model. The mechanical and electrical equations of motion can be written in matrix form, with the mass, damping and stiffness matrices as shown in Eqs. (5.7), (5.8), and (5.9). The impedance of the shaker model can then be determined with a direct frequency response solution using these matrices. The electrical impedance, voltage over current, indicates the presence of two modes in this system. The low frequency mode is the suspension mode, or the armature and stinger masses bouncing relative to the body mass via the flexure stiffness. The second, higher frequency mode is the stinger mass bouncing relative to the armature mass via the stinger stiffness. Some parameters for this model can be obtained from a specification sheet or measured directly. Other parameters, like the stiffness, damping and force factor (BL) have to be inferred from a measurement. The stiffness and damping can be determined by fitting the frequency and amplitude of the peaks in the impedance near the two modes. The force factor affects the entire impedance curve, including the two peaks, but it too can be tuned to match a measurement.

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Table 5.1 Shaker model parameter values

Parameter M1 M2 M3 K12 K13 C12 C13 Re Le BL

Value 0.44 15 0.033 1.10E+05 9.63E+06 9.6 0.42 4 6e−4 −j5e−4 36

Unit kg kg kg N/m N/m N/(m/s) N/(m/s) Ohm Henry –

Table 5.1 gives parameters tuned from a measurement of the impedance of the 100 pound-force shaker of interest in this work. ⎡

M1 ⎢ 0 [Mshk ] = ⎢ ⎣ 0 0 ⎡

0 M2 0 0

0 0 M3 0

⎤ 0 0⎥ ⎥ 0⎦ 0

(C12 + C13 ) −C12 −C13 ⎢ − C12 C12 0 [Cshk ] = ⎢ ⎣ − C13 0 C13 BL −BL 0

(5.7) ⎤ 0 0⎥ ⎥ 0⎦ Le

⎤ (K12 + K13 ) −K12 −K13 −BL ⎢ K12 0 BL ⎥ − K12 ⎥ [Kshk ] = ⎢ ⎣ 0 K13 0 ⎦ − K13 0 0 0 Re

(5.8)

⎡

(5.9)

5.4 Example of Substructuring a Shaker to a Dynamic System With a model of the shaker, the next step is to demonstrate how FBS can be used to integrate the shaker model with a structural dynamics model of some DUT. Here, a simple two-DOF component (A) is attached to the stinger of the shaker model (B) via a spring and damper as shown in Fig. 5.4, with the shaker and DUT sharing DOF x3 . For the FBS process, FRFs are needed at input, output, and connection DOFs. Here, the input DOF is the applied electrical voltage on the shaker component and the outputs are the response of the DUT DOFs, x4 and x5 . The coupling of these two component models is done using a single constraint in the Boolean matrix linking DOF x3 in both components. Then, the coupled system FRF is obtained with Eq. (5.3). Results of the coupled system are shown in terms of the electrical impedance (Fig. 5.5) and the acceleration/voltage and acceleration/stinger force FRFs of the DUT DOFs (Fig. 5.6). The impedance plot compares the FBS model with a model of the coupled system formed as a single set of matrices (no substructuring). This indicates that this FBS model matches the actual system dynamics. The FRFs in Fig. 5.6 show how this coupled model can be used to determine acceleration/voltage FRFs which could then be used to assess the electrical input requirements to achieve some DUT response.

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x5 M5 C45

K45

x4

M4 A

K34 M3

B

C34 x3 C13

K13

x1 M1

K12

fcoil

C12 x2

M2

100

80

Full System FBS

60

40

20

0 200

400

600

Frequency [Hz]

800

1000

Imag. Electrical Impedance, Z=E/I [Ohm]

Real Electrical Impedance, Z=E/I [Ohm]

Fig. 5.4 Example two-DOF component attached to the shaker model 50

0

-50 200

400

600

800

1000

Frequency [Hz]

Fig. 5.5 Real (left) and imaginary (right) parts of the coupled-system electrical impedance using a full system model and a FBS model

5.5 Example of Substructuring a Shaker Model to a Measured System With a shaker model, and the ability to connect it to structural dynamics models of a DUT with FBS, the next step is to apply these techniques to actual hardware. A test article, shown in Fig. 5.7, was driven with a single shaker attached to the base. The shaker model parameters were calibrated to a separate measurement of this shaker. Then, the force and drive point acceleration of the DUT were measured to obtain the DUT FRF matrix. In addition to the acceleration/force FRFs, the force, voltage, current power spectral densities (PSDs), as well as the acceleration/voltage, acceleration/current, force/voltage and force/current FRFs were captured in this measurement. To determine if the substructured electro-mechanical model is predictive, the force, voltage, and current of the measurement were predicted using the various FRFs of the substructured model. Specifically, the drive point acceleration was used to

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10 -4

FRF [(m/s2)/N]

FRF [(m/s2)/V]

a4/e a5/e

10 -6

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10 -2

-8

10 -10

200

400

600

800

Frequency [Hz]

1000

10 -4

200

400

600

800

1000

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Fig. 5.6 Acceleration per input voltage (left) and acceleration per input force (right) from the coupled model

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determine the required input force, voltage, and current using the FRFs of the substructured model. Figure 5.8 shows the voltage and current PSDs predicted by the model compared with those measured in the test. The model does a good job overall up to about 1500 Hz. There is a peak in the model predictions at 1500 Hz which is not present in the test and is likely the result of some model form error, for example this simple three DOF shaker model may not capture all the dynamics in the actual shaker hardware. The electrical impedance prediction, shown in Fig. 5.9, agrees very well with the measurement. Similarly, the FRFs of force and acceleration due to voltage or current inputs, shown in Figs. 5.10 and 5.11, show good agreement between the model and the test. While the substructured model predictions are not perfect, they do match the measurements well overall and are suitable for the task at hand – assessing electrical requirements for various shaker configurations.

5.6 Using the Shaker Electro-Mechanical Model to Choose Shaker Locations The ultimate use for this modeling approach is in shaker location selection. Ideally, multiple shakers would be attached to a device under test such that the desired response is achieved at multiple response locations while being well within the force, voltage, and current requirements of the shaker. By coupling the shaker electromechanical model with measured or simulated acceleration/force FRFs of the DUT, the force, voltage, and current can be assessed.

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A finite element (FE) model of the DUT was developed and used to generate the DUT FRFs, relating the response at 33 output DOFs to inputs at 37 candidate input DOFs. These input locations represent locations where shakers could be realistically mounted and are shown in Fig. 5.12. Each location has a direction associated with it, so the response/force FRFs

5 Vibration Test Design with Integrated Shaker Electro-Mechanical Models

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can be extracted from the FE model. There are axial-direction inputs at the top and bottom, and radial-direction inputs on discs and transverse inputs on the base. The objective is to determine the locations of six shakers to achieve response of the DUT measured at those 33 response DOF in some previously-measured field environment. This is essentially a 33 output, 6 input multiple-input/multiple-output input estimation problem. The first step in that input estimation problem is determining the locations of those 6 inputs from the 37 candidate locations, which is where connecting the shaker model to the DUT model with FBS comes in. Inputs and responses will be predicted given a set of 6 shaker locations sampled from all combinations of 6 locations in 37 candidates. Then, the results for different input locations will be compared in terms of the response accuracy and required input levels. Any input locations which require too much input, in terms of force or current, will be omitted. The locations which provide the best response accuracy within the capabilities of the shaker will be chosen for a future multi-shaker experiment of this DUT. Results of the evaluation of a random set of 2000 of the 2.3 million possible location combinations yields predictions of the response accuracy, and the shaker force, current, and voltage. These predictions are the in the form of CPSD matrices, which results in far too much data to easily view and interpret. Instead, the predictions are condensed down over the frequency band in terms of the root mean square (RMS) response error, shaker force, current, and voltage. Then, those RMS values are condensed down over the gauges or shakers by taking a mean of the RMS values. This results in a single scalar value for each parameter for each set of shaker locations. These condensed results can then be viewed in terms of their distributions to

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Fig. 5.13 Distributions of response accuracy (left), force (center-left), current (center-right) and voltage (right) predicted by the coupled DUTshaker model for 2000 sets of shaker locations

understand the spread of values and how much location matters in terms of each parameter. As seen in the distribution plots in Fig. 5.13, there is a range of inputs, with some sets of shaker locations requiring much greater input than other locations. These inverse cumulative distribution function plots show the parameter value on the y-axis for each of the 2000 sets of shaker locations arranged small to large on the x-axis. It should be noted that the shaker locations which is minimum for one parameter will often not be minimum or even small in terms of the other parameter, so there is a balance of requirements. Ultimately, the objective is to choose a set of locations which are acceptable in terms of the shaker limitations (current and force), while resulting in low response error. With model predictions for the response and input force, voltage, and current, a set of shaker locations was chosen for the multi-shaker test, shown in the diagram in Fig. 5.12. Importantly, the shaker locations could be chosen based not only on the response accuracy or input force, but also in terms of the shaker current or voltage. In this case, shaker force and current were the limiting factors of interest so any shaker locations which resulted in predicted forces or currents above limitations were omitted from consideration. Unfortunately, PSDs of responses and inputs could not be reported in this paper, however the overall response was well matched by these six shaker locations and the full levels were achieved within the shaker capabilities, as predicted with this modeling effort. The integration of the shaker electro-mechanical model into the DUT model with FBS allows the required inputs for any set of chosen locations to be assessed against all the important shaker limitations, ensuring the test can be run at full-level without risking damage to the shaker equipment.

5.7 Conclusions The objective of this work was to provide a predictive capability to assess the force, voltage, and current requirements needed to achieve some DUT response from a set of shakers. This capability could then be used in pre-test design to determine where shakers could be located to ensure a multi-shaker vibration test was achievable in the laboratory. As moving shakers around is very time consuming in the laboratory, it is best to determine shaker locations beforehand using models. Introducing shaker electro-mechanical models to the pre-test design process allows the test design to account for all the important limitations, ensuring the shakers are capable of achieving the test goals. A lumped parameter shaker electro-mechanical model was implemented. The unknown parameters in this model were calibrated with a measurement of the shaker by itself. This electro-mechanical model was then coupled to structural dynamics models of various systems, including a simple two DOF model (to ensure the FBS process was properly implemented). Next, the shaker model was connected to an experimentally-derived model of a more complicated DUT which demonstrated that the coupled DUT-shaker model was adequately predictive. Finally, the shaker model was connected to various candidate input locations on a FE model of the DUT to design a multi-shaker test with limitations on the maximum shaker force and current as design factors.

References 1. Allen, M.S., Rixen, D., Mayes, R. L.: “Short course on experimental dynamic substructuring, module 2: General theory,” Short Course for the International Modal Analysis Conference, Orland, FL (2014) 2. Avitabile, P.: MECH 5150 (Structural dynamic modeling techniques) course notes, system modeling concepts chapter (2017) 3. Tiwari, N., Puri, A., Saraswat, A.: Lumped parameter modeling and methodology for extraction of model parameters for an electrodynamic shaker. J. Low Freq. Noise Vibr. Act. Control. 36(2), 99–115 (2017) 4. Lang, G.F., Snyder, D.: Understanding the physics of electrodynamic shaker performance. Sound Vib. 35(10), 24–33 (2001)

Chapter 6

Reproducing a Component Field Environment on a Six Degree-of-Freedom Shaker Jelena Paripovic and Randall L. Mayes

Abstract Researchers have shown that the dynamic field environment for a component may not be represented well by a component level single Degree-of-Freedom shaker environmental test. Here we demonstrate for a base mounted component, a controlled six Degree-of-Freedom component level shaker test. The field response power spectral densities are well simulated by the component response on the six Degree-of-Freedom shaker. The component is the Removable Component from the boundary condition challenge problem. The field environment was established with the component mounted in the AWE Modal Analysis Test Vehicle during an acoustic test. Interesting mileposts during the process of achieving the controlled component response are discussed. Keywords MIMO testing · 6 DOF testing · Removable component BARC · Modal analysis test vehicle MATV · Acoustic

6.1 Motivation Currently, component qualification testing contains unknown uncertainty. Not enough measurements are provided to define the component motion in the system field environment, system component boundary conditions and impedances are ignored, and laboratory testing may contain different boundary conditions that change the stress fields and failure modes [1]. These uncertainties sometimes go ignored, and it is implicitly assumed that conservative envelopes on the available system measurements provide confidence in component testing [1]. These envelopes are specified in the individual X, Y and Z directions and the hardware is tested one axis at a time on single degree of freedom shakers. Laboratory shaker boundary conditions are constrained in the other 5 degrees of freedom, unlike the field boundary conditions. There is reporting evidence that the response of a structure excited simultaneously in multiple axes differs from sequential single axis tests [2, 3]. There is also evidence in difference in fatigue life and failure modes when comparing MDOF and 1DOF testing [4]. All of these changes provide additional uncertainties, especially when considering that the ‘available measurements’ used to develop these envelopes may be a non-negligible distance from the component base and rotations are completely ignored [1]. Ideally, one has measurements of the system and component of interest and can reduce these uncertainties while improving confidence in the enveloped test specification. Instrumentation, data acquisition, and telemetry technology has advanced to allow a sufficient sensor set to completely define the component motion. This work seeks to show proof of concept that better lab environment testing is possible using a MIMO 6DOF shaker to reproduce component field environment; here we assume the available measurements have been acquired at the system and component locations of interest and use those responses as the target control. This method would remove uncertainties in future testing and provide benefit in understanding accurate stress and damage potential.

J. Paripovic () · R. L. Mayes Center for Engineering Sciences, Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2021 A. Linderholt et al. (eds.), Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47630-4_6

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Fig. 6.1 (a) MATV Finite Element Model Cutaway, (b) MATV Hardware in a Test Setup, (c) target accelerometer locations in large end of the cone

6.2 Introduction and Background The instrumented Removable Component from the Box Assembly and Removable Component (BARC) was the structure of interest in a previous MATV acoustic test and the measured data was recorded and is used as the ‘truth’ field data for this investigation. Here, we use this field data of as the target control on a 6DOF shaker table. The Removable Component (RC) is a base mounted component. The BARC has been a structure of interest for many in the modeling, analytical, and experimental field [5, 6, 7, 8]. The Removable Component (RC) was mounted inside a Modal Analysis Test Vehicle (MATV) and subjected to a 147 dB acoustic “field” test using an acoustic horn in a reverberant chamber at the Institute of Sound and Vibration Research Consulting facility at the University of Southampton. Developed at the Atomic Weapons Establishment (AWE), the MATV is constructed of a conical composite aluminum shell with a foam filled steel pipe in the axial center, as seen in see Fig. 6.1a. Additionally, the MATV consists an aluminum cover plate which is used to close off the larger end, and an aluminum component plate at the top of the steel pipe where the RC mounts to. This whole MATV structure has a mass of 47 kg. To simulate a free-free boundary condition, bungee cords were used to suspend the MATV during the acoustic test, which is shown in Fig. 6.1b. Fourteen internal target control accelerometers were used. The four triaxial accelerometers used are shown in Fig. 6.1c. Here, the focus is on whether the field could be replicated in a MIMO 6-DOF environment and use Nodes 1–4 (triaxial accelerometers on the RC) for the response control. The next section describes the 6-DOF lab environment test conducted to simulate the MATV acoustic test. In this case, we consider the MATV field test data as the ‘truth’ data and use it as the target control.

6.3 Experimental Results and Discussion A nominally identical RC [8] was instrumented using the same 4 triaxial accelerometers (Nodes 1–4) at the same locations and was mounted to the Team Tensor 900 6-DOF shaker. This shaker has 12 servos – 2 on each side of the plate and 4 below the shaker head. Two different adapter plates were used to accommodate the same mounting configuration as the IMMAT test. Nine additional accelerometers were used on the base plate, 4 on the sides on the plate and 4 equidistant on the top surface of the base plate and an additional one in the center, see Fig. 6.2. Two different tests were conducted using this configuration. First, a response control scheme using the Spectral Dynamics MIMO random controller was implemented to control the original 4 nodes (12 responses) to the measured MATV acoustic

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Fig. 6.2 RC attached to Team Tensor 900 instrumented with 13 triaxial accelerometers

test data. The MATV ASDs were sampled at a 1.25 Hz resolution and truncated from 50 Hz to 2000 Hz to accommodate the Spectral Dynamics requirements. The measured ASD responses from the 6 DOF shaker test (red) and the target MATV acoustic responses (blue) are shown in Fig. 6.3. Note, here we are trying to match all 12 responses on the 4 triaxial gauges using a multi-axis base excitation. The measured (red) and target (blue) ASD responses match well over most of the control bandwidth (50–2000 Hz). There are two consistent frequencies where the fit is less precise: 1140 Hz and 1900 Hz. The first discrepancy (1140 Hz) is only in the X direction of the four nodes and corresponds to the third fixed base mode where the RC component lunges side to side in the X-direction [5]. Mathematically, this fixed base mode is controllable, but the Spectral Dynamics control system did not have enough dynamic range to notch the inputs low enough to remove this fixed base resonant response. The second mismatch is near 1900 Hz and can be observed in all 4 nodes in the X, Y and Z direction. This corresponds to a twist mode of the 6 DOF [5] shaker and fixture plate which was not controllable in this experimental set-up with the Team-Tensor 900. We provide these curves as evidence that a controlled 6-DOF test can simulate a field response for a base mounted component. The second test of interest with this configuration was to understand how well one could match the RC response using a single degree of motion test. To do this, first a simple buzz test was conducted using base control. The base accelerometers were used to produce flat broadband spectrum in all three translations (X, Y, Z), and all three rotation (Rx, Ry, and Rz). The six base acceleration inputs were uncorrelated. The responses of nodes 1–4 were measured and a transmissibility matrix was calculated to the six base inputs to understand the dynamic response of the RC on the 6-DOF shaker. The input for this single degree of motion test is an ASD for the X-direction only; this was directly measured on the MATV component plate approximately 4 inches away from the RC. Here the Y-direction and Z-direction translation motions and all three rotations (Rx, Ry and Rz) are ignored, this drastically affects the response. The results of this test are shown in Fig. 6.4. The measured ASD responses from the 1DOF shaker test (red) and the target MATV acoustic responses (blue) is a poor recreation of the field responses. Using a controlled measured X input from the field as the target specification yields a different RC response when Y, Z, Rx, Ry, and Rz are constrained. For the frequency range 50 Hz to 1100 Hz, the measured response and target response match well for all 4 nodes in the X-direction only. However the overall amplitude and response structure for the constrained Y, Z, Rx, Ry, and Rz does not match. MIMO testing can be complex, and this test series presented a few challenges that can be explored in future works to minimize the error between the target and measured ASD responses. Firstly, the reference response matrix must be a positive definite matrix. It is important to note that when using the Spectral Dynamic MIMO random controller, the software truncates all terms to 6 significant digits past the decimal point and this truncation can cause a positive definite system to become nonpositive definite or the coherence to be greater than 1, which is physically impossible. One solution is to truncate the matrix in MATLAB prior to testing while ensuring positive definiteness or multiplying the coherence by 0.9999 (until it is less than 1) which is undesirable. Another important coefficient is the condition number of the matrix. With the current control scheme all 12 response were treated with equal importance and if the condition number is too high then the controller struggles to control to the target, sometimes even aborting prior to reaching steady state at low levels. This single degree of motion test is also related to the dynamic range of the controller software. If the target references span too many decades the drives of the servos will not have enough dynamic range to control and overcome dynamic response and unwanted resonances. For these series of tests, a condition number of 1000 was used, and the matrices were truncated in MATLAB while ensuring positive definiteness. Because there was insufficient dynamic range, the 1140 Hz mode could not be notched enough. If a higher condition number was used, the controller could not run the environment.

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Fig. 6.3 ASD responses from nodes 1–4 on the RC, where blue is the target MATV acoustic test and red is the 6DOF Team Tensor test. This was a 6DOF test were all 12 responses were used as the target control

With these limitations, future work will consist of investigating and identifying better ways to condition the target reference file (matrix) to improve the match between the response and target near 1140 Hz and 1900 Hz. To improve the torsional mode (1900 Hz) mismatch, future work may consider including that mode in the base input in modal space. While this may approach dynamic range limitations, this mode is theoretically controllable.

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6.4 Conclusion The good fit between the target MATV acoustic test, and the 6 DOF measured response shows that MIMO testing is a more suitable testing approach to achieve representative field environment reproductions. Additionally, the significantly poorer fit for the single degree of freedom test reinforces the importance of pursuing MIMO testing for component and system levels. The MIMO testing shows great promise for future investigations of combined axis testing. Understanding how to best identify MIMO specifications, and condition the reference matrices will improve testing and may allow for more complex structures. Investigations into MIMO Swept Sine tests, and MIMO Shocks along with MIMO combined environments should also be investigated in the future. Acknowledgements Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

References 1. Mayes, R.L., Lopp, G.K., Paripovic, J., Nelson, G.D. Schultz, R.A.: Testing a component on a fixture to reproduce system accelerations. In Proceedings of the 65th Institute of Environmental Sciences and Technology Conference, Las Vegas, Nevada, 2019 2. Berman, M.B.: Inadequacies in uniaxial stress screen vibration testing. J. IEST. 44(4), 20–30 (2001) 3. French, R.M., Handy, R., Cooper, H.L.: Comparison of simultaneous and sequential single axis durability testing. Exp. Techn. 30(5), 32–37 (2006) 4. Gregory, D., Bitsie, F., Smallwood, D.O.: Comparison of the response of a simple structure to single axis and multi axis random vibration inputs, Technical report, Sandia National Lab. (SNL-NM), Albuquerque, NM, USA 2009 5. Lopp, G.K., Mayes R.L.: Using the modal Craig-Bampton procedure for the test planning of a six-degree of freedom shaker. In Proceedings of the 38th International Modal Analysis Conference, Houston, TX, February 2020 6. Mayes, R. L., Ankers, L., Daborn, P.: Predicting system response at unmeasured locations. In Proceedings of the 37th international modal analysis conference, Orlando, FL, January 2019 7. Rohe, D.P., Schultz, R.A., Schoenherr, T.F., Skousen, T.J.: Comparison of multi-axis testing of the BARC structure with varying boundary conditions. In Proceedings of the 37th International Modal Analysis Conference, Orlando, FL, January 2019 8. Larsen, W., Blough, J.R., DeClerck J.P., VanKarsen, C.D., Soine, D.E., Jones, R.: Initial modal results and operating data acquisition of shock/vibration fixture. In Proceedings of the 36th International Modal Analysis Conference, Orlando, FL, February 2018

Chapter 7

In-Situ Source Characterization for NVH Analysis of the Engine-Transmission Unit Ahmed El Mahmoudi, Francesco Trainotti, Keychun Park, and Daniel J. Rixen

Abstract Over the past decades, engineers have developed methods to determine critical paths for sound transmission, also known as Transfer Path Analysis (TPA). This demand stems, for instance, from the need to prevent undesired vibrations in modern vehicles. A relatively new method that allows for the characterization of a source in the assembled state is called in-situ TPA (iTPA). Generally, the excitation forces cannot be measured. Although the iTPA makes the development process more complex, it provides insight into the forces generated by active components that are transferred to the receiver structure. The vehicle dynamics as well as its vibration behavior is essentially determined by the behavior of the powertrain. For this more complex system, several challenges affect the method’s applicability, such as the accessibility to the engine mount interfaces and the modeling of coupling degrees of freedom. The aim of this paper is the characterization of the excitation source based on the engine-transmission unit using the iTPA method. It is analyzed and evaluated in terms of its accuracy and applicability. Equivalent forces of two different vehicle configurations, stiff and soft engine rubber mounts, are compared and used to predict the total vehicle vibration responses. Keywords Powertrain NVH · Source characterization · In-situ Transfer Path Analysis

7.1 Introduction In the automotive industry, NVH (Noise, Vibration, and Harshness) technology is becoming more important at all times in order to meet the acoustic performance requirements of the customers. To meet these requirements, methods for predicting and optimizing NVH problems are developed. Transfer Path Analysis (TPA) is a wide-range diagnostic tools for the assessment of noise and vibration. The origin goes back to Verheij’s work on the sound transmission of ship engines in 1982 [1]. The focus is on the assessment of products made of actively vibrating components and the transmission of vibrations to coupled passive structures. It is a useful method if the vibrating structure is too complex to model or measure. It allows for the characterization of excitations through forces and vibrations at the interfaces. In this context, a separation of source excitation and structural/acoustic transmission characteristic is possible. It allows for the troubleshooting of dominant transmission paths to anticipate whether changes should be made to the coupled receiver structure or the source structure. Seijs et al. [2] give a detailed overview of the various TPA methods. Here, TPA is divided into eleven variations classified into three categories, the classical, component-based and transmissibility-based TPA. Classical TPA methods are usually applied to already existing assemblies in more advanced product development phases. It also identifies the interface forces transmitted between the source and receiver. These interface forces, however, are dependent on the overall system of source and receiver. Therefore, an independent characterization of the source structure is not possible and the interface forces cannot be reused for other receivers. If the focus is on an independent characterization of the source structure, the category of the component-based TPA is suitable. Already in the early development phase, a set of forces such as blocked forces or pseudo forces can be determined.

A. El Mahmoudi () · F. Trainotti · D. J. Rixen Chair of Applied Mechanics, Faculty of Mechanical Engineering, Technical University of Munich, Garching, Germany e-mail: [email protected]; [email protected]; [email protected] K. Park Chair of Applied Mechanics, Faculty of Mechanical Engineering, Technical University of Munich, Garching, Germany Hyundai Motor Company, Seocho-gu, Korea e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2021 A. Linderholt et al. (eds.), Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47630-4_7

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The same conditions are ensured by simply using the same transmission path data. One advantage of this is that a much faster and more reliable approach is possible [3]. A further advantage of identifying an independent source is that they can be used as target values. In this industry, active components are often bought from suppliers. Through the use of the interface quantities of a component-based TPA as a target for the suppliers, it is possible to outsource the task of source characterization. In this article, the in-situ block force characterization [4] is particularly discussed. The aim is to investigate the applicability of the method using practical examples. The method is applied on a test vehicle. The focus here will be on characterizing the forces transmitted to the body by the engine-transmission unit. Thereby, Sect. 7.2 will describe the relevant theory behind the in-situ blocked force TPA method. Then, the results from the vehicle tests, in which different rubber mounts were installed in order to change the receiver structure, are shown in Sect. 7.3. Finally, Sect. 7.4 will present some concluding remarks.

7.2 Theory This section gives a brief overview of the required theory. At the beginning, the in-situ blocked forces method from the category of component-based TPA is introduced. Since the identification of the blocked forces requires direct access to the input DoFs at the interface, the virtual point transformation will be briefly explained. Finally, the step-by-step procedure of the iTPA method is described.

7.2.1 In-Situ Blocked Force TPA A method for characterizing the transmitted forces induced by a source structure is the method of in-situ blocked forces from the family of component-based TPA [4]. Different from other TPA methods, the source characterization is performed on the assembled system. Figure 7.1 shows an assembly consisting of source (orange) and receiver (blue) structure. Since source internal structure-borne excitations f 1 are often unknown in practice, interface forces are used in classical TPA to describe sources [1]. However, interface forces do not represent the source independently from the receiver structure. This is where eq the equivalent force f 2 of the component-based TPA comes into play. The term equivalent forces can be understood as a generalization of the more common blocked forces, which derives from the possibility to directly measure them at the interface of the active source if this was mounted perfectly rigid (u2 = 0). A comfortable way to measure equivalent forces is in-situ [4], i.e., in the assembled configuration of source and receiver. On the receiver side, a sufficient number of indicator sensors u4 are required to calculate the blocked forces as follows: 

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uTPA = Y 32 3

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Fig. 7.2 The procedure of Virtual Point Transformation. The substructure is equipped with acceleration sensors (blue), which measure the displacement through excitation (red arrows). The FRFs are then transformed to the virtual point (green)

Note that blocked force calculation requires direct access to the input DoFs at the interface, which is often infeasible to measure and source measurement errors. A geometrical transformation is therefore needed and will be explained in the following.

7.2.2 Virtual Point Transformation Figure 7.2 shows an application of the virtual point transformation (VPT). Helderweirt et al. present the so-called IDM filtering, on whose idea the VPT is based [5]. With the definition of six rigid interface displacement modes (IDM) per interface point and the projection of an admittance matrix with nine degrees of freedom onto this subspace, initially only the dynamics that leave this area rigid will be preserved. Originally, IDM was used for interface deformation modes, but since we assume only rigid displacements, the designation of [6] is chosen. If in the following a substructuring with this filtered admittance is carried out, only those nine degrees of freedom exhibit locally rigid behavior are required to be compatible and balanced. The residual flexibility remains uncoupled. The interface problem is weakened and measurement errors are averaged out due to the reduction. This guiding principle of IDM filtering is now used in the VPT methodology presented in [7, 8]. With the definition of the IDMs for a single interface point, the dynamics is limited to a six degrees of freedom per node kinematics instead of the nine or more degrees of freedom of a measured FRF matrix. If the IDMs are defined to describe three translations (X, Y, Z) and three rotations (φx , φy , φz ) of a single node in a global coordinate system, this kinematic description is equivalent to that of an FE node. In this case, the IDMs can be used to transform translational displacements to the motion of a single point in all six directions. Similarly, this can be used to extract a six-axis excitation to a point from a series of measured excitations. In this minimal example, it can be seen, that in the experimental case, the interface can neither be directly excited nor response signals acquired. Since only the points around the interface are of interest in the following, these are designated u for illustrative purposes. This results in the descriptive equation for the dynamics in the usual admittance form: u = Yf

u ∈ Rn

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The idea now is to represent the n interface displacements u by m < n rigid interface modes IDMs q. The IDMs are contained in the columns of the n × m matrix R, which is a frequency-independent eigenform matrix. Since the number of IDMs is less than the number of interface displacements, a residual term called μ is added to the displacements. This residual term contains all displacements that cannot be mapped with the subspace of the IDMs. These displacements usually correspond to the flexible deformations. Thus the motion can be expressed as follows: u = Rq + μ

q ∈ Rm

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The Moore-Penrose pseudoinverse of R can be used to determine q in the sense of the least squares, which minimizes the norm on residuals and forces R T μ. A comparable relation can be set up for nf > m forces f and virtual point forces/moments m. Thereby, one obtains: u = Ruq

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+ The inverted IDM matrices (R u )+ and R Tf are also known as transformation matrices. Thus, the measured FRFs Y uf can be transformed into a virtual point FRF Y qm :

+ q = Y qm m with Y qm  (R u )+ Y uf R Tf

(7.7)

7.2.3 Procedure of the iTPA In accordance with the proposal for standardization in [9] for the application of the method, the following five steps will be carried out: • • • • •

FRF measurement (including virtual point transformation) Operational measurement under various conditions Blocked force calculation On-board validation Cross validation

Blocked Force Calculation With respect to the in-situ blocked force TPA method, however, no transformation of the displacement is necessary, only of the forces. Thereby, Eq. (7.1) can be converted to the VPT for the force identification as follows: 

eq assembly + op m2 = Y u4 m2 u4

assembly

with Y u4 m2

assembly

 Y u4 f

2

+ R Tf 2

(7.8)

The acceleration DoFs remain the same, while a characterization of forces and moments is created in the virtual point, allowing for the transfer of vibration through the interface.

On-Board Validation The reliability of the identified force is now checked by using the on-board validation. As mentioned in the previous assembly op and target sensor responses u3 are required. If a matrix-vector multiplication is subsection, the target admittances Y 32 assembly performed with the transformed target admittance Y u3 m2 and the identified blocked force, the predicted system response TPA u3 is obtained at the target point: assembly

= Y u3 m2 uTPA 3

eq op m2 ∼ = u3

assembly

with Y u3 m2

assembly

 Y u3 f

2

+ R Tf 2

(7.9)

Cross Validation The cross validation verifies the actual goal of the in-situ TPA, which is to take the identified blocked forces from one assembly (here: soft configuration, see Fig. 7.3) to predict the response in another assembly (here: stiff structure). The

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identification of the blocked forces can be performed both in the soft and in the stiff structure, being understood that the  and its internal excitation remains unchanged. Theoretically, both forces should then be equal

source component eq, soft eq, stiff : = m2 m2 assembly, i

i uTPA, = Y u3 m2 3

eq, j

m2

op, i ∼ = u3

with i, j ∈ {soft, stiff} , i = j

(7.10)

In practice, the blocked forces would not be identified in both test benches. In the context of this work, however, this was done for research purposes.

7.3 Vibration Prediction from Vehicle Measurements In a previous work, we were able to validate the iTPA method at a more advanced academic structure [10]. In this section, the in-situ blocked force TPA procedure will be performed on a more complex system. As shown in Fig. 7.3, a source characterization is performed on a test vehicle. The vehicle is a passenger car with a 4-cylinder diesel combustion engine from Hyundai-Kia Motors Corporation (HKMC). The vehicle assembly is divided into an engine-transmission unit (source structure) and the remainder of the vehicle (receiver structure). An industrial application could be, for instance, first installing soft rubber mounts (configuration 1) to support the powertrain, identifying the blocked forces and afterwards exchanging them with stiff rubber mounts (configuration 2) and redoing the identification. One of the assumptions is that the rubber mounts represent the only interfaces through which the excitation is transferred to the vehicle body. Similar to the test bench measurements [10], both FRF and operational measurements are carried out. The operational measurement campaign was performed by measuring the vibration response caused by no load excitation through a chassis dynamometer and engine full load excitation in different operating points from 1000 to 4000 rpm (discrete and run-up/run-down). As target sensors, accelerometers were mounted on the vehicle body in the engine compartment and on the driver’s seat rail as well as a microphone on the driver’s side.

7.3.1 Force Identification In this subsection, the identified forces from both configurations, soft and stiff rubber mounts, are compared with each other. The calculation was performed as shown in Eq. 7.11: 

eq,i assembly,i + op,speed,i m2 = Y u4 m2 u4 with i ∈ {soft, stiff} , speed ∈ {1000, . . . , 4000} (7.11)

Fig. 7.3 Vehicle setup for TPA. Source excitation through engine-transmission unit (source structure), which is supported by three rubber mounts. Vehicle body and rubber mounts form the receiver structure. Rubber mounts were changed (soft → stiff) for different receiver structures

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In the following, the different operating conditions at which the blocked forces were determined are presented. The first is discrete speeds, at which only a certain speed is maintained and the operational measurements have been carried out. These, in turn, are differentiated according to their type of excitation, drivetrain by internal combustion or by rollers of the chassis dynamometer. Secondly, vehicle run-ups were carried out. All measurements were performed with soft and stiff rubber mounts and compared with each other. As mentioned before, in practice, the blocked forces would not be identified in both configurations. This means that the equivalent forces of the source structure are identified on a test bench, for instance, and then applied to the receiver structure in order to predict the responses of the corresponding structure. In the context of this work, however, this was done for research purposes. Discrete Speed Figures 7.4 and 7.5 show the blocked forces determined at discrete speeds from 1000 to 4000 rpm in steps of 1000 rpm. On the one hand, via excitation by the chassis dynamometer. Here, the vehicle was switched to neutral during the run, i.e. the combustion engine only ran at idle speed. On the other hand, with excitation by full engine load, i.e. full throttle. Run-up Figures 7.6 and 7.7 show the blocked forces determined through a run-up from 1000 to 4000 rpm. All measurements were performed with soft and stiff rubber mounts and compared with each other.

Fig. 7.4 Blocked forces at VP1:+Y (TM), obtained from 1000 to 4000 rpm (read from upper left → lower right) no engine load, driven through chassis dynamometer. Forces estimated from the soft (blue) and stiff (orange) rubber mount configuration

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Fig. 7.5 Blocked forces at VP1:+Y (TM), obtained from 1000 to 4000 rpm (read from upper left → lower right) engine full load excitation. Forces estimated from the soft (blue) and stiff (orange) rubber mount configuration

dB 45

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60

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30 35

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0

10 8

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-10 500 1000 Frequency (Hz)

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Fig. 7.6 Colormap of blocked forces at VP1:+Y (TM), obtained from a run-up from 1000 to 4000 rpm. Forces estimated from the soft (left) and stiff (right) rubber mount configuration

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Fig. 7.7 Blocked forces at VP1:+Y (TM), obtained from a run-up from 1000 to 4000 rpm. Forces estimated from the soft (blue) and stiff (orange) rubber mount configuration

7.3.2 On-Board Validation In order to guarantee the quality of the characterization, it can be checked whether the calculated barrier forces completely describe the source activity. This is done by an on-board validation. This includes a TPA synthesis on the vehicle that calculates the response to a validation sensor. In this case, the on-board validation sensor is located at different positions, for instance, close to the engine but on the receiver side or on the driver seat rail. Since this accelerometer was not used for the characterization itself, it ensures an objective evaluation. The synthesis of the response for the validation sensor is done by applying Eq. 7.9. The synthesized reaction is compared with the originally measured sensor signal, as shown in Figs. 7.8, 7.9, 7.10, and 7.11 for both rubber mount configurations under different operating conditions. The validation provides an accurate prediction of the on-board signal, i.e. the on-board validation is successful. In particular, it indicates that no other source (like tire contact excitation or external noise) significantly contributes to the target output. This means that the calculated blocking forces actually fully describe the active vibrations of the engine-transmission unit and can now be transferred to another receiving structure, such as a vehicle.

Discrete Speed Figures 7.8, 7.9, 7.10, and 7.11 show the on-board validation determined at discrete speeds from 1000 to 4000 rpm in steps of 1000 rpm. The operating conditions are as described above for the blocked forces, on the one hand, via excitation by the chassis dynamometer. Also, here the vehicle was running in idle mode by switching the gear to neutral. On the other hand, with excitation by full engine load, i.e. full throttle. The figures show a high agreement between predicted response and validation response from the operational vibration measurements in all cases.

Run-up Figures 7.12 and 7.13 show the on-board validation determined through a run-up from 1000 to 4000 rpm and run-down. All measurements were performed with soft and stiff rubber mounts and compared with each other.

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Fig. 7.8 On-board validation of the TPA results on the soft rubber mount configuration. Obtained from 1000 to 4000 rpm (left → right) no engine load, excitation through chassis dynamometer. Validation responses (orange) and in-situ TPA predicted responses (blue)

7.3.3 Discussion The identified blocked forces with the two types of rubber mounts compared in Figs. 7.4, 7.5, 7.6, and 7.7 show clear discrepancies. One explanation for the offset between the identified blocked forces might be that the requirement to use the assembly,soft assembly,stiff = f1 is not fulfilled. As a result, a transfer of the identified blocked forces in-situ TPA method of f1 from one structure to another is not possible. Through the on-board validation, a correct execution of the experiment could be shown for the discrete speed case with both rubber bearings, soft and stiff. The on-board validation for the run-up and run-down respectively shifts the attention to a very interesting aspect. Figure 7.12 on the left shows a clear deviation between the predicted and expected response. Interestingly, this is just the case for the run-up of the soft rubber mount configuration. Looking at the run-down case, instead, an almost perfect match can be observed. Considering the fact that the combustion engine fires during run-up, an additional preload on the rubber mounts could be generated. Note that this is not the case for the run-down, since the foot is taken off the accelerator pedal and the vehicle rolls out. This leads to the second hypothesis for why the two identified forces do not coincide. The other reason could be a significant non-linear behavior of the soft rubber mounts. Since the excitation took place on the source’s side of the rubber mounts, the damping characteristics of the rubber influence the result. Looking at the on-board validation for the stiff rubber mounts in Fig. 7.13, one can see a mostly continuous match between predicted and expected response. This is true for both the run-up and the run-down case.

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Fig. 7.9 On-board validation of the TPA results on the soft rubber mount configuration. Obtained from 1000 to 4000 rpm (left → right) engine full load excitation. Validation responses (orange) and in-situ TPA predicted responses (blue)

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Fig. 7.10 On-board validation of the TPA results on the stiff rubber mount configuration. Obtained from 1000 to 4000 rpm (left → right) no engine load, excitation through chassis dynamometer. Validation responses (orange) and in-situ TPA predicted responses (blue)

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Fig. 7.11 On-board validation of the TPA results on the stiff rubber mount configuration. Obtained from 1000 to 4000 rpm (left → right) engine full load excitation. Validation responses (orange) and in-situ TPA predicted responses (blue)

Fig. 7.12 On-board validation of the TPA results on the soft rubber mount configuration. Obtained from a run-up from 1000 to 4000 rpm and run-down (right), vice versa. Validation responses (blue) and in-situ TPA predicted responses (green)

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Fig. 7.13 On-board validation of the TPA results on the stiff rubber mount configuration. Obtained from a runup (left) from 1000 to 4000 rpm and rundown (right), vice versa. Validation responses (blue) and in-situ TPA predicted responses (green)

In future works, it will be necessary to examine at which stiffness the non-linear behavior of the rubber mounts limits the identification of the blocked forces. Furthermore, a closer look at the assumption of the constant input forces is taken. A characterization of rubber mounts under different operating conditions (preload, temperature, etc.) will also be carried out in the future in order to develop an adapted TPA method for rubber mounts with highly behavior.

7.4 Conclusions In this paper, the in-situ blocked force TPA source characterization method was carried out on a complex system, i.e. test vehicle. Hereby, the blocked forces at the three interfaces of the engine-transmission unit to the body were identified. Two different receiver configurations, soft and stiff rubber mounts, were tested. An unexpected offset occurred in the identification of the blocked forces. The authors formulated some plausible hypotheses to explain the observed offset and reveal possible limitation of the blocked in-situ method. A significant non-linear behavior of the soft rubber mounts could lead to strong dependency on the preload. This preload increases in the run-up case, thus making a noticeable difference in the identified blocked forces and the on-board validation. This behavior will be examined.

References 1. Verheij, J.W.: Multi-path sound transfer from resiliently mounted shipboard machinery. Ph.D. thesis. Delft University of Technology (1982) 2. van der Seijs, M.V., de Klerk, D., Rixen, D.J.: General framework for transfer path analysis: history, theory and classification of techniques. Mech. Syst. Signal Process. 68–69, 217–244 (2016). https://doi.org/10.1016/j.ymssp.2015.08.004 3. Moorhouse, A.: Virtual acoustic prototypes: Listening to machines that don’t exist. Acoustics Australia. 33(3), 97 (2005) 4. Moorhouse, A.T., Elliott, A.S., Evans, T.A.: In situ measurement of the blocked force of structure-borne sound sources. J. Sound Vib. 325(4–5), 679–685 (2009) 5. Helderweirt, S., Van der Auweraer, H., Mas, P., Bregant, L., Casagrande, D.: Application of accelerometer-based rotational degree of freedom measurements for engine subframe modelling. In: Proceedings of the XIX International Modal Analysis Conference (IMAC), St. Louis (2004) 6. van der Seijs, M.: Experimental dynamic substructuring. Ph.D. thesis. Delft University of Technology (2016) 7. de Klerk, D., Rixen, D., Voormeeren, S.N., Pasteuning, F.: Solving the RDoF problem in experimental dynamic substructuring. In: 26th International Modal Analysis Conference (IMAC XXVI), Orlando (2008) 8. van der ’Seijs, M.V., van den Bosch, D., Rixen, D.J., de Klerk, D.: An improved methodology for the virtual point transformation of measured frequency response functions in dynamic substructuring. In: COMPDYN (2013) 9. van den Bosch, D., van der Seijs, M., de Klerk, D.: A comparison of two source characterisation techniques proposed for standardisation. Technical report, SAE Technical Paper (2019) 10. El Mahmoudi, A., Trainotti, F., Park, K., Rixen, D.J.: In-situ TPA for NVH analysis of powertrains: an evaluation on an experimental test setup. In: AAC 2019: Aachen Acoustics Colloquium/Aachener Akustik Kolloquium, 26 Nov, Aachen (2019)

Chapter 8

Using Modal Projection Error to Predict Success of a Six Degree of Freedom Shaker Test Tyler F. Schoenherr, Janelle K. Lee, and Justin Porter

Abstract Six degree of freedom shaker tests are becoming more popular as they save testing time because they test a component in multiple directions in one test rather than executing multiple tests in one direction at a time. However, there are several difficulties in conducting a component six degree of freedom shaker test in a way that adequately replicates the component field stress. One difficulty is knowing if a classical rigid test fixture will produce component modes that span the displacement space of the component in the field environment. If the modes of the component while attached to a rigid fixture do not span the space of the component in the field environment, then the test will be unable to replicate that motion and corresponding stresses. This paper will examine the motion of the Removable Component of the BARC hardware in an field assembly and calculate the modal projection error expected by executing a six degree of freedom shaker test on a rigid fixture. The paper will conclude by examining the data and comparing it to the pre-test predictions of error calculated by the modal projection error. Keywords Modal projection error · 6 DOF Shaker · Boundary condition · Testing · BARC

8.1 Introduction The frequency and use cases for multi-axis shaker testing is growing. Multi-axis shaker tests reduce test time testing all axes simultaneously. Multi-axis tests also can increase the fidelity of the result by providing a more realistic stress state [1] for the unit under test. Like all dynamic environment tests, the test fixture and the shaker table need to represent the boundary condition impedance of the unit under test in the field to realize the benefits of the multi-axis shaker test. If the boundary condition of the shaker configuration is different from the field configuration, the unit under test can have a different set of mode shapes or even a different basis set. If the basis set of mode shapes differ between field and test, the stress profile of the unit under test cannot be matched. Quantifying the effect of the test fixture in a dynamic environment test is difficult due to the response of the unit under test being a product of the input forces and the system transfer functions. The modal projection error (MPE) is a newly developed parameter that tests two sets of shapes to determine if one set spans the space of another [3]. In relation to test fixture development, this error determines the lower limit that the test response will exhibit to reproduce the field response. Because the MPE is an examination of the mode shape space, a parameter of the system transfer functions, it is an appropriate method of quantifying the effectiveness of the test fixture. The purpose of this paper is to determine if the MPE is a good predictor of success for a multi-axis shaker test. In order to determine if the MPE can predict multi-axis shaker success, this paper will present two case studies. The first case study uses

Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE- NA-0003525. T. F. Schoenherr () · J. K. Lee Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected] J. Porter Rice University, Houston, TX, USA © The Society for Experimental Mechanics, Inc. 2021 A. Linderholt et al. (eds.), Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47630-4_8

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the Removable Component section of the Box Assembly with Removable Component (BARC) on an aerospace structure. The BARC hardware is intended to examine the effect of boundary conditions [4]. In this case study, the removable component is attached to an aerospace structure and is excited by an acoustic environment in a field configuration. The removable component is then attached via a sufficiently rigid fixture to a six degree of freedom shaker table. To provide contrast to the aerospace structure, the MPE is also calculated using the entire BARC hardware as the field configuration. The removable component is placed on a rigid fixture and the MPE is calculated. The BARC is designed to have a high MPE. The differences between the two case studies are examined.

8.2 Modal Projection Error Theory The MPE is an error metric introduced by Schoenherr [3] to provide insight on the effectiveness of the test fixture. It is derived from the error that occurs when one set of mode shapes is projected onto another set of modes shapes. If the mode shapes involved are truncated, the number of degrees of freedom are higher than the number of modes and there is modal truncation and modal projection error. The expression for MPE is rewritten here ¯

n2 = 1 − φ¯ F+n φ L φ + L φF n ,

(8.1)

where φ¯ F n is the nth field mode shape, φ L is the set of laboratory mode shapes, the superscript + is the Moore-Penrose psuedo-inverse, and n2 is the modal projection error for the nth field mode shape. The MPE is a representation least squared error and is generated through the two pseudo inverses in Eq 8.1. Computing the MPE on a mode by mode basis is what makes the MPE particularly useful and insightful. With respect to boundary conditions, the MPE is a lower bound on the error possible to replicate a specific mode shape and is achieved if the optimal forcing function can be applied at every degree of freedom on the unit under test. The MPE does not provide information on how achievable the error indicated by the MPE is, however, it is assumed that lower errors provide the correct modal basis and make those shapes easier to excite.

8.3 System Configurations 8.3.1 BARC The BARC hardware consists of a box assembly with a removable component (RC) on top. The BARC system is designed to be a platform to design a replacement structure or test fixture for the box assembly and have the RC response match in both systems. This structure was created to study the effects of boundary conditions and the development of test fixtures because the motion of the removable component is coupled to the motion of the box assembly. The BARC hardware shown in Fig. 8.1. The first six modes of the BARC system is computed and can be viewed in Fig. 8.2. An observation of the BARC mode shapes is that the motion of the box assembly and removable component parts are coupled for the first six elastic mode shapes.

8.3.2 Removable Component on a Rigid Fixture The removable component on a rigid fixture is examined. Because the fixture is rigid, the system was modeled by fixing the interface degrees of freedom of the removable component. The mode shapes of this system can be seen in Fig. 8.3 and are referred to as fixed base mode shapes. By comparing the mode shapes of the removable component on a rigid fixture to the BARC system, one can see that the motion of the RC on a rigid fixture does a poor job spanning the space of the RC on the BARC. This poor modal projection is due to the misrepresentation of the boundary condition of the RC.

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Fig. 8.1 Box assembly (red) with removable component (green), dubbed the BARC structure

8.3.3 Removable Component on an Aerospace Structure The other hardware in this paper is referred to as the aerospace structure shown in Fig. 8.4. The aerospace structure is a shelled structure with a center column for support shown in Fig. 8.5. The removable component from the BARC assembly is attached to the back plate of the aerospace structure shown in Fig. 8.6. The mode shapes for the aerospace structure with RC is shown in Fig. 8.7. Of the first six elastic modes of the aerospace structure, the RC is only active in two of them. Of the two active modes, it appears that the first fixed base mode of the RC represents the RC motion in the aerospace modes. However, the sixth mode of the aerospace structure is a drumming mode of the back plate and could be difficult to represent using the modes in Fig. 8.3.

8.4 Environment Field and Laboratory Tests The aerospace structure with removable component (RC) was placed in an acoustic field environment. A picture of the setup for this environmental test is shown in Fig. 8.8. Details of this test setup and execution can be found in Paripovic [2]. The acceleration response of the removable component is measured at several locations in the field environment. From the field measurements on the RC, the environment was reproduced on a 6-degree of freedom (DOF) shaker table with a rigid fixture. A rigid fixture is defined as having no elastic motion in the frequency range of interest. The RC was tested on a 6-DOF shaker table and the response at measured locations from the acoustic environment were compared in Fig. 8.9. Figure 8.9 shows the response in the laboratory well represented the response from the field environment. The discrepancy around 1100 Hz was due to a the shakers difficulty in exciting the third fixed base mode of the RC.

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Fig. 8.2 First elastic modes for the BARC system. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6

8.5 Results This section computes the MPE with respect to different sets of degrees of freedom on the RC and when the RC is on the aerospace structure and the BARC. The MPE is used to predict the success of the laboratory test. The MPE with the analytical mode shapes from the field and laboratory configuration is the quantity used to predict success of the laboratory. Conceptually, if the mode shapes of the RC on a rigid fixture span the space of the RC in the field configuration, then the test will be a success.

8.5.1 Aerospace Structure with RC Base DOFs The first attempt at quantifying if the lab modes span the space of the field mode shapes uses the nodes at the base of the RC. The DOF of these nodes are used in the MPE calculation because the DOFs experience no relative motion in the laboratory due to the rigid fixture. The base DOFs are shown in Fig. 8.10. The MPE values per mode of the aerospace structure are calculated and in Table 8.1. The MPE for the first six aerospace structure modes is effectively zero. This is expected because the first six modes of the aerospace structure are rigid body modes and the rigid body motion of the base of the RC perfectly spans that space. The error for the elastic modes of the aerospace structure are higher than anticipated. It is expected that the first mode of the

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Fig. 8.3 First elastic modes of the removable component on a rigid fixture. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6

Fig. 8.4 Aerospace structure with removable component attached to the back plate

RC on the rigid fixture should span the space of the first elastic mode of the RC in the aerospace structure because the first elastic shape of the aerospace structure visually matches the first fixed base mode of the RC. However, the error is high. This discrepancy is explained through examination of the displacement field in Fig. 8.10. Even though the modes in both systems visually appear the same, the RC base DOFs have rigid body response when attached to a rigid fixture and have relative motion to each other when attached to the aerospace structure as shown in Fig. 8.7.

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Fig. 8.5 Cutaway of the aerospace structure with removable component attached. (Used with permission [2])

Fig. 8.6 Location of the removable component on the back plate of the aerospace structure. (Used with permission [2])

8.5.2 Aerospace Structure with Full Field RC DOFs To provide a more representative MPE, a full field set of DOFs of the RC is chosen. Full field is defined to be enough DOFs to uniquely span the space of all modes in the frequency of interest. The new set of DOFs can be seen in Fig. 8.11. Since the full field DOFs are used, the entire set of shapes of the RC connected to a rigid fixture need to be augmented to the rigid body shapes to provide the full displacement space of the laboratory test. These shapes can be described as rigid body modes and fixed base modes.

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Fig. 8.7 First elastic modes of the aerospace structure with removable component. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6

The MPE was recalculated with the new set of laboratory mode shapes and RC DOFs. The MPE results are shown in Table 8.2. One of the insights MPE provides is the ability to determine if a specific field mode cannot be represented well by a test configuration. Through examination of the sixth field mode in Fig. 8.7, it is thought that the test would be unable to replicate that motion due to the high gradients of displacement on the disk to which the RC is attached. This mode shape is reproduced in Fig. 8.12 and color-mapped with displacement. Figure 8.12 shows the connection degrees of freedom of the RC lying on a nodal line of the mode shape and very little relative displacement between the connection degrees of freedom. This result for the sixth elastic mode agrees with the result of the MPE in Table 8.2.

8.5.3 BARC A separate case study examines the BARC hardware as the field configuration. The case study uses the mode shapes of the RC on a rigid fixture shown in Fig. 8.3 to span the space of the BARC system modes shown in Fig. 8.2. This case study is done to compliment the aerospace structure case study as it is expected that the MPE will be high for the BARC system.

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Fig. 8.8 Setup of the acoustic environment test. (Used with permission [2])

Fig. 8.9 Response of the removable component in a 6-DOF shaker test (Red Dashed) compared to the response of the removable component in the acoustic test (Blue Solid). (Used with permission [2])

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Fig. 8.10 Motion of the removable component and base degrees of freedom used in initial MPE calculations

Table 8.1 Modal projection error calculation using only base DOFs on removable component in the aerospace structure

n2 1 2 3 4 5 6 7

MPE (%) −8.9e-14 −4.4e-14 −2.2e-14 0 1.1e-8 2.4e-8 11.0

n2 8 9 10 11 12 13 14

MPE (%) 17.0 34.0 5.0 11.0 18.0 77.0 27.0

Fig. 8.11 Motion of the removable component and base degrees of freedom used in initial MPE calculations in the BARC assembly

The DOFs and modes used in the MPE calculation are the same as the MPE calculation for the full field analysis done in Sect. 8.5.2. The MPE calculated for the first eight elastic BARC modes is presented in Table 8.3. The MPE for the BARC system are worse than the aerospace system. The higher MPEs are expected because the BARC system is designed to have the RC boundary DOF response to have relative motion, which cannot be reproduced when the RC is attached to a rigid fixture. Even though there isn’t yet a quantifiable comparison between the MPE and a measurable quantity from the environment. The MPE for each BARC mode is compared with respect to the MPE for the other BARC modes. Table 8.3 shows that there is medium level of error for the first two elastic modes, high error for the third mode, and low error for the fourth through sixth elastic modes. This qualitative comparison is corroborated by examining the BARC modes in Fig. 8.2. For the BARC

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Fig. 8.12 Sixth elastic mode of the aerospace structure (left) and zoomed in removable component (right). Colored by displacement Table 8.2 Modal projection error calculation using full field DOFs on removable component in the aerospace structure

n2 1 2 3 4 5 6 7

Table 8.3 Modal projection error calculation using full field DOFs on removable component in the BARC assembly

n2 1 2 3 4 5 6 7

MPE (%) 1.5e-8 7.4e-9 2.9e-9 5.4e-7 1.1e-7 2.0e-7 7.6e-5

MPE (%) −2.4e-13 −8.9e-14 1.7e-13 −2.7e-13 0 −6.7e-14 6.8

n2 8 9 10 11 12 13 14

n2 8 9 10 11 12 13 14

MPE (%) 1.1 0.16 0.22 0.96 0.57 1.9 1.7

MPE (%) 8.8 37 2.1 4.1 4.4 16 26

modes with low MPE error, the fourth and fifth BARC modes can be described by the first fixed base mode and the sixth BARC mode can be described by the third fixed base mode. Low MPE indicates that the laboratory shapes span the space of the field shapes. Because the linear combination of the laboratory shapes produce the overall shape from the field, the displacement and stress response between the two systems have little difference. However, the threshold between “low” and “high” MPE and the related stress values are unknown. The BARC system modes and their corresponding MPEs provide an opportunity to examine the physical correlation between MPE and error in stress. The following comparisons assume that three laboratory modes can be excited and controlled simultaneously. The amplitudes of the top three modes are calculated through a pseudo inverse as ¯ q¯L = φ + L φF n ,

(8.2)

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Fig. 8.13 Stress comparison between the removable component on a rigid fixture to the ninth field mode os the BARC

where q¯L are the modal coordinates or amplitudes of the laboratory mode shapes. Two field modes, seventh and ninth, are examined with corresponding MPEs of 6.8% and 37%, respectively. The seventh field mode and the corresponding laboratory modes used to reconstruct the field mode are shown in Fig. 8.13. From examination of the stress plots, it can be seen that there will be areas of the RC that will experience higher stresses and areas that will experience lower stresses due to the minimization of the least squared error nature of the pseudo inverse in Eq. 8.2. Even with these errors, some parts of the RC will be tested to the correct levels that could have similar stresses depending on the input. The ninth field mode and the corresponding laboratory modes used to reconstruct the field mode are shown in Fig. 8.14. The laboratory modes needed to reconstruct the field mode do not share much resemblance of the field mode when examining the stress. Because of the poor reconstruction, the stresses induced by the ninth field mode will not be properly represented in the laboratory test.

8.6 Conclusion and Future Work The MPE is shown to be able to predict if a shaker test using a rigid fixture can reproduce the response of a component field environment. This conclusion is drawn by calculating low MPE for a laboratory test with a rigid fixture that successfully reproduced a field environment. The high MPE calculated for the BARC structure also shows evidence that MPE can predict the success using a rigid fixture in a laboratory test. The high MPE supports this conclusion because the RC of the BARC is designed to have different mode shapes when attached to a rigid fixture. In order for the MPE to provide insightful results, it is shown that the DOFs chosen from the component is important as choosing only the connection DOFs for the RC provided misleading results. Using a full field DOFs to represent the modes of the RC along with a concatenation of rigid body shapes and fixed base shapes provides useful MPE values.

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Fig. 8.14 Stress comparison between the removable component on a rigid fixture to the ninth field mode os the BARC

The MPE also indicates what modes can and cannot be reproduced for a specific laboratory test. This conclusion is shown specifically in the BARC field example as some BARC modes appeared to look like fixed base modes and other modes not. The MPE corroborated these observations. This knowledge of what field modes can and cannot be represented in the laboratory provides information and can set expectations on what the laboratory test can and cannot achieve. Future work on this topic could include research on tying the MPE to errors in stress by using the relationship between relative displacement and strain. This development would help in quantifying the MPE with respect to acceptable values. In lieu of a quantitative representation of MPE, a larger portfolio of case studies of MPE would aid in its predictive ability on the success of a test. Although MPE was calculated for a six DOF, this process could also be applied to a single axis shaker as well and could be an area of further research.

References 1. Gregory, D.L., Bitsie, F., Smallwood, D.O.: Comparison of the response of a simple structure to single axis and multiple axis random vibration inputs. Technical report, Sandia National Lab.(SNL-NM), Albuquerque (2009) 2. Paripovic, J., Mayes, R.L.: Reproducing a component field environment on a six degree-of-freedom shaker. In: Proceedings of the 38th IMAC, Feb 2020 (2020) 3. Schoenherr, T., Clark, B., Coffin, P.: Improve replication of in-service mechanical environments. Technical Report SAND2018-10187, Sandia National Laboratories, Sept 2018 (2018) 4. Soine, D.E., Jones, R.J., Harvie, J.M., Skousen, T.J., Schoenherr, T.F.: Designing hardware for the boundary condition round robin challenge. In: Topics in Modal Analysis & Testing, Vol. 9, pp. 119–126. Springer International Publishing, Cham (2019)

Chapter 9

On Dynamic Substructuring of Systems with Localised Nonlinearities Thomas Simpson, Dimitrios Giagopoulos, Vasilis Dertimanis, and Eleni Chatzi

Abstract Dynamic substructuring methods encompass a range of techniques, which allow for the decomposing of large structural systems into multiple coupled subsystems. This decomposition of structures into smaller domains has the principle benefit of reducing computational time for dynamic simulation of the system by considering multiple smaller problems rather than a single global one. In this context, dynamic substructuring methods may form an essential component of hybrid simulation, wherein they can be used to couple physical and numerical substructures at reduced computational cost. Since most engineered systems are inherently nonlinear in nature, particular potential lies in incorporating nonlinear treatment in existing sub-structring schemes, which are mostly developed within a linear problem scope. The most widely used and studied dynamic substructuring methods are classical linear techniques such as the CraigBampton and MacNeal-Rubin methods. These methods are widely studied and implemented in many commercial FE packages. However, as linear methods they naturally break down in the presence of nonlinearities. Recent advancements in substructuring have involved the development of enrichments to the linear substructuring methods, which allow for some degree of nonlinearity to be captured. The use of mode shape derivatives has been shown to be able to capture geometrically non-linear effects as an extension to the Craig-Bampton method. Other candidates include the method of Finite Element Tearing and Interconnecting. Linear substructuring methods have been used in several cases for hybrid simulation, rendering additional benefits in removing non-physical high frequency modes which may be excited due to controller tracking errors in hybrid simulations. In this work, a virtual hybrid simulation is presented in which a linear elastic vehicle frame supported on four nonlinear spring damper isolators is decomposed into separate domains. One domain consisting of the finite element model of the vehicle frame, which is reduced using the Craig-Bampton method to only include modes relevant to the structural response. The second domain consists of the nonlinear isolators whose restoring forces are characterised by nonlinear spring and damper forces. Coupling between the models is carried out using a Lagrange multiplier method and time series simulations of the system are conducted and compared to the full global system with regards to simulation time and accuracy. Keywords Dynamic substructuring · Hybrid Simulation · Nonlinearity

9.1 Introduction In this study we investigate implementation of dynamic substructuring schemes with hybrid simulation. Dynamic substructuring (DS) methods are used to decompose dynamic models into several separate regimes or substructures. The global model of the system is always recoverable via use of coupling constraints enforced between substructures. The key benefit of DS methods lies in the capability of analyzing each substructure separately and possibly in parallel, eventually assembling the global solution; this can greatly reduce the computational burden of a given analysis. However, further benefits are generated from the implementation of different analysis schemes on different regions of a structure. A key motivation of the above

T. Simpson () · V. Dertimanis · E. Chatzi Department of Civil, Environmental and Geomatic Engineering, Institute of Structural Engineering, ETH Zürich, Zürich, Switzerland e-mail: [email protected]; [email protected]; [email protected] D. Giagopoulos Department of Mechanical Engineering, University of Western Macedonia, Kozani, Greece e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2021 A. Linderholt et al. (eds.), Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47630-4_9

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being the ability to individually treat nonlinear regions, without the requirement for adoption of a global nonlinear solution method [1]. Dynamic substructuring methods are typically combined with reduction methods, such as Component Mode Synthesis (CMS), whereby reduced order models of the individual substructures are determined, which further decreases the computational load of the solution process. Most traditional techniques such as the Craig-Bampton and Rubin-Macneal methods [2], rely on a modal decomposition of the substructures whereby only a subset of mode shapes is retained -those relevant to the structural response- whilst the remaining modes are ignored. The commonly adopted linear schemes are naturally not suitable for the treatment of nonlinear systems. Recent advancements in substructuring have involved the development of enrichments to the linear substructuring methods, [3–5], which allow for some degree of nonlinearity to be captured. The use of mode shape derivatives has been shown to capture geometrically non-linear effects as an extension to the Craig-Bampton method [3]. An alternative to these methods is the so-called finite element tearing and interconnecting (FETI) technique [6] in which a coupling method making use of Lagrange multipliers is used, whilst the reduction of the substructures can be carried out separately. This allows for the use of any generic meta-modelling technique for the reduction of the non-linear substructures, such as a non-linear autoregressive models or a Gaussian process regression. These techniques can be shown to provide accurate representation of a complex model over a given input range of interest [7–9]. The FETI method can also allow for the coupling of nonlinear substructures or linear to nonlinear substructures. This has previously been used in hybrid simulations [10] and was shown to perform well. Hybrid testing, which forms a focus of the European ITN Project DyVirt, on Dynamic Virtualization, involves the setup of coupled dynamic tests in which a numerical system in a computer is coupled to a physical component in a dynamic testing rig. The system is then solved as a whole by integrating the numerical system, calculating predicted forces at the interface between the structures and applying these to the physical system. The response of the physical system is then measured and fed back into the equations of the numerical system [11]. The benefit of hybrid testing is that simple or well characterised components of a system can be represented numerically, whilst more complex or critical components can be physically tested. This allows for a reduction in cost and complexity of experimentation against full scale testing with an increase in fidelity and reduction in uncertainty with respect to purely numerical modelling. The coupling between the physical and numerical components within hybrid testing is a parallel of the DS problem, which attempts to couple numerical systems; as such, similar methods to those in DS can be used [12]. Within the domain of hybrid testing the “gold standard” is considered to be real time hybrid testing. Real time testing implies that the time scales of the numerical and physical system are the same; and that this time scale matches that of “wall time”. The benefit of real time testing is that it allows for rate dependent behaviour and nonlinearities of the physical system to be accounted for [11]. However as we attempt to approach real time testing the problem of the time required to integrate the numerical substructure becomes an issue. In order to carry out a real time hybrid simulation, each step of the numerical integration must be able to be carried out within a smaller time increment than the integration time step. This becomes a problem when numerical substructures of increasing complexity are considered and leaves the options of either increasing the time step of the integration which can have negative effects on accuracy and stability [13], or finding a reduced order model of the numerical system which can be solved more quickly. However, there remains considerable potential for the use of model order reduction techniques in combination with hybrid simulation. Application of CMS methods in hybrid testing have been made as in [10], although in this case the key purpose of the dynamic substructuring was to remove high frequency modes from the numerical system to yield improved stability characteristics, rather than to improve the speed of calculation. Another issue which arises in hybrid simulation is that as the integration time step is increased to allow for real time simulation, the time step for control of the physical system is also increased to match it in order to couple the systems. This increased time step of the physical system can result in a non-smooth operation of the actuator due to control commands being sent at such a large interval. This can cause issues with regards to the fidelity of the applied loads and issues of stability in the solution [14]. As such, when considering the solution time step not only must the numerical integration be considered but rather also the physical control system. This problem has inspired algorithms which allow for adoption of individual time steps in the physical and numerical subsystems [15, 16].

9.2 Theory The key elements used in the virtual hybrid simulation we present herein are the model order reduction carried out on the linear “numerical” system, and the integration algorithm which solves the systems and imposes coupling between them.

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9.2.1 Craig-Bampton Reduction The Craig-Bampton is one of the most prominent and long standing techniques in CMS and is already implemented in numerous commercial finite element softwares [17]. In the Craig-Bampton method each sub-structure can be reduced individually and then usually the reduced sub-structures are coupled together using a primal assembly. However, in this case only the linear system will be reduced whilst the coupling procedure will be carried out within the integration algorithm. The key aspects of the reduction are that the degrees of freedom (DOFs) of each substructure are partitioned into internal DOFs, xi , and external DOFs, xb , i.e., those which lie on the interface between substructures. 

Mii Mib Mbi Mbb

     x¨i Kii Kib xi + =0 x¨b Kbi Kbb xb

(9.1)

The internal DOFs, which in most cases are the majority, can be significantly reduced using a modal decomposition from the eigenvalue problem in equation 9.2. Given that for structural dynamics the lowest eigenmodes are dominant, the majority of these modes can be discarded, with only the lower modes, r , retained. These are known as the fixed interface normal modes. Mii−1 Kii = Σ T ,

  = r d

(9.2)

In addition to the fixed interface normal modes, constraint modes, , are also used to characterise the boundary DOFs and the static response. Constraint modes correspond to the static deformation shape due to a unit displacement applied at a boundary DOF. As such there exist as many, as there exist boundary DOFs. These are calculated as follows.

= −Kii−1 Kib

(9.3)

Using both the fixed interface and constraint modes, the original high dimensional coordinate set can be approximated on a significantly lower coordinate set, with the reduction of DOFs dependant upon the number of fixed interface modes that are discarded. The reduced coordinate set consists of the generalised internal coordinates q and the retained boundary coordinates xb .        xi r

q q ≈ = CB xb 0 I xb xb

(9.4)

The mass and stiffness matrices of the sub-structures can then be reduced by projection onto these reduced coordinate sets. Kˆ = CB T KCB

Mˆ = CB T MCB

(9.5)

In the Craig-Bampton method the reduction basis consists of the generalised internal coordinates, whilst the boundary coordinates are retained as physical; this simplifies the coupling procedure.

9.2.2 Integration and Coupling The coupled integration scheme used herein is inspired by that presented in [16], this comprises a technique designed for hybrid simulation in which each of the substructures is solved as a free solution over the time interval with coupling enforced at the end of the time interval using a method similar to the FETI method [6]. The partitioned design of this integration method allows it to be applied in a hybrid test in that, the 2 systems are solved in parallel with coupling only enforced at the end of each time step. This means that in the context of a hybrid test, displacements can be enforced on the physical system, with the resulting restoring force being measured and being used to solve the free problem in the physical system and finally coupling being enforced at the end of the time step. The coupled equations of motion for the numerical and physical systems to be integrated are shown in equations 9.6 and 9.7. These present the typical equations of motion of a nonlinear mechanical system in state space form. However, these

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equations are augmented with the term L, this represents the coupling forces between the two systems with  being the Lagrange multipliers giving the interface force intensities and L a boolean matrix which locates the interface forces on the overall system. N N N M N Y˙n+1 + R N (Yn+1 ) = LN n+1 + Fn+1

(9.6)

P P P M P Y˙n+1 + R P (Yn+1 ) = LP n+1 + Fn+1

(9.7)

Each of the physical and numerical subsystems is solved over a given time interval. This is initially carried out as a free problem, i.e., ignoring the coupling of the other substructures within this interval. For the free solution of the substructures a trapezoidal integration rule was used which corresponds to a γ parameter = 0.5. The trapezium rule integration uses a prediction and a correction step in order to step forward in the integration. The prediction step as indicated in equation 9.8 gives the predicted state at the next integration step based on the previous state and state rate values, the integration parameter γ and the time step T . F Y˜n+1 = Yn + (1 − γ )T Y˙n

(9.8)

F indicates the predicted state of the free solution at the next time step which will be corrected to Within equation 9.8, Y˜n+1 F . The Predicted state is made based on the solutions from the previous coupled time step: Y find the final free solution Yn+1 n and Y˙n . The predicted state rate is then calculated using the predicted state, according to equation 9.9 making use of the matrix D assembled as in equation 9.10 F F Y˙n+1 = D −1 (Fn+1 − R(Y˜n+1 ))

D = M + γ T R0 ;

(9.9) (9.10)

In the above equations F is the external force vector, R(Y ) evaluates the potentially nonlinear restoring force vector, consisting of the elastic and damping forces, evaluated at state Y and D is a matrix as formed in equation 9.10 wherein M is the mass matrix, and R0 the tangential restoring force vector linearised at zero state. This tangential restoring force vector was found using the ADIGATOR toolbox for automatic differentiation [18]. This toolbox allows the tangential stiffness matrix to be found numerically inclusive of systems with nonlinear restoring forces. The correction step is then made as in equation 9.11 making use of the predicted state and state rates found previously. F F F = Y˜n+1 + γ T Y˙n+1 Yn+1

(9.11)

Having found the free solution for each of the substructures individually and in parallel, the coupling can then be enforced between them. The coupling is enforced using a dual assembly procedure, by enforcing equal and opposite forcing at the boundary DOFs using Lagrange multipliers. Firstly, a matrix is assembled, as in equation 9.12, which forms the so called Steklov-Poincaré operator [19]. This relates the discrepancy in state at the boundary nodes to the Lagrange multiplier intensities. −1 N H = GN DN L + GP DP−1 LP

(9.12)

In equation 9.12 the matrices GN and GP are boolean matrices, which enforce compatibility of the substructures [2] for the numerical and physical systems respectively, whilst LN and LP are boolean matrices, which locate the interface forces for the numerical and physical structures [2]. Having derived the Steklov-Poincaré operator, which is assembled only once at the beginning of the integration procedure, the Lagrange multipliers can be identified based on the free solutions of each of the substructures as in equation 9.13. n+1 = −H −1 (GN Yn+1 + GP Yn+1 ) N,f

P ,f

(9.13)

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The link solutions for each of the substructures are then found, these represent the effect on the state of the substructure of the interface forces due to coupling. These link solutions are found by first finding the state rate due to the interface forces as in equation 9.14 and then the link state using this state rate as in equation 9.15. P ,L Y˙n+1 = DP−1 LP n+1 P ,L P ,L Yn+1 = γ T Y˙n+1

N,L −1 N = DN L n+1 Y˙n+1

(9.14)

N,L N,L Yn+1 = γ T Y˙n+1

(9.15)

Finally the global solution for each of the substructures taking into account the coupling between them is found as in equation 9.16 by summing the free and link solutions for each of the substructures. P ,F P ,L P = Yn+1 + Yn+1 Yn+1

N,F N,L N Yn+1 = Yn+1 + Yn+1

(9.16)

9.2.3 With Sub-cycling In order to ameliorate the problem of actuator smoothness in hybrid simulations, wherein the numerical integration time step is necessarily larger than the control frequency of the actuator, a set of methods exist in which the physical system is controlled at a smaller time step than the numerical system, with coupling enforced at the larger numerical time step [16]. In connection with the above described algorithm, the sub-cycling method can be implemented in the calculation of the free solution of the physical substructure. Firstly, a number of sub-cycles is decided upon, defined as the ratio of the numerical time step to the physical time step. The integration scheme with sub-cycling operates similarly as described previously except that the free solution for the physical substructure, as described by equations 9.8, 9.9, 9.10, and 9.11 above, is now replaced by the following procedure. The sub-cycling loop runs over the number of sub-cycles ss. It is initiated by the coupled solution at the previous numerical system time step. The loop ends by finding the free solution at the next numerical time step. Equation 9.17 shows the prediction step of the trapezium rule in which T P is the time step of the physical system. Y˜ P ,Fj = Y P ,Fj −1 + (1 − γ )T P Y˙ P ,Fj −1 n+ ss

n+

n+

ss

(9.17)

ss

Equation 9.18 calculates the state rate prediction in which it is worth noting that the Lagrange multipliers from the last coupling time step are now included in the solution and that the D matrix is formed as in equation 9.10 albeit with the finer time step of the physical system used in the assembly. Y˙ P ,Fj = DP−1 (F P

j n+ ss

n+ ss

j − R(Y˜ P ,Fj ) + LP λ(1 − )) n+ ss ss

(9.18)

Finally, the corrected free solution is calculated as in equation 9.19. Y P ,Fj = Y˜ P ,Fj + γ T P Y˙ P ,Fj n+ ss

n+ ss

n+ ss

(9.19)

The sub-cycling loop then ends after ss sub-cycles whereby the n + 1 prediction of the free state of the physical system is found and the coupling procedure can then be followed as described in the previous section.

9.3 Case Study The case study considered herein is a steel vehicle-like frame considered to be linear elastic, which is mounted on four nonlinear suspension structures. Computer models of the frame and suspension structures were created based on physical specimens. The frame structure model, along with the mounting points are presented in Fig. 9.1. The frame structure is considered to be linear elastic and is meshed using a combination of quadrilateral shell elements and hexahedral solid elements with a total of 45,564 DOFs. The material properties of the frame are a Youngs modulus of

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Fig. 9.1 Vehicle frame structure model with attachment points of suspension indicated

210 GPa, a Poissons’ ratio of 0.3 and a density of 7800 kgm−3 . This model has previously been calibrated to a physical structure, details of which along with dimensions of the frame can be seen in [20]. The nonlinear suspension systems consist of a base excited mass representing the wheel and its interaction with the road surface, alongside a linear spring and nonlinear damping element which connect the wheel mass with the frame structure. These nonlinear suspension elements have also been identified from a physical component connected to the physical frame in a previous study [21]. The best model found to approximate the non linearity in the damper was found to be a dry friction type damping force [20]. The restoring force equations of the nonlinear suspension system are shown in equation 9.20, in which k1 represents the linear stiffness coefficient, c1 the linear damping coefficient and, c2 and c3 the nonlinear damping coefficients. The coefficients are summarised below. fr (x) = k1 x

fd (x) ˙ = c1 x˙ +

c2 x˙ c3 + |x| ˙

m = 0.160, k1 = 35, c1 = 0.65, c2 = 10, c3 = 0.55

(9.20) (9.21)

The form of the nonlinearity present in the damper is illustrated in Fig. 9.2 where it is seen that the frictional nonlinearity delivers a more pronounced effect around the zero point, whilst further from zero more linear behaviour is observed. Figure 9.3 presents the linear FE mesh of the frame structure and a diagrammatic representation of the nonlinear suspension systems. In the case of this virtual hybrid simulation, the linear frame is considered to be the numerical system which is to be reduced in order, whilst the suspension systems are considered to be the physical systems to be solved in full order.

9.4 Virtual Hybrid Simulation A virtual hybrid simulation was carried out in which the case study described above was considered. The linear frame structure was considered to be the numerical system with the four suspension substructures considered as the physical systems. The linear system was first reduced using a Craig-Bampton reduction, and then the partitioned integration procedure was implemented on the substructures.

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Fig. 9.2 Restoring force curve of the nonlinear damper elements

Fig. 9.3 Linear and nonlinear substructures of the vehicle frame and suspension. (a) Linear FE mesh of the vehicle frame substructure. (b) Nonlinear suspension subsystem model

9.4.1 Reduction of the Linear Frame The linear substructure was reduced according to the methodology described in Sect. 9.2.1. As described earlier, a subset of fixed interface mode shapes were retained, which represent the dynamic behaviour of the structure within the frequency region of interest. The reduction basis was formed on the 30 lowest fixed interface normal mode shapes of the frame, along with 4 constraint modes each corresponding to the vertical DOFs at the 4 mounting points of the suspension systems. Figures 9.4 and 9.5 demonstrate the fidelity of the reduction basis relative to the full model of the frame structure. In Fig. 9.4 a comparison is made between the 20 lowest natural frequencies and modal shapes between the full model and the CB reduction. Figure 9.4a shows the cross modal assurance criterion [22] between the mass orthogonalised mode shapes of

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Fig. 9.4 Mode shape and natural frequency fidelity of the CB reduced system

Fig. 9.5 Time series response fidelity of the CB reduced system

the full system and the CB reduced system. This gives a metric for the orthogonality of the mode shapes to one another and is calculated as in equation 9.22. MAC( CB , F ) =

# # # T F #2 CB ( TCB CB )( TF F )

(9.22)

For identical mode shapes the diagonal of the plot should show one whilst all off diagonal elements should be zero. The plot demonstrates the fidelity of the mode shapes, as all diagonal elements are very close to one whilst all off diagonals are close to zero. Figure 9.4b compares the first 20 natural frequencies of the reduced and full systems, along with the normalised mean square error (NMSE) between them. The natural frequencies of the systems are very similar with none of the first 20 natural frequencies exhibiting an error larger than 0.1%. Figure 9.5 compares the time history response of both the reduced and full order systems to the same input. The input used in this case was a band limited white noise signal between 0 and 200 Hz with a variance of 1 which was applied in the vertical direction at the four nodes to which the suspension systems connect. This frequency region was chosen as the region of interest as ISO 8608, detailing the classification of road surface profiles, considers the region of interest to be up to a spatial frequency of 17.77 rad/m, at an assumed vehicle speed of 100 km/h. This corresponds to a maximum frequency of

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Fig. 9.6 White noise corrupted multi sine input signal

interest of 78.6 Hz [23]. The frequency range of interest was then selected to be comfortably above this value so as to ensure fidelity within the region. The reconstruction of the time series response is near perfect for this input with a mean square error value of 1.04e − 08. According to both the eigenvalue and time series analyses the Craig-Bampton reduced system shows excellent fidelity to the full order model when excited. In both cases a Newmark integration scheme was used in the time series simulation.

9.4.2 Comparison of Monolithic and Partitioned Solutions To assess the performance of the partitioned solution of the reduced order system, it was compared to a monolithic solution of the full system with regards to both fidelity and simulation time. The inputs chosen for the simulation were all multi sine excitations corrupted by white noise. The inputs were applied as base motion to the masses in the suspension substructure. Whilst each of the suspension substructures were excited with the same multi-sine content, they differed in the white noise corruption. Figure 9.6 shows an exemplar signal applied to one of the four suspension systems. Figure 9.7 compares the response of the DS reduced system to the monolithic solution produced in Abaqus. The solutions show good agreement although some discrepancy can be seen particularly around the peaks. The mean squared error between the signals was 2.63e − 11 showing the good fidelity. Some discrepancy is expected between the responses firstly due to the reduced linear system not performing identically to the full linear system. Additionally, the tangential stiffness matrix which is used for the coupling procedure introduces errors as it is effectively a first order linearisation of the coupling process. In both the monolithic and partitioned solutions displayed, a time step of 1e − 3 was used and 1 s of time history was simulated. This simulation from the monolithic solver required 1557 s compared to 0.15 s for the DS reduced system. This demonstrates the massive time savings possible using the reduction scheme. Additionally, it is noteworthy that the DS solution was found in faster than real time hence demonstrating its potential for real time hybrid simulation. It is worth noting that there are additional offline calculations required for the DS solution such as, the CB reduction and the forming of tangential stiffness matrices, the total offline time for these operations was 109 s. As such the overall time was still considerably less for the DS reduced system and furthermore the offline times are not of great concern with regards to hybrid simulation. Figure 9.8 shows the solution of one of the boundary DOF of the numerical system and the boundary DOF of the physical system to which it is coupled. There is very close agreement between the two solutions showing that the coupling procedure was successful. However, Fig. 9.8b shows that there is a slight discrepancy indicating that the coupling procedure does not

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Fig. 9.7 Comparison of the monolithic and partitioned solution of the time series response of the reduced system

Fig. 9.8 Response of physical and numerical boundary DOF

enforce hard compatibility between the two substructures but rather that a slight discrepancy is allowed as a result of the dual assembly [24].

9.4.3 Subcycling The effect of the addition of the sub-cycling of the physical system in the algorithm was also investigated. Firstly a comparison was made between the non sub-cycling algorithm with a time step of 1e − 4 and the sub-cycling algorithm with 10 sub-cycles and a time step of 1e − 3 this comparison is presented in Fig. 9.9. The overall response of the two algorithms are similar however, it is clear that there is some loss in fidelity. Overall, the sub-cycling algorithm exhibits stability issues.

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Fig. 9.9 Response integrated at T = 1e − 4 and T = 1e − 3 with 10 sub-cycles

Fig. 9.10 Response of physical and numerical boundary DOF with 10 sub-cycles

Previous implementations of the method in [16] and [15] adopted an algorithm with numerical damping, as opposed to the undamped trapezium rule method used here. It is thought that the use of a damped algorithm could ameliorate these issues. The algorithm shows potential however given that the sub-cycling algorithm can allow the time step of the physical system to be decreased by order 10 with an increase of computational time of approximately 2 times. It is also evident from Fig. 9.9b that the sub-cycled system did exhibit a smoother response in the physical system, which should improve the control of the actuator. Figure 9.10 presents the response of a coupled boundary DOF on the numerical and physical substructures. Compared to the case without sub-cycling there is a clearly discrepancy, this discrepancy again highlights the capacity of the dual coupling procedure to allow for a soft coupling procedure. As seen in Fig. 9.10b the response of the physical system is initially very uneven which seems to cause the discrepancy which is maintained throughout. This unevenness is once again thought to be a result of the instability in the sub-cycling procedure, as highlighted previously.

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9.5 Conclusions A dynamic sub-structuring process has been presented, which can be used to couple linear and nonlinear systems in a manner suitable to hybrid simulation. It was demonstrated that this method can considerably decrease the required computational time and was also shown to reduce the integration time sufficiently to implement a real time hybrid simulation. The reduced system was also shown to deliver good fidelity when compared to the full monolithic system. The dual assembly procedure carried out was shown to enforce coupling between the substructures without enforcing hard compatibility and allowed an amount of discrepancy. The DS integration algorithm was also implemented using a sub-cycling procedure whereby the physical system was integrated at a time step smaller than that of the numerical system. This method had considerable problems with instability of the physical system but it is thought that with the use of an algorithm with numerical damping these problems could be improved. It was shown that the use of the sub-cycling method resulted in a smoother response in the physical system which has advantages for the control of the actuator related to the physical system. Acknowledgments This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 764547.

References 1. Voormeeren, S.N., van der Valk, P.L.C., Rixen, D.J.: Practical Aspects of Dynamic Substructuring in Wind Turbine Engineering, pp. 163–185. Springer, New York (2011) 2. Van der Valk, P.L.C.: In: TU Delft (ed.) Model Reduction & Interface Modeling in Dynamic Substructuring: Application to a Multi-Megawatt Wind Turbine., Masters Thesis, Delft Netherlands (2010) 3. Wu, L.: In: TU Delft (ed.) Model Order Reduction and Substructuring Methods for Nonlinear Structural Dynamics., Doctoral Thesis, Delft Netherlands (2018) 4. Kim, J.-G., Lee, P.-S.: An enhanced Craig-Bampton method. Int. J. Numer. Methods Eng. 103(2), 79–93 (2015) 5. Kim, J.-H., Kim, J., Lee, P.-S.: Improving the accuracy of the dual Craig-Bampton method. Comput. Struct. 191, 22–32 (2017) 6. Farhat, C.: A method of finite element tearing and interconnecting and its parallel solution algorithm. Technical report (1991) 7. Costas, C., Papadimitriou, E.N. (eds.): Identification Methods for Structural Health Monitoring. Springer International Publishing, (2016). ISBN:978-3-319-32075-5 8. Mai, C.V., Spiridonakos, M.D., Chatzi, E.N., Sudret, B.: Surrogate modelling for stochastic dynamical systems by combining NARX models and polynomial chaos expansions. Technical report (2016) 9. Gruber, F.M., Rutzmoser, J.B., Rixen, D.J.: Comparison between primal and dual Craig-Bampton substructure reduction techniques. In: Proceedings of the 11th International Conference on Engineering Vibration, pp. 1245–1254, Ljubljana (2015) 10. Miraglia, G., Petrovic, M., Abbiati, G., Mojsilovic, N., Stojadinovic, B.A.: Model-order reduction framework for hybrid simulation based on component-mode synthesis. Earthquake Engng Struct Dyn. 49, 737–753 (2020). https://doi.org/10.1002/eqe.3262 11. Benson Shing, P.: Real-Time Hybrid Testing Techniques, pp. 259–292. Springer, Vienna (2008) 12. Wagg, D.J., Gardner, P., Barthorpe, R.J., Worden, K.: On key Technologies for Realising Digital Twins for structural dynamics applications. In: Barthorpe, R. (ed.) Model Validation and Uncertainty Quantification, vol. 3. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham (2020) 13. Pegon, P.: Continuous PsD Testing With Substructuring, pp. 197–257. Springer, Vienna (2008) 14. Ahmadizadeh, M., Mosqueda, G., Reinhorn, A.M.: Compensation of actuator delay and dynamics for real-time hybrid structural simulation. Earthq. Eng. Struct. Dyn. 37(1), 21–42 (2008) 15. Abbiati, G., La Salandra, V., Bursi, O.S., Caracoglia, L.: A composite experimental dynamic substructuring method based on partitioned algorithms and localized Lagrange multipliers. Mech. Syst. Signal Process. 100, 85–112 (2018) 16. Abbiati, G., Lanese, I., Cazzador, E., Bursi, O.S., Pavese, A.A.: Computational framework for fast-time hybrid simulation based on partitioned time integration and state-space modeling. Struct. Control. Health Monit. 26, e2419 (2019). https://doi.org/10.1002/stc.2419 17. Craig, R., Bampton, M.: Coupling of substructures for dynamic analyses. AIAA J. Am. Inst. Aeronaut. Astronaut. 6(7), 1313–1319 (1968) 18. Patterson, M.A., Weinstein, M., Rao, A.V.: An efficient overloaded method for computing derivatives of mathematical functions in matlab. ACM Trans. Math. Softw. 39(3), 17:1–17:36 (2013) 19. Quarteroni, A., Valli, A.: Theory and application of Steklov-Poincaré operators for boundary-value problems. In: Spigler, R. (ed.) Applied and Industrial Mathematics. Mathematics and its Applications, vol. 56. Springer, Dordrecht (1991) 20. Giagopulos, D., Natsiavas, S.: Hybrid (numerical-experimental) modeling of complex structures with linear and nonlinear components. Nonlinear Dyn. 47(1–3), 193–217 (2006) 21. Tsotalou, M., Giagopoulos, D., Dertimanis, V., Chatzi, E.N.: Model updating of a nonlinear experimental vehicle using substructuring and unscented Kalman filtering. Proceedings of the 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering, pp. 66–75 (2017) 22. Pastor, M., Binda, M., Hararik, T.: Modal assurance criterion. Proc. Eng. 48, 543–548 (2012) 23. Múˇcka, P.: Simulated road profiles according to ISO 8608 in vibration analysis. J. Test. Eval. 46, 405–418 (2017) 24. Rixen, D.J.: A dual Craig–Bampton method for dynamic substructuring. J. Comput. Appl. Math. 168(1–2), 383–391 (2004)

Chapter 10

Source Characterization for Automotive Applications Using Innovative Techniques J. Harvie and D. de Klerk

Abstract Transfer path analysis (TPA) and source characterization using the in-situ blocked force methodology is becoming increasingly common in the automotive world. While robust techniques exist for this type of characterization in general, there are certain conditions where the analysis is more straight-forward than others. In this work, several techniques are presented to help improve the characterization across different frequency ranges. At the very low frequencies, where structures should behave rigidly, TPA results can be improved by filtering out any non-rigid body motion from a set of measured FRFs. In the mid-frequency range, testing can be simplified using a volume source to capture reciprocal FRFs and then predict sound levels at the driver’s ear. In the mid- and high- frequency ranges, the addition of rotational FRFs can help improve TPA predictions. These techniques are demonstrated using recent test results on various components and vehicles in this paper. Keywords Transfer path analysis · Automotive · Frequency response functions · Blocked forces

Nomenclature u f g Y AB A ;B 1 2 3 4 DoF IDM FRF NTF SNR TPA VP

Dynamic displacements/rotations Applied forces/moments Interface forces/moments Admittance FRF matrix Pertaining to the assembled system Pertaining to the active/passive component Source excitation DoF Interface DoF Receiver DoF Indicator DoF Degree of freedom Interface displacement mode Frequency response function Noise transfer function Signal to noise ratio Transfer path analysis Vvirtual point

J. Harvie () VIBES.technology, Delft, The Netherlands e-mail: [email protected] D. de Klerk VIBES.technology, Delft, The Netherlands Delft University of Technology, Faculty of Mechanical, Maritime and Material Engineering, Department of Precision and Microsystems Engineering, Section Engineering Dynamics, Delft, The Netherlands Müller-BBM VibroAkustik Systeme B.V, Hattem, The Netherlands © The Society for Experimental Mechanics, Inc. 2021 A. Linderholt et al. (eds.), Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47630-4_10

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Fig. 10.1 The transfer path problem: (a) based on the admittance of assembly AB and (b) based on the admittances of subsystems A and B [1]

10.1 Background Transfer path analysis is used to analyze the vibration transmission paths between an active source and a receiving structure. An in-depth overview of transfer path analysis techniques is provided in [1], and an overview is shown schematically in Fig. 10.1. Essentially, assembly AB consists of an active vibrating subsystem A and a passive receiving subsystem B. The system is excited by some force f1 that cannot easily be directly measured. It is desired to understand the response at the locations of interest u3 on the passive structure through an interface u2 due to this excitation. The response at the locations of interest on the passive component can therefore be described using the system FRFs YAB 31 with u3 = YAB 31 f1

(10.1)

While the internal excitation forces f1 are typically hard to measure and numerous, active sources are typically mounted with discrete points to their receivers. Hence finding an equivalent loadcase at the interface reduces the passive side loading to a manageable size / number. Such is the objective of component-based TPA, described next. Other families of TPA include classical TPA and transmissibility-based TPA, which are detailed in [1].

10.1.1 Component-Based TPA Component-based TPA involves determining a set of equivalent forces that are a property of the source alone and can therefore be applied to an assembly AB with any receiving side B. When applied to system AB, the equivalent forces feq will yield the same responses u3 as the original source f1 . Equivalent forces at the interface can be calculated using an in-situ approach [2, 3] with +

eq f2 = YAB u4 42

(10.2)

where u4 indicator sensors around the interface are used to improve the conditioning of the FRF matrix YAB 42 . This FRF matrix contains responses at the indicator sensors and inputs at the interface degrees of freedom (DoF). The equivalent force in Eq. 10.2 is often referred to as a blocked force in the literature, as the forces acquired will ideally be equal to the forces that would be measured if subsystem A was rigidly fixed or blocked at the interface. The equivalent forces can be validated by applying them to the assembly AB with eq

u3 = YAB 32 f2

(10.3)

and comparing the synthesized responses u3 to validation measurements. Equation 10.3 can also be used to predict responses of interest for new assemblies containing the active component A, as the equivalent forces are a property of only the source.

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Fig. 10.2 Schematic showing the virtual point transformation [4]

10.1.2 Virtual Point Transformation Additionally, to obtain the FRFs in Eqs. 10.2 and 10.3 with references exactly at the interfaces, the virtual point (VP) transformation can be used. With respect to TPA, the virtual point transformation can be used to provide a full 6-DoF description of the forces and moments acting exactly at the interface. The VP transformation is shown schematically in Fig. 10.2 and further detailed in [4]. Using this technique, forces are applied in a typical manner using an impact hammer or shaker, and a geometry-based transformation matrix is used to transform the FRF to have inputs exactly at the desired interface node(s), including both translations and rotations. Essentially, this transformation can be written using interface displacement modes (IDMs) as u = Ru q

⇒

q = (Ru )+ u

(10.4)

m = Rf T f

⇒

+

f = Rf T m

(10.5)

where the measured responses u are transformed to the q virtual point responses using the response IDM matrix Ru , and similarly the measured forces f are transformed to the m virtual point forces and moments using the force IDM matrix Rf . The IDM matrices are constructed to accurately characterize the problem at hand, often using six degrees of freedom (three translations and three rotations). The sensors and excitations are typically placed close to the virtual point such that rigid IDMs are used; however flexible IDMs can also be included. The consistency functions detailed in [5] can be used to assess whether the chosen IDMs accurately represent the observed dynamics. The sensor consistency is essentially calculated by transforming the measured responses to the virtual point, projecting the virtual point responses back to the original coordinate system, and comparing the “filtered” responses with the original raw signals. This produces a frequency-dependent metric that is equal to 1 if all sensor responses are accurately described by the chosen IDMs. A similar calculation can be done for the impacts. The sensor and impact consistencies are typically assessed for the measurements made around each of the attachment points of an active source.

10.1.3 Techniques Presented Here In-situ blocked force TPA using the virtual point transformation is becoming increasingly common in the automotive world. Under typical testing conditions where impact testing performs well, the TPA results are generally quite good. However there are certain conditions where the results can be improved using the techniques described below. Rigidness Correction for Low Frequency TPA At low frequencies (less than 20 or 50 Hz), it is often challenging to get enough consistent energy into large structures using a regular impact hammer. Data quality issues at the low frequencies are generally not uncommon when performing impact testing; for an overview of practical test considerations see [6]. These issues can be accepted in many applications where higher frequencies are the source of noise and vibration problems and the low frequency data quality is not paramount to the success of the analysis. However, for certain applications such as ride comfort, the lower frequencies can be of high interest. For such cases, the consistency of the data at the low frequencies should be reviewed. At the very low frequencies, all sensors should move together, exhibiting rigid body motion. The sensor consistency can be used to assess the rigidness of

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the data at these low frequencies. If the data are not consistent at the low frequencies, the virtual point transformation can be used to rigidify the FRFs. Using Eqs. 10.4 and 10.5, a set of FRFs can be transformed to a single 6-DoF virtual point near the center of an object, and then projected back to the original set of degrees of freedom. However, this new set of FRFs will only be accurate in the frequency range where the test object behaves rigidly. Therefore, prior knowledge of the first flexible mode of the test object should be used to select a cutoff frequency where the rigidified FRFs can be cross-faded into the original set of FRFs. Additionally, the FRF matrix will now be very poorly conditioned at the low frequencies where the data has been rigidified, containing only six substantial singular values to represent whatever FRF size has been measured. To use the new FRF matrix for in-situ TPA, or any activities involving a matrix inverse, only the first six singular values should be included in a truncated inverse. The new hybrid set of FRFs should produce more accurate TPA results at the lower frequencies while leaving the higher frequencies unaffected. Reciprocal FRFs for Mid-Frequency TPA Predictions As noted in Eq. 10.3, component-based TPA can be used to predict responses of interest (accelerations or sound pressure levels) using equivalent forces and the FRFs of the assembled system. The FRFs YAB 32 contain the responses at the locations of interest relative to the interface degrees of freedom. This FRF matrix is usually obtained by making several impacts around each interface and measuring the responses of interest. However, these FRFs can be obtained in an alternative way due to the property of reciprocity. Due to reciprocity, we can instead measure YAB 32 , with inputs at the locations where responses are desired, and outputs at the interface degrees of freedom. A review of reciprocity in this context is provided in [7, 8]. Practically, this means installing sensors around the interfaces, and then exciting the structure at the locations of interest using either an impact hammer for structural interests or a volume source for acoustic interests. The actual testing time will be greatly reduced for a test of this form. These FRFs can then be used in TPA predictions across whatever frequency range has good coherence, which is typically best in the mid-frequency range. Rotational FRFs for Mid- to High-Frequency TPA In general, it is easier to measure translational forces than rotational moments for both classical and component-based TPA. While 6-DoF force gauges exist that can be used to directly measure forces and moments, those force gauges can often be substantial in size, and therefore may not be able to measure the desired quantities exactly at the interfaces. With FRF-based TPA techniques, impact hammers are naturally only able to measure translational force inputs. Thus, TPA is often performed using only translational forces. TPA using only translational forces is likely accurate at low frequencies in most cases where the behavior of the components and systems is fairly rigid. However, with the current trends in the automotive industry shifting toward electric vehicles and components, higher frequencies are becoming more and more relevant and troublesome. At the higher frequencies where the dynamics are more complex, translational forces alone are likely inadequate for source characterization. It will be shown that including rotational degrees of freedom in the virtual point transformation can improve the results at those higher frequencies. The advantages of including rotations in source characterization have previously been discussed in [9] where the recommendation to include rotations in two upcoming ISO standards [10] [11] was introduced. Additionally, the use of an on-board validation sensor to validate the accuracy of the calculated blocked forces is also presented in [9] and will be used in this work.

10.2 Analysis 10.2.1 Rigidness Correction for Low Frequency TPA An active component was recently tested on a rigid test bench to determine blocked forces using various methods. The structure was equipped with 32 tri-axial accelerometers (96 channels) during both FRF testing and operational measurements, in order to calculate in-situ blocked forces with Eq. 10.2. The consistency of these sensors during FRF testing is shown in the left side of Fig. 10.3. As seen in the figure, the consistency is close to 100% from around 20 to 40 Hz. This means that all sensors are moving rigidly together in this frequency range. There are modes of the structure above 40 Hz (related to both the component and the test bench), so the consistency of all sensors is not expected to be high in that frequency range. However, at frequencies below 20 Hz, the structure should be moving rigidly and the sensor consistency should be higher than it is. It is believed that the poor consistency in this frequency range is primarily due to poor signal-to-noise ratio (SNR) on the very stiff test bench.

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Fig. 10.3 Sensor consistency of (left) raw FRF data and (right) rigidified FRF data

The FRFs were then transformed to a single 6-DoF virtual point and projected back to the original degrees of freedom, as described above. This set of rigidized FRFs was merged with the original FRFs with a cutoff frequency centered at 25 Hz. The sensor consistency of the new set of FRFs is presented on the right side of Fig. 10.3. As expected, the “filtered” FRFs behave very rigidly and consistently at the low frequencies, while leaving the sensor consistency at higher frequencies unaffected. An operational test with broadband content was then performed to do in-situ source characterization and TPA. Of the 32 sensors used in the test, 30 were used as indicators (u4 ) and 2 were used as validation sensors (u3 ). A total of 40 interface DoF were considered. The blocked forces were calculated using both the raw FRFs and the rigidified FRFs. These blocked forces were applied to the system using Eq. 10.3, and TPA results are compared to the measured response at one of the validation sensors in Fig. 10.4. As seen, both predictions match the measured response well at higher frequencies, but there is a discrepancy between the measured response and the raw FRF TPA at very low frequencies (below 10 Hz). By rigidifying the FRFs, the TPA results are greatly improved at those low frequencies.

10.2.2 Reciprocal FRFs for Mid-Frequency TPA Predictions A separate test was recently done on a full vehicle for source characterization of the tire noise. Sensors were placed around each of the wheel hubs during FRF and operational tests, again for in-situ blocked force TPA. Impacts were made around each wheel hub and transformed to virtual points, and these virtual point FRFs were used to calculate blocked forces during a constant speed test with Eq. 10.2. For this test series, the response of interest was the sound at the driver’s ear. To calculate this response due to the blocked forces, two sets of FRFs were measured. First, the FRFs YAB 32 between a microphone at the driver’s ear and the impacts at the wheel hubs were measured, as is usually done. Additionally, a low frequency volume source (LFS) was placed at the driver’s ear position, and the accelerometers around the wheel hubs were used to measure YAB 23 . A sample noise transfer function (NTF) from the volume source measurement is shown in Fig. 10.5 where the coherence looks reasonable up to approximately 500 Hz. A few comparisons between the impact test and the volume source NTFs are shown in Fig. 10.6 and Fig. 10.7. Most of the NTFs compare quite well, with similar quality as in Fig. 10.6. Some of the comparisons are not quite as good, as in Fig. 10.7; the worse comparisons are generally for rotational DoF. There are a couple possible reasons for the discrepancies. There may be inaccuracies in the impact measurements that are more pronounced in the rotational DoF. There could also be inaccuracies in the accelerometer responses during the volume source measurement, as the discrepant region is near the region of low coherence in Fig. 10.5. The primary cause of the discrepancies was not further investigated.

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Fig. 10.4 Blocked force TPA validation using raw FRFs and rigidified FRFs

Fig. 10.5 Sample FRF data quality between the volume source and an accelerometer

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Fig. 10.6 Comparison between noise transfer functions using impact hammer and volume source, “best” comparison

Fig. 10.7 Comparison between noise transfer functions using impact hammer and volume source, “worst” comparison

The calculated blocked forces were applied to both sets of NTFs to predict the sound at the driver’s ear during a constant speed environment, with results shown in Fig. 10.8. Both predictions compare well with the actual sound measured during the operational test. In general, the reciprocal volume source predictions do not perform notably better or worse than the impact test predictions. This suggests that there may be some small errors in the initial blocked force calculation that are propagating through to the validation, regardless of the technique used. This confirms that the reciprocal measurement can be used for TPA predictions in place of the impact test measurement.

10.2.3 Rotational FRFs for Mid- to High-Frequency TPA Finally, another series of tests was done on a different active component on a test bench to perform in-situ blocked force TPA. Tests were run using three tri-axial accelerometers as indicators around each of the three attachment points. Impacts were

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Fig. 10.8 Blocked force TPA in a constant speed roller dyno environment, comparison between impact and volume source FRFs

Fig. 10.9 Blocked force TPA validation using only forces as well as forces and moments

made around each attachment point and transformed to virtual points. Two sets of virtual points were considered. In the first set, each connection point was assumed to have three translational degrees of freedom. In the second set, each connection point was assumed to have three translational degrees of freedom as well as three rotational degrees of freedom. Blocked forces were calculated for a controlled load case, using an impact on the component rather than an actual operational load condition to ensure reliable data. The response at a validation sensor was then calculated by applying the blocked forces to the system. In Fig. 10.9, the validation sensor responses are compared between the measured data and the TPA simulations with and without moments included. At lower frequencies, below ~600 Hz, both TPA simulations compare very well with the measured response. However at higher frequencies, the TPA using only translational forces deviates from the measured response, while the TPA using both translational forces and rotational moments continues to

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match the measured response. This suggests that while translational measurements may produce accurate results at lower frequencies, it is important to include rotational effects as the analysis goes to mid- and high-frequency regions.

10.3 Conclusion Transfer path analysis is used extensively in the automotive industry to characterize actively vibrating components. While current techniques provide reasonable analysis of the vibrations, there are several recent advances that can further improve the analyses. Here, it was shown how rigid body corrections can improve the very low frequency TPA results. It was also shown how using a volume source rather than impact testing can produce accurate TPA results in the mid-frequency range, and will also speed up test time. Finally, it was shown how including moments in a blocked force calculation can improve TPA results at higher frequencies over using just translational forces alone. In the cases shown, it was very obvious which frequency ranges benefited from using the newer techniques. However, for different test structures and test setups, the appropriate frequency ranges to employ each technique will vary. It is important to think about the dynamics involved for any specific test case and use the appropriate methods to get an accurate characterization.

References 1. van der Seijs, M., de Klerk, D.: Rixen, D.J.: General framework for transfer path analysis: History, theory and classification of techniques. Mechanical Systems and Signal Processing, pp. 68–69, August 2015 2. Elliott, A.S., Moorhouse, A.T.: Characterisation of structure borne sound sources from measurement in-situ. J. Acoust. Soc. Am. 123(5), 3176 (2008). https://doi.org/10.1121/1.2933261 3. Moorhouse, A.T., Elliott, A.S., Evans, T.A.: In situ measurement of the blocked force of structure-borne sound sources. J. Sound Vib. 325(4–5), 679–685 (2009). https://doi.org/10.1016/j.jsv.2009.04.035 4. van der, Seijs, M., et al.: An improved methodology for the virtual point transformation of measured frequency response functions in dynamic substructuring. COMPDYN 2013, Kos Island, Greece, 12–14 June 2013 5. van der Seijs, M.: Experimental Dynamic Substructuring: Analysis and Design Strategies for Vehicle Development. PhD Thesis, Technische Universiteit Delft, June 2016 6. Allen, M.S., Rixen, D., van der Seijs, M., Tiso, P., Abrahamsson, T., Mayes, R.L.: Substructuring in Engineering Dynamics: Emerging Numerical and Experimental Techniques. CISM International Centre for Mechanical Sciences, Courses and Lectures, Volume 594, 2020, Section 4.5 7. Fahy, F.J.: Some applications of the reciprocity principle in experimental vibroacoustics. Acoust. Phys. 49(2), 217–229 (2003) 8. ten Wolde, T.: Reciprocity measurements in acoustical and mechano-acoustical systems. Review of theory and applications. Acta Acust. United AC. 96(1), 1–13 (2010) 9. van den Bosch, D., van der Seijs, M., de Klerk, D.: A Comparison of Two Source Characterisation Techniques Proposed for Standardisation. SAE Technical Paper 2019-01-1540, 05 June 2019 10. ISO/TC43/SC1: ISO/AWI 21955 Vehicles – Experimental Method for Transposition of Dynamic Forces Generated by an Active Component from a Test Bench to a Vehicle. International Organization for Standardization, Under Development, https://www.iso.org/standard/72288.html 11. ISO/TC43/SC1: ISO/DIS 20270 Acoustics – Characterization of Sources of Structure-Borne Sound and Vibration – Indirect Measurement of Blocked Forces. International Organization for Standardization, Under Development, https://www.iso.org/standard/67456.html

Chapter 11

Impact of Junction Properties on the Modal Behavior of Assembled Structures Jean-Baptiste Chassang, Adrien Pelat, Frédéric Ablitzer, Laurent Polac, and Charles Pezerat

Abstract This communication is part of the study of the acoustic radiation of assembled structures and presents a preliminary work that consists in analyzing the links between the junction parameters and the dynamic behaviour of the structure, without considering its radiation. The study concerns a system of two beams connected by stiffeners that represent the junction and coupled compressional and flexural motions. The damping at the junction is described by a Kelvin-Voigt model. The coupled motions in each beam are developed by a modal decomposition based on the decoupled free/free beams. The model is solved in the frequency domain by a standard inversion method in the case of linear stiffenesses. The method is validated by comparison with reference results from finite element simulations obtained from Comsol. The analysis of the results aims to evaluate the phenomena of stiffening, damping and wave conversion induced by the junction that may significantly modify the vibroacoustic properties. Keywords Force response · Modal approach · Kelvin-Voigt model · Stick · SDM

11.1 Introduction Powertrains, regardless of their technology (thermal or electrical), are presented as mechanical assemblies with complex geometries. Attention is focused here on the oil sump assemblies that play a central role in the acoustic radiation of the whole structure. These assemblies involve interfaces and junctions of different natures (bolted, glued, crimped, clipped,. . . ) and having various mechanical properties. The faithful modeling of these junctions is an open scientific problem leading to research work. The description of the induced damping and stiffness added by these junctions are key points in this work. In many books and articles, the study is focused on modelling the bolted joint, from the rheological model to the constitutive model [1]. Internal studies at Renault have shown that accurate modelling of a junction does not necessarily have the expected impact on numerical obtained data. A hammer test is performed on an assembly of a cylindrical block and an oil pan connected by ten bolted joints and a frequency response is measured at the excitation point using an accelerometer. The experimental data are compared using a numerical model with a variant of the bolted joint modelling. In each of the three models, the contact areas of the 2 substructures are connected by “glue” elements (mechanical connection of a stiffness type) [2]. In the simplest model, screws are not represented. In a second, more sophisticated model, the screws are represented by equivalent point stiffness, refers as coupled model [3]. In a last model, the 3D geometry of the screws is represented in detail, calls as solid model [3]. The results obtained, Fig. 11.1, shows that these three models of increasing complexity lead to transfer functions very close but quite far from the experimental realization, especially at medium and high frequency.

J.-B. Chassang () Laboratoire de l’Université du Mans, LAUM – UMR 6613 CNRS, Le Mans Université, LE MANS, France RENAULT, Lardy, France e-mail: [email protected] A. Pelat · F. Ablitzer · C. Pezerat Laboratoire de l’Université du Mans, LAUM – UMR 6613 CNRS, Le Mans Université, LE MANS, France e-mail: [email protected]; [email protected]; [email protected] L. Polac RENAULT, Lardy, France e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2021 A. Linderholt et al. (eds.), Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47630-4_11

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Fig. 11.1 Frequency response comparison between experimental data (thick black solid line) and numerical results with no bolted junction (black solid line), 10 bolted coupled models (gray solid line) and 10 bolted joints in three dimensions (black dash line)

This indicates that it is not necessary to represent the geometry of the junction in detail. However, a better calculation/test correlation can be achieved by describing more realistically the nature of the contact between the substructures,

11.2 Modelling In order to simplify the problem and study only the vibratory impact of a junction, the study is about an assembly of two beams connected by a single junction. Radiation is not revelant in this case. Then the impact of damping and induced stiffness are studied using a modal approach. Since the jonction modelling has slight impact on transfer function data, the junction is not represented only by his own physical integrity in the model but also by the transmission forces it represents at the surface contact between two substructures. The first step consists in modelling three viscoelastic pads placed between the two beams and simulating geometric roughnesses of surface condition. This first step is obviously very simplifying but it allows to consider a controlled tribological environment. Indeed, number of pads is known as well as their normal stiffness, kN , tangential stiffness, kT and their geometric characteristics. To describe the behaviour of each pad, it have been chosen the Kelvin-Voigt model with both elastic and viscous properties [4]. There are as many models as there are pads and the stiffness and damping values are the same for each pad. The following figure shows the beam model with a viscoelastic pad schematic (Fig. 11.2). With this model, the objective is to study the emergence of induced stiffness and damping as well as intermodal couplings within and between bending and tensile-compression of the two beams. These couplings have two origins based on the same principle. Indeed, the dynamic behaviour of beams is studied from the Euler-Bernoulli theory, which assumes that the beams are thin enough to work only on neutral lines. However, the two beams are offset, resulting in a lever arm moment. In addition, the viscoelastic pads are not located on the neutral line of each beam, so there is a lever arm to take into account. The first step is to study the dynamic of each substructure alone by treating the homogeneous problem in free – free in flexion and tensile-compression. Then, junction efforts are added to dynamically link the substructures. In the physical space, the equations of motion are represented as, ˜ ˙ t)) = fe (t), Mx(t) ¨ + Kx(t) + fJ CT (δ(x, t), δ(x,

(11.1)

with the displacement, velocity and acceleration vectors denoted accordingly as x(t), x(t) ˙ and x(t). ¨ These vectors are combinations of the displacements of each substructure, whether in flexion or in tensile-compression and defined accordingly  ˙ t) described micro-displacements at the two as, x = w(1) w (2) u(1) u(2) , super scripts refering to the beam. δ(x, t)andδ(x, ˜ are the linear mass and stiffness beams junction and are expressed in function of displacements described earlier. M and K matrices which are obtained from Euler-Bernoulli beam theory. Beam damping is introduced into stiffness matrix by Young’s modulus, E ∗ = E(1 + iη). Its imaginary part is a frequency damping coefficient. It was decided to leave it set at 0.1% over the entire frequency domain studied. All efforts at the junction are described by fJ CT which depends directly of the micro-

11 Impact of Junction Properties on the Modal Behavior of Assembled Structures

a)

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b)

c)

Beam 2

Beam 1

Beam 1 Fig. 11.2 The system under study (a) and the equivalent model (b) which are three viscoelastic pads describe by Kelvin-Voigt models for flexural and compression waves (c)

displacement (normal or tangent direction). The physical displacement is thus projected on a set of modal displacements by Ritz transformed, x = q

(11.2)

 is the modal shape matrix composed by rigid and free-free modes that are known analytically and q is the generalized ¯ = T K. The ¯ = T M and stiffness matrices generalisation K displacement. The reduction is completed by a mass M transformed system of equations of motion becomes, ¯ q¨ + Kq ¯ + T fN (δ(q, t), δ(q, ˙ t)) = T fe (t). M

(11.3)

To these models are considered normal and tangential forces at the junction. These are elastic as well as dissipative and are expressed in function of micro-displacement. To simplify writing, δ(x, t) is defined as δ. The assembled structure is also excited by a punctual effort. These forces are expressed in physical coordinates. For flexion dynamic, the normal force is expressed by fN(1) (δN , δ˙N )

=

ppads $ x+ dn

2 n=1



x− d2n

 (1) hb ∂ (1) kN δN + cN δ˙N + f (δT , δ˙T ) dx, 2 ∂x T

(11.4)

(1) refers to half-width pad in the contact area of the two beams. h(1) b and hp are beam and pad height respectively. In the same way, for tensile-compression, the tangential force is, dn 2

fT (δT , δ˙T , t) = (1)

ppads $ x+ dn

2 n=1

x− d2n

  kT δT + cT δ˙T dx,

(11.5)

with, δN and δT expressed in function of relative flexion and tensile-compression movements, ⎧ (2) (1) ⎪ ⎨ δN = w (x, t) − w (x, t), ⎪ ⎩ δ = u(2) (x, t) − T

h(2) ∂w(2) (x,t) 2 ∂x

− u(1) (x, t) −

h(1) ∂w(1) (x,t) 2 ∂x



(11.6) .

The second beam efforts are expressed using principle of action-reaction. Considering junction effort expressions, it can be directly include as stiffness and damping matrices in Eq. 11.3. The following equation takes into account bending and tensile-compressive strength and also intermodal couplings in bending, φ , tensile-compression, ψ , and both movements, φψ ,

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⎤

⎤

⎡

⎡

⎤

⎢+M ⎥ ⎢ +C ⎢K ¯ φ + +C ¯ φψ +⎥ ¯ φ K ¯φ +K ¯ φψ ⎥ ⎢ ¯ φ + +0+ ⎥  ⎥ ⎥ ⎢ ⎢  ⎢ ⎥ ⎥ ⎥ ⎢ ⎢ F¯ F (t) ⎢ ⎥ q¨ φ (t) ⎥ q˙ φ (t) ⎥ qφ (t) ⎢ ⎢ +⎢ +⎢ = ¯ . ⎢ ⎥ ⎥ ⎥ ⎢ ⎥ q¨ ψ (t) ⎥ q˙ ψ (t) ⎥ qψ (t) ⎢ ⎢ FT C (t) ⎢ ⎥ ⎥ ⎥ ⎢ ⎢ ¯ ψ +⎦ ¯ ψφ + +C ¯ ψ + ⎦ ¯ ψ ⎦ ¯ ψφ K ¯ψ +K ⎣ +0+ +M ⎣+C ⎣ K (11.7) Then, the eigenvector problem is resolved,

 ˆ − Ω 2M ¯  = 0. K

(11.8)

ˆ = K ¯ +K ¯  where K ¯  is known as the gradient of joint forces or joint tangent stiffness near The stick matrix is K ∂ ¯ ¯ equilibrium and directly equal to, K = k ∂x Fj ct where k is a normal or tangential stiffness. The modal angular frequency, Ω, is directly impacted by damping ratio. A harmonic excitation is applied√on the first beam, both in flexion and tensilecompression, fe (t) = fe eiωt , where ω is the driving angular frequency, i = −1. The steady state of the modal response is ˙ = iωHeiωt and assumed to be reached with the displacement, velocity and acceleration having the forms q(t) = Heiωt , q(t) 2 iωt ¨ = −ω He . Substituting these forms into, Eq. 11.7, after a mass normalisation and solving for H yields, q(t)  H = diag

 1 T f e . Ω 2 − ω2 + iη(Ω)ωΩ

(11.9)

11.3 Conclusion The first part consists in studying a bolted joint connection of two beams in order to understand the dissipation, stiffness added and wave conversion mechanisms. According to studies conducted within Renault, the modelling of the bolted joint does not have a significant impact on vibration responses leading to the interaction between the substructures’s surfaces in contact using Kelvin-Voigt models. The purpose of this first part is therefore to study the linear viscoelastic interaction of the two beams and their impact on the assembled beams eigenvalues (frequencies and modal shapes). Each result is compared to numerical and experimental results. The main goal is to define the most important sensitivities to be taken into account for vibroacoustic models.

References 1. Bograd, S.: Modeling the dynamics of mechanical joints. Mech. Syst. Signal Process. (Reading) 25, 2801–2826 (2011) 2. Release Guide: MSC NASTRAN (2019) 3. Hammami, C.: Intégration de modèles de jonctions dissipatives dans la conception vibratoire de structures amorties. (french), [Integration of dissipative junction models in the vibration design of damped structures]. Cnam Ph.D. Thesis (2014) 4. Love, A.E.H.: The Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (1927)

Chapter 12

Quantifying Joint Uncertainties for Hybrid System Vibration Testing Nadim A. Bari, Manuel Serrano, Safwat M. Shenouda, Stuart Taylor, John Schultze, and Garrison Flynn

Abstract Controlled laboratory vibration testing has been the preferred method to experimentally quantify a system’s governing dynamic response. However, dynamic testing of full-scale commercial structures in service environments can be an expensive and challenging endeavor to perform due to structural size and complexities. Hybrid substructuring is a method that involves performing tests on components of interest coupled with a numerical model simulating behavior of the primary system. The physical test and the numerical model are combined to represent the dynamic response of the complete system. Hybrid substructuring could also be implemented in the opposite fashion: the primary system as the physical substructure and a component of interest as the numerical substructure. Such an approach enables predictions of the dynamic response of a critical component when it cannot be directly measured during a system test. Within the numerical model uncertainties in the coupling of substructures can be accounted for through probabilistic model parameters. Using this approach, the effects of uncertainties in the component of interest and its interface to the primary system, such as those due to jointed connections, are captured. This paper aims to quantify a system’s uncertainties by incorporating a suite of experimental data with inverse analysis to determine distributions of uncertain parameters, thus allowing a hybrid substructuring scheme to make statistically bounded predictions. The approach is demonstrated through an analysis of the bolted joints uncertainties in the Box and Removable Component test structure. Keywords Bolted joints · Modal analysis · Model calibration · Uncertainty quantification · Vibration testing

12.1 Introduction Determining the complete dynamic response of large-scale engineering systems is critical for assessing performance of complex systems in their real-world operating environments. However, performing full-scale testing is difficult due to size, weight, and cost. Accurate testing and qualification of a full system, as well as each of its components, becomes increasingly difficult with system complexity. Test scenarios meant to replicate real-world conditions are rarely ideal. Examples of common challenges in testing include (1) inability to instrument a component directly (2) variability in jointed connections coupling the component to its system (3) laboratory test configurations unable to represent complexity of true service environment. New methods to increase reliability, reduce uncertainty, and limit over testing are of great interest to the structural dynamics community. Herein, we seek to alleviate such physical constraints to the greatest extent possible by representing critical components as a numerical substructure coupled to a physical substructure of the larger system.

N. A. Bari Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA Los Alamos National Laboratory, Los Alamos, NM, USA M. Serrano Department of Aerospace & Mechanical Engineering, New Mexico State University, Las Cruces, NM, USA S. M. Shenouda Department of Mechanical Engineering, North Carolina A&T State University, Greensboro, NC, USA S. Taylor · J. Schultze · G. Flynn () Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2021 A. Linderholt et al. (eds.), Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47630-4_12

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Rather than full-scale testing, the behavior of the complete structure is commonly inferred from test results obtained from experiments on a scaled model of the entire structure or testing of a critical component of the structure [6]. One approach to characterizing a system’s dynamic response is pseudo-dynamic testing (PsD), in which only a vital element of the entire structure (typically a complex, difficult to model component) is tested physically while the remainder of the structure is modeled numerically [7, 12, 13]. With PsD methods, the physical element is excited statically under loading conditions and not by dynamic excitations, such as ground acceleration, which leads to several experimental inaccuracies. Recent technological advances in signal processing have allowed for full-scale structural testing to be performed on components of interest, with the remainder of the structure being numerically modeled. Hybrid Substructuring (HS) is a recently developed technique to investigate the dynamic behavior of structural systems [1, 2]. These advances have led to a new, innovative approach to computing displacement, velocity, and acceleration of components under dynamic excitation during experimental settings [8]. This testing technique, referred to as Real-Time Hybrid Substructuring (RTHS), is widely used for performance evaluations of structural systems subjected to dynamic loading [5]. RTHS is similar to the PsD testing method; however, with RTHS, both physical and numerical substructure partitions are simultaneously integrated into a realtime loop while the physical structure is dynamically excited [3]. Once the theoretical model is validated as an accurate mathematical representation of the structure, HS is typically the simplest method of the two to perform because no transfer system or real-time control loop algorithm is required. Both HS and RTHS are methods that rely heavily on a numerical model of a substructure to represent the dynamic response of the complete system accurately. Consequently, these approaches permit for impacts of uncertainties in the system, such as those due to jointed connections, to be incorporated into the numerical substructure as probabilistic variables. This paper aims to quantify a system’s uncertainties by incorporating experimental data with an optimization algorithm to find probabilistic values of uncertain parameters so that system uncertainty may be propagated into a hybrid substructuring scheme. The methodology will be demonstrated through an analysis of the bolted joints uncertainties in the Box and Removable Component (BARC) test structure.

12.2 Experimental Procedure 12.2.1 Test Component The Box Assembly with Removable Component (BARC) test structure, developed at Sandia National Laboratories and Kansas City National Security Campus for the Boundary Condition Round Robin Challenge [11], was used in this study. This structure was explicitly designed as a standard system for researchers to integrate into a testbed when designing environmental shock and vibration tests, focusing on the issue of uncertain boundary conditions [10, 11]. The BARC consists of two substructures. The base substructure is a cut box frame and the removable component is a beam, with the two substructures connected by two C-channels. The BARC substructures are aluminum 6061 and the C-channels are aluminum 6063. The cut box substructure is 3 (7.62 cm) wide and 6 (15.24 cm) tall with a top cut of 0.5 (1.27 cm) and thickness of 0.25 (0.635 cm). The removable component is 1 (2.54 cm) wide and 5 (12.7 cm) long with a thickness of 0.125 (0.3175 cm). C-clamps are 1 (2.54 cm) wide and 2 (5.08 cm) tall. The BARC was assembled onto the shaker table using four button cap screws with its flat faces normal to the direction of motion. The four mounting screws were hand tightened until secure. Then they were tightened using a socket wrench. The C-channel brackets were attached to the box frame using eight stainless steel socket cap screws, which were each torqued to 20 in-lbs [9]. The beam was then attached using two hex bolts, which were each torqued to 50 in-lbs. Although the specified assembly configuration is asymmetric, the BARC was assembled symmetrically for the results presented in this paper, as illustrated in Fig. 12.1.

12.2.2 Experimental Setup The BARC was instrumented with eight single-axis accelerometers measuring along the axis of excitation, two on each leg of the box and two on each C-channel. A bi-axial accelerometer was placed at the center of the removable component measuring acceleration vertically as well as in the direction of excitation. Locations of the accelerometers are illustrated in Fig. 12.2. Accelerometers were also placed at the base of the shaker table to measure the input acceleration and controlling the excitation. Due to hardware constraints the tests were run using two data acquisition systems with limited channels. Two

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Fig. 12.1 BARC structure in a symmetric test configuration with inset showing bolted connection of the box to the removable component

Fig. 12.2 Acceleration measurement setup during BARC testing. A schematic for the accelerometer locations and node labels for test sessions A and B are shown on the left figure. Measurements at the top and base nodes were collected during both sessions. Uniaxial accelerometers measured acceleration in the direction of excitation symmetrically on the BARC vertical components and a bi-axial accelerometer measured vertical acceleration as well as acceleration in the direction of excitation on the top removable component

sessions (referred to as session A and session B) were completed for every test to allow for data acquisition at all measurement locations. For each test, a random vibration with RMS of 0.005 g’s was input as a base excitation for 30 seconds.

12.2.3 Test Procedure A total of 30 tests were completed where each test consisted of both sessions A and B. Between each test, the top two hex cap screws connecting the removable component to the C-clamps were loosened and then retorqued to 50 in-lbs. In addition to retorquing the top bolts, the eight socket screws used to fasten the brackets to the box frame were also loosened and retorqued to 20 in-lbs.

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Fig. 12.3 The BARC represented as an Euler-Bernoulli beam model with uncertain joint connections (highlighted in red) represented as spring elements

12.3 Numerical Model The BARC system is modeled using Euler-Bernoulli beam elements. These elements exhibit three degrees of freedom at each node: axial, rotational, and translational. Because the BARC was assembled in a symmetric configuration of this paper, the out-of-plane dynamics of the BARC were assumed to be negligible. The BARC is modeled as a system containing nine beams, with each beam being modeled independently using finite elements. The local models are then appropriately coupled to form the global finite element model of the BARC. Uncertainties in the numerical substructure would change the natural frequencies of the system. Thus only the natural response of the BARC was modeled as the natural frequency was used for validation. The equation of motion for the BARC is derived to be − → − → → γ =0 [M] γ¨ + [C] γ˙ + [K] −

(12.1)

where [M] is the global consistent mass matrix, [C] is the damping matrix, and [K] is the global Euler-Bernoulli stiffness matrix. The global mass and stiffness matrices were derived from elemental mass and stiffness matrices (beam theory). The damping matrix was evaluated from a known modal damping matrix and was not experimentally determined [4]. The natural frequencies were evaluated as the squareroot of the eigenvalues of the mass and stiffness matrices. These bolted joints are subject to self-loosening under vibrations; this may alter the natural frequencies of the system. Consequently, the uncertainties in the system come from the bolted joints. The bolted joints were modeled as single, massless, Euler-Bernoulli beam elements (springs). The three degrees of freedom at each node (axial, rotational, and translational) correspond to axial, rotational, and shear stiffness parameters. The BARC is modeled such that only the stiffness parameters of the bolted joints are associated with uncertainties of the system while all welded joints are assumed perfectly stiff. Figure 12.3 illustrates the location of the modeled spring elements.

12.4 Analysis 12.4.1 Joint Stiffness Calibration Parameters representing uncertainties present within the bolted joints of the removable component and the C-clamps were calibrated to each of the 30 tests (with bolts loosened and tightened between) independently, forming a distribution of calibrated joint stiffness values. These uncertainties were represented as varying axial, shear, and rotational stiffness values. The calibration was completed for each of the experimental tests, producing a distribution of calibrated stiffness values.

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Calibration of parameter values for each test seeks to minimize the root mean square error between normalized natural frequency predictions, η, and normalized natural frequencies measured experimentally, y. Natural frequency predictions are normalized according to: ωexp − ωsim,i η=  ωexp + ωsim,i /2

(12.2)

where ωexp is the experimental natural frequencies and ωsim, i is the predicted natural frequencies given the ith parameter sample values. Experimentally measured natural frequencies are normalized according to: ωexp − ωsim,0 y= ωexp + ωsim,0 /2

(12.3)

where ωsim, 0 is the model prediction with a set of nominal parameter values. Frequencies are normalized within the objective function so that all frequencies have equal weight rather than allowing high frequencies skewing the calibration.

12.5 Results and Discussion Both HS and RTHS are methods that rely heavily on a numerical model of a substructure to represent the dynamic response of the complete system accurately. Consequently, these approaches permit for impacts of uncertainties, such as those due to jointed connections, to be represented numerically as probabilistic variables. Uncertainties in the numerical substructure would change the natural frequencies of the system. Thus only the natural response of the BARC was modeled and the natural frequencies were used for validation. This section presents the experimentally determined natural frequencies of the BARC with quantified uncertainties.

12.5.1 Experimentally Determined Natural Frequencies The frequency response function (FRF) of the BARC was derived globally with respect to all measured degrees of freedom by combining acceleration output at each sensor location per acceleration input at the base of the structure. Figure 12.4 shows the global FRF for each of the 30 tests completed. The BARC is assumed to be a lightly damped system and a parameterestimation method was used to determine the system’s natural frequencies one mode at a time. Variability of the measured natural frequency resulting from the bolt uncertainty is illustrated in Fig. 12.5. In this figure each of the first three modes is normalized by its nominal value such that we can see the test-to-test variation. Test set A includes measurements at degrees of freedom 1, 4, 5, 6, 7 and 10. Test set B includes measurements at degrees of freedom 2, 3, 5, 6, 8 and 9 (Fig. 12.2).

Fig. 12.4 Transmissibility plots collected for 30 tests where the bolted joints were loosened and retorqued between each test

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Fig. 12.5 Variability in the first three experimentally measured natural frequencies due to joint uncertainty

The simple mathematical model does not capture out of plane (torsional) motion. Therefore, more complex finite element model simulations were used to determine that the first four natural frequencies of the BARC which are correlated to in-plane mode shapes and any natural frequencies corresponding to torsional model shapes were removed from the analysis.

12.5.2 Stiffness Uncertainty Quantification The optimization algorithm was developed to take in the experimental natural frequency values and output a distribution of stiffness values. Stiffness values are optimized such that the sum of squares residual of the experimentally and numerically modeled natural frequencies are minimized. The univariate and bivariate distributions of the stiffness values, kaxial , kshear , krotational , are shown in Fig. 12.6 and statistics of the calibrated distributions are provided in Table 12.1. Correlations between the three stiffness parameters were not found to be statistically significant, which is a welcome observation to avoid trade-offs between the stiffness parameters. Distributions of the stiffness parameters are estimated using a kernel density estimate. Calibrated parameters distributions are then propagated forward to the numerical model to predict the BARC’s first four natural frequencies corresponding to in-plane modes. Distributions of the predicted frequencies are shown in Fig. 12.7 and summary statistics are provided in Table 12.2.

12.6 Conclusions and Future Work This report investigates uncertainty in bolted joints by determining distributions of joint stiffness parameters with respect to experimentally captured variations in a structure’s natural frequencies. Uncertainties in the system response were assumed to be related to the system’s bolted connections (represented as stiffness parameters) which were loosened and tightened to the same torque between every test. Joint stiffness parameters were calibrated to every test independently resulting in a distribution of calibrated values. These calibrated values were then propagated forward in the numerical model to make new predictions of BARC frequencies with statistical bounds. This paper served to demonstrate the ability to quantify and propagate uncertainties in a system that may be represented as substructures. The next step in this study is to incorporate the calibrated parameter distributions into a hybrid substructuring, and eventually real time hybrid substructuring, scheme. Simulations would be advanced to predict time series response, such as displacement, velocity, and acceleration with uncertainty bounds. Ultimately, the numerical model would be reduced to only the component and probabilistic joint parameters. Measurements could then be collected at the base of the joints

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Fig. 12.6 Distributions of the stiffness values. Along the diagonal are histogram plots of each stiffness values. The off-diagonal subplots of the matrix are scatter plots indicating the density of two independent stiffness values with respect to each other Table 12.1 Statistics of calibrated stiffness parameters Model parameter kaxial (N/m) kshear (N/m) krotational (N/rad)

Minimum 1000 100 25

Maximum 10,796 1999 199

Mean 5861 1242 112

Std. dev. 3016 671 58

and used as inputs to the numerical substructure, which in turn makes probabilistic predictions of the component. This substructured model would enable predictions of a component response after a full system test where the component could not be directly measured. Similarly, such a substructured model could be used to control a system vibration test in a scenario where the component is not accessible for instrumentation, presenting new opportunities for system qualification. Finally, implementation of the methodology with a more complex model capable of predicting out-of-plane and torsional modes should be completed. Quantification of uncertainties with more advanced systems, such as a nonsymmetrical BARC are necessary to determine limitations of the method due to system complexities.

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Fig. 12.7 Distributions of the predicted frequencies given calibrated distributions of joint stiffness values Table 12.2 Statistics of forward model predictions considering joint uncertainty

Mode 1 Mode 2 Mode 3 Mode 4

Frequency statistics (Hz) Minimum 27 102 278 462

Maximum 372 266 431 551

Mean 105 251 408 537

Std. dev. 13.7 15.6 26.7 8.19

References 1. Blakeborough, A., Williams, M.S., Darby, A.P., Williams, D.M.: The development of real–time substructure testing. Philos. Trans. R. Soc. London, Ser. A. 359(1786), 1869–1891 (2001) 2. Chen, C., Ricles, J.M.: Large-scale real-time hybrid simulation involving multiple experimental substructures and adaptive actuator delay compensation. Earthq. Eng. Struct. Dyn. 41(3), 549–569 (2012) 3. Dermitzakis, S.N., Mahin, S.A.: Development of Substructuring Techniques for on-Line Computer Controlled Seismic Performance Testing. UCB/EERC-85/04. University of, California, Berkeley, CA (1985) 4. Farrar, C., Nishio, M., Hemez, F., Stull, C., Park, G., Cornwell, P., Figueiredo, E., Luscher, D.J., Worden, K.: Feature Extraction for Structural Dynamics Model Validation. Los Alamos National Lab, Los Alamos (2016) 5. Fermandois, G.A.: Application of model-based compensation methods to real-time hybrid simulation benchmark. Mech. Syst. Signal Process. 131, 394–416 (2019) 6. Lee, S.K., Park, E.C., Min, K.W., Lee, S.H., Chung, L., Park, J.H.: Real-time hybrid shaking table testing method for the performance evaluation of a tuned liquid damper controlling seismic response of building structures. J. Sound Vib. 302(3), 596–612 (2007) 7. Mahin, S.A., Shing, P.S.B., Thewalt, C.R., Hanson, R.D.: Pseudodynamic test method—current status and future directions. J. Struct. Eng. 115(8), 2113–2128 (1989) 8. Nakashima, M., Kato, H., Takaoka, E.: Development of real-time pseudo dynamic testing. Earthq. Eng. Struct. Dyn. 21(1), 79–92 (1992) 9. Rohe, D.P.: Modal data for the BARC challenge problem Test Report (No. SAND-2018-0640R). Sandia National Lab.(SNL-NM) (2018) 10. Skousen, T.J., Harvie, J.M., Schoenherr, T.F., Soine, D., Jones, R.: Designing hardware for the boundary condition round robin challenge (No. SAND2017-11293C). Sandia National Lab.(SNL-NM) (2017) 11. Soine, D.E., Jones, R.J., Harvie, J.M., Skousen, T.J., Schoenherr, T.F.: Designing hardware for the boundary condition round robin challenge. Top. Modal Anal. Test. 9, 119–126 (2019) 12. Takanashi, K., Udagawa, K., Seki, M., Okada, T., Tanaka, H.: Nonlinear earthquake response analysis of structures by a computer-actuator on-line system. Bull. Earthq. Resist. Struct. Res. Cent. 8, 1–17 (1975) 13. Takanashi, K., Taniguchi, H., Tanaka, H.: Inelastic response of H-shaped columns to two dimensional earthquake motions. Bull. Earthq. Resis. Struct. Res. Cent. (13), 15–28 (1980)

Chapter 13

Damping Identification and Model Updating of Boundary Conditions for a Cantilever Beam Nimai Domenico Bibbo and Vikas Arora

Abstract Model updating is a crucial tool for dynamically loaded structures as natural frequencies and damping can have a vital impact on normal operation and fatigue life. Finite element models always deviate from experimentally obtained results due to variations in mass and stiffness and the lack of a damping matrix. Thus, it is vital for many large structures, that the dynamic properties be determined experimentally and subsequently apply model updating to “tune” the finite element model. In this paper, model updating techniques are tested on a small cantilever beam where the fixed boundary condition is assumed to be flexible and the main source of damping. The methods are tested using both simulated data and experimental data for updating the stiffness and damping matrices. The results of the analytical model are compared both before and after updating with its experimental counterpart. The model updating procedure follows a two-step process. In step one the stiffness matrix is updated using the iterative eigensensitivity approach where selected stiffness related parameters, in this case the boundary conditions, are updated based on the sensitivities to eigenfrequencies and mode shapes. In the second step, the damping of the structure is updated. Keywords Model updating · Damping identification · Modal expansion/Reduction · Simulated model updating · Experimental model updating

13.1 Introduction The advent of the finite element method [1] has brought about the ability for more advanced and refined analysis of engineering structures. Stress distributions and modal properties can be analyzed for geometry far more complex than what is possible otherwise. The benefit of this is the ability to create structures optimized to the environmental factors they are subjected to. There is however one caveat, a finite element model is only ever as good as the users ability to capture the physical properties of the system. The stiffness and mass of a system are often fairly well known, however differences often exist due to production tolerances, joints and boundary conditions. These differences can significantly impact modal results and stress distributions. With the advances in experimentally measuring modal parameters [2], methods were in turn developed to update finite element models using the results from experimental analysis. Development started in the 1960/1970s, however much of the use and further development first came around in the 1990s with the increasing performance of computational hardware. These early methods and the model updating procedure is well described in [3]. The updating of mass and stiffness matrices can generally be split into two categories, direct updating methods and iterative updating methods. The direct methods have the advantage of updating the mass and stiffness matrices in one step and are able to match the experimentally obtained results very accurately. However for this article, only the iterative methods are of interest as the direct methods update the system matrices without consideration of the physical connectivity of the finite element model. Using iterative methods, updating parameters are selected where it is thought that error in the finite element model exists. This could be joints, boundary conditions, youngs modulus, density, etc. Many iterative methods exist, one of the most popular being the inverse eigensensitivty approach [4] which utilizes the eigenvalues and modeshapes of a

N. D. Bibbo () Department of Technology and Innovation, University of Southern Denmark, Odense, Denmark Lindø Offshore Renewables Center, Munkebo, Denmark e-mail: [email protected] V. Arora Lindø Offshore Renewables Center, Munkebo, Denmark © The Society for Experimental Mechanics, Inc. 2021 A. Linderholt et al. (eds.), Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47630-4_13

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system for updating. Others have sought to improve the inverse eigensensitivity method, such as Lin [5]. other types use the frequency response functions for updating, such as the normal response function method seen in [6, 7] and [8]. Alternative iterative updating methods exist such as constrained optimization methods [9] or genetic algorithms [10] and neural networks [11] for improving the optimization methods. Another more recent approach, known as the cross-model cross-mode method [12] shows a lot of promise as the method is able to update selected updating parameters in one step. Thus providing the computational efficiency of a direct method but still maintaining the physical connectivity of the system. The method has also been tested on a small experimental structure [13]. For this article, the inverse eigensensitivity approach is used as it is a very proven and robust method of model updating. This is important for the subsequent damping identification as these require accurate stiffness and mass matrices. Updating of damping is a very complex problem and to a large extent the mechanisms of the force dissipation from a system are unknown. Many damping models have been proposed to capture the energy dissipation from a vibrating system, one of the most common being the viscous Rayleigh/proportional damping model [14], although many other damping models exist. Much like the updating of the mass and stiffness matrices, damping can also be updated either using an iterative or direct approach. Lancaster [15] proposed an iterative method using updated mass and stiffness matrices and complex eigendata. Other methods determine the damping matrix using frequency response functions, such as the work by Lee and Kim [16]. In this article, a modified version of the Lancaster method [17] is used, where the damping matrix is found directly. One of the issues that arises often when performing model updating, is the incompleteness of measurement data with respect to the finite element model. Finite element models can have millions of DOFs where as with an experimental measurement, you are lucky if you have 100 measurement DOFs. Further, rotational degrees of freedom are challenging to measure. For this reason, it is often necessary to either expand the measured modal data to the DOFs of the finite element model or reduce the finite element model down to the DOFs of the measured modal data. Many methods exist for expansion/reduction, some better then others. One of the best known and earliest methods is the guyan reduction method[18], which is a static reduction technique that closely replicates the modes, however there is some error due to neglecting mass effects. Another common method used is the kidder approach [19], this method is an improvement over the Guyan reduction method at higher frequencies as it includes mass effects. The method used in this article is the SEREP(system equivalent expansion reduction process) method [20].

13.2 Theory 13.2.1 Model Reduction and Modal Expansion Model reduction and modal expansion are mathematical tools that have been developed to deal with the spatial incompleteness that occurs during experimental modal analysis testing. For the case of model updating, the limited number of measurements during testing presents a problem, as the number of degrees of freedom for a finite element system is often many orders of magnitude higher than that of the experimental setup. Many updating algorithms only work if the DOF size of the experimental model is equal to the DOF size of the finite element model. Thus a form of mapping is required between the experimental DOFs (master DOFs) and the non-measured analytical DOFs (slave DOFs) [21]. All modal expansion/model reduction techniques are based on the same concept. A transformation matrix is identified that can be used for either expanding an experimental data set from a set of master DOFs to the slave DOFs or to reduce the analytical model down to the slave DOFs. In general, the following relationships for the techniques are thus defined in equations (13.1), (13.2) and (13.3).        Xm = T Xm (13.1) Xn = Xs       where Xn is the expanded mode, Xm are the mode shape coefficients for the master modes, Xs are the mode shape   coefficients for the slave nodes and the transformation matrix between them is given by T . For model reduction, the transformation matrix is multiplied before and after on the mass and stiffness matrices, thus reducing the mass and stiffness matrices down to the master degrees of freedom as seen in equations (13.2) and (13.3).    T     Mm = T Mn T

(13.2)

   T     Km = T Kn T

(13.3)

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13.2.2 SEREP Modal Expansion/Model Reduction The SEREP method utilizes the analytical models mode-set in order to interpolate the experimental mode shapes or to reduce the analytical mass and stiffness matrices [20].The transformation matrix is shown in equation (13.4). [TU ] = [Un ]([UmT ][Um ])−1 [Um ]T

(13.4)

where [TU ] is the SEREP transformation matrix, [Un ] is the analytical mode shape matrix and [Um ] is the analytical modeshape matrix containing only master DOFs. This method of modal expansion and model reduction is used in all subsequent analyses when expansion or reduction is required.

13.2.3 Inverse Eigensensitivity Approach The inverse eigensensitivity approach [4] is an iterative model updating technique that uses experimental and analytical eigenvalues and modeshapes for updating selected updating parameters. The method is based on the rate of change of the eigenvalues and modeshapes [22] and utilizes a least-squares updating approach. It can be used for updating both the analytical mass and stiffness matrices. Results obtained from the method are highly dependent upon the choice of updating parameters, this is one of the most crucial steps involved. If incorrect parameters are selected, the updating will not be able to converge on a satisfactory result. If to many updating parameters are selected, the calculation time will increase and the method can potentially become unstable [4]. The parameter sensitivity to eigenvalues

(∂ωA )2r ∂pi

is seen in equation (13.5) and

(∂φA )r ∂pi

in equation (13.6). to modeshapes Sensitivity of eigenvalues: (∂ωA )2r ∂[KA ] ∂[MA ] = {φA }Tr {φX }r − (ωx )2r {φA }Tr {φX }r ∂pi ∂pi ∂pi

(13.5)

(∂ω )2

In equation (13.5), the parameters eigenvalue sensitivity ∂pAi r for mode r is dependent on the analytical modeshape for mode r {φA }r , the experimental modeshape for mode r {φX }r , the experimental eigenvalue for mode r (ωx )2r and the partial derivatives of the analytical stiffness [KA ] and mass matrix [MA ] towards the updating parameter pi . Sensitivity of modeshapes: (∂φA )r = ∂pi

N

{φA }j {φA }Tj

j =1;j =r

(ωX )2r − (ωA )2j



 ∂[KA ] ∂[MA ] ∂[MA ] 1 {φX }r − {φA }r {φA }Tr − (ωX )2r {φX }r ∂pi ∂pi 2 ∂pi

(13.6)

A )r The parameter sensitivity of modeshapes (∂φ ∂pi in equation (13.6) is dependent upon the same input parameters as for the sensitivity to eigenvalues in equation (13.5). It is further also dependent on the analytical eigenvalue (ωx )2r . The parameter updating is then solved using a least squares updating approach as seen in equation (13.7). The sensitivities of eigenvalues and modeshapes are added

{p} = [S]+ {ξ }

(13.7)

Thus the change in updating parameters {p} is found taking the pseudo inverse of the sensitivity matrix [S] (where the + symbol denotes the pseudo inverse) and then multiplying the pseudo inverse sensitivity matrix on the vector {ξ } containing the differences in eigenvalues and modeshapes. The method requires multiple iterations to reach a converged solution.

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13.2.4 Non-proportional Damping Proportional damping is defined from a linear combination of the mass and stiffness matrices [2]. The formulation of proportional damping can be seen in equation (13.8). It is seen that the damping matrix [C] is a linear combination of [M] and [K], where the coefficients a and b govern the amount of damping contribution from the mass and stiffness matrices. [C] = a[M] + b[K]

(13.8)

The damping matrix can be made non-proportional by taking equation (13.8) and applying it element-wise, with different values of a and b. Thus each element will have a damping matrix [Ce ] formulated by corresponding damping coefficients ae and be and element stiffness and mass matrices [Ke ] and [Me ]. Thus each element damping matrix can be formulated as in equation (13.9). The damping matrix is then assembled based on element connectivity. [Ce ] = ae [Me ] + be [Ke ]

(13.9)

13.2.5 Direct Damping Updating The damping updating method used is based on Lancasters iterative method. Pilkey [17] describes how this method can be used as a direct updating method. The approach is based on a viscously damped system as seen in equation (13.10). The modeshapes must be scaled to the eigenvalues as in equation (13.11), a damping matrix can then be calculated as seen in equation (13.12). The mass and stiffness matrix must be accurate in order for this method to produce an adequate damping matrix. [[M]λ2i + [C]λi + [K]]{φi } = 0

(13.10)

{φi }T ([M]λ2i − [K]){φi } = λi

(13.11)

¯  ¯ 2 ][Φ]∗ )[M] [C] = −[M]([Φ][2 ][Φ]T + [Φ][

(13.12)

For equations (13.10), (13.11) and (13.12): [M], [C] and [K] are the mass, damping and stiffness matrices. λi and {φi } ¯ and are the i’th eigenvalue and modeshape. [Φ] is the modeshape matrix, [2 ] the diagonalized eigenvalue matrix and [Φ] 2 ∗ ¯ [ ] their complex conjugate counterparts. [Φ] is the complex conjugate transpose of the modeshape matrix.

13.3 Simulated Beam A simulated “experimental” beam is initially used for testing the inverse eigensensitivity approach and the direct updating method. Using a simulated beam allows for testing the accuracy of the methods, as the true stiffness and damping values are known.

13.3.1 Model Setup The simulated beam is modeled as a fixed-fixed beam as seen in Fig. 13.1. The finite element modal analysis is implemented in Matlab using 39 beam elements. The boundary conditions are made soft, using a rotational and translational spring at each end. Two beams are simulated initially, a simulated analytical beam and a simulated experimental beam. Both beams have identical dimensions and material properties, however the spring stiffnesses are different between the two models. The simulated experimental beam is measured at 8 equidistant points along the beam, simulating the use of accelerometers. Beam properties for both beams can be seen in Table 13.1. For simulating damping, non proportional damping is applied.

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Fig. 13.1 Simulated beam setup. Rotational and translational springs are used at each end for boundary conditions. The modal results are extracted at 8 equidistant points along the structure Table 13.1 Fixed-fixed beam properties



L [mm]

W [mm]

t [mm]

Youngs modulus [GP a]

Density

910

50

5

200

7850

Table 13.2 Fixed-fixed beam spring stiffnesses

(a)

Springs   k1_t N m  N ◦ k1_r m /   k2_t N m  N ◦ k2_r m /

(b)

10-2

Analytical

5 · 105

7.5 · 105

2 · 103

3 · 103

4 · 105

5.2 · 105

4 · 103

5 · 103

10-2 Analytical [C] updated Simulated experimental

10-4

Dynamic flexability

10-4

Dynamic flexability



Simulated experimental

Analytical Analytical [K] updated Simulated experimental

10-6

10-6

10-8

10-8

10-10

kg m3

0

50

100

150

Frequency [Hz]

200

250

10-10

0

50

100

150

200

250

Frequency [Hz]

Fig. 13.2 (a): FRF’s before and after updating the stiffness matrix versus the simulated experimental FRF. (b): Frequency response function after updating the damping matrix versus the simulated experimental FRF

Before using the eigensensitivity approach, the simulated experimental modeshapes are expanded up to the same number of DOFs as the analytical modes and when using the Pilkey method for damping updating, the analytical mass and stiffness matrices are reduced down to the number of simulated experimental DOFs (Table 13.2).

13.3.2 Results This section shows the results for the analytical fixed-fixed beam both before and after updating the [K] and [C] matrices. Figure 13.2 shows the FRF’s of the analytical beam compared to the simulated experimental beam both before and after updating. In Table 13.3, the natural frequencies and MAC values of the simulated experimental, analytical and the analytical after [K] updating are compared and in Table 13.4, the simulated experimental and analytical model after [C] matrix updating

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Table 13.3 Natural frequency and MAC value comparisons between the simulated experimental beam, the analytical beam and the analytical beam after [K] matrix updating Simulated exp.

Analytical model

[K] updated

Mode [no]

Freq. [Hz]

Freq. [Hz]

Error [%]

MAC-value

Freq. [Hz]

Error [%]

MAC-value

1 2 3 4

26.75 71.849 133.75 205.62

27.782 74.856 140.69 218.69

3.86 4.19 5.21 6.33

1.00 1.00 1.00 0.99

26.463 71.513 134.08 207.57

−1.07 −0.47 0.27 0.92

1.00 1.00 1.00 1.00

Table 13.4 Natural frequency and MAC value comparisons between the simulated experimental beam and the analytical beam after [C] matrix updating Mode [nr]

Simulated exp. Frequency [Hz]

[C] updated Frequency [Hz]

Error [%]

MAC-value

1 2 3 4

26.75 71.849 133.75 205.62

26.463 71.513 134.08 207.57

−1.07 −0.47 0.27 0.92

1.00 1.00 1.00 1.00

Table 13.5 Fixed-fixed beam spring stiffnesses after updating Springs   k1_t N m  ◦ k1_r N m/ N  k2_t m  ◦ k2_r N m/

Simulated experimental

Analytical updated

Error [%]

5 · 105

5.41 · 105

8.2

2 · 103

1.79 · 103

−10

4 · 105

4.20 · 105

5

4 · 103

3.59 · 103

−10.3

are compared. Finally in Table 13.5, the spring stiffnesses after updating are compared with the simulated experimental spring stiffnesses. From Fig. 13.2a it is seen that after updating the stiffness matrix, that the FRFs of the updated model almost coincide, though with a slight shift in frequencies. Likewise in Fig. 13.2b it is seen that after damping identification, that the peeks are nearly identical in amplitude. From Table 13.3 it is found that the MAC-values for the analytical model are good, however as expected there is some error of the natural frequencies. After updating the stiffness matrix, the error percentages of the natural frequencies decrease and their is a slight improvement in the MAC values. In Table 13.4 the frequencies and MAC values between the simulated model and the [C] matrix updated model are compared. Comparing the frequency error values for the [C] matrix updated model with the frequency error values of the [K] matrix updated model in Table 13.3, it is seen that the frequency error does not change. This is logical, as the damping matrix does not affect the natural frequency of the system, only the damped frequency. The MAC values remain the same. After updating the stiffness matrix, the spring values of the simulated experimental beam and the analytical model after updating are compared in Table 13.5. Comparing the analytical updated stiffness values to the simulated experimental, it is seen that there still is some error in the spring stiffness’s. This explains why in Fig. 13.2a and Table 13.3 we see discrepancies in the natural frequencies. Comparing the analytical updated values with the analytical model before updating from Table 13.2, it is seen that the updated model values are still an improvement over those before updating.

13.4 Experimental Beam After verifying the methods on the simulated setup, the inverse eigensensitivity approach and Pilkey damping updating method are tested on an experimental beam setup. This tests the robustness of the algorithm and method when the “true” updating parameters are unknown and must be user selected.

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Table 13.6 Cantilever beam experiment properties



Length [mm]

Height [mm]

Thickness [mm]

Youngs modulus [GP a]

Density

700

70

2

200

7850

kg m3



Fig. 13.3 Experimental beam setup: DOF placements and overall dimensions of the beam

13.4.1 Model Setup The experimental beam data used comes from a masters thesis [23] where EMA was preformed on a slender beam structure. The beam was excited using an impact hammer at the tip of the beam and the response was measured at 8 points along the structure using piezoelectric accelerometers. The beam dimensions and accelerometer positions can be seen in Table 13.6 and Fig. 13.3. As the focus of the article does not cover EMA approaches, for more details on the EMA setup and methods, details are given in the thesis. For finite element analysis of the beam, the beam was modeled in Matlab using 39 beam elements, a rotational spring element and translational spring element. Before using the eigensensitivity approach, the experimental modeshapes are expanded up to the same number of DOFs as the analytical modes and when using the Pilkey method for damping updating, the analytical mass and stiffness matrices are reduced down to the number of experimental DOFs.

13.4.2 Results This Section shows the results for the experimental cantilever beam both before and after updating. Figure 13.4 shows the FRF’s of the analytical beam versus the experimental beam both before and after updating the analytical model. In Table 13.7, the natural frequencies and MAC values of the experimental, analytical and the updated [K] and [C] matrix analytical models are compared. The results shown in Table 13.8 compare the natural frequencies and MAC values after the [C] matrix is formulated.

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(b) 102 Analytical [C] updated Experimental

Analytical Analytical [K] updated Experimental

100

Response [m/N]

Response [m/N]

100

10-2

10-4

10-6

10-2

10-4

10-6

10-8

10-8 0

50

100

150

200

250

300

350

400

450

0

Frequency [Hz]

50

100

150

200

250

300

350

400

450

Frequency [Hz]

Fig. 13.4 (a): FRF’s before and after updating the stiffness matrix versus the experimental FRF. (b): Frequency response function after updating the damping matrix versus the experimental FRF Table 13.7 Natural frequency and MAC value comparisons between the experimental beam, the analytical beam and the stiffness matrix updated analytical beam Mode [no] 1 2 3 4 5 6 7

Measured Freq. [Hz] 3.0752 19.845 56.062 112.19 182.99 272.78 380.28

Analytical model Freq. [Hz] Error [%] 3.2243 5.44 20.344 2.51 57.017 1.70 111.83 −0.30 185.03 1.11 276.64 1.42 386.69 1.69

MAC-value 1.00 1.00 1.00 1.00 1.00 1.00 1.00

[K] updating Freq. [Hz] 3.1334 19.748 55.545 109.28 181.29 271.66 380.46

Error [%] 1.89 −0.49 −0.92 −2.58 −0.93 −0.41 0.05

MAC-value 1.00 1.00 1.00 0.99 0.99 0.99 1.00

Table 13.8 Natural frequency and MAC value comparisons between the experimental beam and the damping matrix updated analytical beam Mode [nr] 1 2 3 4 5 6 7

Measured Frequency [Hz] 3.0752 19.845 56.062 112.19 182.99 272.78 380.28

[C] Updated model Frequency [Hz] 3.133 19.748 55.552 109.28 181.29 271.68 380.47

Error [%] 1.88 −0.49 −0.91 −2.59 −0.93 −0.40 0.05

MAC-value 1.00 1.00 1.00 0.99 0.99 0.99 1.00

From Fig. 13.4a it is seen that before updating, there is only a slight discrepancy in the natural frequencies in the FRF. After updating the stiffness matrix, the peaks shift slightly and match better with the experimental results. Though still with some error of the peaks lining up. Likewise in Fig. 13.4b it is seen that after damping identification, that the peeks are nearly identical in amplitude. From Table 13.7 it is found that the MAC-values for the analytical model are good, however there is some error of the natural frequencies. After updating the stiffness matrix, the error percentages of the natural frequencies decrease but the MAC-values slightly decrease. In Table 13.8 the frequencies and MAC values between the simulated model and the [C] updated model are compared. Comparing the frequency error values for the [C] updated model with the frequency error values of the [K] updated model in Table 13.7, it is seen that the frequency error does not change. Just as for the simulated experiment, this is as expected, as the damping matrix should not affect the natural frequencies.

13 Damping Identification and Model Updating of Boundary Conditions for a Cantilever Beam Table 13.9 Cantilever beam spring stiffnesses after updating

Springs   k1_t N m  ◦ k1_r N m/

147 Analytical

[K] updated

Change [%]

5 · 1010

4.19 · 1014

–

1 · 103

4.16 · 102

−58.4

In Table 13.9, the boundary condition spring stiffnesses are compared before and after updating the stiffness matrix. The translational spring sees a huge increase in stiffness, this is probably due to the rigid bolting of the beam and a low stiffness value of the translational spring initially. The rotational spring decreases by 58%, this reduction in stiffness is what accounts for the natural frequency changes in Table 13.7.

13.5 Discussion Based on the results of the simulated and the experimental beams, it is seen that the inverse-eigensensitivity method and direct damping updating method proprosed by Pilkey [17] provide good results when updating boundary conditions, albeit still with some error in the updated parameters. For the simulated case, the error in stiffness updated parameters is likely due to the modeshape expansion. According to Jung [4], the inverse eigensensitivity approach can be used on incomplete modal data without modeshape expansion. This is something that must be tested further to see if improvements can be made to the parameter estimation. Further, alternate methods of updating the stiffness matrix could be tested, as the Pilkey damping updating method [17] requires accurate mass and stiffness matrices in order to obtain the best possible results. For the experimental case, the natural frequencies generally improved well, however the MAC-values slightly decreased. This could indicate that more than just the boundary conditions require updating. The material stiffness can be subjected to slight variations from batch to batch, thus the elastic modulus would be a good candidate for further updating. Also, the beam is fairly slender and quite a few accelerometers are attached. Thus mass loading of the beam could be affecting the natural frequencies and modeshapes. An analytical model with point masses at measurement locations may further improve the results.

13.6 Conclusion The inverse-eigensensitivity approach and the direct damping updating method proposed by Pilkey where tested for updating boundary conditions and damping. The methods were both tested on a simulated set of data and corresponding FE model and on an experimentally tested cantilever beam. In both cases, the methods were able to update the stiffness and damping, thus providing more accurate modal results over their non-updated counterparts.

References 1. Chandrupatla, T.R., Belegundu, A.D.: Introduction to Finite Elements in Engineering. Pearson, Noida (2014) 2. Brandt, A.: Noise and Vibration Analysis – Signal Analysis and Experimental Procedures. Wiley, Chichester (2011) 3. Friswell, M.I., Mottershead, J.E.: Finite Element Model Updating in Structural Dynamics. Springer, Dordrecht/London (1995) 4. Jung, H.: Structural dynamic model updating using eigensensitivity analysis. Ph.D thesis, Imperial College of Science, Technology and Medicine (University of London) (1992) 5. Lin, R.M., Lim, M.K., Du, H.: Improved inverse eigensensitivity method for structural analytical model updating. J. Vib. Acoust. 117(2), 192–198 (1995) 6. Pradhan, S., Modak, S.V.: A two-stage approach to updating of mass, stiffness and damping matrices. Int. J. Mech. Sci. 117(2), 192–198 (2018) 7. Pradhan, S., Modak, S.V.: Normal response function method for mass and stiffness matric updating using complex frfs. Mech. Syst. Signal Process. 32, 232–250 (2012) 8. Lin, R.M., Ewins, D.J.: Analytical model improvement using frequency response functions. Mech. Syst. Signal Process. 8, 437–458 (1994) 9. Modak, S.V., Kundra, T.K., Nakra, B.C.: Model updating using constrained optimization. Mech. Res. Commun. 27, 543–551 (2000) 10. Levin, R.I., Lieven, N.A.J.: Dynamic finite element model updating using simulated annealing and genetic algorithms. Mech. Syst. Signal Process. 12, 91–120 (1998) 11. Atalla, M.J., Inman, D.J.: On model updating using neural networks. Mech. Syst. Signal Process. 12, 135–161 (1997)

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12. James Hu, S-L., Li, H., Wang, S.: Cross-model cross-mode method for model updating. Mech. Syst. Signal Process. 21, 1690–1703 (2007) 13. Wang, S., Li, Y., Li, H.: Structural model updating of an offshore platform using the cross model cross mode method: an experimental study. Elsevier Ocean Eng. 97, 57–64 (2015) 14. Rayleigh, L.: Theory of Sound (two volumes). Dover Publications, New York (1897) 15. Lancaster, P.: Expression for damping matrices in linear vibration. J. Aerosol Sci. 28 (1961) 16. Lee, J-H., Kim, J.: Development and validation of a new experimental method to identify damping matrices of a dynamic system. J. Sound Vib. 246(3), 505–524, (2001). ISSN 0022-460X 17. Pilkey, D.F.: Computation of a damping matrix for finite element model updating. Ph.D. thesis, Virginia Polytechnic Institute and State University (1998) 18. Guyan, R.J.: Reduction of stiffness and mass matrices. Am. Inst. Aeronaut. Astronaut. 3, 380 (1966) 19. Kidder, R.L.: Reduction of structural frequency equations. Am. Inst. Aeronaut. Astronaut. 11, 892 (1973) 20. Avitable, P.: Model reduction and model expansion and their applications part 1 – theory 21. Mendes Maia, N.M., Motalvão e Silva, J.M., He, J., John Lieven, N.A., Lin, R.M., Lin, G.M., To, W-M., Urgueir, A.P.V.: Theoretical and Experimental Modal Analysis. Research Studies Press LTD., Baldock (1997) 22. Fox, R.L., Kapoor, M.P.: Rates of change of eigenvalues and eigenvectors. Am. Inst. Aeronaut. Astronaut. 6, 2426–2429 (1968) 23. Bibbo, N., Berntsen, J.: Operational modal analysis: time variance in wind turbine towers. Master’s thesis, University of Southern Denmark (2019)

Chapter 14

An Experimental Substructure Test Object: Components Cut Out From a Steel Structure Andreas Linderholt

Abstract Substructuring is the topic of the Society of Experimental Mechanics’ Technical Division on Dynamic Substructures. During a number of the most recent IMAC conferences, a lot of studies of coupling and de-coupling of substructures have been presented. In addition, frequency response-, modal- and state-space based techniques for coupling of components represented by numerical or experimental models have been developed. In many such studies, the dynamics of the numerically coupled structures are compared with test data stemming from measurements of the physically assembled counterparts. An embedded issue when assembling components is the interfacing between the substructures, introducing dry friction in the form of micro slip and varying contact areas. These introduced sources of deviation between the numerically formed assembly and its real world counterpart blend with sources, of deviation, such that test data being incomplete, biased or having random errors. Here, the initial test object is manufactured as a one piece solid structure. After that, the structure will be cut to form two substructures. Finally, the substructures will be assembled again. Vibrational tests will be made on the solid structure, the substructures as well as on the assembly. The aim is to compare vibrational data and differences in dynamical properties; especially damping, eigenfrequencies and linearity between the solid structures, the numerically formed assembled and the physical re-assembled structure are studied. The purpose of the study is to isolate the causes of possibly deviations by removing the issues stemming from unknown interfaces. Here, the structure together with synthetic modes are presented. Keywords Experimental substructuring · One piece · Test object · Wire cutting · Coupling

14.1 Introduction Analytical/numerical substructuring techniques, developed largely by Hurty, Craig and Bampton decades ago, are frequently used in industry. That is not the situation for experimental substructuring which is lagging behind. During the last years, the need for research focusing on experimental substructuring has been made. One problem is to get appropriate interfaces between substructures. The transmission simulator method became a tool to handle that. The need for more accurate test data, than needed for other structural dynamics exercises, constitutes another problem. In 2012, the Society of Experimental Mechanics’, SEM’s, substructuring focus group introduced a modified version of the AmpAir 600 (A600) wind turbine as a benchmark structure [1]. Seven such benchmark structures were manufactured. A lot of research on that structure or on components of it have been made. The A600 is an industrial (not purely academic) structure that is interesting and challenging to work on. However, it is a complex structure, which introduces many uncertainties. In practice, there was also a limited access to them since only seven of them exist. In 2018, the SEM focus group, which organizes courses, IMAC sessions and tutorials, became the SEM Technical Division (TD) on Dynamic Substructures [2]. During some years, the need for a simpler benchmark structure has been a topic for discussion. A proposal of such a structure was presented at IMAC XXXVIII, 2020 [3]. One idea has been to set out from a structure manufactured as one-piece. By cutting the structure, its components/substructures are formed. That procedure enables experimental substructuring exercises with a known true answer. A beam together with a few discs build up such a structure, see the example in Fig. 14.1a. Randy Mayes thought of such a structure several years ago. Another benchmark

A. Linderholt () Department of Mechanical Engineering, Linnaeus University, Växjö, Sweden e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2021 A. Linderholt et al. (eds.), Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47630-4_14

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Fig. 14.1 Ideas of substructures that can be manufactured in one piece and then cut to form their components/substructures. (a) a beam together with a few discs, (b) and (d), a structure similar to the BARC (Box Assembly with Removable Component), (c) a structure with small/discrete coupling points

structure, addressing dynamic environment testing is the BARC (Box Assembly with Removable Component). The results from a round robin of that structure was presented at IMAC XXXVII in 2019. Manufacturing a structure, similar to the BARC but in one piece, see the example in Fig. 14.1b and d, was one option in the work presented here. A decision on the contact has to be made. Having discrete contact points, small contact surfaces, see the example in Fig. 14.1c, may enable a Frequency based Substructuring (FBS) approach whereas larger contact surfaces may make Component Mode Synthesis the obvious choice.

14.2 The Test Object The structure presented here was designed with the following objectives. The structure should be possible to manufacture in one piece. The mechanical engineering workshop at Linnaeus University in Växjö, Sweden, has access to a wire cutter that is used. To start with, the maximum dimension in x, y and z was set to 220 mm to allow cutting along all axes. When the structure, later on, is cut, one of the substructures should have a simple form that is easy to represent with an analytical numerical model. Furthermore, the intension is that the structure should be easy to make measurements on; that means that is should be easy to excite the structure and to measure on with lasers as well as accelerometers. The structure is shown in Fig. 14.2. The one-piece structure is supposed to consist of a 172 by 50 by 2.5 mm top plate, the purple part in Fig. 14.2a, as one of its substructures. That will be cut off using the wire cutter after measurements on the complete structure. The remaining part has base length and width equal to 212 and 160 mm. The height of the remaining part is 166 mm. The structure is formed as steps and it has piecewise more massive parts. The latter enables larger areas for attachments of accelerometers and excitations and it should also be possible to equip the structure with several one degree-of-freedom accelerometers on the massive parts to capture rotations. The steps makes it easy to measure responses with a scanning laser. The two substructures, the top plate and the remaining part, are joined at the two 50 by 50 mm contact areas, which are brown in Fig. 14.2e. It is also

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Fig. 14.2 The one-piece structure. (a) an ISO view, (b) front view of the x–z plane, (c) top view of the x–y plane, (d) side view of the y–z plane, (e) the substructure after cutting off the top plate, (f) a possible design having discrete connections between the substructures

possible to make cut outs so that the connections in the one-piece structure consists of small, discrete, areas such as shown in Fig. 14.2f. Eight M6 screws will join the two substructures when re-assembled. The screws will be added also to the one piece structure. The manufacturing was mainly made with a wire cutter. The wire sits in an upper and a lower support. The two supports can be controlled independently of each other, which enables manufacturing of very complex shapes. As an example, the upper support can move in a path forming a square at the same time as the lower support follows a circle form. A few pictures from the manufacturing are shown in Fig. 14.3. After the pictures were taken, the steps will be formed and the holes for the screws will be drilled.

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Fig. 14.3 Manufacturing of the test object. A wire cutter, shown in the upper left and, partly, in the upper right pictures, is used. The structure after the inner and outer cuts are shown Table 14.1 The elastic eigenmodes, below 1000 Hz, of the FE-model representing the one piece structure, with the plate on top

Mode # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Natural frequency (Hz) 104.5 117.1 190.1 228.4 304.9 331.9 395.4 510.6 556.0 675.5 691.0 784.0 887.2 887.8 984.9

Plane of deformation Within the x–z plane Within the x–z plane Within the x–z plane Within the x–z plane Within the x–z plane Within the x–z plane Out of the x–z plane Out of the x–z plane Within the x–z plane Within the x–z plane Out of the x–z plane Out of the x–z plane Out of the x–z plane Within the x–z plane Within the x–z plane

14.3 Finite Element Analyzes of the One Piece Structure An MSC Nastran finite element model representing the structure was made. Solid, MSC Nastran CHEXA, elements build up the model. The frequencies of the elastic eigenmodes below 1 kHz are shown in Table 14.1 and the associated mode shapes are shown in Fig. 14.4.

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Fig. 14.4 The 15 elastic eigenmode shapes, associated with frequencies below 1000 Hz, of the one-piece structure, with the plate on top. The mode shapes stem from an MSC Nastran FE-model

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Table 14.2 The elastic eigenmodes, below 1000 Hz, of the FE-model representing the structure without the plate on top

Mode # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Natural frequency (Hz) 31.8 50.7 118.2 177.7 210.3 225.6 253.2 324.3 387.5 498.0 557.3 674.9 735.4 811.3 884.9

Plane of deformation Within the x–z plane Within the x–z plane Within the x–z plane Within the x–z plane Out of the x–z plane Within the x–z plane Within the x–z plane Within the x–z plane Out of the x–z plane Out of the x–z plane Within the x–z plane Within the x–z plane Out of the x–z plane Out of the x–z plane Within the x–z plane

Table 14.3 The elastic eigenmodes, below 4000 Hz, of the FE-model representing the top plate Mode # 1 2 3 4 5 6

Natural frequency (Hz) 449.6 929.8 1245.1 1942.3 2447.8 3112.4

Mode shape First bending around the y-axis First torsion Second bending around the y-axis Second torsion Third bending around the y-axis Third torsion

14.4 Finite Element Analyzes of the Two Components Two substructures are formed by cutting off the top plate. Inevitably, some material is lost during the cutting. Using the wire cutter, the thickness of the cut out is roughly 0.2 mm. However, the top plate is a simple structure that is easy to represent analytically or with a finite element model. The frequencies of the elastic eigenmodes below 1 kHz of the remaining structure, without the top plate and below 4 kHz of the top plate, are listed in Tables 14.2 and 14.3. The associated mode shapes are shown in Figs. 14.5 and 14.6.

14.5 Future Work The manufacturing of the structure is about to be finalized. Then, vibrational tests of the one-piece structure including the screws will be made. Both accelerometers and a scanning laser Doppler will be used as sensors in turn. After that, the structure will be cut and thereby form the two substructures. The same kinds of vibrational tests as above will be made on the substructure that remains after removing the top plate. The test data together with data stemming from the finite element representation of the top plate constitute the necessary information to make coupling exercises and the true solution from a coupling with ideal joints is known. A new top plate will be manufactured since some material is removed in the cutting process. Measurement data of the slightly thinner, from cutting, top plate as well as the new top plate will be used together with the finite element model to verify that the new top plate has the same properties as the one removed. Finally, the new top plate will be assembled with the remaining substructure. Thereby, the dynamics of a structure with ideal joints between its components can be compared with the same structure where the components are assembled with screw joints. The aim is to make the data available to the ones that are interested to use them.

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Fig. 14.5 The elastic eigenmode shapes, below 1000 Hz, of the structure without the plate on top. The mode shapes stem from an MSC Nastran FE-model

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Fig. 14.6 The elastic eigenmode shapes, below 4000 Hz, of the top plate. The mode shapes stem from an MSC Nastran FE-model

14.6 Conclusion With a wire cutter, it is possible to manufacture structures with advanced shapes. The purpose in the present work is to manufacture a one-piece structure that later is divided into substructures. This allows for comparison of the dynamics of a structure with ideal joints between its components and the same structure where the components are assembled with screw joints. The objective is also to manufacture a structure suited for vibrational tests. This means a structure on which it is easy to attach sensors and excitations.

References 1. Mayes, R.L.: An introduction to the SEM substructures focus group test bed – the AmpAir 600 wind turbine. In: Conference Proceedings of the Society for Experimental Mechanics Series. 30th IMAC, A Conference on Structural Dynamics (2012) 2. SEM/IMAC Dynamic Substructuring Technical Division’s web space [online]. Available at http://substructure.engr.wisc.edu 3. Roettgen, D., et al.: Technical Division Benchmark Structure for Dynamic Substructuring. IMAC XXXVIII, Houston (2020)

Chapter 15

Frequency Based Model Mixing for Machine Condition Monitoring Samuel Krügel and Daniel J. Rixen

Abstract Creating holistic, efficient models for vibration-based monitoring applications containing rotor systems is still challenging. Reasons for these difficulties are application-dependent housing peripheries and inaccessible measurement points. Due to the complexity of these systems, numerical modeling is cumbersome and the application of experimental techniques only is restricted. With this contribution, we propose a solution approach for combining the experimental determined housing dynamics with a numerical rotor model. The method performs in the frequency domain, based on Lagrange Multiplier Frequency Based Substructuring (LM FBS) and System Equivalent Model Mixing (SEMM) as a closely related method. Our technique rests upon three parts: Firstly, a finite elements rotor model with reduced degrees of freedom (DoF) is created and the complete Frequency Response Function (FRF) matrix for all interface and input DoF is calculated. Secondly, the entire FRF-matrix is coupled with the simulatively determined transfer functions of the housing. Thirdly, FRFs of a collocated subset DoF of the rotor-assembly are experimentally measured. These are expanded to the FRFmatrix of the coupled model using the SEMM method. As a result, we get the complete FRF-matrix being full rank and containing dynamics of the entire, coupled system. Finally, the proposed methodology is experimentally validated based on an exemplary transfer function. Keywords Dynamic substructuring · Experimental frequency based substructuring · System equivalent model mixing · Hybrid modelling · Condition monitoring

15.1 Introduction Low cost condition monitoring systems increase in popularity for industry. A common approach for system identification and fault scenarios are data based systems. Though, this data is not available in the development process. Another problem are differences between boundary conditions at the test rig and the final application. The impact of this effect is all the more, if measurement positions are placed at the outside housing. Due to wireless sensor data transmission, sensor placement at the outside housing is mandatory. Integrated sensor systems in form of wireless sensor nodes regularly use these standards. At this point, model-based systems are favorable. With known transfer paths between the rotor system DoF and the measurement DoF, quantitative fault identification is thought to be possible. Our technique is demonstrated on an industrial application, particularly a cooling blower for an electrical drivetrain (see Fig. 15.1). During operation, it sucks the air axially, and blows it on the drivetrain by means of the vertically supported impeller.

S. Krügel () · D. J. Rixen Chair of Applied Mechanics, Faculty of Mechanical Engineering, Technical University of Munich, Garching, Germany e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2021 A. Linderholt et al. (eds.), Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47630-4_15

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Air in

a

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a

b

15.2 Numerical Model For the numerical model description, we use a reduced finite elements rotor model in combination with a simplified model of the housing. The underlying method for rotor modeling is described in [1]. Here, free-free boundary conditions are applied and the complete FRF-matrix for all input and output DoF (136 × 136) is calculated. The housing is approximated with a hollow, cylindrical Timoshenko-beam (component a at Fig. 15.2). Its wall thickness and support stiffnesses are tuned to match the first eigenfrequencies of the blower test rig. This test set up is needed for the measurements in Sect. 15.3. The first regarded modes (experimental modal analysis reported in [2]) occur at 9 and 11 Hz. By means of this simplified model, a coarse assumption of the housing FRFs can be performed. Based on the tuned model, a 4 × 4 transfer function matrix of the housing at the connection point with the rotor can be calculated, considering the displacements along the x, y axis and rotations around the same axes. Compatibility and equilibrium condition between that point and the bearing positions are assumed performing LM FBS method. These locations of the two systems a and b are marked by grey arrows in Fig. 15.2.

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Thus, the solution for the coupled system can be formulated as: Y coupled = Y − Y B T (BY B T )−1 BY . (ab)

(15.1)

Equation 15.1 uses the complex admittances Y = fu with system responses u to external forces f of the uncoupled subsystems a and b in block diagonal form. Here, B contains the coupling DoF between rotor and housing expressed by a signed Boolean matrix. For detailed information about LM FBS, the authors refer to [3]. Thus, we get a coupled FRF matrix describing the blower system (ab), comprising rotor and housing.

15.3 Modal Expansion In order to refine the model approximation, the numerical solution is mixed with experimentally determined transfer functions. Therefore, we perform frequency response measurements using the test set up illustrated in Fig. 15.3. Force impacts are applied with the automatic modal hammer AMimpact [4] and system responses sensed by piezoelectric accelerometers (Kistler 8688A). Siemens LMS Test Lab is used for data acquisition to set up an experimental FRF-matrix with 6 × 4 DoF (x- and y-direction only). This FRF-matrix represents our overlay model within performing the SEMM method. In this case, all these measured DoF are considered as boundary DoF for mixing with the numerical model. One ability of SEMM is to expand the experimental determined dynamics to internal DoF at the rotor, which cannot be measured. Herewith, we close the gap between accessible housing measurement locations and all rotor positions of interest via the hybrid SEMM model. The numerical FRF-matrix is used as parent model with all DoF kept for SEMM. Here, the numerical subset FRF-matrix with 6 × 4 DoF contains the boundary DoF between numerical simulation and experimental model. The 8 × 4 DoF subset FRF-matrix contains 6 “measurement” DoF (4 at the rotor and 2 at the housing, see Fig. 15.3). Furthermore, it implies 4 “excitation” DoF (2 at the rotor and 2 at the housing, see Fig. 15.3) The remainder is kept as internal DoF subset. According to SEMM implementation, we perform the fully extended interface method with the associated single-line 

Y

SEMM

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Y ik Y ib − Y bk Y bb

(ab)

 rem +  rem Y bb − Y ov Y bk Y bb bb



Y kb Y bb

rem + 

Y ki Y kb Y bi Y bb

(ab) (15.2)

with internal DoF i, boundary DoF b and kept DoF k, removed model “rem” and the overlay model “ov”.

Z Y X

Fig. 15.3 Partial blower representation with measurement set up. Blue force impacts (arrows) and triaxial accelerometers are used for modal expansion. The orange schematic accelerometer represents the validation measurement point at the rotor shaft end

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Frequency / Hz Fig. 15.4 FRF magnitude in y-direction based on impact excitation on top of the rotor and frequency response at the reference measurement point. Simulated FRF (grey), FRF-prediction by the hybrid SEMM model (blue) and experimental validation (orange). Top: Full bandwidth up to 1.6 kHz, bottom: Lower frequency range

For detailed information about SEMM, the authors refer to [5]. The results for the FRF magnitude at the reference ymeasurement point with y-excitation at the upper rotor shaft end are shown in Fig. 15.4 (orange validation measurement point). Here, the numerical approximation is opposed with the hybrid SEMM model prediction and validated by the experimental solution. The large deviation of the numerical approximation is striking and except the first rigid body mode, no relevant resonances are visible in the depicted frequency range. Moreover, the amplitude amplification is overestimated over the entire frequency range. This effects may be owed to the coarse modeling assumptions and the neglection of the connecting struts between rotor and housing. In contrast to that, the hybrid model predicts a reasonable transfer function. Especially in the exemplary selected lower frequency range (Fig. 15.4 above), the result is quite acceptable. The predicted admittance in this frequency range could thus possibly serve as a base for monitoring of rotor faults.

15.4 Conclusion The numerical approximation of an exemplary industrial application using a coupled approach of models with a low amount of DoF leads to FRF overestimation. Even system-relevant resonances are not represented. Mixing the numerical model with experimental data using the SEMM method is therefore a promising technique for significant improvement of predicting the real transfer function. In future work, we intend to evaluate and optimize the method for operational excitation conditions.

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References 1. Krügel, S., Maierhofer, J., Thümmel, T., Rixen, D.J.: Rotor model reduction for wireless sensor node based monitoring systems. In: International Conference on Dynamics of Rotating Machines, Copenhagen (2019) 2. Mulser, T.: Experimentelle Transferpfadanalyse und Substrukturierung am Fahrmotorlüfterprüfstand. Masterarbeit. Technische Universität München (2019) 3. de Klerk, D., Rixen, D.J., Voormeeren, S.: General framework for dynamic substructuring: history, review and classification of techniques. AIAA J. 46(5), 1169–1181 (2008). https://doi.org/10.2514/1.33274 4. Maierhofer, J., El Mahmoudi, A., Rixen, D.J.: Development of a low cost automatic modal hammer for applications in substructuring (2019). https://doi.org/10.13140/RG.2.2.21408.43527 5. Klaassen, S.W., van der Seijs, M.V., de Klerk, D.: System equivalent model mixing. Mech. Syst. Signal Process. 105, 90–112 (2018). https:// doi.org/10.1016/j.ymssp.2017.12.003

Chapter 16

Using a Machine Learning Approach for Computational Substructure in Real-Time Hybrid Simulation Elif Ecem Bas, Mohamed A. Moustafa, David Feil-Seifer, and Janelle Blankenburg

Abstract Hybrid simulation (HS) is a widely used structural testing method that combines a computational substructure with a numerical model for well-understood components and an experimental substructure for other parts of the structure that are physically tested. One challenge for fast HS or real-time HS (RTHS) is associated with the analytical substructures of relatively complex structures, which could have large number of degrees of freedoms (DOFs), for instance. These large DOFs computations could be hard to perform in real-time, even with the all current hardware capacities. In this study, a metamodeling technique is proposed to represent the structural dynamic behavior of the analytical substructure. A preliminary study is conducted where a one-bay one-story concentrically braced frame (CBF) is tested under earthquake loading by using a compact HS setup at the University of Nevada, Reno. The experimental setup allows for using a small-scale brace as the experimental substructure combined with a steel frame at the prototype full-scale for the analytical substructure. Two different machine learning algorithms are evaluated to provide a valid and useful metamodeling solution for analytical substructure. The metamodels are trained with the available data that is obtained from the pure analytical solution of the prototype steel frame. The two algorithms used for developing the metamodels are: (1) linear regression (LR) model, and (2) basic recurrent neural network (RNN). The metamodels are first validated against the pure analytical response of the structure. Next, RTHS experiments are conducted by using metamodels. RTHS test results using both LR and RNN models are evaluated, and the advantages and disadvantages of these models are discussed. Keywords Dynamic substructuring · Real-time hybrid simulation · Machine learning · Linear regression · Recurrent neural network

16.1 Introduction Hybrid simulation (HS) is developed to answer the need for realistic dynamic testing of structures that combines experimental and analytical models and benefits from their advantages simultaneously. HS was first introduced by Takanashi et al. [1], where it was defined as “on-line testing” and the structural system modeled as a discrete spring-mass model within the time domain. Since then, many researchers were studied in different areas to enlarge the HS capabilities such as developments in numerical integration methods [2–6], substructuring techniques [7–9], delay compensation, and error mitigation [10– 12]. With these developments, this experimental technique became more reliable, accurate, efficient, and cost-effective for large-scale and full-scale real-time dynamic testing [13]. During the dynamic analysis in HS/RTHS, the equation of motion for the coupled experimental-computational simulations is usually solved by direct numerical integration algorithms. However, there are still some limitations that exist for these integration algorithms for complex structures where larger degrees of freedoms are involved along with numerical and/or experimental nonlinearities. These limitations could affect the performance and reliability of the HS/RTHS. To avoid these difficulties and improve the performance of the HS/RTHS, machine learning algorithms could be an alternative way to represent the analytical substructure. Machine learning is the science of programming computers so that they can learn from data [14]. The use of machine learning algorithms or metamodels became popular recently in engineering problems, with

E. E. Bas () · M. A. Moustafa Department of Civil and Environmental Engineering, College of Engineering, University of Nevada, Reno, Reno, NV, USA e-mail: [email protected] D. Feil-Seifer · J. Blankenburg Department of Computer Science and Engineering, College of Engineering, University of Nevada, Reno, Reno, NV, USA © The Society for Experimental Mechanics, Inc. 2021 A. Linderholt et al. (eds.), Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47630-4_16

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the growing complexity of finite element (FE) models [15]. Even with the computational powers that we have today, the computational cost for FE models, which have large numerical systems, could be very high, and this led researchers to develop alternative representations of these FE models to predict the dynamic response of the simulation. The main idea of this paper is to represent an entire concentric braced frame (CBF) computational substructure, i.e. columns, beam, mass, and damping, with a metamodels that is developed using a machine learning algorithm to define the analytical substructure response in HS. Moreover, the inherent servo-hydraulic dynamics in the HS system could lead to a time delay in response to the command of displacement, which generates inaccurate results, especially in RTHS [16]. This time delay is usually eliminated by using a proper delay compensator. In this paper, time delay with the actuator input and the feedback are eliminated within the metamodel instead of using a time delay compensator. To do these, a dataset for the training model is obtained by the pure analytical model time history analysis of a one-bay one-story CBF, under earthquake excitation. The model is considered as batch learning since all the dataset are provided offline. A linear regression (LR) model and a recurrent neural network (RNN) model are considered to develop the metamodels, which predicts the input displacement value for the actuator. Once the metamodels are trained, these are first compared with the pure analytical FE model response. The FE model response is considered to be the exact solution of the system. Then, the metamodels are incorporated into the HS loop to conduct preliminary linear tests, where the HS test results and model responses can be compared against the exact values from pure FE analysis as well.

16.2 System Components and Capabilities A compact HS setup is designed and constructed in the Large-Scale Structures Laboratory (LSSL) at the University of Nevada, Reno. This small-scale setup is developed for CBF demonstrations, educational purposes, and tackle new research problems pertaining to computational challenges for HS/RTHS. The analytical substructure can be modeled using either Simulink or OpenSees [17] platform by using proper FE techniques. Moreover, the HS system is capable of running both real-time and pseudo-dynamic (slow) experiments. More details about the HS system development and verification can be found in [18]. The system consists of: (1) load frame with a dynamic actuator run by an isolated hydraulic pump; (2) MTS STS controller with 4-channels with 2048 Hz clock speed; (3) real-time high performance Simulink machine (Speedgoat xPC Target); (4) Windows machine (Host PC) for MATLAB, OpenSees, and the HS middleware OpenFresco [19]; and (5) SCRAMNet ring that provides shared memory locations for real-time communication. The xPC Target provides a high-performance host-target prototyping environment that enables the researcher to connect the Simulink and Stateflow models to physical systems. It sends and receives data from the controller. The controller (STS 493 Hardware Controller) has 4-channels with 2048 Hz clock speed and controls the motion of the actuator. Currently, it has only one channel connected since it is only controlling one actuator, but it is capable of controlling four actuators. STS Host PC is where the basic controller properties are controlled with the graphical user interface of the MTS 493 controller [20]. The load frame is the experimental setup for the HS system. The dynamic actuator of the system has 7 kips (31.14 kN) maximum load capacity, which has (±1 in.) stroke. The peak velocity of the actuator at no load is 338.84 mm/s (13.34 in/s). The isolated hydraulic power supply system has 8.71 lt/min (2.3 gpm) pumps, and the reservoir capacity of oil volume is 56.78 lt (15 gallons).

16.3 Modeling Assumptions The one-story one-bay steel braced frame is selected for both verifications and evaluations of the system challenges. CBFs are ideal for HS testing since the columns and beams are not expected to be damaged during an earthquake and can be accurately modeled in the computer along with the mass and the damping forces (i.e. analytical substructure), while the braces can be physically tested to capture buckling accurately and low-cycle fatigue induced rupture (i.e., experimental substructure). Figure 16.1 illustrates the CBF substructuring for HS testing. The experimental setup allows for using a smallscale brace as the physical substructure along with a steel frame at the prototype full-scale for the analytical substructure. The HS of this model has been verified against the pure analytical solution of frame, and these results are also included in this section. Two machine learning algorithms are pursued in this study to develop metamodels and explore possibilities to compile a successful HS test. First, the linear regression (LR) method is used to train a model. Secondly, a recurrent neural network

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Fig. 16.1 Model and substructuring of CBFs for HS testing

(RNN) with different hidden layers and time step delays is modeled and tested for performance. Both algorithms are compared with the pure analytical models first. Then, following the assessment of their performance in the pure analytical model predictions, the metamodels are further tested in the HS loop with real feedback from the actuator that is free to move without specimen attached as explained later. In general, an LR model predicts by simply computing a weighted sum of the input features, plus a constant called bias term [14], shown in Eq. (16.1). Here yˆ is the predicted value, n is the number of features. xi is the ith feature value, and θ j is the jth model parameter (including the bias term θ 0 and the feature weights term). yˆ = θ0 + θ1 x1 + θ2 x2 + · · · + θn xn

(16.1)

The LR model is trained by using a pure analytical solution of the CBF. For the training, five features are selected to be the input of this metamodel to predict the output as a command displacement of the experimental substructure of the HS system. The features for training are selected as: ground motion acceleration, displacement feedback value of the brace, force feedback value of the brace, one step behind the value of the predicted displacement, and two steps behind the value of the predicted displacement. First, the pure analytical solution is used as training data, generated by using the pure analytical solution responses and assuming that there is a 28-time step time delay for the feedback values. Moreover, since it is hard to explore the exact delay from the actuator feedback, this model is run first in the HS loop without including the displacement feedback from the actuator. However, the displacement feedback of the actuator is recorded to be used in the next training phase. Then, a more refined model is generated by using the “real” displacement and force feedback, and the other three features that are used for the training phase. For the RNN model, three different models are trained and generated. An RNN is an artificial neural network that allows exhibiting temporal dynamic behavior since the model uses its memory to process the sequence of inputs [21]. For the RNN models, the inputs are ground motion acceleration and force feedback. Basic RNN models are generated by using different hidden layers to find the optimum hidden layer number for an accurate response. Once the models are trained, each method is evaluated, and the results are shown. Root mean square error (RMSE) is calculated to evaluate the performance of the model predictions. The first verification is done for a pure analytical machine learning algorithm to see the model capabilities offline. Here, the prediction is made for each step while running the simulation. Once all the models are compared for the pure analytical solution, the one with the least RMSE error will be put in the HS loop. The first validation for the HS system is where the metamodel is compiled in the xPC and is tried with “fake” feedback with actuators on the system (online) to see the actuator performance. After this, the “real” feedback is fed into the metamodel, and the system response is evaluated against the pure analytical system response. The RMSE values are calculated for these cases as well as discussed next.

16.4 Model Parameters for HS The analytical substructure for the HS model consists of two columns (W14x311) with fixed end conditions and a beam (W36x150), which also has moment connections to the columns. Both columns and the beam are considered linear elastic elements, wherein braced frames the nonlinearity is commonly limited only to the braces. The mass and the damping of the system are considered to be part of the analytical substructure. The one-bay one-story braced frame is simplified as single degree of freedom (SDOF) for validation and modeling purposes. The dynamic properties of the SDOF model is as defined

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Fig. 16.2 (a) El-Centro ground motion acceleration; and brace results from pure analysis: (b) displacement history, and (c) force history

where M is the mass matrix, and K is the stiffness matrix given by Eqs. (16.2) and (16.3). M = [m]

(16.2)

K = [ka + ke ]

(16.3)

For this study, the story mass m is selected to be 1.75 kN-s2 /mm, and the frame stiffness (ka ) is determined from the frame sections to be 176.75 kN/mm. The axial brace stiffness, along its local axis, is 1224.1 kN/mm. The resulting natural period of the frame is 0.294 s. The structure is assumed to have an inherent damping of 2% modeled using Rayleigh damping. The equation of motion of the full system can be written as shown in Eq. (16.4). mx¨ + cx˙ + kx = −mu¨ g

(16.4)

c is the inherent viscous damping of the structure; x(t), x(t), ˙ and x(t) ¨ are the displacements, velocity, and acceleration response, respectively, and u¨ g is the ground motion acceleration. For HS, the equation is rewritten such that ka is the frame stiffness representing the analytical substructure, and ke is the experimental stiffness which represents the brace stiffness. The brace stiffness is transformed into global coordinates. In the HS case, the feedback from the experimental substructure replaces the ke xi term in Eq. (16.5). mx¨i + cx˙i + (ka + ke ) xi = −mu¨g

(16.5)

The pure analytical model that is developed in Simulink is using the Chang integration algorithm and uses a 1/2048 s time step [22]. The time step of the integration is selected to be the same as the controller time step to synchronize the data transfer. For the purpose of dynamic analysis, and later the HS seismic testing, a typical California ground motion is selected and used, which is the 1940 El-Centro ground motion excitation shown in Fig. 16.2a. From pure dynamic analysis, the brace displacement and brace force histories are obtained and shown in Fig. 16.2b and c, respectively. It is noted that such time history data is modified and used as the training data, to represent the feedback data, for the machine learning approach. To do so, a 28-time step delay is considered along with the pure analytical model data when used for the training and validation phases of pure analysis of the metamodel. As previously mentioned, metamodel is updated with the “real” feedback data that is obtained from free moving actuator.

16.5 Validation for RTHS with FE Model In this section, before getting into the HS test with metamodel, a brief summary of the validation tests that are conducted to validate the SCRAMNetGT card connection between the computers and the controller are presented. These tests were conducted in real-time, i.e. RTHS. For the offline validation test, the feedback from the experimental model was considered as it is coming from the command of the system. The displacement command was multiplied with the constant stiffness value of the specimen, and these validation tests were considered for linear elastic experimental material. Fig. 16.3a shows

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Fig. 16.3 HS validation test results: (a) for the offline case with FE model, (b) for the online case with FE model without specimen

the brace displacement time history comparison between the pure analytical model and offline RTHS response. In this case the RTHS response has an RMSE value as 5.4%. Moreover, the system capabilities were also verified, where an actual feedback from the actuator is fed into the analytical model. In this validation phase, instead of using a linear specimen, the feedback displacement from the free-to-move actuator is multiplied with a constant value equals to the stiffness of the brace, i.e. experimental substructure, in order to get the force feedback for the HS experiment. This case also represents a hypothetical linear-elastic brace material since the stiffness was used as a constant value. The actuator delay was corrected and compensated for using the Adaptive Time Series (ATS) compensator [10]. Figure 16.3b shows the displacement history comparisons for the online validation case of RTHS. When the actuator is online, and the real feedback is considered in the HS system (as it should be), the RMSE value is calculated to be 9.4%.

16.6 Methodology for Linear Regression Algorithm Firstly, the LR method is used in order to train a machine learning metamodel. The input features of this metamodel are selected to be: earthquake excitation, displacement feedback of the brace, force feedback of the force, and one- and twoprevious steps of the history of predicted displacement value of the brace. Once the model is trained, the model parameters are obtained, and a Simulink model with Matlab Function that contains the LR model is prepared as illustrated in Fig. 16.4. This model is compared with the pure analytical model solution that is presented above, i.e. FE model.  xprediction,i+1 = f x¨g , xf eedback,i , Ff eedback,i , xprediction,i , xprediction,i−1 = θ1 x¨g + θ2 xf eedback,i + θ3 Ff eedback,i + θ4 xprediction,i + θ5 xprediction,i−1

(16.6)

In Fig. 16.5, the brace displacement response of the FE model and metamodel with LR are compared when both models are used in a pure analytical setting, i.e. not from a HS test. The FE model response is considered as the exact solution of the system. The RMSE value for this comparison using the LR model is 0.15%, which indicates that the LR model performs well for the pure analytical case. It is noted that this model is trained with “real” displacement feedback that is obtained from the HS setup as previously discussed. However, in this pure analytical metamodel, displacement feedbacks are generated by using a 28-time step delay from the calculated (predicted) displacement. Once the pure analytical model is verified for the metamodel, the metamodel is compiled in the xPC Target so that it can be tested on a different hardware. For this model, again the calculated (predicted) value is modified with a 28-time step delay and fed back into the model, which means there is no feedback from the actuator. However, the hydraulics of the HS system are on and the actuator is free to move with the given input command, which is the predicted displacement of the brace. Figure 16.6 shows the brace displacement response as obtained from RHTS offline tests incorporating the LR metamodel, which is compared with the pure analytical FE results. The results are comparable as before which further validates the LR model. It can be seen from the above verifications that the metamodel is giving adequate predictions when compared with the exact solution. Next, the actuator behavior is also investigated that no unexpected behavior is observed during actual tests. The metamodel is required to be validated with the real feedback from the actuator. Thus, no specimen is used here but the actuator displacement feedback is multiplied with a constant stiffness value of the brace to produce an equivalent force

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Fig. 16.4 Machine learning model (pure analytical model)

Fig. 16.5 Comparison of results from pure analysis using FE model versus LR metamodel

Fig. 16.6 Comparison of brace displacement from offline HS validation test with the LR model (w/o actuator feedback) and pure analysis

feedback of a linear elastic specimen. The Simulink configuration that is used for this validation phase is shown in Fig. 16.7. This model is also compared with the pure analytical FE model solution, which can be seen in Fig. 16.8. The comparison shown in the figure corresponds to a RMSE value of 0.066%, which concludes that the RTHS test with the LR model is acceptable and the performance is very comparable to pure analytical cases.

16.7 Methodology for Recurrent Neural Network Algorithm As a second metamodeling algorithm, an RNN is used since this problem is a time series prediction. RNN is a class of nets that can predict the future, which can analyze time series [14]. The RNN model representation is shown in Fig. 16.9, which shows two inputs and one output. In this paper, the two input features are defined as: the earthquake excitation (x1 ) and the force feedback of the brace (x2 ) where the output for a HS test setting is the command signal for the brace displacement (y). For this method, since it has a connection pointing backward by default, there is no need to define an input for either oneor/and two-step behind predicted displacement value of the brace.

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Fig. 16.7 Simulink configuration of HS with the LR model

Fig. 16.8 Comparison of brace displacement from online HS validation test with the LR model (w/ actuator feedback) and pure analysis

Fig. 16.9 RNN architecture

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Table 16.1 RNN model parameters

Model name Model-1 Model-2 Model-3

Hidden layers 5 10 20

Fig. 16.10 Response comparison of FE model and pure analytical RNN model with five hidden layers (Model 1)

Fig. 16.11 Response comparison of FE model and pure analytical RNN model with ten hidden layers (Model 2)

The methodology in an RNN is defined as follows. Each recurrent neuron has two sets of weights where one for the inputs x(t) and the other for the outputs of the previous time step, y(t−1) , and these weight vectors are wx and wy , respectively. The output of a single recurrent neuron for a single instance can be defined as shown in Eq. (16.7) [14]. In the following equation b is the bias term and ∅(.) is the activation function.

 y(t) = ∅ x(t) T · wx + y(t−1) T · wy + b

(16.7)

Basic RNN is used in order to investigate the method. Three models with different number of hidden layers are generated to evaluate the performance. The model properties and names are summarized in Table 16.1. Five hidden layers were selected to start with and increased to 10 and 20 at last. These models are compared with the pure analytical FE model response, and the RMSE values are obtained as discussed next. In Fig. 16.10, the RNN model response that is trained with five hidden layers is shown. It can be seen from the responses that the RNN model cannot catch the peak displacement responses at the beginning. Moreover, a zoomed-in view of the response show that there are serious overfitting problems for some data points. On the other hand, that metamodel can capture the dynamics adequately. The RMSE value for this model is 4.23%, which can be considered to be overall adequate. After assessing the performance of Model 1, the hidden layer number is increased to 10 and results from Model 2 are shown in Fig. 16.11. As it can be seen in Fig. 16.11, the model performance is enhanced considerably when compared to Model 1. The dynamics of the structure are captured well, and the RNN model catches the peak displacements. The RMSE value is decreased to 3.74%. However, when the initial points are investigated, there is a severe oscillation in the behavior that can be observed as shown in Fig. 16.11. Moreover, along the displacement history, there is still overfitting in some data points. Lastly, an RNN model with 20 hidden layers is trained, i.e. Model 3. The brace displacement response comparison for Model 3 against FE model is shown in Fig. 16.12. The RMSE value has not been improved as compared to Model 2 and slightly increased to 3.88%. The performance also is not improved where this model still do not capture the peaks accurately. Moreover, the noise level at the beginning of the movement is also increased. The RNN models are shown here as potential

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Fig. 16.12 Response comparison of FE model and pure analytical RNN model with 20 hidden layers (Model 3)

alternative metamodels but given that RNN models performance is not significantly better than LR, no HS tests are conducted here using the RNN and only the analytical results above are provided for completeness.

16.8 Summary and Conclusions In this paper, machine learning algorithms are used for first time to model structural response of potential HS analytical substructure as a new alternative way to replace finite element models. The idea is to represent all analytical substructure components, i.e. frame columns, beam, mass, and damping in this case, with a metamodel, to reduce the computational time spent on the analytical substructure. Two different machine learning algorithms are used, i.e. LR and a basic RNN. Besides the computational time benefits, using metamodels for HS testing could eliminate the delay between the displacement command and feedback in the actuator within the model without using any time delay compensator. The HS setup and the validation of the setup using a FE model is first presented. The dataset to train the machine learning algorithms are obtained from a one-bay one-story braced frame response under the El-Centro earthquake. The columns and brace are defined as an analytical substructure where the brace is an experimental substructure. In this study, both analytical and experimental substructures are defined to be linear elastic. Firstly, an LR model is generated. For this model, five inputs features are used which are: earthquake excitation, displacement feedback of the brace, force feedback of the brace, and one- and two-previous steps from history of predicted displacement value of the brace. The FE model responses from pure analysis are used as dataset. Force and displacement feedbacks are generated by including a 28-time step delay to the analytical model response. However, it was observed that measuring the “online” actuator delay and the amplitude error is not possible. Thus, a preliminary metamodel should be generated initially and used in a HS setting in order to get “real” feedback. Once the “real” feedback is obtained, the final metamodel is generated to use for more representative HS tests. The metamodel with the LR algorithm is verified with a pure analytical FE model response. When the RMSE value of the predicted response history is found to be satisfactory, the model was tested within the HS loop but without including the “real” feedback data in the machine learning algorithm to check the actuator response offline. Once this gives accurate results, the “real” feedback is also included in the machine learning algorithm. The LR model has performed well and provided adequate and comparable results as pure analytical response. Secondly, RNN is also trained to represent the analytical substructure in the HS system. Three different models are trained where the hidden layers are changed to be 5, 10, and 20, respectively. The RNN model with ten hidden layers, has given the most accurate result, which has the least RMSE value when compared to pure FE analytical response. The RNN models are able to capture the dynamics of the structure accurately. However, when the response is investigated in more detail, it can be seen that there are some overfitting on some data points. In addition, lots of oscillations were observed in the initial part of the response. Since these errors can cause critical problems for the actuator, the RNN model was not considered further to use in the HS loop but yet presented here to outline its limitations. In conclusion, for a linear-elastic brace and linear-elastic analytical substructure, the new concept of HS testing using a machine learning algorithm is validated, for the first time, using actual HS tests. The analytical substructure represented with an LR algorithm gives satisfactory results which is very promising as compared to FE models. On the other hand, RNN models are not yet validated to use within the HS loop and more work is needed on RNN to reduce the noise and overfitting issues. For future work, a crucial step is to generalize the machine learning concept for HS for models with nonlinear responses. Moreover, different machine learning algorithms can also be investigated in the search for more effective and accurate methods for HS testing.

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References 1. Takanashi, K., Udagawa, K., Seki, M., Okada, T., Hisashi, T.: Non-linear earthquake response analysis of structures by a computer-actuator on-line system. Bull. Earthq. Resist. Struct. Res. Cent. 8, 1–17 (1975) 2. Ahmadizadeh, M., Mosqueda, G.: Hybrid simulation with improved operator-splitting integration using experimental tangent stiffness matrix estimation. J. Struct. Eng. 134, 1829–1838 (2008). https://doi.org/10.1061/(ASCE)0733-9445(2008)134:12(1829) 3. Jinting, W., Liqiao, L., Fei, Z.: Efficiency analysis of numerical integrations for finite element substructure in real-time hybrid simulation. Earthq. Eng. Eng. Vib. 17, 73–86 (2018). https://doi.org/10.1007/s11803-018-0426-0 4. Kolay, C.: Parametrically dissipative explicit direct integration algorithms for computational and experimental structural dynamics. Ph.D. Thesis (2016) 5. Bonnet, P., Williams, M.S., Blakeborough, A.: Evaluation of numerical time-integration schemes for RTHS. Earthq. Eng. Struct. Dyn. 37, 1467–1490 (2008). https://doi.org/10.1002/eqe.821 6. Chen, C., Ricles, J.M.: Development of direct integration algorithms for structural dynamics using discrete control theory. J. Eng. Mech. 134, 676–683 (2008). https://doi.org/10.1061/(ASCE)0733-9399(2008)134:8(676) 7. Shao, X., Mueller, A., Mohammed, B.A.: Real-time hybrid simulation with online model updating: methodology and implementation. J. Eng. Mech. 142, 1–19 (2015). https://doi.org/10.1061/(ASCE)EM.1943-7889.0000987 8. Mettupalayam, S., Reinhorn, A.: Real time dynamic hybrid testing using force-based substructuring. In: Structures Congress 2006 17th Analysis and Computation Specialty Conference, pp. 1–10. American Society of Civil Engineers, Reston (2006) 9. Zhou, M.X., Wang, J.T., Jin, F., Gui, Y., Zhu, F.: Real-time dynamic hybrid testing coupling finite element and shaking table. J. Earthq. Eng. 18, 637–653 (2014). https://doi.org/10.1080/13632469.2014.897276 10. Chae, Y., Kazemibidokhti, K., Ricles, J.M.: Adaptive time series compensator for delay compensation of servo-hydraulic actuator systems for real-time hybrid simulation. Earthq. Eng. Struct. Dyn. 42, 1697–1715 (2013). https://doi.org/10.1002/eqe.2294 11. Gunay, S., Mosalam, K.M.: Enhancement of real-time hybrid simulation on a shaking table configuration with implementation of an advanced control method. Earthq. Eng. Struct. Dyn. 44, 657–675 (2015). https://doi.org/10.1002/eqe.2477 12. Schellenberg, A.H., Mahin, S.A., Fenves, G.L.: Advanced implementation of hybrid simulation. Ph.D. Thesis (2009) 13. McCrum, D.P., Williams, M.S.: An overview of seismic hybrid testing of engineering structures. Eng. Struct. 118, 240–261 (2016). https:// doi.org/10.1016/j.engstruct.2016.03.039 14. Géron, A.: Hands-on Machine Learning with Scikit-Learn and TensorFlow. O’Reilly, Sebastopol (2017) 15. Spiridonakos, M.D., Chatzi, E.N.: Metamodeling of dynamic nonlinear structural systems through polynomial chaos NARX models. Comput. Struct. 157, 99–113 (2015). https://doi.org/10.1016/j.compstruc.2015.05.002 16. Chen, C., Ricles, J.M.: Servo-hydraulic actuator control for real-time hybrid simulation. In: 2009 American Control Conference, St. Louis, MO, pp. 5222–5227 (2009). https://doi.org/10.1109/ACC.2009.5160186 17. OpenSees. Open system for earthquake engineering simulation. From http://opensees.berkeley.edu (2008) 18. Bas, E.E., Moustafa, M.A., Pekcan, G.: Compact hybrid simulation system: validation and applications for braced frame seismic testing. J. Earthq. Eng. 1–30 (2020). https://doi.org/10.1080/13632469.2020.1733138 19. Schellenberg, A.H., Kim, H.K., Mahin, S.A.: OpenFresco. Universtiy of California, Berkeley (2009) 20. MTS. Civil engineering testing solutions. From http://www.mts.com (2014) 21. Sak, H., Senior, A., Beaufays, F.: Long short-term memory recurrent neural network architectures for large scale acoustic modeling. Int. J. Speech Technol. 22, 21–30 (2019). https://doi.org/10.1007/s10772-018-09573-7 22. Chang, S.-Y.: An explicit method with improved stability property. Proc. 2011 Am. Control Conf. 1885–1891 (2009). https://doi.org/10.1002/ nme

Chapter 17

On the Stability of a Discrete Convolution with Measured Impulse Response Functions of Mechanical Components in Numerical Time Integration Wolfgang Witteveen, Lukas Koller, and Florian Pichler

Abstract The relationship between one input and one output degree of freedom of a linear structure is completely described by the impulse response function (IRF). By convolution of this IRF with any input, the output can be computed. Several publications show in principle how components can be considered in numerical time integration of an overall system based on their IRF. The convolution integral is approximated as a sum, usually with the trapezoidal rule. However, it has been observed that a measured IRF may lead to an unstable behavior in the simulation of the overall system. This is expressed by an increasing amplitude of the involved quantities with increasing simulation time. The cause is suspected to be noise. In this paper, the real reason for the instability is shown which is somehow more general and noise is just a very relevant example which triggers the instability. After a theoretical derivation and discussion of this error, countermeasures for stabilization are explained. In the concluding chapter of the numerical examples, the findings from the theoretical chapter are confirmed. The stabilization strategies lead to the desired success. In the last example, an impulse response is generated based on measured data. After the impulse response is stabilized, the simulation results are compared with the measurement results and show good agreement. Keywords Time domaine substructuring · Impulse response function · Simulation · Substructuring · Experimental substructuring

17.1 Introduction The relationship between input and output of a linear system can be described by the use of differential or integral equations. In the latter case, the input variable (e.g. a force) is convoluted with the impulse response function (IRF) in order to obtain the output signal (e.g. displacement). Although this knowledge is very old, it plays a minor role in numerical simulation in the time domain. A generalization of the IRF for nonlinear systems is a Volterra series. However, this work focuses on the first term of a Volterra series, which is the IRF. In the 1990s, Gordis and his colleagues took up this integral way of thinking and carried out numerical simulations based on IRFs. In [1] the basic concept of this approach is presented. One key idea is that for the numerical simulation the convolution integral is approximated by a sum. As emphasized in this paper, this approach can also be interpreted as a form of model reduction. This is because an arbitrarily complex linear system between two degrees of freedom can be described with the IRF. In [2] this method was applied to large, mainly linear structures with local nonlinearities. The publications of Gordis et al. confirm the basic applicability of this integral method for numerical simulations. Starting in 2010, Rixen and his staff published several papers dealing with this topic in depth. In [3] the convolution is also approximated with a sum and the subsystems are coupled by constraint equations, explicitly demanding compatibility at the interface points. In the literature, this is often described as “dual assembly”. One problem with the convolution is that as long as the impulse response is not zero, the entire past has to be integrated. However, most systems are characterized by the fact that they are exposed to ongoing excitation. In this case, the first part of the IRF dominates the response. Therefore, Rixen and Haghighat consider only the first part of the impulse response in [4]. A logical consequence is the application to damped

W. Witteveen () · L. Koller · F. Pichler Study Program “Mechanical Engineering”, University of Applied Sciences Upper Austria, Wels, Austria e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2021 A. Linderholt et al. (eds.), Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47630-4_17

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high-frequency oscillators (=modes), since their impulse response quickly approaches zero, see [5, 6]. An investigation on the coupling of components described by the Finite Element Method (FEM) with components characterized by IRFs can be found in [7]. An obvious field of application would be the direct use of measured IRFs in numerical simulation. Note, that the term “measured IRF” is used throughout the paper even the correct term would be “IRF based on measured data”. The main advantage of using IRFs is that no mathematical model in terms of mass, damping and stiffness is needed. This means that no parameters need to be known or identified. Another big advantage is the topic of secrecy. Companies do not have to hand over detailed models of their components but can only provide IRFs. Moreover, this approach allows an “as it is” description of the measured components, at least in a linearized sense. The group of Rixen published some papers on this topic as well, see [8–10]. The latter publications show that the consideration of measured IRFs in a numerical simulation can lead to considerable challenges regarding stability. As soon as an IRF is no longer ideal but noisy, the simulations tend to become unstable after a sufficiently long simulation time. The coupling forces grow beyond all limits. This can happen even with very small noise amplitudes. In the latter mentioned publications different strategies are considered to improve the situation. In most cases the aim is to minimize the noise. Original scientific contribution of the work: In the former publications, the cause of the unstable behavior is suspected to be noise. In this paper the real reason for the instability is shown to be the approximation of the convolution with the trapezoidal rule. As a consequence, high frequency content of a signal are causing the instability, even in the absence of noise. Noise of course, raises the higher frequency content of a signal and is therefore a very relevant cause, which triggers the instability. Content of this contribution: In the next chapter all equations are recalled that are needed for the following theoretical considerations. As references are made to the relevant literature, no derivation is made in detail. In the actual key chapter it is shown that a systematic error is made by the sum which approximates the convolution. To our knowledge, this error is not discussed in the former mentioned literature and its effect has not been systematically investigated. Based on these insights, a remarkably simple method of stabilization is proposed. The theoretical findings are numerically investigated using an two-mass oscillator. In a concluding example, the transfer function of a real bearing arrangement is computed based on different measurements. The measured IRF is coupled with an unbalance rotor in a simulation and the results are compared with measurements.

17.2 Error of the Discrete Convolution 17.2.1 Discrete Fourier Transformation The discrete Fourier Transformation (DFT) transforms a signal from the time domain into the frequency domain. In time domain, a signal is described by the (N × 1) vector t and the (N × 1) vector y t = [t0 t1 · · · tN −1 ]

  y = y0 y1 · · · yN −1

(17.1)

The vector t holds the sampling times and y the signal values at that particular time. Using the sampling time t the i-th entry of the time vector can be given as ti = it. The DFT transforms the data of (17.1) into a (N × 1) frequency vector f and a (N × 1) vector c containing the Fourier coefficients.   f = f0 f1 · · · fN −1

  c = c0 c1 · · · cN −1

(17.2)

The i-th entry of f can be given as fi = if. The quantity f is denoted as frequency resolution and is defined as the reciprocal value of the measurement time T (=Nt). For the considerations in this paper it is important to recall, that the original time signal can be reconstructed at the sampling times tj (0 ≤ j ≤ N − 1) using N

  |ci | cos ωi tj + arg (|ci |) y tj = c0 + i=1

(17.3)

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where the equality ωi = 2π fi is used. If an odd number N is assumed and the symmetry of the vector of Fourier coefficients is taken into account the time signal can be reconstructed as well with N−1

2

  |ci | cos ωi tj + arg (|ci |) y tj = c0 + 2

(17.4)

i=1

If N is a straight number, Eq. (17.4) would have a slightly different form.

17.2.2 Error Due to the Approximation of an Integral with the Trapezoidal Rule Let us consider a function f (x) in the interval [a,b], see Fig. 17.1. For the trapezoidal rule, the integral of f (x) between a and b is approximated by the integral of a straight line with the endpoints f (a) and f (b). In numerous mathematics books it can be verified that an upper bound of the error can be given as |eab | ≤

# Δx 3 ##  f (xab )# 12

(17.5)

where f holds the second derivative of f with respect to x and xab leads to the maximum value of f in the interval [a,b]. The symbol x is an abbreviation for the length of the interval b − a. If it is assumed, that an interval [a,b] is subdivided into M equidistant subintervals, each with length x, an upper bound of the error can be given as |eAB | ≤ M

# Δx 3 ##  f (xAB )# 12

(17.6)

where xAB leads to the maximum value of f in the interval [a,b].

17.2.3 Discrete Convolution: Error Due to Trapezoidal Rule For a linear time invariant system the output y(t) can be computed as a function of the input x(t) by means of the convolution $t y(t) =

$t h (t − τ ) x (τ ) dτ =

0

h (τ ) x (t − τ ) dτ

(17.7)

0

where h(t) is the IRF between the considered degrees of freedom. For the following considerations a time invariant test signal x(t) = 1 is assumed so that the convolution simplifies to $t y(t) =

h (τ ) dτ 0

Fig. 17.1 Trapezoidal rule in the interval [a,b]

(17.8)

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Note, that the numerical examples given below indicate that the conclusions based on a step response are of general nature and hold although another type of force is applied. The impulse response of a damped mechanical system tends towards zero with ongoing time. It is assumed that it falls below a certain limit at time TC so that further numerical integration no longer makes sense. Therefore, Eq. (17.8) can be written as $t  y(t) =

t  = t t < TC t  = TC t ≥ TC

h (τ ) dτ 0

(17.9)

See publication [4, 5] for some more thoughts on truncated convolution where t < t. The DFT transforms the time depended IRF into the frequency depended Frequency Response Function (FRF). The complex coefficients are denoted as Hi (0 < i < N − 1) and can be used to reconstruct the IRF along Eq. (17.4). Inserting this reconstruction into Eq. (17.9) leads to $t  y(t) =

N−1

H0 dτ + 2 0

t

2 $

|Hi | cos (ωi τ + arg (|Hi |)) dτ

(17.10)

i=1 0

When each integral in the convolution (17.10) is approximated by a sum and it is assumed, that the interval length for trapezoidal rule is equal to the sampling rate t an upper error bound can be given by the application of (17.6) to each integrand in (17.10) as N−1

|e| ≤ 2

2

i=1

M

Δt 3 |Hi | ωi 2 12

(17.11)

When the relations t = Mt, ωi = 2π fi , fi = if, f = 1/T and T = Nt are applied to Eq. (17.11) |e| ≤ t  k

(17.12)

with N−1

2 2π 2

|Hi | i 2 k= 3N 2

(17.13)

i=1

is obtained. A closer look to Eqs. (17.12) and (17.13) reveals some observations: First: The error is growing with the convolution time t as long as t < TC . This is well known from numerical observations. Argued differently, this means that a short time convolution with a very small t is always advantageous with respect to stability. This is probably the reason for a stable convolution of stiff differential equations as proposed in [6]. In the latter publication, it is suggested to use the convolution for the numerical time integration of stiff modes. If, for example, such a mode has an eigenfrequency of 1000 Hz and a critical damping of 0.01, the amplitude decreases in 0.07 s to a value less then 1% of the maximum amplitude. If TC is set to this value, t would be limited by this (small) number. Second: A small value of k would be good in terms of stability. The sum in (17.13) contains a fundamental problem in this regard. Each summand, which is associated with a certain frequency, contains the absolute value of the corresponding Fourier coefficient multiplied with the square of the index i. Thus, higher frequencies are weighted more than lower frequencies. Consequently, instability is caused by a high frequency content, leading to a high value of k, causing an unacceptable error e. In other words: If a signals Fourier coefficients associated with higher frequencies are too high, the convolution becomes unstable. Therefore noise is not the primarily cause of the observed instabilities. Even ideal signals without noise lead to an unstable convolution when the frequency content is too high. This is confirmed by the numerical studies below. The presence of noise naturally aggravates the situation as the absolute values of higher frequency Fourier coefficients tend to be raised.

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Third: It is observed in numerical tests, that the convolution becomes more stable when the resolution N is decreased. This can be demonstrated, when an artificial signal with constant Fourier coefficients Hi = H is assumed. Using the approximation n ) (N − 1)/2 ≈ N/2, the relationship j 2 = n(n+1)(2n+1) and the assumption 1/N  1 the factor k can be simplified to 6 j =1

k = H

π2 18



N 3 1 + + 2 2 N



≈ H

π2 (N + 3) 36

(17.14)

Equation (17.14) reveals, that a lower resolution, which is equal to a smaller number of N, leads to a smaller k, which is advantageous with respect to stability.

17.3 Possibilities for Stabilization 17.3.1 Modal Fit An obvious noise reduction strategy is the approximation of the measured data with noise-free analytical functions. The parameters of those functions must be fitted. There is some literature available under the keyword “Modal Parameter Estimation”, for example [11, 12]. A successful application of this strategy is reported in [9]. However, this process is not trivial, requires experience and/or software support. In this way, the noise problem can be avoided, but not the fundamental limitation that the IRF must not exceed a certain frequency content. The “modal fit” strategy is not further pursued in this work.

17.3.2 Filtering in the Frequency Domain The disadvantageous sum in Eq. (17.13) can be positive influenced if the Fourier coefficients are set to zero (or reduced) above a certain cut-off frequency fc . This is a basic idea of filtering. Afterwards an inverse DFT (IDFT) can be applied and a filtered IRF is obtained. This obvious strategy is visualized in Fig. 17.2. In this example the Fourier Coefficients Hi are set to zero for f > 50 Hz. The advantages and disadvantages of this simple method are obvious. The information above the cut-off frequency fc is lost. On the other hand, the error is limited because the summands inside the sum in Eq. (17.13) evaluate to zero for frequencies higher fc . As a conclusion it can be stated, that this method seems promising in case the measured component contains mainly low frequency information. Note that other and more sophisticated filters can be used to remove frequency content from the IRF. However, the later one is simple to implement.

17.3.3 Decreasing High Frequency Content by the Use of Artificial Mass Additional mass on the involved degrees of freedom (dof) reduces the frequency content of the signal. The resonance peaks shift to lower values and the Fourier coefficients in the higher frequency range decrease. Thus, it can be recommended for the measurement to consider as much mass as possible in the component for which an IRF is determined. However, “artificial” mass can also be included in the measured IRF in order to stabilize the subsequent time integration. The idea of this procedure is illustrated in Fig. 17.3. In the upper image a situation is depicted where a measured (driving point) IRF is coupled via a constraint equation to a dof of a simulation model with an associated mass m. Let us assume that the time integration of the overall system is unstable because of the frequency content of the measured IRF. The idea is to reduce the mass m by the value mS and to add mS to the IRF. The modified IRF should have less tendency to destabilize the overall time simulation. If such a strategy is successful no system information is lost. This is a clear advantage in comparison to the former approach. In numerical investigations it is observed, that the mass mS cannot be arbitrarily high. When the difference between m and mS gets negative, the time integration does not deliver good results anymore. Consequently, we restricted ourselves to

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Fig. 17.2 IRFs based on filtered FRFs

cases where m > mS holds. Following Fig. 17.3, the mass mS is in series to the measured IRF. This leads to HS (ω) =

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where H(ω) represents the original (unstable) FRF and HS (ω) holds the modified and hopefully stable one. An inverse Fourier transformation leads to hs (t) which can be used in the time integration. Once again it is emphasized that the mass m of the coupling dof has to be decreased by mS because it is already considered in hs (t).

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Fig. 17.3 IRF with artificial mass

Fig. 17.4 Stability test scenario

17.3.4 Systematic Stabilization Approach The maximum error emax results from Eq. (17.12) by setting t = TC . In the case of a step response, one possible requirement would be that this error remain small against the maximum absolute value of the impulse response itself. This results in an error measure r of r=

emax hmax

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whereby hmax holds the Impulse Responses absolute maximum within the interval 0 < t < TC . From the experience of the numerical investigations, we recommend the following stabilization procedure in two steps. In the first step, the IRF is modified so that r is somehow smaller than 1. In the second step, the stability is checked by testing the impulse response with a single degree of freedom oscillator. Step 1: Mass modification or frequency filtering in order to decrease r It is recommended to put as much mass as possible into the IRF. Either in the real testing or artificial as suggested in former subsection c. When this strategy leads to a value of r somehow smaller than 1, step 2 can be tried. This approach should be tried first since no information of the signal is lost. If stabilization is successful, the results remain exact. If the artificial mass approach is not successful, the frequency filtering from subsection b can be applied. When r is small enough, step 2 can be tried. Step 2: Testing the modified IRF with a Single-Degree-Of-Freedom Oscillator This step is necessary to test the stability of the numerical time integration. We observed no case, where this test has been successful and the final implementation into a more complex system was unstable. It seems that general stability can be assumed when the IRF passes the suggested test. Figure 17.4 holds a schematic representation of the test scenario. The IRF is simply coupled to a mass m2 which is loaded by a unit step force f (t). The simple set of equations which needs to be time integrated is m2 x¨2 − λ2 = f (t) x2 + h ◦ λ2 = 0

(17.17)

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where the compatibility condition x1 = x2 and the equilibrium condition λ1 = −λ 2 are already incorporated. The simulation time needs to be at least TC . In case of a stable test the IRF can be used for further simulations. Otherwise the IRF has to be modified again (→ step 1).

17.4 Examples This chapter is devoted to the numerical verification of the former conclusions. The first example is a two-degree-of-freedom oscillator and the second example is an unbalance rotor where its support is characterized by measured IRFs.

17.4.1 Two-Degree-of-Freedom Oscillator A two-degree-of-freedom oscillator with attached mass according to the left hand side of Fig. 17.5 is considered. In the numerical simulation the mass-damper-spring system m1 , d1 , c1 and m2 , d2 , c2 are considered as IRF (=h) while the mass m3 is a separate body, see the right hand side of Fig. 17.5. The interface forces acting between m2 and m3 are denoted as λ2 and λ3 and the displacements as x1 , x2 and x3 . The equilibrium and compatibility conditions are λ2 = −λ 3 and x2 = x3 , respectively. Trivial boundary conditions are applied and as excitation, a force step with amplitude 1 is considered. The final set of equations for the numerical time integration can be given as m3 x¨3 − λ3 = f (t) x3 + h ◦ λ3 = 0

(17.18)

The first line holds the dynamics of the attached mass while the second line holds the compatibility condition and –x2 is substituted by the convolution – (h ◦ λ3 ). The parameters have been set to m1 = 1 kg, d1 = 1.5 Ns/m, c1 = 1000 N/m, m2 = 0.1 kg, d2 = 0.5 Ns/m, c2 = 5000 N/m and m3 = 1 kg. The IRF h(t) has been determined with N = 100,000 and t = 0.001 s (T = 100 s). The numerical time integration is performed using the HHT integration scheme [13] which has been implemented in Scilab [14]. For all simulations the HHT Parameter has been set to α = −0.3. The time resolution is t = 0.001 s. More details on the time integration with a discretized convolution can be found in the publications of Gordis’ and Rixen’s groups, see [1–3, 5, 9]. The numerical time integration has been done up to 100 s and the convolution has been performed without truncation (TC = 100 s). The results of “stable” simulations have been verified by a reference computation with a common model using no IRF, whereby the displacement x3 matches the reference computation over the entire simulation time. The theoretical considerations of the previous chapter show that in general the frequency content of a signal is the cause of the observed instabilities and not primarily noise. The following simulations without any noise confirm this. Figure 17.6 contains the FRFs on which the IRFs are based. The solid curves represent the unmodified and ideal signals based on analytical formulas. The dashed curves hold the signals when a frequency filter is applied and the dotted curves

Fig. 17.5 Two-degree-of-freedom oscillator with attached mass

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Fig. 17.6 FRFs and IRFs of the two-degree-or freedom oscillator Table 17.1 k and r for different IRFs k r

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gives the quantities when the IRF is modified by an artificial additional mass of mS = 0.9 kg. The values of k (Eq. 17.13) and r (Eq. 17.16) for the different IRFs are given in Table 17.1. The simulation without frequency filtering and additional mass is unstable, even in the absence of noise. Unstable means, that the amplitude for x3 is increasing with simulation time. This is in accordance with Eq. (17.12) because the error grows with the time t . The upper image in Fig. 17.7 contains the results of the stable simulations. The solid line holds the reference computation while the dashed curve gives the result for the simulation with the frequency filtered IRF and the dotted curve represents the result with a mass modified IRF. The lower image contains the time evolution of the error, defined as the relative deviation with respect to the reference solution at each time instant. As expected, the calculation with the frequency-filtered IRF shows deviations. By setting Fourier coefficients above a certain frequency to zero, information is lost. The results of the mass-modified IRF are excellent. As expected, no information is lost by this modification.

17.4.2 Unbalance Rotor Mounted on a Beam The aim of this section is the application of the above considerations to an IRF that is determined based on measurements. It is examined whether the procedure leads to stable time integration and whether the result accuracy is good enough. An unbalance rotor is mounted on a beam clamped on both sides. In simulation, the unbalance rotor is modelled as a concentrated parametric system, whereas the beam is considered as a measured IRF only. In addition to the measurement of the beam for the IRF, the overall system (unbalance rotor + beam) is also measured and the results are compared to the simulation. Figure 17.8 shows the implementation of the complete system. A circular foundation mass is mounted in the middle of a double-

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Fig. 17.7 Result and error of two-degree-of-freedom oscillator

Fig. 17.8 Unbalance rotor mounted on a double-sided clamped beam

sided clamped beam. The vertical direction of this middle point is the degree of freedom of interest for which a driving point IRF is determined. The electrical motor with an unbalance is coupled to this degree of freedom. Therefore, it is called coupling degree of freedom (cdof).

Measurement of cdof Driving Point IRF In order to determine the IRF, the unbalance rotor is removed and the response and excitation of this point are measured. The IRF are determined based on those data. Two different methods were used to determine the transfer function. Direct measurement of the transfer function: It is well known from literature that the velocity of the output dof is equal to the IRF when a unit step is applied to the input. In order to measure the IRF in this way, a negative step was generated. For this purpose, the beam was prestressed at the cdof with a defined load, see Fig. 17.9. The prestress has been realized via a mass whose weight force was applied to the beam with a thin string. The string is cut with a lighter and the speed of the oscillation of cdof was measured with a laser vibrometer. The final IRF is obtained by scaling the measurement with the negative weight force. Alternatively, the impulse response was calculated with the H1 estimator: For this purpose the cdof was excited with an impulse hammer. The introduced force as well as the resulting oscillation is measured. An inductive sensor is used for the vibration measurement. The measured time data are transformed into the frequency domain and the FRF is calculated using

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Fig. 17.9 Measurement setup for inverse step response Amplitude Response in Frequency Domain

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the H1 estimator. The back transformation into the time domain results in the searched IRF. For the sake of comprehensibility no filtering or other treatment has been applied to the measured data. Figure 17.10 contains a plot of the IRFs obtained with both methods and the corresponding FRF up to 500 Hz. It can be seen, that the direct measured IRF is of very good quality while the other one contains a lot of noise. This is not surprising as the first one is not determined based on calculation, but measured directly.

IRF Treatment for the Sake of Stabilization As it can be seen in Fig. 17.10 the IRF based on a direct measurement with a laser vibrometer has a very good quality. However, if it is used directly in the simulation, no stable time integrations can be performed. According to the suggested

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Fig. 17.11 IRFs after the application of a 100 Hz filter to the Fourier coefficient

Fig. 17.12 Mass modified Laser-IRF Table 17.2 k and r of different IRF Laser IRF not modified (unstable) Laser IRF fc = 100 Hz H1 IRF not modified (unstable) H1 IRF fc = 100 Hz Laser IRF mass modified k 8·10−4 4·10−7 81.9 3·10−7 3·10−5 −4 −4 r 0.38 2·10 38,950 1.5·10 0.02

stabilization procedures, the Fourier coefficients for frequencies higher than 100 Hz have been set to zero and transformed back into the time domain. The resulting IRF can be seen in Fig. 17.11. While there is almost no visible difference between the original and filtered IRF based on the direct measurement there is a significant difference between the original H1 IRF (Fig. 17.10, lower left picture) and the filtered one. The second stabilization strategy has been applied to the IRF obtained by the vibrometer. A stabilization mass of mS = 0.46 kg has been moved from the rotor to the beam. The resulting IRF can be seen in Fig. 17.12. It can be assumed from Fig. 17.10 that the IRF decreases to zero with increasing time. In order to save computational time the convolution cut off time TC is set to 3.8 s for the final simulations. The values of k (Eq. 17.13) and r (Eq. 17.16) for the different IRFs are listened in Table 17.2. A closer look to Tables 17.1 and 17.2 indicates that r is more a useful indicator than a clear criterion.

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Fig. 17.13 Voltage supply of the electric motor

Reference Measurement of the Complete System Beam + Unbalance Rotor During the measurement of the overall system beam + unbalance rotor, the deflection of the cdof was measured with an inductive sensor. First, the electric motor is supplied with a constant voltage so that a quasi-stationary state is achieved. A smooth voltage ramp is then applied so that the rotational speed passes through the range of the system’s first natural frequency. Since the sensor has a limitation of ±2 mm, the slope of the ramp is chosen so that these values are not reached. Furthermore, it is assured, that no relevant geometric nonlinearity takes place in the former mentioned range of deflection. The voltage used to drive the motor is shown in Fig. 17.13.

Simulation The model on which the simulation is based can be taken from Fig. 17.14. The unbalance rotor is modelled with a rotating point mass mU . This mass is connected to a massless rod of length rU . The rod is mounted via a revolute joint on a mass m2 . This mass has only one degree of freedom (x2 ) and is connected to the ground with the measured IRF. The dof of the IRF element is x1 and the interface forces are referred to as λ1 and λ2 . The massless rod is driven with an imposed torque M(t) and a rotary damper with the parameter dU prevents the system for an uncontrolled run up. The final system of equations can be taken from Eq. (17.19) where λ1 and λ2 are replaced by λ with the correct sign. 

m2 + mU −mU rU sin ϕ mU rU 2 − mU rU sin ϕ x2 − h ◦ λ = 0



x¨2 ϕ¨



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The symbol h stands for the measured impulse response. By comparing simulation and test bench at several constant speeds, rU , mU and dU are identified. The finally used values are rU = 0.03 m, mU = 0.0035 kg, m2 = 0.496 kg and dU = 0.2 Nms. The torque M(t) is given analogously to the voltage in Fig. 17.13 and it starts at 15.44 Nm and is increased to 18.63 Nm. Those values have again been identified by comparing measurement and simulation results at a constant speed.

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Fig. 17.14 Simulation model

The time integration was performed in Scilab [14] with a self-written HHT integrator corresponding to [13]. More details concerning the numerical time integration and the approximation of the convolution integral via a sum can be found in the publications of Gordis’ and Rixen’s groups, see [1–3, 5, 9]. The former mentioned HHT integrator has been implemented with a constant time step size which is set to t = 1e - 4. The HHT parameter is set α = −0.3 and the maximum allowed local error for the physical dof as well as for the constraint equation is 1e-8.

Comparison of Measurement and Simulation As already mentioned, all simulations are unstable when measured IRF are used without additional treatment. This is also true for the direct measured IRF, even it looks very good, see Fig. 17.10. The simulations with the post-treated IRFs are all stable. These results can be seen in Fig. 17.15. The upper picture shows the course over the entire simulation time. In the lower picture the time section can be seen when the moment is increased from the smaller to the larger value. The figure also contains the measured deflection of the cdof according to the excitation by the real unbalance rotor. Figure 17.15 indicates a very good result accuracy. The amplitudes and frequencies before and after the ramp fit very well. Also in the area of the ramp a very good agreement is given. Concerning the deviations it has to be considered that the mathematical image of the electric motor is very simple. In addition, only one movement in the vertical direction was examined. On the test bench, however, the unbalance rotor also causes excitation in the transverse direction. Although the transverse direction is considerably stiffer, differences could also caused by this. The computed amplitudes match the measured data better for the laser based IRF than for the H1 estimator based ones. This is not surprising, due to the better quality of the direct measured IRF, see Fig. 17.10.

17.5 Summary and Conclusion The description of subcomponents inside a complex overall model simulation with measured or calculated impulse responses has already been discussed in detail in the literature. Unfortunately, unstable behavior has often been observed in practice, especially in connection with measured impulse responses. In this publication it is shown that the cause lies in the systematic error of the discrete convolution. The decisive point is that the signal content must be low enough at higher frequencies.

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Fig. 17.15 Comparison of experiment and simulation (full time range and detail)

Noise is therefore a practically highly relevant case that triggers this instability, but not the primary cause. Two easy-toimplement stabilization strategies were presented to lower the frequency content of a signal. With two concluding examples, the theoretical findings are confirmed and the stabilization proposals successfully implemented. Finally, it is pointed out again that this method delivers a model-free representation of a structure based on experimental data. Therefore, classical parameter identification, like for spring parameters or masses is not necessary. Acknowledgements The Austrian Research Promotion Agency (FFG) under Grant number 866851 supported this work.

References 1. Gordis, J.H.: Integral equation formulation for transient structural synthesis. AIAA J. 33(2), 320–324 (1995) 2. Gordis, J.H., Radwick, J.: Efficient transient simulation for large locally nonlinear structures. Shock. Vib. 6, 1–9 (1999) 3. Rixen, J.D.: Substructuring using impulse response functions for impact analysis. In: Proceedings of the IMAC-XXVIII, Jacksonville, FL, USA, Feb 1–4, 2010, ©Society for Experimental Mechanics Inc 4. Rixen, J.D., Habhighat, N.: Truncating the impulse responses of substructures. In: Proceedings of the IMAC-XXX, Jacksonville, FL, USA, Jan 30 – Feb 2, 2012, ©Society for Experimental Mechanics Inc 5. van der Seijs, M.V., Rixen, J.D.: Efficient impulse based sub structuring using truncated impulse response functions. In: Proceedings of ISMA2012, Leuven, Belgium, Sept 17–19, 2012 6. Geradin, M., Rixen, D.J.: Impulse-based substructuring in a floating frame to simulate high frequency dynamics in flexible multibody dynamics. Multibody Syst. Dyn. 42, 47–77 (2018) 7. van der Valk, P.L.C., Rixen, J.D.: An Impulse Based Substructuring method for coupling impulse response functions and finite element models. Comput. Methods Appl. Mech. Eng. 275, 113–137 (2014) 8. Rixen, J.D.: A substructuring technique based on measured and computed impulse response functions of components. In: Proceedings of ISMA2010, Leuven, Belgium, 2010

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9. van der Horst, T.: Experimental dynamic substructuring using direct time-domain deconvolved impulse response functions. In: Proceedings of ISMA2010, Leuven, Belgium, 2010 10. van der Seijs, M.V., van der Valk, P.L.C., van der Horst, T., Rixen, J.D.: Towards dynamic substructuring using measured impulse response functions. In: Proceedings of the IMAC-XXXII, Orlando, FL, USA, Jan 30 – Feb 2, 2014, ©Society for Experimental Mechanics Inc 11. Ewins, D.J.: Modal Testing: Theory, Practice and Applications, 2nd edn. Research Studies Press LTD, Baldock (2009). ISBN-10: 0863802184 12. Maia, N.M.M. (ed.): Theoretical and Experimental Modal Analysis. Research Studies Press, Baldock, ISBN 9780863802089 (1998) 13. Negrut, D., Rampalli, R., Ottarsson, G., Sajdak, A.: On an implementation of the Hilbert-Hughes-Taylor method in the context of index 3 differential-algebraic equations of multibody dynamics. J. Comput. Nonlinear Dyn. 2, 73–85 (2007). https://doi.org/10.1115/1.2389231 14. www.scilab.org

Chapter 18

Development of an Electrodynamic Actuator for an Automatic Modal Impulse Hammer Johannes Maierhofer and Daniel J. Rixen

Abstract In Experimental Dynamic Substructuring, automatic modal impulse hammers have been shown to be a very useful tool. The Automatic Modal Impulse Hammer AMimpact (described in an earlier publication (Maierhofer et al, Development of a low cost automatic modal hammer for applications in substructuring. In: Conference proceedings of the society for experimental mechanics series. Springer International Publishing, June 2019, pp 77–86. https://doi.org/10.1007/978-3-03012184-6_9)) developed at the Chair of Applied Mechanics (TUM) uses an electromagnetic actuator with the principle of reluctance forces. This has some disadvantages regarding the power density, and therefore makes it difficult to render the automatic hammer compact. Also, the force is very nonlinear with respect to the position of the piston. This results in quite some effort in positioning the impulse hammer at the right distance from the object. This contribution shows the process of developing a new electromagnetic actuator in combination with a permanent magnet. The physical principle is now based on the Lorentz forces. The goal is to find a configuration with minimal packaging and a sufficiently adjustable impulse peak. Using a modern 3D-FEM approach, the multiphysical system is simulated and optimized. A special control strategy is developed to overcome the disadvantage of unknown distance to the structure. Therefore, a hall sensor is used to monitor the actual position of the hammer. The system is built, then experimentally tested on an academic benchmark system. Furthermore, test series were carried out to prove repeatability. Keywords Testing equipment · Automatic modal hammer · AMimpact · Dynamic substructuring · Frequency based substructuring · Experimental substructuring · High quality FRF

18.1 Introduction Modal testing is a well established method that has been around since the 1970s. It became the mainstream testing technique for a broad band of applications due to the high portability of its excitation [3]. For many applications it is sufficient for obtaining the right eigenfrequencies and mode shapes. In product development they are often only evaluated qualitatively. [6] With modern methods of substructuring, the demands on the quality and accuracy of measurements have increased significantly. To produce high quality FRFs, one would normally recommend shaker measurements. This has the major disadvantage of requiring very complex constructions to place the shaker on the right position, if it is even possible. Also there are minor added mass effects due to the stinger of the shaker and due to the impedance head between the stinger and the structure. Seeing this, impact hammer measurements provide fast and quite good measurements. To increase repeatability and to reduce the dispersion of the excitation position as well as the variation of the excitation strength, an automatic impulse hammer AMimpact was developed [7]. This showed very good results in comparison with manual hammering. This paper shows the evolution of the impact hammer, the AMimpact Pro, with a more complex actuator and an advanced control system.

J. Maierhofer () · D. J. Rixen Chair of Applied Mechanics, Faculty of Mechanical Engineering, Technical University of Munich, Garching, Germany e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2021 A. Linderholt et al. (eds.), Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47630-4_18

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18.1.1 What Is an Ideal Impact? To decide on the direction of further development of the hammer, it is important to understand why bad impacts actually matter and what characterizes an impact as good or bad. The ideal peak is the Dirac peak, i.e. infinitely short but also infinitely high. The integral over an ideal Dirac peak is defined as 1. Therefore, the time function of the impact is modeled as a normal (or Gaussian) distribution. The parameter a is actually the standard deviation σ and is used to modulate the width of the peak. The area under this function is exactly 1 [1]. Seen from the perspective of mechanics this means that we are looking at an impact with an impulse of 1 N s. 2 1 − (t−μ) f (t|μ, a) = √ e 2a2 a 2π

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The Fourier transform of the normal distribution is again a normal distribution in the frequency domain [4]. 1

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For practical reasons, we set μ = 0 and only consider the plot for positive frequencies f . With decreasing a (i.e. a tendency towards a slimmer peak), the wider the FFT of the peak becomes. Although the impulse remains constant (at 1), the energy of the impulse (i.e. the integral over time of the force squared) increases with increasing slenderness of the peak in time. This can be seen using the Parseval Theorem and looking at the area under the FFT of the signal. The extreme case shows up in the ideal Dirac impulse, whose energy is infinite. The influence of different tip materials was examined by many other sources, for example [3], so this will not constitute a topic of discussion in this contribution. The following explains how a double pulse is described mathematically. The double impacts in Fig. 18.1 are constructed from two normal distributions with the same parameters as the single impacts are generated with a different mu, i.e. a time shift. Then they are summed up with a weighting factor in such a way that the integral over the whole signal still is presevered 1. The orange curve is weighted with k1 = 0.3, k2 = 0.7 and the green with k1 = k2 = 0.5. The energy of both double impact signals is nearly the same (although it is nearly half of the energy of the single impact). The starting amplitude of the FFT is only dependent on the area of the time signal. Thus all signals start at the same value. The frequency position of the amplitude’s drop is dependent on the distance between the two peaks of the normal distributions. The further apart they are, the earlier the drop. The strength of the drop is dependent on the similarity of the two distributions. The more similar they are, the deeper the drop. As a rule of thumb, the dashed line shows the maximum allowed drop of the amplitude of −20 dB to still achieve a good result for the FRF [2]. Consequently, there are three different possible control strategies for an automatic impulse hammer: Firstly, the absolute amount of force should be kept as constant as possible, and secondly the energy introduced into the system should be

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Fig. 18.2 Different Sampling Rates of one generic impulse result in different PSD-Curves

constant. Third, the impulse is constant. All three do not have to be fulfilled at the same time. Depending on the task, one or the other has priority.

18.1.2 A Comment About Sampling Rate During Impact Testing A frequently asked question is the question of the appropriate sampling rate for the determination of a high quality Impact FRF. Seeing it from the sensors perspective, one could think that the sampling rate should be little more than two times the frequency needed for the later evaluation. This will result in FRFs that are not correct in their magnitude. Coming from the hammer side, the Shanon-theorem has to be fulfilled for the hammer’s input spectrum. To decide what frequency content of the impulse is necessary can be tricky. It depends on the material pairings, i.e. the material and form of the hammer tip and the target structure, as they determine the shape of the impulse time signal. In real measurements it is more useful to look at the Power Spectral Density (PSD) of the signal. It is the Fourier transform of the Signal squared [2]. The calculation is performed using Welch’s method [10]. From these PSDs the FRFs are then constructed. The following numerical study in Fig. 18.2 shows the effect. The trigger moment is called T0 = 0 and all the signals have the same value at that moment. The impulse function is modeled with a normal distribution function that is evaluated at different steps (depending on the given sample rate). For the PSD calculation, Welch’s method is used. The spectral density is plotted here in a linear scale to set the focus on the frequency band with the highest energy. It is easy to see that too low sampling frequency gives wrong PSD values. Related to where the sampling points are, the area under the time signal is different. This results in a different starting point in the PSD. In most of the cases the PSD will be underestimated, but there are cases where it is in fact overestimated as can be seen from the curve with 2.5 kHz. In practice it is necessary to try different sampling rates and compare the PSDs of the force signal to see if the values converge towards a constant value.

18.1.3 Development Potential of the AMimpact The main advantage of the AMimpact, as developed in [7], is its absolute simplicity. Its main goal was to produce reproducable impacts for one situation. This is ideal for linear and nonlinear structures to achieve good results during the averaging process when 5–10 hits are necessary. Nevertheless, there are a few points that offer potential for improvement. The first point is the fact that the impact strength is dependent on the distance between the target structure and the impulse hammer in its rest position. This makes it a cumbersome procedure to position the hammer in different places with regard to the structure, whilst maintaining the same distance. The second aspect is the controllability in special situations.

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18.2 Multibody Simulation In this chapter, a more detailed multibody simulation is set up to get a better understanding of the system behavior. The simulation is completely set up with Python, in combination with NumPy and the SciPy integrator solveivp. The overall simulation is similar to the idea to the one described in [7], but is now capable of multiple dof structures (Fig. 18.3). The simulation consists of two instances of the class mechanical structure, where one is the hammer structure and the other one the target structure. Both can be initialized as MCK-model with pre-built matrices or a few simple finite element bar elements. On a selected node, the hammer structure is accelerated with the force of the magnetic actuator for a specific time period (dashed line in Fig. 18.4). An event is fired when the distance between the tip (again a nodal point of the hammer structure) and the surface of the target structure is smaller or equal to zero. Then the differential equation is switched and a coupling spring damper element is inserted between the two structures. The state vector of the step (where the event was fired) is the initial starting vector for the new equation after the event. It is important to choose a time step that is small enough to have a few samples during the impact event. In this case, a fixed timestep of 1 × 10−5 s was a good choice, and results in converged impulse data. For the impact itself, the most critical parameters are the values for the contact spring-damper element. Here they are calculated with the Hertzian theory. It is also possible to implement a nonlinear force function. These parameters strongly correlate with the material and the geometry of the tip and the target point. For the simulation in Fig. 18.4 the Hertzian parameters of a steel sphere that penetrates an aluminum plate were taken. Whereas Fig. 18.4 is made with very low frequencies for system B and high frequencies for system A, for Fig. 18.5 the frequency of the hammer was lowered. This shows a double impact due to the system dynamics of the hammer. The phenomenon of double impacts can have different origins. It can be from the control strategy, i.e. the force is still acting towards the structure even if there still is contact, or it can be from the ratio of the system dynamics of the hammer and the structure. The next step is to also integrate the controller dynamics into the simulation to increase the value of the simulation.

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Fig. 18.5 Force from the simulation with unfavorable ratio of system eigenfrequencies that result in a kind of double impact

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18.3 Actuator Design The general requirement of the desired actuator for an automated impulse hammer is controllability. It should be possible to move the end-effector forth and back and generate a reproducible force peak by hitting the structure.

18.3.1 Physical Principle In the selected principle of action, the Lorentz force is used on moving charges in a magnetic field. Therefore, a permanent magnet is positioned above two rectangular coils. The current in the coils is such that the adjacent cross-sections have the current flowing in the same direction. The flow of charges is now perpendicular to the magnetic field which results in a force in the normal direction between the magnetic field and charge flow (Fig. 18.6). Apart from scattering effects, the force is linear to the current. The force direction can be easily changed by reversing the current.

18.3.2 Finite Element Simulation of the Actuator In order to find the right dimensions for the purpose, a finite element simulation is carried out. In terms of software, the open source FEM-Software NGsolve [8] with Paraview [9] for postprocessing is used. The equations for magneto-statics will be skipped here, as this has no relevance to the topic of this paper. For this study a completely linear approach is used. As there are no ferromagnetics for the prototype, this is a sound assumption. The force on the permanent magnet is the target of the simulation. The finite difference method using the system energy is a very simple method to calculate the force on a mobile part. The whole system energy (or co-energy) is calculated. Then the mobile part is displaced by a small displacement δx and after recomputing the magnetic field, the system energy is calculated again. The change in energy is then assumed to come from the variation of position using a specific amount of force Eq. (18.3). F =

W2 − W1 δx

(18.3)

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Fig. 18.7 B-field of the solved actuator

Fig. 18.8 B-field of the solved actuator with iron core parts

The value of the displacement δx is a compromise between a principle error (due to the linearization of the force function over the displacement), and a numerical error. In general δx should be as small as possible. One drawback of this method is the fact that the field equations have to be calculated twice. However, using the solution of the first calculation as the starting solution for the second computation can reduce the computation time. As current, we assume a uniform current density in the copper coil of 15 A mm−2 . This allows an almost heatingfree operation of the actuator. For the permanent magnet a neodymium (NdFeB) magnet of grade N52 is chosen. The magnetization direction is in the y-direction (Fig. 18.7). To increase the force it would be possible to bring in ferromagnetic parts to guide the field lines in a more concentrated way around the permanent magnet, Fig. 18.8. This was done in a numerical test and it showed that it could be possible to increase the force by up to one power of ten.

18.3.3 Electrical Circuit One goal of the new actuator is the capability to drive the system with only one Li-Ion Cell (i.e. a voltage range of 2.8–4.2 V). This poses no significant problem for the microcontroller which runs at 3.3 V, but for the actuator, this results in two main requirements: A low inductivity and high current loads in the coil.

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Fig. 18.9 PCB for the DVR8850 to control the electromagnetic actuator with low voltage

Fig. 18.10 CAD rendering

To drive the actuator, a special Low-Voltage H-Bridge IC (DRV8850 from Texas Instruments) is used. This device integrates a NMOS H-Bridge with a current regulation circuitry. To drive the internal N-channel power MOSFETs, a charge pump generates a voltage which is greater than VCC. This results in an FET having a resistance RDS(on) of around 30 m to 35 m [5]. Therefore a electric board Fig. 18.9 was designed to drive the circuit with a microcontroller.

18.3.4 Mechanical Design The mechanical design is designed to allow a relative movement of 10 mm with as little backlash as possible. The coils are wound around a 3D-printed core and then screwed into a U-shaped aluminum part. The slider is made from brass, so as to ensure good gliding between the slider and the housing. The tip is made of steel and can be screwed into the slider. It is important to make the tip exchangeable. On this tip the force sensor is glued with acrylic adhesive. The CAD rendering is shown in Fig. 18.10. A major drawback of the precise slider mechanism is the fact that it is quite sensitive to environmental influences.

196 Fig. 18.11 Position sensing with an analog hall sensor and an auxiliary permanent magnet. (a) Hall sensor with secondary permanent magnet (b) Calibration curve (orange) over the measured points (blue)

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18.4 Control Strategy In the first attempt, a controller was implemented as a way to constantly adjust the energy of the slider for each stroke. Therefore the force is applied over a constant length of travel. This approach does not consider the energy that is lost due to friction effects during the way from position A to position B.

18.4.1 Position Sensing To measure the position of the slider, an analog hall sensor is used. For this purpose, a little cylindrical axial magnetized permanent magnet is attached to the slider and points towards the hall sensor, Fig. 18.11a. Due to the field distribution around the magnet, the output of the hall sensor is in a nonlinear relation to the position of the slide. The signal is then amplified with an operational amplifier and digitalized with the microcontroller AD unit. Using an exponential function, it was possible to calibrate the voltage of the hall sensor to the position. In this case, the focus is on the positions between 5 and 10 mm.

18.4.2 Control Sequence The control strategy for the AMimpact Pro is divided into different parts and displayed in Fig. 18.12. After positioning the device in front of the structure a so called initial homing procedure is started. There, the actuator is driven to touch the structure. Using the hall sensor this value is saved. After that, the shaft is moved back for a certain (user-defined) distance using a PI-Controller. When the position is reached within a small tolerance, the controller is shut off and the system is at rest. Then the user is able to trigger an impact. For that the coil is under current until short before the sensor hits the structure. The current is turned off and the shaft impacts the structure ballistically. A short time after the impact has occurred, the PI-controller is turned on and it brings the shaft back to the rest position. A new impact can then be triggered.

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Fig. 18.12 Sequence chain of the controller

18.5 Verification Measurements 18.5.1 High-Speed Camera To measure the motion of the hammer during an impact, we decided to use a high-speed camera. The advantage is that it is a non-intrusive measurement, i.e. the system will not be perturbed. The sample rate was 40,000 Hz with a resolution of 512 pixels × 512 pixels Fig. 18.13. The images are monochrome with 12 bit resolution. The hammer was configured with the PCB 086E80 force sensor (standard steel tip) and full power, which results in a peak force of around 150 N. The target structure was a massive steel block with an attached impedance-sensor with a brass plate. Using the Kanade-Lucas-Tomasi (KLT) feature tracker algorithm, the position of the force sensor tip is measured and plotted in Fig. 18.14. The first image is taken as reference image. The respective distances between the four feature markers and the reference positions are then measured. To achieve a smooth result a gliding average over 10 samples is taken. The velocity is generated through simple finite differences, which causes some noise. In the velocity plot of Fig. 18.14 one can observe the control strategy during one impact. The increasing velocity before the impact comes from the active acceleration by the electromagnet. At around 40 ms the current is switched off and the impact occurs. After that, the full force in the opposite direction acts on the slide, increasing the negative velocity. The kink at around 65 ms is the moment when the PI controller starts to slow down the motion to find the rest position.

18.5.2 Force The forces that were achieved with this actuator and the prescribed control scheme where in the range of 0–150 N. This fits quite well with the simulation. The problem was that the repeatability was not as high as expected. This could have several reasons. Certainly, the most important one is that the microcontroller used was only capable of a cycle frequency of 1 ms. As the maximum expected speed of the slider is approximately 0.5 m s−1 this means that the shut-off point can vary over a range of 0–0.5 mm. That means that the level of energy could not be equal for every impact. Also the friction of the slider varied with the temperature of the device.

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Fig. 18.13 Pictures from the high-speed camera. (a) 12.5 ms (b) 25 ms (c) 37.5 ms (d) 42.625 ms

Fig. 18.14 Position tracking of the force sensor tip

18.6 Conclusion To summarize the current work, it has been shown that knowledge around the theory of impacts is essential to build automated impulse hammers that fit the requirements of today’s impulse-based testing. The need for an advanced electromechanical actuator that is highly controllable and has a fast system dynamics is obvious. But this work also shows that it is not easy to implement such a complete system with reliable results. This applies especially to the field of microcontroller software where very fast scans of position measurement and then force measurement are necessary. The next steps will focus on packaging the whole system in a complete device. Increasing the force density of the actuator is also a further development topic as there might be heavier structures that need a more powerful impact. Another simplification for the user is the testing of an electronic spirit level, which permits the exact orientation of the impact direction. This will be done using a MEMS-based Inertial Measurement Unit (IMU), together with a processing of the orientation using quaternions.

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One open issue is the task of generating impacts with the same peak force. Therefore, the hammer would need some force-feedback and a learning algorithm to find the right parameters. After the learning process, the real measurements can be taken. Technically this is not easy, due to the requirements for a high sampling rate and the accurate force measurement. Acknowledgments The lead author would like to thank the students Karl Benkler, Christoph Dietz and Jonas Koch for their helpful work in the lab and their constructive comments.

References 1. Arens, T., Hettlich, F., Karpfinger, C., Kockelkorn, U., Lichtenegger, K., Stachel, H.: Mathematik. Springer, Berlin/Heidelberg (2018). https:// doi.org/10.1007/978-3-662-56741-8 2. Brandt, A.: Noise and Vibration Analysis: Signal Analysis and Experimental Procedures. Wiley (2011). https://ebookcentral-proquest-com. eaccess.ub.tum.de/lib/munchentech/detail.action?docID=792461 3. Brown, D.L., Allemang, R.J., Phillips, A.W.: Forty years of use and abuse of impact testing: a practical guide to making good FRF measurements. In: Experimental Techniques, Rotating Machinery, and Acoustics, vol. 8, pp. 221–241. Springer International Publishing (2015). https://doi.org/10.1007/978-3-319-15236-321 4. Bryc, W.: The Normal Distribution. Springer, New York (1995). https://doi.org/10.1007/978-1-4612-2560-7 5. DRV8850 Low-Voltage H-Bridge IC With LDO Voltage Regulator. Texas Instruments (2016) 6. Ewins, D.J.: Modal testing: theory and practice. Research Studies Press Wiley, Letchworth, Hertfordshire, England, New York (1984). ISBN:9780863800177 7. Maierhofer, J., Mahmoudi, A.E., Rixen, D.J.: Development of a low cost automatic modal hammer for applications in substructuring. In: Conference Proceedings of the Society for Experimental Mechanics Series, pp. 77–86. Springer International Publishing (2019). https://doi. org/10.1007/978-3-030-12184-6_9 8. NGSolve.: version: 6.2.1908. (2019) 9. ParaView.: version: 5.6.0. (2019) 10. Welch, P.: The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans. Audio Electroacoust. 15(2), 70–73 (1967). ISSN: 0018-9278. https://doi.org/10.1109/TAU.1967.1161901. http:// ieeexplore.ieee.org/document/1161901/.