Dynamic stability and bifurcation in nonconservative mechanics 9783319937212, 9783319937229

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CISM International Centre for Mechanical Sciences 586 Courses and Lectures

Davide Bigoni · Oleg Kirillov Editors

Dynamic Stability and Bifurcation in Nonconservative Mechanics International Centre for Mechanical Sciences

CISM International Centre for Mechanical Sciences Courses and Lectures Volume 586

Series editors Executive Editor Paolo Serafini, Udine, Italy The Rectors Elisabeth Guazzelli, Marseille, France Franz G. Rammerstorfer, Vienna, Austria Wolfgang A. Wall, Munich, Germany The Secretary General Bernhard Schrefler, Padua, Italy

For more than 40 years the book series edited by CISM, “International Centre for Mechanical Sciences: Courses and Lectures”, has presented groundbreaking developments in mechanics and computational engineering methods. It covers such fields as solid and fluid mechanics, mechanics of materials, micro- and nanomechanics, biomechanics, and mechatronics. The papers are written by international authorities in the field. The books are at graduate level but may include some introductory material.

More information about this series at http://www.springer.com/series/76

Davide Bigoni Oleg Kirillov •

Editors

Dynamic Stability and Bifurcation in Nonconservative Mechanics

123

Editors Davide Bigoni Department of Civil, Environmental and Mechanical Engineering University of Trento Trento Italy

Oleg Kirillov Department of Mathematics, Physics and Electrical Engineering Northumbria University Newcastle upon Tyne UK

ISSN 0254-1971 ISSN 2309-3706 (electronic) CISM International Centre for Mechanical Sciences ISBN 978-3-319-93721-2 ISBN 978-3-319-93722-9 (eBook) https://doi.org/10.1007/978-3-319-93722-9 Library of Congress Control Number: 2018944340 © CISM International Centre for Mechanical Sciences 2019 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, express 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. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Nonconservative mechanical systems have been known about since the end of the nineteenth century when Greenhill posed a problem on the buckling of a screw shaft of a steamer subject to both an end thrust and an axial torque. In the 1920s, Nicolai introduced a follower torque into the Greenhill problem and demonstrated a dynamic instability (flutter) of the shaft. In the 1930s, Theodorsen presented a model of flutter of flexible structures in a flow and derived circulatory lift and drag forces. In the 1950s, Ziegler proposed a classification of conservative and nonconservative loads, distinguishing nonpotential positional forces that produce nonzero work on a closed contour, and, inspired by aerodynamics, called them circulatory. Nearly at the same time as Greenhill published his work, Kelvin and Tait, studying models of formation of binary stars, discovered the destruction by viscosity of gyroscopic stabilization of rotating ellipsoidal masses of fluid. This was the first example of dissipation-induced instabilities. Nowadays dissipative and circulatory forces are recognized as the two fundamental nonconservative forces in a growing number of scientific and engineering disciplines including physics, fluid and solid mechanics, fluid–structure interactions, and modern multidisciplinary research areas such as biomechanics, micro- and nanomechanics, optomechanics (for instance, optical tweezers generate a circulatory force field), robotics, energy harvesting, and material science. Nonconservative systems display unusual and counter-intuitive dynamics and stability properties. The occurrence of flutter and divergence instabilities is usually analyzed to be avoided in mechanical structures, although sometimes these become desirable, for instance, to harvest energy. However, the determination of these instabilities is a challenging mechanical problem. This is due to the nonself-adjoint (non-Hermitian) character of the governing equations that, as a rule, depend on multiple parameters. Traditional university curricula do not offer a coherent collection of modern mathematical tools for the analysis of multiparameter families of nonself-adjoint differential equations combined with a firsthand demonstration of how they actually work in practical applications.

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This monograph is the collection of the Lecture Notes for the CISM-AIMETA Advanced School Dynamic Stability and Bifurcation in Nonconservative Mechanics held at the International Centre for Mechanical Sciences (CISM) in Udine, Italy, April 10–14, 2017. The course was given by six lecturers (D. Bigoni from the University of Trento, O. Kirillov from the University of Northumbria, O. Doaré from ENSTA Paris Tech, E. Hemingway from the University of California at Berkeley, A. Metrikine from Delft University, and A. Ruina from Cornell University) and attended by participants from European and extra European countries. The chapters are devoted to flutter and divergence instability in structures and solids (D. Bigoni), to dissipation-induced instabilities in fluid–structure interactions (O. Doaré), to perturbation theory of the Ziegler destabilization paradox and general stability theorems for nonconservative systems (O. Kirillov) and to new results on conservative and nonconservative moments in the dynamics of rods and rigid bodies (E. Hemingway and O. O’Reilly). We wish to thank the Rectors of the CISM Profs. Elisabeth Guazzelli, Franz G. Rammerstorfer, and Wolfgang A. Wall, the Secretary-General Prof. Bernhard A. Schrefler, and all the staff for the warm hospitality and kind assistance during the course. Finally, financial support from the FP7-PEOPLEIDEAS-ERC-2013-ADG-340561-INSTABILITIES is gratefully acknowledged. Trento, Italy Newcastle upon Tyne, UK

Davide Bigoni Oleg Kirillov

Contents

Flutter from Friction in Solids and Structures . . . . . . . . . . . . . . . . . . . . Davide Bigoni Dissipation Induced Instabilities of Structures Coupled to a Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Olivier Doaré

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Some Surprising Conservative and Nonconservative Moments in the Dynamics of Rods and Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . 103 Evan G. Hemingway and Oliver M. O’Reilly Classical Results and Modern Approaches to Nonconservative Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Oleg N. Kirillov

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Flutter from Friction in Solids and Structures Davide Bigoni

Abstract The theory of flutter instability in structures and solids is presented, starting from the illuminating case of the Ziegler double pendulum, continuing with the Beck and Pflüger columns, and ending with the conditions for flutter in solids, considering in particular nonassociative elastoplastic models for granular and rocklike materials. The role of dissipation, leading to the so-called ‘Ziegler paradox’ is presented in detail. It is explained how to obtain a tangential follower load in a structure by exploiting Coulomb friction and it is shown that structures working in a flutter condition can reach a limit cycle, in which they behave as self-oscillating devices.

1 Introduction Friction during sliding contact between solids has been usually advocated as a source of self-excited vibrations and dynamical instabilities (Den Hartog 1956; Ibrahim 1994a, b); examples are the violin string being excited by a bow, the brake ‘squeal’ (in other words, high frequency noise), the ‘chatter’ (low-frequency noise) produced by the cutting tool of a machine, the ‘song’ of a fingertip moved upon the rim of a glass of water, and the unstable fault slip in the Earth’s crust, which generates an earthquake. These mechanical instabilities are often undesirable and lead to excessive damage or wear of the pieces involved in sliding, so that the principal motivation for their study is to ensure their better elimination. However, a recent approach to the mechanics of structures is their exploitation as compliant mechanisms for soft robot arms or energy harvesting devices, even in the range of large displacements and beyond critical loads. In this line of research, flutter instability could be profitably used, see for instance (Doaré and Michelin 2011).

D. Bigoni (B) Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano 77, 38123 Trento, Italy e-mail: [email protected] © CISM International Centre for Mechanical Sciences 2019 D. Bigoni and O. Kirillov (eds.), Dynamic Stability and Bifurcation in Nonconservative Mechanics, CISM International Centre for Mechanical Sciences 586, https://doi.org/10.1007/978-3-319-93722-9_1

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D. Bigoni

The most common and older explanation of these friction-related instabilities is through the introduction of stick and slip behaviour (an alternate switch between static and kinetic friction), modelled as a drop of the friction coefficient with the relative velocity of the two solids in contact. This approach was initiated by Den Hartog (1931, 1956) and generalized in different ways, see among others, Rice and Ruina (1983). Without invoking variable friction, but rather assuming constant, timeindependent, Coulomb friction, an unstable behavior has been theoretically proven (but only with reference to the linearized equations governing the problem, see Adams 1995; Martins et al. 1999; Simões and Martins 1998; Nguyen 2003), so that instability appears to be intrinsically related to even the simplest model of frictional sliding. In parallel to these works, elastoplastic continua characterized by flow rule nonassociativity, the counterpart of Coulomb friction for solids, have been demonstrated to display blowing-up unstable waves, as related to the frictional behaviour of the material (Rice 1977; Loret 1992; Bigoni and Willis 1994; Bigoni 1995; Bigoni and Loret 1999; Loret et al. 2000; Piccolroaz et al. 2006). In all the above-mentioned works, in which instability is proven for constant friction, the concept of flutter is introduced, a nomenclature borrowed from the instability occurring in elastic structures subject to follower forces. The best known of these structures are the so-called ‘Ziegler double pendulum’, a 2 d.o.f. system with concentrated elasticity, and the ‘Beck’s column’, an elastic cantilever rod subject to a tangential force at its free end. The analogy between flutter in a continuum and in a structure is more strict than it may appear at a first glance, in fact, in both cases: (i) flutter initiates when a complex conjugate eigenvalue solution for vibrations emerges, which corresponds to an oscillation of increasing amplitude; (ii) a necessary condition for flutter is the lack of symmetry (or self-adjointness) in the mechanical system; (iii) in a space of parameters, flutter occurs usually within a region, which separates stability from divergence, the latter being an exponentially blowing-up motion; (iv) flutter ultimately leads to a self-sustaining oscillation, which absorbs energy from a steady source. In solids1 flutter instability has been only theoretically predicted, but never experimentally detected. This was also true for the structural flutter, where the practical realization of the tangential force necessary for the instability was considered the major unsolved problem,2 until Bigoni and Noselli (2011) (see also Bigoni et al. 2014, 2018) experimentally and theoretically showed how to generate 1 Kröger

et al. (2008) and Neubauer et al. (2005) report examples of self-excited vibrations related to fluctuating orthogonal forces at the contact between two elements or to geometrical effects and non-conservative restoring forces. In the former case, the system is already oscillating (while the focus of this book chapter is on systems subject to a steady source of energy) and in the latter case the instability is observed in a system with finite degrees of freedom. 2 Koiter (1996) proposed the ‘elimination of the abstraction of follower forces as external loads from the physical and engineering literature on elastic stability’ and concluded with the warning: ‘beware of unrealistic follower forces’. In an attempt of realizing these forces, experiments with water or air flowing from a nozzle were conducted by Herrmann et al. (1966) and Paidoussis (2014), while a solid motor rocket was fixed at the end of an elastic column by Sugiyama et al. (1995, 2000). In the former case, there are hydrodynamical effects affecting the motion, while in the latter case the

Flutter from Friction in Solids and Structures

3

a tangentially follower force in the Ziegler double pendulum through an element in sliding contact against a plate under constant Coulomb friction. The aim of the present chapter is to review the problem of flutter instability in both structures and solids, from the specific perspective pursued by the author in the last twenty-five years. In particular, it is believed that solids and structures are akin from the point of view of flutter instability, so that they display several common features and that this instability is possible when an element draws energy from frictional sliding against another element. The chapter is organized as follows. • The illuminating case of the Ziegler double pendulum (Sect. 2.1) will show: (i) how lack of symmetry of the geometrical stiffness matrix is responsible for flutter (Sect. 2.1); (ii) why flutter cannot be detected with a quasi-static bifurcation analysis (Sect. 2.3); (iii) how viscosity decreases the flutter load and (iv) what is the so-called ‘Ziegler paradox’, for which a vanishing small viscosity yields a strong decrease in the flutter load (Sect. 2.4). • The equations governing the dynamics of the Beck’s column will be obtained from a general setting, namely, from the planar Euler elastica, in which restrictions on the magnitude of the deflection are not included (Sect. 2.5). The Beck’s column, analyzed in a generalization given by Pflüger and including also internal and external dissipation (Sect. 2.7), will reveal that a continuous system displays essentially the same mechanical behaviour found for the Ziegler double pendulum, namely, that lack of symmetry (called now ‘self-adjointness’, Sect. 2.8) is a necessary condition for flutter and that the load producing the instability is significantly lowered by the presence of viscosity. • Numerical analyses will show that the dynamic motion of both the Ziegler double pendulum and the Beck’s column reach a limit cycle (Sect. 2.9). Therefore, these structures are examples of self-oscillating mechanisms (Jenkins 2013), in which a source of steady energy produces an oscillatory motion of given frequency. This frequency can be varied by changing the geometry and stiffness of the system. • The possibility is shown to generate a follower force from sliding Coulomb friction (Sect. 2.10), a concept introduced by Bigoni and Noselli (2011), which is based on an extreme form of orthotropic friction, null in the direction orthogonal to the sliding. It is also anticipated that a device similar to that employed for the Ziegler double pendulum can be designed to produce flutter instability in the Beck and Pflüger rods (Bigoni et al. 2018). • The behaviour of a continuous elastoplastic material, in which the yielding is pressure-sensitive and the plastic flow nonassociative, is introduced (Sect. 3) from a simple 2 d.o.f. model of contact with Coulomb friction (Sect. 3.1). • With reference to nonassociative and pressure-sensitive elastoplastic solids, the problem of plane waves is solved with reference to the loading branch of the

rocket has a non-negligible variable mass and burns so fast that a long-term analysis of the motion is prevented. Therefore, in both cases the follower tangential force invented by Ziegler (1952) is not properly realized, see the exhaustive discussion by Elishakoff (2005).

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D. Bigoni

constitutive operator3 and is shown to be governed by the eigenvalues of the acoustic tensor (Sect. 3.3). Strain localization into planar bands is explained in terms of vanishing of an eigenvalue of the acoustic tensor (Sect. 3.4). The conditions for flutter instability in the above-mentioned solids are finally analyzed (Sect. 3.5) and it is explained that this instability has to be considered more frequent than it is usually believed. • As conclusions, a discussion on the state-of-the-art and the perspectives in the research on flutter instability in solids and structures is reported (Sect. 4).

2 Flutter for Structural Systems Flutter in structural systems is analyzed under the hypothesis that follower forces are present. In general, these forces do not admit a potential and therefore are nonconservative, so that work can be extracted in a closed path of deformation. This can easily be understood with reference to the two structures shown in Fig. 1. One of these structures is made up of two rigid bars jointed together with a hinge and constrained with another hinge at one end, while at the other end a load is applied, which remains coaxial with the bar to which it is applied. The other structure is a clamped elastic rod subject to a force, which remains tangential to the elastica at the free end. The load, as illustrated in the figure caption, is capable of producing a positive work in a closed deformation loop.

2.1 The Ziegler Double Pendulum Flutter and divergence instability can be vividly and simply illustrated with reference to a structure, invented more than sixty years ago by Ziegler (1952, 1977). The structure is the double pendulum shown in Fig. 2, made up of two rigid bars, connected through an elastic hinge and fixed with another elastic hinge at one end. The structure is subject to a tangential follower load on the free end, which remains coaxial to the second rigid rod and can produce positive work in a closed loop (as explained in Fig. 1). Subject to the follower load, the Ziegler double pendulum (Fig. 2) is considered, where two rotational springs of stiffnesses k1 and k2 provide the elasticity and three concentrated masses the inertia. The generic configuration of the system remains determined by the two Lagrangean parameters α1 and α2 . The concentrated masses m 1 and m 3 are located at the points D and E, at a distance d from A and h from B, while the concentrated mass m 2 is located at C. The tangential follower load P, applied at C and taken positive when compressive, maintains the direction parallel 3 The

treatment is limited to the linearized version of contact with friction and to the linearized incremental plasticity, so that the complex role of nonlinarity is not considered for simplicity. The interested reader is addressed to Bigoni and Petryk (2002) for further details on this delicate issue.

Flutter from Friction in Solids and Structures

5

P

2

P

3

1

P Fig. 1 A tangential follower force is applied: (left) at the right end of a two-degree-of-freedom system composed by two hinged rigid bars; (right) at the free end of a clamped elastic rod. The load is not conservative, since in the closed loops 1 −→ 2 −→ 3 (for the structure on the left) and 1 −→ 2 −→ 3,−→ 4,−→ 1, (for the structure on the right) a positive work is produced by the applied force. The deformation loop sketched in the deformable rod shown on the right is based on a small displacement assumption and consists in a rotation of the loaded end (1 −→ 2, during which the load does not work), which is followed by a translation of the end (2 −→ 3, during which a positive work is produced), by another rotation at fixed position (3 −→ 4, during which the load does not work), and, finally, by an horizontal translation back to the initial position (4 −→ 1, during which work is not produced)

to the rod BC. In addition to this force, a second load, produced by a conservative dead force H, is also applied at C. The analysis of a mechanical system similar to that under consideration (in which the mass m 3 is not present) can be found in Herrmann (1971), Ziegler (1952), Ziegler (1953), Ziegler (1956), Ziegler (1977), Nguyen (1995). Governing equations for the double pendulum A simple static analysis of the structure shown in Fig. 2 is sufficient to conclude that only the trivial (straight) configuration satisfies equilibrium when only the follower load P is applied (in fact equilibrium of the rod BC is only possible if α1 = α2 , and equilibrium of the complex ABC additionally requires α1 = 0), so that quasi-static bifurcations are excluded. Therefore, flutter and divergence instabilities, which will be found to occur in the Ziegler double pendulum, necessary represent dynamical instabilities, the former will be shown to consist in an oscillatory vibration of increasing amplitude, while the latter in an exponentially growing motion. Note that the situation changes when only the dead load H is applied, in which case non-trivial equilibrium configurations and therefore quasi-static bifurcations are possible. The equations of motion for the system can be obtained starting from the position vectors of the three concentrated masses m 1 , m 2 and m 3 D − A = d cos α1 e1 + d sin α2 e2 , E − A = (l1 cos α1 + h cos α2 ) e1 + (l1 sin α1 + h sin α2 ) e2 , C − A = (l1 cos α1 + l2 cos α2 ) e1 + (l1 sin α1 + l2 sin α2 ) e2 ,

(1)

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Fig. 2 The Ziegler double pendulum, namely, a two-degree-of-freedom system subject to a follower (non-conservative) load (the force P, applied at C, remains always parallel to the rod BC). In addition, a dead (conservative) load (the force H, applied at C, remains always parallel to e1 ) is also included for reference. The rods are rigid and massless and connected with two springs of stiffness k1 and k2 ; three concentrated masses m 1 , m 2 , and m 3 are present. Two cases will be analyzed, H = 0 corresponding to purely follower load and P = 0 corresponding to purely conservative load. In the former case, quasi-static bifurcation is excluded, but flutter and divergence instability will be shown to be always possible for sufficiently high P. In the latter case, quasi-static bifurcation occurs and only divergence instability is possible

where e1 and e2 are the two unit vectors singling out the horizontal and vertical directions respectively, so that the forces P and H, of moduli P and H , can be expressed as (2) P = −P cos α2 e1 − P sin α2 e2 , H = −H e1 . The velocities of points D, C, and E (the time derivative is denoted by a superimposed dot) are ˙ = −d (α˙ 1 sin α1 e1 − α˙ 1 cos α1 e2 ) , D ˙ C = (−l1 α˙ 1 sin α1 − l2 α˙ 2 sin α2 ) e1 (3) + (l1 α˙ 1 cos α1 + l2 α˙ 2 cos α2 ) e2 , E˙ = (−l1 α˙ 1 sin α1 − h α˙ 2 sin α2 ) e1 + (l1 α˙ 1 cos α1 + h α˙ 2 cos α2 ) e2 , ¨ C ¨ and E ¨ of the masses m 1 , m 2 and m 3 can be evaluated so that the accelerations D, as

Flutter from Friction in Solids and Structures

    ¨ = −d α˙ 12 cos α1 + α¨ 1 sin α1 e1 − d α˙ 12 sin α1 − α¨ 1 cos α1 e2 , D   ¨ = −l1 α˙ 12 cos α1 − l1 α¨ 1 sin α1 − l2 α˙ 22 cos α2 − l2 α¨ 2 sin α2 e1 C   + −l1 α˙ 12 sin α1 + l1 α¨ 1 cos α1 − l2 α˙ 22 sin α2 + l2 α¨ 2 cos α2 e2 ,   ¨ = −l1 α˙ 12 cos α1 − l1 α¨ 1 sin α1 − h α˙ 22 cos α2 − h α¨ 2 sin α2 e1 E   + −l1 α˙ 12 sin α1 + l1 α¨ 1 cos α1 − h α˙ 22 sin α2 + h α¨ 2 cos α2 e2 .

7

(4)

Noting that the moments transmitted by the rotational springs to the rods are k1 α1 and k2 (α2 − α1 ), the principle of virtual power writes as ¨ · δD − m 2 C ¨ · δC − m 3 E ¨ · δE (P + H) · δC − m 1 D − k1 α1 δα1 − k2 (α2 − α1 )(δα2 − δα1 ) = 0,

(5)

where ‘ · ’ denotes the scalar product and the virtual velocities δC, δD, δE have the same expressions provided by Eq. (3) with the ‘˙’ replaced by ‘δ’. The virtual power of the external loads is (P + H) · δC = Pl1 sin(α1 − α2 )δα1 + H (l1 δα1 sin α1 + l2 δα2 sin α2 ) .

(6)

It is noted that, while the conservative load H admits the potential W (α1 , α2 ) = H (l1 cos α1 + l2 cos α2 − l1 − l2 ) , so that H · δC = −

∂W ∂W δα1 − δα2 , ∂α1 ∂α2

(7)

(8)

the nonconservative force P does not admit one, as demonstrated by the condition ∂0 ∂ sin(α1 − α2 ) = . ∂α2 ∂α1

(9)

Imposing now the virtual power Eq. (5) and invoking the arbitrariness of δα1 and δα2 yields the two equations   m 1 d 2 + (m 2 + m 3 )l12 α¨ 1 + (m 2 l2 + m 3 h)l1 α¨ 2 cos (α1 − α2 ) +(m 2 l2 + m 3 h)l1 α˙ 22 sin (α1 − α2 ) + k1 α1 + k2 (α1 − α2 ) − Pl1 sin (α1 − α2 ) − Hl1 sin α1 = 0, (m 2 l2 + m 3 h)l1 α¨ 1 cos (α1 − α2 ) + (m 2 l22 + m 3 h 2 )α¨ 2 −(m 2 l2 + m 3 h)l1 α˙ 12 sin (α1 − α2 ) − k2 (α1 − α2 ) − Hl2 sin α2 = 0,

(10)

governing the (nonlinear) dynamics of the system. The differential equation (10), linearized near the trivial (equilibrium) configuration α1 = α2 = 0, can be written, in matrix form, as

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  α¨ 1 m 1 d 2 + (m 2 + m 3 )l12 (m 2 l2 + m 3 h)l1 α¨ 2 (m 2 l2 + m 3 h)l1 m 2 l22 + m 3 h 2

  mass matri x ⎛ sti f f ness matri ⎞ x geometric matri x

 

      ⎜ ⎟ −l1 (P − H ) Pl1 ⎟ α1 ⎜ k + k2 −k2 +⎜ 1 + = 0. ⎟ 0 −Hl2 ⎠ α2 ⎝ −k2 k2

(11)

Note that the mass matrix and the stiffness matrix are both real, symmetric, and positive definite, while the geometric matrix is real, but unsymmetric, thus introducing the only source of unsymmetry. More specifically, the unsymmetry comes only from the presence of the follower load P, but not from the dead load H . Time-harmonic vibrations of the double pendulum Looking for time-harmonic vibrations near the equilibrium configuration, the Lagrangean parameters α j are assumed to be harmonic functions of time α j = a j e−i t ,

j = 1, 2,

(12)

where a j are (complex) amplitudes, √  is the (possibly complex) circular frequency, and i is the imaginary unit (i = −1), so that a substitution of Eq. (12) into (11) yields   k1 + k2 − l1 (P + H ) −k2 + Pl1 −k2 k2 − Hl2 (13)     2 a1 m 1 d + (m 2 + m 3 )l12 (m 2 l2 + m 3 h)l1 2 − = 0. a2 (m 2 l2 + m 3 h)l1 m 2 l22 + m 3 h 2 The algebraic system (13) represents a generalized eigenvalue problem for 2 , that after introducing the mass M, the stiffness K, and geometric G matrices, can be written as   (14) K + G − 2 M a = 0, which would be a standard eigenvalue problem if M would be equal to the identity. However, the mass matrix M is real, symmetric and positive definite, so that its square root M1/2 (defined in such a way that M1/2 M1/2 = M) is invertible. Therefore, the generalized eigenvalue problem (14) can be rewritten as   (K + G) M−1/2 − 2 M1/2 M1/2 a = 0.

(15)

Therefore a multiplication by M−1/2 transforms the nonstandard eigenvalue (14) into a standard one   −1/2 (16) M (K + G) M−1/2 − 2 I M1/2 a = 0,

Flutter from Friction in Solids and Structures

9

where the matrix M−1/2 (K + G) M−1/2 is not symmetric because of the unsymmetry of G, which in turn follows from the presence of the follower load P. It can be concluded from Eq. (16) that: a necessary condition for the eigenvalues 2 to be complex conjugate (a situation which will be identified with the flutter instability) is that the geometric matrix G be unsymmetric, otherwise, if G is symmetric, the eigenvalues will always be real.

A generalization to n d.o.f. The generalized eigenvalue problem (14) applies to all mechanical systems (with smooth and bilateral constraints) with finite, say n, degrees of freedom. The mass matrix M becomes n × n, but remains symmetric and positive definite, so that the generalized eigenvalue problem can be cast in the standard form (16), where now all matrices are n × n. The stiffness matrix is also symmetric and positive definite, therefore the only possible source of unsymmetry remains G, so that it may be concluded that in all cases where G is symmetric, complex conjugate eigenvalues (and therefore flutter instability) are excluded.

2.2 The Standard Case as a Reference: The Dead Load H on the Double Pendulum Assuming that the double pendulum is loaded only with the dead loading H, so that P = 0, and introducing now the dimensionless variables, m1 m3 d h , ρ2 = , y1 = , y2 = , m2 m2 l1 l2 2 2 l1 l  k1 m Hl1 2 2 λ = , k = , ω2 = , η= , l2 k2 k2 k2

ρ1 =

(17)

matrix G becomes symmetric (so that complex eigenvalues are excluded) and the generalized eigenvalue problem (13) can be compacted so that the matrix multiplying vector [a1 , a2 ] is now 

 (ρ1 y12 + ρ2 + 1)λ2 ω 2 − 1 − k + η λ ω 2 (1 + ρ2 y2 ) + 1 , ω 2 (1 + ρ2 y22 ) − 1 + η/λ λ ω 2 (1 + ρ2 y2 ) + 1

(18)

and nontrivial solutions become possible when its determinant vanishes, a condition providing √ β1 − ηδ2 ± (η) , (19) ω2 = 2a where the discriminant is (η) = (β1 − ηδ2 )2 − 4ac(η),

(20)

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and   a = λ2 y12 ρ1 + (1 − y2 )2 ρ2 + y12 y22 ρ1 ρ2 > 0,   β1 = λ 2 + 2y2 ρ2 + λ(1 + y12 ρ1 + ρ2 ) + (k + 1)(1 + ρ2 y22 ) > 0, δ2 = 1 + λ(1 + y12 ρ1 + ρ2 ) + ρ2 y22 > 0,

(21)

c(η) = η 2 /λ − η(1 + 1/λ + k/λ) + k, Now the problem (13) is symmetric, because follower forces are absent, so that (η) cannot be negative. Therefore, two solutions for ω 2 remain positive for loads inferior to the critical load, namely, to the smaller, Hcr , of the two buckling loads ⎫ Hcr l1 ⎪ ⎬ 1 + k + λ ∓ (λ − k)2 + 1 + 2k + 2λ k2 > 0. Hsup l1 ⎪ = 2 ⎭ k2

(22)

At Hcr , and also at Hsup , one of the solutions for ω 2 vanishes, so that the critical load coincides with that calculated using the static method, because at Hcr and at Hsup the matrix (18) becomes singular for ω = 0 and its determinant is c(η). For loads H higher than the critical load, the solutions ω 2 become:   • one positive and one negative when: H ∈ Hcr , Hsup , • two negative when: H > Hsup .

(23)

As a consequence, the four solutions  

 1 =± l2

k2 m2

β1 − ηδ2 ±



(β1 − ηδ2 )2 − 4ac(η) , 2a

(24)

are all four real for loads smaller than Hcr , two real and two forming one purely imaginary complex conjugate pair for loads higher than Hcr , but smaller than Hsup , and four split into two purely imaginary complex conjugate pairs at a load higher than Hsup . The purely imaginary complex pair always contains an element which, multiplied by −i, provides an exponentially blowing-up solution, which corresponds to divergence instability. Therefore, for dead load, either stability or divergence instability may only occur. Note that, as a particular case, the critical load (22) for k1 = k2 = K and l1 = l2 = L (λ = 1) simplifies to √ (3 − 5)K K Hcr = ≈ 0.382 . 2L L

(25)

The effect of viscosity on the buckling of the double pendulum subject only to dead loading H can be investigated assuming that the two elastic hinges become

Flutter from Friction in Solids and Structures

11

viscoelastic. Viscosity reacts to an angular relative velocity of the connected rods with a couple, so that, denoting with c1 and c2 the two coefficients of viscosity of the hinges, Eq. (11) is modified through the addiction of the term 

c1 + c2 −c2 −c2 c2



 α˙ 1 , α˙ 2

(26)

which has exactly the same form of the elastic stiffness matrix. Equation (14) becomes therefore   K + G − iC − 2 M a = 0,

(27)

where the viscosity matrix C contains the ci coefficients in Eq. (26) and the dynamics is governed by the nontrivial solutions of 

(ρ1 y12 + ρ2 + 1)λ2 ω 2 − 1 − k + iω(1 + c) + η λ ω 2 (1 + ρ2 y2 ) + 1 − i ω   λ ω 2 (1 + ρ2 y2 ) + 1 − i ω a1 = 0, a2 ω 2 (1 + ρ2 y22 ) − 1 + i ω + η/λ

(28)

where the following dimensionless constants have been introduced c=

c2 c1 , = √ . c2 l2 k2 m 2

(29)

The characteristic equation, obtained from the matrix (28) and written with the notation ω˜ = −iω, becomes p0 ω˜ 4 + p1 ω˜ 3 + p2 ω˜ 2 + p3 ω˜ + p4 = 0,

(30)

  p0 = λ2 ρ2 + ρ1 y12 − ρ2 y2 + ρ22 y2 + ρ1 ρ2 y12 y2 − ρ22 y22 , p1 = 1 + c + 2λ + λ2 + λ2 ρ2 + λ2 ρ1 y12 + ρ2 y2   + cρ2 y2 + 2λρ2 y2 ) , p2 = 1 + c 2 − η + k + λ 2 − η + λ − (η − λ) ρ2 + ρ1 y12 + ρ2 y2 (1 − η + k + 2λ) ,  η(1 + c) , p3 = c − η + k − λ p4 = −η + k + λη (η − 1 − k) .

(31)

where

˜ = −i, so that the time harNote that the notation ω˜ = −iω, corresponds to  ˜ and the instability occurs monic assumption, Eq. (12), becomes α j = a j exp t, when Re[ω] ˜ > 0.

12

D. Bigoni

A quasi-static solution of Eq. (30) corresponds to ω˜ = 0 and can be found when p4 = 0, which provides the two buckling loads (22). Therefore, the presence of the viscosity does not alter the quasi-static bifurcations of the double pendulum with dead load H ; it will be shown that the situation changes when follower load are present.

2.3 The Follower Force P on the Double Pendulum Assuming that the Ziegler double pendulum is loaded only with the follower force P, so that H = 0, and using again the dimensionless variables (17), except that η is now replaced by Pl1 , (32) γ= k2 the generalized eigenvalue problem (13) can be compacted and leads to the condition of vanishing of the determinant of the matrix 

 (ρ1 y12 + ρ2 + 1)λ2 ω 2 − 1 − k + γ λ ω 2 (1 + ρ2 y2 ) + 1 − γ , ω 2 (1 + ρ2 y22 ) − 1 λ ω 2 (1 + ρ2 y2 ) + 1

(33)

a condition which immediately provides the two solutions for ω 2 ω = 2

β1 − γβ2 ±



(β1 − γβ2 )2 − 4ka , 2a

(34)

where a and β1 are the same coefficients as in the list (21), again reported below to facilitate reading   a = λ2 y12 ρ1 + (1 − y2 )2 ρ2 + y12 y22 ρ1 ρ2 > 0,   β1 = λ 2 + 2y2 ρ2 + λ(1 + y12 ρ1 + ρ2 ) + (k + 1)(1 + ρ2 y22 ) > 0, β2 = 1 + λ + y22 ρ2 + y2 λρ2 > 0.

(35)

From the pair of solutions (34) for ω 2 , four solutions for  follow 

 1 =± l2

k2 m2

β1 − γβ2 ±



(β1 − γβ2 )2 − 4ka . 2a

(36)

Note that: • It can be shown that both the the discriminant  = (β1 − γβ2 )2 − 4ka and the coefficient β1 are strictly positive when γ = 0, in other words, (0) > 0 and β1 (0) > 0, so that all solutions  are real, which is coherent with the fact that the structure has to be stable when unloaded.

Flutter from Friction in Solids and Structures Fig. 3 Graphical study of Eq. (34), showing a ‘competition’ between a linear β1 − γβ2 and a parabolic (β1 − γβ2 )2 − 4ka term. Note that the magnitude of the follower load P is included in the variable γ, which determines the stability behaviour of the structure shown in Fig. 2 in the case H = 0

13

ka

stability

flutter

divergence

• For tensile load, γ < 0, all solutions  are real, which means that the structure is stable. • As graphically represented in Fig. 3, the following conditions can be established: β1 , β2 √ β1 2 ka − (β1 − γβ2 )2 − 4ka < 0 ⇐⇒ β2 β2 √ β1 2 ka 0

⇐⇒ γ
0; vibrations are sinusoidal. • Flutter instability: two complex conjugate values for ω 2 , which correspond to two complex conjugate pairs for  and occur when (β1 − γβ2 )2 − 4ka < 0; four exponential solutions exist, namely, two unstable (which blow-up) and the other two decaying with time. • Divergence instability: two real and negative values for ω 2 , which correspond to two purely imaginary conjugate pairs for  and occur when γ > β1 /β2 and (β1 − γβ2 )2 − 4ka > 0; vibrations become exponential functions of time, two of which amplify (denoting unstable behaviour) and two decay. It should be noted that, while stability is always verified at sufficiently small load, flutter and divergence always occur when the load is sufficiently high, independently of the geometry and stiffness of the system. Moreover, flutter instability determines an interval of load separating stability from divergence.

14

D. Bigoni

As a conclusion, the response is stable when: P < P f lu ,

(38)

where P f lu l1 = k2 √   λ 2 + 2y2 ρ2 + λ(1 + y12 ρ1 + ρ2 ) + (k + 1)(1 + ρ2 y22 ) − 2 k a 1 + λ + y22 ρ2 + y2 λρ2

(39) ,

is the critical load for flutter instability, which occurs when the load P falls within the interval: (40) P f lu ≤ P < Pdiv where Pdiv l1 = k2 √  λ 2 + 2y2 ρ2 + λ(1 + y12 ρ1 + ρ2 ) + (k + 1)(1 + ρ2 y22 ) + 2 k a 1 + λ + y22 ρ2 + y2 λρ2

(41) ,

is the critical load for divergence instability, which occurs for loads P higher than or equal to Pdiv . Note that at the onset of flutter and divergence instabilities only two values of  are found from Eq. (36), both real and with opposite signs in the case of flutter, pure imaginary and with opposite signs in the case of divergence. In the particular case in which there are only two masses, m 3 = 0, namely, ρ1 = ρ, ρ2 = 0, y1 = y, the flutter load, Eq. (39), becomes √ √ (1 + λ)2 + (λy ρ − k)2 P f lu l1 . = k2 1+λ Therefore the load for flutter can be minimized in the situation where   √ k1 d m1 √ k = λy ρ, ⇐⇒ = , k2 l2 m 2 which corresponds to

 P f lu = k2

while the divergence load becomes

1 1 + l1 l2

(42)

(43)

 ,

(44)

Flutter from Friction in Solids and Structures

 Pdiv = k2

1 1 + l1 l2

15

 + 4k1

l2 l1



1 l1 + l2

 .

(45)

If it is assumed for simplicity l1 = l2 = L, d = l1 , k1 = k2 = K , and m 1 = m 2 , the critical loads for flutter and divergence become simply P f lu = 2

K , L

Pdiv = 4

K . L

(46)

It can finally be concluded that, in a linearized context, while divergence instability corresponds to a motion growing exponentially in time, flutter instability corresponds to a blowing-up oscillation. Note that both these two instabilities cannot be detected with a quasi-static analysis. The above statement is confirmed in Fig. 4, where results are reported as numerical solution of the linear differential system (11) for H = 0 and with the initial conditions α1 = α2 = 0.5◦ (α1 = α2 = −0.5◦ for divergence) and α˙ 1 = α˙ 2 = 0. For the numerical solution, the following parameters (taken to be representative of the structural model that will be presented in Sect. 2.10) have been selected: l = 3 d = 3 h = 100 mm, m 1 = 12 m 2 = 4 m 3 = 552 g, k1 = k2 = 0.189 Nm,

(47)

which correspond from Eqs. (39) and (41) to a flutter load P f lu ≈ 4.8 N and to a divergence load Pdiv ≈ 8.8 N, so that P = 6.8 N (P = 15.4 N) has been assumed for the simulation of flutter (of divergence). A sequence 0.44 (0.2) s long of configurations at different instants of time is reported in Fig. 4, where each configuration is drawn at fixed intervals of time (0.04 s). The oscillatory blow-up (The exponential growth) of the solution is clearly visible in the case of flutter (of divergence). Flutter cannot be detected via quasi-static bifurcation analysis Equation (42) shows that the flutter load for the Ziegler double pendulum with two masses, depends on the mass distribution of the system, through parameters y and ρ. This distribution, which does not influence the quasi-static behaviour, alters the load for flutter instability. In fact, if Eq. (42) is employed for the same set of parameters (stiffness and entity of masses) which yield the critical loads (46), but with a different disposition of masses, namely, y1 = d/l1 = 1/2 (instead than y1 = 1), the following critical loads for flutter and divergence instability are obtained 

1 P f lu (y1 = 1/2) = 2 + 8



K , L

Pdiv (y1 = 1/2) =

25 K . 8 L

(48)

The fact that two different critical loads for flutter and divergence instability, namely, (46) and (48) are calculated for two mechanical systems differing only in the their mass distribution, which would not influence results calculated with the quasi-static criterion for bifurcation, shows that

16

D. Bigoni

Fig. 4 A sequence (0.44 s for flutter and to 0.2 s for divergence) of deformed configurations at consecutive time intervals of 0.04 seconds of the Ziegler double pendulum (sketched in Fig. 2 with H = 0) and exhibiting flutter (upper part) and divergence (lower part) instability. Results have been obtained through a linearized analysis, Eq. (11), with initial conditions α1 = α2 = 0.5◦ (α1 = α2 = −0.5◦ for divergence) and α˙ 1 = α˙ 2 = 0, at the load P = 6.8 N inside the flutter region (upper part) and at the load P = 15.4 N inside the divergence region (lower part). The values of parameters employed for the analysis are reported in the list (47)

Flutter from Friction in Solids and Structures

17

the quasi-static criterion for bifurcation is inadequate to calculate critical loads of systems subject to follower loads.

2.4 Surprising Effects Related to the Viscosity: The Ziegler Paradox The effect of viscosity on the flutter and divergence instability of the Ziegler double pendulum subject to the follower load P can be investigated by assuming that the two elastic hinges become viscoelastic, thus adding to Eq. (11) the term (26), to yield again Eq. (27), where now the geometric matrix G contains P. The dynamics of the double pendulum is governed by the nontrivial solution of 

(ρ1 y12 + ρ2 + 1)λ2 ω 2 − 1 − k + iω(1 + c) + γ λ ω 2 (1 + ρ2 y2 ) + 1 − i ω   λ ω 2 (1 + ρ2 y2 ) + 1 − i ω − γ a1 = 0, a2 ω 2 (1 + ρ2 y22 ) − 1 + i ω

(49)

where the dimensionless constants (29) have been used. Assuming for simplicity l1 = l2 = L, k1 = k2 = K , ρ1 = 1, ρ2 = 0, y1 = 1, λ = 1, k = 1, c = 1, the problem (49) leads to the following characteristic equation 

2ω 2 − 2 + 2i ω + γ ω 2 + 1 − i ω − γ det ω 2 + 1 − i ω ω 2 − 1 + i ω

 = 0,

(50)

which, written with the notation ω˜ = −iω, becomes p0 ω˜ 4 + p1 ω˜ 3 + p2 ω˜ 2 + p3 ω˜ + p4 = 0,

(51)

where p0 = 1,

p1 = 6 ,

p2 = 6 + 2 − 2γ,

p3 = 2 ,

p4 = 1.

(52)

˜ = −i, so that the time harNote that the notation ω˜ = −iω, corresponds to  ˜ and the instability occurs monic assumption, Eq. (12), becomes α j = a j exp t, when Re[ω] ˜ > 0 (divergence when Im[ω] ˜ = 0 and flutter when Im[ω] ˜ = 0). Note also that, differently from the case of dead loading, the characteristic equation (51) does not admit quasi-static solutions, ω˜ = 0, because p4 cannot vanish. The analysis of the nature of the solutions to the characteristic equation (51) can be performed using the Routh–Hurwitz criterion for a fourth-degree polynomial (see for instance Ziegler 1977), so that stability is assured when p1 > 0,

p1 p2 − p0 p3 > 0,

( p1 p2 − p3 ) p3 − p12 p4 > 0,

p4 > 0,

(53)

18

D. Bigoni

providing the following limit condition for stability P ∗f lu

 =

4 2 + 3 2



K . L

(54)

The behaviour of the Ziegler double pendulum with viscoelastic hinges is shown in Fig. 5, where the real and imaginary parts of the eigenvalue ω˜ are reported for three cases: • The case in which the viscosity is absent ‘from the beginning’, which is governed by the characteristic equation provided by the determinant of the matrix (33); • The viscoelastic case, which is governed by the characteristic equation (51) for two coefficients of viscosity, = 0.1 and 0.5. Figure 5 shows that in the undamped case the real part of ω˜ remains null, until the critical load for flutter (46) is reached, namely, γ = 2. After this value is met, the solution displays a positive real part, denoting an unstable character. The instability corresponds to flutter (and not to divergence) because, in addition to the positive real part, ω˜ displays also an imaginary part. In the cases of viscoelastic hinges, the critical value for flutter decreases to γ = 1.338 (to γ = 1.458), for = 0.1 (for = 0.5). It should be noted that at decreasing viscosity the curves in the figure tend to the undamped case, but the critical load for flutter tends to that obtained from Eq. (54), in the limit of → 0, namely, γ = 4/3 ≈ 1.333. The critical load for flutter in the ‘undamped system’ (46), namely, in the case when the viscosity of the hinges is absent ‘from the beginning’, is notably higher than the value (54), so that a comparison between the flutter loads (46) and (54) leads to the following conclusions: • the critical load for flutter is an increasing function of the viscosity parameter , • but this critical load at sufficiently small viscosity is smaller than the critical load calculated under the hypothesis that viscosity is absent, so that for instance if P f lu = 2K /L for the undamped system, P ∗f lu decreases to 1.338K /L and 1.458K /L for equal to 0.1 and 0.5, respectively. Despite the fact that this behaviour is counter-intuitive, because adding an extra viscous ‘constraint’ to a system would apparently seem not lower a critical load, the conclusion is that the introduction of a small viscosity in the Ziegler double pendulum reduces the critical load; • in the limit of vanishing viscosity, → 0, Eq. (54) yields P ∗f lu = 4/3 K /L, which does not coincide with (and is noticeably lower than) the flutter load for the undamped system (46), calculated for absent viscosity. This result is so surprising that it is known as ‘the Ziegler paradox’ and implies that the flutter load calculated in the absence of viscosity is meaningless, as for a real structure some small, but never null, viscosity does always exist. The paradox was first discovered by Ziegler (1952), later framed in the general context of instability theory by Bottema (1956), and quoted by Bolotin (1963) as one of the most important theoretical aspects in stability when nonconservative

Flutter from Friction in Solids and Structures

19

Re[ ]

Im[ ]

undamped

Fig. 5 Behaviour (at varying the dimensionless load parameter γ) for the viscoelastic Ziegler double pendulum of the eigenvalue ω, ˜ solved from Eq. (51). Two values, 0.1 and 0.5, of the viscosity dimensionless parameter are considered, together with the undamped case, where the viscosity is absent ‘from the beginning’. The structure with = 0.1 (with = 0.5) suffers flutter instability at γ = 1.338 (at γ = 1.458), showing that the inclusion of viscosity lowers the flutter load from the value γ=2 obtained in the absence of viscosity. The Ziegler paradox is evident, because a vanishing small viscosity produces a drop in the flutter load from the value γ = 2 of the undamped system, to γ = 4/3, corresponding to the damped system in the limit → 0

forces are present. A modern and general discussion on the paradox can be found in Kirillov (2005), Kirillov (2013). The destabilizing effect of viscous forces is now an accepted concept in mechanics and leads to the concept of ‘dissipation instabilities’ (Krechetnikov and Marsden 2007; Kirillov and Verhulst 2010, see also the chapter written by O. Doaré in the present book).

2.5 The Deformation of an Elastic Rod: The Euler’s Elastica The determination of the static and dynamic behaviour of an elastic inextensible rod subject to large deflection is a problem of great interest, which is here addressed in a two-dimensional context (in a reference system denoted through the two unit vectors e1 and e2 ). The purpose of this section is to provide the Euler elastica theory, including

20

D. Bigoni

Fig. 6 The kinematics of an elastic inextensible rod of length l, rectilinear in the reference configuration. Displacement of a point of coordinate x0 is u(x0 ) = x − x0 e1 . Note that inextensibility implies that the curvilinear coordinate s is equal to the coordinate x0 , namely, s = x0

t

n

x

s e2

u(x0)

x0

0

e1

the case in which the material behaviour of the rod is viscoelastic and various external loadings are present, thus providing a generalization of Bigoni (2012) and Bigoni et al. (2015). Classical references are Love (1927), Reiss (1969), Audoly and Pomeau (2010). Kinematics An inextensible rod of length l is considered, rectilinear in the reference configuration and smoothly deformed, as shown in Fig. 6. In the undeformed and deformed configurations, the generic point can be picked up using a coordinate x0 ∈ [0, l] in the reference configuration and a curvilinear coordinate s ∈ [0, l] in the current configuration, so that inextensibility implies x0 = s and therefore d x0 = ds. The displacement u of the point x0 e1 (where e1 is the unit vector singling out the axis of the undeformed rod) from the reference configuration is u = u 1 (x0 )e1 + u 2 (x0 )e2 = x − x0 e1 ,

(55)

which, introducing the (twice-continuously differentiable) deformation x = g(x0 ),

(56)

u = g(x0 ) − x0 e1 .

(57)

becomes

Equation (56) is the parametric representation of the curve describing the elastica. Two neighbor points are considered of the reference configuration at coordinates x0 and x0 + ω0 , defining the vector t0 = ω0 e1 . This vector is mapped to g(x0 + ω0 ) − g(x0 ),

(58)

Flutter from Friction in Solids and Structures

21

so that, assuming ω0 small and performing a Taylor series expansion of the deformation around ω0 = 0, yields the transformed vector (tangent to the deformed line at x0 ) as    ∂g ω0 = u 1 + 1 e1 + u 2 e2 ω0 , (59) ∂x0 where the superscript denotes differentiation with respect to the coordinate x0 = s. Since the elastica is assumed inextensible, the transformed vector ω0 ∂g/∂x0 must maintain the same length of the initial vector ω0 e1 , a constraint which from Eq. (59) can be expressed as    ∂g    (60)  ∂x  = 1, 0

which, using Eq. (59) yields   u 1 + 1 = sgn u 1 + 1 1 − (u 2 )2 .

(61)

A derivative of Eq. (61) with respect to s, finally provides the inextensibility constraint in the form   u u

(62) u

1 = −sgn u 1 + 1  2 2 . 1 − (u 2 )2 Since the inextensibility constraint is enforced, the unit vector t, tangent to the elastica at x, is given by     t = u 1 + 1 e1 + u 2 e2 = sgn u 1 + 1 1 − (u 2 )2 e1 + u 2 e2 ,

(63)

and the angle θ of inclination of the tangent t to the elastica at x can be implicitly provided through the expressions sin θ = x2 = u 2 ,

  cos θ = x1 = u 1 + 1 = sgn u 1 + 1 1 − (u 2 )2 .

(64)

The signed length d of the projection of the elastica onto the e1 axis is 

l

d=



l

cos θds = l + u 1 (l) − u 1 (0) =

0

0

  sgn u 1 + 1 1 − (u 2 )2 ds,

(65)

while the signed projection onto the e2 axis is  h= 0

l

sin θds = u 2 (l) − u 2 (0).

(66)

22

D. Bigoni

The unit vector n normal to the elastica at x can be obtained through differentiation (with respect to s) of the scalar product t · t, so that t is found to be orthogonal to t in the form   u u

t = −sgn u 1 + 1  2 2 e1 + u

2 e2 , or 1 − (u 2 )2 (67)

t = −θ sin θe1 + θ cos θe2 , and therefore the unit normal n can be obtained from Eq. (67)1 or (67)2 , through division by the modulus |t |, which is the so-called ‘curvature’ |t | = 

|u

2 | 1 − (u 2 )2

= |θ |,

(68)

thus obtaining    

  

2 or, n = sgn u 2 −sgn u 1 + 1 u 2 e1 + 1 − (u 2 ) e2 n = sgn{θ } (− sin θe1 + cos θe2 ) .

(69)

The signed curvature is   θ = sgn u 1 + 1 

u

2 1 − (u 2 )2

,

(70)

Finally, the following unit vector m is introduced m = sin θe1 − cos θe2 ,

(71)

which is orthogonal to t and rotated of π/2 anticlockwise from it, so that it is always parallel to n, but may differ in sign, and it satisfies the following relations t = −θ m,

m = θ t.

(72)

Force and internal action The elastica is subject to external forces and couples, which are assumed to be applied at points (‘concentrated forces’) or diffused along the line (‘forces per unit length’) of the elastica. It is also assumed that forces and moments are transmitted internally to the elastica, so that when this is ideally cut at a point and the internal forces given evidence, they represent the internal action, and provide the equilibrium of the rod, ideally separated into two parts, Fig. 7. In the plane e1 –e2 the internal action is comprised of a force a(s) and a moment vector M(s)e3 , where e3 is the unit vector orthogonal to the plane. The Fig. 7 also shows that the internal action obeys the Newton’s law of action-reaction, so that if on the left of the cut (characterized by the tangent vector t ‘exiting’ from the rod) the internal action is given by the pair a(s) and M(s), on the right hand (vector t ‘entering’ in the rod) the internal action is −a(s) and −M(s).

Flutter from Friction in Solids and Structures

23

Fig. 7 The action, force a(s) and moment M(s), internal to a deformed rod element. The Newton’s law of action-reaction requires that if on the left side of the ideal cut, dividing the rod into two parts, the action is represented by the pair a(s) and M(s), on the right side the internal action becomes −a(s) and −M(s). The unit vectors m and t single out the normal and the tangential directions, respectively Fig. 8 The equilibrium of a part of a deformed rod free of concentrated forces, but subject to a distributed load q, defined per unit length. The internal action is represented through vector a(s) and moment M(s)

The internal force a is split into a normal, N (s), and a shear, T (s), component in the reference system t and m as a(s) = N (s)t(s) + T (s)m(s),

(73)

so that considering the equilibrium of a generic part of the rod free of concentrated forces, but subject to a generic distributed load q, Fig. 8, the following conditions can be written down  s2 q(s) ds = 0, −a(s1 ) + a(s2 ) + s1 (74) !s2  s2 M(s)e3 + [P(s) − O] × a(s) + [P(s) − O] × q(s) ds = 0, s1

s1

where P(s) denotes a generic point on the elastica comprised between the two end points s1 and s2 .

24

D. Bigoni

Since for every continuously differentiable function f (s) the following identity holds true  s 2

− f (s1 ) + f (s2 ) =

f (s) ds,

(75)

s1

Equation (74)1 can be rewritten as 

s2

  a (s) + q ds = 0,

(76)

s1

an equation which can be localized, because it holds true for every interval (s1 , s2 ) free of concentrated forces, so that the following equation is obtained a (s) + q = 0.

(77)

On application again of the property (75) to Eq. (74)2 yields 

s2

  M (s)e3 + P (s) × a(s) + (P(s) − O) × (a (s) + q(s)) ds = 0,

(78)

s1

where P (s) = t, so that Eq. (77) leads to the following condition 

s2

  M (s)e3 + t × a(s) ds = 0,

(79)

s1

which can be localized and, keeping into account Eq. (73), provides M (s) = T (s).

(80)

The Eqs. (77) and (80) are the equilibrium equations holding for the internal action along the rod. These equations can be written in components as N + θ T + q · t = 0, T − θ N + q · m = 0, M (s) = T (s),

(81)

which hold true regardless the nature of the material of which the rod is made up. Finally, it can be noticed that Eqs. (81)1 and (81)3 can be combined to obtain N + θ M = −q · t.

(82)

Flutter from Friction in Solids and Structures

25

P2

Fig. 9 The clamped elastica subject to two forces P1 and P2 at the free end and to the forces per unit length q1 and q2

e2

P1

q2 q1

s

u2

u2 s e1

0

u1

s u1 s

2.6 Constitutive Equation and Dynamics For simplicity the elastica is assumed to be clamped at its left end, subject along the axis to forces per unit length of components q1 and q2 and to a load P (with components −P1 and −P2 ) at the other end (Fig. 9). The constitutive equation used for the elastica is the following viscoelastic generalization of the Jacob Bernoulli’s assumption that the deflection curvature is linearly proportional to the bending moment M(s) = Bθ (s) + D θ˙ (s),

(83)

in which a superimposed dot denotes the time derivative, B is the bending stiffness, assumed constant for simplicity (in the linear beam theory B equals the product between the Young modulus of the rod and the moment of inertia of its cross section), and D, assumed constant, accounts for the viscosity of the rod.4 Note that for a purely elastic rod (D = 0) with constant bending stiffness B, a substitution of Eqs. (83) into (82) provides the two equivalent conditions

a linearized theory, assuming for a rod a uniaxial σ1 − 1 constitutive law with an elastic term (singled out by the elastic modulus E) and a viscous term (characterized by a constant η) in the form σ1 = E 1 + η˙ 1 , (84)

4 In

the bending moment is given by    M= σ1 y d A = E

1 y d A + η

˙ 1 y d A, A

A

(85)

A

where A is the cross section of the rod and y is the position of a point measured orthogonally to the neutral axis. In a linearized theory = yθ and ˙ = y θ˙ , so that the bending moment becomes M = E I θ + η I θ˙ ,

where I =

" A

y 2 d A is the moment of inertia of the rod’s transverse section.

(86)

26

D. Bigoni

  (θ )2 = −q · t, N+B 2

  M2 N+ = −q · t. 2B

(87)

With reference to two arbitrary points s and σ of the elastica, the following geometrical (thus holding for every constitutive equation of the rod) relations can be written  σ  σ sin θ(ξ)dξ, σ + u 1 (σ) − s − u 1 (s) = cos θ(ξ)dξ, (88) u 2 (σ) − u 2 (s) = s

s

so that the bending moment at the generic point s can be calculated as generated by the following external loads • (i) the end force P of components −P1 and −P2 , ¨ • (ii) the inertia force −ρu, • (iii) the external dissipative force −κu˙ (defined per unit length of the rod, for instance the force generated by the air drag during vibrations), • the diffused load q(s), with components q1 and q2 , in the form

 l  l M(s) = P1 sin θ(σ)dσ − P2 cos θ(σ)dσ s  sl # $# $ ρu¨ 2 (σ) + κu˙ 2 (σ) − q2 (σ) σ + u 1 (σ) − s − u 1 (s) dσ − s l # $# $ ρu¨ 1 (σ) + κu˙ 1 (σ) − q1 (σ) u 2 (σ) − u 2 (s) dσ. +

(89)

s

Equating the ‘external’ moment (89) produced by the loads to the ‘internal’ moment generated by the curvature, Eq. (83), yields  l  l sin θ(σ)dσ + P2 cos θ(σ)dσ Bθ (s) + D θ˙ (s) − P1 s s  l# $# $ (90) ρu¨ 2 (σ) + κu˙ 2 (σ) − q2 (σ) σ + u 1 (σ) − s − u 1 (s) dσ + s  l# $# $ ρu¨ 1 (σ) + κu˙ 1 (σ) − q1 (σ) u 2 (σ) − u 2 (s) dσ = 0. − s

A calculation of the first derivative of Eq. (90) with respect to s and use of Eq. (64) yields D ˙

θ (s) B   l# $  1 P1 + ρu¨ 1 (σ) + κu˙ 1 (σ) − q1 (σ) dσ sin θ(s) + B s l # $  1 P2 + ρu¨ 2 (σ) + κu˙ 2 (σ) − q2 (σ) dσ cos θ(s) = 0, − B s

θ

+

(91)

Flutter from Friction in Solids and Structures

27

which is the equation of the Euler’s elastica, generalized to include (in addition to the two loads P1 and P2 ): internal viscosity (parameter D), external dissipation ¨ (parameter κ), external loading per unit length (parameter q), inertial forces (ρu, where ρ is the mass density per unit length). In the special case of a purely elastic rod D = 0, subject only to end loads P1 and P2 , Eq. (91) reduces to the well-known form of the elastica (Bigoni 2012; Bigoni et al. 2015) P1 P2 θ

(s) + sin θ(s) − cos θ(s) = 0. (92) B B Linearization. Assuming small oscillations about a rectilinear configuration, the rotation θ is small, so that Eq. (64) provide u 2 (s) ≈ θ(s) u 1 (s) ≈ 0.

(93)

Equation (93)2 implies that u 1 = 0 for the clamped rod and therefore Eq. (91) simplifies to    l

(s) + D u ˙ (s) + P − q (σ) dσ u 2 (s) Bu

1 1 2 2 − P2 −

 l#

s

$

(94)

ρu¨ 2 (σ) + κu˙ 2 (σ) − q2 (σ) dσ = 0,

s

which can be derived with respect to s to finally achieve the usual equation of the linearized theory (I V and roman numerals denote derivatives with respect to the variable x, which singles out points of the rod’s axis)  Bu 2I V (x)

+

D u˙ 2I V (x)



l

+ P1 − x

 q1 (σ) dσ u 2I I (x) + q1 (x)u 2I (x)

(95)

+ ρu¨ 2 (x) + κu˙ 2 (x) − q2 (x) = 0, Equation (95) governs the small vibrations of a straight viscoelastic rod of bending stiffness B and internal viscosity D, of mass ρ per unit length, prestressed with axial loads P1 and q1 , subject to transverse load q2 per unit length, and external viscosity κ (a simplified version of this equation is reported by Graff 1975). Boundary Conditions At the clamped end of the rod, s = 0, the following geometrical constraints have to be imposed: u 1 (0) = u 2 (0) = 0, θ(0) = 0, (96) while at the loaded end of the rod, s = l, the following conditions on the internal action hold: N (l) = P · t(l),

T (l) = P · m(l),

M(l) = 0,

(97)

28

D. Bigoni

Fig. 10 The Beck (left) and the Pflüger (right) rods. Note that the difference between the two structures is the presence in the latter of a concentrated mass M at its end

M

where P = −P1 e1 − P2 e2 , so that N (l) = −P1 cos θ(l) − P2 sin θ(l), T (l) = −P1 sin θ(l) + P2 cos θ(l), M(l) = 0.

(98)

Linearization. Assuming small oscillations about a rectilinear configuration, the conditions (96) and (98) simplify to u 1 (0) = u 2 (0) = u 2 (0) = 0, N (l) = −P1 − P2 θ(l), T (l) = −P1 θ(l) + P2 ,

M(l) = 0.

(99)

Note that in a linearized setting, the term M2 has to be disregarded in Eq. (87)2 , so that N = q · t. Therefore, the linearized boundary conditions (99) are consistent with the internal action only if the term P2 θ(l) is negligible, so that N (l) = −P1 . The term can be neglected for instance in the case of tangentially follower force, treated in the next Section, in fact in this case P2 = P sin θ(l) ≈ Pθ(l), so that N (l) = −P1 − P2 θ2 (l) ≈ −P1 .

2.7 The Beck and Pflüger Rods Beck (1952) and Pflüger (1950, 1955) have introduced two schemes of cantilever rod, which are clamped at one end and subject to a tangential load at the other, Fig. 10. The Pflüger rod is a generalization of the Beck rod, because in the former structure there is a concentrated mass positioned at the end where the tangential load is applied. These mechanical systems can be considered as the continuous elastic realization of the Ziegler double pendulum. It should be noted that non-trivial static equilibrium, where θ˙ = 0, can be shown to be impossible for both the Beck and Pflüger rods, regardless the magnitude of displacements (but for deformed configurations in which the normal force inside the

Flutter from Friction in Solids and Structures

29

rod remains compressive). In fact, Eq. (87)2 , valid for D = 0 or θ˙ = 0, in the absence of tangential diffuse load, q · t = 0, implies N+

M2 = constant, 2B

(100)

so that, since M = 0 and N = −P at the loaded end of the rod, the condition N+

M2 = −P, 2B

(101)

has to hold true along the axis of the Beck and Pflüger rods in a static solution. The condition (101) can only be satisfied in the rectilinear configuration, because M2 ≥ 0 and |N | ≤ |P|, so that only the undeformed configuration is a possible static solution. Linearized Analysis of the Pflüger Rod Denoting the rod’s deflection u 2 as −v (so that the transverse displacement is positive when opposite to e2 ), Eq. (95) governs the linearized dynamics of a straight rod, in which the relation θ(x) = −v I (x) holds true. The moment-curvature viscoelastic constitutive relation (83) becomes now M(x, t) = −Bv I I (x, t) − D v˙ I I (x, t),

(102)

where a superimposed dot denotes the time derivative. The shear force T (x) can be computed from Eq. (81)3 to be the derivative of the bending moment, so that, for constants moduli B and D, it can be written as T (x, t) = −Bv I I I (x, t) − D v˙ I I I (x, t).

(103)

For the Pflüger column of length l with a concentrated mass M (rotational inertia of the mass is neglected), the boundary conditions can easily be deduced from Eq. (99) and are clamped end, v(0, t) = v I (0, t) = 0, loaded end, M(l, t) = −Bv I I (l, t) − D v˙ I I (l, t) = 0, ¨ t), loaded end. T (l, t) = −Bv I I I (l, t) − D v˙ I I I (l, t) = −M v(l,

(104)

The linearized differential equation of motion (95) which governs the dynamics of a rod subject to small displacements, to an an axial force P (positive when compressive), and to a distributed external damping κ is ˙ t) + ρv(x, ¨ t) = 0, Bv I V (x, t) + D v˙ I V (x, t) + Pv I I (x, t) + κv(x,

(105)

30

D. Bigoni

which, introducing the dimensionless quantities    x t B Pl 2 M , ξ= , τ= 2 , p= , α = arctan l l ρ B ρl  κl 2 D B , γ=√ , η= 2 Bl ρ ρB

(106)

can be rewritten as ˙ τ ) + v(ξ, ¨ τ ) = 0, v I V (ξ, τ ) + η v˙ I V (ξ, τ ) + pv I I (ξ, τ ) + γ v(ξ,

(107)

where now a roman numeral denotes differentiation with respect to ξ and a dot differentiation with respect to τ . Governing Equations Assuming time-harmonic vibrations of pulsation ω˜ ωτ ˜ , v(ξ, τ ) = v(ξ)e ˜

(108)

Equation (107) yields a linear differential equation for v(ξ), ˜ which can be written as ˜ L[v] ˜ = −ω˜ 2 v,

(109)

where the differential operator L is defined as L[Y ] = (1 + ωη) ˜

d2 Y d4 Y + p + ωγ ˜ Y, dξ 4 dξ 2

(110)

and has to be complemented by the boundary conditions (104) now rewritten as v(0) ˜ = v˜ (0) = 0, clamped end,

loaded end, v˜ (1) = 0, ˜ = 0, loaded end. (1 + η ω) ˜ v˜

(1) − ω˜ 2 tan(α)v(1)

(111)

The differential problem (109) with the boundary conditions (111) always admits the trivial solution v˜ = 0. The characteristic equation of the differential equation (109) is λ4 (1 + η ω) ˜ + λ2 p + γ ω˜ + ω˜ 2 = 0, which admits the two solutions for λ2  ˜ ω˜ + ω˜ 2 ) − p ± p 2 − 4(1 + η ω)(γ 2 . λ1,2 = 2(1 + η ω) ˜ Therefore, the solution for v˜ becomes

(112)

(113)

Flutter from Friction in Solids and Structures

v(ξ) ˜ = A1 sinh(λ1 ξ) + A2 cosh(λ1 ξ) + A3 sin(λ2 ξ) + A4 cos(λ2 ξ),

31

(114)

where Ai (i = 1, . . . , 4) are arbitrary constants. A substitution of the solution (114) into the boundary conditions (111) yields an algebraic system of equations which admits non-trivial solutions at the vanishing of the matrix of coefficients ⎡ ⎤ 0 1 0 1 ⎢ ⎥ 0 λ2 0 ⎢ 2 λ1 ⎥ (115) ⎣ λ1 sinh λ1 λ21 cosh λ1 −λ22 sin λ2 −λ22 cos λ2 ⎦ a41 a42 a43 a44 where

a41 = (1 + η ω)λ ˜ 31 cosh λ1 − ω˜ 2 tan α sinh λ1 , a42 = (1 + η ω)λ ˜ 31 sinh λ1 − ω˜ 2 tan α cosh λ1 , a43 = −(1 + η ω)λ ˜ 32 cos λ2 − ω˜ 2 tan α sin λ2 , ˜ 32 sin λ2 − ω˜ 2 tan α cos λ2 . a44 = (1 + η ω)λ

(116)

Noting that the λi ’s are functions of the applied load p, the pulsation ω, ˜ the viscosity η, and the damping γ, nontrivial solutions for the vibrations of the Pflüger rod correspond to the fulfillment of the condition f ( p, ω, ˜ α, γ, η) = 0, where

(117)

f ( p, ω, ˜ α, γ, η) = (1 + η ω)(λ ˜ 41 + λ42 ) + 2(1 + η ω)λ ˜ 21 λ22 cosh λ1 cos λ2 + λ1 λ2 (1 + η ω)(λ ˜ 22 − λ21 ) sinh λ1 sin λ2 λ2 + λ22 − ω˜ 2 tan α 1 [λ2 sinh λ1 cos λ2 λ1 λ2 − λ1 cosh λ1 sin λ2 ].

(118)

For a given elastic system, the dimensionless parameters α, γ, and η are fixed. Therefore, for a fixed value of the dimensionless load p, Eq. (117) can be solved for ω. ˜ If the real part of ω˜ is positive, the system is unstable and in this case, if ω˜ has also a complex part, flutter instability occurs, otherwise divergence instability occurs. Note that both flutter instability and divergence instability occur for the Pflüger column (the former is achieved at critical loads smaller than those inducing the latter), except in the limit case of the Beck column (α = 0), where the divergence load tends to infinity and it is therefore not found. An example of determination of flutter instability for the Beck column in the presence of internal damping, but not external, is reported in Fig. 11, where the following values of parameters have been used: l = 0.350 m, ρ = 0.0546 kg/m, B = 0.0332 Nm2 , D = 7.078 × 10−6 Nsm2 , P = 5.5 N.

(119)

32

D. Bigoni

Fig. 11 Branches of the real (Re[ω]) ˜ and imaginary (Im[ω]) ˜ parts of the pulsation for vibration of the Beck column as functions of the dimensionless load p. The undamped case (in which damping is absent ‘from the beginning’) is reported on the upper part, where flutter occurs at p = 20.05. The case in which internal damping is present (but there is no external damping) is shown on the lower part. Here, the flutter load decreases to p = 10.94. Flutter occurs when a real branch of the pulsation ω˜ becomes positive (with non-null values of its imaginary part). Note the detrimental effect of dissipation on the critical load for flutter

The branches of the real (Re[ω]) ˜ and imaginary (Im[ω]) ˜ parts of the pulsation for vibration of the Beck column are reported in Fig. 11 as functions of the dimensionless load p. The undamped case (in which damping is absent ‘from the beginning’) is reported on the upper part of Fig. 11, where flutter occurs at p = 20.05 a value found by Beck (1952), which is much higher than the value (2.467) associated with Euler’s buckling of the same structure subject to dead loading. For the Beck’s column divergence is not found. The case in which the only dissipation source is the internal damping is shown on the lower part, where the flutter load decreases to p = 10.94. Flutter occurs when a real branch of the pulsation ω˜ becomes positive (with non-null values of its imaginary part). This figure shows clearly the strong detrimental effect of dissipation on the flutter critical load, which decreases from p = 20.05 to p = 10.94. An example of determination of flutter for the Pflüger column in the presence of internal damping, but not external, is reported in Fig. 12, where, in addition to the values of the parameter list (119), M/(ρl) = 1 has been assumed, so that tan α = 1.

Flutter from Friction in Solids and Structures

33

Fig. 12 Branches of the real (Re[ω]) ˜ and imaginary (Im[ω]) ˜ parts of the pulsation for vibration of the Pflüger column (with tan α = M/(ρl) = 1) as functions of the dimensionless load p. The undamped case (in which damping is absent ‘from the beginning’) is reported on the upper part, where flutter (marked with the subscript f ) occurs at p = 16.51 and divergence (marked with the subscript d) at p = 30.22. The case in which internal damping is present (but there is no external damping) is shown on the lower part. Here, the flutter load decreases to p = 7.78, while the divergence load increases to p = 30.52. The flutter load strongly decreases with the introduction of the internal viscosity, while the divergence load slightly increases

The most important difference between results reported in Figs. 11 and 12 is that for the Pflüger column divergence instability also occurs in addition to flutter, so that the two critical loads for these instabilities determine the flutter region. In the undamped case (where damping is not introduced) the critical loads for flutter and divergence are respectively p = 16.51 and p = 30.22 (on the upper part of Fig. 12), which become p = 7.78 and p = 30.52 when internal damping is present (on the lower part of Fig. 12). It can be therefore concluded that the presence of internal damping increases the size of the flutter region by strongly decreasing the flutter load and by slightly increasing the divergence load. It is finally noticed that rotary damping (in other words a damping connected to the rotation of the rod’s cross section) has also been considered, the interested reader is addressed to Lottati and Elishakoff (1987).

34

D. Bigoni

2.8 Self-adjointness, an Exclusion Condition for Flutter For discrete systems, unsymmetry of the geometric matrix in Eq. (11) is a necessary condition for flutter instability, which is in fact impossible for dead loading, ruled by a symmetric matrix. It is important now to obtain a condition similar to symmetry for discrete systems, which excludes flutter instability for continuous systems. It is assumed, for simplicity, a null internal and external viscosity, η = γ = 0, and the Beck rod (where α = 0) is considered. Moreover, for comparison, the ‘standard’ case of dead loading is also treated. In both cases of follower and dead load, the differential operator is the same, which is the following reduction of the operator (110), L[Y ] =

d2 Y d4 Y + p , dξ 4 dξ 2

(120)

so that the differential equation (109), complemented with the boundary conditions v(0) ˜ = v˜ I (0) = 0, clamped end, II loaded end, v˜ (1) = 0,+ 0, loaded end for follower load, III v˜ (1) = − p v˜ I (1), loaded end for dead load,

(121)

becomes a Sturm–Liouville problem (Broman 1970) for a fourth-order differential equation (note that the only difference between dead and follower load is in the boundary condition involving the shear force). For this differential problem selfadjointness excludes complex eigenvalues ω˜ 2 , so that flutter instability is a-priori ruled out. The condition of self-adjointness (or symmetry) of the differential operator L [defined by Eq. (120)] is 

1



1

L[Y (ξ)]X (ξ) dξ =

0

L[X (ξ)]Y (ξ) dξ,

(122)

0

which has to hold for every pair of functions X (ξ) and Y (ξ), both satisfying the boundary conditions (121). A repeated use of integration by parts, namely, I  I  I  Y IV X = Y III X − Y II X I + Y I X II   I − Y X III + Y X IV , I    I Y II X = Y I X − Y XI + Y XII, allows to obtain the following identity

(123)

Flutter from Friction in Solids and Structures



1 0

 L[Y (ξ)]X (ξ) dξ =

35

1

L[X (ξ)]Y (ξ) dξ 1 + p YI X − Y XI 0 1  + Y III X − Y II XI + Y I XII − Y XIII 0 , 0

(124)

which makes evident that the self-adjointness condition (122) involves the boundary conditions and is equivalent (for the problem under consideration) to 

Y III X − Y II XI + Y I XII − Y XIII

1 0

1  + p Y I X − Y X I 0 = 0.

(125)

Imposition of the boundary conditions (121)1,2 at the clamp and the null moment condition at the loaded end (121)3 yields the following self-adjointness condition   Y I I I (1)X (1) − Y (1)X I I I (1) + p Y I (1)X (1) − Y (1)X I (1) = 0,

(126)

valid for both the structures subject to the follower load and to the dead load, because the two structures differ only in the boundary condition on shear, Eq. (121)4 . A consideration of the boundary condition (121)4 shows that the self-adjointness condition (126) is satisfied for dead loading, in which case flutter instability is excluded, but is not satisfied for follower load, so that flutter instability becomes possible in that case.

2.9 Beyond the Linearized Solution: Limit Cycle Behaviour The stability analysis developed so far for the Beck and Pflüger columns is based on the time-harmonic solution, which is valid for the linearized equations of motion (95). This analysis is valid therefore only in a neighborhood of the instability point, but neglecting nonlinear terms becomes unacceptable when the motion starts to grow. For the Ziegler double pendulum, the linearized problem is governed by Eq. (11) and the nonlinear dynamics by the Eq. (10), which can be integrated in time to provide the behaviour of the Ziegler double pendulum in the flutter and divergence regions, where large displacements and rotations are allowed. This integration has been numerically performed (using the function NDSolve of Mathematica 10.0) for the following values of constants: l1 = l2 = 0.1 m, k1 = k2 = 0.189 Nm, c1 = c2 = 0.006 Nms, m 1 = m 2 = 0.2 Kg, m 3 = 0, d = l1 , P = 1.5 P ∗f lu ,

with P ∗f lu evaluated through Eq. (54) and an initial imperfection α1 = α2 = 0.1◦ . Numerical integration leads to the trajectory plotted in the phase plane reported in Fig. 13. It is clear from Fig. 13 and from Fig. 14, which completes the phase portrait by showing the angular velocities α˙ 1 and α˙ 2 (plotted as functions of the rotations α1 and α2 ), that the Ziegler double pendulum reaches a stable limit cycle (see also D’Annibale et al. 2015).

36

D. Bigoni

Fig. 13 Trajectory in the α1 –α2 phase plane for the Ziegler double pendulum (with viscoelastic hinges), which attains a stable limit cycle

When the structure reaches a limit cycle, self-sustained vibrations occur and the system behaves as a self-oscillating structure (Jenkins 2013). The attainment of a limit cycle is a consequence of the dissipation, which is represented by the viscosity of the hinges for the Ziegler double pendulum. For the Beck’s column in the absence of external damping, but in the presence of internal damping, Fig. 15, shows the trajectory of the end of the rod, while Figs. 16 and 17 complete the phase portrait by reporting the velocities of the end of the rod, plotted, respectively, as functions of the displacement components and in a u˙ 1 –u˙ 2 representation. Values from the list (119) of parameters have been used. Figures 15, 16 and 17 have been obtained with a nonlinear computational model, implemented in the finite element software ABAQUS Standard 6.13-2. Specifically, 2-nodes linear elements of type B21 (in the ABAQUS nomenclature) were employed to discretize the viscoelastic rod of constant, rectangular cross section. A number of 20 elements was found to be sufficient to adequately resolve for the rod dynamics. A linear viscoelastic model of the Kelvin–Voigt type was implemented for the constitutive response of the rod by means of a UMAT user subroutine, such that the bending moment was proportional to the rod curvature and its time derivative respectively through the elastic and viscous moduli. In the analysis, a rod of length l = 0.35 m, B = 0.0332 Nm2 , D = 7.07810-6 Nsm2 , and density ρ = 0.0546 kg/m was subject at its tip to a tangential follower force P = 5.5 N (inside the flutter region). The dynamic analysis was performed by exploiting the default settings of ABAQUS Standard 6.13–2 and with a time increment of 10–4 s. Figures 15, 16 and 17 suggest that the Beck’s column reaches a limit cycle, even if the attainment of this limit cycle has not been formally proven, because the system has infinite degrees of freedom and only a numerical analysis was performed.

Flutter from Friction in Solids and Structures

37

Fig. 14 Angular velocities α˙ 1 (upper part) and α˙ 2 (lower part) as functions of the rotations α1 and α2 , completing the phase portrait (see also Fig. 13), of the Ziegler double pendulum (with viscoelastic hinges), which reaches a stable limit cycle

2.10 Follower Forces from Coulomb Friction The way to generate a follower force from Coulomb friction is shown in Fig. 18, referred to the Ziegler double pendulum. The idea is to mount a little wheel (of negligible rotational inertia) on the end of the pendulum and to force it to slide

38

D. Bigoni

ti p

Fig. 15 Trajectory of the loaded end of the Beck’s column (the displacement components u 1 ti p and u 2 of the rod’s end are reported) with internal, but not external, dissipation, showing the achievement of a limit cycle. Note that this limit cycle is a closed loop, which is rather flat and therefore hardly visible

against a plate with a contact force which can be calibrated. Assuming that the sliding is governed by a simple Coulomb law, the force transmitted at the rod end is: (i) coaxial to the rod (because the wheel is free of rotating and cannot transmit any orthogonal force) and (ii) proportional, through a friction coefficient, to the load creating the contact force between the wheel and the plate. In this way an extremely orthotropic friction is introduced, which transmits only a highly directional force, coaxial to the second rod of the Ziegler double pendulum. The scheme to generate a tangentially follower force is an idealization, so that its practical realization can introduce difficulties. However, Bigoni and Noselli (2011) have designed, manufactured and tested a device and have shown that it works in reality, with negligible discrepancies with respect to the conceptual scheme. Without entering into details (the interested reader is referred to Bigoni and Noselli 2011), it was possible to measure the onset of flutter and divergence instability in terms of applied loads (Fig. 19) and to measure the time variation of the accelerations at the end of the double pendulum (Fig. 20). Results shown in Fig. 19 demonstrate that a critical load for flutter and divergence instability is found and that the experiments support the conclusion that viscosity is a destabilizing factor. Figure 20 shows that the measured acceleration at the end of the structure displays an oscillation initially blowing-up and later converging into a limit cycle. The scratch left by the sliding wheel on the plate (highlighted with red spots) is visible in a photo reported in Fig. 21, where also the nonlinear solution is indicated (with a white curve). It can be pointed out in general that a definite agreement is observed between all experimental results and the model predictions. The experimental apparatus designed by Bigoni and Noselli can be developed to test continuously deformable structures. A new device has been recently designed, manufactured, and tested by Bigoni et al. (2018). This device allows for new testings on the Beck’s and Pflüger rods, shows that the Beck’s model of tangentially follower force can be realized in practice, and provides the first experimental evidence that the viscous dissipation decreases the flutter load.

Flutter from Friction in Solids and Structures

39

Fig. 16 Velocities u˙ 1 (upper part) and u˙ 2 (lower part) as functions of the displacements u 1 and u 2 , showing the achievement of a limit cycle for the Beck’s column (with internal, but not external) dissipation

2.11 Self-oscillating Systems The realization of the Ziegler double pendulum through frictional sliding against an external plate is an example of self-oscillating system,5 in which a input of steady 5 Self-oscillators are distinct from forced and parametric resonators, in which the power that sustains

the motion must be externally modulated.

40

D. Bigoni

Fig. 17 Velocity u˙ 2 as function of the velocity u˙ 1 , showing the achievement of a limit cycle for the Beck’s column (with internal, but not external) dissipation

energy (the frictional force transmitted from the plate to the structure in the present case), lacking periodicity, induces and maintains a self-oscillation of constant frequency (Jenkins 2013). In other words, the complex ‘Ziegler double pendulumexternal plate’ displays, in the flutter region, self-excited vibrations, in which the oscillating system draws energy from the plate, a mechanism similar to wind induced oscillations of suspension bridges and iced telephone wires.

3 Flutter in Frictional Solids Until now the presentation was limited to structures. Now elastoplastic solids will be addressed with the purpose of showing that, when frictional behaviour is taken into account, phenomena akin to flutter and divergence in structural systems may occur even in a continuous medium. Micromechanisms such as sliding between grains or at micro-fissures are typical of granular or rock-like materials and introduce effects of friction in the inelastic constitutive modelling of solids. As a consequence, this modelling is characterized by pressure-sensitivity of yielding (in other words, an increase with the mean pressure of the shear stress needed to produce yielding) and dilatant/contractant inelastic deformation. Consideration of these constitutive features leads to the so-called ‘nonassociative elastoplasticity’, introduced by Mróz (1963, 1966), Mandel (1962, 1966) and Maier (1970), which can be thought as the counterpart of the model of contact with Coulomb friction in continuum mechanics.

Flutter from Friction in Solids and Structures

41

Fig. 18 The way to produce a force coaxial to a rod from sliding friction; a freely rotating wheel of negligible mass is mounted at the end of the Ziegler double pendulum and is constrained to slide against a rigid plate (upper part). The way of calibrating the force which compresses the wheel against the plate is to use the Ziegler double pendulum itself as a lever subject to a load W generating a contact force R and thus the follower force P by friction (lower part)

stability

divergence

Fig. 19 Experimental investigation on flutter and divergence instability with the apparatus sketched in Fig. 18. An increasing vertical load (producing the follower force via friction) is applied for a fixed sliding velocity of the plate against the wheel (50 mm/s). Experimental results are shown with spots and theoretical predictions are reported with horizontal bars. The model in which viscosity is absent ‘from the beginning’ predicts flutter (divergence) instability to occur at a load higher (smaller) than the load calculated when viscosity is present. In particular, the flutter (divergence) load with viscosity is 0.70 (1.49) times the value calculated in the absence of viscosity. Therefore the experimental results provide evidence of the destabilizing effect of viscosity

42

D. Bigoni

Fig. 20 Measured acceleration (at the end of the Ziegler double pendulum) versus time (6 s are reported on the left) during a flutter test, performed with the apparatus sketched in Fig. 18 for a load in the middle of the flutter region and a plate velocity of 75 mm/s. The part of the figure on the right is a detail of the part on the left, referred to the initial 1.5 s time interval. Results of a numerical nonlinear (and linear) simulation at the same load and with initial conditions α1 = α2 = 0.5◦ are reported in red (in blue). The solution of the linear equations (blue curve) has been interrupted at 1.5 s since the blow-up was too high. Note the initial increase in the amplitude of the acceleration denoting flutter (well captured even by the linearized viscoelastic analysis), and the following stabilization (well captured by the nonlinear analysis) into a limit cycle

Fig. 21 The scratch left by the wheel on the plate of the apparatus shown in Fig. 18, compared with the nonlinear solution (obtained numerically), for a load in the middle of the flutter region and a plate velocity of 100 mm/s. Initial conditions α1 = α2 = 0◦ and α˙ 1 = α˙ 2 = 0.5 rad/s have been used to produce the numerical results. The red spots along the scratches are positions of the wheel corresponding to photos taken with a high speed camera; the scratch left during the experiment is clearly visible in the initial part of the test, not evidencing detachments. The whole sequence corresponds to a 2.04 s interval of time

Flutter from Friction in Solids and Structures

43

pN

pT a

p N, u N

kN pT

T

s T

u

b

.s T

. .

p slip

uN kT

uT

p stick

pT

pN tan

Fig. 22 Contact with friction between two ‘bricks’ (left); a simple model of this, involving Coulomb friction (center); the Coulomb criterion of friction (right). Note the normal a to the Coulomb criterion and direction of slip b (the symbol ‘ ’ means ‘parallel’). Force ‘points’ {p} = { p N , pT } inside the criterion (grey zone) correspond to stick, while for points at the boundary of the criterion stick or slip may occur, depending on the direction of the rate p˙ = { p˙ N , p˙ T }; in particular, stick (slip) corresponds to p˙ · a < 0 (= 0). Note that the non parallelism of a to b implies that the model does not obey the so-called ‘normality (or associativity) rule’, which is ab, so that the model is ‘nonassociative’

3.1 Contact with Coulomb Friction Versus Nonassociative Elastoplasticity The close analogy between the equations governing contact between solids with Coulomb friction and the constitutive equations of nonassociative elastoplasticity can be appreciated by the simple model illustrated below with reference to a masonry-like material. A typical contact between two ‘bricks’ is sketched on the left in Fig. 22, which is idealized with the simple model reported on the center in the same figure. In particular, a point mass capable of moving only in the horizontal direction is attached to two springs of stiffness k N and k T . The vertical displacement u N of the end of the vertical spring (subject to the compressive force p N ) is purely reversible (and due to the spring deformation only), while the displacement u T of the end of the horizontal spring (subject to the tangential force pT ) is the sum of the reversible deformation of the spring itself, plus a possible slip u sT of the point mass on the rigid horizontal constraint. The analysis is limited to the condition of contact (so that separation and the so-called ‘grazing’ are not considered, see Radi et al. 1999), corresponding to a compressive (assumed positive) normal force p N > 0. The Coulomb friction condition, playing the role of the yield condition in plasticity, sets a limit to the possible components pT , p N of the force vector p in the form f ( p N , pT ) ≤ 0,

(127)

f ( p N , pT ) = | pT | − μ p N ,

(128)

where

44

D. Bigoni in which μ is the friction coefficient. A geometrical interpretation of criterion (127) is given in Fig. 22 on the right: force states p, represented as points of coordinates p N , pT , cannot lie outside the region bounded by the two inclined lines, which graphically represent the Coulomb yield criterion.

Inside the Coulomb criterion: stick When condition (127) is satisfied with the strict inequality ‘ 0 slip < 0 stick

(139)

while the transition condition a · Eu˙ = 0 represents the so-called ‘neutral loading’. Rate equations for slip in final, incrementally-nonlinear form The following rate constitutive equations for contact with friction are finally deduced , p˙ =

˙ Eu,

if | pT | − μ p N < 0, ˙ a · Eu Eb if | pT | − μ p N = 0, Eu˙ − a · Eb

(140)

which, in the case considered in Fig. 22, become the rate equations of contact with Coulomb friction

46

D. Bigoni

p˙ N = k N u˙ N ,

p˙ T = k T u˙ T − −μ k N u˙ N + k T u˙ T sgn pT sgn pT .

(141)

The operator · in Eqs. (140) and (141) is the so-called ‘Macaulay bracket’, defined for every α ∈ R as α = (|α| + α)/2. The Macaulay bracket operator provides the rate piecewise linearity (a simple form of incremental nonlinearity, which distinguishes between slip and stick) typical of elastoplasticity (where ‘stick’ is replaced by ‘elastic unloading’ and ‘ slip’ by ‘plastic loading’).

Equation (140) are formally identical to the rate constitutive equations of nonassociative ideal (i.e. with null hardening) elastoplasticity. In the equations of elastoplasticity p˙ is replaced by the rate of stress, u˙ by the deformation rate, a by the yield function gradient, b by the plastic flow mode tensor and E by the fourth-order elastic tensor. In the problem of contact with friction, exactly as for the rate constitutive equations of nonassociative elastoplasticity, it turns out that: • The contact condition (140) is written in a rate form and cannot be resolved into equations involving finite quantities (to understand this important point it suffices to consider that the knowledge of a finite displacement u T at given value of vertical force p N does not determine the tangential force pT , since the irreversible part of displacement u sT is not known and this can be obtained only through integration in time of the rate equations). • The rate equations are incrementally nonlinear and characterized by an elastic ˙ p˙ = Eu,

(142)

and a plastic ˙ p˙ = Cu,

C=E−

1 Eb ⊗ Ea, a · Eb

(143)

branch [the symbol ‘⊗’ is the dyadic product]. • The constitutive tensor C characterizing the plastic branch is not symmetric; therefore the structure of problems involving friction is not self-adjoint. Note that the fact that C is not symmetric follows from the difference between a and b. The former vector is normal to the friction criterion (see Fig. 22), while the latter is not. Therefore, the model lacks ‘normality’ or in other words is ‘nonassociative’, in the sense that the slip rule ‘associated’ to the friction criterion requires b to be parallel to a.

3.2 The Rate Equations of Nonassociative Elastoplasticity for Frictional Solids Elastoplasticity is based on the concept of yield function f (σ, K) ≤ 0,

(144)

depending on the stress σ and on a set of internal variables K governing the inelastic deformation of the material. Negative values of f determine elastic states, for which only elastic deformation is possible, while plastic flow may occur only when f = 0. Positive values of f are excluded.

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The rate equations of incremental elastoplasticity can be derived from the following four assumptions: • 1. Additive decomposition of elastic, ˙ e , and plastic, ˙ p , strain rates ˙ = ˙ e + ˙ p ,

(145)

where ˙ is the rate of strain, so that the superimposed dot denotes derivative with respect to a time-like parameter, governing the loading program. • 2. The rate of stress σ˙ is related to the rate of elastic strain σ˙ = E[˙e ], through a fourth-order elastic tensor E. • 3. The plastic flow rule ˙ p = λ˙ P,

λ˙ ≥ 0,

(146)

(147)

where λ˙ is a non-negative plastic multiplier and P is a symmetric, second-order tensor, which rules the direction of the rate of plastic strain. • 4. The hardening rule ∂f ˙ · K˙ = λH, (148) − ∂K where H is the hardening modulus, positive for hardening, null for ideal plasticity, and negative for softening. Imposing the Prager consistency, namely, f˙ = 0 for plastic flow, yields ∂f ∂f ˙ · E[P] − λH ˙ = 0, f˙ = · σ˙ + · K˙ = Q · E[˙] − λQ ∂σ ∂K

(149)

from which the plastic multiplier is obtained < Q · E[˙] > , λ˙ = H + Q · E[P]

(150)

where denotes the Macaulay brackets. As a conclusion, the rate equations of elastoplasticity can be written as σ˙ =

⎧ ⎨ E[˙],

if f < 0, < Q · E[˙] > E[P], if f = 0, ⎩ E[˙] − g

(151)

where the plastic modulus g (assumed strictly positive) is defined as g = H + Q · E[P].

(152)

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D. Bigoni

The elastic branch of the constitutive equation (151) is simply the elastic fourthorder tensor, E, while the plastic branch can be written in a form similar to (143), namely 1 (153) C = E − E[P] ⊗ E T [Q], g which shows the following interesting features: • Assuming the major symmetry of E, the operator C is symmetric if and only if P is proportional to Q, in other words when (for a scalar α) P = αQ; • Defining the stiffness of C in terms of the second-order work ˙ · C[˙], it is evident that the  elastic stiffness can be smaller than the elastoplastic stiffness, when (˙ · E[P]) QE T [˙] < 0. The normality rule corresponds to P = Q, which determines the associative elastoplastic model, and corresponds to a symmetric operator (153). Generally speaking, the choice of P as related to Q should be based on experimental results. These show that many solids (and in particular granular materials) exhibit a peculiar kind of non-associativity, involving only the volumetric part of plastic deformation. This case of special interest corresponds to so-called deviatoric associativity, where the deviatoric parts of P and Q are aligned, so that χ2 P = χ1 Sˆ + I, 3

ψ2 Q = ψ1 Sˆ + I, 3

(154)

where Sˆ ∈ Sym is traceless, χ1 and ψ1 are assumed strictly positive. The parameters ψ2 and χ2 respectively describe the pressure-sensitivity and the dilatancy (when χ2 > 0) or contractility (when χ2 < 0) of the material. In the case of the Drucker– Prager model and the flow rule nonassociativity used, among many others, by Bigoni and Loret (1999), the parameters can be rewritten as χ1 = cos χ, χ2 = devσ , Sˆ = |devσ|

√ √ 3 sin χ, ψ1 = cos φ, ψ2 = 3 sin φ, (155)

where devσ = σ − I (trσ/3) is the deviatoric stress.

3.3 The Propagation of Incremental Plane Waves The rate equations of elastoplasticity are incrementally nonlinear, in the sense that the rate response is different for plastic loading or elastic unloading, a property evidenced by presence of the Macaulay brackets in Eq. (151). Therefore, every solution, even in rate form, is the solution of a nonlinear problem. Under this nonlinearity assumption, the usually simple problem of sinusoidal wave propagation in an infinite

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49

body becomes complicated (Bigoni and Petryk 2002). Therefore, wave propagation in plasticity is usually analyzed for acceleration waves, viewed as propagating discontinuity surfaces for the acceleration (Hill 1962; Mandel 1962; Raniecki 1976; Bigoni 2012). However, the acceleration wave approach for plastic waves provides exactly the same result that it is found for sinusoidal incremental waves restricted to the loading branch of the constitutive operator. Therefore, to simplify the treatment, incremental disturbances in the form of plane sinusoidal waves will be considered in the following with reference to the plastic branch of the constitutive equation (151), so that ‘rates’ will be identified with ‘increments’, namely, σ˙ becomes the increment of stress σ and ˙ becomes the gradient of an incremental displacement, ∇w. An infinite, homogeneously deformed and stressed elastoplastic material is considered, so that equilibrium and compatibility are trivially satisfied. Any incremental dynamic solution must satisfy the incremental equations of motion (when body forces are absent) ¨ divσ = ρw, (156) ¨ is an incremental acceleration. where ρ is the mass density of the material and w Incremental solutions are sought in the following sinusoidal wave form w = Re {aeik(n · x±ct) },

(157)

√ where i = −1, n is the unit vector of propagation, a is the (possibly complex) wave amplitude vector, k is the (positive) wave number, c is the (possibly complex) wave speed. Adopting the complex notation, the gradient and the time derivative of Eq. (157) are ˙ = ±ick w, ∇w = ik w ⊗ n, w (158) while the second gradient and second time-derivative are ¨ = −c2 k 2 w. ∇ (∇w) = −k 2 w ⊗ n ⊗ n, w

(159)

Inserting Eq. (159) into the momentum balance (156) leads to the propagation condition (160) (A(n) − c2 I)a = 0, where A(n) is the acoustic tensor defined, for every vector g, as A(n)g =

g · E T [Q]n 1 E[g ⊗ n]n − E[P]n, ρ ρg

(161)

so that the squared propagation velocities c2 are the eigenvalues of the acoustic tensor corresponding to the loading branch of the constitutive elastoplastic operator.

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D. Bigoni

On the basis of the nature of the three eigenvalues of the acoustic tensor, the following nomenclature can be introduced (Rice 1977). • Stability occurs when all values for c2 are strictly positive, so that waves propagate sinusoidally with a finite speed; • divergence instability corresponds to a real and negative value for c2 , so that waves grow exponentially during propagation; • flutter instability corresponds to two complex conjugate eigenvalues c2 , so that an oscillation blowing-up in time is predicted. The following conclusions can be drawn. • In the case when E is hyperelastic (and thus possesses all symmetries) and for the associative flow rule P = Q, the eigenvalues c2 are always real and the corresponding eigenspaces orthogonal, so that flutter instability is excluded. • Therefore, a necessary condition for flutter instability is that the plastic flow be nonassociative, P = Q. It may also be easily seen from the form of the acoustic tensor (161) that the nonassociativity of the flow rule opens the possibility that a wave involving plastic loading can travel faster than an elastic unloading wave, characterized by the acoustic tensor Ael (n) defined, with reference to an arbitrary vector g, as Ael (n)g =

1 E[g ⊗ n]n. ρ

(162)

This possibility, called ‘achronic state’, occurs when a plastic eigenvalue is larger than any of the elastic eigenvalues and leads to possible dynamical instabilities (Sandler and Rubin 1987; Pucik et al. 2015; Burghardt and Brannon 2015; Brannon 2007).

3.4 Strain Localization into Planar Bands Considering an infinite solid body, subject to a continued path of uniform strain, the condition that an incremental (or rate) strain localizes into a planar band of infinite extent can be analyzed in terms of vanishing speed of a planar plastic wave, c = 0, which corresponds to the singularity of the acoustic tensor A(n), Eq. (161), for at least one direction n, namely det A(n) = 0,

strain localization condition.

(163)

Assuming that the elastic fourth-order tensor E be positive definite and assuming ρ = 1, the condition (163) can be rewritten as   1 (n)p ⊗ q = 0, det I − A−1 g el

(164)

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where p = E[P]n, q = E T [Q]n.

(165)

Using the algebraic property det(I + a ⊗ b) = 1 + a · b, the strain localization condition (164) can be written in terms of a critical value of the plastic modulus gcrE = q · A−1 el (n)p.

(166)

The fact that condition (163), and therefore (166), corresponds to the possibility of a localization of strain can be understood by considering that the emergence of a discontinuity surface (of unit normal vector n) for the strain rate during a continued homogeneous deformation is conditioned by the fulfillment of the following two conditions. • Incremental equilibrium across the discontinuity surface, which requires continuity of the traction rate [[σ]]n ˙ = 0, (167) where the bracket [[·]] denotes the jump across the surface of the relevant argument. • Validity of the Maxwell compatibility conditions, which specify a form for the jump in the gradient of the velocity ∇ u˙ across the discontinuity surface ˙ = g ⊗ n, [[∇ u]]

(168)

where g represents the jump in the normal derivative of the gradient of velocity. The condition (167) has a simple mechanical interpretation, while the condition (168) is more complicated and merits a detailed explanation. A field quantity which is prescribed to remain continuous across a surface, but may admit a discontinuity in its gradient, has to remain continuous when the directional derivatives are taken tangential to the discontinuity surface. Consideration of the tangential derivative leads to Eq. (168). This point can be exemplified with the following two examples. Example 1 (Twinning shear deformation) A deformation such as that sketched in Fig. 23 is often found in crystals and is composed of two mirror-like simple shears, where the symmetry line defines the ‘twinning plane’. An incremental displacement u˙ such as that reported in Fig. 23 has the following form (169) u˙ = |x · f2 |f1 , where f1 is the unit vector defining the twinning line and f2 is its orthogonal complement to define a basis. The gradient of the incremental displacement is ∇ u˙ = sgn(x · f2 )f1 ⊗ f2 ,

(170)

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D. Bigoni

e2

Fig. 23 Twinning shear deformation of plane f1 , across which the incremental displacement u˙ is continuous, but not its gradient

f2 u

f1

x

e1 where sgn(x · f2 ) provides the jump in the derivative orthogonal to the twinning plane. Therefore, the jump in the gradient of incremental displacement is ˙ = 2f1 ⊗ f2 , [[∇ u]]

(171)

which is in the form (168), because f2 is the normal to the discontinuity plane and 2f1 is the jump of the normal derivative of the incremental displacement across the same plane. Note that in the reference system e1 –e2 f1 = cos αe1 + sin αe2 , f2 = − sin αe1 + cos αe2 ,

(172)

condition (170) becomes  ˙ = 2 −e1 ⊗ e1 sin α cos α + e1 ⊗ e2 cos2 α [[∇ u]] −e2 ⊗ e1 sin2 α + e2 ⊗ e2 sin α cos α ,

(173)

an expression which ‘hiddens’ the dyadic structure of the Maxwell compatibility (168). Example 2 An incremental deformation is considered where the incremental displacement depends on the variable x2 only through its absolute value, y = |x2 |, so ˙ 1 , y, x3 ) and its gradient in components is that u˙ = u(x ⎡ ⎤ u˙ 1,1 sgn(x2 )u˙ 1,y u˙ 1,3 ∇ u˙ = ⎣ u˙ 2,1 sgn(x2 )u˙ 2,y u˙ 2,3 ⎦ . (174) u˙ 3,1 sgn(x2 )u˙ 3,y u˙ 3,3 The jump in the gradient across the plane x2 = 0 of unit normal e2 can be written as ⎡

⎤ 0 2u˙ 1,y 0 ˙ = ⎣ 0 2u˙ 2,y 0 ⎦ , [[∇ u]] 0 2u˙ 3,y 0

(175)

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so that the mathematical structure (168) is recovered, namely, ˙ = g ⊗ e2 , [[∇ u]]

(176)

g = 2u˙ 1,y e1 + 2u˙ 2,y e2 + 2u˙ 3,y e3 .

(177)

with

The condition for strain localization (163) can now be derived by combining Eqs. (167) and (168) with the plastic branch of the rate constitutive equation (153). Therefore, the mechanical meaning of the condition (163), implying the possibility of a localization of deformation into a planar band, becomes clear. It should be noted that in the ‘standard’ case of the associative flow rule, P = Q, the acoustic tensor is symmetric. Therefore, the usual situation is that, during a uniform strain path of a material, the hardening modulus evolves from a positive value, vanishes for perfectly plastic behaviour, and finally becomes negative for strain softening. During this evolution, the eigenvalues of the acoustic tensor are initially positive and decreasing functions of the strain hardening, so that when a critical value of the hardening modulus is met, one of these eigenvalues vanishes and the plastic deformation starts to localize into a planar band. After plastic localization occurs, usually (the situation can be much more complicated, see Gajo et al. 2004) the material outside the band starts to elastically unload, while the material inside the band continues to deform plastically. For associative elastoplasticity at small deformation, the critical hardening modulus for strain localization is never positive, so that localization occurs for perfectly plastic or softening behaviour. However, the situation is different both when large deformation effects are taken into account or when the flow rule is nonassociative, in which cases strain localization may occur even during hardening.

3.5 The Analysis of the Acoustic Tensor and Flutter Instability The most important results on flutter instability relative to small strain nonassociative elastoplasticity, based on isotropic elasticity, are due to Loret (1992) and Loret et al. (1990) and are now presented following Bigoni (2012) and Bigoni and Zaccaria (1994). Results relative to nonassociative plasticity with anisotropic elastic behaviour were provided by Bigoni and Loret (1999) and Bigoni et al. (2000). An isotropic elastic tensor E is assumed in the following for a material with unit mass density, ρ = 1, so that the elastic acoustic tensor is Ael (n) = (λ + μ)n ⊗ n + μI,

(178)

where λ and μ are the Lamé constants. The acoustic tensor corresponding to the plastic branch of the constitutive equation (151) is given by

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D. Bigoni

1 A(n) = (λ + μ)n ⊗ n + μI − p ⊗ q, g

(179)

q = λ(trQ)n + 2μQn, p = λ(trP)n + 2μPn,

(180)

where are linear functions of n. Assuming that n × q = 0, the following non-orthogonal dual bases for the three-dimensional Euclidean space are employed e1 = n, e1 =

e2 = q,

e3 =

n×q , |n × q|

(q2 )n − (q · n)q q − (q · n)n , e2 = 2 , e3 = e3 , q2 − (q · n)2 q − (q · n)2

(181)

which satisfy the property ei · e j = δ ij , where δ ij is the Kronecker delta. Projected onto the bases (181), the acoustic tensor writes as ⎤ e1 · Ae1 e1 · Ae2 e1 · Ae3 [A] = ⎣ e2 · Ae1 e2 · Ae2 e2 · Ae3 ⎦ , e3 · Ae1 e3 · Ae2 e3 · Ae3 ⎡

(182)

so that the eigenvalue problem for (179) yields the characteristic equation ⎛

λ + 2μ − η

− g1 p · n

⎜ 1 det ⎜ ⎝ (λ + μ)q · n μ − g p · q − η 0

− g1 p

· e3

0 0 μ−η

⎞ ⎟ ⎟ = 0, ⎠

(183)

where η is the generic eigenvalue of the acoustic tensor (179). The three solutions to the characteristic equation (183) are the eigenvalue μ and the two roots of the polynomial equation:     1 1 η 2 − λ + 3μ − p · q η +(λ + 2μ) μ − p · q g g 1 + (λ + μ)(p · n)(q · n) = 0. g

(184)

Strain localization into a planar band of unit normal n, Eq. (166), occurs when η = 0 in Eq. (184), which corresponds to the following critical condition for the plastic modulus gcrE (n) = −

p·q λ+μ (p · n)(q · n) + . μ(λ + 2μ) μ

(185)

Flutter instability is equivalent to the condition that the discriminant  of the second-order polynomial in Eq. (184) assumes negative values. The discriminant

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can be written as    2 1 gcrE (n) ,  = λ + 3μ − p · q − 4μ(λ + 2μ) 1 − g g

(186)

where gcrE (n) represents the critical plastic modulus for strain localization at fixed n, Eq. (185). Therefore, it can be concluded that (Bigoni and Zaccaria 1994): For a given direction n, flutter is always excluded for values of the plastic modulus less than or equal to the critical plastic modulus for strain localization in a band orthogonal to that direction n.

Straightforward manipulation of the discriminant (186) yields the necessary and sufficient conditions for flutter (n · p)(n · q) > 0, & (n · p)(n · q) − p · q > 0, & g ∈ (g1 , g2 ),

(187)

where g1 g2

0 =

$2  1 # (n · p)(n · q) ± (n · p)(n · p) − p · q . λ+μ

(188)

If deviatoric associativity (154) is assumed, a simple calculation shows that condition (187)2 is never satisfied, which leads to the conclusion (Loret et al. 1990; Brannon and Drugan 1993): For elastic-plastic solids in the presence of isotropic elasticity and deviatoric associativity (154) with parameters χ1 and ψ1 being strictly positive, complex eigenvalues of the acoustic tensor are excluded.

However, coincident eigenvalues are possible. These may be determined by requiring that the discriminant (186) be null, which occurs when one of the following two conditions is satisfied C (n) = − (n · p)(n · q) = 0, and g = gcr

or C (n · p)(n · q) = p · q, and g = gcr (n) =

p·q , λ+μ p·q . λ+μ

(189)

(190)

Assuming isotropic elasticity and deviatoric associativity (154), it is easy to obtain that # $ ˆ · Sn ˆ − (n · Sn) ˆ 2 ≥ 0. (191) p · q − (n · p)(n · q)4μ2 χ1 ψ1 Sn

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D. Bigoni

Therefore C (n) ≤ 0, (n · p)(n · q) = 0 =⇒ p · q ≥ 0 =⇒ gcr

(192)

so that Case (189) is not interesting. Examining Case (190), Eq. (184) provides two coincident solutions equal to μ. Therefore: The acoustic tensor (for certain n) has an eigenvalue equal to μ, with multiplicity 3, when condition (190) is satisfied.

Now the critical plastic modulus for such a coalescence can be determined, noting that the following condition holds true at coalescence ˆ · Sn ˆ = (n · Sn) ˆ 2, Sn

(193)

ˆ but in this case n is also an and is verified if and only if n is an eigenvector of S, eigenvector of E[P] and E[Q]. Therefore, the critical plastic modulus for coalescence of eigenvalues is (E[P])i (E[Q])i , i =1,2,3 λ+μ

C gcr = max

(194)

where E has the isotropic form and the index i, not summed, denotes principal components of E[P] and E[Q] (in the same reference system). To summarize with the above specific example at hand, for deviatoric associativity, complex eigenvalues of the acoustic tensor are excluded, but coalescence of three eigenvalues may be verified. When this coalescence occurs and with reference to the above example, Bigoni and Loret (1999) have shown that a perturbation in terms of a small (appropriate) elastic anisotropy superimposed on the isotropic elastic law is sufficient to trigger flutter. Therefore, even if for the considered model complex eigenvalues are excluded, flutter as induced by physically motivated perturbations is possible and the critical condition corresponds to coalescence of the three eigenvalues of the acoustic tensor. When coalescence is considered, Bigoni and Loret (1999) have shown that this situation may occur even if the constitutive operator is positive definite, a situation similar to what happen with the Ziegler double pendulum, where flutter occurs in the absence of any static bifurcation.

4 Concluding Remarks on Flutter Instability in Structures and Solids Flutter instability has been shown to be related to friction in both structures (where a follower force may be induced by higly anisotropic Coulomb friction) and

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elastoplastic solids (where frictional terms introduce a lack of symmetry in the constitutive equations). For structural systems, numerical simulations which keep into account all nonlinearities (included those related to frictional contact) show that flutter initially corresponds to an oscillation blowing-up in time (as the linearized solution correctly predicts), but later this oscillation reaches a limit cycle, so that the structure behaves as a self-oscillating device. This behaviour is fully confirmed by experiments and can be considered indisputable. For solids the situation is more complex. Available analytical solutions based on linearization predict the possibility of flutter as induced by nonassociativity of the plastic flow. In this context, flutter is understood in terms of waves blowing-up during propagation. Numerical evidences of flutter are still inconclusive both in showing a clear blowing-up and in predicting the achievement of a limit cycle. The blowing-up of a signal during its propagation in a solid should not be too surprising, as materials obeying elastoplasticity with nonassociative flow rule have been shown to produce useful energy in closed loading cycles (Petryk 1985). This circumstance does not necessarily violate conservation of energy, because a release of energy can be produced at the expense of the strain energy stored in the material in connection with the presence of initial prestress (always needed to generate plastic flow). Analysis of constitutive models describing the behaviour of granular materials (Gajo et al. 2004) reveals that flutter instability should be considered more common than one could expect, a fact in agreement with the observation that granular matter is prone to unstable releases of energy. However, due to the effects of nonlinearities, the instability of flutter could be less ‘explosive’ than the exponential blow-up predicted by the linearized analysis. Although experimental results indisputably showing flutter in a continuum are not available, there is evidence of oscillatory instabilities occurring in different situations involving the mechanics of granular materials and pointing to flutter instability. One of these instabilities is responsible of the so-called ‘singing (or squeaking) sand’, a phenomenon known since a long time (it was reported by Marco Polo in his Il milione and by Charles Darwin in his Voyage of the Beagle) and consisting in the emission of an audible sound when certain types of sand are subject to shearing deformation. The grains of singing sands are coated with a silica layer and are of quartz or calcareous nature; the grain-size distribution is often uniform, and the grains roughly rounded. All these elements suggest that the squeaking of the sand is strictly related to intergranular friction, which is indeed the key element accounted for in the current models of this phenomenon (Andreotti and Bonneau 2009; DagoisBohy et al. 2012). Sound emissions have also be noticed during straining of other granular materials such as snow (Patitsas 2015). In all cases the frictional nature of the material and the vibrational origin of the phenomenon suggest a connection to flutter instability in a continuum. Another phenomenon which can be thought to be in relation with flutter instability in solids is the so-called ‘silo music’ and ‘silo quake’, occurring during discharge of granular matter stored in silos. These structures fail with a much higher frequency

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D. Bigoni

Fig. 24 Local buckling in two silos (near Bazzano, Italy)

than other industrial equipments (Carson and Holmes 2003), frequently develop localized buckling (Fig. 24) and may also collapse, leading to loss of use. During the operational life of silos, vibrations and repeated quakes occur at characteristic frequencies, leading to noise and even strong acoustic emission (nicknamed ‘music’). These vibrations have relations with the failure of the structure and therefore have been thoroughly analyzed (Muite et al. 2004; Tejchman and Gudehus 1993; Wilde et al. 2010). Several mechanical effects have been evidenced: (i) resonant interactions between granular matter and silo structure, (ii) formation of arch mechanisms in the granular body and (iii) stick-slip motion at the interface between container and contained. Flutter instability may certainly play a role in this phenomenology. Acknowledgements Financial support from the ERC advanced grant ERC-2013-ADG-340561INSTABILITIES is gratefully acknowledged.

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Q.S. Nguyen, Instability and friction. Comptes Rendus Mécanique 331, 99–112 (2003) Q.S. Nguyen, Stabilité des structures élastiques (Springer, Berlin, 1995) M.P. Paidoussis, Fluid-Structure Interactions, vol. 1, 2, 2nd edn. (Elsevier, London, 2014) A.J. Patitsas, Snow sounds when rubbing or impacting a snow bed. Can. J. Phys. 93, 1302–1309 (2015) H. Petryk, On stability and symmetry conditions in time-independent plasticity. Arch. Mech. 37, 503–520 (1985) A. Pflüger, Stabilitätsprobleme der Elastostatik (Springer, Berlin, 1950) A. Pflüger, Zur stabilität des tangential gedruckten stabes. Z. Angew. Math. Mech. 5(191) (1955) A. Piccolroaz, D. Bigoni, J.R. Willis, A dynamical interpretation of flutter instability in a continuous medium. J. Mech. Phys. Solids 54, 2391–2417 (2006) W. Prager, Recent developments in the mathematical theory of plasticity. J. Appl. Phys. 20, 235–241 (1949) T. Pucik, R.M. Brannon, J. Burghardt, Nonuniqueness and instability of classical formulations of nonassociated plasticity, i: case study. J. Mech. Mater. Struct. 10, 123–148 (2015) E. Radi, D. Bigoni, A. Tralli, On uniqueness for frictional contact rate problems. J. Mech. Phys. Solids 47, 275–296 (1999) B. Raniecki, Ordinary Waves in Inviscid Plastic Media (Springer, Wien, 1976) E.L. Reiss, Column Buckling: An Elementary Example of Bifurcation (W.A. Benjamin Inc, New York, 1969), pp. 1–16 J.R. Rice, The Localization of Plastic Deformation (North-Holland, Amsterdam, 1977), pp. 207– 220 J.R. Rice, A.L. Ruina, Stability of steady frictional slipping. J. Appl. Mech. 50, 343–349 (1983) I. Sandler, D. Rubin, The Consequences of Non-associated Plasticity in Dynamic Problems (Elsevier Science Publishing Co, Amsterdam, 1987), pp. 345–353 F.M.F. Simões, J.A.C. Martins, Instability and ill-posedness in some friction problems. Int. J. Eng. Sci. 36, 1265–1293 (1998) Y. Sugiyama, K. Katayama, S. Kinoi, Flutter of a cantilevered column under rocket thrust. J. Aerosp. Eng. 8, 9–15 (1995) Y. Sugiyama, K. Katayama, K. Kiriyama, B.J. Ryu, Experimental verification of dynamic stability of vertical cantilevered columns subjected to a sub-tangential force. J. Sound Vib. 236, 193–207 (2000) J. Tejchman, G. Gudehus, Silo music and silo-quake experiments and a numerical cosserat approach. Powder Tech. 76, 201–212 (1993) K. Wilde, J. Tejchman, M. Rucka, M. Niedostatkiewicz, Experimental and theoretical investigations of silo music. Powder Tech. 198, 38–48 (2010) H. Ziegler, Die stabilitätskriterien der elastomechanik. Ingenieur-Archiv, XX: 49–56 (1952) H. Ziegler, Linear elastic stability a critical analysis of methods. Z. Angew. Math. Phys. 4, 89–121 (1953) H. Ziegler, On the concept of elastic stability. Adv. Appl. Mech. 4, 351–403 (1956) H. Ziegler, Principles of Structural Stability (Birkhäuser, Basel und Stuttgart, 1977)

Dissipation Induced Instabilities of Structures Coupled to a Flow Olivier Doaré

Abstract Different coupling mechanisms between a flow and a structure are presented in this chapter. They are derived in the context of structural linear dynamics by identifying fluid-induced added masses, dampings or stiffnesses. These dynamical effects may induce different energy transfers between the fluid and the solid that can lead to different instabilities. The objective of this chapter is firstly to classify the different instability mechanisms, secondly to show how some of these mechanisms are triggered by the addition of a classical viscous damping, and finally to use them to convert the kinetic energy of a flow into electricity through a simple example.

In these lecture notes, we are concerned with the instabilities to which are subjected some structures when they are coupled to a flow. These instability phenomena are very common, they can be observed in many engineering domains as in aeronautics, where wings or panels can bear strong deformations in flows at large velocities (Theodorsen 1979), maritime engineering, where long cables, risers, or slender towed structures may oscillate (Païdoussis 2004; Païdoussis et al. 2011), nuclear industry, where one designs structures in the core and heat exchangers to prevent unwanted vibrations (Guo and Paidoussis 2000), civil engineering, where some bridges may be prone to strong oscillations (Païdoussis et al. 2011), as proven by the famous accident of the Takoma-Narrows bridge (Billah and Scanlan 1991). Instabilities due to fluidstructure couplings is also a natural phenomenon observed on plants (De Langre 2008; Gosselin and Langre 2009). The manifestation of these instabilities are hence numerous and can be due to different coupling mechanisms. In a first part of these lecture notes, a review of the different instability mechanisms will be done. A particular attention will be given to the influence of dissipation on the stability. It can be observed that some of these instabilities can be triggered earlier when dissipation is added to the system. This phenomenon may seem counter-intuitive O. Doaré (B) IMSIA, ENSTA ParisTech, CNRS, CEA, EDF, Université Paris-Saclay, 828 bd des Maréchaux, 91762 Palaiseau cedex, France e-mail: [email protected]; [email protected] © CISM International Centre for Mechanical Sciences 2019 D. Bigoni and O. Kirillov (eds.), Dynamic Stability and Bifurcation in Nonconservative Mechanics, CISM International Centre for Mechanical Sciences 586, https://doi.org/10.1007/978-3-319-93722-9_2

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from the point of view of someone working in structural mechanics. The phenomenon is however very common in fluid-structure interaction but also in systems involving gyroscopic forces. In the second part we will address the main features of such phenomena in two simple and representative systems: the two degrees of freedom airfoil and the fluid-conveying pipe. Because of the eventual damage they could cause, most of the research effort has been put on the prediction of these instability in order to prevent their apparition. Recently these instabilities have seen a renewed interest in the context of energy harvesting from flows (Allen and Smits 2001; Taylor et al. 2001; Tang and Païdoussis 2009; Barrero-Gil et al. 2010; Doaré and Michelin 2011). This leads us to consider the opposite objective. In the third part of this chapter, the basic principles of energy harvesting from flow-induced instabilities will then be presented.

1 Dynamics and Instabilities of Structures Coupled to a Flow The analysis of a system coupling a flow and a compliant structure involves two complex physical domains, the fluid and solid domains, and their interaction at their common boundary. The later can be additionally moving, leading to even more complexity. There is active research aiming at developing efficient computational methods in a general case of a large deformations of a solid in a turbulent flow (Bazilevs et al. 2013). In the present lectures, we pursue a different objective. We indeed want to evidence the fundamental instability phenomena and their sensibility to damping. For that purpose we only need to consider the properties of the equilibrium positions of the structure in the context of small perturbations in the fluid and the induced efforts on the structure. Two fundamental systems will serve as model systems in the following: • A rigid body mounted on springs, such that its rigid body dynamics is characterized by one or two degrees of freedom. • A compliant panel or a compliant pipe, modelled as a beam or plate, interacting with an axial flow. With these two systems we will be able to cover most of the fundamental flow-induced instability phenomena one may encounter.

1.1 Cross-Flow Instabilities Rigid-body models In Fig. 1a, c are shown two typical structures in presence of flow. These are slender structures whose axis of slenderness is perpendicular to the flow. From this property these problems derive their name “cross flow” problems. The first example of Fig. 1 consists of a wing oscillating along a given mode of deformation – here a first beam-like mode. An equivalent simplified problem consists of a 2D

Dissipation Induced Instabilities of Structures Coupled to a Flow

(a)

(b)

(c)

(d)

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Fig. 1 Examples of real structures in flows (left) and their equivalent 2D simplified models (right)

profile at a median position on the wing oscillating perpendicularly to a 2D-flow at the same velocity. This approximation becomes questionnable as soon as the profile geometry varies a lot along the span or boundary effects at the clamp or the free end of the wing introduce significant variations of the fluid forces. However, this modeling is able to capture most of the coupling physical phenomena occuring in its 3D counterpart. The second example is a beam or cable that can oscillate in two directions. Its equivalent 2D simplified model is represented by a 2D profile having the same shape as the cross-section of the structure, whose displacements are constrained by the presence of springs. The stiffnesses of the later model the modal stiffnesses of the structure along two particular modes. Here again, this simplification makes sense if we are interested in the main flow-structure physical phenomena arising in this configuration. In Fig. 2 is represented a tentative of an exhaustive set of 2D rigid-body models of any cross section. These are one or two degrees of freedom (DOF) systems, involving three kinds of stiffnesses: a beam-like stiffness for a displacement perpendicular to the flow (cases b, c, e), a beam-like stiffness for a displacement in the the same direction as the flow (cases d, e) or a beam-like stiffness for a torsional deformation (cases a, c). These five examples allow to cover most of the encountered cross-flow-induced instabilities that will be evidenced hereafter. Buckling Instability due to Negative Flow-Induced Stiffness We consider first the case sketched in Fig. 2a. It presented with more details on Fig. 3 in the case

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(a)

(d)

(b)

(c)

(e)

Fig. 2 Five different generic models of oscillating rigid bodies in flows. a One DOF oscillator rotating around the elastic center E, b one degree of freedom (DOF) oscillator translating perpendicularly to the flow, c two DOFs oscillators having translation perpendicularly to the flow and rotation around the elastic center E, d one DOF oscillator translating along the flow direction, e two DOFs oscillator having translations alongside and perpendicularly to the incident flow

Fig. 3 Schematic view of a wing profile in a flow, mounted on a torsionnal spring

of a 2D wing profile. The later is secured to the ground via a torsionall spring of stiffness c, at the point E on the profile, referred to as the elastic center. The moment of inertia around the point E is noted J . The flow pressure distribution around the airfoil induces a force on the profile. This force is decomposed into a drag force FD , in the direction of flow, and a lift force FL , perpendicular to the flow. It is exerted at the aerodynamic center A, which is the reference point where the resultant force created by the aerodynamic pressure exert no moment on the solid. The angle of attack is noted θ, and the reference θ = 0 is chosen such that it corresponds to the static equilibrium of the system when the flow velocity equals zero. The distance AE is noted d, counted positive when A is upstream the elastic center E. For small variations of θ, the lift force contributes to a moment d FL and the contribution of the drag force can be neglected, so that the dynamic equilibrium

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equation governing small variations of the angle of attack is: J θ¨ + cθ = d FL .

(1)

We now make the three following assumption: • The lift force can be determined only through the knowledge of the lift coefficient CL , 1 FL = ρU 2 SC L , (2) 2 where ρ is the density of the fluid, S is the equivalent surface of the object, its chord length times its spanwise length. This is a good approximation for large Reynolds numbers, and if it does not vary too much in the range of considered velocities. • The velocity of the fluid-solid boundary remains small compared to the velocity of the fluid. The dynamics of the fluid hence occurs on small timescales compared to the solid dynamics. One can then consider that the lift coefficient, which is a consequence of the fluid dynamics, depends only on the solid’s position θ. • Small angle of attack variations are considered, hence the lift coefficient can be approached by C L ∼ θ ∂C L /∂θ. In the following, we note C L = ∂C L /∂θ. Equation (1) then writes,   1 J θ¨ + k − ρU 2 SdC L θ = 0. 2

(3)

The influence of the fluid can hence be summarized as an added stiffness, whose sign depends on the sign of C L and d. Consider for instance the classical case of a thin aerodynamic profile, where C L  2π (Dowell et al. 1995). The sign of the added stiffness is only determined by the position of the elastic center with respect to the aerodynamic center. If the former is upstream the aerodynamic center, d > 0, the added stiffness is positive and the presence of flow induces an increase of the system’s natural frequency. Conversely, if the elastic center is downstream, d < 0, the added stiffness is negative, and the total stiffness vanishes when the flow velocity reaches the critical velocity,  2k . (4) Ub = ρSdC L Above this limit value, the equilibrium position θ = 0 is unstable and any perturbation is exponentially amplified with time until the profile reaches a new equilibrium position. This instability is referred to as buckling instability.

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1.2 Dynamic Instability by Negative Flow-Induced Damping The den Artog criterion Let us now consider the case sketched in Fig. 2b. The profile displacement is now a translation perpendicular to the flow direction, noted y. A more detailed sketch of the system of interest is provided in Fig. 4. The mass of the solid is m, and the stiffness of the spring is k. There is also a damping acting on the structure dynamics, modeled by a linear damping coefficient cd . If the same asumptions as in the previous case are made, it is not possible to show that the fluid forces have any influence on the dynamics, because they do not depend on y. We now raise the hypothesis that the velocity of the fluid-solid boundary is negligible compared to the velocity of the fluid, and consider instead that from the point of view ot the flow, the solid is driven by a constant vertical translation speed y˙ . In the reference axis of the moving solid, the flow seen by the solid is now at the velocity U1 = and the angle of attack

 U 2 + y˙ 2 ∼ U,

  y˙ y˙ ∼− . θ = arctan − U U

(5)

(6)

Provided y˙ is not small compared to U , but y˙ 2 is small compared to U 2 , the force coefficient in the vertical direction C y can be approximated by Cy ∼ θ

∂C y y˙ ∂ y˙   ∼− CL + CD . (C L cos θ + C D sin θ) ∼ − ∂θ U ∂θ U

(7)

Inserting the resulting force in the dynamical oscillator equation, one obtains the following:   1  m y¨ + cd + ρU S(C L + C D ) y˙ + ky = 0. (8) 2 The flow influence takes hence the form of an added damping. Its sign is the same as the sign of C L + C D . If the later is positive, the consequence is a positive added damping. If it is negative, the system can present exponentially growing oscillations once the flow velocity is greater than Ug = −

2cd . ρU S(C L + C D )

(9)

This instability is generally referred to as galloping. The criterion for instability is called the Den Artog criterion (Blevins 1990). Drag Crisis Instability We now consider the case of a structural displacement in the direction of the flow, as sketched in Fig. 2d. The dynamical equations governing the displacement x of the profile is

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Fig. 4 Sketch of a profile of any shape moving perpendicularly to the flow

m x¨ + c x˙ + kx = FD ,

(10)

1 FD = − ρ(U + x) ˙ 2 SC D (R Ec ) , 2

(11)

where the drag force FD reads

S being the characteristic section offered to the flow. R Ec is the equivalent Reynolds number in the referential of the moving solid, R Ec =

(U + x)L ˙ , ν

(12)

with ν the kinematic viscosity of the fluid. Considering that x˙  U , the drag force can be expressed at first order as,   ∂C D 1 2 x˙ x˙ FD = − ρU S C D (R E ) + 2 C D (R E ) + R E . 2 U U ∂ RE

(13)

The flow hence induces an added damping which can be negative if, 2C D (R E ) + R E

∂C D < 0. ∂ RE

(14)

If the later condition is satisfied and if the induced negative damping has a greater amplitude than the structural damping c, a galloping type instability can occur. For circular profiles, condition (14) is true in a particular range of Reynolds numbers, as represented in Fig. 5. This phenomenon is called drag crisis and the associated instability is called drag crisis instability.

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Fig. 5 Drag coefficient of a rigid cylinder in cross flow as function of the Reynolds number. Drag crisis occurs between R E = 105 and R E = 106

1.3 Wing Instabilities Due to Mode Coupling The case of a two degrees of freedom profile that can translate and rotate around the elastic center is now considered. This case corresponds to the generic configuration of Fig. 2c, and is sketched with more details in the case of a thin wing profile in Fig. 6. The general case where the aerodynamic center A, the elastic center E and the center of gravity G are three distinct points is considered. As in Sect. 1.1, we consider that the solid’s velocity is negligible compared to the flow velocity, such that the aerodynamic efforts depend only on the solid’s position. The dynamical equations governing small variations of y and θ read then m y¨ + ky + kxθ = FL ,

(15)

J θ¨ + (c + kx )θ + kx y = (x + d)FL ,

(16)

2

where x is the distance G E, counted positively when the center of gravity is downstream the elastic center. In the context of small perturbations, the lift coefficient is again approximated by θC L and the dynamical system (15)–(16) writes 

m 0

0 J

   y¨ k + kx θ¨

kx − 21 ρU 2 SC L c + kx 2 − 21 ρU 2 SdC L

    y 0 = . θ 0

(17)

It can be written in the following classical form, M q¨ + K q = 0,

(18)

where M is referred to as the mass matrix, K the stiffness matrix and q the state vector. Flow forces appear then in the stiffness matrix in the form of one diagonal

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Fig. 6 Schematic view of a wing profile in a flow, mounted on a spring and a torsionnal spring, modeling the dynamics of a wing whose deformation can be along torsional and a flexural modes

Fig. 7 Evolution of ω 2 as function of the flow velocity U . Typical values of the parameters are chosen. Third plot: phase difference between rotational and translational modal components

term and one off-diagonal term. Both terms can be positive or negative, depending on the sign of d and C L . This equations set has a similar form as the classical equations derived by Ziegler (1952) in the case of a double pendulum submitted to a follower force. This case is treated in details in the chapter written by Davide Bigoni. In the the present airfoil model the flow couples flexural and torsional wing modes whereas in the double pendulum a follower force couples the two rotation angles of the pendulum. Looking for harmonic solutions of this system in the form [y, θ]t = V e−iωt , one obtains the following eigenvalue problem, M −1 K V = ω 2 V .

(19)

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If U = 0, one can easily observe that the resolution of this problem consists of calculating the eigenvalues of a symmetric positive definite matrix. Hence, in absence of flow ω 2 is real and positive. The eigenfrequencies are real, and the solution of the problem hence takes the form of undamped oscillations. If U = 0, the matrix M −1 K is not symmetric anymore and the eigenvalues may be positive, negative or complex. An example of the evolution of the eigenfrequencies of this system is presented on Fig. 7. At U = 0, the square of the two eigenfrequencies are real and positive. In a general case where E and G are not the same, the associated modes (eigenvectors) couple movements of rotation and translation. When the velocity is increased, the two eigenfrequencies approach until they become equal at a critical velocity U f ∼ 0.41. Above this velocity, the eigenfrequencies of the system form complex conjugate pairs. The dynamics associated to one eigenfrequency and its corresponding eigenvector reads,  

 y (20) = eωi t V e−iωr t , θ where V is the eigenvector associated to the eigenvalue ω 2 and the subscripts r and i refer to the real and imaginary parts respectively. In Eq. (20), one recognizes an oscillation at the frequency ωr whose amplitude varies as eωi t . Hence, a positive imaginary part of the eigenfrequency indicates an instability. The amplitude of the oscillations grows exponentially with time, until eventually non linear effects not taken into account in the present model saturate this growth phenomenon. This instability arises when two structural modes are coupled by the flow in the stiffness matrix if this coupling is skew-symmetric. It is hence referred to as coupled mode flutter. It is also sometimes called frequency coalescence instability because theoretically and experimentally, it is preceded by a rapprochement of two eigenfrequencies. In the third plot Fig. 7 the phase difference between the rotation component and the translation component of the eigenvetor is plotted for the two modes as function of the flow velocity. Below the critical velocity, the phase difference equals zero or π. Hence translation and rotation are in phase or in phase opposition. Above the critical value we see that the phase varies slowly. The unstable eigenmode has a corresponding positive phase difference, while the damped mode has a negative phase difference. In order to get insight on the link between the phase and the instability, let us analyse the energy transfers between the flow and the structure by comparing the work done by the lift force on the system on both modes: the temporally amplified mode and the temporally damped mode. The kinematics of the system along these two modes is given by the real part of Eq. (20). It its sketched in Fig. 8. In case (a), the airfoil oscillates along a mode that is temporally amplified. It can be observed on this figure that the lift force F p and the velocity of the profile y˙ are in the same direction. The work done by the fluid on the structure is hence positive if integrated on a period. Conversely, for a damped mode, the work done by the flow on the structure is negative. In case (a), the rotation θ is ahead of phase compared to the displacement

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(a)

(b)

Fig. 8 Typical kinematics of the airfoil when oscillating along an unstable mode (a) and a stable mode (b)

Fig. 9 Sketch view of a fluid-solid interface. Locally, it consists of a wall which is plane when unperturbed, below a flow at the velocity U aligned with the wall

y, while in case (b), it is lagging behind. This is in agreement with the phase plot in Fig. 7.

1.4 Axial Flow Problems In Sect. 1.1, focus has been put on slender structures coupled to a flow in a direction perpendicular to its slenderness direction. This was molelled considering discrete structures coupled to a 2D-flow. The aerodynamic effects have been described only through the knowledge of lift and drag coefficients, which eventually depend on the discrete kinematic variables and the Reynolds number. We want now to address another generic case which consist of a compliant slender structure interacting with a flow in the direction of the slenderness. Flow effects on a moving boundary As a first approach we will consider the deformation of a simple surface in contact of a potential and irrotational flow. We will show that the flow can induce added inertia, damping and stiffness.

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Consider the system represented in Fig. 9. It consists of a compliant boundary that locally deforms below a potential and irrotationnal flow. The velocity and pressure fields in the flow are U and P respectively. In the reference configuration where the boundary is undeformed, it is plane and the fluid has locally an homogeneous and constant flow velocity U0 e x . Consider a solid’s displacement of small amplitude of the form, (21) w = εwe y , where ε  1 is a small amplitude parameter. At the fluid-solid interface, the normal velocity of the fluid has to be equal to the normal velocity of the solid, U .n =

∂w .n ∂t

(22)

Each quantity in this boundary condition may be decomposed as a Taylor expansion at order one, ∂w e , ∂x x U = U0 e x + εu, ∂w ∂w =ε e . ∂t ∂t y n = ey − ε

(23) (24) (25)

At order zero, the kinematic boundary condition U0 (e x .e y ) = 0 is readily satisfied, while at order one, it takes the following form, u.e y =

∂w ∂w + U0 , ∂t ∂x

(26)

which is also known as the impermeability condition (Païdoussis 2004). The velocity potential  is now introduced, grad  = U . (27) The later may be also decomposed between a stationary part and a fluctuation of small amplitude, (28)  = U0 x + εψ, with grad ψ = u.

(29)

Introducing (29) in the incompressibility condition divu = 0, we show that the velocity potential satisfies a Laplace equation, ψ = 0,

(30)

Dissipation Induced Instabilities of Structures Coupled to a Flow

75

with the following boundary equation: ∂w ∂w ∂ψ = + U0 . ∂y ∂t ∂x

(31)

We now make use of the momentum conservation equation for an inviscid and incompressible flow, ρ

 ∂U

+ grad.U U + gradP = 0. ∂t

(32)

At order zero, this equation becomes,

 ρ grad U0 e x . gradψ0 + grad p0 = 0.

(33)

After integrating with respect to space, this law reads, 1 ρ U02 + p0 = ct, 2

(34)

ct being an integration constant. This is the Bernoulli equation satisfied by the permanent quantities of the flow. In our particular case of an homogeneous flow, this is readily satisfied since both velocity and pressure are constant in the whole fluid domain. At order one, the momentum conservation law can then be expressed in the following form, ρ grad

 ∂ψ + ρ grad U0 e x . gradψ + grad p = 0. ∂t

(35)

The later expression can be integrated with respect to space, which leads to the unsteady Bernoulli equation, p = −ρ

∂ψ ∂ψ − ρU0 . ∂t ∂x

(36)

This equation links the pressure in the fluid to the velocity potential. It has to be noted that no integration constant has been kept here, since we are only dealing with the fluctuations. We are then faced to a Laplace problem for the velocity potential (30) with the boundary condition (31). Once the potential ψ is obtained for a prescribed displacement, the pressure is calculated using the instationary Bernoulli equation (36). Let us consider a displacement as the product of a time and space functions, w(x, t) = q(t)φ(x).

(37)

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O. Doaré

Thanks to the linearity of the fluid-solid problem, the solution for the velocity potential is looked for as a sum of two contributions, ψ = U 0 ψ1 + ψ2 ,

(38)

ψ1 and ψ2 satisfying the Laplace equation, with the following boundary conditions, ∂ψ1 ∂w = , ∂y ∂x

∂ψ2 ∂w = . ∂y ∂t

(39)

The form of the boundary conditions indicates that ψ1 and ψ2 can be chosen of the form, ∂q(t) ϕ2 (x). ψ1 = q(t)ϕ1 (x) , ψ2 = (40) ∂t Finally, after applying the unsteady Bernoulli equation (36) on the potential (38), the pressure is found in the form of the sum of terms proportional to the acceleration, the velocity and the displacement of the fluid-solid interface,   ∂ϕ2 q˙ − ρU02 ϕ1 q. p = −ρϕ2 q¨ − ρU0 ϕ1 + ∂x

(41)

The pressure in the fluid appears as the sum of three contributions, whose time dependence are respectively proportional to the acceleration, the velocity and the displacement of the plate. Hence, once eventually projected on the solid’s displacement these terms will induce added mass, damping and stiffness. When the kinematics is decomposed along several modes φn , n = 1 . . . N , the linearity of the fluid problem allows to decompose the pressure perturbation in the fluid on pressure modes pn , N = 1 . . . N . Each pressure mode may exert a forcing on each displacement mode. As a consequence, the flow will couple the different modes of the structure through stiffness, damping and inertia terms. Slender structures in presence of axial flow The general deformation introduced in Sect. 1.4 can be sought as a local deformation of the interface of any compliant solid. From now on, we will focus on this particular geometry, which is representative of a large number of systems, for instance plates, flags, cylinders in axial flows, pipes with internal flows. Two examples will be treated: a vibrating plate with a flow on one side (Fig. 10a) and a pipe conveying fluid (Fig. 10b). For these two systems we make the assumption that the solid part is a slender beam or plate subjected to a transversal deflection w(x) of small amplitude, such that it can be modeled under the Euler–Bernoulli beam approximation, B

∂2w ∂4w + m = f (x, t), + boundary conditions. ∂x 4 ∂t 2

(42)

Dissipation Induced Instabilities of Structures Coupled to a Flow

77

(a)

(b)

Fig. 10 Slender structures interacting with an axial flow. a Infinite span plate below an inviscid and irrotationnal flow at velocity U . b Pipe conveying fluid

For the plate sketched in Fig. 10a, B is the flexural rigidity of a plate of thickness h, made of a material of Young’s modulus E and Poisson’s coefficient ν, B = Eh 3 / (1 − ν 2 ), m is the surface density of the plate and f (x, t) is the opposite of the fluid pressure at y = 0: f = − p(x, y = 0, t). (43) For the fluid-conveying pipe sketched in Fig. 10b, B is the flexural rigidity of a beam of annular cross section (Blevins 1990) and m is the lineic density of this beam. The righthand term f (x, t) is now a force per unit length which reads, f =



− pn.e y dL ,

(44)

where  is the fluid-solid boundary at x, represented by a closed line, and n is the unitary vector normal to the boundary, directed from the solid to the fluid. For a pipe of circular section of radius R,  would be the circle of radius R, centered a (x, y = 0, z = 0). Dispersion relation The study of stability properties of slender structures like beams and plates is first performed with respect to the waves propagating in such structures. Considering a disturbance in the form of a plane harmonic wave w = w0 ei(kx−ωt) ,

(45)

solving the Laplace problem (30) for the velocity potential with boundary condition (31) and using the unsteady Bernoulli equation (36) gives the pressure in the fluid generated by the solid’s perturbation,

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O. Doaré

p=

w0 (ω − U k)2 e−|k|y ei(kx−ωt) . |k|

(46)

The fluid-solid force f in Eq. (42) is then f = − p(x, y = 0, t). Inserting this force and the displacement in the form of the harmonic plane wave in Eq. (42) leads to the dispersion relation, an algebraic equation that links the wavenumber k to the frequency ω, ρ (ω − U k)2 = 0. (47) D(k, ω) = Dk 4 − mω 2 − |k| The same calculation can be performed in the case of the circular geometry of the fluid-conveying pipe, where the same fluid equations are satisfied, with the same kind of boundary conditions, but on an interface of different geometry ( contour of Eq. (44). Solving this equation leads to the following expression for the pressure in the fluid domain, p = ρ(ω − U k)2

2I1 (kr ) sin θei(kx−ωt) . k[I0 (k R) + I2 (k R)]

(48)

Using Eq. (44), we obtain the following expression for f , f = ρ(ω − U k)2

2π R I1 (k R) w0 ei(kx−ωt) , k[I0 (k R) + I2 (k R)]

(49)

which takes the following form if the wavelength is large compared to the radius of the pipe (k R  1): f = ρS(ω − U k)2 w0 ei(kx−ωt) . (50) The dispersion relation for the slender (k R  1) fluid-conveying pipe finally writes, D(k, ω) = Bk 4 − mω 2 − ρS(ω − U k)2 = 0.

(51)

If one considers the deformation of a slender structure of any other cross section, with internal or external homogeneous potential flow, the fluid force f is still of the form f (x, t) = a(k)(ω − U k)2 ei(kx−ωt) , the function a(k) being a characteristic function of the cross section geometry of the system. In the three-dimensional cases, the function a tends to a constant value A when k R → 0, R being the characteristic dimension of the cross section. In the case of the fluid-conveying pipe, a(k) can be deduced from Eq. (49) and A = ρS. Other systems can be considered, giving different expressions for a(k) and A, a 3D flag (Eloy et al. 2007), a 2D confined flag (Guo and Paidoussis 2000), a 3D flag confined in the spanwise direction (Doaré et al. 2011), a cylinder with external flow (de Langre et al. 2007). Instability of waves propagating in infinite 1D media: Local analysis We want now to perform a stability analysis of these media with respect to wave propagation. An infinite length medium is said to be stable if, for any sinusoidal wave of infinite

Dissipation Induced Instabilities of Structures Coupled to a Flow

79

extent in the x-direction and associated to a real wavenumber k ∈ R, the corresponding frequencies given by D(k, ω) = 0 are such that the displacement remains finite in time. The local instability criterion is then, Instability if ∃ k ∈ R \ Im[ω(k)] > 0.

(52)

This approach is said to be temporal, since it consists of examining the temporal evolution of waves in time.1 The behaviour of ω when k is varied is represented in Fig. 11 for the plate and pipe cases and for different values of the flow velocity U . For both systems, as soon as the flow velocity is not zero, there exist a range [0, kc ] of wavenumbers for which one corresponding frequency has a positive imaginary part. Consequently, these media bear unstable waves at any flow velocity. In fact decreasing or increasing the flow velocity do not change the stability properties, but has an effect on the characteristic wavelengths of the unstable waves. kc increases as U increases; the higher is the velocity, the smaller is the smallest unstable wave. This will have practical consequences when we will study the stability of finite length systems. These aspects will be regarded in more details in Sect. 2 at the same time we will address the influence of structural dissipation. Unstable modes in finite length systems: Global analysis The global analysis considers the same local medium, but in a finite domain  = [0, L], associated with a set of boundary conditions, denoted as Bi (y) = 0, i = 1 . . . N , where N is the maximal order of the spatial derivatives in the local equation. In the present elastic beam or plate problem, N = 4 and the boundary conditions are classical beam theory boundary conditions, which, among other, can be clamped,

pinned,

or free,

∂w

, w(x0 , t) = ∂x (x0 ,t)

(53)

∂ 2 w

, w(x0 , t) = ∂x 2 (x0 ,t)

(54)

∂ 3 w

∂ 2 w

= , ∂x 2 (x0 ,t) ∂x 3 (x0 ,t)

(55)

1 If one is interested in the response of the medium to localized disturbances, the spatial approach is

more relevant. Conversely to the temporal approach it consists of the calculation of wavenumbers associated to one frequency. For plates or beams in presence of flows, the dispersion relation takes the form of a fourth-order polynomial in k in case of slender structures or more complicated forms in more general cases (see Eq. 49 for example). The response of unstable media to localized disturbances involves studies of the dispersion relation using the spatial approach in the complex ω− and k− planes which are beyond the scope of the present lecture notes.

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O. Doaré

Fig. 11 Real and imaginary parts of ω as function of k for different values of the flow velocity U (blue:0.7, green:1, red:1.5) in the plate and pipe cases

Dissipation Induced Instabilities of Structures Coupled to a Flow

81

where x0 = 0 or L is the location of the boundary condition. The linear dynamics of this slender structure is governed by a wave equation of the form, ∂ ∂2 M [w(x, t)] + C [w(x, t)] + K [w(x, t)] = 0 on  = [0, L], ∂t 2 ∂t

(56)

where M, C and K are mass, damping and stiffness operators respectively. In the fluid-conveying pipe case, their respective expression can be easily recovered after Eq. (51): M = m + ρS , C = ρSU

∂ ∂x

K=B

2 ∂4 2 ∂ + ρSU . ∂x 4 ∂x 2

(57)

From now on, we will focus on the fluid-conveying pipe model system. Other systems such as plates, flags or cylinders in axial flow are governed by very similar equations and present very similar stability properties. Considering ansatz solutions of the form y = φ(x)e−iωt and inserting this in Eq. (56), one obtains a Sturm-Liouville problem which solution is represented by an infinite set of eigenfunctions φn (x) and eigenfrequencies ωn .2 The instability condition reads then: (58) Instability if ∃ ωn \ Im[ωn ] > 0. In most cases, numerical methods are necessary to solve this kind of problems. We present here a so-called Galerkin method to compute approximate solutions. The solution w(x, t) is decomposed on a truncated function basis that satisfies the boundary conditions, N  φn (x)q(t). (59) w(x, t) = n=1

In practice, a good choice of eigenfunctions can be a subset of operators in Eq. (56) with the same boundary conditions, for which an analytical solution exists. A natural choice for slender beams in flows are the eigenmodes of the same beam in vacuum. After defining a scalar product,  f, g =

f gdx,

(60)



the approximated form of w defined in Eq. (59) is introduced in (56), which is next projected on a mode φm (x). One finally obtains a discrete set of coupled oscillator 2 Note

that most of the systems considered here are non self-adjoint. Hence, a complete solution set consists of a biorthogonal set of eigenfunctions and adjoint eigenfunctions. This has a practical importance when one wants to analyse the response of such non conservative problems to external forcings.

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equations,

M q¨ + C q˙ + K q = 0.

(61)

The coefficients of the matrices M, C, and K result from the projection of the inertia, damping and rigidity operators of Eq. (56), Mmn = M(φn ), φm , Cmn = C(φn ), φm ,

K mn = K(φn ), φm .

(62)

Hence, one ends up with an approximate equivalent of the continuous, finite system, in the form of a finite degree of freedom mechanical system. The discrete mechanical equations presented in Sect. 1.1 and that obtained by Galerkin discretisations of Eq. (56) all fall into the category represented by the general discrete equation (61). The analysis of such systems is done by looking for solutions in the form of time-harmonic solutions of the form q = q0 e−iωt . This leads to a second order eigenvalue problem for the eigenfrequency. The criterion for global instability is then given by (58). If the studied discrete system comes from a Galerkin discretisation (Sect. 1.4), the corresponding eigenvector represents the eigenmode in the form of a combination of functions φn (x). Similarly to the discrete structures in flows presented in Sect. 1.1, the fluid conveying pipe is prone to buckling or flutter instabilities, which are intrinsically related to the symmetries of matrices C and K . In particular, depending on the boundary conditions, the flow added stiffness term of Eq. (57) can contribute to symmetric or skew-symmetric parts in Eqs. (61)–(62). This will be regarded in more details in the next section at the same time we consider the effect of damping on stability.

2 Damping Induced Instabilities of Structures Coupled to a Flow The effect of damping in the fluid-structure system is now addressed. It is a conventional linear damping, as opposed to the flow-induced damping evidenced in some cases of the previous section. It can be easily predicted that only one type of flow-induced instabilities can be eventually enhanced after introduction of damping: • In the negative flow-induced damping instabilities, the conventional damping counteracts the flow effects and prevents instability. • The negative flow-induced stiffness instability is a static instability that is not affected by velocity dependent forces. • The skew-symmetric coupling in the stiffness matrix induces a dynamic instability that might be triggered earlier by the addition of damping. This is the case detailed hereafter. Hence, wing flutter results will be presented. Next, the fluid conveying-pipe will be considered as a model problem of a continuous system that bears unstable waves, but

Dissipation Induced Instabilities of Structures Coupled to a Flow

83

Fig. 12 Evolution of ω as function of the flow velocity U for the two-DOF wing with added dissipation. Typical values of the other parameters are chosen

also bears unstable modes when boundaries are added. This system will be used to travel from waves to modes, and next from modes to discrete oscillators and see the link between the local and global stability properties with respect to the dissipationinduced instabilities.

2.1 Damping Induced Instabilities of Wings The two degrees of freedom wing in flow has been introduced in Sect. 1.3. It was shown that the flow couples the two degrees of freedom of the system. If this coupling is skew-symmetric, eigenfrequencies form complex conjugate pairs and induce an instability of the flutter type. The equations governing the dynamics of this system are identical to that of the Ziegler pendulum (Ziegler 1952). It’s stability is addressed in details in the chapter written by Davide Bigoni. Mathematical aspects are developed in the chapter written by Oleg Kirillov. The damping induced instability of wings is hence only briefly presented here in a typical example. Consider that a dissipative force is added to the system on the translation degree of freedom. A dissipation matrix is added to system, which now writes, 

m 0

0 J

   y¨ γ + 0 θ¨

0 0

  y˙ θ˙

(63)

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 +

k kx

kx − 21 ρU 2 SC L c + kx 2 − 21 ρU 2 SdC L

    y 0 = . θ 0

(64)

As discussed in Sect. 1.3, the instability is linked to the energy transfers between the flow and the structure, which is in turn related to the respective phases of translation and rotation. Adding damping modifies the relative phases of these degree of freedom an may hence have a destabilising effect. The evolution of the eigenfrequencies when the flow velocity is increased from zero is shown in Fig. 12 for the same parameters as in Fig. 7 with an additional damping γ = 0.1. The occurence of a positive imaginary part eigenfrequency is observed at a velocity U ∼ 0.18 which is much lower than that of the case without dissipation, U ∼ 0.41 (see Sect. 1.3).

2.2 The Fluid-Conveying Pipe Model System We consider again the dynamic equation of the pipe with internal flow presented in Eqs. (56)–(57). As already mentioned, this is a simple and general equation describing the linear dynamics of a slender structure interacting with an axial flow. Many other systems share the exact same equations, provided typical lengthscales flexural deformations are large compared to the transverse dimensions of the structure (see discussion after Eq. 50). It consists of an Euler–Bernoulli beam with additional terms taking into account an internal flow of an inviscid fluid. It is written again here using capital letters for all the variables in order to allow a future non dimensionalisation of the equations, B

2 ∂2Y ∂2Y ∂2Y ∂4Y 2∂ Y =0 + m + ρS + ρSU + 2ρSU ∂X4 ∂T 2 ∂T 2 ∂X2 ∂ X ∂T

(65)

The first two terms in this equation are the flexural rigidity and inertia terms of the linearized Euler–Bernoulli equation. The third term is an inertia term that comes from the presence of the fluid inside the pipe. The fourth term may be understood as a centrifugal term that arises as soon as the beam experiences a local curvature. Finally the fifth term is generally referred to as a Coriolis force and may be interpreted by considering a portion of the pipe moving at a constant velocity. Due to the presence of a moving mass inside, a force is exerted on this portion of the pipe when it rotates. When damping is to be considered, one may add one or both of these two additional forces to the wave equation, D f (Y ) = C

∂5Y ∂Y , Ds (Y ) = B ∗ . ∂T ∂ X 4 ∂T

(66)

The first case is referred to as viscous damping and is generally a consequence of the presence of a viscous fluid at rest around the pipe. The second is called

Dissipation Induced Instabilities of Structures Coupled to a Flow

85

structural damping and is the consequence of a viscoelastic behavior of the material that constitutes the pipe. 1/2

B and time τ = After introducing the characteristic length η = ρSU 2 

1/2 (m+ρS)η 4 and rescaling all dimensional quantities using these two parameters, B the wave equation without dissipation takes a form that depends only on one independent parameter β,  ∂2 y ∂4 y ∂2 y ∂2 y = 0, + + + 2 β ∂t 2 ∂x 4 ∂x 2 ∂x∂t where β is the mass ratio, β=

ρS ∈ [0, 1]. m + ρS

(67)

(68)

Note that the case β = 0 is strictly equivalent to a cantilevered beam with a follower force, referred to as the Beck’s column (Beck 1952), which is a continuous equivalent to the Ziegler’s pendulum (Ziegler 1952). The opposite case β = 1 is strictly equivalent to the problem of travelling webs or chains (Asokanthan and Ariaratnam 1994). In their dimensionless forms, the terms operators now write, d f (y) = c

∂y ∂5 y , ds (y) = α 4 . ∂t ∂x ∂t

(69)

When considering a finite length system, the non-dimensional length has to be introduced,  L ρS l = = UL , (70) η B and two boundary conditions have to be specified at each boundary. They read y(x = x0 , t) = y  (x = x0 , t) = 0 ,

y  (x = x0 , t) = y  (x = x0 , t) = 0,

(71)

for a clamped end and a free end respectively, where the primes (.) denote derivation with respect to x and x0 takes the value 0 or l. Local stability In our study of the stability properties of the fluid-conveying pipe system, let us first address the stability of waves in the infinite medium, an approach referred to as local. It directly follows from Eq. (67) that the dispersion relation of the undamped fluid-conveying pipe reads,  D(k, ω) = k 4 − ω 2 + k 2 + 2 βkω = 0. The frequency associated to a real wavenumber k then reads,

(72)

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ω± =



 βk ± k β + k 2 − 1.

(73)

√ For β ∈ [0, 1[ and k ∈ [0, 1 − β], frequencies ω± are complex conjugate and the positive imaginary part of one of them √ gives rise to a wave with an amplitude exponentially growing in time. For k > 1 − β, ω(k) ∈ R and waves are said neutral. Thus the medium is locally unstable ∀β ∈ [0, 1[. Conversely for β ≥ 1, the medium is neutrally stable.3 In various studies on the effect of damping on wave propagation, the key role of the wave energy has been evidenced. Although introduced in the context of shear layer waves between two non miscible fluids (Cairns 1979), the definition is generic and can be readily used in any mechanical system. Consider an harmonic wave with ω ∈ R and k ∈ R and D(k, ω) = 0. The wave energy E is defined as the work done on the system to establish this neutral wave from t = −∞ to t = 0, and reads, E =−

ω ∂D 2 y . 4 ∂ω 0

(74)

If E is negative, it means that energy has to be removed from the system to establish the wave. The latter is then referred to as a Negative Energy Wave (NEW) (von Laue 1905). Now consider that a small amount of viscous damping is added in the medium, so that the dispersion relation takes the form, D1 (k, ω + δω) = D(k, ω + δω) − ic(ω + δω) = 0,

(75)

where δω  ω is a small perturbation to the frequency introduced by the damping, which satisfies at order one,

∂ D

 icω. (76) δω ∂ω (k,ω) We readily deduce from this expression the perturbation on the growth rate δσ = Im(δω), cω . (77) δσ  ∂ D/∂ω This quantity has the opposite sign of the wave energy E. A NEW is hence destabilized by viscous damping. The same calculation performed with viscoelastic damping gives: αk 4 ω δσ  , (78) ∂ D/∂ω which leads us to the same conclusions. 3 However, it has to be noted that β > 1 has no physical meaning in the present context (see the definition of β in Eq. 68).

Dissipation Induced Instabilities of Structures Coupled to a Flow 1 Unstable range with damping

0.8

0.6

k

Fig. 13 Range of unstable wavenumbers as function of β. This illustrates the fact that the range of √ wavenumbers 1 − β is stabilized by the Coriolis force and this range is then destabilized when damping is added in the medium

87

0.4 Unstable range without damping

0.2

0 0

0.2

0.4

0.6

0.8

1

β

In the fluid-conveying pipe case, the wave energy has for expression,

   1 2 2 k2 + β − 1 ± β , (79) k k +β−1 2 √ and E − has negative values in the range k ∈] 1 − β, 1[. Hence, the range of temporally √ unstable waves becomes [0, 1[ when damping is added, whereas it was k ∈ [0, 1 − β] in the conservative case. Damping enlarged the range of unstable wavenumbers. Moreover, the system is now temporally unstable for any value of β, when it was for β ∈ [0, 1[ in the conservative case. In Fig. 13, the ranges of unstable wave in the damped and undamped cases are compared when the parameter β, quantifying the Coriolis force, varies from 0 to 1. It can be concluded from this figure that Coriolis force stabilizes waves, which are in turn destabilized by a small amount of damping. The same kind of behavior was observed in discrete gyroscopic systems by Thomson and Tait (1879). Global stability Boundary conditions and finite length parameter l are now introduced. The dimensionless parameter l in Eq. (70) is proportional to both L and U , indicating that it can be seen as a dimensionless length or flow velocity. Although the limit l = 0 has no meaning when it is sought as a length, it can be achieved by letting the flow velocity vanish. In the finite length global approach, it is then more convenient to use L to rescale the lengths, so that the dimensionless wave equation becomes, 2  ∂2w ∂2w ∂4w 2∂ w = 0, (80) + + u + 2 βu ∂t 2 ∂x 4 ∂x 2 ∂x∂t E± =

where the length of the dimensionless problem is the unity and with, u = l.

(81)

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Fig. 14 Evolution in the complex plane of the eigenfrequencies of the clamped-clamped and clamped-free pipe when the velocity parameter v is increased from zero, for different values of the mass ratio β

The Galerkin method presented in Sect. 1.4 is used to obtain the results presented hereafter. The chosen test functions are the eigenmodes of the pipe without flow. The functions φn (x) are hence the eigenfunctions of Eq. (80) with the same set of boundary conditions and u = 0. These eigenfunctions are basically the eigenmodes of a beam and are known and well documented analytic functions (see for instance the book by Blevins 1979). Equation (80) is then projected on each mode φm , leading to N ordinary differential equations for the time variable, which read,  q = 0, q¨ + 2 βv C q˙ + (K + v 2 A)

(82)

where q is the vector containing modal displacement, as defined in Eq. (59). The coefficients of the matrices C, K and A result from the projection of the Coriolis, flexural stiffness and centrifugal operators respectively. Note that K is diagonal because the choosen test functions diagonalise the flexural stiffness ∂4 operator ∂x 4 . The coefficients of these matrices can be found in the litterature (Gregory and Païdoussis 1966; Blevins 1979; Païdoussis et al. 2011). They are re-

Dissipation Induced Instabilities of Structures Coupled to a Flow Fig. 15 Real and imaginary parts of the two first eigenfrequency of the pipe as function of the non dimensional flow velocity u for 3 different values of β

89

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Table 1 Coefficients of the added damping matrix C and the added stiffness matrix A of Eq. (82). More details on these coefficients, for various sets of boundary conditions can be found in the reference book of Blevins (1979) Clamped-clamped Clamped-free Cii Ci j

0

2 4λi2 λ2j 

λ4j

4

− λi

 (−1)i+ j − 1

Aii

λi σi (2 − λi σi )

Ai j

4λi2 λ2j [λ j σ j λ4j − λi4

λi σi

4 (λi /λ j )2 + (−1)i+ j λi σi (2 − λi σi )

  − λi σi ] (−1)i+ j + 1

4.73004, 7.85320, 10.99560, 14.13717, 17.27876, (2i + 1) π2 for i > 5 0.982502, 1.000777, 0.999966, 1.000001, 1 for i > 4

4(λi σi − λ j σ j ) (λi /λ j )2 + (−1)i+ j 1.87510, 4.59091, 7.85476, 10.99554, 14.13717, (2i − 1) π2 for i > 5 0.73, 1.018, 0.9992, 1.00003,0.99999, 1 for i > 5

ported in Table 1 for two particular sets of boundary condition: clamped at both ends, and clamped uptream, free downstream. It is interesting to note here that the symmetries of the fluid-structure coupling matrices depend on the boundary conditions. In the clamped-clamped case, C is skew-symmetric and K f is symmetric, while in the clamped-free case, C and K have both symmetric and skewsymmetric parts. This has consequences on the bifurcations properties of these two systems. To illustrate the influence of the boundary conditions and the matrix symmetries, the evolution of the eigenfrequencies when v is increased from 0 is plotted in Fig. 14 in four typical cases: a pipe clamped at both ends, a clamped-free pipe at β = 0, and a clamped-free pipe at β = 0.2 and β = 0.95. These graphs illustrate the typical behaviors of the eigenfrequencies when the flow velocity is increased. Different bifurcations are evidenced. In the case of a clamped-clamped pipe, instability always arises through a saddle-node bifurcation, also called pitchfork bifurcation. The instability is called static instability, or buckling. In the case of a clamped-free pipe, the bifurcation depends on the value of the mass ratio β. If β = 0, the dissipation matrix vanishes in Eq. (82) and the instability occurs via a Hopf bifurcation after the merging of two eigenfrequencies on the real axis. In the fluid-elastic community, this instability is referred to as flutter instability, as it results in self-sustained oscillations of the structure once the amplitude of the solution has been saturated by the nonlinear effects. When β = 0, the increase of the flow velocity has at first a stabilizing effect, related to the flow-induced damping (C matrix): all the eigenfrequencies travels towards the negative imaginary part half plane. When further increasing the flow velocity, one eigenfrequency changes its trajectory and crosses the real axis, giving rise to a flutter instability. The damping effect of the Coriolis force is further illustrated in Fig. 15, where the evolution of the first two eigenfrequencies is plotted using a

Dissipation Induced Instabilities of Structures Coupled to a Flow

(a) 30 25

91

(b) 20 Global instability (flutter)

Global stability

15 c = 10 Marginal stability without damping

15

l

l

20 10

c = 100

10 5 5 0 0

c = 1000

Local Local instability stability

1

2

3

0 0

0.5

1

β

1.5

2

β

Fig. 16 a Marginal global stability curve of the pipe conveying fluid in the parameter plane (β, l) (thick black) compared with the local stability criterion; b Marginal global stability curves for increasing viscous damping 20

15

l

Fig. 17 Global stability curves of the pipe conveying fluid in the (β, l) plane for increasing values of the viscous damping (plain and dashed lines), compared with the length criteria defined in Eq. (83) (dash-dotted lines)

10

5

0

0

0.5

1

1.5

2

β

different representation as in Fig. 14. For β = 0, it is clearly visible with both representations that the bifurcation occurs through a pinching of two eigenfrequencies on the real axis, and form a complex conjugate pair after the bifurcation. When β > 0, Fig. 14 shows that the instability occurs when a single eigenfrequency crosses the real axis. Figure 15 indicates that the eigenfrequencies are still approaching but their trajectories are deformed and they never form a dooble-root pair at the occurence of the instability. Let us now address the effect of dissipation on the marginal stability of the clamped-free pipe. The global stability of the system is now characterized by plotting in the (β, l) plane the marginal stability curve in Fig. 16a for c = 0 (no damping). This curve corresponds to the line in the (β, l) plane where the maximum growth rate maxn [Im(ωn )] = 0. In the same figure, the local stability criterion β = 1 is plotted. It appears then that the long system limit for global instability is the local stability criterion. In Fig. 16b, different values of the dimensionless damping c from 0 to 1000

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are considered. The resulting marginal stability curves move continuously from the undamped limit to an horizontal limit. For β ∈ [0, 0.2], the damping appears to have a stabilizing effect while it has a destabilizing effect for β > 0.2. Hence also for the finite length clamped-free pipe, damping can have a destabilizing effect. While in absence of damping, the global instability criterion of the finite length tends to the local one when l is increased, no such limit can be observed in the damped case because the damped medium is locally unstable ∀β. However, the horizontal limit observed at high values of the damping cannot be predicted by a local criterion, as addressed in the next section. Lengthscale criterion We have discussed the emergence of instabilities and their dependence on the boundary conditions, through the matrix symmetries. We want here to show that provided we know the system can be destabilised by flutter, a simple lengthscale criterion can be invoked to predict the critical instability parameter. Firstly, let us note that in Fig. 16a, for the values of the parameters comprised between the local and global stability curves, and below the dashed lines in Fig. 16b the system is globally stable although locally unstable. In this situation, the confinement induced by considering a short system has for consequence to prevent unstable waves to play a role in the dynamics. This confinement effect can be quantified and can give an approximate criterion of stability. Let us state that an unstable wave can give rise to an unstable mode of the finite system only if its wavelength is smaller that the length of the system. The smallest unstable wavelengths are, λ= √

2π , λd = 2π, 1−β

(83)

in the undamped and damped cases respectively. Plotted against the marginal stability curves in Fig. 17, these criteria show a good agreement. The marginal stability curve goes continuously from the length criterion without damping to that with damping.

2.3 Conclusion This section and the previous one were devoted to the description of flow-induced instabilities and the effect of dissipation on these instabilities. Cases of destabilization by dissipation have been identified for the 2-DOF wing system and the fluid-conveying pipe: in some particular ranges of the parameters, a stable system without dissipation can become unstable by the addition of a small amount of dissipation. For both systems, the key ingredient is the existence of non conservative fluid forces which induce skew-symmetric components in the stiffness matrix. In the fluid-conveying pipe system, we also identified a phenomenon of gyroscopic restabilization, which was cancelled by the addition of damping in the system. In the next section, we will consider that the dissipation comes from a mechanical to electrical energy transfer.

Dissipation Induced Instabilities of Structures Coupled to a Flow

93

Fig. 18 Sketch of two generic electromechanical converters. On the left, a piezoelectric material is deformed, which creates an electric field in the material. The presence of electrodes on the opposite faces generates a voltage. For small deformations, this voltage is proportional to the displacement that induced the deformation. On the right, a coil is moving at a velocity y˙ in a magnetic field. This generates a voltage proportionnal to the velocity in the neighbour of an equilibrium position

3 Applications in Energy Harvesting In recent years, the use of flow-induced vibrations to harvest kinetic energy from environnemental flows has been the focus of a incrediblely large amount of scientific papers. Although the engineering interest and applicability in terms of economic or durability are still open questions, the concepts have raised numerous of interesting fundamental questions that are still open: what efficiency can we theoretically expect from these designs? How to optimize this efficiency? For a particular design or at a particular scale is piezoelectricity or induction the more efficient electromechanical coupling? In this section, the basic modelling of flow-induced vibrations coupled to energy harvesting circuits is presented. Linear stability is addressed as well as non linear saturation of the instabilities.

3.1 Energy Converters The idea behind any energy converter is to convert a mechanical work into an electrical work. Two generic electromechanical converters are sketched on Fig. 18. The first one consists of a piezoelectric material that is stretched or compressed consecutively to the displacement y of one of its boundaries. Due to its particular crystallographic arrangement, deformation induces an electric field that can be converted as an electric voltage v if electrodes are placed on its boundaries. A second electromechanical converter that can be considered is based on the electromagnetic induction. A coil moving at a velocity y˙ can generate a voltage at its outlets if it moves in a magnetic field.

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Fig. 19 Top: an energy harvesting device based on the vibrations generated by negative flow-induced damping. Bottom: electrical circuits of the piezoelectric and inductive energy converted coupled to a resistance modelling the energy harvesting

3.2 Models of Energy Harvesting Systems Based on Flow-Induced Vibrations In order to introduce the principles of flow-induced vibrations based energy harvesters, we will consider here a simple system that becomes unstable by negative added damping. The system and the den Artog criterion for instability are presented in Sect. 1.2. Equation (8) describing the dynamics of this system can be simplified to m y¨ + (cd + AU ) y˙ + ky = 0,

(84)

where A is an aerodynamic coefficient depending on the profile geometry and its reference position (the equilibrium position around which there may exist an oscillation). If A is negative, there is a possibility of negative damping once a critical velocity is reached, hence exponentially growing oscillations may appear. The coupled systems are sketched on Fig. 19 where an energy harvesting device is added to the mechanical system studied in Sect. 1.2. If it consists of a piezoelectric element, it behaves electrically as a voltage generator in series with a capacitive device. The charge displacement q in the piezoelectric device reads q = χy + Cv,

(85)

where χ is a coupling coefficient which depends on the material, its cristallographic properties, and the geometry. If the converter consists of a coil in a magnetic field, it

Dissipation Induced Instabilities of Structures Coupled to a Flow

95

behaves electrically as a voltage generator in series with an inductance.4 The charge displacement in the coil satisfies, L q¨ + χ y˙ = v.

(86)

The coupling coefficient χ here depends on the magnetic field, the coil geometry and the direction of the displacement. We have hence modelled the so-called direct effect, which concerns the coupling from the mechanical part to the electrical part. The converse effect is also present. A voltage applied at the outlets of the piezoelectric material induces a force on the device, F = −χv. (87) Similarly, a current in the coil induces a force on itself, F = Lχq. ˙

(88)

These electrical devices are connected to a resistance R modelling the energy harvesting, as sketched in Fig. 19. Making use of the Ohm’s law we can provide an additionnal constitutive relation to the electrical model: v + R q˙ = 0.

(89)

Linear Models Using Eqs. (84), (85), (87) and (89), one obtains a set of coupled linear equations modelling the dynamics of the profile coupled with an electrical circuit through piezoelectric coupling,   χ2 χ y + q = 0, (90) m y¨ + (cd + AU ) y˙ + k + C C χ 1 (91) R q˙ + q + y = 0. C C Equation (90) can be seen as a damped oscillator equation coupled to the electrical part through the term (χ/C)q while Eq. (91) is the dynamical equation of an RC circuit coupled to the mechanical part through the term (χ/C)y. Hence, piezoelectricity couples the dynamics of the mechanical system and the electrical circuit through two symmetrical terms involving structural and charge displacements. Additionally, the fact that electrical energy can be stored in the equivalent capacity induces an additionnal stiffness χ2 /C in the structural dynamics part. In the same manner, using Eqs. (84), (86), (88), (89), the dynamical system modelling the dynamics of the profile coupled with an electrical circuit through inductive coupling takes the form 4 In

practice, the coil has also a DC-resistance, but it is neglected in this introductory approach.

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m y¨ + (cd + AU ) y˙ + ky − χq˙ = 0,

(92)

L q¨ + R q˙ + χ y˙ = 0.

(93)

Here, the mechanical oscillator equation is coupled to the electrical part through a term proportional to q˙ and the dynamical equation of an LC circuit is symmetrically coupled to the mechanical part through a term proportionnal to y˙ . Let us first analyse the linear dynamics and stability of piezoelectric energy harvester. In order to simplify the parametric analysis, the equations will be put in non dimensional form. The following dimensionless variables are introduced:  y 1 k , y¯ = , q¯ = q √ , (94) t¯ = t m D kC D where D is a characteristic length of the system.5 The dimensionless set of equations, equivalent to Eqs. (90)–(91) is then obtained as, y¨¯ + (γ − δ) y˙¯ + (1 + α2 ) y¯ + αq¯ = 0, β q˙¯ + q¯ + α y¯ = 0, with

χ α= √ , β = RC kC



cd k AU , γ=√ , δ = −√ . m km km

(95) (96)

(97)

α is the piezoelectric coefficient.  β is the ratio between the electrical timescale RC

and the mechanical timescale mk . γ is the dimensionless structural damping and δ can be sought as the dimensionless flow velocity. In order to study the linear dynamics of the system, it is written in matrix form, at order one for the time derivatives: ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ 1 0 0 y˙ 0 −1 0 y ⎣ 0 1 0 ⎦ ⎣ y¨ ⎦ + ⎣ 1 + α2 γ − δ α ⎦ ⎣ y˙ ⎦ = 0 (98) 0 0 β q˙ α 0 1 q Looking for solutions in the form ⎡ ⎤ y ⎣ y˙ ⎦ = V eλt = V eλr t eiλi t , q

(99)

one obtains an eigenvalue problem of the form λV = −M1−1 M2 V , where M1 and M2 are the first and second matrices of Eq. (98) respectively. The real and imaginary 5 It

can be a typical length of the cross section profile for instance. But in this context, the choice of the length as no incidence on the results of the analysis

Dissipation Induced Instabilities of Structures Coupled to a Flow γ=0.1

α=0(dashed),0.75(plain) β=1.5 γ=0.1

Real part of λ (growth rate)

97

0.25

0.2 0 −0.2

α=0.5

−0.4 −0.6 −0.8

0.2

−1 0

0.1

0.2

0.3

δcrit

Imaginary part of λ (frequency)

α=0.4

0.4

δ 2

0.15

α=0.3

1 α=0.2

0 −1 0.1 −2 0

0.1

0.2

δ

0.3

0

0.4

10

β

Fig. 20 Galloping system of figure (19) with a piezoelectric energy harvester. Left: Real and imaginary parts of λ as function of the dimensionless velocity δ. The circles are a guide for the eyes indicating the crossing of the curve with the horizontal axis. Right: critical value of the dimensionless velocity δ as function of the timescale ratio β

parts of the eigenvalues are plotted as function of the dimensionless velocity δ in Fig. 20 for α = 0 (no piezoelectric coupling) and α = 0.75 for typical values of the other parameters. For α = 0, the eigenvalues are that of two uncoupled systems: • A mechanical oscillator. Two of the three eigenvalues belong the the oscillator dynamics. They are complex conjugate. Their imaginary part correspond to the circular frequency of the oscillations (see Eq. 99), while their real part correspond to the growth rate. A positive real part leads to exponentially growing oscillations (instability). In the particular case of the plot, the critical value δcrit for instability is ∼0.1. • An RC circuit. One real negative eigenvalue is found. Its value is the inverse of the characteristic time decrement of a capacity discharging in a resistance. When piezoelectric coupling is added to the system, one can make the following observations. • The imaginary part of the eigenvalues of the mechanical part increases. The additionnal stiffness induced by the energy stored in the capacity (α2 term in Eq. 95) increases the frequency of the oscillations.

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• Damping induced by the energy dissipation in the resistance increases the apparent damping seen by the mechanical oscillator. It hence increases the critical velocity δcrit . • The dynamics of the electrical circuit is influenced by the mechanical part. The characteristic time hence varies with δ. In the right plot of Fig. 20 the value of δcrit is plotted as function of β for different values of α. It appears on this plot that the critical velocity equals ∼0.1 for small and large values of β. If β is sought as the dimensionless resistance, small values corresponds to a closed circuit. In this situation, the voltage is cancelled and the force induced by the reverse piezoelectric effect vanishes. The opposite case of large β (or large resistance) corresponds to an open circuit. Here, there is no charge displacement in the circuit which do not dissipate any energy in the resistance. There is no additional induced damping and the critical velocity is not modified by piezoelectric coupling. However, there is a voltage at the outlets of the piezoelectric material, and the additionnal stiffness term still exists. Between these two limit cases, the influence of piezoelectric coupling is maximized around β = 1. For this value of β, the electrical circuit synchronizes with the mechanical system. This gives us a condition for maximizing the influence of the electrical part on the mechanical part: synchronization. The same analysis can be performed with an energy haverster of the inductive type. The electrical charge displacement is now non dimensionalized using 1 q¯ = q D



L , m

(100)

while the characteristic time and displacement used for non dimensionalization are the same as in the piezoelectric case. One can finally write the dimensionless dynamical equations as, y¨¯ + (γ − δ) y˙¯ + y¯ − αq˙¯ = 0, 1 q¨¯ + q¯˙ + α y¯˙ = 0, β with

χ L α= √ , β= R kL



k , m

(101) (102)

(103)

and γ and δ unchanged. In the same manner as in the case of piezoelectric coupling, this linear system can be written at order one for the time derivatives. The system writes then ⎡

1 ⎣0 0

0 1 0

⎤⎡ ⎤ ⎡ 0 y˙ 0 0 ⎦ ⎣ y¨ ⎦ + ⎣ 1 1 q¨ 0

−1 γ−δ α

⎤⎡ ⎤ 0 y −α ⎦ ⎣ y˙ ⎦ = 0 1/β q˙

(104)

99 γ=0.1

α=0(dashed),0.75(plain) β=1.5 γ=0.1 0.25

0.2 0 −0.2

α=0.5

−0.4 −0.6 −0.8

0.2

−1 0

0.1

0.2

0.3

δ Imaginary part of λ (frequency)

α=0.4

0.4

δcrit

Real part of λ (growth rate)

Dissipation Induced Instabilities of Structures Coupled to a Flow

2 0.15

α=0.3

1 α=0.2

0 −1 0.1 −2 0

0.1

0.2

0.3

0

0.4

δ

10

β

Fig. 21 Galloping system of figure (19) with an inductive energy harvester. Left: Real and imaginary parts of λ as function of the dimensionless velocity δ. Right: critical value of the dimensionless velocity δ as function of the timescale ratio β

The stability analysis is again performed through an eigenvalue analysis. Typical results are plotted on Fig. 21. It is remarkable here that the stability results are almost identical in the inductive coupling case. All the most salient features evidenced in the piezoelectric case can also be evidenced in the inductive case. Non Linear Saturation and Efficiency The linear system presented in the previous section still needs some improvement to satisfactorly model a real energy harvesting system. Indeed, when an instability is evidenced, the model predicts exponential growth of the oscillations. Of course, oscillations in real systems never grow to infinity. Instead, the system reach a saturated regime at a finite amplitude, caused by non linear forces that start to play a role when amplitudes reach finite values. These forces may originate from aerodynamic effects at large oscillation amplitudes, non linear material behaviour or structural geometrical non linearities. In the present approach, this will be modelled using a non linear cubic damping. A force f n l is then added on the right side the dynamics of the mechanical system: (105) f nl = −knl x˙ 3 .

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This non linear stifness can be seen as a low order approximation of non linearities present in the system, or in other words, as the term of a Taylor expansion of the force next to the linear term. The resulting non linear system can then be integrated in time for a given set of parameters and initial conditions, using any numerical method. If the parameters are such that the system is linearly unstable, its dynamics consists of an oscillation with exponentially growing amplitude as long as amplitude is small and nonlinear terms are negligible. At larger amplitude, nonlinearities start to play a role in the dynamics and amplitude saturate. This is illustrated in the top graph of Fig. 22a for a typical set of parameters. On this graph, cases α = 0 (no piezoelectric coupling) and α = 0.3 are presented. It is visible that the coupling between the mechanical system and the energy harvesting circuit induce additionnal damping in the system that reduce the growth rate as well as the displacement amplitude in the saturated regime. Now that the system once instability is triggered is able to saturate, we are interested in the power transfered to the electrical circuit. The instantaneous power of heat conversion in the resistance, modelling energy harvesting is P = R q˙ 2 , or in dimensionless form, (106) p = β q˙¯ 2 . This quantity is plotted as function of time in the bottom graph of Fig. 22a. In the saturated regime, the power fluctuates between zero and a maximum value pmax with a period half of the mechanical oscillation period. On Fig. 22b, the amplitude of oscillation and pmax are ploted as function of the value of β. For each explored value of β, one nonlinear integration is performed until a limit cycle is obtained. This graph shows that for values of β around unity, synchronization enhance energy transfer from the oscillating structure to the harvesting circuit: the vibration amplitude is lowered and at the same time, pmax is maximized. Hence, synchronization between the mechanical oscillations and the electrical conversion circuit are also found to enhance the energy transfer.

3.3 Conclusion In this section, some basic concepts of energy harvesting from flow induced instabilities have been introduced with the help of a simple galloping system. We have in particular shown that the key parameter for an efficient energy transfer is synchronization fluid-solid dynamics with the electrical circuit, whatever the electromechanical coupling is, inductive or piezoelectric. Of course, when a practical design has to be realized, the question of synchronization can become a different challenge if one use induction or piezoelectric coupling. The concepts introduced here with a very simple system of a galloping oscillator could be readily applied to other systems such as 2-DOF fluttering wing profiles, fluttering flags or pipes. As well as with classical dissipation, energy harvestinginduced dissipation can destabilize the system. Also, more sophisticated circuits can

Dissipation Induced Instabilities of Structures Coupled to a Flow

(a)

α=0(dashed),0.2(plain)

β=1

γ=0.1

101 δ=0.2

ε=0.01

4

y

2 0 −2 −4

0

20

40

60

80

100

120

140

160

180

200

120

140

160

180

200

t 0.25

Pdiss

0.2 0.15 0.1 0.05 0

0

20

40

60

80

100

t

(b)

3.7 3.6

m

y ax

3.5 3.4 3.3 3.2 −2 10

−1

10

0

1

10

10

2

10

3

10

0.25

Pmax

0.2 0.15 0.1 0.05 0 −2 10

−1

10

0

1

10

10

2

10

3

10

β

Fig. 22 a Displacement (top) and instantaneous dissipated power in the resistance (bottom) as function of time for typical values of the dimensionless parameters. b In the saturated regime, maximum displacement (top) and maximum dissipated power for different values of β in the range [10−2 , 2 × 102 ]

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be considered. For instance, by adding an inductance in the piezoelectric energy harvester, one creates a resonant circuit that can improve the harvested power.

References J.J. Allen, A.J. Smits, Energy harvesting eel. J. Fluids Struct. 15, 629–640 (2001) S.F. Asokanthan, S.T. Ariaratnam, Flexural instabilities in axially oving bands. J. Vib. Acoust. 116, 275 (1994) A. Barrero-Gil, G. Alonso, A. Sanz-Andres, Energy harvesting from transverse galloping. J. Sound Vib. 329(14), 2873–2883 (2010) Y. Bazilevs, K. Takizawa, T.E. Tezduyar, Computational Fluid-Structure Interaction: Methods and Applications (Wiley, New York, 2013) M. Beck, Die Knicklast des einseitig eingespannten tangential gedruckten stabes. Z. Angew. Math. Phys. 3, 225–229 (1952) K.Y. Billah, R.H. Scanlan, Resonance, Tacoma narrows bridge failure, and undergraduate physics textbooks. Am. J. Phys. 59(2), 118–124 (1991) R.D. Blevins, Formulas for Natural Frequency and Mode Shape (Von Nostrand Reinhold, New York, 1979) R.D. Blevins, Flow-Induced Vibration (1990) R.A. Cairns, The role of negative energy waves in some instabilities of parallel flows. J. Fluid Mech. 92, 1–14 (1979) E. De Langre, Effects of wind on plants. Ann. Rev. Fluid Mech. 40, 141–168 (2008) E. de Langre, M.P. Païdoussis, O. Doaré, Y. Modarres-Sadeghi, Flutter of long flexible cylinders in axial flow. J. Fluid Mech. 571, 371–389 (2007) O. Doaré, S. Michelin, Piezoelectric coupling in energy-harvesting fluttering flexible plates: linear stability analysis and conversion efficiency. J. Fluids Struct. 27(8), 1357–1375 (2011) O. Doaré, M. Sauzade, C. Eloy, Flutter of an elastic plate in a channel flow: confinement and finite-size effects. J. Fluids Struct. 27(1), 76–88 (2011) E.H. Dowell, E.F. Crawley, H.C. Curtiss Jr., D.A. Peters, R.H. Scanlan, F. Sisto, A Modern Course in Aeroelasticity (Kluwer Academic Publishers, Dordrecht, 1995) C. Eloy, C. Souilliez, L. Schouveiler, Flutter of a rectangular plate. J. Fluids Struct. 23(6), 904–919 (2007) F. Gosselin, E. de Langre, Destabilising effects of plant flexibility in air and aquatic vegetation canopy flows. Eur. J. Mech. - B/Fluids 28(2), 271–282 (2009) R.W. Gregory, M.P. Païdoussis, Unstable oscillation of tubular cantilevers conveying fluids. I. Theory. Proc. R. Soc. Lond. A 293, 512–527 (1966) C.Q. Guo, M.P. Paidoussis, Stability of rectangular plates with free side-edges in two-dimensional inviscid channel flow. J. Appl. Mech. 67(1), 171–176 (2000) M.P. Païdoussis, Fluid-Structure Interactions. Slender Structures and Axial Flow, vol. 2 (Academic Press, Cambridge, 2004) M.P. Païdoussis, S.J. Price, E. de Langre, Fluid-Structure Interactions - Cross-Flow-Induced Instabilities (2011) L. Tang, M.P. Païdoussis, The coupled dynamics of two cantilevered flexible plates in axial flow. J. Sound Vib. 323(3–5), 790–801 (2009) G.W. Taylor, J.R. Burns, S.M. Kammann, W.B. Powers, T.R. Welsh, The energy harvesting eel: a small subsurface ocean/river power generator. IEEE J. Ocean. Eng. 26(4), 539–547 (2001) Theodore Theodorsen, General theory of aerodynamic instability and the mechanism of flutter Technical report, NASA (1979) W. Thomson, P.G. Tait, Treatise on Natural Philosophy (Cambridge University Press, Cambridge, 1879) M. von Laue, The propagation of radiation in dispersive and absorbing media. Ann. Physik (1905) H. Ziegler, Die stabilitätskrieterien der elastomechanik. Ing.-Arch 20, 49–56 (1952)

Some Surprising Conservative and Nonconservative Moments in the Dynamics of Rods and Rigid Bodies Evan G. Hemingway and Oliver M. O’Reilly

Abstract Representations for conservative and nonconservative moments in classical mechanics are discussed in this expository article. When the rotation is parameterized by a set of Euler angles, a particularly transparent representation can be found which has ties to classic works in mechanics dating to Lagrange in 1780 and joint coordinate systems that are commonly used in orthopaedic biomechanics. The article also surveys connections between Lagrange’s equations of motion and the Newton–Euler equations of motion. A variant on the Lagrange top and a satellite dynamics problem are presented to illustrate some of the key concepts discussed in the paper.

1 Introduction In a remarkable paper, Lagrange (1780) presented a dynamic model to explain the oscillations (librations) in the attitude of the Moon as seen by an Earth-based observer. The starting point for his model featured for the first time his celebrated equations of motion (cf. Lagrange (1780, Sect. 11)): d dt



∂T ∂ q˙ K

 −

∂T ∂V =− K, K ∂q ∂q

(K = 1, . . . , 6) ,

(1)

where V and T are the respective potential and kinetic energies of the Moon. Later in this work (cf. Lagrange (1780, Sect. 21)), Lagrange used a set of 3-1-3 Euler angles to parametrize the rotation of the Moon.

E. G. Hemingway (B) · O. M. O’Reilly Department of Mechanical Engineering, University of California, Berkeley, CA 94720-1740, USA e-mail: [email protected] O. M. O’Reilly e-mail: [email protected] © CISM International Centre for Mechanical Sciences 2019 D. Bigoni and O. Kirillov (eds.), Dynamic Stability and Bifurcation in Nonconservative Mechanics, CISM International Centre for Mechanical Sciences 586, https://doi.org/10.1007/978-3-319-93722-9_3

103

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A few years prior to Lagrange’s work, Euler published a series of seminal works on the dynamics of rigid bodies (cf. Euler 1752, 1775). Among the contributions from Euler’s works that have had a lasting influence are the Euler angle parameterization of rotations and the Newton–Euler equations of motion for a rigid body of mass m: F = m x¨¯ ,

˙ M = H,

H = Jω,

(2)

where F is the resultant force acting on the rigid body, M is the resultant moment relative to the center of mass of the rigid body, J is the moment of inertia tensor of the rigid body relative to its center of mass, x¯ is the position vector of the center of mass, H is the angular momentum relative to the center of mass of the rigid body, and ω is the angular velocity vector of the rigid body. Lagrange’s treatment of the libration problem is surprising. First, although he employs Euler angles and is completely comfortable with mass moments of inertia, he uses an entirely different formulation of the problem than the Newton–Euler form (2). While he calculates gravitational forces and approximations to the gravitational potential energy, it is not clear what the corresponding gravitational moments are from his work. Indeed the moment in question, known as a gravity-gradient torque (cf. Eq. (83)2 ), is credited to James Mac Cullagh (1809–1847) following the posthumous publication of his lecture notes by Allman (1855). Assuming that the dynamic equations of motion formulated using Lagrange’s equations (1) and the Newton–Euler equations (2) are equivalent, it is natural to ask if ∂V some of the partial derivatives − ∂q K can be interpreted as moment components. Using results on dual Euler basis vectors from O’Reilly (2007) we are able to show how these partial derivatives are specific components of force and moment vectors and thus ∂V facilitate physical interpretations of the partial derivatives − ∂q K . Our discussion is illuminated with examples from rigid body dynamics and orthopaedic biomechanics and also highlights the simplest known representation for a conservative moment. An outline of this expository article is as follows. Background on the Euler angle parameterization of a rotation is collected in Sect. 2. The notation we use for the three angles follows Lagrange (1780). We supplement his treatment with a discussion of the Euler basis and dual Euler basis vectors and the representation of vectors using these distinct bases. In Sect. 3, the relationship between Lagrange’s equations of motion and the Newton–Euler equations of motion for a single rigid body are examined. Particular attention is paid to the incorporation of ideal integrable constraints and potential energies. A simple representation for a conservative moment that features the dual Euler basis vectors is discussed in Sect. 4. To dispel possible confusion, the case where the motion of a rigid body is constrained to have a fixed axis of rotation is presented in Sect. 5. We close the paper with two examples. First, a derivation of the equations of motion of a Lagrange top subject to applied forces and moments is discussed. Then, in Sect. 7, we return to the problem of a satellite in a gravitational field that was the subject of Lagrange (1780). Among other matters, we are able ∂V to demonstrate how the partial derivatives − ∂q K in Eq. (1) can be considered as components of a moment vector.

Some Surprising Conservative and Nonconservative Moments …

105

The notation and terminology employed in this paper closely follows the textbook by O’Reilly (2008). We appeal to a recent expository article by O’Reilly and Srinivasa (2014) when discussing constraint forces and constraint moments. Additional complementary background on rotations can be found in the exceptional survey article by Shuster (1993) and the online resource http://rotations.berkeley.edu/.

2 Background on Euler Angles and Bases Central to our discussion is the method of Euler angles to parameterize a rotation tensor. Of the twelve available sets of Euler angles, we focus our attention here on the 3-1-3 set. We define a fixed right-handed orthonormal basis {E1 , E2 , E3 } and use the rotation tensor Q to define a basis {e1 , e2 , e3 }: ei = QEi .

(3)

It is straightforward to show using the facts that QQT = I and det (Q) = 1 that the basis {e1 , e2 , e3 } is right-handed and orthonormal. In addition, the rotation tensor Q has the representations Q=

3  k=1

ek ⊗ Ek =

3  3  i=1 k=1

Q ik Ei ⊗ Ek =

3  3 

Q ik ei ⊗ ek .

(4)

i=1 k=1

That is, the components Q ik = ek · Ei of Q can be considered as direction cosines. In the 3-1-3 set of Euler angles, the tensor Q is decomposed into the product of three simple rotations: Q = QE (ϕ, g3 ) QE (ω, g2 ) QE (ψ, g1 ) ,

(5)

where the function QE (θ, i) describes a rotation about an axis described by a unit vector i through a counterclockwise angle θ: QE (θ, i) = cos(θ) (I − i ⊗ i) + sin(θ)skew (i) + i ⊗ i.

(6)

In the above representation for a rotation tensor, the operator skew(i) transforms i into a skew-symmetric tensor such that i × b = skew(i)b for any vector b. The associated operator ax transforms a skew-symmetric tensor into a vector: ax (A) × b = Ab where A = −AT is a skew-symmetric tensor. The basis {g1 , g2 , g3 } is known as the Euler basis. This basis is not orthogonal, and, for the 3-1-3 set of interest here,

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e∗2

E2

e∗∗ 3

e∗1

ψ

ψ

E3 ω

E1

e2

e∗∗ 2 ω

e∗∗ 2 ϕ

e∗2

e1 ϕ

e∗1

Fig. 1 Schematic of a set of 3-1-3 Euler angles that are used to parameterize a rotation from the basis {E1 , E2 , E3 } to the basis {e1 , e2 , e3 }. The three axes of rotation are g1 = E3 = e3∗ , g2 = e1∗ = e1∗∗ , and g3 = e3∗∗ = e3 and the inset image is Handmann’s portrait of Leonhard Euler (1707–1783) from 1753. For details on the intermediate bases used to construct the figure, see Eq. (9)

g1 = E3 = cos (ω) e3 + sin (ω) (cos (ϕ) e2 + sin (ϕ) e1 ) , g2 = cos (ψ) E1 + sin (ψ) E2 = cos (ϕ) e1 − sin (ϕ) e2 , g3 = cos (ω) E3 + sin (ω) (sin (ψ) E1 − cos (ψ) E2 ) = e3 .

(7)

The angles ψ and ϕ range from 0 to 2π. Because (g1 × g2 ) · g3 = − sin (ω) ,

(8)

in order to ensure that the Euler basis is a basis for E3 , we restrict the second angle ω ∈ (0, π). Other perspectives on the singularity when ω = 0, π include noting that ψ and ω are polar coordinates for the axis of rotation g3 = e3 . Thus, the singularity arises when multiple values of ψ + ϕ (when ω = π) and ψ − ϕ (when ω = 0) are possible for a given rotation tensor Q. Referring to Fig. 1, it is straightforward to transform from  ei to Ei and viceversa  with the help of two pairs of intermediate bases e1∗ , e2∗ , e3∗ and e1∗∗ , e2∗∗ , e3∗∗ : ⎡

⎤ e1∗ ⎣ e2∗ ⎦ = e3∗ ⎡ ∗∗ ⎤ e1 ⎣ e2∗∗ ⎦ = e3∗∗ ⎡ ⎤ e1 ⎣ e2 ⎦ = e3

⎡

⎤⎡ ⎤ cos (ψ) sin (ψ) 0 E1 ⎣ − sin (ψ) cos (ψ) 0 ⎦ ⎣ E2 ⎦ , 0 0 1 E3 ⎤⎡ ∗⎤ ⎡ e1 1 0 0 ⎣ 0 cos (ω) sin (ω) ⎦ ⎣ e2∗ ⎦ , 0 − sin (ω) cos (ω) e3∗ ⎡ ⎤ ⎡ ∗∗ ⎤ cos (ϕ) sin (ϕ) 0 e1 ⎣ − sin (ϕ) cos (ϕ) 0 ⎦ ⎣ e2∗∗ ⎦ . 0 0 1 e3∗∗

(9)

Some Surprising Conservative and Nonconservative Moments …

(a) e3 = g3

(b) E3 = g1

ω

107

E3 = g1 e3 = g3 ω

ϕ

ϕ E2

E1

g1

ψ

g2

E2 g

3

E1

ψ

g2 = g 2

Fig. 2 Schematic of the Euler and dual Euler basis vectors associated with a 3-1-3 set of Euler angles. a The Euler basis vectors, g1 , g2 , and g3 , and their relation to the Euler angles (cf. Eq. (7)) and b the corresponding dual Euler basis vectors, g1 , g2 , and g3 (cf. Eq. (14))

We note that the three matrices in Eq. (9) can be combined to provide expressions for the components Q ik = (QEk ) · Ei of Q: ⎡

⎤ ⎡ ⎤ Q 11 Q 12 Q 13 −s1 c2 s3 + c1 c3 −c1 s3 − c3 c2 s1 s1 s2 ⎣ Q 21 Q 22 Q 23 ⎦ = ⎣ c2 c1 s3 + c3 s1 −s1 s3 + c2 c1 c3 −c1 s2 ⎦ . Q 31 Q 32 Q 33 s3 s2 s2 c3 c2

(10)

In writing expressions for the components Q ik , we have used the helpful abbreviations c1 = cos (ψ), s1 = sin (ψ), c2 = cos (ω), s2 = sin (ω), c3 = cos (ϕ), and s3 = sin (ϕ). The Euler basis vectors feature in the representation of the angular velocity vector ω associated with the rotation tensor Q. In particular,  ˙ 1 ˙ T = ϕg ˙ 3 + ωg ˙ 2 + ψg ω = ax QQ  = ψ˙ sin (ω) sin (ϕ) + ω˙ cos (ϕ) e1   + ψ˙ sin (ω) cos (ϕ) − ω˙ sin (ϕ) e2 + ψ˙ cos (ω) + ϕ˙ e3 .

(11)

˙ using This representation can be established by direct, but lengthy computation of Q Eq. (10) or, more rapidly, using two relative angular velocity vectors as in Casey and Lam (1986).

2.1 The Euler and Dual Euler Bases In addition to the Euler basis, we also have a companion dual Euler basis. Given an Euler basis, the corresponding dual Euler basis vectors are defined by the nine relations (12) gk · gi = δik , (i = 1, 2, 3, and k = 1, 2, 3) ,

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where δik is the Kronecker delta: δik = 1 if i = k and is otherwise 0. The solution to these nine equations is known in differential geometry1 : g2 × g3 , (g1 × g2 ) · g3 g2 = g2 , g1 × g2 . g3 = (g1 × g2 ) · g3 g1 =

(13)

The expression for g2 is greatly simplified because the Euler basis vector g2 is perpendicular to the other two: g1 ⊥ g2 and g3 ⊥ g2 . With the help of Eq. (13), we now compute the dual Euler basis vectors for the 3-1-3 set of Euler angles: g1 = cosec (ω) (sin (ϕ) e1 + cos (ϕ) e2 ) , g2 = cos (ϕ) e1 − sin (ϕ) e2 , g3 = cot (ω) (− cos (ϕ) e2 − sin (ϕ) e1 ) + e3 .

(14)

The Euler and dual Euler bases are sketched in Fig. 2. We observe from Eq. (14) that the dual Euler basis vectors are not defined when ω = 0, π. The dual Euler basis vectors were first introduced in O’Reilly and Srinivasa (2002) and O’Reilly (2007). They are also related to the dual vectors described by Howard et al. (1998) and Žefran and Kumar (2002) in their discussion of screw motions for rigid bodies.

2.2 Vector Representations As discussed in Nichols and O’Reilly (2017) and O’Reilly et al. (2013), the dual Euler basis feature in representations for the joint moment vector that is commonly used in orthopaedic biomechanics. To elaborate, a vector b has multiple representations including b = b1 g1 + b2 g2 + b3 g3 = b1 g1 + b2 g2 + b3 g3 ,

(15)

where b k = b · gk ,

1 We

bk = b · gk .

(16)

are exploiting the correspondence between the Euler and dual Euler basis sets and covariant and contravariant sets of basis vectors in differential geometry and are able to use a well-known result. See, e.g., Green and Zerna (1968), Eq. (1.9.13) or Simmonds (1982), Exercise 2.11).

Some Surprising Conservative and Nonconservative Moments … Fig. 3 Schematic of a joint coordinate system for the human knee joint. The axis j1 corotates with the femur and j3 corotates with the tibia. The angles α, β, γ and the axes {j1 , j2 , j3 } can be identified with a set of 3-2-1 (or 1-2-3) Euler angles and their associated Euler basis vectors

109

j3 γ Femur

j1

α j1

C2

C1

β j2

j3

j1 × j3

Tibia

Referring to Fig. 3, in biomechanics of anatomical joints, the first and third Euler basis vectors are identified with landmarks on the respective bones and moment components M · gk are computed.2 Consequently, in order

to reconstruct the moment vector, the dual Euler basis vectors are needed: M = 3k=1 (M · gk ) gk .

3 Lagrange’s Equations of Motion and the Newton–Euler Equations of Motion Consider a rigid body of mass m which has an inertia tensor J relative to its center of mass. We assume that a set of six coordinates are used to characterize the kinematics of the rigid body:  x¯ = x¯ q 1 , . . . , q 6 ,  Q = Q q 1, . . . , q 6 .

(17)

These coordinates are usually chosen to readily accommodate the integrable constraints on the system. Referring to Fig. 4, for the Lagrange top q 1 , q 2 , and q 3 are usually chosen to be a set of Euler angles while q 4 , q 5 , and q 6 are chosen to be the Cartesian coordinates of the fixed point O of the top. In satellite dynamics problems 2 See,

for example, Desroches et al. (2010), Grood and Suntay (1983), and Schache and Baker (2007).

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(b)

(a)

e3

g E1

O

E3 e2

¯ X

E3

e1

e3

e1

E2

M E1

O ϑ

¯ x

¯ X

m e2

E2 er

Fig. 4 a Schematic of a rigid body freely rotating about a fixed point O. This rigid body is commonly known as a Lagrange top and the inset image is a portrait of Joseph-Louis Lagrange (1736–1813). b Schematic of a rigid body of mass m in motion about a fixed rigid body of mass M

where a steady motion of the satellite involves a 1-1 locking of the orbital angular speed and the angular velocity, a set of cylindrical polar coordinates are chosen for q 1 , q 2 = ϑ, and q 3 and by a set of Euler angles and the  the rotation is parameterized polar angle: Q = Q ϑ, q 4 = ψ, q 5 = ω, q 6 = ϕ . It is straightforward to show that v¯ =

6 

q˙ K

K =1

Further, ω=

6 

q˙ w K , K

K =1

∂ x¯ , ∂q K

∂ x¯ ∂ v¯ = . ∂q K ∂ q˙ K

(18)

  ∂ω ∂Q T wK = = ax Q . ∂ q˙ K ∂q K

(19)

If a set of Euler angles are used to parameterize Q,  Q = Q q 6 = ϕ, q 5 = ω, q 4 = ψ ,

(20)

then w3+k = gk and w1,2,3 = 0. The kinetic energy of the rigid body has the representation T =

1 m v¯ · v¯ + ω · Jω. 2 2

(21)

Typically the corotational (or body-fixed) basis vectors ei are chosen to be the principal axis of the body. In this case, J can be expressed as J = λ1 e1 ⊗ e1 + λ2 e2 ⊗ e2 + λ3 e3 ⊗ e3 ,

(22)

Some Surprising Conservative and Nonconservative Moments …

111

where λk are the principal mass moments of inertia. It can be shown that Lagrange’s equations of motion are equivalent to a linear combination of the Newton–Euler balance laws3 :   ∂T ∂T ∂ v¯ ∂ω d − =F· +M· . (23) K K K dt ∂ q˙ ∂q ∂ q˙ ∂ q˙ K That is, ∂ v¯ ∂ω ∂T = m v¯ · +H· , ∂ q˙ K ∂ q˙ K ∂ q˙ K     d ∂ v¯ ∂ω ∂T d ¯ + H · . = m v · ∂q K dt ∂ q˙ K dt ∂ q˙ K

(24)

It is an interesting exercise to establish these results first for a single particle of mass m - where the angular momentum terms can be ignored. Indeed this case is discussed in the classic textbook (Synge and Griffith 1959).

3.1 A Force F A Acting at a Material Point X A The right hand side of Lagrange’s equations (23) have several simplifications. First, suppose that a force F A acts at a material point X A which has a position vector x A and velocity vector v A : (25) v A = v¯ + ω × (x A − x¯ ) . The position vectors x A and x¯ can be expressed as functions of the six coordinates q 1 , . . . , q 6 . Unlike v¯ , ω, and v A , this pair of position vectors do not depend on the velocities q˙ 1 , . . . , q˙ 6 . For the force F A , a simple differentiation of Eq. (25) with respect to q˙ K can be used to show that FA ·

∂ v¯ ∂v A ∂ω + ((x A − x¯ ) × F A ) · = FA · . K K ∂ q˙ ∂ q˙ ∂ q˙ K

(26)

This identity is helpful in several respects. First, it enables a direct comparison of treatments of Lagrange’s equations where a virtual work argument is used to prescribe nonconservative generalized forces on the right-hand side of Lagrange’s equations.4 In addition, if F A is a conservative force with a potential energy function U = U (x A ) , 3A

(27)

proof of this correspondence can be found in Casey (1995). Casey’s proof is discussed in the textbook O’Reilly (2008) where additional examples are presented. 4 See, for example, the lucid discussion in Baruh (1999, Chap. 4, Sect. 9).

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then FA = −

∂U . ∂x A

(28)

With the help of the identity v A = x˙ A =

6  K =1

q˙ K

∂x A , ∂q K

(29)

it is straightforward to show using the chain rule that FA ·

∂v A ∂x A = FA · ∂ q˙ K ∂q K ∂U ∂x A =− · ∂x A ∂q K ∂U =− K. ∂q

(30)

Finally, Eq. (26) allows one to easily write down Lagrange’s equations in cases where a nonconservative follower force or a dynamic Coulomb friction force acts on the mechanical system. The most celebrated instance of the former case is Ziegler’s pendulum.

3.2 Ideal Integrable Constraints For the second simplification to the right-hand side of Lagrange’s equations of motion, suppose that an integrable constraint is imposed on the system: q 6 − f (t) = 0.

(31)

Then, if the constraint force Fc and constraint moment Mc satisfy Lagrange’s prescription, it can be shown that5 Fc ·

∂ v¯ ∂ω + Mc · = μδ6K , K ∂ q˙ ∂ q˙ K

(32)

where μ is a scalar function (Lagrange multiplier) and δ6K is the Kronecker delta. This remarkable result enables Lagrange’s equations of motion to decouple into two sets: one for the unconstrained (or generalized) coordinates and the other for the function μ. 5 This

constraint in this case is sometimes known as ideal because Fc and Mc have no frictional components. For further details on constraint forces and constraint moments, see the expository paper by O’Reilly and Srinivasa (2014).

Some Surprising Conservative and Nonconservative Moments …

113

In most treatments of Lagrange’s equations of motion interest is restricted to the former set and the equations of motion for the generalized coordinates are formulated. For multibody systems involving rigid bodies connected by pin joints the savings in algebraic computations when attention is restricted to the generalized coordinates can be considerable. For instance, for the planar double pendulum, the number of equations of motion for the generalized coordinates is two while the remaining ten equations give expressions for the reaction forces and reaction moments at the two pin joints.

3.3 Potential Energies The third and final simplification we wish to mention occurs if a conservative force Fcon acting at the center of mass and a conservative moment (relative to the center of mass) Mcon act on the rigid body. The combined mechanical power of these quantities is assumed to be equal to the negative rate of change of a potential energy function: Fcon · v¯ + Mcon · ω = −U˙ , where U = U (¯x, Q) ,

U˙ =

6  ∂U K q˙ . ∂q K K =1

(33)

(34)

Assuming that Fcon and Mcon are independent of the rates q˙ 1 , . . . , q˙ 6 , it follows that Fcon ·

∂ v¯ ∂ω ∂U + Mcon · =− K. ∂ q˙ K ∂ q˙ K ∂q

(35)

Thus, we find that the partial derivatives of U with respect to the coordinates q K are linear combinations of the components of the conservative force and conservative moment. Expressions for Fcon and Mcon can be established as gradients of U with respect to x¯ and Q, respectively. We shall examine one such representation for Mcon shortly.

3.4 A Canonical Form, Equilibria, and Linearization Lagrange’s equations of motion reveals a canonical form of the equations of motion that is pervasive in mechanics. To elaborate, consider a system with N degreesof-freedom whose kinetic energy can be expressed as a quadratic form and whose potential energy function depends on the N generalized coordinates:

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1  a I K q˙ I q˙ K , 2 I =1 K =1 N

T =

N

 U = U q 1, . . . , q N ,

(36)

 where a I K = a I K q 1 , . . . , q N . The N × N matrix formed by the components of a I K is known as the mass matrix M. This matrix is symmetric and, apart from instances where there are singularities in the coordinate system, will be positive definite.6 For such a system, it is known that7 d dt



∂T ∂ q˙ K



  ∂T = a K I q¨ I + [S J, K ] q˙ S q˙ J . K ∂q I =1 S=1 J =1 N

−

N

N

(37)

Here, we have used the Christoffel symbols of the first kind [S J, K ] to collect the quadratic velocity terms: 1 [S J, K ] = 2



∂a K J ∂a K S ∂a S J + − S J ∂q ∂q ∂q K

 ,

(J, K , S = 1, . . . , N ) .

(38)

If we assume that the only generalized forces acting on the system are conservative, then the equations of motion can be expressed as N  I =1

a K I q¨ I +

N N  

[S J, K ] q˙ S q˙ J = −

S=1 J =1

∂U , ∂q K

(K = 1, . . . , N ) .

(39)

An equilibrium of these equations satisfies the following 2N conditions: q˙ K = 0,

q K = q0K ,

(K = 1, . . . , N ) .

(40)

Examining the equations of motion (39), we observe that at an equilibrium the potential energy is extremized: ∂U  1 q0 , . . . , q0N = 0. K ∂q

(41)

To establish the linearized equations of motion in the neighborhood of an equilibrium, we consider the following asymptotic expansions: q 1 = q01 + η1 ,

...,

q N = q0N + η N .

(42)

After substituting into (39), performing Taylor series expansions of U , a I K , and [S J, K ], using the equilibrium conditions, and ignoring terms of order 2 and higher, 6 For

additional perspectives on this matter see Hemingway and O’Reilly (2018). for example, the classic text by McConnell (1947).

7 See,

Some Surprising Conservative and Nonconservative Moments …

115

we find the following equations governing the linearized dynamics in a neighborhood of the equilibrium: (43) M0 η¨ + K0 η = 0. The mass matrix M0 and stiffness matrix K0 are both symmetric: T  η = η1 , . . . , η N ,   ⎤ ⎡ a11 q01 , . . . q0N · · · a1N q01 , . . . q0N ⎢ ⎥ .. .. .. M0 = ⎣ ⎦, . .  1.  N N 1 a1N q0 , . . . q0 · · · a N N q0 , . . . q0 ⎡ 2   1 ⎤ 2 ∂ U U q01 , . . . q0N · · · ∂q∂1 ∂q q0 , . . . q0N N ∂q 1 ∂q 1 ⎥ ⎢ .. .. .. ⎥. K0 = ⎢ . . . ⎦ ⎣   2 2 ∂ U ∂ U N N 1 1 q · · · q , . . . q , . . . q 1 N N N 0 0 0 0 ∂q ∂q ∂q ∂q

(44) (45)

(46)

Thus, for many mechanical systems, Lagrange’s equations of motion allows us to infer the equilibria of the system and the equations governing the linearized dynamics by simply computing the kinetic and potential energies and the derivatives of the latter energy.

4 Simple Conservative Moments As remarked by Ziegler (1968), p. 30, the simplest nonconservative moment is a constant moment. He showed that a constant moment was nonconservative by examining the work done in rotating a rigid body through 180◦ . Such a motion can be accomplished in two equivalent manners. The first method is direct while the second involves successively rotating the body through 180◦ about two perpendicular axis. The work done by the moment in the latter case is zero and in the former case is non-zero. Whence, the constant moment is not conservative. Ziegler’s conclusion is surprising and involves an ingenious use of the HamiltonRodrigues theorem on finite rotations. His work also begs the question that if a constant moment isn’t conservative, then which moment is? This question was also posed by Simmonds and answered by Antman (1972) and later by Simmonds (1984) himself. Antman’s solution uses the Euler representation for a rotation featuring an axis of rotation and an angle of rotation. A simpler answer can be found using Euler angles.8

8 Our

developments and discussion are based on the works O’Reilly and Srinivasa (2002) and O’Reilly (2007, 2008).

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4.1 A Simple Representation for a Conservative Moment Consider a potential energy U that depends solely on the orientation of the rigid body. Thus, we can express U as a function of the Euler angles and ∂U ∂U ˙ ∂U ω˙ + ϕ. ˙ U˙ = ψ+ ∂ψ ∂ω ∂ϕ

(47)

However, using the dual Euler basis vectors ˙ ω · g1 = ψ,

ω · g2 = ω, ˙

ω · g3 = ϕ. ˙

Substituting into the expression for U˙   ∂U 1 ∂U 2 ∂U 3 ˙ U= g + g + g · ω. ∂ψ ∂ω ∂ϕ

(48)

(49)

Paralleling the case of a conservative force, we define a conservative moment Mcon by postulating that. (50) U˙ = −Mcon · ω. Assuming in addition that Mcon is independent of ω, we find the representation Mcon = −

∂U 1 ∂U 2 ∂U 3 g − g − g . ∂ψ ∂ω ∂ϕ

(51)

This is the simplest known representation of a conservative moment.9 It has evident parallels to a representation of a conservative force acting on a particle that features a gradient expressed using contravariant basis vectors.

4.2 Ziegler’s Example Revisited Returning to Ziegler’s example, suppose that M0 E3 (where M0 is a constant) is conservative. Then, the potential energy function associated with this moment would have to satisfy the following set of partial differential equations: ∂U = M0 E3 · g1 = M0 E3 · E3 = M0 , ∂ψ ∂U = M0 E3 · g2 = M0 E3 · e1∗ = 0, − ∂ω ∂U = M0 E3 · g3 = M0 E3 · e3 = M0 cos (ω) . − ∂ϕ

−

9A

(52)

compilation of representations for the gradient of U for various representations of the rotation Q can be found in O’Reilly (2008, Sect. 6.10).

Some Surprising Conservative and Nonconservative Moments …

117

However, the statements ∂U = 0 and ∂U = −M0 cos (ω) are contradictory and we ∂ω ∂ϕ conclude that no such U can exist. Thus, M0 E3 is nonconservative.

4.3 Torsional Springs We can use the representation for a conservative moment to establish an expression for the moment provided by a torsional spring. Suppose that the torsional spring’s potential energy function is Uspring =

K (ψ − ψ0 )2 , 2

(53)

where ψ0 is a constant and K is the spring constant. Invoking Equation (51), we find that Mspring = −K (ψ − ψ0 ) g1 = −K (ψ − ψ0 ) (E3 − cot (ω) (− cos (ψ) E2 − sin (ψ) E1 )) .

(54)

Observe that the spring moment has components orthogonal to the E3 direction. The fact that these components become unbounded as ω → 0, π is a manifestation of the singularity in the 3-1-3 set of Euler angles at these values of ω. Unless the rigid body is constrained, it is necessary to switch to a complementary set of Euler angles such as the 3-2-1 set as ω approaches these singular values.

5 The Case of a Fixed Axis of Rotation The singular behavior of the torsional spring moment in the previous section begs the question of how to deal with the case when the axis of rotation is constrained to be fixed. In this situation, choosing E3 , say, to be parallel to the axis of rotation, we ˙ 3 . In addition, the find that the angular velocity vector has the representation ω = ψE motion of the rigid body is subject to a pair of constraints that can be expressed as ω · E1 = 0,

ω · E2 = 0.

(55)

A constraint moment with two independent components is needed to enforce these constraints: (56) Mc = μ1 E1 + μ2 E2 . The moment Mc does no work and is nonconservative. If a pin joint is used to ensure that the axis of rotation stays constant, then the constraint moment Mc can be considered as a reaction moment provided by the pin joint. If the body is sliding on

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a flat surface, then Mc is the resultant moment provided by the normal forces acting on the surface of the body contacting the plane. The potential energy of a torsional spring in this case is again given by Uspring =

K (ψ − ψ0 )2 . 2

(57)

Now, however, we seek solutions Mspring to the equation U˙ = −Mspring · ω where ˙ 3 . The resulting solution is ω = ψE Mspring = −K (ψ − ψ0 ) E3 .

(58)

Fortunately, this expression has none of the issues associated with its threedimensional counterpart (54). On a related note, as remarked in Ziegler (1968, Chap. 5), in the dynamics of rods where terminal moments of the form M0 E3 are applied, the boundary conditions often restrict the end rotation of the rod to be along E3 . In this case, M0 E3 is a conservative moment. We refer the reader to O’Reilly (2017, Sect. 5.15) for further discussion of this case.

6 The Lagrange Top To illustrate many of the previous developments, we now consider the classic example of an axisymmetric rigid body which is free to rotate about a point O. The rigid body is under the action of a vertical gravitational force. This mechanical system is known as the Lagrange top and its celebrated dynamics have a long and storied history.10 For the purposes of exposition, we establish the equation of motion for this system and include the effects of a constant moment Ma E3 acting on the body and a follower force Fa e1 acting at the tip of the top. Further, we allow the point O to be given a prescribed vertical motion f (t)E3 . This motion can be imagined by assuming that the top is freely spinning about a point O and then the point O is oscillated in a vertical manner.

6.1 Kinematical Considerations To establish the equations of motion for the top, we assign a set of 3-1-3 Euler angles to describe its orientation. The translational motion of the rigid body is characterized by a set of coordinates to describe the motion of O: 10 For additional references and discussion, see Baruh (1999), Lewis et al. (1992), and Marsden and

Ratiu (1999).

Some Surprising Conservative and Nonconservative Moments …

q 1 = ψ,

119

q 2 = ω, q 3 = ϕ,

q 4 = x O · E1 , q 5 = x O · E2 , q 6 = x O · E3 .

(59)

We assume that the position vector of the center of mass X¯ relative to O is x¯ − x O = e3 .

(60)

The mass of the top is denoted by m and its inertia tensor is J = λt (I − e3 ⊗ e3 ) + λa e3 ⊗ e3 .

(61)

The velocity vector of the center of mass of the top has the representation v¯ = q˙ 4 E1 + q˙ 5 E2 + q˙ 6 E3 + ω2 e1 − ω1 e2 .

(62)

In this expression, the components ωk = ω · ek can be easily read off from Eq. (11). For future purposes, we note that ∂ω = gk , ∂ q˙ k

∂ω = 0, ∂ q˙ k+3

∂v O = 0, ∂ q˙ k

∂v O = Ek , ∂ q˙ k+3

(63)

where k = 1, 2, 3.

6.2 Constraints and Constraint Forces The motion of the top is subject to three constraints. We have chosen the six coordinates in anticipation of these constraints being imposed. The three constraints can be expressed as follows: q 4 = 0,

q 5 = 0,

q 6 − f (t) = 0.

(64)

These constraints can also be expressed in terms of a velocity vector: v O · E1 = 0,

v O · E2 = 0,

v O · E3 − f˙ = 0.

(65)

Referring to Sect. 3.2, this representation of the constraints allows us to easily appeal to Lagrange’s prescription to write down a representation for the constraint force acting on the the top: Fc = μ1 E1 + μ2 E2 + μ3 E3 acting at O.

(66)

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This force is none other than the reaction force at O. With the help of the earlier results (63) we find that Fc · Fc ·

∂v O = 0, ∂ q˙ 1

∂v O = μ1 , ∂ q˙ 4

∂v O = 0, ∂ q˙ 2 ∂v O Fc · = μ2 , ∂ q˙ 5 Fc ·

∂v O = 0, ∂ q˙ 3 ∂v O Fc · = μ3 . ∂ q˙ 6 Fc ·

(67)

Thus, as anticipated, the constraint force Fc will be absent from three of the six Lagrange’s equations of motion.

6.3 Kinetic and Potential Energies The unconstrained kinetic energy of the top is m Tˆ = 2

 3 

q˙

3+i

  3   3+k Ei + ω2 e1 − ω1 e2 · q˙ Ek + ω2 e1 − ω1 e2

i=1

k=1

2 λt  2 λa  ϕ˙ + ψ˙ cos (ω) + ω˙ + ψ˙ 2 sin2 (ω) . + 2 2

(68)

The unconstrained potential energy of the top is Uˆ = mgE3 · (x O + e3 ) .

(69)

This potential energy is a function of q 6 and the Euler angle q 2 = ω. We ornament T and U with hats ˆ· to distinguish them from their constrained counterparts. Imposing the constraints (65), the constrained kinetic and potential energies can be found: 2 λt + m2  2 λa  ϕ˙ + ψ˙ cos (ω) + ω˙ + ψ˙ 2 sin2 (ω) 2 2 m ˙2 + f − m ω˙ f˙ sin (ω) , 2 U = mg f + mg cos (ω) . T =

(70)

Observe that λtO = λt + m2 is a mass moment of inertia relative to O.

6.4 The Equations of Motion The rigid body is subject to an applied moment Ma E3 and a force Fa e1 which follows e1 and acts at the tip of the top. The latter point, which we denote by X t is assumed

Some Surprising Conservative and Nonconservative Moments …

121

to have a position vector 1 e3 relative to O. The equations governing the generalized coordinates ψ, ω, and ϕ are obtained from Lagrange’s equation of motion: d dt



∂T ∂ q˙ k

 −

As vt = v O + ω × 1 e3 ,

∂T ∂U ∂ω ∂vt = − k + Ma E3 · k + Fa e1 · k . ∂q k ∂q ∂ q˙ ∂ q˙ ∂vt = 1 gk × e3 , ∂ q˙ k

(71)

(72)

where k = 1, 2, 3. Whence, the right-hand side of Lagrange’s equations can be simplified: d dt



∂T ∂ ψ˙



d dt d dt

∂T ∂U − =− + (Ma + 1 Fa sin (ω) cos (ϕ)) , ∂ψ ∂ψ   ∂T ∂T ∂U − =− − 1 Fa sin (ϕ) , ∂ ω˙ ∂ω ∂ω   ∂T ∂T ∂U − =− + Ma cos (ω) . ∂ ϕ˙ ∂ϕ ∂ϕ

(73)

Notice that the nonconservative follower force and nonconservative moment both contribute terms that are coordinate dependent to the right-hand sides of Lagrange’s equations of motion. Evaluating the derivatives of T and U , we find that ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ β1 α1 λa cos2 (ω) + λtO sin2 (ω) 0 λa cos(ω) ψ¨ O ⎦ ⎣ ⎣ 0 0 λt ω¨ ⎦ + ⎣ α2 ⎦ = ⎣ β2 ⎦ , α3 β3 0 λa λa cos(ω) ϕ¨    ⎡

(74)

M

where ⎤ ⎡  O ⎤ 2 λt − λa ω˙ ψ˙ cos α1  O(ω) sin (ω)2 − λa ϕ˙ ω˙ sin (ω) ⎣ α2 ⎦ = ⎣ λa ϕ˙ ψ˙ sin (ω) − λt − λa ψ˙ cos (ω) sin (ω) ⎦ , α3 −λa ψ˙ ω˙ sin (ω) ⎤ ⎡ ⎤ ⎡ Ma + 1 Fa sin (ω) cos (ϕ) β1 ⎣ β2 ⎦ = ⎣ m g + f¨  sin(ω) −  F sin (ϕ) ⎦ . 1 a β3 Ma cos (ω) ⎡

(75)

The mass matrix M on the left-hand side of Eq. (74) can be inferred from the expression for the kinetic energy function (70)1 . Note that this matrix is not positive definite when ω = 0, π. This is in line with our earlier remarks on singularities in

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Fig. 5 Equilibrium configuration of the Lagrange top subject to a follower force Fa e1 and a nonconservative moment M a E3

E3 g O E1

Fa e1

E2

M a E3

π 2

ψ0 +

¯ X e3 e2

Sect. 3.4. The α1,2,3 terms are related to sums of Christoffel symbols featuring the derivatives of the components of M with respect to the coordinates. We observe from the equations of motion that the effect of the vertical, timevarying motion of the point O is equivalent to changing the gravitational constant from g to g + f¨.

6.5 Equilibria and Linearized Equations of Motion Suppose that the point O is fixed. A static equilibrium of the top corresponds to stationary values of ψ, ω, and ϕ. The static values, which are distinguished by a subscript 0, are Ma + 1 Fa sin (ω0 ) cos (ϕ0 ) = 0, mg sin (ω0 ) − 1 Fa sin (ϕ0 ) = 0, Ma cos (ω0 ) = 0.

(76)

Whence, ψ0 is arbitrary, and Fa and Ma must satisfy the latter pair of the following relations for a static equilibrium to exist: ω0 =

π , 2

sin (ϕ0 ) =

mg , Fa 1

cos (ϕ0 ) = −

Ma . Fa 1

(77)

Thus, an infinite number of equilibrium states exist. As can be seen in Fig. 5, the top . is tilted at 90◦ to the vertical, ψ0 is arbitrary, and tan (ϕ0 ) = − mg Ma We now consider perturbations from an equilibrium position: ψ = ψ0 + η1 ,

ω = ω0 + η2 ,

ϕ = ϕ0 + η3 .

(78)

Some Surprising Conservative and Nonconservative Moments …

123

Inserting these expressions into the equations of motion (74), using the equilibrium conditions (77), performing Taylor series expansions, and ignoring terms of order 2 , we find the linearized equations ⎡

λtO ⎣ 0 0 

⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ η¨1 0 0 mg η1 0 0 0 λtO 0 ⎦ ⎣ η¨2 ⎦ + ⎣ 0 0 −Ma ⎦ ⎣ η2 ⎦ = ⎣ 0 ⎦ . 0 η¨3 0 Ma 0 η3 0 λa      M0

(79)

K0

The linearized system has six eigenvalues:

0,

0,

    M2 ±± − aO . λa λt

(80)

The pair of zero eigenvalues are a reflection of the arbitrariness of ψ0 . For non-zero values of Ma , the remaining four eigenvalues form a quartet. As two of the quartet have positive real parts, we conclude that the equilibrium is unstable. Observe that the mass matrix M0 is symmetric and can be readily deduced from the mass matrix M associated with the nonlinear equations of motion (74). In contrast to the conservative case presented in Eq. (43), the stiffness matrix K0 is asymmetric. This asymmetry can be attributed to the follower force load Fa e1 and the nonconservative moment Ma E3 acting on the top. The asymmetry of the stiffness matrix in the linearized equations (79) is a generic feature of equilibria of nonconservative mechanical systems including models for brake squeal.11 The asymmetry of the stiffness matrix also makes the equilibrium susceptible to dissipation-induced destabilization.12

6.6 Solving for the Reaction Force The constraint force Fc acting on the top can be determined from the equations of motion   ∂ Tˆ ∂ Uˆ ∂ω d ∂ Tˆ − k+3 = − k+3 + Ma E3 · k+3 k+3 dt ∂ q˙ ∂q ∂q ∂ q˙ + Fa e1 ·

11 For

∂vt ∂v O + Fc · k+3 . ∂ q˙ k+3 ∂ q˙

(81)

additional details on brake squeal, the reader is referred to the review article Kinkaid et al. (2003). 12 For a pseudospectral perspective on this topic see Kessler et al. (2007).

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These equations are computed using the unconstrained kinetic and potential energy functions. Using the identity vt = v O + ω × 1 e3 and

Eqs. (63) and (67), we find the following expressions for the components of Fc = 3k=1 μk Ek : d μ1 = dt d μ2 = dt d μ3 = dt

  

∂ Tˆ ∂ q˙ 4 ∂ Tˆ ∂ q˙ 5 ∂ Tˆ ∂ q˙ 6

 −

∂ Uˆ ∂ Tˆ + − Fa e1 · E1 , ∂q 4 ∂q 4

−

∂ Uˆ ∂ Tˆ + 5 − Fa e1 · E2 , 5 ∂q ∂q

−

∂ Uˆ ∂ Tˆ + 6 − Fa e1 · E3 . 6 ∂q ∂q

 

(82)

In other words, these three Lagrange’s equations of motion are simply the projections of F = m v˙¯ onto the Cartesian basis vectors.

7 The Satellite Dynamics Problem We now return to the problem considered by Lagrange. Suppose a rigid body of mass m is orbiting a fixed spherically symmetric rigid body of mass M (cf. Fig. 4b). We locate the origin of our coordinate system at the center of the body of mass M. Mac Cullagh’s approximate expressions for the gravitational force Fn , moment Mn , and potential energy Un on the rigid body of mass m are Fn ≈ mg,   3G M e R × (Je R ) , Mn ≈ R3     3G M GM G Mm tr(J) + − Un ≈ − (e R · (Je R )) , R 2R 3 2R 3 where mg = −

3G M G Mm eR − (2J + (tr(J) − 5e R · Je R ) I) e R , R2 2R 4

and R = ||¯x|| ,

eR =

x¯ . ||¯x||

(83)

(84)

(85)

Notice that the unit vector e R points from the center of mass of the spherically symmetric body to the center of mass of the body of mass m. In addition, if the moment of rotational inertia of the rigid body is ignored, then the moment vanishes and these expressions reduce to the familiar expression for a gravitational force on a particle of mass m. The moment Mn is often known as a gravity gradient torque

Some Surprising Conservative and Nonconservative Moments …

125

in the satellite dynamics literature and features in studies of the precession of the equinoxes and librations of the Moon.13 Suppose a set of cylindrical polar coordinates (r, ϑ, z) are used to parameterize the position vector x¯ and a set of 3-1-3 Euler angles are used to parameterize Q: q 1 = r, q 2 = ϑ, q 3 = z, q 4 = ψ, q 5 = ω, q 6 = ϕ. That is x¯ = r er + zE3 ,

˙ ϑ + z˙ E3 , v¯ = r˙ er + r ϑe

(86)

(87)

where er = cos (ϑ) E1 + sin (ϑ) E2 ,

eϑ = cos (ϑ) E2 − sin (ϑ) E1 .

(88)

Then, with the help of Eq. (35), we can readily identify the terms on the right-hand side of Lagrange’s equations of motion (1) with the conservative force and conservative moment acting on the rigid body of mass m: ∂Un = Fn · er , ∂r ∂Un = Mn · g1 , − ∂ψ −

∂Un = Fn · r eϑ , ∂ϑ ∂Un = Mn · g2 , − ∂ω

−

∂Un = Fn · E3 , ∂z ∂Un = Mn · g3 . − ∂ϕ −

(89)

Explicit expressions for Un , Fn , and Mn in terms of the Euler angles and the cylindrical polar coordinates can be readily obtained but they are very lengthy and disguise the important relations (89). We note that ∂Un 1 ∂Un ∂Un er − eϑ − E3 , ∂r r ∂ϑ ∂z ∂Un 1 ∂Un 2 ∂Un 3 g − g − g . Mn = − ∂ψ ∂ω ∂ϕ

Fn = −

(90)

These relations follow readily from Eqs. (15) and (89). Thus, we have been able to present transparent representations for the forces and moments featuring in Lagrange’s equations (1). After substituting for the Euler angles and the cylindrical polar coordinates into the expression for Un , one finds the well-known result that Un can be expressed as a function of ψ + ϑ. With the help of Eq. (89)2,4 , we can conclude that Fn · r eϑ = Mn · E3 . As is often assumed in examining the dynamics of artificial and natural satellites, if the body is axisymmetric with λ1 = λ2 then one finds that Un is independent of ϕ. Using Eq. (89)6 , we deduce that Mn has no component along the axis of symmetry e3 and lies entirely in the plane spanned by e1 and e2 . Additionally, we can then conclude from Lagrange’s equations of motion that the angular momentum component H · e3 is conserved. 13 See,

e.g., Goldstein (1980, Sect. 5–8), Hughes (1986), or Kane et al. (1983).

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Acknowledgements The work of Evan Hemingway was supported by a Berkeley Fellowship from the University of California at Berkeley and a U.S. National Science Foundation Graduate Research Fellowship.

References G.J. Allman, On the attraction of ellipsoids with a new demonstration of Clairaut’s theorem, being an account of the late Professor Mac Cullagh’s lectures on those subjects. Trans. R. Ir. Acad. 22, 379–395 (1855), http://www.jstor.org/stable/30079834 S.S. Antman, Solution to problem 71–24: angular velocity and moment potentials for a rigid body, by J.G. Simmonds. SIAM Rev. 14, 649–652 (1972). https://doi.org/10.1137/1014112 H. Baruh, in Analytical Dynamics (McGraw-Hill, Boston, 1999) J. Casey, On the advantages of a geometrical viewpoint in the derivation of Lagrange’s equations for a rigid continuum. ZAMP 46, S805–S847 (1995). https://doi.org/10.1007/978-3-0348-92292_41 J. Casey, V.C. Lam, On the relative angular velocity tensor. ASME J. Mech. Transm. Autom. Des. 108, 399–400 (1986). https://doi.org/10.1115/1.3258746 G. Desroches, L. Chèze, R. Dumas, Expression of joint moment in the joint coordinate system. ASME J. Biomech. Eng. 132(11), 114503 (2010). https://doi.org/10.1115/1.4002537 L. Euler, Nova methodus motum corporum rigidorum determinandi. Novi Commentari Academiae Scientiarum Imperalis Petropolitanae 20, 208–238 (1775). Reprinted in pp. 99–125 of Euler (1968) L. Euler, Découverte d’un nouveau principe de méchanique. Mémoires de l’Académie des Sciences de Berlin 6,185–217 (1752). The title translates to “On the discovery of a new principle of mechanics.” Reprinted in pp. 81–108 of Euler (1957) L. Euler, in Leonhardi Euleri Opera Omnia, volume 5 of II, ed. by J.O. Fleckenstein (Orell Füssli, Zürich, 1957) L. Euler, in Leonhardi Euleri Opera Omnia, volume 9 of II, ed. by C. Blanc (Orell Füssli, Zürich, 1968) H. Goldstein, in Classical Mechanics, 2nd edn. (Addison-Wesley, Reading, 1980) A.E. Green, W.T. Zerna, in Theoretical Elasticity, 2nd edn. (Clarendon Press, Oxford, 1968) E.S. Grood, W.J. Suntay, A joint coordinate system for the clinical description of three-dimensional motions: application to the knee. ASME J. Biomech. Eng. 105(2), 136–144 (1983). https://doi. org/10.1115/1.3138397 E.G. Hemingway, O.M. O’Reilly, Perspectives on Euler angle singularities, gimbal lock, and the orthogonality of applied forces and applied moments. Multibody Syst. Dyn. 1–26 (2018). https:// doi.org/10.1007/s11044-018-9620-0 S. Howard, M. Žefran, V. Kumar, On the 6 × 6 Cartesian stiffness matrix for threedimensional motions. Mech. Mach. Theory 33(4), 389–408 (1998). https://doi.org/10.1016/ S0094-114X(97)00040-2 P.C. Hughes, in Spacecraft Attitude Dynamics (Wiley, New York, 1986) T.R. Kane, P.W. Likins, D.A. Levinson, in Spacecraft Dynamics (McGraw-Hill, New York, 1983) P. Kessler, O.M. O’Reilly, A.-L. Raphael, M. Zworski, On dissipation-induced destabilization and brake squeal: a perspective using structured pseudospectra. J. Sound Vib. 308(1–2), 1–11 (2007). https://doi.org/10.1016/j.jsv.2007.06.066 N.M. Kinkaid, O.M. O’Reilly, P. Papadopoulos, Automotive disk brake squeal. J. Sound Vib. 267(1), 105–166 (2003). https://doi.org/10.1016/S0022-460X(02)01573-0 J.L. Lagrange, Théorie de la libration de la Lune, et des autres phénomènes qui dépendent de la figure non sphèrique de cette Planète. Nouveaux Mémoires de l’Académie Royale des Sciences et des Belles-Lettres de Berlin 30, 203–309 (1780). Reprinted in pp. 5–122 of Lagrange (1870)

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J.L. Lagrange. in Oeuvres de Lagrange, vol. 5, ed. by J.-A. Serret (Gauthier-Villars, Paris, 1870) D. Lewis, T. Ratiu, J.C. Simo, J.E. Marsden, The heavy top: a geometric treatment. Nonlinearity 5(1), 1–48 (1992), http://stacks.iop.org/0951-7715/5/i=1/a=001 J.E. Marsden, T.S. Ratiu, in Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, vol. 17, 2nd edn., Texts in Applied Mathematics (Springer, New York, 1999). https://doi.org/10.1007/978-0-387-21792-5 A. J. McConnell, in Applications of the Absolute Differential Calculus (Blackie and Son, London, 1947). Corrected reprinted edition J.K. Nichols, O.M. O’Reilly, Verifying the equivalence of representations of the knee joint moment vector from a drop vertical jump task. Knee J. 24(2), 484–490 (2017). https://doi.org/10.1016/j. knee.2016.10.019 O.M. O’Reilly, The dual Euler basis: constraints, potentials, and Lagrange’s equations in rigid body dynamics. ASME J. Appl. Mech. 74(2), 1–10 (2007). https://doi.org/10.1115/1.2190231 O.M. O’Reilly, in Intermediate Dynamics for Engineers: A Unified Treatment of Newton-Euler and Lagrangian Mechanics (Cambridge University Press, Cambridge, 2008) O.M. O’Reilly, in Modeling Nonlinear Problems in the Mechanics of Strings and Rods (Springer, New York, 2017). https://doi.org/10.1007/978-3-319-50598-5 O.M. O’Reilly, A.R. Srinivasa, On potential energies and constraints in the dynamics of rigid bodies and particles. Math. Probl. Eng. Theory Methods Appl. 8(3), 169–180 (2002). https://doi.org/10. 1080/10241230215286 O.M. O’Reilly, A.R. Srinivasa, A simple treatment of constraint forces and constraint moments in the dynamics of rigid bodies. ASME Appl. Mech. Rev. 67(1), 014801 (2014). https://doi.org/10. 1115/1.4028099 O.M. O’Reilly, M. Sena, B.T. Feely, J.C. Lotz, On representations for joint moments using a joint coordinate system. J. Biomech. Eng. 135(11), 114504 (2013). https://doi.org/10.1115/1.4025327 A.G. Schache, R. Baker, On the expression of joint moments during gait. Gait Posture 25(3), 440– 452 (2007). https://doi.org/10.1016/j.gaitpost.2006.05.018 M.D. Shuster, A survey of attitude representations. Am. Astronaut. Soc. J. Astronaut. Sci. 41(4), 439–517 (1993) J.G. Simmonds, in A Brief on Tensor Analysis (Springer, New York, 1982) J.G. Simmonds, Moment potentials. Am. J. Phys. 52, 851–852 (1984). https://doi.org/10.1119/1. 13525. Errata published on p. 277 of Vol. 53 J.L. Synge, B.A. Griffith, in Principles of Mechanics, 3rd edn. (McGraw-Hill, New York, 1959) M. Žefran, V. Kumar, A geometrical approach to the study of the Cartesian stiffness matrix. ASME J. Mech. Des. 124(1), 30–38 (2002). https://doi.org/10.1115/1.1423638 H. Ziegler, in Principles of Structural Stability (Blaisdell, Waltham, MA, 1968)

Classical Results and Modern Approaches to Nonconservative Stability Oleg N. Kirillov

Abstract Stability of nonconservative systems is nontrivial already on the linear level, especially, if the system depends on multiple parameters. We present an overview of results and methods of stability theory that are specific for nonconservative applications. Special attention is given to the topics of flutter and divergence, reversible- and Hamiltonian-Hopf bifurcation, Krein signature, modes and waves of positive and negative energy, dissipation-induced instabilities, destabilization paradox, influence of structure of forces on stability and stability optimization.

1 Introduction 1.1 “It was Greenhill who Started the Trouble... ...though he never knew it,” remarked Gladwell (1990) in his historical account of the genesis of the field of nonconservative stability. As many of his scientific contemporaries, Greenhill successfully combined his interest to pure mathematical subjects, such as elliptic functions, with contributions to applied problems of ballistics (Greenhill 1879), hydrodynamics (Greenhill 1880), and elasticity (Greenhill 1881) coming from the flourishing industries of the British Empire. In particular, motivated by the problem of buckling of propeller-shafts of steamers he analyzed in Greenhill (1883) stability of an elastic shaft of a circular cross-section, length L, and mass per unit length m under the action of a compressive force, P, and an axial torque, M. Figure 1 taken from Gladwell (1990) illustrates five possible in this system boundary conditions: I. Symmetric clamped-clamped shaft II. Asymmetric clamped-clamped shaft III. Clamped-free shaft O. N. Kirillov (B) Northumbria University, Newcastle upon Tyne NE1 8ST, United Kingdom e-mail: [email protected] © CISM International Centre for Mechanical Sciences 2019 D. Bigoni and O. Kirillov (eds.), Dynamic Stability and Bifurcation in Nonconservative Mechanics, CISM International Centre for Mechanical Sciences 586, https://doi.org/10.1007/978-3-319-93722-9_4

129

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Fig. 1 Five realizations of Greenhill’s elastic shaft loaded by a compressing force, P, and an axial torque, M, corresponding to five different boundary conditions (from Gladwell 1990)

IV. Clamped-hinged shaft V. Hinged-hinged shaft In the absence of the axial torque (M = 0), the Greenhill problem reduces to the famous Euler’s buckling under compression of 1757. The critical load at the onset of the static instability can be found by the equilibrium method, which seeks values of the axial force, for which there are nontrivial equilibrium configurations. This yields the Euler formula for the critical buckling force Pcr = k

π2 E I BC I II III IV V , , where 2 k 4 1 1/4 2.046 1 L

(1)

E is the Young modulus and I is the moment of inertia of a (circular) cross-section of the shaft. In contrast to the Euler buckling case, Greenhill set P = 0 and tried to find the critical torque that causes buckling of the shaft. Using the equilibrium method, he managed to find the critical torque for the boundary conditions I, II, and V (Greenhill 1883; Ziegler 1953a, b; Gladwell 1990) Mcr = k

πE I BC I II III IV V , where . k 2.861 2 ? ? 2 L

(2)

The cases III and IV have not been analyzed by Greenhill and remained untreated until Nicolai (1928) reconsidered a variant of the case IV, in which the axial torque

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131

is replaced with the follower torque, M, such that the vector of the torque is directed along the tangent to the deformed axis of the shaft at the end point (Gladwell 1990). Nicolai (1928) had established that no nontrivial equilibrium configuration of the shaft exists different from the rectilinear one, meaning stability for all magnitudes, M, of the follower torque and thus k = ∞ in (2). Being unsatisfied with this overoptimistic result, Nicolai realized that the equilibrium method does not work properly in the case of the follower torque. He decided to study small oscillations of the shaft about its rectilinear configuration using what is now known as the Lyapunov stability theory (Lyapunov 1992) that, in particular, can predict instability via eigenvalues of the linearized problem. Surprisingly, it turned out that there exist eigenvalues with positive real parts (instability) for all magnitudes of the torque, meaning that the critical value of the follower torque for an elastic shaft of a circular cross-section is actually Mcr = 0, i.e. k = 0 in (2). Because of its unusual behavior, this instability phenomenon received a name “Nicolai’s paradox” (Nicolai 1928; Gladwell 1990). In 1951-56 Ziegler re-considered the five original Greenhill problems with the Lyapunov approach and found that at P = 0 the shaft is unstable in cases III and IV for all values of the axial torque M, just as in Nicolai’s problem with the follower torque (Ziegler 1951a, b, 1953a, b, 1956). Mcr = k

πE I BC I II III IV V . , where k 2.861 2 0 0 2 L

(3)

Moreover, Ziegler realized that “Stability problem for a shaft loaded by an axial torque M, is generally non-conservative, as in the cases III, IV, and V, where the end slope is unconstrained. Only in exceptional cases the work of such torques in a virtual deformation can be represented as a variation of an integral” and the problem is conservative, as in cases I and II, where the equilibrium method gives the correct critical torque. “In any case”, concluded Ziegler, “the results show that even very simple models are not conservative and, if they occur as stability problems, they should be treated dynamically”, i.e. with the use of the Lyapunov approach (Ziegler 1951a, b). Note that already Nicolai (1929) realized that the cases III and IV do not represent generic situations because it is possible to modify the end conditions, or consider a shaft with unequal stiffness (non-circular cross-section) yielding a nonzero critical torque (Bolotin 1963; Gladwell 1990). These conclusions were later confirmed by Ziegler (1956) and developed further in the recent works on the Nicolai paradox by Seyranian and Mailybaev (2011) and Luongo et al. (2016).

1.2 Greenhill’s Shaft as a Non-self-adjoint Problem Small vibrations of the Greenhill’s shaft near its non-deformed rectilinear configuration are described by the following partial differential equation (Bolotin 1963)

132

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 l0 ∂z4 w + l1 ∂z3 w + l2 ∂z2 w + m∂t2 w = 0, z ∈ [0, L], w =

w1 w2



where the matrices l0 , l1 , and l2 are       EI 0 0 M P 0 l0 = , l1 = , l2 = 0 EI −M 0 0 P

(4)

(5)

The nonconservative clamped-free case (III) is characterized by the following boundary conditions w(0) = w (0) = 0, l0 w (L) + l1 w (L) = 0, l0 w (L) + l1 w (L) + l2 w (L) = 0,

(6)

corresponding to the constrained deflection and slope at the clamped end (z = 0) and vanishing axial force and axial torque at the free end (z = L). Separating time with w = ueλt , and introducing the matrix  l4 (λ) = λ

2

m 0 0 m

 ,

we come to the boundary eigenvalue problem L(λ)u = l0 ∂z4 u + l1 ∂z3 u + l2 ∂z2 u + l4 (λ)u = 0

(7)

with the boundary conditions u(0) = u (0) = 0, l0 u (L) + l1 u (L) = 0, l0 u (L) + l1 u (L) + l2 u (L) = 0, 

(8)

where prime denotes partial differentiation with respect to z. The equilibrium state is unstable if there is a value of λ with positive real part. Integrating by parts the inner product (Lu, v) = v T L(λ)u, where the bar indicates complex conjugation, we obtain (Kirillov 2010) 

L 0

T

 v T L(λ)ud x =

L

T

(L (λ)v)T ud x + vT Lu.

0

Here L (λ)v =: L† (λ)v is the adjoint differential expression (Kirillov 2010)

(9)

Classical Results and Modern Approaches to Nonconservative Stability

L† (λ)v =

133

4  T (−1)4−q ∂z4−q (lq v) = l0 ∂z4 v + l1 ∂z3 v + l2 ∂z2 v + l4 (λ)v,

(10)

q=0

the vectors u and v are T T T  T  T uT = (uT (0), uz T (0), uz T (0), u z (0), u (L), uz (L), uz (L), uz (L)) T T T  T  T vT = (vT (0), vz T (0), vz T (0), v z (0), v (L), vz (L), vz (L), vz (L))

and the block matrix L := (li j )  L=

⎛

l00 ⎜ l10 −L(0) 0 , L(z) = ⎜ ⎝ l20 0 L(L) l30 

l01 l11 l21 0

l02 l12 0 0

⎞ l03 0 ⎟ ⎟. 0 ⎠ 0

The matrices li j are expressed through the matrices of the differential expression as (Kirillov 2010, 2013a) li j =



3− j  (−1)k Mikj ∂zk−i l3− j−k ,

Mikj :=

k=i

which yields

i+ j ≤3 k≥i ≥0

0, i + j > 3 k < i

k! , (k−i)!i!

⎛

⎞ 0 l2 l1 l0 ⎜ −l2 −l1 −l0 0 ⎟ ⎟ L(z) = ⎜ ⎝ l1 l0 0 0 ⎠ , −l0 0 0 0

where 0 denotes the 2 × 2 zero matrix. Boundary conditions (8) can be written in the matrix form as Uk u =

3  j=0

Ak j u(z j) (z = 0) +

3 

Bk j u(z j) (z = L) = 0, k = 1, . . . , 4

j=0

where A10 = A21 = I, B32 = l1 , B33 = l0 , B42 = l2 , B43 = l1 , B44 = l0  and all of other matrices Ak j and Bk j are zero. Introducing the matrices A = (Ak j )z=0  and B = (Bk j )z=L and composing the block matrix U = [A, B] we can finally write the boundary conditions (8) in the compact matrix form (Kirillov 2010, 2013a) Uu = [A, B]u = 0.

(11)

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Fig. 2 (Left) Greenhill-III problem with the axial torque described by the problem (14). (Right) Nicolai’s variant of the Greenhill-III problem with the follower torque described by the problem (13) which is adjoint to (14) (from Ziegler 1951a)

Extend the original matrix U to a square non-degenerate matrix U by an appropriate  choice of the auxiliary matrices  A and B  U = [A, B] → U =

 AB   , det U = 0. AB

Then, we can obtain the formula for calculation of the matrix V of the boundary conditions for the adjoint differential expression (10) Vv = 0  and the auxiliary matrix V 

Choosing

 −V V

T = LU ⎛

0 ⎜0  A=⎜ ⎝0 0

−1

0 0 0 0

I 0 0 0

 =

−L(0) 0 0 L(L)

⎞ ⎛ 0 0 ⎜0 I⎟ ⎟, B =⎜ ⎝0 0⎠ 0 I



0 0 I 0

0 0 0 0

AB   AB

−1 (12)

⎞ 0 0⎟ ⎟, 0⎠ 0

where I is the 2 × 2 identity matrix and 0 denotes the 2 × 2 zero matrix, we find that det U = (E I )4 = 0. Then, the differential expression (10) and the relation (12) yield the adjoint boundary eigenvalue problem:

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l0 ∂z4 v + l1 ∂z3 v + l2 ∂z2 v + l4 (λ)v = 0, v(0) = v (0) = 0, v (L) = 0,  l0 v (L) + l2 v (L) = 0,

(13)

which is instructive to compare with the original boundary eigenvalue problem (7), (8): l0 ∂z4 u + l1 ∂z3 u + l2 ∂z2 u + l4 (λ)u = 0, u(0) = u (0) = 0, l0 u (L) + l1 u (L) = 0, l0 u (L) + l1 u (L) + l2 u (L) = 0.

(14)

It is easy to see that the differential expressions of the problems (14) and (13) are identical and the difference  comesfrom the terms in the boundary conditions 0 M that is non-zero at nonzero torque M. Only that contain the matrix l1 = −M 0 if M = 0 the matrix l1 = 0 and the boundary conditions of the original boundary eigenvalue problem and the adjoint boundary eigenvalue problem coincide. Therefore, only in the absence of the torque (M = 0), the problem (14) as well as its adjoint (13), is self-adjoint and represents a conservative system, which is not surprising in view that it is the Euler buckling problem for an elastic shaft. In case when M = 0 the boundary conditions of the adjoint problem (13) do not coincide with the boundary conditions of the original problem (14), manifesting the non-self-adjoint nature of the non-conservative Greenhill-III problem (Ziegler 1951a, b, 1956). It is well-known that adjoint problems have the same characteristic equation that determines eigenvalues. Hence, stability properties of (14) and (13) are identical despite they have different mechanical meaning. The boundary value problem (14) corresponds to the original Greenhill-III clamped-free shaft loaded by the axial force and the axial torque, Fig. 2(left). It turns out that its adjoint given by (13) corresponds to the Nicolai’s variant of the Greenhill-III problem with the axial force and the follower torque, Fig. 2(right), Bolotin (1963). Both mechanical systems shown in Fig. 2 are nonconservative but have the same spectrum and, therefore, the same stability properties. In the both problems the critical value of the torque at P = 0 is Mcr = 0 (Nicolai’s paradox) no matter whether the torque is axial or follower.

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(b)

(c)

(a)

Fig. 3 a Pflüger’s hinged-hinged column loaded by the distributed follower force (static instability, or divergence), b Pflüger’s clamped-free column of the mass per unit length, m, carrying the end mass, M, and loaded by the concentrated follower force at the tip, c Beck’s column loaded by the concentrated follower force is a particular case of b with the end mass M = 0 (dynamic instability, or flutter), from Gladwell (1990)

1.3 From Follower Torques to Follower Forces A remarkable property of the Greenhill’s five problems established by Nicolai and Ziegler is that, depending on boundary conditions, they could be both conservative and nonconservative. In conservative cases I and II, the Greenhill’s shaft loses stability of the rectilinear equilibrium statically, i.e. without vibrations (divergence instability). In the nonconservative cases III and IV (and their Nicolai’s variants with the follower torque), however, the mechanism of instability involves growing oscillations about the rectilinear equilibrium and is called flutter. Whereas divergence is the only possible type of instability in conservative systems, the nonconservative systems possess both flutter and divergence. For instance, the nonconservative Greenhill-V shaft loses its stability by divergence (Greenhill 1883; Ziegler 1951a; Gladwell 1990). In 1950 Pflüger established divergence instability of a nonconservative hinged-hinged elastic column loaded by a distributed follower force, Fig. 3a. Note that columns loaded by distributed follower forces provide a basis for mathematical modeling of some biomechanical objects. We mention, for instance, recent works on the human spine (Rohlmann et al. 2009), centipede locomotion (Aoi et al. 2013), and flutter of flagella under the action of distributed tangential follower forces caused by cytoskeletal motor proteins (Bayly and Dutcher 2016). Immediately after the Pflüger’s work, Beck (1952) has found flutter of a clampedfree elastic column of length, L, and mass per unit length, m, loaded by the

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137

Fig. 4 (Left) Stability map for the elastic Pflüger column in the “load” - “mass ratio” plane (from 2 Ryu and Sugiyama 2003). (Right) Load parameter p = PELI versus dimensionless squared vibration 2 4

frequency ξ = mωE IL for the Pflüger column at different mass ratios μ when 1/μ is close to zero (from Sugiyama et al. 1976)

Fig. 5 Molecular motors (kinesin) transporting membranes along microtubules (cytoskeletal filaments) inside a cell cause tangential follower forces acting on the microtubules (from Vale Lab web site https://valelab4.ucsf.edu/ external/moviepages/ moviesMolecMotors.html)

concentrated follower force at its tip, Fig. 3c. In 1955 Pflüger re-considered the Beck’s column with an end mass, M, (see Fig. 3b) and found it flutter-unstable for almost all mass ratios μ = mML , except for the case when m = 0 or μ → ∞, Fig. 4(left). Figure 4(right) shows the load parameter of the Pflüger column as a function of the squared dimensionless eigenfrequency at small values of μ−1 . The lower hyperbolic branch has its maximum at the critical flutter value of the load. The interval of loads

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Fig. 6 Linear reversible-Hopf bifurcation: (Left) eigenvalues of a stable reversible system are all imaginary and semi-simple; (centre) a pair of two simple imaginary eigenvalues (as well as the complex conjugate pair) merges into a pair of double imaginary eigenvalues with the Jordan block at the flutter threshold; (right) the pair of the double non-semi-simple eigenvalues unfolds into a complex quadruplet inside the flutter domain (from Lamb and Roberts 1998)

corresponding to flutter is between the minimum of the upper hyperbolic branch and the maximum of the lower hyperbolic branch in Fig. 4(right). As μ increases, the size of the flutter interval tends to zero so that in the limit μ → ∞ the two hyperbolic branches merge and form a crossing at the load p ≈ 20.19 (Sugiyama et al. 1976). Exactly at the crossing the eigenfrequency is double zero with the Jordan block, which corresponds to the onset of the divergence instability. In the μ → ∞ limit the Pflüger column is weightless and is known as the Dzhanelidze column (Bolotin 1963). The opposite limit, μ → 0, of the Pflüger column is known as the Beck column with the critical flutter load p ≈ 20.05. It is instructive to note that the critical load reaches its local maxima exactly in these two limiting cases, Fig. 4(left). Connection of a maximum of the critical load and a crossing in the load-frequency plane (Fig. 4(right)) is not a coincidence. Already Mahrenholtz and Bogacz (1981) emphasized that “In the case of complicated structures there may appear different shapes of characteristic curves, and only an analysis in the [load-frequency] plane may assure the correct results for the design of structures subjected to nonconservative loads”. A general perturbation approach to local extrema associated with the crossings of characteristic curves has been developed in Kirillov and Seyranian (2002a, b). The follower force problems of 1950-s are increasingly popular nowadays in the mathematical modeling of mechanics underlying complex cellular phenomena caused by molecular motors that translocate along cytoskeletal filaments, carrying cargo, Fig. 5. It turns out that molecular motors produce piconewton tangential follower forces acting on filaments and resulting in their flutter, which is well described by the classical continuous models of Beck and Pflüger and their discrete analogue — the Ziegler pendulum (Ziegler 1952; Saw and Wood 1975) — as is shown in the recent work by De Canio et al. (2017). Note that the Ziegler pendulum has been realized experimentally by Bigoni and Noselli (2011) and the Pflüger column by Bigoni et al. (2018).

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139

Fig. 7 Steady-state bifurcation in a reversible system: (left) eigenvalues of a stable reversible system are all imaginary and semi-simple; (centre) a conjugate pair of simple imaginary eigenvalues merges into a double zero eigenvalue with the Jordan block at the divergence threshold; (right) the double zero non-semi-simple eigenvalue splits into two real eigenvalues of opposite signs inside the divergence domain

2 Reversible and Circulatory Systems O’Reilly, Malhotra and Namachchivaya (1995, 1996) observed that the governing equations of the classical structures with nonconservative follower loads possess a special type of symmetry, which largely determines their stability properties. This symmetry, known as the reversible symmetry, can be defined with reference to the differential equation (Lamb and Roberts 1998) dx = g(x), x ∈ Rn dt which is said to be R-reversible (R−1 = R) if it is invariant with respect to the transformation (x, t) → (Rx, −t), implying that the right hand side should satisfy Rg(x) = −g(Rx). If x = x0 is a reversible equilibrium such that Rx0 = x0 , and A = ∇g is the linearization matrix about x0 , then A = −RAR, and the characteristic polynomial det(A − λI) = det(−RAR − RλR) = (−1)n det(A + λI), implies that ±λ, ±λ are eigenvalues of A (Lamb and Roberts 1998). Due to the spectrum’s symmetry with respect to both the real and imaginary axes of the complex plane, stability requires that all the eigenvalues of A stay on the imaginary axis, Fig. 6(left). Transition from stability to flutter instability occurs through the reversible-Hopf bifurcation (Lamb and Roberts 1998) that requires the generation of a non-semisimple double pair of imaginary eigenvalues and its subsequent separation into a complex quadruplet, Fig. 6. Transition from stability to divergence instability is accompanied by the steadystate bifurcation in which two simple imaginary eigenvalues merge at zero and then split into a real couple with the opposite signs, Fig. 7.

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An important for applications fact is that reversible are all equations of second order (Lamb and Roberts 1998): d 2x = f(x). dt 2 Indeed, denoting x1 = x and x2 =

dx dt

(15)

we can write the first-order system

x˙ 1 = x2 , x˙ 2 = f(x1 ), which is invariant under the transformation x1 → x1 , x2 → −x2 , t → −t. The system (15) is reversible also in the case when the positional force f(x) has a non-trivial curl ∇ × f(x) = 0, which makes the reversible system nonconservative. Such nonconservative curl forces (Berry and Shukla 2016) that cannot be derived from any potential appear in modern opto-mechanical applications, including optical tweezers (Wu et al. 2009; Simpson and Hanna 2010; Sukhov and Dogariu 2017) and light robotics (Phillips et al. 2017). In mechanics these nonconservative positional forces are known as circulatory forces for producing non-zero work along a closed circuit (Ziegler 1953a, b). A circulatory force acting on an elastic structure and remaining directed along the tangent line to the structure at the point of its application during deformation is the already familiar to us follower force (Ziegler 1952; Bolotin 1963). We notice that in aeroelasticity the term ‘circulatory’ is frequently associated with the lift force in the Theodorsen lift model (Theodorsen 1935) that was developed to explain flutter instability occurring in aircrafts at high speeds. The Kutta–Joukowski theorem relates the lift on an airfoil to a circulatory component (circulation) of the flow around the airfoil. The circulation is the contour integral of the tangential velocity of the air on a closed loop (circuit) around the boundary of an airfoil. Hence the name circulatory lift force, see Pigolotti et al. (2017). Remarkably, the Theodorsen model is nonconservative and the non-potential positional forces arising in it due to the circulatory lift are simultaneously the circulatory forces in the sense of Ziegler (Pigolotti et al. 2017).

2.1 Zubov-Zhuravlev Decomposition of Non-potential Force Fields Zubov (1970) established the following instructive result:

Classical Results and Modern Approaches to Nonconservative Stability

141

Fig. 8 (Left) The non-potential force field f = (x, x y)T = f + f ; (right) its circulatory part f = T  2 − y3 , x3y

Theorem 2.1 (Zubov 1970) Let f(t, x) : R+ ×  Rn → Rn be a real-valued conn T xi f i (t, x) be a continuously tinuous vector-function and let w(t, x) = f x = i=1 differentiable function with respect to components of x. Then, (a) there exists a real-valued function V (t, x) : R+ × Rn → R, which is continuous and continuously differentiable with respect to components of x; (b) f(t, x) possesses the following representation f(t, x) = −∇x V (t, x) + Px,

(16)

where P(t, x) is an n × n skew-symmetric matrix (PT = −P) with the elements that are continuous functions of t and components of x. Example: Let

 f(t, x) =

x xy

 , x=

  x y

(17)

According to Theorem 2.1, there exists the following decomposition ⎛ ∂V ⎞ f(t, x) = − ⎝ ⎛ =⎝

∂x

∂V ∂y

x+ 2x y 3

⎠+ y 3 y2 3

⎞



⎛

⎠+⎝

0 −1 1 0 2

− y3 xy 3

  x y

⎞ ⎠=



x xy

 ,

(18)

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O. N. Kirillov 2

2

where V (t, x) = − x2 − x3y , see Fig. 8. Notice that many examples of nonconservative force fields and their curls can be found in the modern literature on optical tweezers, see e.g. Wu et al. (2009), Simpson and Hanna (2010), Sukhov and Dogariu (2017), and light robotics (Phillips et al. 2017). Zhuravlev (2007, 2008) proposed an algorithm for constructing the Zubov decomposition, in particular, of nonlinear generalized forces in the Lagrange equations. Here we are interested in positional forces only. Let T denote kinetic energy of a mechanical system. Consider the Lagrange equations   ∂T d ∂T − = f i (t, q1 , . . . , qn ), i = 1, . . . , n. dt ∂ q˙i ∂qi We assume that the generalized forces f i have positional character, being functions of time and generalized coordinates only. Let us first assume that the generalized forces f are linear f = −Aq, A = AT . Recall that the n × n matrix A can be uniquely represented as the sum A=

A − AT A + AT + = K + N, 2 2

where K = KT is a real symmetric matrix and N = −NT is a real skew-symmetric matrix. Then, we can write the generalized positional force as f = −Kq − Nq, where the force f = −Kq is derived from the potential V (q) = 21 qT Kq: f = −∇V (q) and the circulatory force f = −Nq is orthogonal to the vector of generalizes coordinates qT f = 0. Indeed,

qT f = −qT Nq = (qT NT q)T = qT Nq

⇒

qT Nq = 0.

A linear circulatory system is thus defined as (Ziegler 1953a, b, 1956) q¨ + Kq + Nq = 0. This is a reversible system (O’Reilly, Malhotra and Namachchivaya 1996).

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143

Let us calculate the work of the linear positional force f on the displacement q with the frozen time  1  1  1 1 qT f(sq)ds = − qT Kqsds − qT Nqsds = − qT Kq W = 2 0 0 0 Therefore, the potential component of the linear positional force f is f = ∇W = −∇V and the circulatory component is just f = f − f . Zhuravlev (2007, 2008) employs this idea for the decomposition of nonlinear non-potential force fields into a potential and circulatory parts. Following Zhuravlev (2007, 2008), we define the potential part of f as f = −∇V ,  1 where qT f(sq)ds. V =− 0

Then, the circulatory part of the nonlinear force f is f = f − f , f · q = 0. Example: Decompose the non-potential vector field f into the potential and circulatory parts   x f= = f + f . xy First, construct the potential function V of the potential part of the field 

1

V =−



1

[x f x (sx, sy) + y f y (sx, sy)]ds = −

0

(x 2 s + x y 2 s 2 )ds

0

=−

x y2 x2 − . 2 3

Then, find the potential part of f ⎛ ∂V ⎞ f = − ⎝

∂x

∂V ∂y

⎛

⎠=⎝

x+

y2 3

⎞ ⎠.

2x y 3

Finally, determine the circulatory part of f ⎛ y2 ⎞    −3 y 0 −1 x   ⎝ ⎠ f =f−f = = , f · q = 0, 1 0 y 3 xy 3

in agreement with Theorem 2.1. Note that ∇ × f = yez = 0. The decomposition is unique up to the class of potential forces that are simultaneously orthogonal to the vector of coordinates: qT ∇V = 0. For instance, the force derived from the potential V (x, y) = x/(x + y) belongs to this class (Zhuravlev 2007, 2008)

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O. N. Kirillov Flutter

Stability

Fig. 9 (Left) Rotating shaft by Shieh and Masur (1968). (Right) Stability map of the model (21) with k1 = 1 and m = 1

⎛ ∂V ⎞ f = −⎝

∂x

∂V ∂y

⎠=

1 (x + y)2



−y x

 , f · q = 0.

In this case, obviously, ∇ × f = −∇ × (∇V ) = 0.

2.2 Circulatory Forces in Rotor Dynamics Non-potential circulatory forces historically originated in equations of rotor dynamics when dissipation both in rotor and stator was taken into account. The two types of damping were introduced by Kimball (1925) in order to explain a new type of instability observed in built-up rotors at high speeds in the early 1920s. Smith (1933) implemented this idea in a model of a rotor carried by a flexible shaft in flexible bearings with the linearization given by the equation z¨ + D˙z + 2G˙z + (K + (G)2 )z + βNz = 0

(19)

where zT = (x, y) is the position vector in the framerotating  with the shaft’s angular 0 −1 velocity , D = diag(δ + β, δ + β), G = J, J = , K = diag(k1 , k2 ), and 1 0 N = J. In Smith’s model (19) the stationary (in the laboratory frame, and thus external with respect to the shaft) damping coefficient β > 0 represents the effect of viscous damping in bearing supports while the rotating damping coefficient δ > 0 represents the effect of viscous damping in the shaft itself (internal damping). The term βNz in Eq. (19) corresponds to circulatory forces.

Classical Results and Modern Approaches to Nonconservative Stability

145

In a more general model of the rotating shaft by Shieh and Masur (1968), the diagonal elements of the damping matrix in Eq. (19) are allowed to be different. In fact, Shieh and Masur (1968) model the shaft as the point mass m which is attached by two springs with the stiffness coefficients k1 and k2 = k1 + κ and two dampers with the coefficients μ1 and μ2 to a Cartesian coordinate system Ouv rotating at constant angular velocity , Fig. 9 (left). A non-conservative positional force which is proportional to the radial  distance  −βv  of the mass from the origin and perpendicular to the radius vector f = acts βu on the mass. Such a force on the shaft in the bearings may arise in a rotating fluid or in an electromagnetic field. The linearized equations of motion of the shaft have the form (Shieh and Masur 1968; Kirillov 2013a, 2011a, b) m u¨ + μ1 u˙ − 2mv˙ + (k1 − m2 )u + βv = 0, m v¨ + μ2 v˙ + 2mu˙ + (k2 − m2 )v − βu = 0.

(20)

Assuming that damping is absent (μ1 = 0, μ2 = 0) and that the shaft is not rotating  = 0 we reduce the model (20) to the motion of the planar oscillator under the action of a nonconservative circulatory force m u¨ + k1 u + βv = 0, m v¨ − βu + k2 v = 0.

(21)

˜ λt , introducing the stiffness Separating time in (21) with u = ue ˜ λt and v = ve anisotropy κ = k2 − k1 , and writing the solvability condition for the resulting system of two algebraic equations we end up with the quadratic equation in λ2 . Its solutions  λ = ±i

2m(2k1 + κ ±



−4β 2 + κ2 )

2m

are imaginary (stability) if κ2 > 4β 2 and form a complex quadruplet with negative and positive real parts (flutter) if κ2 (22) β2 > . 4 This conical flutter domain is shown in Fig. 9(right) in the (κ, β)-plane of the stiffness anisotropy, κ, and magnitude of the circulatory force, β. Note that flutter instability occurs already at β > 0 if the stiffness is symmetric (κ = 0), similarly to the Nicolai paradox for the cantilever rod of circular cross-section under a follower or axial torque. However, stiffness anisotropy (κ = 0), no matter how small, increases the flutter threshold as |β f | = |κ|/2. Again, similar to the disappearance of the Nicolai’s paradox in rods of non-circular cross-section (Nicolai 1929). This is not just a coincidence. Indeed, the linearization of a two-degrees-of-freedom model of the

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Greenhill-Nicolai problem considered recently by Luongo and Ferretti (2016) is described exactly by Eq. (21).

2.3 Stability Criteria for Circulatory Systems Let us consider a circulatory system x¨ + (K + N)x = 0

(23)

where K = KT and N = −NT are real m × m matrices. Separating time in (23) with the standard substitution x = ueλt , write the characteristic polynomial p(λ) = det(λ2 + K + N) p(λ) = a0 λ2m + a1 λ2m−2 + a2 λ2m−4 + . . . + λ2 am−1 + am . Write the 2m × 2m discriminant matrix for p(λ) ⎛

a0 ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 =⎜ ⎜··· ⎜ ⎜··· ⎜ ⎝ 0 0

a1 a2 a3 · · · an ma0 (m−1)a1 (m−2)a2 · · · am−1 a1 a2 · · · am−1 a0 0 ma0 (m−1)a1 · · · 2am−2 ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· 0 0 ··· 0 a0 0 0 ··· 0 0

0 0 am am−1 ··· ··· a1 ma0

⎞ 0 0 0 0 ⎟ ⎟ 0 0 ⎟ ⎟ 0 0 ⎟ ⎟ ··· ··· ⎟ ⎟ ··· ··· ⎟ ⎟ · · · am ⎠ · · · am−1

(24)

Consider a sequence of determinants of all even-order submatrices along the main diagonal of  starting from the upper left corner  det 1 = det

a0 a1 0 ma0

 , det 2 , · · · , det m = det 

(25)

Theorem 2.2 (Gallina criterion Gallina 2003) A necessary and sufficient condition for all the eigenvalues λ of the eigenvalue problem for the undamped circulatory system (23) to be imaginary is that the elements of the discriminant sequence corresponding to the discriminant matrix  are all nonnegative and that the coefficients of the polynomial p(λ) are either all non-positive or all non-negative: det 1 ≥ 0, det 2 ≥ 0, · · · , det m = det  ≥ 0, a0 ≥ 0, a1 ≥ 0, a2 ≥ 0, . . . , am ≥ 0.

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147

With the use of the Leverrier-Barnett algorithm, see e.g. Kirillov (2013a), one can write the characteristic polynomial of the system (23) as p(λ) = λ2m + trKλ2m−2 +

 1 (trK)2 − trK2 − trN2 λ2m−4 + . . . 2

(26)

Since for the polynomial (26) we have det 1 = m > 0, then, Gallina criterion gives a sufficient condition for instability if a02 ((a12 − a2 a0 )m − a12 ) < 0.

(27)

With the explicit expressions for the coefficients of the polynomial from (26), we re-write (27) as 1 trK2 + trN2 < (trK)2 . (28) m Taking into account that trK2 = tr(KT K) = K 2 , and trN2 = tr(−NT N) = − N 2 , where the norm is understood as the Frobenius norm, we represent (28) in the form

N 2 > K 2 −

1 (trK)2 . m

(29)

The inequality (29) is known as the Bulatovic flutter condition. Theorem 2.3 (Bulatovic flutter condition Bulatovic 2011, 2017) If

N 2 > K 2 −

1 (trK)2 m

the equilibrium of the circulatory system x¨ + (K + N)x = F(x, x˙ ), where K = KT , N = −NT , and F is a collection of terms of no lower than second order, is unstable. In a particular case when the stiffness matrix is proportional to the identity matrix, K = κI, we have trK = κm and trK2 = K 2 = κ2 m. With this, the flutter condition (29) reduces to the inequality N 2 > 0, which is always fulfilled if N = 0. Instability in this degenerate case occurs at arbitrary small circulatory forces. This statement is the famous Merkin theorem, see e.g. Krechetnikov and Marsden (2007); Udwadia (2017).

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Fig. 10 Geometrical interpretation of the Bulatovic flutter condition and Merkin theorem for m = 2 degrees of freedom

Flutter

0

Stabilty

k 11 - k22 2

k 12

Theorem 2.4 (Merkin Theorem (Merkin 1956)) Perturbation by arbitrary linear circulatory forces of a stable pure potential system with eigenfrequencies coinciding into one with the algebraic multiplicity equal to the dimension of the system destroys the stability of the equilibrium regardless of the form of the nonlinear terms.

2.4 Geometrical Interpretation for m = 2 Degrees of Freedom Let us now assume that m = 2 in Eq. (23). Notice that the 2 × 2 matrix A = K + N has the following decomposition     1 k11 − k22 2k12 k11 + k22 1 0 + 2k12 k22 − k11 01 2 2   0 −ν + = C + H + N, ν 0

A=

(30)

where the matrix C corresponds to potential forces of spherical type, H to potential forces of hyperbolic type, and N to circulatory forces (Zhuravlev 2007, 2008). When H = 0 we are in the conditions of the Merkin theorem. Calculating the eigenvalues of the corresponding eigenvalue problem, which are the roots of the polynomial det(λ2 I + A), we find

Classical Results and Modern Approaches to Nonconservative Stability

λ2 = −

 1 k11 + k22 2 ± (k11 − k22 )2 + 4k12 − 4ν 2 . 2 2

149

(31)

which is complex (flutter) if ν2 >

(k11 − k22 )2 2 + k12 . 4

(32)

This condition determines an interior of a double cone in the space of parameters k11 −k22 , k12 , and ν, see Fig. 10. 2 Let us establish a connection between the stability diagram of Fig. 10 and already known to us Bulatovic’s flutter condition and Merkin’s theorem. Observing that 2 2 2 + k22 + 2k12 , (trK)2 = (k11 + k22 )2 , N 2 = 2ν 2

K 2 = k11

we find

1 (k11 − k22 )2 2 + 2k12 .

K 2 − (trK)2 = 2 2

Hence, ν2 >

(k11 − k22 )2 1 2 + k12 ⇔ N 2 > K 2 − (trK)2 . 4 m

and we establish the equivalence of the Bulatovic flutter condition (29) and (32). Therefore, the Bulatovic flutter condition determines the conical flutter domain in Fig. 10. The axis of the cone passing through the origin at k11 − k22 = 0, k12 = 0 and ν = 0 lies in the flutter domain, corresponding to the condition ν 2 > 0 or N 2 > 0 given by the Merkin theorem. The apex of the cone at k11 − k22 = 0, k12 = 0 and ν = 0 corresponds to the potential system under the action of potential forces of spherical type, which is stable. Potential forces of spherical type and circulatory forces imply Merkin’s instability at all values of ν = 0. Potential forces of hyperbolic type stabilize the Merkin-unstable 2 2 22 ) + k12 . This is equivalent to the finite threshold for a system at ν < νcr = (k11 −k 4 torque in the Nicolai shaft with a non-circular cross-section (Nicolai 1929; Ziegler 1951a, b; Bolotin 1963; Seyranian and Mailybaev 2011; Luongo and Ferretti 2016).

2.5 Approximating Flutter Cone by Perturbation of Eigenvalues Consider the matrix A defined by Eq. (30) as a function of three parameters A = A(k22 , k12 , ν), whereas the parameter k11 is fixed, and the eigenvalue problem for it A(k22 , k12 , ν)u = σu,

(33)

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O. N. Kirillov

(a)

(b)

Fig. 11 Conical flutter domain of a circulatory system in the vicinity of a point in the parameter space corresponding to a semi-simple eigenvalue of the matrix A = K + N. a Given by Eq. (39). b Given by Eq. (40)

where σ = −λ2 . Let A0 = A(k22 = k11 , k12 = 0, ν = 0). Then  A0 =

k11 0 0 k11

 .

(34)

This matrix has a semi-simple real eigenvalue σ0 = k11 with the two linearlyindependent right eigenvectors u1 and u2 and two linearly-independent left eigenvectors, v1 and v2 . In general, left and right eigenvectors of a non-symmetric matrix differ but in our example A0 is real and symmetric and we can choose u1 = v1 =

    0 1 , u2 = v2 = . 1 0

(35)

Let us introduce the vector of parameters p = (k22 , k12 , ν) and denote p0 = (k11 , 0, 0). Then, A(k22 , k12 , ν) = A(p) and A0 = A(p0 ). In the following, we briefly consider a perturbative approach to the study of stability of circulatory systems following (Kirillov 2010, 2013a). We introduce a scalar parameter ε and consider a smooth path in the parameter space p(ε) and consider it in the vicinity of p0 = p(ε = 0) p(ε) = p0 + ε

dp + o(ε). dε

Then, the matrix family A(p(ε)) takes an increment A(p(ε)) = A0 + εA1 + o(ε),

Classical Results and Modern Approaches to Nonconservative Stability

151

 dps where A1 = ns=1 ∂∂A . In our example n = 3, p1 = k22 , p2 = k12 , and p3 = ν. ps dε It can be shown by perturbation argument (Kirillov 2010, 2013a) that the double semi-simple eigenvalue σ0 splits into two simple eigenvalues as follows σ(ε) = σ0 + ε

(A1 u1 , v1 ) + (A1 u2 , v2 ) ε √ D + o(ε), ± 2 2

(36)

where D = x 2 + y 2 − z 2 and x = f∗ , e,

y = f+ , e, z = f− , e.

(37)

T  1 n , . . . , dp . The components of the vectors f∗ , f+ and f− are The vector e = dp dε dε given by the expressions f ∗,s = (∂ ps Au1 , v1 ) − (∂ ps Au2 , v2 ), f ±,s = (∂ ps Au1 , v2 ) ± (∂ ps Au2 , v1 ).

(38)

The brackets  ,  in (37) denote the inner product of vectors in n-dimensional space and the brackets (, ) in (38) denote the inner product of vectors in m-dimensional space. Recall that in our example m = 2 and n = 3. The perturbed eigenvalues (36) are complex if z2 > x 2 + y2,

(39)

that is, inside the conical surface in the (x, y, z)-space, see Fig. 11a. In order to describe this conical flutter domain in the space of parameters p, we introduce the vectors a = f∗ × f+ , b = f∗ × f− , c = f− × f+ and the polar angle ϕ through the relations x = z cos ϕ and y = z sin ϕ. Then we can describe the flutter cone at the point p0 as the tangent cone to the flutter domain, i.e. as a set of directions e in which from the given point one can send a curve that lies in the flutter domain: {e : e = t (a + d(b sin φ + c cos φ)), t ∈ R, ϕ ∈ [0, 2π], d ∈ [0, 1)}.

(40)

Taking into account the eigenvectors (35) of the matrix (34) and constructing the gradient vector ⎛ ⎞ k22 − k11 e = ⎝ k12 ⎠ ν we find the vectors

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(b)

(c)

Fig. 12 Flutter instability of the shaft (41) at weak damping and weak stiffness anisotropy for k1 = 1, m = 1 and β = 0.05. a Stability domain (42) with two Whitney umbrella singular points in the (μ1 , μ2 , κ)-space. b Instability at weak damping and zero stiffness anisotropy (κ = 0). c Stabilization by weak damping at large stiffness anisotropy (κ = 2β = 0.1)

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 f∗ = ⎝ 0 ⎠ , f+ = ⎝ 2 ⎠ , f− = ⎝ 0 ⎠ . 0 0 −2 Substituting these vectors into the flutter condition f∗ , e2 + f+ , e2 − f− , e2 < 0, we reproduce the flutter cone (32). Note that the conical singularity is one of the eight generic singularities of codimension 3 that can occur on stability boundaries of circulatory systems with at least three parameters (Kirillov 2013a). In case of two parameters the number of generic singular points reduces to four, and in one-parameter families of circulatory systems we have only two singular points, corresponding to the reversible-Hopf bifurcation and to the steady-state bifurcation shown in Figs. 6 and 7, respectively.

3 Perturbing Circulatory Systems 3.1 Shieh–Masur Shaft with Dissipative Forces Let us return to the model (20) of a rotating shaft by Shieh and Masur (1968) in the case when the shaft is non-rotating ( = 0) and take into account damping m u¨ + μ1 u˙ + k1 u + βv = 0 m v¨ + μ2 v˙ + k2 v − βu = 0

(41)

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153

Separating time with u = ue ˜ λt and v = ve ˜ λt and applying to the characteristic polynomial of the resulting system of two algebraic equations the Hurwitz stability criterion, we find that the trivial solution u = 0, v = 0 is stable asymptotically, if and only if (μ1 + μ2 )2 (μ1 μ2 k1 − mβ 2 ) + μ1 μ2 κ(κm + μ1 (μ1 + μ2 )) > 0, μ1 + μ2 > 0.

(42)

The stability conditions (42) ensure the exponential decay with time of all no-trivial solutions u(t) and v(t) of the Eq. (41). The conditions (42) have a complicated form in contrast to the undamped case corresponding to μ1 = 0 and μ2 = 0 when the shaft is stable at β 2 < κ2 /4. How the damped and undamped cases are connected? Does the undamped flutter condition always follow from the damped one in the limit of vanishing damping coefficients? Let us investigate. Equate the left side of Eq. (42)1 to zero and solve the resulting equation with respect to κ. Then assume in the result μ1 = bμ2 and consider its limit as μ2 → 0. This yields   √ μ1 1 b+ √ , b= . (43) κ(b) = ±β μ2 b The function κ(b) has a minimum equal to 2β and a maximum equal to −2β at b = 1. This means that the threshold of stability of the dissipative system coincides with the threshold of the undamped system (κ2 = 4β 2 ) in the limit of vanishing dissipation only if μ1 = μ2 , or b = μ1 /μ2 = 1. Let us expand κ(b) in a Taylor series in the vicinity of b = 1 κ = ±2β ± β

  (b − 1)2 + o (b − 1)2 . 4

(44)

Truncating the series and taking into account that b = μ1 /μ2 , we write κ(μ1 , μ2 ) = ±2β ± β

(μ1 − μ2 )2 . 4μ22

(45)

Equation (45) is in the form zy 2 = x 2 , which is the normal form of a surface in the O x yz-space that has the Whitney umbrella singular point at the origin (Bottema 1956; Arnold 1972; Langford 2003; Kirillov and Verhulst 2010). The function z(x, y) = x 2 /y 2 > 0 at all x, y except for the specific line x = 0, where z(0, y) = 0. In our case the line x = 0 is the line μ1 = μ2 in the (μ1 , μ2 )-plane, see Fig. 12 where the stability domain (42) is shown with the two Whitney umbrella singular points situated on the κ-axis at κ = ±2β. It is remarkable that a weak stiffness anisotropy in the presence of weak damping does not prevent the system from flutter when circulatory forces are acting, Fig. 12b. Indeed, at κ = 0 criterion (42)

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Fig. 13 Stability map of the rotating shaft with k1 = 1, m = 1 (green lines) without dissipation and (red curves) with dissipation when (left) dissipation coefficients are equal, μ1 = μ2 = 0.05, (right) when μ1 = 0.07 and μ2 = 0.01. The asymptotic dashed lines are given by Eqs. (48) and (49), respectively

yields stability beyond a hyperbolic branch in the first quadrant of the (μ1 , μ2 )plane (46) μ1 μ2 k1 − mβ 2 > 0, at some distance form the origin. Notice that stability condition (46) traces back to Kapitsa (1939) who derived it in his study of transition to supercritical speeds in a special high-frequency expansion turbine that he developed for liquefaction of air. As soon as the absolute value of the stiffness anisotropy increases, the stability domain comes closer to the origin and touches it in a cuspidal point exactly when κ = ±2β, Fig. 12c, i.e. at the Whitney umbrella singular points. We observe that at κ = ±2β there exists only one direction pointing to the stability domain from the origin, and this direction is along the line μ1 = μ2 , in full agreement with (45). Decreasing dissipation along this line yields tending the critical flutter load smoothly to its values κ = ±2β for the undamped shaft. However, this is not true for all other directions, i.e. damping ratios b = μ1 /μ2 different from 1. In fact, near the Whitney umbrella points the stability boundary behaves much like a ruled surface, which has exactly two rulers μ1 = b± μ2 , where  κ2 − 4β 2 κ2 − 4β 2 b± = 1 + ± κ 2β 2 2β 2 at every κ such that κ2 > 4β 2 . Consequently, tending damping to zero along either of the two directions, μ1 = b± μ2 , will result in the value of κ that does not coincide with the undamped values ±2β. The flutter load of the damped shaft has therefore a singular zero-dissipation limit at the Whitney umbrella points. At every damping

Classical Results and Modern Approaches to Nonconservative Stability

155

ratio, except for 1, the flutter load in the limit of vanishing dissipation differs by a finite value from the flutter load of the undamped system. This is the famous Ziegler–Bottema destabilization paradox (Ziegler 1952; Bottema 1956). Now we are prepared to answer how dissipation affects the conical flutter domain of the undamped shaft given by the Bulatovich flutter condition that is shown in Fig. (9)(right). From (43) an expression for the two lines in the (κ, β)-plane follows √

β=±

μ1 μ2 κ. μ1 + μ2

(47)

The slope of the lines depends on the damping ratio in the manner dictated by the ruled surface geometry near the Whitney umbrella singularities. Indeed, for equal damping coefficients, μ1 = μ2 , the lines (47) are β=±

√ μ1 μ2 1 κ = ± κ. μ1 + μ2 2

(48)

They coincide with the flutter boundaries of the undamped system, Fig. 13(left). If we plot the stability domain (42) in the (κ, β)-plane for different damping coefficients that satisfy the constraint μ1 = μ2 , we will see that the stability boundary is a hyperbolic curve with the asymptotes (48). In the limit of vanishing dissipation such that μ1 = μ2 the stability boundary of the dissipative system degenerates into the cone κ2 = 4β 2 . However, taking the limit of vanishing dissipation at any other constraint on the damping coefficients, say, μ1 = 7μ2 , results in the different conical domain with the boundaries √ √ μ1 μ2 7 κ. (49) κ=± β=± μ1 + μ2 6 The flutter domain in the limit of vanishing dissipation given by the inequality 36β 2 > 7κ2 is therefore larger than the flutter domain of the undamped shaft, κ2 < 4β 2 , Fig. 13(right), providing an instructive example of a dissipation-induced instability (Bloch et al. 1994; Krechetnikov and Marsden 2007).

3.2 A Circulatory System Perturbed by Dissipative Forces The Shieh and Masur (1968) shaft is a non-conservative system with two degrees of freedom illustrating the properties summarized in the remark by Leipholz (1987): “Independent works of Bottema (1956) and Bolotin (1963) for second-order systems has shown that in the non-conservative case and for different damping coefficients the stability condition is discontinuous with respect to the undamped case.”

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O. N. Kirillov

Let us build a general theory proving this effect in a finite-dimensional mechanical system of arbitrary order under the action of positional conservative forces represented by a real symmetric matrix K = K T and positional non-conservative (or circulatory) forces with the real skew-symmetric matrix N = −NT : M¨x + (K + N(q))x = 0.

(50)

The matrix of circulatory forces smoothly depends on a parameter q. Assuming solution to the problem (50) in the form x = u exp λt, we arrive at the eigenvalue problem L(λ, q)u := (K + N(q))u + λ2 Mu = 0.

(51)

Let at the value of the parameter q = q0 there exist an algebraically double imaginary eigenvalue λ0 = iω0 with the Jordan block that satisfies the following equations (K + N(q0 ))u0 − ω02 Mu0 = 0 (K + N(q0 ))u1 − ω02 Mu1 = −2iω0 Mu0 ,

(52)

where u0 is an eigenfunction and u1 is an associated function at λ0 . Note that the eigenfunction v0 and the associated function v1 at the eigenvalue λ0 = −iω0 are governed by the adjoint equations (K − N(q0 ))v0 − ω02 Mv0 = 0 (K − N(q0 ))v1 − ω02 Mv1 = 2iω0 Mv0 .

(53)

Let us perturb the parameter q in the vicinity of q0 as q(ε) = q0 + εq1 + o(ε2 ).

(54)

N(q(ε)) = N(q0 ) + εN1 + o(ε)

(55)

Then,

where N1 =



∂N dq  ∂q dε ε=0

and λ(ε) = λ0 + λ1 ε1/2 + λ2 ε + o(ε), u(ε) = u0 + z1 ε1/2 + z2 ε + o(ε).

(56)

Substituting the expansions (55) and (56) into (51), we get (K + N(q0 ) + εN1 + o(ε))(u0 + z1 ε1/2 + z2 ε + o(ε)) + (λ20 + 2ε1/2 λ0 λ1 + ε(2λ0 λ2 + λ21 ) + o(ε))M(u0 + z1 ε1/2 + z2 ε + o(ε)) = 0. (57)

Classical Results and Modern Approaches to Nonconservative Stability

157

Collecting terms at ε0 , ε1/2 , and ε1 we obtain the equations (K + N(q0 ))u0 + λ20 Mu0 = 0 (K + N(q0 ))z1 + λ20 Mz1 = −2λ0 Mλ1 u0 (K + N(q0 ))z2 +

λ20 Mz2

(58)

= −2λ0 λ1 Mz1 − N1 u0 − (2λ0 λ2 +

λ21 )Mu0 .

T

Let (a, b) = b a be an inner product of vectors a and b. Taking the inner product of the last of the Eq. (58) with the vector v0 , we find ((K + N(q0 ))z2 , v0 ) + λ20 (Mz2 , v0 ) = −2λ0 λ1 (Mz1 , v0 ) − (N1 u0 , v0 ) − (2λ0 λ2 + λ21 )(Mu0 , v0 ). (59) In view of the property (Lu, v) = (u, L† v), where the adjoint matrix polynomial 2 is just L† = K − N + λ M, and taking into account that L† v0 = 0, we find 2λ0 λ1 (Mz1 , v0 ) + (N1 u0 , v0 ) + (2λ0 λ2 + λ21 )(Mu0 , v0 ) = 0.

(60)

Observing that z1 = λ1 u1 + C1 u0 and (Mu0 , v0 ) = 0 we arrive at the equation 2λ0 λ21 (Mu1 , v0 ) + (N1 u0 , v0 ) = 0.

(61)

Hence,

i(N1 u0 , v0 ) . 2ω0 (Mu1 , v0 )   dq  ∂N  In these conditions with εN1 = ∂N ε =  ∂q dε ∂q  λ21 =

ε=0

 λ(q) = iω0 ±

q

q=q0

i(Nq u0 , v0 ) 2ω0 (Mu1 , v0 )

 u(q) = u0 ± u1 q  v(q) = v0 ± v1 q

(62)

i(Nq u0 , v0 ) 2ω0 (Mu1 , v0 ) i(Nq u0 , v0 ) 2ω0 (Mu1 , v0 )

q = Nq q we obtain

 + o( |q|),

(63)

 + o( |q|),

(64)

 + o( |q|).

(65)

Therefore, we have approximations to the eigenvalues and eigenvectors of the undamped circulatory system in the vicinity of q = q0 , i.e. in the vicinity of the flutter boundary corresponding to the reversible-Hopf bifurcation. Assume that at q < q0 the eigenvalues of the circulatory system are imaginary and at q > q0 the eigenvalues are complex-conjugate (instability).

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Let us study how simple imaginary eigenvalues of a circulatory system change due to dissipative perturbation with the matrix D(p) where p = ( p1 , p2 , . . . , pn )T and D(p = 0) = 0. Write the dissipatively perturbed eigenvalue problem (51) L(λ, q, p)u := (K + N(q))u(q) + λ(q)D(p)u(q) + λ2 (q)Mu(q) = 0.

(66)

as well as its adjoint 2

L† (λ, q, p)v := (K − N(q))v(q) + λ(q)D(p)v(q) + λ (q)Mv(q) = 0.

(67)

We assume in the above equations that q is fixed such that q < q0 . Let at p = 0 the eigenvalue problem (66) has a simple eigenvalue λ(q) = iω(q) with an eigenvector u(q). Assuming p = p(ε), where p(ε) = εp1 + o(ε), we obtain D(p(ε)) = εD1 + o(ε) with D1 =

n

∂D s=1 ∂ ps

 dps  dε 

ε=0

(68)

. Then, the eigenvalues of (66) are

λ(ε) = λ(q) −

(D1 u(q), v(q)) ε + o(ε). 2(Mu(q), v(q))

(69)

In other words n λ(q, p) = λ(q) −

 s=1 (D ps u(q), v(q))ps

2(Mu(q), v(q))

+ o( p ).

(70)

Following Andreichikov and Yudovich (1974) we require n 

(Dps u(q), v(q))ps = 0

(71)

s=1

as a condition for the imaginary eigenvalue remain imaginary after a dissipative perturbation. This means that we approximately stay on the neutral stability surface after the dissipative perturbation. Eq. (71) gives an exact linear approximation to the neutral stability surface at every q < q0 , if we know exactly the dependencies u(q), v(q) and λ(q). Usually, however, these functions are determined numerically, see e.g. Andreichikov and Yudovich (1974); Luongo et al. (2016). Kirillov (2007, 2013a) proposed to use in the method of Andreichikov and Yudovich (1974) approximations to u(q), v(q) and λ(q) in the vicinity of q = q0 such as those given by Eqs. (63), (64) and (65). Substituting them into (71), we express the approximate critical flutter load explicitly as

Classical Results and Modern Approaches to Nonconservative Stability

159

Fig. 14 The Kelvin gyrostat (Thomson 1880)

1 q = q0 + 2 λ1



n

 s=1 (D ps u0 , v0 )ps n   s=1 [(D ps u0 , v1 ) + (D ps u1 , v0 )]ps

2 .

(72)

In the particular case of n = 2 parameters we assume that p1 = βp2 . Introducing the quantity (Dp u0 , v0 ) , (73) β0 = −  2 (D p1 u0 , v0 ) we can write q(β) retaining only the terms of order (β − β0 )2 and lower: q = q0 +

 2 2 λ−2 1 (D p1 u0 , v0 ) (β − β0 )

[(Dp1 u0 , v1 )β0 + (Dp1 u1 , v0 )β0 + (Dp2 u0 , v1 ) + (Dp2 u1 , v0 )]2

. (74)

Therefore, we have derived a general analogue of the expression (44), which gives the quadratic approximation to the vanishing-dissipation limit of the critical flutter load, q(β), in the vicinity of β = β0 in a rigorous sense. This approximation is sufficient to capture the Whitney umbrella singularity that is responsible for the Ziegler–Bottema destabilization paradox.

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4 Krein Signature and Stability of Hamiltonian Systems An attempt to spin a hard-boiled egg always ends up successfully: when spun sufficiently rapidly, its symmetry axis can even rise to the vertical position demonstrating a gyroscopic stabilization. The mathematical model of this effect is the rotating solid prolate spheroid known as Jellett’s egg, see e.g. Kirillov (2013a). In contrast, trying to spin a raw egg containing a yolk inside, surrounded by a liquid, will generally lead to its slow wobbling motion. Thomson (1880) experimentally demonstrated that a thin-walled and slightly oblate spheroid completely filled with liquid remains stable if rotated fast enough about a fixed point, which does not happen if the spheroid is slightly prolate, Fig. 14. In the same year this observation was confirmed theoretically by Greenhill (1880), who found that rotation around the center of gravity of the top in the form of a weightless ellipsoidal shell completely filled with an ideal and incompressible fluid, is unstable when a < c < 3a, where c is the length of the semiaxis of the ellipsoid along the axis of rotation and the lengths of the two other semiaxes are equal to a (Greenhill 1880). Quite similarly, bullets and projectiles fired from the rifled weapons can relatively easily be stabilized by rotation, if they are solid inside. In contrast, the shells, containing a liquid substance inside, have a tendency to turn over despite seemingly revolved fast enough to be gyroscopically stabilized. Motivated by such artillery applications, in 1942 Sobolev, then director of the Steklov Mathematical Institute in Moscow, considered stability of a rotating heavy top with a cavity entirely filled with an ideal incompressible fluid (Moiseyev and Rumyantsev 1968; Ramodanov and Sidorenko 2017)—a problem that is directly connected to the classical XIXth century models of astronomical bodies with a crust surrounding a molten core (Stewartson 1959). For simplicity, the solid shell of the top and the domain V occupied by the cavity inside it, can be assumed to have a shape of a solid of revolution. They have a common symmetry axis where the fixed point of the top is located. The velocity profile of the stationary unperturbed motion of the fluid is that of a solid body rotating with the same angular velocity  as the shell around the symmetry axis. Following Sobolev, we denote by M1 the mass of the shell, M2 the mass of the fluid, ρ and p the density and the pressure of the fluid, g the gravity acceleration, and l1 and l2 the distances from the fixed point to the centers of mass of the shell and the fluid, respectively. The moments of inertia of the shell and the ‘frozen’ fluid with respect to the symmetry axis are C1 and C2 , respectively; A1 (A2 ) stands for the moment of inertia of the shell (fluid) with respect to any axis that is orthogonal to the symmetry axis and passes through the fixed point. Let, additionally, L = C 1 + C 2 − A1 − A2 −

K , 2

K = g(l1 M1 + l2 M2 ).

(75)

The solenoidal (div v = 0) velocity field v of the fluid is assumed to satisfy the no-flow condition on the boundary of the cavity: vn |∂V = 0.

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Stability of the stationary rotation of the top around its vertically oriented symmetry axis is determined by the system of linear equations derived by Sobolev in the frame (x, y, z) that has its origin at the fixed point of the top and rotates with respect to an inertial frame around the vertical z-axis with the angular velocity of the unperturbed top, . If the real and imaginary part of the complex number Z describe the deviation of the unit vector of the symmetry axis of the top in the coordinates x, y, and z, then these equations are, see e.g. Kopachevskii and Krein (2001); Kirillov (2013a): dZ = iW, dt dW (A1 +ρκ2 ) = iL Z + i(C1 −2 A1 +ρE)W dt    ∂χ ∂χ vx − vy d V, + iρ ∂y ∂x V ∂t vx = 2v y − ρ−1 ∂x p + 2i2 W ∂ y χ, ∂t v y = −2vx − ρ−1 ∂ y p − 2i2 W ∂x χ, ∂t vz = −ρ−1 ∂z p,

(76)

    where 2κ2 = V |∇χ|2 d V , E = i V ∂x χ∂ y χ − ∂ y χ∂x χ d V , and the function χ is determined by the conditions ∇ 2 χ = 0,

∂n χ|∂V = z(cos nx + i cos ny) − (x + i y) cos nz,

(77)

with n the absolute value of a vector n, normal to the boundary of the cavity. Sobolev realized that some qualitative conclusions on the stability of the top can be drawn with the use of the bilinear form  ρ Q(R1 , R2 ) = LZ 1 Z 2 + (A1 + ρκ2 )W1 W 2 + v T v1 d V (78) 22 V 2 on the elements R1 and R2 of the space {R} = {Z , W, v}. The linear operator B defined by Eq. (76) that can be written as ddtR = i B R has all its eigenvalues real when L > 0, which yields Lyapunov stability of the top. The number of pairs of complex-conjugate eigenvalues of B (counting multiplicities) does not exceed the number of negative squares of the quadratic form Q(R, R), which can be equal only to one when L < 0. Hence, for L < 0 an unstable solution R = eiλ0 t R0 can exist with Imλ0 < 0; all real eigenvalues are simple except for maybe one (Kopachevskii and Krein 2001). In the particular case when the cavity is an ellipsoid of rotation with the semi-axes a, a, and c, the space of the velocity fields of the fluid can be decomposed into a direct sum of subspaces, one of which is finite-dimensional. Only the movements from this subspace interact with the movements of the rigid shell, which yields a finite-dimensional system of ordinary differential equations that describes coupling between the shell and the fluid.

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Calculating the moments of inertia of the fluid in the ellipsoidal container C2 = denoting m =

c2 −a 2 , c2 +a 2

8πρ 4 a c, 15

A2 = l22 M2 +

4πρ 2 2 a c(a + c2 ), 15

and assuming the field v = (vx , v y , vz )T in the form

vx = (z − l2 )a 2 mξ, v y = −i(z − l2 )a 2 mξ, vz = −(x − i y)c2 mξ, one can eliminate the pressure in Eq. (76) and obtain the reduced model dx = iA−1 Cx = iBx, dt

(79)

where x = (Z , W, ξ)T ∈ C3 and ⎛

⎞ 1 0 0 2 2 2 ) A = ⎝ 0 A1 +l22 M2 + 4πρ a 2 c (cc2−a 0 ⎠, 15 +a 2 2 0 0 c + a2 ⎛ ⎞ 0 1 0 C = ⎝ L C1 −2 A1 −2l22 M2 − 8πρ a 2 c3 m 2 − 8πρ a 4 c3 m 2 ⎠ . 15 15 0 −2 −2a 2

(80)

The matrix B = BT in Eq. (79) after multiplication by a symmetric matrix ⎛

L 0 ⎜ 0 A +l 2 M + 4πρ a 2 c (c2 −a 2 )2 G=⎝ 1 2 2 15 c2 +a 2 0 0

0 0 4πρ 4 3 (c2 −a 2 )2 a c c2 +a 2 15

⎞ ⎟ ⎠

(81)

T

yields a Hermitian matrix GB = (GB) , i.e. B is a self-adjoint operator in the space C3 endowed with the metric [u, u] := (Gu, u) = uT Gu, u ∈ C3 ,

(82)

which is definite when L > 0 and indefinite with one negative square when L < 0. If λ is an eigenvalue of the matrix B, i.e. Bu = λu, then uT GBu = λuT Gu. On the other hand, uT (GB)T u = λ uT Gu = λ uT Gu. Hence, (λ − λ)uT Gu = 0, implying uT Gu = 0 on the eigenvector u of the complex λ = λ. For real eigenvalues λ = λ and uT Gu = 0. The sign of the quantity uT Gu can be different for different real eigenvalues.

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(a)

163

(b)

Fig. 15 a Simple real eigenvalues (83) of the Sobolev’s top in the Greenhill’s case for a = 1 with (red) uT Gu > 0 and (green) uT Gu < 0. b At simple complex-conjugate eigenvalues (black) and at the double real eigenvalue λd we have uT Gu = 0

For example, when the ellipsoidal shell is massless and the supporting point is at the center of mass of the system, then A1 = 0, C1 = 0, M1 = 0, l2 = 0. The matrix B +T + has thus one real eigenvalue (λ+ 1 = −1, u1 Gu1 > 0) and the pair of eigenvalues λ± 2

 1 1 32πρ ca 4 , =− ± 1+ 2 2 15 L

L=

4πρ 2 2 a c(a − c2 ), 15

(83)

which are real if L > 0 and can be complex if L < 0. The latter condition together with the requirement that the radicand in Eq. (83) is negative, reproduces the Greenhill’s instability zone: a < c < 3a (Greenhill 1880). With the change in c, +T + the real eigenvalue λ+ 2 with u2 Gu2 > 0 collides at c = 3a with the real eigenvalue −T − λ− 2 with u2 Gu2 < 0 into a real double defective eigenvalue λd with the algebraic multiplicity two and geometric multiplicity one, see Fig. 15. Note that ud T Gud = 0, where ud is the eigenvector at λd . Therefore, in the case of the ellipsoidal shapes of the shell and the cavity, the Hilbert space {R} = {Z , W, v} of the Sobolev’s problem endowed with the indefinite metric (L < 0) decomposes into the three-dimensional space of the reduced model (79), where the self-adjoint operator B can have complex eigenvalues and real defective eigenvalues, and a complementary infinite-dimensional space, which is free of these complications. The very idea that the signature of the indefinite metric can serve for counting unstable eigenvalues of an operator that is self-adjoint in a functional space equipped with such a metric, turned out to be a concept of a rather universal character possessing powerful generalizations that were initiated by Pontryagin in 1944 (Yakubovich and Starzhinskii 1975; Kopachevskii and Krein 2001).

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4.1 Canonical and Hamiltonian Equations Following Yakubovich and Starzhinskii (1975), we consider a complex vector space Cn with the inner product (x, y) = yT x. Define an indefinite inner product in Cn as [x, y] = (Gx, y) = yT Gx,

(84)

T

where G = G (det G = 0) is an arbitrary (neither positive nor negative definite) Hermitian n × n matrix. Hence, [x, x] is real but in contrast to (x, x) it can be positive, negative, or zero for x = 0. The matrix A+ with the property [Ax, y] = [x, A+ y]

(85)

is said to be G-adjoint to A. From Eq. (85) it follows that T

A+ = G−1 A G.

(86)

dz = Hz, dt

(87)

A differential equation i −1 G

where H is Hermitian, is called Hamiltonian equation. The matrix A = iG−1 H yields [Ax, y] = −[x, Ay], (88) i.e. A+ = −A, and is called the G-Hamiltonian matrix (Yakubovich and Starzhinskii 1975; Zhang et al. 2016). In terms of the G-Hamiltonian matrix A, the Hamiltonian system (87) takes the form dz = Az. (89) dt T

T

Since A = −G−1 A G, the matrices −A and A have the same spectrum. Consequently, if λ is an eigenvalue of A, then so is −λ. Hence, the spectrum of a G-Hamiltonian matrix is symmetric about the imaginary axis. The eigenvalue λ lies on the imaginary axis if and only if λ = −λ (Yakubovich and Starzhinskii 1975). Let I be the unit k × k-matrix and   0 −I (90) J= = −J−1 , I 0 the canonical symplectic matrix. The n × n matrix G = iJ, where n = 2k, is T Hermitian: G = iJT = −i(−J) = iJ = G. With G = iJ and H = HT real, the Hamiltonian equation (87) reduces to

Classical Results and Modern Approaches to Nonconservative Stability

J

dx = Hx dt

165

(91)

that is referred to as the canonical equation, whereas the indefinite inner product takes the form (Yakubovich and Starzhinskii 1975). [x, y] = yT (iJ)x = iyT Jx.

(92)

The canonical Hamiltonian linear equation (91) describe motion of a system with k degrees of freedom ∂H d xs = , dt ∂xk+s

∂H d xk+s =− , s = 1, . . . , k, dt ∂xs

(93)

where xs are generalized coordinates and xk+s are generalized momenta. The quadratic form H = 21 (Hx, x), where xT = (x1 , . . . , x2k ), is referred to as a Hamiltonian function. The real symmetric 2k × 2k-matrix H of the quadratic form H is called the Hamiltonian (Yakubovich and Starzhinskii 1975). Seeking for the solution to Eq. (91) in the form x = u exp(λt), we find Hu = λJu.

(94)

From Eqs. (92) and (94) it follows that if λ is a pure imaginary eigenvalue with the eigenvector u of the (iJ)-Hamiltonian matrix J−1 H, then (Hu, u) = Imλ [u, u].

(95)

Since J and H are real matrices and the eigenvalues of a (iJ)-Hamiltonian matrix are symmetric with respect to the imaginary axis, the spectrum of the matrix J−1 H is symmetric with respect to both real and imaginary axes of the complex plane. Theorem 4.1 Let λ be an eigenvalue of the eigenvalue problem (94). Then so is its complex conjugate, λ, and −λ. Hence, for a canonical Hamiltonian linear equation (91) the eigenvalues come in singlets {0}, doublets {λ, −λ} with λ ∈ R or λ ∈ iR, or quadruplets {λ, −λ, λ, −λ}. The algebraic multiplicity of the eigenvalue λ = 0 is even. Consequently, the equilibrium x = 0 of the system (91) is Lyapunov stable, if and only if the eigenvalues λ of the eigenvalue problem (94) are pure imaginary and semi-simple (Yakubovich and Starzhinskii 1975).

4.2 Krein Signature of Eigenvalues Let λ (Reλ = 0) be a simple pure imaginary eigenvalue of a G-Hamiltonian matrix A and u be a corresponding eigenvector:

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Au = λu.

(96)

Definition: A simple pure imaginary eigenvalue λ = iω with the eigenvector u is said to have positive Krein signature if [u, u] > 0 and negative Krein signature if [u, u] < 0. Let, further, λ (Reλ = 0) be a multiple pure imaginary eigenvalue of a GHamiltonian matrix A, and let Lλ be the eigensubspace of A belonging to the eigenvalue λ, i.e. the set of all u ∈ Cn satisfying Eq. (96). If [u, u] > 0 for any u ∈ Lλ (u = 0), then λ is a multiple eigenvalue with positive Krein signature and the eigensubspace Lλ is positive definite; if [u, u] < 0, λ is a multiple eigenvalue with negative Krein signature and the eigensubspace Lλ is negative definite. In such cases the multiple eigenvalue is said to have definite Krein signature. If there exists a vector u ∈ Lλ (u = 0) such that [u, u] = 0, the multiple pure imaginary eigenvalue λ is said to have mixed Krein signature (Yakubovich and Starzhinskii 1975). Note that in case when a multiple pure imaginary eigenvalue λ0 of A has geometric multiplicity that is less than its algebraic multiplicity, then there is an eigenvector u0 at λ0 such that [u0 , u0 ] = 0, i.e. λ0 has mixed Krein signature. Indeed, there exists at least one associated vector u1 : Au1 = λ0 u1 + u0 , where Au0 = λ0 u0 . Taking into account the property (88), we obtain (Kirillov 2013a)   2 [Au0 , Au1 ] = − u0 , A2 u1 = −λ0 [u0 , u1 ] − 2λ0 [u0 , u0 ] = λ0 λ0 [u0 , u1 ] + λ0 [u0 , u0 ] ,

(97)

which yields [u0 , u0 ] = 0

(98)

since λ0 = −λ0 (Yakubovich and Starzhinskii 1975). On the other hand, if λ0 = −λ0 then [u0 , u0 ] = 0 for any eigenvector u0 at λ0 , which follows from the identity   2 [Au0 , Au0 ] = λ0 λ0 [u0 , u0 ] = − u0 , A2 u0 = λ0 [u0 , u0 ] . Therefore, a multiple pure imaginary eigenvalue can have definite Krein signature only if it is semi-simple.

4.3 Krein Collision or Linear Hamiltonian-Hopf Bifurcation Let in the eigenvalue problem (94) the matrix H smoothly depend on a vector of real parameters p ∈ Rm : H = H(p). Let at p = p0 the matrix H0 = H(p0 ) has a double pure imaginary eigenvalue λ = iω0 (ω0 ≥ 0) with the Jordan chain consisting of the eigenvector u0 and the associated vector u1 . Hence, H0 u0 = iω0 Ju0 , H0 u1 = iω0 Ju1 + Ju0 .

(99)

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167

Transposing Eq. (99) and applying the complex conjugation yields u0T H0 = iω0 u0T J, u1T H0 = iω0 u1T J − u0T J.

(100)

As a consequence, u1T Ju0 + u0T Ju1 = 0, i.e. [u0 , u1 ] = −[u1 , u0 ].

(101)

Varying the vector of parameters along the curve p = p(ε) (p(0) = p0 ) and applying the perturbation formulas for double eigenvalues that can be found e.g. in Kirillov (2013a, 2017), we obtain √ √ λ± = iω0 ± iω1 ε + o(ε1/2 ), u± = u0 ± iω1 u1 ε + o(ε1/2 ) under the assumption

 ω1 =

where

u0T H1 u0 u1T Ju0

> 0,

 m  ∂H dps  . H1 = ∂ ps dε ε=0 s=1

(102)

(103)

(104)

When ε > 0, the double eigenvalue iω0 splits into two pure imaginary ones according to the formulas (102). Calculating the indefinite inner product for the perturbed eigenvectors u± by Eq. (92) and taking into account the conditions (98) and (101), we find (Kirillov 2013a, 2017) √ [u± , u± ] = ±2ω1 u1T Ju0 ε + o(ε1/2 ),

(105)

i.e. the simple pure imaginary eigenvalues λ+ and λ− have opposite Krein signatures. When ε decreases from positive values to negative ones, the pure imaginary eigenvalues of opposite Krein signatures merge at ε = 0 to the double pure imaginary eigenvalue iω0 with the Jordan chain of length 2 that further splits into two complex eigenvalues, one of them with the positive real part. When ω0 = 0, this process is known as the linear Hamiltonian-Hopf bifurcation (Langford 2003), the onset of flutter, non-semi-simple 1 : 1 resonance or the Krein collision (Kirillov 2013a). When ω0 = 0, a pair of pure imaginary eigenvalues of opposite Krein signatures colliding at zero and splitting then into a pair of real eigenvalues of different sign means the onset of the non-oscillatory instability or divergence known also as the linear steady-state bifurcation.

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5 Dissipation-Induced Instabilities of Hamiltonian Systems 5.1 The Kelvin-Tait-Chetaev Theorem Potential system of the form M¨x + Kx = 0 with the mass matrix, M = MT , and the stiffness matrix, K = KT , can be transformed to the Hamiltonian form (91). Furthermore, this can be done also in the presence of velocity-dependent gyroscopic forces with the matrix G = −GT for the gyroscopic system M¨x + G˙x + Kx = 0. Gyroscopic forces can stabilize the otherwise unstable static equilibrium. This gyroscopic stabilization can be lost in the presence of dissipation, as we all know from observing the behavior of rotating tops. This dissipation-induced instability of gyroscopic systems is formalized by the Kelvin-Tait-Chetaev theorem (Thomson and Tait 1879; Bloch et al. 1994; Krechetnikov and Marsden 2007). Theorem 5.1 (Kelvin-Tait-Chetaev Theorem) Stability of solutions of the equation M¨x + (G + D)˙x + Kx = 0,

(106)

where M > 0, D = DT > 0 and K nondegenerate is the same as the stability of solutions of the corresponding potential system, M¨x + Kx = 0. In particular, if all the eigenvalues of the real symmetric matrix K are positive (negative) then the system (106) is asymptotically stable (unstable). The number of eigenvalues with positive real parts of the system (106) is equal to the number of negative eigenvalues of the matrix K (Zajac Theorem, 1964). If the number of negative eigenvalues of K (known also as the Poincaré instability degree) is even, then the equilibrium of the corresponding potential system can be stabilized by the gyroscopic forces. However, this gyroscopic stabilization is destroyed when dissipative forces with full dissipation (D > 0) are added, no matter how weak they are Kirillov (2013a). Remarkably, the origin of the Kelvin-Tait-Chetaev theorem is in the centuriesold problem (going back to Newton) on the stability of rotating and self-gravitating masses of fluid motivated by the question of the actual shape of the Earth (Lebovitz 1998; Borisov et al. 2009).

5.2 Secular Instability of the Maclaurin Spheroids In 1742 Maclaurin has found that an oblate spheroid y2 z2 x2 + 2 + 2 = 1, a3 < a2 = a1 2 a1 a2 a3

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Fig. 16 (Left) Families of Maclaurin spheroids and Jacobian ellipsoids in the plane of angular velocity versus eccentricity with the common point at e ≈ 0.8127. (Right) Sequence of bifurcations proposed by the fission theory of binary stars (Lebovitz 1987)

is a shape of relative equilibrium of a self-gravitating mass of inviscid fluid in a solid-body rotation about  the z-axis, provided that the rate of rotation, , is related to the eccentricity e =

1−

a32 a12

through the formula (Lebovitz 1998)

 2 (e) = 2e−3 (3 − 2e2 ) sin−1 (e) 1 − e2 − 6e−2 (1 − e2 ).

(107)

A century later, Jacobi (1834) has discovered less symmetric shapes of relative equilibria in this problem that are tri-axial ellipsoids y2 z2 x2 + + = 1, a3 < a2 < a1 . a12 a22 a32 Later on Meyer (1842) and Liouville (1846) have shown that the family of Jacobi’s ellipsoids has one member in common with the family of Maclaurin’s spheroids at e ≈ 0.8127, see Fig. 16. The equilibrium with the Meyer-Liouville eccentricity is neutrally stable, Fig. 17. In 1860 Riemann established neutral stability of inviscid Maclaurin’s spheroids on the interval of eccentricities (0 < e < 0.9529..). At the Riemann point with the critical eccentricity e ≈ 0.9529 the Hamilton-Hopf bifurcation sets in and causes dynamical instability with respect to ellipsoidal perturbations beyond this point. A century later Chandrasekhar (1969) proposed a virial method to reduce the problem to a finite-dimensional system, which stability is governed by the eigenvalues of the matrix polynomial

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 Li (λ) = λ2

10 01



 +λ

0 −4  0



 +

0 4b − 22 0 4b − 22

 ,

(108)

where (e) is given by the Maclaurin law (107) and b(e) is as follows √

π !   1 − e2 2 −1 −1 2 + (4e2 − 3) 2) − tan e(3 − 2e ) 1 − e (e 1 − e . 4e5 2 (109) The eigenvalues of the matrix polynomial (108) are b=

   λ = ± i ± i 4b − 2 .

(110)

Requiring λ = 0 we can determine the critical Meyer-Liouville eccentricity by solving with respect to e the equation (Chandrasekhar 1969) 4b(e) = 22 (e). The critical eccentricity at the Riemann point follows from requiring the radicand in (110) to vanish: 4b(e) = 2 (e). Remarkably, when 2 (e) < 4b(e) < 22 (e)

(111)

both eigenvalues of the stiffness matrix 

0 4b − 22 0 4b − 22



are negative, i.e. the Poincaré instability degree of the equilibrium is even and equal to 2. Hence, the interval (111) corresponding to 0.8127.. < e < 0.9529.., which is stable according to Riemann, is, in fact, the interval of gyroscopic stabilization of the Maclaurin spheroids, Fig. 17. According to the Theorem 5.1 the gyroscopic stabilization of the equilibrium with nonzero Poincaré instability degree can be destroyed even by the infinitely small dissipation with the positive-definite damping matrix. In the words by Thomson and Tait (1879), “If there be any viscosity, however slight, in the liquid, the equilibrium [beyond e ≈ 0.8127] in any case of energy either a minimax or a maximum cannot be secularly stable”. The prediction made by Thomson and Tait (1879) has been verified quantitatively only in the XX-th century by Roberts and Stewartson (1963). Using the virial approach Chandrasekhar (1969) reduced the linear stability problem to the study of eigenvalues of the matrix polynomial

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Fig. 17 (Left) Frequencies and (right) growth rates of the eigenvalues of the inviscid eigenvalue problem Li (λ)u = 0 demonstrating the Hamilton-Hopf bifurcation at the Riemann critical value of the eccentricity, e ≈ 0.9529 and neutral stability at the Meyer-Liouville point, e ≈ 0.8127

Fig. 18 (Left) Frequencies and (right) growth rates of the (black lines) inviscid Maclaurin spheroids and (green and red lines) viscous ones with μ = aν2 = 0.01. Viscosity destabilizes the gyroscopic 1

stabilization of the Maclaurin spheroids on the interval 0.8127 . . . < e < 0.9529 . . ., which is stable in the inviscid case (Roberts and Stewartson 1963; Chandrasekhar 1969; Chandresekhar 1984)

 Lv (λ) = λ

2

10 01





10μ −4 +λ  10μ



 +

0 4b − 22 0 4b − 22

 ,

(112)

where μ = aν2 and ν is the viscosity of the fluid. The operator Lv (λ) differs from the 1 operator of the ideal system, Li (λ), by the matrix of dissipative forces 10λμI, where I is the 2 × 2 unit matrix.

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Fig. 19 Paths of the eigenvalues in the complex plane for (left) viscous Maclaurin spheroids with μ = aν2 = 0.002, (centre) Maclaurin spheroids without dissipation, and (right) inviscid Maclaurin 1

spheroids with radiative losses for δ = 0.05. The Krein collision of two modes of the non-dissipative Hamiltonian system shown in the centre occurs at the Rieman critical value e ≈ 0.9529. Both types of dissipation destroy the Krein collision and destabilize one of the two interacting modes at the Meyer-Liuoville critical value e ≈ 0.8127

The characteristic polynomial written for Lv (λ) yields the equation governing the growth rates of ellipsoidal perturbations in the presence of viscosity: 252 μ2 + (Reλ + 5μ)2 (2 − Reλ2 − 10Reλμ − 4b) = 0.

(113)

The right panel of Fig. 18 shows that the growth rates (113) become positive beyond the Meyer-Liouville point. Indeed, assuming Reλ = 0 in (113), we reduce it to 50μ2 (2 − 2b) = 0, meaning that the growth rate vanishes when 2 = 2b no matter how small the viscosity coefficient μ is. But, as we already know, the equation 2 (e) = 2b(e) determines exactly the Meyer-Liouville point, e ≈ 0.8127. It turns out, that the critical eccentricity of the viscous Maclaurin spheroid is equal to the Meyer-Liouville value, e ≈ 0.8127, even in the limit of vanishing viscosity, μ → 0, and thus does not converge to the inviscid Riemann value e ≈ 0.9529. This is nothing else but the Ziegler–Bottema destabilization paradox in a near-Hamiltonian dissipative system (Langford 2003; Krechetnikov and Marsden 2007; Kirillov 2007, 2013a). Viscous dissipation destroys the Krein interaction of two modes at the Riemann critical point and destabilizes one of them beyond the Meyer-Liouville point, showing a typical for the destabilization paradox avoided crossing in the complex plane, Fig. 19(left). Thomson and Tait (1879) hypothesised that the instability, which is stimulated by the presence of viscosity in the fluid, will result in a slow, or secular, departure of the system from the unperturbed equilibrium of the Maclaurin family at the MeyerLiouville point and subsequent evolution along the Jacobi family, as long as the latter is stable (Lebovitz 1998). Therefore, a rotating, self-gravitating fluid mass, initially symmetric about the axis of rotation, can undergo an axisymmetric evolution in which it first loses stability

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Fig. 20 Critical eccentricity in the limit of vanishing dissipation depends on the damping ratio, X , and attains its maximum (Riemann) value, e ≈ 0.9529 exactly at X = 1. As X tends to zero or infinity, the critical value tends to the Meyer-Liouville value e ≈ 0.8127, (Lindblom and Detweiler 1977; Chandresekhar 1984)

to a nonaxisymmetric disturbance, and continues for a while evolving along a nonaxisymmetric family toward greater departure from axial symmetry, Fig. 16; then it undergoes a further loss of stability to a disturbance tending toward splitting into two parts (Lebovitz 1998). Rigorous mathematical treatment of the fission theory of binary stars proposed by Thomson and Tait (1879) by Lyapunov and Poincaré has laid a foundation to modern nonlinear analysis. In particular, it has led Lyapunov to the development of a general theory of stability of motion (Borisov et al. 2009). As we remember, it is the Lyapunov stability theory that helped Nicolai and Ziegler to shed light on stability of nonconservative systems under circulatory forces. Chandrasekhar (1970) demonstrated that there exists another mechanism making the Maclaurin spheroid unstable beyond the Meyer-Liouville point of bifurcation, namely, the radiative losses due to emission of gravitational waves. However, the mode that is made unstable by the radiation reaction is not the same one that is made unstable by viscosity, Fig. 19(right). In the case of the radiative damping mechanism stability is determined by the spectrum of the following matrix polynomial (Chandrasekhar 1970) Lg (λ) = λ2 + λ(G + D) + K + N that contains the matrices of gyroscopic, G, damping, D, potential, K, and nonconservative positional, N, forces 5 G= 2



   −3/2 0 − δ162 (6b − 2 ) , D= −3/2 δ162 (6b − 2 )  0 

K=

0 4b − 2 0 4b − 2



 , N=δ

2q1 2q2 −q2 /2 2q1

 ,

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where (e) and b(e) are given by Eqs. (107) and (109). Explicit expressions for q1 and q2 can be found in Chandrasekhar (1970). Lindblom and Detweiler (1977) studied the combined effects of gravitational radiation reaction and of viscosity on the stability of the Maclaurin spheroids. As we know, each of these dissipative effects induces a secular instability in the Maclaurin sequence past the Meyer-Liouville point of bifurcation. However, when both effects are considered together, the sequence of stable Maclaurin spheroids therefore reaches past the bifurcation point to a new point determined by the ratio of the strengths of the viscous and the radiative forces. Figure 20 shows the limit of the critical eccentricity as a function of the damping ratio in the limit of vanishing dissipation. This limit coincides with the inviscid Riemann point only at a particular damping ratio. At any other ratio, the critical value is below the Riemann one and tends to the Meyer-Liouville value as this ratio tends either to zero or infinity. Lindblom and Detweiler (1977) correctly attributed the cancellation of the secular instabilities to the fact that viscous dissipation and radiation reaction cause different modes to become unstable, see Fig. 19. Andersson (2003) relates the mode destabilized by the fluid viscosity to the prograde moving spherical harmonic that appears to be retrograde in the frame rotating with the fluid mass and the mode destabilized by the radiative losses to the retrograde moving spherical harmonic when it appears to be prograde in the inertial frame. This gives a link to destabilization of positive- and negative energy modes (Ostrovsky et al. 1986; Kirillov 2009, 2013a) as well as to the theory of the anomalous Doppler effect (Nezlin 1976; Ginzburg and Tsytovich 1979; Vesnitskii and Metrikin 1996). It is known (Nezlin 1976) that to excite the positive energy mode one must provide additional energy to the mode, while to excite the negative energy mode one must extract energy from the mode. The latter can be done by dissipation and the former by the nonconservative positional (curl) forces. Both are presented in the model by Lindblom and Detweiler (1977). The destabilization of a Hamiltonian system in the presence of two different types of non-Hamiltonian perturbations can be understood on the example of the general two-dimensional system x¨ (t) + (δD + G)˙x(t) + (K + νN)x(t) = 0, x ∈ R2

(114)

where δ, , ν are scalar coefficients and matrices D > 0, K > 0 are real and symmetric, while matrices G and N are skew-symmetric as follows  G=N=

 0 −1 . 1 0

This system is a conservative Hamiltonian system if δ = 0,  = 0, and ν = 0, which is statically unstable for K < 0 with the even Poincaré instability degree equal to 2. Adding gyroscopic forces√with  >√0, keeps this system Hamiltonian and yields its stabilization if  >  f = −κ1 + −κ2 , where κ1,2 < 0 are eigenvalues of K.

Classical Results and Modern Approaches to Nonconservative Stability

(a)

175

(b)

Fig. 21 Given  = 0.3, the green lines depict (a) imaginary and (b) real parts of the eigenvalues of the PT - symmetric problem with indefinite damping (116) as functions of the parameter 2 = μ1 − μ2 = 2μ when k = 1. Red lines correspond to the eigenvalues of the problem (41) with k1 = 1, κ = k2 − k1 = 0.1 and 1 = μ1 + μ2 = 0.1

Owing to the ‘reversible’ symmetry of its spectrum (MacKay 1991; Bloch et al. 1994), the Hamiltonian system displays flutter instability via the collision of imaginary eigenvalues at  =  f and their subsequent splitting into a complex quadruplet as soon as  decreases below  f . This is the already familiar to us linear HamiltonHopf bifurcation. If δ > 0, ν > 0 the gyroscopic stability is destroyed at the threshold of the classical-Hopf bifurcation (Kirillov 2007, 2013a) 2 f H ≈  f + (ω f trD)2



tr(KD + (2f − ω 2f )D) ν − δ 2 f

2 ,

√ where ω 2f = κ1 κ2 and D > 0. The dependency of the new gyroscopic stabilization threshold just on the ratio ν/δ implies that the limit of  H as both ν and δ → 0 is higher than  f for all ratios except a unique one. Similarly to the case of nonconservative reversible systems, this happens because the classical Hopf and the Hamilton-Hopf bifurcations meet in the Whitney umbrella singularity that exists on the stability boundary of a nearlyHamiltonian dissipative system and corresponds to the onset of the Hamilton-Hopf bifurcation (Bottema 1956; Arnold 1972; Langford 2003; Kirillov 2007, 2013a; Krechetnikov and Marsden 2007; Kirillov and Verhulst 2010).

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6 Stability in the Presence of Potential, Circulatory, Gyroscopic and Dissipative Forces Beletsky (1995) remarked that when potential, circulatory and gyroscopic forces are present simultaneously, it becomes nontrivial to judge about stability. “The pairwise interaction of arbitrary two of these [forces] results in the existence of stable domains in the parameter space. However, the simultaneous action of all three effects always results in instability!” (Beletsky 1995). Addition of dissipation entangles stability analysis even more (Kirillov 2013a; Hagedorn et al. 2014; Kliem and Pommer 2017). Here we present several examples illustrating these statements.

6.1 Rotating Shaft by Shieh and Masur (1968) Let us return once again to the model (20) of a rotating shaft by Shieh and Masur (1968) with damping but without circulatory forces m u¨ + μ1 u˙ − 2v˙ + (k1 − 2 )u = 0 m v¨ + μ2 v˙ + 2u˙ + (k2 − 2 )v = 0.

(115)

Although the literal meaning of the word ‘damping’ prescribes the coefficients μ1 and μ2 to be nonnegative, it is instructive to relax this sign convention Kirillov (2013b). Therefore, we consider the gyroscopic system (115) where the negative sign of the damping coefficient corresponds to a gain and the positive one to a loss (Karami and Inman 2011; Schindler et al 2011). In mechanics, negative damping terms enter the equations of motion of moving continua in frictional contact when the dependence of the frictional coefficient on the relative velocity has a negative slope, which can be observed already in the tabletop experiments with the singing wine glass (Kirillov 2009, 2013a). In physics, a pair of coupled oscillators, one with gain and the other with loss, can naturally be implemented as an LRC-circuit (Schindler et al 2011). When μ1 = −μ2 = μ > 0 the gain and loss in Eq. (41) are in perfect balance. Let us further assume that k1 = k2 = k: m u¨ + μu˙ − 2v˙ + (k − 2 )u = 0 m v¨ − μv˙ + 2u˙ + (k − 2 )v = 0.

(116)

Let us look at what happens with these equations when we change the direction of time, assuming t → −t. Then, m u¨ − μu˙ + 2v˙ + (k − 2 )u = 0 m v¨ + μv˙ − 2u˙ + (k − 2 )v = 0

(117)

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(b)

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(c)

(a)

Fig. 22 Stability domain of the rotating shaft by Shieh and Masur for k1 = 1,  = 0.3, and β = 0. a The Plücker conoid in the (μ1 , μ2 , κ)-space and its slices in the μ1 , μ2 -plane with b κ = 0 and c κ = 0.1. Open circles show locations of exceptional points (EPs) where pure imaginary eigenvalues of the ideal PT -symmetric system (116) experience the nonsemisimple 1 : 1 resonance; green lines are locations of the exceptional points where double nonsemisimple eigenvalues have negative real parts

and we see that Eq. (116) are not invariant to the time reversal transformation (T). The interchange of the coordinates as x ↔ y in Eq. (116) results again in Eq. (117), which do not coincide with the original. Hence, the Eq. (116) are not invariant with respect to the parity transformation (P). Nevertheless, two negatives make an affirmative, and the combined PTtransformation leaves the Eq. (116) invariant despite the T-symmetry and P-symmetry not being respected separately. The spectrum of the PT-symmetric system (116) with indefinite damping is symmetrical with respect to the imaginary axis on the complex plane as it happens in Hamiltonian and reversible systems. To see this, let us consider the eigenvalues λ of the problem (41) introducing the new parameters 1 = μ1 + μ2 , 2 = μ1 − μ2 and κ = k2 − k1 . At 1 = 0 and κ = 0 they represent the spectrum of the problem (116)   1 222 − 16k1 − 162 ± 2 (162 − 22 )(16k1 − 22 ) λ=± 4 where k1 = k and 2 = 2μ. In Fig. 21 the eigenvalues are shown by the green lines. They are pure imaginary when |2 | < 4||. At the exceptional points (EPs), 2 = ±4, the pure imaginary eigenvalues collide into a double defective one which with the further increase in 2 splits into a complex-conjugate pair (flutter instability). PT - symmetry can be violated by the asymmetry both in the stiffness distribution κ = 0 and in the balance of gain and loss 1 = 0. In such a situation, the merging of eigenvalues that was perfect for the PT-symmetric system (116) is destroyed. The red eigencurves in Fig. 21 demonstrate the imperfect merging of modes that causes a decrease of the stability interval with respect to that of the symmetric system (the effect similar to the Ziegler–Bottema destabilization paradox in circulatory systems).

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Fig. 23 Imaginary and real parts of the roots of the characteristic equation (118) as a function of the damping coefficient μ1 under the constraints (119) for k1 = 1,  = 0.03 and β = 0.03

The Routh–Hurwitz conditions applied to the characteristic polynomial of the system (1.60) yield the domain of the asymptotic stability μ1 μ2 κ2 + (μ1 + μ2 )(μ1 μ2 + 42 )(μ1 κ + (μ1 + μ2 )(k1 − 2 )) > 0 μ1 + μ2 > 0, shown in Fig. 22a in the (μ1 , μ2 , κ)-space. The surface has a self-intersection along the κ-axis that corresponds to a marginally stable conservative gyroscopic (Hamiltonian) system. More intriguing is that in the (κ = 0) - plane there exists another self-intersection along the interval of the line μ1 + μ2 = 0 with the ends at the exceptional points (μ1 = 2, μ2 = −2) and (μ1 = −2, μ2 = 2), see Fig. 22b. This is the interval of marginal stability of the oscillatory damped (PT-symmetric) gyroscopic system (116) with the perfect gain/loss balance. At the exceptional points, the stability boundary has the Whitney umbrella singularities. In the (κ = 0) - plane the range of stability is growing with the increase of the distance from the line μ1 + μ2 = 0, which is accompanied by detuning of the gain/loss balance, Fig. 22b. Indeed, in this slice the boundary of the domain of asymptotic stability is the hyperbola (μ1 − μ2 )2 − (μ1 + μ2 )2 = 162 . At μ1 + μ2 = 0 it touches the two straight lines μ1 − μ2 = ±4, every point of which corresponds to a pair of double defective complex-conjugate eigenvalues with real parts that are negative when μ1 + μ2 > 0, positive when μ1 + μ2 < 0, and zero when μ1 + μ2 = 0: λ=−

μ1 + μ2 1 ± (μ1 + μ2 )2 − 16(k1 − 2 ) 4 4

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(a)

(b)

179

(c)

Fig. 24 Stability domain of the rotating shaft by Shieh and Masur for k1 = 1,  = 0.03, β = 0.03. a The ‘Viaduct’ in the (μ1 , μ2 , κ)-space and its slices in the (μ1 , μ2 )-plane with b κ = 0.06 and c κ = 0.03 (Kirillov 2011a, b)

The two lines of exceptional points stem from the end points of the interval of marginal stability of the PT - symmetric system and continue inside the asymptotic stability domain of the near-PT-symmetric one (green lines in Fig. 22b). The proximity of a set of defective eigenvalues to the boundary of the asymptotic stability, that generically is characterized by simple pure imaginary eigenvalues, plays an important role in modern nonconservative physical and mechanical problems. Near this set the eigenvalues can dramatically change their trajectories in the complex plane. For this reason, encountering double eigenvalues with the Jordan block and negative real parts is considered as a precursor to instability. The full model of Shieh and Masur (20) provides even more non-trivial example. Indeed, its characteristic equation has the form λ4 + (μ1 + μ2 )λ3 + (μ1 μ2 + k1 + k2 + 22 )λ2

(118)

+(k1 μ2 + μ1 k2 + 4β − (μ1 + μ2 )2 )λ + (2 − k1 )(2 − k2 ) + β 2 = 0. Equation (118) is biquadratic in the case when μ1 + μ2 = 0, κ = −

4β , μ1

(119)

with κ = k2 − k1 . If k1 > 2 and β > 0 then all the roots of Eq. (118) are imaginary when 4β(k1 − 2 ) < μ1 ≤ 2. (120) 2 ≤ μ1 < 0, β 2 + (k1 − 2 )2 In Fig. 23 the imaginary eigenvalues are shown by black lines as functions of the damping parameter μ1 . At μ1 = μd :=

4β(k1 − 2 ) β2 2 , κ = κ := −k +  − d 1 β 2 + (k1 − 2 )2 k1 − 2

(121)

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there exists a double zero eigenvalue with the Jordan block, see Figs. 23 and 24a. In the interval 0 < μ1 < μd there exist one positive and one negative real eigenvalue. In Fig. 23 the eigenvalues with non-zero real parts are shown in red. In the (μ1 , μ2 , κ)space the exceptional points (EPs) (−2, 2, 2β), (2, −2, −2β) correspond to the double imaginary eigenvalues with the Jordan block   λ−2 = ±i k1 − 2 + β, λ2 = ±i k1 − 2 − β, for μ1 = −2 and μ1 = 2, respectively. We see in Fig. 23 that changing the damping parameter μ1 we migrate from the marginal stability domain to that of flutter instability by means of the collision of the two simple pure imaginary eigenvalues as it happens in gyroscopic or circulatory systems without dissipation. It is remarkable that such a behavior of eigenvalues is observed in the gyroscopic system in the presence of dissipative and non-conservative positional forces. Let us now establish how in the (μ1 , μ2 , κ)-space the domain of marginal stability given by the expressions (119) and (120) is connected to the domain of asymptotic stability of the Eq. (118). Writing the Liénard and Chipart conditions for asymptotic stability of the polynomial (118) we find p1 := μ1 + μ2 > 0, p2 := μ1 μ2 + k1 + k2 + 22 > 0, p4 := (2 − k1 )(2 − k2 ) + β 2 > 0, H3 := (μ1 + μ2 )(μ1 μ2 + k1 + k2 + 22 )   × k1 μ2 + μ1 k2 + 4β − (μ1 + μ2 )2 − (μ1 + μ2 )2 ((2 − k1 )(2 − k2 ) + β 2 ) − (k1 μ2 + μ1 k2 + 4β − (μ1 + μ2 )2 )2 > 0.

(122)

The surfaces p4 = 0 and H3 = 0 are plotted in Fig. 24a. The former is simply a horizontal plane that passes through the point of the double zero eigenvalue with the coordinates (μd , −μd , κd ) and thus bounds the stability domain from below. The surface H3 = 0 is singular because it has self-intersections along the portions of the hyperbolic curves (119) selected by the inequalities (120). The curve of selfintersection that corresponds to κ > 0 ends up at the EP with the double pure imaginary eigenvalue λ−2 . Another curve of self-intersection has at its ends the EP with the double pure imaginary eigenvalue λ2 and the point of the double zero eigenvalue, 02 . In Fig. 24a the curves of self-intersection are shown in red and the EP and 02 are marked by the black and white circles, respectively. At the point 02 the surfaces p4 = 0 and H3 = 0 intersect each other forming a trihedral angle singularity of the stability boundary

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with its edges depicted by red lines in Fig. 24a. The surface H3 = 0 is symmetric with respect to the plane p1 = 0. Thus, a part of it that belongs to the subspace p1 > 0 bounds the domain of asymptotic stability. At the EPs, the boundary of the asymptotic stability domain has singular points that are locally equivalent to the Whitney umbrella singularity. Between the two EPs the surface H3 = 0 has an opening around the origin that separates its two sheets. This window allows the flutter instability to exist in the vicinity of the origin for small damping coefficients and small separation of the stiffness coefficients κ. In Fig. 24b a cross-section of the surface H3 = 0 by the horizontal plane that passes through the lower exceptional point is shown. The domain in grey indicates the area of asymptotic stability. Its boundary has a cuspidal point singularity at the EP. Although the very singular shape of the planar stability domain is typical in the vicinity of the EP with the pure imaginary double eigenvalue with the Jordan block, the unusual feature is the location of the EP that corresponds to non-vanishing damping coefficients, Fig. 24b. According to the theorems of Bottema (1955), Lakhadanov (1975) the undamped gyroscopic system with non-conservative positional forces is generically unstable, see e.g. Beletsky (1995), Kirillov (2013a). By examining the slices of the surface H3 = 0 at various values of κ one can see that the origin is indeed always unstable, Fig. 24b, c. At κ = 0 the origin is unstable in the presence of the non-conservative positional forces even when the rotation is absent ( = 0) according to the Merkin theorem. Contrary to the situation known as the Ziegler–Bottema destabilization paradox, in the Shieh–Masur model the tending of the damping coefficients to zero along a path in the (μ1 , μ2 )-plane cannot lead to the set of pure imaginary spectrum of the undamped system because in this model such a set corresponds to the nonvanishing damping coefficients. Therefore, the Shieh–Masur model provides a nontrivial example of a gyroscopic system that can have all its eigenvalues pure imaginary in the presence of dissipative and circulatory forces. The highly non-trivial shape of the discovered stability boundary illustrates the peculiarities of stability of a system loaded by non-conservative positional forces in their interplay with the dissipative, gyroscopic and potential ones.

6.2 Two-Mass-Skate (TMS) Model of a Bicycle Kooijman et al. (2011) considered a reduced model of a bicycle with vanishing radii of the wheels (that are replaced by skates), known under the name of the two-massscate (TMS) bicycle. The deviation from the straight vertical equilibrium is described by the leaning angle of the frame and the steering angle of the front wheel/skate that are governed by the following system of two linear equations M¨x + vD˙x + gKx + v 2 Nx = 0, where dot denotes time differentiation,

(123)

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Fig. 25 The two-mass-skate bicycle model (Kooijman et al. 2011)

 M=  D= 

m B z 2B + m H z 2H −m H u H z H −m H u H z H m H u 2H



0 −(m B x B z B + m H x H z H )/wˆ 0 (m H u H x H )/wˆ

, 

−m H u H mBzB + mH zH −m H u H −m H u H sinλs   0 −(m B z B + m H z H )/wˆ , N= 0 (m H u H )/wˆ

K=

,  , (124)

u H = (x H − w) cos λs − z H sin λs , wˆ = w/ cos λs and g denotes the gravity acceleration. The model (123), (124) is nonconservative, containing dissipative, gyroscopic, potential and circulatory forces. Curiously enough, Eq. (123) has a form that is typical in many fluid-structure interactions problems, where the parameter v would correspond to the velocity of the flow either inside of a flexible pipe or around a flexible structure (Mandre and Mahadevan 2010; Paidoussis 2016). This similarity in the mathematical description suggests an analogy between the weaving bicycle and fluttering flag, which is not very obvious. In fact, Eq. (123) depends on 9 dimensional parameters: w, v, λs , m B , x B , z B , m H , x H , z H that represent, respectively, the wheel base, velocity of the bicycle, steer axis tilt, rear frame assembly (B) mass, horizontal and vertical coordinates of the rear frame assembly center of mass, front fork and handlebar assembly (H ) mass, and horizontal and vertical coordinates of the front fork and handlebar assembly center of mass. Choosing the wheelbase, w, as a unit of length, and introducing the Froude number, Fr, we find that, actually, the model depends on the following seven dimensionless parameters:

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mH v xB xH zB zH , ξH = , ζB = , ζH = , λs . Fr = √ , μ = , ξB = gw mB w w w w We can assume that for realistic bicycles 0 ≤ μ ≤ 1. Notice that ζ B ≤ 0 and ζ H ≤ 0 due to choice of the system of coordinates, Fig. 25.  Assuming the solution ∼ exp(σt) and introducing  the dimensionless time τ = g t w

such that the dimensionless eigenvalue is s = σ

w , we write the characteristic g

polynomial of the TMS bicycle model: p(s) = a0 s 4 + a1 s 3 + a2 s 2 + a3 s + a4 , with the coefficients a0 = −(ζ H tan λs − ξ H + 1)ζ B2 , a1 = Fr(ζ B ξ H − ζ H ξ B )ζ B , a2 = Fr 2 (ζ B − ζ H )ζ B − ζ B (ζ B + ζ H ) tan λs − (ξ H − 1)(μζ H − ζ B ), a3 = −Fr(ξ B − ξ H )ζ B , a4 = −ζ B tan λs − μ(ξ H − 1). (125) Notice that in the case when the coordinates of the masses m B and m H coincide: ξH = ξB , ζH = ζB the characteristic polynomial simplifies and factorizes as p(s) = −(s 2 ζ B + 1)(ζ B (ζ B tan λs − ξ B + 1)s 2 + ζ B tan λs + μ(ξ B − 1)). Since ζ B < 0 by definition, this immediately yields static instability (growth of the leaning angle yielding capsizing of the bike). Asymptotic Stability and Critical Froude Number We study linear stability of the TMS bicycle with the Lienard–Chipart version of the Routh–Hurwitz criterion (Kirillov 2013a). First, compute the Hurwitz determinants of the characteristic polynomial h 1 = Fr(ζ B ξ H − ζ H ξ B )ζ B , h 2 = Frζ B f, h 3 = −Fr 2 ζ B2 (ζ B − ζ H )h, h 4 = Fr 2 ζ B2 (ζ B − ζ H )(tan(λs )ζ B + μξ H − μ)h,

(126)

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where f = −ζ B (ζ B2 ξ H − ζ H2 ξ B ) tan λs − ζ H (ξ H − 1)(ζ B ξ H − ζ H ξ B )μ + ζ B (ζ B − ζ H )(ζ B ξ H − ζ H ξ B )Fr 2 + ζ B ξ B (ξ H − 1)(ζ B − ζ H )

(127)

h = −ζ B ξ B ξ H (ζ B − ζ H ) tan λs − ξ H (ξ H − 1)(ζ B ξ H − ζ H ξ B )μ + ζ B (ξ B − ξ H )(ζ B ξ H − ζ H ξ B )Fr 2 + ζ B ξ B (ξ H − 1)(ξ B − ξ H ).

(128)

and

The Lienard–Chipart criterion requires that a4 > 0, a3 > 0, a1 > 0, a0 > 0, h 1 > 0, h 3 > 0. The relation h 1 = a1 eliminates one of the inequalities and in view of that μ > 0, ζ B < 0, and ξ B > 0 yields the following explicit conditions ξ H > 1 + ζ H tan λs ζB tan λs ξH < 1 − μ ξH < ξB ζH > ζB Fr > Fr c > 0,

(129)

where the critical Froude number at the stability boundary is given by the expression Fr 2c =

ξB ξH ζB − ζH ξH − 1 ξH (ξ H − 1)ξ B tan λs + μ− ξB − ξH ζB ξH − ζH ξB ξB − ξH ζB ζB ξH − ζH ξB

(130)

that follows from the condition h = 0. At 0 ≤ Fr < Fr c the bicycle is unstable by flutter demonstrating the weaving motion (Kooijman et al. 2011) Critical Fr for the Benchmark Bikes of Kooijman et al. (2011) For the design determined by w = 1 m, λs =

5π rad, m H = 1 kg, m B = 10 kg, 180

x B = 1.2 m, x H = 1.02 m, z B = −0.4 m, z H = −0.2 m the critical Froude number is Fr 1 = 0.9070641497,

(131)

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5π Fig. 26 For w = 1 m, λs = 180 rad, m H = 1 kg, m B = 10 kg, x B = 1.2 m, z B = −0.4 m (left) stability diagram at Fr = Fr 1 = 0.9070641497 with the circle corresponding to x H = 1.02 m and z H = −0.2 m; (right) stability diagram at Fr = Fr min = 0.6999527422. Black circle denotes a point with the coordinates (0.9716634870,–0.3238878290) given by (135)

which corresponds to the critical velocity of weaving v1 = 2.841008324 m/s

(132)

in accordance with the original result by Kooijman et al. (2011). For the alternative design determined by w = 1 m, λs = −

5π rad, m H = 1 kg, m B = 10 kg, 180

x B = 0.85 m, x H = 1 m, z B = −0.2 m, z H = −0.4 m the critical Froude number is Fr 2 = 0.8415708896,

(133)

which corresponds to the critical velocity of weaving v2 = 2.635877411 m/s

(134)

in accordance with the original result by Kooijman et al. (2011). Notice that careful analysis of the Lienard-Chipart criteria for the TMS bicycle proves the existence of just two classes of self-stable TMS bikes that differ by the sign of λs , see Austin Sydes (2018).

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Finding Designs that Minimize the Critical Fr Let us fix λs , ξ B , ζ B , and μ and plot the stability domain specified by Eq. (129) in the (ξ H , ζ H ) - plane at different values of Fr, Fig. 26. This yields a vertical line ξ H = ξ B , a horizontal line ζ H = ζ B and an inclined line ξ H = 1 + ζ H tan λs that form a rectangular triangle in the (ξ H , ζ H ) - plane, Fig. 26. There is no stability outside of this triangle. On the other hand the condition Fr = Fr c defines two hyperbola-like curves, one of which always passes through the right lower corner of the triangle and the other one always passes through a point on the hypotenuse of the triangle shown by a black circle in Fig. 26. Solving simultaneously equations ξ H = 1 + ζ H tan λs and Fr = Fr c we find the coordinates of this point to be ξH =

−ξ B −ζ B , ζH = . ζ B tan λs − ξ B ζ B tan λs − ξ B

(135)

If we take, for instance w = 1 m,

5π λs = 180 rad, x B = 1.2 m,

m H = 1 kg, z B = −0.4 m,

m B = 10kg,

(136)

the branch of the curve Fr = Fr c passing through the point (135) with the coordinates (0.9716634870, −0.3238878290) lies partially inside the triangle, Fig. 26(left). The area between this part and the hypotenuse is the stability domain, which for the TMS bicycle is further restricted by the condition ζ H < 0. Can we change the design in order to minimize the critical Froude number? If we plot the curve Fr c (ξ H , ζ H ) = Fr at different values of Fr, we will see that the portion of its branch passing through the point (135) and lying in the triangle tends to get smaller as Fr decreases. At some Fr min the branch is tangent to the hypotenuse at the point (135), and the stability domain disappears, Fig. 26(right). Therefore, the design specified by the conditions (135) gives the minimum possible Froude number, beyond which the TMS bike becomes stable: Fr 2min =

(ζ B tan λs − ξ B )2 + μ tan λs . (ζ B tan λs − ξ B )(ζ B tan λs − ξ B + 1)

For instance, if we take parameters as in (136) and use (135) to find x H = 0.9716634870 m and z H = −0.3238878290 m, then we obtain the minimal Froude number and the corresponding velocity of weaving Fr min = 0.6999527422, vmin = 2.192316351 m/s that indeed are smaller then that given by (131) and (132) for the benchmark TMS bike in Kooijman et al. (2011). Further results on stability-optimized TMS bicycle one can find in the recent work by Kirilov (2018).

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