Компьютерные вычисления в пакете MathCAD 100-100-100-1

Настоящее учебно-методическое пособие подготовлено по материалам учебного курса ''Компьютерная обработка инфор

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2. « 4. « 6. « ( )».

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».

, ,

,

,

,

1.

.

MathCAD , .

MathCAD,

1.1.

-

,

.

.

.

. .

,2

3

15.7 .

/

3

, /

3

,

/ 3.

: V := 2 ⋅cm

3

M := 15.7 ⋅gm : ρ :=

M V

γ := ρ ⋅g −3

ρ = 7.85 × 10

−3

gm⋅mm

−3

ρ = 7.85 gm⋅cm

ρ = 7.85 × 10 kgm 3

-3

γ = 7.698 × 10 m newton 4 -3

400

1.2. [9].

.

3

298

: kJ := 10 ⋅J 3

n := 3 ⋅mol

R := 8.3143 ⋅

T1 := 298 ⋅K

J mol⋅K

T2 := 400 ⋅K : W := n⋅R⋅( T2 − T1) W = 2.544 kJ

200°

1.3. 20°

. 220° ,

,

Q

1.5 0.54

/ · .

. : kJ := 10 ⋅J 3

M := 1.5kg

T1 := ( 20 + 273)K

T2 := ( 220 + 273)K C := 0.54

kJ

kg⋅K

:

Q := C ⋅M ⋅( T2 − T1)

Q = 162 kJ

t2 := T1 , T1 + 20 ⋅K .. T2 Q ( t2) := C ⋅( t2 − T1) ⋅M Q ( t2) =

0 16.2 32.4 48.6 64.8 81 97.2 113.4 129.6 145.8 162

200

kJ

180 160 140 120 Q( t2) kJ

100 80 60 40 20 300 325 350 375 400 425 450 475 500 t2 K

1.4.

α

.

, [6].

v0

: x0 := 0 ⋅m

α := 45 ⋅deg

y0 := 0 ⋅m a := −g

v0 := 100 ⋅

m sec

:

t := 0 ⋅sec , 0.1 ⋅sec .. 14.5 ⋅sec

vx0 := v0⋅cos (α )

vy0 := v0⋅sin(α )

vx( t) := vx0

vy( t) := vy0 + a ⋅t

x( t) := x0 + vx0⋅t

y( t) := y0 + vy0⋅t +

a 2

⋅( t)

2

200 y( t) m

0

500

1000

x( t ) m

⎛ vy( t) ⎞ ⎟ ⎝ vx( t) ⎠

α ( t) := atan⎜

-

50

α ( t) 0

deg

5

10

15

50 t sec

1.5.

.

,

σ.

. 1.1, [17].

. 1.1.

1.5

:

kN := 10 ⋅ N

MPa := 10 ⋅ Pa

3

ORIGIN ≡ 1

6

n := 2

i := 1 .. n

H1 := 10 ⋅ mm

1

B1 := 10 ⋅ mm

2 1

H2 := 20 ⋅ mm B2 := 10 ⋅ mm

2

L1 := 2 ⋅ m

1

L2 := 1 ⋅ m

2

E1 := 2 ⋅ 10 ⋅ MPa 5

1

E2 := 2 ⋅ 10 ⋅ MPa 5

2

F1 := 10 ⋅ kN

1

F2 := −40 ⋅ kN

2

:

→ ⎯ A := ( B ⋅ H)

⎛ 1⎞ 2 ⎟ cm ⎝ 2⎠

A= ⎜

⎛ 10 ⎞ ⎟ kN ⎝ −40 ⎠

F = ⎜

∑ i

σ i :=

i

=1 Ai

Fi

⎛ 100 ⎞ ⎟ MPa ⎝ −150 ⎠

σ = ⎜ δ i :=



n + 1−i

i

⎛ σ n + 1−i ⋅ Ln + 1−i ⎞ ⎜ ⎟ En + 1−i ⎝ ⎠ =1

⎛ 0.25 ⎞ ⎟ mm ⎝ − 0.75 ⎠

δ = ⎜

σ 2 ⋅ L2 E2 σ 1 ⋅ L1 E1

1.6.

= − 0.75 mm

2

+

1

σ 2 ⋅ L2 E2

= 0.25 mm

.

, .

U( x , y) :=

q

(2

2 ⋅π ⋅λ ⋅ x + y

)

1

2 2

: q := 1000

,

λ := 0.4

,

U ( x , y) :=

q

(2

2 ⋅π ⋅λ ⋅ x + y

)

/(c * )

1

2 2

: 1.6.1. i := 0 .. 10

j := 0 .. 10

xi := −100 + 20.00000000000001 ⋅i

yj := −100 + 20.00000000000001 ⋅ j Mi, j := U ( xi , yj)

M

1.6.2.

CreateMesh

M := CreateMesh( U , −100 , 100 , −100 , 100 , 100 , 100)

M

1.7. s ( t)

. [12]. g⋅

v

,

a

s

t

t + v0⋅t + s0 2 . 2

1.7.1.

t : g⋅

s ( t)

2

t

2

+ v0⋅t + s0 :

" d s ( t) dt v( t) dd dt dt

/

"

g⋅t + v0

g⋅t + v0

s ( t)

a ( t)

1.7.2.

/

g

g

, , :

t := 0 ⋅sec , 0.1 ⋅sec .. 2 ⋅sec v0 := 1 ⋅

s0 := 0 ⋅m

m sec

s ( t) := g⋅

t + v0⋅t + s0 2 2

v( t) := g⋅t + v0

:

9.8066500000002 g 9.80665

d v( t) dt d2 dt

2

s ( t ) 9.8066499999998

9.8066499999996 0

0.5

1

1.5

2

t sec

1.8.

1 2 ⋅a ⋅π. 4

π.

.

(

(0, 0), (a, 0), (a, a), (0, a)) ,

a

[12]

,

S = ∫ f ( x )dx . a

0

4.

π.

1.8.1. : ⌠ ⎮ f ( x) dx ⌡0 a

S f ( x)

a a −x 2

f1( x)

2

:

⌠ 2 ⎮ a dx → a ⌡0 a

⌠ 1 ⎮ 2 ⎮ a2 − x2 dx → 1 ⋅ a2 ⋅ π ⌡0 4 a

(

)

π

⎤ ⎡ ⌠a 1 ⎢ ⎥ ⎮ ⎢⎮ 2 2 2 ⎥ ⎢ ⌡ ( a − x ) dx ⎥ ⎣ 0 ⎦ ⋅ 4 → π = 3.142 ⌠ ⎮ a dx ⌡0 a

1.8.2. [18] a := 5

N := 1000

: c

a

:

i := 0 .. N

Xi := 0 + i ⋅

a N

m := length( X) − 1 Xm − X0 m

Yi := a

−3

= 5 × 10

Y1i := a − ( Xi) 2

2



m−1 ⎞ Xm − X0 ⎛⎜ X0 + Xm I := ⋅ + Yi⎟ ⎜ 2 ⎟ m i =0 ⎝ ⎠



m−1 ⎞ Xm − X0 ⎛⎜ X0 + Xm I1 := ⋅ + Y1i⎟ ⎜ 2 ⎟ m i =0 ⎝ ⎠

⌠ ⎮ a dx → 25 = 25 ⌡0

I = 25.012

a

⌠ 1 ⎮ 2 ⎮ a2 − x2 dx → 25 ⋅ π = 19.635 ⌡0 4

(

)

PI :=

I1 ⋅4 I

a

π

π = 3.142 π − PI π

2. 2.1. 60 ?

I1 = 19.66

PI = 3.144

= −0.076 %

. .

. 40

/ ;

: V0 := 40 ⋅

h := 60 ⋅m

m sec

−g = −9.807 msec

-2

: 2.1.1. t := 0 ⋅sec , 2 ⋅sec .. 10 ⋅sec

F ( t) := V0⋅t −

g 2 ⋅t 2

t := 0 ⋅sec .. 10 ⋅sec

F ( t) =

0 m 35.097 60.387 75.87 81.547 77.417 63.48 39.737 6.187 -37.169 -90.332

100

50 F( t ) m h

0

m 50

100

0

5 t sec

"

/ / " : (2, 60), (6, 60)

10

2.1.2. C

t. "

/

/

a ⋅t + b ⋅t + c 2

"

0

⎛ 1⎞⎤ ⎡ ⎡ ⎜ 2⎟⎥ ⎢ 1 ⎢ ⎝ ⎠ 2 ⋅ ⎣ − b + ( b − 4 ⋅ a ⋅c ) ⎢ ⎦ ( 2 ⋅ a ) ⎢ ⎢ ⎛ 1⎞⎤ ⎡ ⎜ 2⎟⎥ ⎢ 1 ⎢ ⎝ ⎠ 2 ( ) − ⋅ − b b − 4 ⋅ a ⋅ c ⎣ ⎦ ⎢ ⎣ ( 2 ⋅a ) a :=

−g 2

b := V0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

c := − h

⎛ 1⎞⎤ ⎡ ⎜ ⎟ ⎝ 2⎠⎥ 1 ⎢ 2 t1 := ⋅ ⎣ − b + b − 4 ⋅a ⋅ c ⎦ ( 2 ⋅a)

(

⎛ 1⎞⎤ ⎡ ⎜ ⎟ ⎝ 2⎠⎥ 1 ⎢ 2 ⋅⎣ − b − b − 4 ⋅ a ⋅ c t2 := ⎦ ( 2 ⋅a)

)

(

t1 = 1.981 sec 2.1.3.

)

t2 = 6.177 sec

"root"

TOL = 1 × 10

−3

f ( t) := − h + V0 ⋅t −

t := 0 ⋅sec

g 2 ⋅t 2

t_1 := root ( f ( t) , t) t_1 = 1.981 sec

t := 10 ⋅sec

t_2 := root ( f ( t) , t) t_2 = 6.177 sec

2.1.4.

"polyroot" − h + V0 ⋅t −

g 2 ⋅t 2

0

−h a 0 := m

a 1 :=

V0

a 0 ⋅ t + a 1 ⋅t + a 2 ⋅ t 0

1

⎛ −g ⎞ ⎜ ⎟ ⎝ 2 ⎠ a 2 := m

m sec

t0 = 1.981

t := polyroots( a)

2.1.5.

h := 60

TOL = 1 × 10

−3

sec

"Find"

g := 9.81

Given

0

−3

− h + V0 ⋅t −

f ( t) := Find( t)

g 2 ⋅t 2

f ( 0) = 1.981

f ( 10 ) = 6.173 2.1.6.

"Find"

Given 0

−h + V0 ⋅ t −

g 2 ⋅t 2

2

t1 = 6.177

V0 := 40

CTOL = 1 × 10

2

0 − 60 ⎞ ⎜⎛ ⎟ a = ⎜ 40 ⎟ ⎜ − 4.903 ⎟ ⎝⎝ .903⎠ ⎠

⎛ 1⎞⎤ ⎛ 1⎞⎤ ⎡ ⎡ ⎡ ⎜ 2⎟⎥ ⎜ 2⎟⎥ ⎢ 1 ⎢ ⎢ ⎝ ⎠ ⎝ ⎠ 1 2 2 Find( t) → ⎢ ⋅ ⎣ 2 ⋅ V0 + 2 ⋅ (V0 − 2 ⋅ g ⋅ h) ⋅ ⎣ 2 ⋅ V0 − 2 ⋅ ( V0 − 2 ⋅ g ⋅ h) ⎦ ⎦ ( 2 ⋅ g ) ( 2 ⋅ g ) ⎣

2.2.

.

40 .

/ .

,

,

.

V0 := 40 ⋅

h := 60 ⋅m

: m sec

−g = −9.807 msec

-2

: 2.2.1.

t := 0 ⋅sec .. 10 ⋅sec

F ( t) := V0 ⋅t −

g 2 ⋅t 2

100

F( t ) 0

m

100

0

5

10

t sec

" 2.2.2.

t := 0 ⋅sec Given F ( t)

/ / : (4, 81.547) "Minerr"

100 ⋅m

t_max := Minerr ( t) t_max = 4.079 sec

F ( t_max) = 81.577 m 2.2.3.

t= [0, 10] "Maximize"

t := 0 ⋅sec Given

0 ⋅sec ≤ t ≤ 10 ⋅sec

tmax := Maximize( F , t)

tmax = 4.079 sec F tmax = 81.577 m

(

)

⎤ ⎥ ⎥ ⎦

"

2.2.4.

F(t) >= 0 "Minimize"

t := 0 ⋅ sec Given

F ( t) ≥ 0

t_begin := Minimize ( F , t) t_begin = 0 sec

F ( t_begin ) = 0 m t := 10 ⋅ sec Given

F ( t) ≥ 0

t_end := Minimize ( F , t) t_end = 8.158 sec

F ( t_end ) = − 1.652 × 10

−5

t := 0 ⋅ sec .. 10 ⋅ sec

m

100 F ( t) m

50

F ( t_max ) m F ( t_begin ) m F ( t_end ) m

0

5

10

50

100 t t_max , , t_begin , t_end sec sec

1 2

2.3.

. ;

АВ В Q=1 α = 30°, h = 1 , L = 3 [2].

P=2 . 2.1).

(

А

, –

Y

B L C

A

α

Q h

X . 2.1.

2.3

.

. 2.2.

А

, ,

Y

B Q

N C

P

Ya Xa

A

X

. 2.2.

2.3

: kN := 10 ⋅N 3

P := 2 ⋅kN

N := Q

Q := 1 ⋅kN

L := 3 ⋅m

XA := P

YA := P

h := 1 ⋅m

: 2.3.1.

"Find"

Given XA − N ⋅cos ( 60 ⋅deg)

0

YA − P + N ⋅cos ( 30 ⋅deg) − Q

0

L N ⋅h −P ⋅ ⋅cos ( 30 ⋅deg) + − Q ⋅L⋅cos ( 30 ⋅deg) 2 sin( 30 ⋅deg) 1.299 ⎞ ⎜⎛ ⎟ Find( XA , YA , N) = ⎜ 0.75 ⎟ kN ⎜ 2.598 ⎟ ⎝ ⎠

0

2.3.2.

"Find"

Given XA − N ⋅cos ( 60 ⋅deg)

0

YA − P + N ⋅cos ( 30 ⋅deg) − Q

0

L N ⋅h −P ⋅ ⋅cos ( 30 ⋅deg) + − Q ⋅L⋅cos ( 30 ⋅deg) 2 sin( 30 ⋅deg)

0

Find( XA , YA , N) "

/

/

"

1 ( P + 2 ⋅Q) ⎡⎢ ⋅L⋅cos ( 30 ⋅deg) ⋅sin( 30 ⋅deg) ⋅ ⋅cos ( 60 ⋅deg) 2 h ⎢ ⎢ −1 (−2 ⋅P ⋅h + L⋅cos ( 30 ⋅deg) 2⋅sin( 30 ⋅deg) ⋅P + 2 ⋅L⋅cos ( 30 ⋅deg) 2⋅sin( 30 ⋅deg) ⋅Q − 2 ⋅Q ⋅h) ⎢ ⋅ h ⎢ 2 ⎢ ( P + 2 ⋅Q ) 1 ⋅L⋅cos ( 30 ⋅deg) ⋅sin( 30 ⋅deg) ⋅ ⎢ h 2 ⎣

⎥⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

2.3.3. :

P := 2

Q := 1

N := Q

XA := P

L := 3

h := 1

YA := P

:

XA − N ⋅ cos ( 60 ⋅ deg)

0

YA − P + N ⋅ cos ( 30 ⋅ deg) − Q −P ⋅

0

N⋅h L ⋅ cos ( 30 ⋅ deg) + − Q ⋅ L ⋅ cos ( 30 ⋅ deg) 2 sin( 30 ⋅ deg)

⎛ 1 0 − cos ( 60 ⋅ deg) ⎜ ⎜ 0 1 cos ( 30 ⋅ deg) A := ⎜ h ⎜ 0 0 sin ( 30 ⋅ deg) ⎝ X := A

−1

⎞ ⎟ ⎟ D := ⎟ ⎟ ⎠

0

0 ⎛ ⎜ Q+P ⎜ ⎜ L ⎜ P ⋅ ⋅ cos ( 30 ⋅ deg) + Q ⋅ L ⋅ cos ( 30 ⋅ deg) 2 ⎝

⋅D

1.299 ⎞ ⎜⎛ ⎟ X = ⎜ 0.75 ⎟ ⎜ 2.598 ⎟ ⎝ ⎠

X := lsolve( A , D)

⎛⎜ 1.299 ⎞⎟

X = ⎜ 0.75 ⎟

⎜ 2.598 ⎟ ⎝ ⎠

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

2.3.4. XA − N ⋅cos ( 60 ⋅deg)

0

YA − P + N ⋅cos ( 30 ⋅deg) − Q

0

L N ⋅h −P ⋅ ⋅cos ( 30 ⋅deg) + − Q ⋅L⋅cos ( 30 ⋅deg) 2 sin( 30 ⋅deg)

⎛ 1 0 −cos ( 60 ⋅deg) ⎜ ⎜ 0 1 cos ( 30 ⋅deg) ⎜ h ⎜ 0 0 sin ( 30 ⋅deg) ⎝

0

0 ⎞ ⎛ ⎟ ⎜ Q+P ⎟ ⋅⎜ ⎟ ⎜ L ⎟ ⎜ P ⋅ ⋅cos ( 30 ⋅deg) + Q ⋅L⋅cos ( 30 ⋅deg) ⎠ ⎝ 2

⎞ ⎟ ⎟→ ⎟ ⎟ ⎠

−1

cos ( 60 ⋅deg) ⎞ ⎛1 ⎡ ⋅sin( 30 ⋅deg) ⋅⎜ ⋅P ⋅L⋅cos ( 30 ⋅deg) + Q ⋅L⋅cos ( 30 ⋅deg) ⎟ ⎢ h ⎝2 ⎠ ⎢ ⎢ Q + P − cos ( 30 ⋅deg) ⋅sin( 30 ⋅deg) ⋅⎜⎛ 1 ⋅P ⋅L⋅cos ( 30 ⋅deg) + Q ⋅L⋅cos ( 30 ⋅deg) ⎟⎞ ⎢ h ⎝2 ⎠ ⎢ 1 ⎛1 ⎞ ⎢ ⋅sin( 30 ⋅deg) ⋅⎜ ⋅P ⋅L⋅cos ( 30 ⋅deg) + Q ⋅L⋅cos ( 30 ⋅deg) ⎟ h ⎣ ⎝2 ⎠

2.4.

.

(

[9]:

( . ). 0

=

log( P)

12.486 −

log( P)

7.884 −

3160 T

1860 T

0

: log( P)

12.486 −

log( P)

7.884 −

3160

1860 T

: P := 1

T := 1

Given log( P)

12.486 −

log( P)

7.884 −

3160 T

1860 T

⎛ P⎞ ⎜ ⎟ := Find( P , T) ⎝ T⎠

P = 19.934

P

= 20

.

. (2.666·103

).

T

T = 282.486

.

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .)

2.5.

.

, [9].

.

,

2%

: P1_0 − P1 P1_0

x2

n2 ( n2 + n1)

P1_0 := 100 P1 := 98 n1 :=

1000 18

-

n2

: n2 := 0

x2 := 0

Given P1_0 − P1 P1_0

x2

n2 ( n2 + n1)

⎛ x2 ⎞ ⎜ ⎟ := Find( x2 , n2) ⎝ n2 ⎠

2.6. ,

. – 50 %

n2 = 1.134

. , 10 %

70 % 40 %

30 %

.

,

– 80 % ,

15 %

20 % .

[10]? ш

.

u–

,v– .

,w– 15 %

,

,

0.3u + 0v + 0.1w = 0.15 . u+v+w 0.7u + 0.8v + 0.5w . u+v+w

. :

x=

(

u v w , y= , z= . u+v+w u+v+w u+v+w : 0.3x+0.1z–0.15=0 x+y+z–1=0, ), ц

0.7x+0.8y+0.5z. :

x, y, z ц .

.

, :

⎧0.7 x + 0.8 y + 0.5 → max ⎪ 0.3 x + 0.1z − 0.15 = 0 ⎪ ⎨ x+ y + z =1 ⎪ ⎪⎩ x ≥ 0, y ≥ 0, z ≥ 0

⎧0.7 x + 0.8 y + 0.5 → min ⎪ 0.3x + 0.1z − 0.15 = 0 ⎪ ⎨ x+ y+ z =1 ⎪ ⎪⎩ x ≥ 0, y ≥ 0, z ≥ 0

: F ( x , y , z) := 0.7 ⋅ x + 0.8 ⋅ y + 0.5 ⋅ z : x := 0

y := 0

z := 0

Given 0.3 ⋅ x + 0.1 ⋅ z − 0.15 x+ y+ z

0

1

x≥ 0

y≥ 0

z≥ 0

P := Minimize( F , x , y , z)

⎛⎜ 0.25 ⎞⎟ P = ⎜ 0 ⎟ ⎜ 0.75 ⎟ ⎝ ⎠ x := 0

y := 0

z := 0

Given 0.3 ⋅ x + 0.1 ⋅ z − 0.15 x+ y+ z

0

1

x≥ 0

y≥ 0

P := Maximize( F , x , y , z)

z≥ 0

⎛⎜ 0.5 ⎞⎟ P = ⎜ 0.5 ⎟ ⎜ 0 ⎟ ⎝ ⎠

3. 3.1. (

π [10]. (0, 0), (a, 0), (a, a), (0, a)).

. , ,

, «

». .

4.

a

1 2 ⋅a ⋅π. 4 π.

3.1.1.

(

.

1.8)

⌠ 2 ⎮ a dx → a ⌡0 a

⌠ 1 ⎮ 2 ⎮ a2 − x2 dx → 1 ⋅ a2 ⋅ π ⌡0 4 a

(

)

π

⎡ ⌠a ⎤ 1 ⎢ ⎥ ⎮ ⎢⎮ 2 2 2 ⎥ ⎢ ⌡ (a − x ) dx ⎥ ⎣ 0 ⎦ ⋅ 4 → π = 3.142 ⌠ ⎮ a dx ⌡0 a

3.1.2.

:

ORIGIN := 1 a := 5

c

N := 1000 :

i := 1 .. N

x := 0 , 0.01 .. a

y ( x) :=

a −x 2

2

X i := rnd ( a )

Y i := rnd ( a )

6

4

y ( x) Yi

2

0

n :=



2

x, X i

4

6

"

⎡⎣⎡⎣ ( Xi) 2 + ( Y i) 2 ⎤⎦ ≤ a2⎤⎦

N

i

0

=1

π

PI :=

n ⋅4 N

PI = 3.14

"

n = 785

π = 3.142 -

π − PI π

N = 0.051 %

a

3.2.

.

)

24 . ~ 600

II (CuSO4),

(V = 20.0 . ,

250 25

5–6

.

,

, (



; ,

2Cu 2 + + 4 J − (

Na2S2O4 HCl H2 SO4 . : ) → Cu2 J 2 ↓ + J 2 ,

).



J 2 + 2 S 2O32 − → 2 J − + S 4O32 − .

Cu2+

N T ⋅ VT ⋅ ЭCu ⋅ 250 = 63,54 ⋅ 10 ⋅ N T ⋅ VT , 25 = 0.09132); VT – , 0.5 . 100 20 , . 3.1, mi –

qCu 2+ =

NT –

( NT 5 ,

,

. . , qi –

. 3.1

qi 600,0 601.5 602.5 603 604 604.5 605

mi 1 1 1 1 2 2 4

qi 605.5 606 606.5 607 607.5 608 608.5

mi 3 1 3 4 4 5 4

qi 609 609.5 610 610.5 611 611.5 612

mi 3 6 5 5 6 4 4

qi 612.5 613 613.5 614 614.5 615 615.5

mi 5 4 3 3 3 2 2

[19]. 3.2.1.

i := 0 .. 50

qi 616 616.5 617 617.5 618 618.5 621

mi 2 1 2 1 1 1 1

data1 i := 600 600.5 601 601.5 602 602.5 603 603.5 604 604.5 605 605.5 606 606.5 607 607.5 608 608.5 609 609.5 610 610.5 611 611.5 612 612.5 613 613.5 614 614.5 615 615.5 616 616.5 617 617.5 618 618.5 619 619.5 620 620.5 621

data2 i := 1 0 0 1 0 1 1 0 2 2 4 3 1 3 4 4 5 4 3 6 5 5 6 4 4 5 4 3 3 3 2 2 2 1 2 1 1 1 0 0 0 0 1

〈〉 data 0

:= data1

〈〉 data 1

3.2.2.

:= data2

(

)

〈〉 n := length data 0 n = 43 N :=

∑ (data 〈 〉 ) n −1

1

=0 N = 100

i

i

3.2.3.

(

) − min(data

〈〉 ω := max data 0

ω = 21 3.2.4.

)

〈0〉

∑ (data 〈 〉 ) ⋅(data 〈 〉 ) n −1

mx :=

=0

i

1

0

i

i

N

mx = 610.235 3.2.5. j := 0 .. n − 1 E j :=

∑ (data 〈 〉 ) j

j

=0

E20 = 50 med :=

1

(data

med = 610.25

〈0〉

j

E21 = 55

)20 + (data 2

〈0〉

)21

N = 100

3.2.6.

∑ ⎡⎣⎡⎣(data 〈 〉 ) − mx⎤⎦ ⎤⎦ ⋅(data 〈 〉 ) n −1

D :=

k

=0

1

k

k

N−1

D = 16.098 σ := ( D)

2

0

1 2

σ = 4.012 3.2.7.

h := 0.5 i := 0 ..

(max(data 〈0〉 ) − min(data 〈0〉 )) h

(

〈〉 xi := min data 0 3.2.8. P :=

) + (i⋅h)

〈1〉

data

N

3.2.9. 0.1

Pi

0.05

0 600

605

610

615

620

625

xi

3.2.10.

∑ ⎡⎣⎡⎣(data 〈 〉 ) − mx⎤⎦ ⎤⎦ ⋅(data 〈 〉 ) n −1

sk :=

k

3

0

=0

1

k

σ

sk = 0.03579

3

k



N

( N − 1 ) ⋅( N − 2 )

3.2.11.

∑ ⎡⎣⎡⎣(data 〈 〉 ) − mx⎤⎦ ⎤⎦ ⋅(data 〈 〉 ) n −1

ku :=

k

=0

4

0

1

k

σ

4

k



N ⋅( N + 1 )

( N − 1) ⋅( N − 2) ⋅( N − 3)



3 ⋅ ( N − 1)

2

( N − 2) ⋅( N − 3)

ku = −0.11443 3.2.12. X := min( x) , min( x) + h .. max( x)

mx = 610.235

⎡ −( X − mx) 2⎥⎤ exp⎢ ⎢ 2 ⋅σ 2 ⎥ ⎣ ⎦ G ( X) :=

σ = 4.012

σ ⋅ 2 ⋅π

med = 610.25

0.15

0.1

G( X) P h

0.05

0 600

X =

G ( X) =

600 600.5 601 601.5 602 602.5 603 603.5 604 604.5 605 605.5 606 606.5 607 607.5

3.841·10 -3 5.238·10 -3 7.033·10 -3 9.296·10 -3 0.012 0.016 0.02 0.024 0.03 0.036 0.042 0.05 0.057 0.064 0.072 0.079

605

610

0 1 2 3 4 5 6 x= 7 8 9 10 11 12 13 14 15

X, x

615

0 600 600.5 601 601.5 602 602.5 603 603.5 604 604.5 605 605.5 606 606.5 607 607.5

620

625

0 1 2 3 4 5 6 P = 7 h 8 9 10 11 12 13 14 15

0 0.02 0 0 0.02 0 0.02 0.02 0 0.04 0.04 0.08 0.06 0.02 0.06 0.08 0.08

3.2.13. 0,9

qchisq( 0.9 , n − 1 − 2) = 51.805

⎡⎢ ⎛ Pi ⎞ 2 ⎥⎤ ⎜ ⎟ ⎢⎝ h ⎠ ⎥ 3 ⎢ G ( x ) − N⎥ = −4.298 × 10 i ⎣ ⎦ =0

∑ n −1

i

⎡⎢ ⎛ Pi ⎞ 2 ⎥⎤ ⎜ ⎟ ⎢⎝ h ⎠ ⎥ ⎢ G ( x ) − N⎥ < qchisq( 0.9 , 43 − 3) = 1 i ⎣ ⎦ =0

∑ n −1

i

3.3.

.

. 3.2, .

READPRN

«tit.txt»,

. 3.2 600.0 601.5 602.5 603.0

607.0 607.0 607.0 607.5

609.5 609.5 609.5 609.5

611.0 611.5 611.5 611.5

613.5 614.0 614.0 614.0

604.0 604.0 604.5 604.5 605.0 605.0 605.0 605.0 605.5 605.5 605.5 606.0 606.5 606.5 606.5 607.0 3.3.1.

607.5 607.5 607.5 608.0 608.0 608.0 608.0 608.0 608.5 608.5 608.5 608.5 609.0 609.0 609.0 609.5

read := READPRN ( "tit.TXT")

(

data := stack read

3.3.2.

3.3.3.

〈0〉

, read

〈1〉

611.5 612.0 612.0 612.0 612.0 612.5 612.5 612.5 612.5 612.5 613.0 613.0 613.0 613.0 613.5 613.5

, read

〈2〉

614.5 614.5 614.5 615.0 615.0 615.5 615.5 616.0 616.0 616.5 617.0 617.0 617.5 618.0 618.5 621.0

, read

〈3〉

N := length( data ) N = 100

ω := max( data ) − min( data )

ω = 21 3.3.4.

609.5 610.0 610.0 610.0 610.0 610.0 610.5 610.5 610.5 610.5 610.5 611.0 611.0 611.0 611.0 611.0

mx := mean( data )

mx = 610.235

min( data ) = 600 max( data ) = 621

, read

〈4〉

)

3.3.5.

med := median( data) med = 610.25

3.3.6.

D := Var( data) D = 16.098

3.3.7.

Stdev( data) = 16.098 2

σ := Stdev( data)

σ = 4.012 3.3.8.

k := 5 ⋅ log( N)

max( data) − min( data)

h := 2

k

max( data) − min( data) h

n := 11

i := 0 .. n

xi := 600 + i ⋅ h

V := hist( x , data)

3.3.9. P :=

V N

= 2.1 = 10.5

k = 10

3.3.10. 0 600 602 604 606 608 610 612 614 616 618 620 622

0 1 2 3 4 x = 5 6 7 8 9 10 11

0 2 2 11 12 18 20 16 10 6 2 1

0 1 2 3 V = 4 5 6 7 8 9 10

0 1 2 3 P = 4 5 6 7 8 9 10

0 0.02 0.02 0.11 0.12 0.18 0.2 0.16 0.1 0.06 0.02 0.01

20

V

0 600

610

620

x

3.3.11.

3.3.12.

3.3.13.

skew ( data ) = 0.03579 kurt ( data ) = − 0.11443 G := dnorm (x , mx , σ )

mx = 610.235 σ = 4.012

med = 610.25

0.1

G P

0.05

h

0 600

610

620 x

0 1 2 3 4 x= 5 6 7 8 9 10 11

0 600 602 604 606 608 610 612 614 616 618 620 622

0 0 3.841·10 -3 1 0.012 2 0.03 3 0.057 4 0.085 G = 5 0.099 6 0.09 7 0.064 8 0.035 9 0.015 10 5.144·10 -3 11 1.35·10 -3

0 0 0.01 1 0.01 2 0.055 3 0.06 P 0.09 = 4 h 5 0.1 6 0.08 7 0.05 8 0.03 9 0.01 10 5·10 -3

3.3.14. 0,9

qchisq( 0.9 , n − 1 − 2) = 13.362 j := 0 .. n − 1

∑ n −1

j

=0

∑ n −1

j

3.4.

.

=0

8

⎛ Pj ⎞ ⎜ − G j+ 1⎟ h ⎝ ⎠ = 0.072 2

Pj h

⎛ Pj ⎞ ⎜ − G j+ 1⎟ h ⎝ ⎠ < qchisq( 0.9 , 10 − 3) = 1 2

Pj h

1.76 %.

0.08 %.

, [19].

S8 95-

: n := 8

mx := 1.76 ⋅ %

S_8 := 0.08 ⋅ %

γ := 0.95

: α := 1 − γ

α = 0.05

⎛ ⎝

tx := qt ⎜ 1 −

α ⎞ , n − 1⎟ 2 ⎠

(

)

tx = 2.365 δ := tx ⋅

S_8 n

δ = 0.067 % mx − δ = 1.693 % mx + δ = 1.827 %

4. 4.1.

y = φ( x )

. ,

. 4.1 [12]. 4.1

X Y

1 3

2 4

y = ax + b .

3 2.5

5 0.5 .

y = φ( x )

4.1.1. a

b.



Given

⎤ 2 ⎡⎣Yi − ( a ⋅Xi + b)⎤⎦ ⎥ ⎥ ⎣ i =0 ⎦ ⎡ N−1

d ⎢ da ⎢



⎡ N−1

d ⎢ db ⎢

⎣ i =0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Find( a , b) → ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎛ ⎜ X := ⎜ ⎜ ⎜ ⎝

⎤ 2 ⎡⎣Yi − ( a ⋅Xi + b)⎤⎦ ⎥ 0 ⎥ ⎦ N − 1 N−1 N−1 ⎛ ⎞ ⎜ Yi⋅Xi⋅N − Xi⋅ Yi⎟ ⎜ ⎟ i =0 i =0 ⎝ i =0 ⎠ 2 ⎡ N−1 ⎛ N−1 ⎞ ⎤ ⎢ ⎥ 2 ⎜ Xi⎟ ( Xi) ⋅N − ⎜ ⎢ ⎟ ⎥ ⎣ i =0 ⎝ i =0 ⎠ ⎦ N−1 N−1 N−1 ⎡ N−1 ⎤ 2 ⎢ Yi⋅Xi⎥ Xi) − Xi⋅ Yi⋅ ( ⎢ ⎥ i =0 i =0 i =0 ⎣ i =0 ⎦ 2 ⎡ N−1 ⎛ N−1 ⎞ ⎤ ⎢ ⎥ 2 ⎜ Xi⎟ ( Xi) ⋅N − ⎜ ⎢ ⎟ ⎥ ⎣ i =0 ⎝ i =0 ⎠ ⎦



∑ ∑





∑ ∑

∑ ∑ ∑

⎛ ⎜ Y := ⎜ ⎜ ⎜ ⎝

1 ⎞

⎟ 2 ⎟ 3 ⎟ ⎟ 5 ⎠

N := length( X)

N = 4

0







3 ⎞



4 ⎟ 2.5 ⎟



0.5 ⎠



N− 1 N−1 ⎛ N− 1 ⎞ ⎜ Yi ⋅Xi ⋅N − Xi ⋅ Yi⎟ ⎜ ⎟ i =0 i =0 i =0 ⎝ ⎠ a := 2⎤ ⎡ N− 1 N − 1 ⎛ ⎞ ⎢ ⎥ 2 Xi ⎟ Xi) ⋅N − ⎜ ( ⎢ ⎜ ⎟ ⎥ ⎣ i =0 ⎝ i =0 ⎠ ⎦







⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦







N− 1 N− 1 N−1 ⎡ N− 1 ⎤ 2 ⎢ Xi) + Yi ⋅ Yi⋅Xi⎥ Xi ⋅ − − ( ⎢ ⎥ i =0 i =0 i =0 i =0 ⎣ ⎦ b := 2⎤ ⎡ N− 1 N − 1 ⎛ ⎞ ⎢ ⎥ 2 Xi⎟ Xi) ⋅N − ⎜ ( ⎢ ⎜ ⎟ ⎥ ⎣ i =0 ⎝ i =0 ⎠ ⎦





a = − 0.743

b = 4.543

x := 1 , 1.1 .. 5

y( x) := a ⋅x + b 4 y ( x) Y

2

0

0

2

x, X

4

approximation exptriment_point

6

4.1.2. a

a := 0

b := 0

b

Given



N−1

=0

∑ i

N−1

i

=0

( Yi⋅Xi) − a ⋅ ∑ ( Xi) N−1

Yi − a ⋅

∑ N−1

i

=0

i

=0

2

− b⋅

Xi − b ⋅N

∑ N−1

i

=0

0

0

⎛ a⎞ ⎜ ⎟ := Find( a , b) ⎝ b⎠ x := 1 , 1.1 .. 5

Xi

a = −0.743

b = 4.543

y( x) := a ⋅x + b

4 y ( x) Y

2

0

0

2

x, X

4

6

approximation exptriment_point

4.1.3. a

i := 0 .. N − 1

b

m := 0 .. 1

k := 0 .. 1 Ak , m :=



N−1

i

=0

( Xi)

k+ m

Dk :=

⎛ 4 11 ⎞ ⎟ ⎝ 11 39 ⎠

A=⎜

⎛ 10 ⎞ ⎟ ⎝ 21 ⎠

i

D=⎜

⎛ b⎞ −1 ⎜ ⎟ := A ⋅D a ⎝ ⎠ x := 1 , 1.1 .. 5



N−1

a = −0.743

=0

Yi⋅( Xi)

k

b = 4.543

y( x) := a ⋅x + b

4 y ( x) Y

2

0

0

2

x, X

4

6

approximation exptriment_point

4.1.4. a b

"intercept"

b := intercept ( X , Y )

"slope"

a := slope ( X , Y )

a = − 0.743 b = 4.543

x := 1 , 1.1 .. 5

y ( x) := a ⋅x + b 4 y ( x) Y

2

0

0

2

x, X

4

x, X

4

6

approximation exptriment_point 4.1.5. a b

"line"

⎛ b ⎞ ⎜ ⎟ := line( X , Y ) ⎝ a ⎠ a = − 0.743 b = 4.543

x := 1 , 1.1 .. 5

y ( x) := a ⋅x + b 4 y ( x) Y

2

0

0

2

approximation exptriment_point

6

4.1.6.

[4]

p := 1

N = 4



N−1

SSf :=

=0

∑ i

N−1

SSr :=

=0

∑ i

N−1

SS :=

=0

i

( y ( X i) − mean ( Y) ) 2 ( Yi − y ( Xi) ) 2 ( Yi − mean ( Y) ) 2

SS = 6.5

R_2 :=

R :=

SSf + SSr = 6.5

R_2 = 0.743

SSf SS

a ⋅corr ( Y , X ) ⋅stdev ( X ) stdev ( Y )

R = 0.743 2

0,683 F :=

R ⋅( N − p − 1 )

( 1 − R 2) ⋅ p

2

F-

F = 5.778

qF ( 0.683 , p , N − p − 1 ) = 1.749

F > qF ( 0.683 , p , N − p − 1 ) = 1 F-

, -

,

4.2.

.

X

Y

X Y

X [15].

Y

,

X i :=

Y i :=

1 2 3 4 5 6 7 8 9 10 11 12

145 111 135 130 122 98 100 85 90 79 15 68

i := 0 .. 11

x := min( X ) , min( X ) + 0.01 .. max ( X )

p := 1

n := length ( X ) 4.2.1. Y

b ⋅a

log ( Y ) b1

X

log ( b ) + X ⋅ log ( a )

(

log ( b )

)

→ ⎯ → b1 := intercept X , log ( Y ) b := 10 a1

(

(

)

log ( a ) → ⎯ → a1 := slope X , log ( Y )

a := 10

)

→ ⎯ → intercept X , log ( Y )

(

)

→ ⎯ → slope X , log ( Y )

a = 0.891

b = 185.732

y( x) := b ⋅a

x

2.5

2

log( y ( x) ) log( Y)

1.5

1

5

x, X

10



( log( y( Xi) ) − mean(log(Y))) 2



( log( Yi) − mean(log(Y))) 2

n −1

SSf :=

i

SS :=

0

=0

n −1

i

R_2 :=

=0

15

SSf SS

R_2 = 0.478

F :=

0,9

R_2⋅( n − p − 1) ( 1 − R_2) ⋅p

FF = 9.151

qF ( 0.9 , p , n − p − 1) = 3.285

F > qF ( 0.9 , p , n − p − 1) = 1 F-

, -

X

Y

,

4.2.2.

"linfit"

⎜⎛ ⎜ f ( x) := ⎜ ⎜ ⎜ ⎝

1 ⎞



x ⎟



x ⎟ 2



x ⎠ 3

⎛ ⎜ a = ⎜ ⎜ ⎜ ⎝

a := linfit( X , Y , f )

y1 ( x) := f ( x) ⋅ a 150

130.091 ⎞

⎟ ⎟ − 1.944 ⎟ ⎟ 0.079 ⎠ 5.103

100

Y y1 ( x)

50

0

0

5

10

X,x

15

∑ (a ⋅corr (Y , X ) ⋅stdev (X )) 3

R :=

j

=1

j

j

j

stdev ( Y )

R = 0.873

R = 0.761 2

F :=

R ⋅( n − p − 1 )

( 1 − R 2) ⋅ p

0,9

2

F-

F = 31.89

qF ( 0.9 , p , n − p − 1 ) = 3.285

F > qF ( 0.9 , p , n − p − 1 ) = 1 F-

, -

X

Y

,

⎛ ⎜ ⎜ ⎜ ff ( x) := ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

1 ⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

1 x 1 x

2

1 x

3

⎛ − 33.131 ⎜ ⎜ 1.253 × 10 3 a = ⎜ 3 ⎜ − 2.795 × 10 ⎜ 3 ⎝ 1.721 × 10

a := linfit( X , Y , ff )

y2 ( x) := ff ( x) ⋅ a

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

150

100

Y y2 ( x)

50

0

0

5

X,x

10

15

∑ (a ⋅corr (Y , X ) ⋅stdev (X )) − j

3

R :=

j

=1

j

− j

stdev ( Y )

R = 0.896

R = 0.802 2

F :=

0,9

R ⋅( n − p − 1 )

( 1 − R 2) ⋅ p

2

F-

F = 40.542

qF ( 0.9 , p , n − p − 1 ) = 3.285

F > qF ( 0.9 , p , n − p − 1 ) = 1 F-

, -

X

Y

⎛ ⎜ ⎜ ⎜ fff( x) := ⎜ ⎜ ⎜ ⎜ ⎝

,

1 ⎞



1 ⎟ x ⎟



x ⎟ 1 ⎟ 2

⎛ 141.471 ⎜ ⎟ a := linfitX ( , Y , fff) 3 − 44.111 x ⎠ a = ⎜ ⎜ − 0.676 ⎜ y3 ( x) := fff( x) ⋅a ⎝ 47.454

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

150

Y

100

y3 ( x) 50

0

R :=

0

5

X, x

10

15

(a1⋅corr(Y , X− 1) ⋅stdev(X− 1)) + (a2⋅corr(Y , X2) ⋅stdev(X2)) + (a3⋅corr(Y , X− 3) ⋅stdev(X− 3)) stdev( Y)

R = 0.876

R = 0.767 2

F :=

R ⋅( n − p − 1 )

(1 − R2) ⋅p

0,9

2

F-

F = 32.908

qF ( 0.9 , p , n − p − 1) = 3.285

F > qF ( 0.9 , p , n − p − 1) = 1 F-

, -

X

4.3.

.

,

Y

4.1

y = φ( x )

y = a1 ⋅ a0 x .

4.3.1.

(

)

n −1 ⎡ n−1 2⎤ X X Xi ⎢ ⎡Yi − (a1 ⋅ a0 Xi)⎤ ⎥ → −2 ⋅ Yi − a1 ⋅ a0 i ⋅ a1 ⋅ a0 i ⋅ ⎣ ⎦ ⎥ ⎢ a0 da0 ⎣ i =0 ⎦ i =0



d

.



(

)

n −1 2⎤ Xi Xi ⎡Yi − (a1 ⋅ a0 Xi)⎤ ⎥ → −2 ⋅ Yi − a1 ⋅ a0 ⋅ a0 ⎣ ⎦ ⎥ ⎣ i =0 ⎦ i =0



⎡ n−1

d ⎢ da1 ⎢

⎛ ⎜ X := ⎜ ⎜ ⎜ ⎝

1 ⎞

⎟ 2 ⎟ 3 ⎟ ⎟ 5 ⎠



⎛ ⎜ Y := ⎜ ⎜ ⎜ ⎝

3 ⎞



4 ⎟ 2.5 ⎟

x := min( X) , min( X) + 0.01 .. max( X) n := length( X) n=4



0.5 ⎠

a

a0 := 1



Given n −1

i =0 n −1

∑ i

=0

a1 := 2

(

−2 ⋅ Yi − a1 ⋅ a0

Xi

−2 ⋅ Yi − a1 ⋅ a0

Xi

(

b

) ⋅ a1 ⋅ a0X ⋅ Xa0i i

) ⋅ a0X

i

⎛ a0 ⎞ ⎜ ⎟ := Find( a0 , a1) ⎝ a1 ⎠

0

0

a0 = 0.773

y( x) := a1 ⋅ a0

x

a1 = 4.824

4

Y 2

y ( x)

0

0

2

4.3.2.

4

X, x

"genfit" a1 ⋅ a0

x

y( x)

"

/

/

". :

a0

a1 ⋅ a0 ⋅ x

a1 x a0

x

a0

6

x ⎡ ⎤ ⎢ a1 ⋅ ( a0) ⎥ ⎢ ⎥ x x f ( x , a) := ⎢ a1 ⋅ ( a0) ⋅ ⎥ a0 ⎢ ⎥ x ⎢ ⎥ ( a0) ⎣ ⎦ ⎛ a0 ⎞ ⎜ ⎟ := genfit( X , Y , v , f ) ⎝ a1 ⎠

y ( x) := a1 ⋅ ( a0)

⎛ 1 ⎞ ⎟ ⎝ 2 ⎠

v := ⎜ a0 = 0.775

a1 = 4.799

x 4

Y 2

y ( x)

0

4.3.3.

0

2

X, x

"Minimize" a0 := 1 f ( a0 , a1 ) :=

∑ n −1

i

=0

4

6

a1 := 1

⎡Yi − ( a1 ⋅ a0 Xi)⎤ ⎣ ⎦

2

Given 0 < a0 < 10

0 < a1 < 10

⎛ a0 ⎞ ⎜ ⎟ := Minimize( f , a0 , a1 ) ⎝ a1 ⎠ y ( x) := a1 ⋅ a0

a0 = 0.773

a1 = 4.824

x 4

Y 2

y ( x)

0

4.4.

0

2

X, x

4

6

.

298 K . 4.2 [9]: 4.2

1 2

0.0363 0.0184

0.0668 0.0504

0.0940 0.0977

c1 – ,

/ .

0.126 0.146

0.210 0.329

, n

K

/ , c2 –

0.283 0.553

c1n =K. c2

0.558 0.650

0.756 2.810

0.912 4.340

c1i :=

c2i :=

0.0363 0.0668 0.0940 0.126 0.210 0.283 0.558 0.756 0.912

0.0184 0.0504 0.0977 0.146 0.329 0.553 0.650 2.810 4.340

i := 0 .. 8

C1 := min( c1) , min( c1) + 0.01 .. max( c1) 4.4.1.

c1

K K i :=

K

c2

c1i c2i 1/ 2 = K

1/ 2

,

2 Ki

1

0

5

10

i n

4.4.2.

n

K

⎛ 1 ⎞ + n ⋅ log( c1) ⎟ ⎝K⎠ ⎛1⎞ log⎜ ⎟ ⎝K⎠ 1

10 b1 K :=

(

b1

)

→ ⎯ → ⎯ intercept log( c1) , log( c2)

(

1

)

→ ⎯ → ⎯ intercept log ( c1) , log ( c2)

(

)

→ ⎯ → ⎯ n := slope log( c1) , log( c2) 10

C2 ( C1 ) :=

1 K

⋅ C1

n = 1.572

6

log ( C2( C1) )

C2( C1) 1.5

1

K = 0.278

n

2

log ( c2)

K

log⎜

log( c2) b1

c1 c2

K

0.5

0

2 log ( C1) , log ( c1)

c2

4 2

0

0.5

C1, c1

1

4.4.3.

"genfit" n

c1 c2

K 1

a0

K

a1

n

C2

a0 ⋅ C1

"

a1

/

/

". :

a0 C1

a1

a1

a1 ⎛ a0 ⋅ C1 ⎜ ⎜ a1 f ( C1 , a) := C1 ⎜ a1 ⎜ ⎝ a0 ⋅ C1 ⋅ ln( C1 )

⎛ a0 ⎞ ⎜ ⎟ := genfit( c1 , c2 , v , f ) ⎝ a1 ⎠

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

a0 ⋅ C1

a1

⋅ ln( C1 )

⎛ 1 ⎞ ⎟ ⎝ 1 ⎠

v := ⎜

n := a1 K :=

1 a0

C2 ( C1 ) :=

1 K

⋅ C1

n

6

4

c2 C2( C1)

2

0

4.5.

.

0

0.5

1

c1 , C1

. 4.3 [1].

– 11.3

.

– 100

. 4.3

№№

ΔL,

P, 1 2 2 4 5 6 7 8 9

1.5 2.0 2.5 3.0 3.5 3.7 3.9 4.1 4.3 σ– (

-

0.0 0.2 0.3 1.9 1.7 2.6 3.8 5.6 9.2 ).

.

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ΔL := ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0.0 ⎞ 0.2 0.3 0.9 1.7 2.6 3.8 5.6 9.2

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

,

d := 0.0113 d

ε :=

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ σ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

F ⋅10

L

146.728 ⎞ 244.547 293.456 342.366 361.929 381.493 401.057 420.621

3.5 3.7 3.9 4.1

,

−4

1000 ⋅9.81

195.638

^2

3

2

4

ΔL

,

2.5

,

F = 1.003 × 10 σ := P ⋅

4.3

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

2

,

L := 100

F := π ⋅

1.5 ⎞

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ P := ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

,

6

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ε = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0

2 × 10

3 × 10

9 × 10

−3 −3 −3

0.017 0.026 0.038 0.056 0.092

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

4.5.1.

N

N := length(ε ) N =9

i := 0 .. N − 1

m := 0 .. N − 1

k := 0 .. N − 1

Ak , m :=

∑ N−1

∑ i

Dk :=

N−1

i

=0

=0

(ε i)k+ m

σ i⋅(ε i)

a := lsolve( A , D)

k

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ a= ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞ ⎟ −3.052 × 10 ⎟ ⎟ 7 4.139 × 10 ⎟ 9 ⎟ −8.76 × 10 ⎟ 11 7.789 × 10 ⎟ ⎟ 13 ⎟ −3.441 × 10 ⎟ 14 7.827 × 10 ⎟ 15 ⎟ −8.639 × 10 ⎟ 16 ⎟ 3.6 × 10 ⎠ 147.222

4

x := min(ε ) , min(ε ) + 0.0001 .. max( ε ) y( x) :=

∑ N−1

k

ak ⋅x

k

=0

5 .10

5

σ

0

y ( x)

0.05

0.1

5 .10

5

1 .10

6

ε, x

N

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ y(ε ) = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ σ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

147.222 ⎞ 193.062 246.902 293.087 342.494 361.893 381.498 401.057 420.621

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

146.728 ⎞ 195.638 244.547 293.456 342.366 361.929 381.493 401.057 420.621

4.5.2.

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

3

i := 0 .. N − 1 N := 3

m := 0 .. N − 1

k := 0 .. N − 1 Ak , m :=



N−1

i

=0

(ε i)

a := lsolve( A , D)

k+ m

Dk :=



N−1

i

=0

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ a= ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

σ i⋅(ε i)

k

141.083

9.693 × 10

3

−7.442 × 10 5.183 × 10

8

−7.355 × 10 3.952 × 10

10

12

−9.945 × 10 1.163 × 10

4

13

15

−5.016 × 10

15

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

y( x) :=

∑ N−1

k

=0

ak ⋅x

k

600 σ

400

y ( x) 200

0

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ y( ε ) = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0.05

141.083 ⎞ 160.172 169.494 222.296 284.363 342.802 401.965 450.521 402.953

4.5.3.

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ σ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

-

⎛ ⎜ ⎜ ⎜ u := ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

146.728 ⎞ 195.638 244.547 293.456 342.366 361.929 381.493 401.057 420.621

0 0.004 0.005 0.015 0.02 0.03 0.092

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

, B-

y( x) := interp(W , ε , σ , x) 600

400

y ( x) y( u)

200

0

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

B-

. W := bspline(ε , σ , u , 3)

σ

0.1

ε, x

0.05

ε , x, u

0.1

4.6.

.

– –

298 K

. 4.4 [9]: 4.4

XC2H5OC2H5 PC2H5OC2H5 ·10-4, PCHCl2 ·10-4,

0 0.000 1.933

0.2 0.460 1.480

0.4 1.287 0.920

0.6 2.666 0.460

0.8 4.093 0.165

1.0 5.333 0.000 .

, ,

. ,

. . .

.

, (ΔH < 0).

xC2H5OC2H5 := i 0 0.2 0.4 0.6 0.8 1.0

;

i := 0 .. 5

PC2H5OC2H5 := i

0.000 0.460 1.287 2.666 4.093 5.333

PCHCl3 := i

1.933 1.480 0.920 0.460 0.165 0.000

6

PC2H5OC2H5

4

PCHCl3 PC2H5OC2H5+ PCHCl3

2

0

0

0.5

1

xC2H5OC2H5

4.6.1.

(

) + 0.01 .. max(xC2H5OC2H5 ) Z1 ( x) := linterp( xC2H5OC2H5 , P C2H5OC2H5 , x) Z2 ( x) := linterp( xC2H5OC2H5 , P CHCl3 , x) Z3 ( x) := linterp( xC2H5OC2H5 , P C2H5OC2H5 + P CHCl3 , x) "linterp"

)

(

x := min xC2H5OC2H5 , min xC2H5OC2H5

6 PC2H5OC2H5 PCHCl3 PC2H5OC2H5+ PCHCl3

4

Z1 ( x) Z2 ( x)

2

Z3 ( x)

0

0

0.2

0.4

0.6

0.8

xC2H5OC2H5 , xC2H5OC2H5 , xC2H5OC2H5 , x, x, x

1

"linterp"

xC2H5OC2H5 = i 0

Z1 ( 0.101 ) = 0.232

0.2 0.4

Z1 ( 0.3 ) = 0.873

Z2 ( 0.101 ) = 1.704

0.6

Z2 ( 0.3 ) = 1.2

Z3 ( 0.101 ) = 1.937

0.8 1

4.6.2.

(

"interp"

W1 := lspline xC2H5OC2H5 , P C2H5OC2H5

(

Z3 ( 0.3 ) = 2.073

)

)

Z4 ( x) := interp W1 , xC2H5OC2H5 , P C2H5OC2H5 , x

(

W2 := cspline xC2H5OC2H5 , P C2H5OC2H5

(

)

)

Z5 ( x) := interp W2 , xC2H5OC2H5 , P C2H5OC2H5 , x

(

W3 := pspline xC2H5OC2H5 , P C2H5OC2H5

(

)

)

Z6 ( x) := interp W3 , xC2H5OC2H5 , P C2H5OC2H5 , x

"linterp" "interp"

0..0.2

0.4 Z1 ( x) Z4 ( x)

0.2

0

0

0.1

linterp interp

(

"cspline", "pspline"

)

0.2

x

(

"interp" "lspline",

)

(

)

x := min xC2H5OC2H5 − 1 , min xC2H5OC2H5 − 1 + 0.01 .. max xC2H5OC2H5 + 1 20

Z1( x)

10

Z4( x) Z5( x) Z6( x)

1

0

1

2

10

20 x

4.7.

.

. 4.5

, (

αHCOOH – pH)

pH, pH 1.00 6.00. pH 1.00 – 6.00. ( αHCOO¯ – pH) [5]. 4.5

pH 1.00 2.00 3.00 4.00

HCOOH 100 98 85 36

HCOO¯ 0 2 15 64

5.00 6.00

5 0 i := 0 .. 5

pH i :=

95 100

HCOO_ i :=

HCOOH i := 100 98 85 36 5 0

1.00 2.00 3.00 4.00 5.00 6.00

0 2 15 64 95 100

100

HCOOH HCOO_

50

0

2

4

6

pH

"interp"

"lspline"

x := min( pH ) , min( pH ) + 0.01 .. max ( pH )

W1 := lspline( pH , HCOOH )

Z1 ( x) := interp ( W1 , pH , HCOOH , x)

W2 := lspline( pH , HCOO_ )

Z2 ( x) := interp ( W2 , pH , HCOO_ , x)

150

HCOOH 100 HCOO_ 50

Z1( x) Z2( x)

0

50

0

2

pH , pH , x, x

"

4

6

/ / " : (3.75 , 51.225 )

"interp" Z1( 3.75) = 48.775

Z2( 3.75) = 51.225 "Find" x := 0 Given Z1( x)

Z2( x)

Find( x) = 3.727

5. 5.1.

( .

M ,

) (R = M·k2·v2,

k–

. ) [3].

:

V0 := 0

, /

k := 0.16

g :=

g

, / −2

m⋅sec

, / ^2

M := 1

, :

t := 0 .. 15 v( t) :=

, 2⋅ k⋅ t ⋅ g

g e −1 ⋅ k e2⋅ k⋅ t⋅ g + 1

5.1.1.

I

"odesolve"

Given M ⋅V' ( t) V ( 0)

M ⋅g − M ⋅k ⋅V ( t) 2

2

, /

0

vv := odesolve( t , 15 , 100)

15 -

t := 0 , 1 .. 15

20 vv ( t) v ( t)

10

0

5

10

15

t

5.1.2.

V0 := V0

I

N := 15

D( t , V) := g − k ⋅( V0) 2

t1 := 0

vv( t) =

v( t) =

0 9.061 14.923 17.727 18.874 19.313 19.477 19.537 19.559 19.567 19.57 19.572 19.572 19.572 19.572 19.572

0 9.061 14.923 17.727 18.874 19.313 19.477 19.537 19.559 19.567 19.57 19.572 19.572 19.572 19.572 19.572

"rkfixed"

t2 := 15

2 0

V_ := rkfixed( V , t1 , t2 , N , D)

20 v ( t) V_

〈1〉

V_ =

10

0

0

1

1

9.055

2

2

14.9

3

3 17.696

4

4 18.851

5

5 19.299

6

6

7

7 19.534

8

8 19.558

9

0

( v⎡⎣( V_

〈〉 v⎡⎣ V_ 0 〈0〉

5

)10⎤⎦ = 19.57 )15⎤⎦ = 19.572

t , V_

〈0〉

10

(V_ (V_

〈1〉

〈1〉

15

)10 = 19.57 )15 = 19.572

1

0

19.47

9 19.567

10

10

11

11 19.571

19.57

12

12 19.572

13

13 19.572

14

14 19.572

15

15 19.572

5.1.3.

I

N = 15

T0 := t1

V0 = 0

[18]

TN := t2 i := 1 .. N

F ( T , V) := g − k ⋅V 2

2

TN − T0

h :=

N Ti := T0 + h⋅i

Vi := Vi−1 + F ( Ti−1 , Vi−1) ⋅h

20 v( t) V

VN = 19.572

10

0

5

t,T

10

15

v( T3) = 17.727

T3 = 3

V3 = 19.427

v( T10) = 19.57

T10 = 10

V10 = 19.572

v( T15) = 19.572

T15 = 15

V15 = 19.572

5.2.

[13]. –

k,

Q,

.

.

Q –

, ,



y_0. .

,

. –

, – ,

-k·y,

k –



»). ,

,

Q ,

(

− λ ⋅ v = −λ ⋅

,

). Q⋅

k, ≥ 0.

II

2

d y dt 2

= −k ⋅ y − λ ⋅

: dy , dt

: Q := 20

λ := 40

k := 100

y_0 := 0.01

V_0 := 0

, , / , / , , /

dy , dt

= const ≥ 0

: 5.2.1.

λ

y''( t) +

Q

II ⋅y'( t) +

k Q

⋅y( t)

0

5.2.1.1. II

p +

λ

2

"

Q

/

⋅p +

k

/

" p

0

Q

.

⎛ 1⎞⎤ ⎡ ⎡ ⎜ ⎟ ⎢ 1 ⎢ ⎝ 2⎠⎥ 2 ( ) ⎢ ⋅⎣ −λ + λ − 4 ⋅Q ⋅k ⎦ ⎢ ( 2 ⋅Q ) ⎢ ⎛ 1⎞⎤ ⎡ ⎜ 2⎟ ⎢ 1 ⎢ ⎝ ⎠⎥ 2 ( ) ⋅⎣ −λ − λ − 4 ⋅Q ⋅k ⎢ ⎦ ⎣ ( 2 ⋅Q )

⎤ ⎥ ⎥ ⎥ = ⎜⎛ −1 + 2i ⎞⎟ ⎥ ⎝ −1 − 2i ⎠ ⎥ ⎥ ⎦

5.2.1.2.

, ,

y( t)

II

exp(α ⋅t) ⋅(C1 ⋅cos (β ⋅t) + C2 ⋅sin(β ⋅t))

α = -1 5.2.1.3.

β =2 II

"

.

/

/

"

t

d dt

y( t)

α ⋅exp( α ⋅t) ⋅(C1 ⋅cos( β ⋅t) + C2 ⋅sin(β ⋅t) ) + exp(α ⋅t) ⋅(−C1 ⋅sin(β ⋅t) ⋅β + C2 ⋅cos (β ⋅t) ⋅β )

5.2.1.4.

C1

y( 0)

y_0

y'( 0)

V_0

exp(α ⋅0) ⋅(C1 ⋅cos (β ⋅0) + C2 ⋅sin(β ⋅0) )

C2

y(0)

y'(0)

Given

y_0

α ⋅exp(α ⋅0) ⋅( C1 ⋅cos ( β ⋅0) + C2 ⋅sin(β ⋅0)) + exp(α ⋅0) ⋅(−C1 ⋅sin(β ⋅0) ⋅β + C2 ⋅cos (β ⋅0) ⋅β )

V_0

⎡ 1.0000000000000000000 ⋅10-2 ⎢ Find( C1 , C2) → ⎢ -2 α ⎢ −(1.0000000000000000000 ⋅10 ) ⋅ β ⎣ 5.2.1.5.

II

α := −1

C1 := 1 ⋅10

β := 2 -2

C2 := −1 ⋅10 ⋅ -2

t := 0 , 0.1 .. 6 α⋅t

y( t) := e

α β

⋅(C1 ⋅cos (β ⋅t) + C2 ⋅sin(β ⋅t)) 0.01

0.005 y( t) 0

0.005

0

2

4 t

6

⎤ ⎥ ⎥ ⎥ ⎦

5.2.2.

II

Given

Q ⋅y'' ( t)

"odesolve"

− k ⋅y ( t) − λ ⋅y' ( t) ,

y( 0)

y_0

y' ( 0 )

V_0

, /

y_ := odesolve ( t , 6 , 100 )

6-

0.01

y_ ( t )

0.005

y ( t) 0

0.005

0

2

4

6

t

5.2.3.

II

Y0 := y_0

"rkfixed"

Y1 := V_0 II

dY 0

I

Y1

dt

λ −k ⋅ Y0 − ⋅ Y 1 Q Q

dY 1 dt

⎛ Y0 ⎞ ⎟ ⎝ Y1 ⎠

Y := ⎜

t1 := 0

II

t2 := 6

Y1 ⎛ ⎜ D ( t , Y) := ⎜ − k ⋅ Y0 − λ ⋅ Y 1 Q ⎝ Q Z := rkfixed( Y , t1 , t2 , N , D)

⎞ ⎟ ⎟ ⎠

N := 100

0

0.01

Z

〈1〉

0

0.005

Z=

y ( t) 0

0.005

0 Z

(Z (Z

〈0〉 〈1〉

)10 = 0.6

〈0〉

)10 = 4.546 × 10 − 3

,t

5

(

〈〉 y ⎡⎣ Z 0

⎤ ⎡⎛ 0.01 ⎞ ⎟ , t1 , t2 , N , D⎥ ⎣⎝ 0 ⎠ ⎦

Z_01 := rkfixed⎢⎜

⎤ ⎡⎛ 0.02 ⎞ ⎟ , t1 , t2 , N , D⎥ ⎣⎝ 0 ⎠ ⎦

Z_02 := rkfixed⎢⎜

⎡⎛ 0.04 ⎞ ⎤ ⎟ , t1 , t2 , N , D⎥ ⎣⎝ 0 ⎠ ⎦

Z_04 := rkfixed⎢⎜

⎤ ⎡⎛ 0.08 ⎞ ⎟ , t1 , t2 , N , D⎥ ⎣⎝ 0 ⎠ ⎦

Z_08 := rkfixed⎢⎜

1 0

2

0.01

0

1

0.06 914·10

-3

819·10

-3

2

0.12 669·10

-3

271·10

-3

3

0.18 288·10

-3

356·10

-3

4

0.24 794·10

-3

081·10

-3

5

0.3 206·10

-3

-0.01

6

0.36 545·10

-3

-0.012

7

0.42 832·10

-3

-0.012

8

0.48 083·10

-3

-0.013

9

0.54 316·10

-3

-0.013

10

0.6 546·10

-3

-0.013

11

0.66 786·10

-3

-0.013

12

0.72 048·10

-3

-0.012

13

0.78 341·10

-3

-0.011

14

0.84 675·10

-3

15

0.9 056·10

-3

-0.011 898·10

-3

)10⎤⎦ = 4.546 × 10 − 3

Z_01 Z_02 Z_04 Z_08

〈2〉 〈2〉 〈2〉 〈2〉

0

0.05

0.1

0.15

0.02

0 Z_01

D ( t , Y) :=

⎡⎛ ⎣⎝ ⎡⎛ Z_02 := rkfixed⎢⎜ ⎣⎝ ⎡⎛ Z_04 := rkfixed⎢⎜ ⎣⎝ ⎡⎛ Z_08 := rkfixed⎢⎜ ⎣⎝

0.04 〈〉 , Z_04 1

0.06 〈〉 , Z_08 1

0.08

⎞ ⎟ ⎟ ⎠

⎛ Y1 ⎜ ⎜ − k ⋅Y0 ⎝ Q

Z_01 := rkfixed⎢⎜

0.02 〈〉 , Z_02 1

〈1〉

0.01 ⎞

⎤ ⎟ , t1 , t2 , N , D⎥ ⎦ ⎤ ⎟ , t1 , t2 , N , D⎥ 0 ⎠ ⎦ 0.04 ⎞ ⎤ ⎟ , t1 , t2 , N , D⎥ 0 ⎠ ⎦ 0.08 ⎞ ⎤ ⎟ , t1 , t2 , N , D⎥ 0 ⎠ ⎦ 0 ⎠ 0.02 ⎞

0.2

Z_01 Z_02 Z_04 Z_08

〈2〉 〈2〉 〈2〉

0

〈2〉

0.2

5.3. V

α

. (

0.1 Z_01

0.05 〈1〉

, Z_02

0

〈1〉

, Z_04

〈1〉

0.05

, Z_08

〈1〉

0.1

,

. 5.1).

[14].

F=kV

y(t)

N(x, y)

V0 α

P=mg

0

x(t)

. 5.1. II

N(x, y) :

5.3 :

P = M·g

F = k·V.

x( 0) M⋅ M⋅

2

d

2

x

dt

d2 2

y

dt

d −k ⋅ x dt

d

d −k ⋅ y − M ⋅g dt

y( 0)

dt

d dt

0

x( 0) 0

y( 0)

V⋅cos (α )

V⋅sin(α )

:

α := 45 ⋅deg

, ,

V0 := 10

, /

k := 0.16

,

M := 1

,

g := 9.81

, / ^2

x ( 0) M⋅ M⋅

d

2 2

x

dt d

2 2

y

dt

d −k ⋅ x dt

d

d −k ⋅ y − M ⋅g dt

y ( 0)

dt

d dt

: II I dx dt

z1

dz1

−k

dt

M

dy dt

⋅z1

z3

dz3

−k

dt

M

⋅z3 − g⋅

M M

0

x( 0)

y( 0)

0

V⋅cos (α )

V⋅sin(α )

/

II "rkfixed"

"Bulstoer"

0 ⎛ ⎜ V0⋅cos (α ) z := ⎜ ⎜ 0 ⎜ ⎝ V0⋅sin(α )

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

z1 ⎛ ⎞ ⎜ ⎟ ⎜ −k ⎟ ⎜ M ⋅z1 ⎟ D( t , z) := ⎜ ⎟ z3 ⎜ ⎟ ⎜ −k ⎟ ⋅z3 − g ⎟ ⎜ ⎝ M ⎠

Z := rkfixed( z , 0 , 1.5 , 15 , D)

Z1 := Bulstoer( z , 0 , 1.5 , 15 , D) t := 0 , 0.1 .. 1.5



−k



⋅t ⎜ M M ⎟ ⋅V0⋅cos (α ) ⋅⎝ 1 − e x( t) := ⎠ k

− k ⎞⎤ ⎡ ⎛ ⋅t ⎢M ⎜ g⋅M M ⎟⎥ y( t) := ⎢ ⋅(g⋅M + k ⋅V0⋅sin(α )) ⋅⎝ 1 − e ⋅t ⎠⎥ − 2 k ⎣k ⎦

4 y( t) Z Z1

〈3〉

2

〈3〉

0

5

2 x( t) , Z

〈1〉

10

, Z1

〈1〉

(Z1 〈0〉 〈0〉 x⎡⎣(Z (Z1 〈1〉 〈〉 y⎡⎣(Z 0 (Z1 〈3〉

)10 = 1 )10⎤⎦ = 6.534 )10 = 6.534 )10⎤⎦ = 1.881 )10 = 1.881

(Z 〈0〉 )10 = 1 x( 1) = 6.534

(Z 〈1〉 )10 = 6.534 y( 1) = 1.881

(Z 〈3〉 )10 = 1.881

(Z 〈1〉 )10 − x⎡⎣(Z 〈0〉 )10⎤⎦ −8 = −5.104 × 10 % 〈0〉 ) ⎤ ⎡ ( x⎣ Z 10⎦ 〈 〉 1 (Z1 )10 − x⎡⎣(Z1 〈0〉 )10⎤⎦ −9 = −1.186 × 10 % 〈 〉 0 )10⎤⎦ x⎡⎣(Z1 (Z 〈3〉 )10 − y⎡⎣(Z 〈0〉 )10⎤⎦ −6 = −1.715 × 10 % 〈0〉 ) ⎤ ⎡ ( y⎣ Z 10⎦ 〈 〉 3 (Z1 )10 − y⎡⎣(Z1 〈0〉 )10⎤⎦ −8 = −3.983 × 10 % 〈 〉 0 )10⎤⎦ y⎡⎣(Z1 5.4.

.

1

, /( · ),

2

= 0.23

/( · ),

3

: 1,3– 2 = 60 = 45 /( · ).

2

, –



1 3

T1 = 780 °C,

=8

α1 = 70

= 100

, 2 –

. /( 2· ), α2 = 12

/( 2· ).

– : 1 = 0.81 T2 = 20 °C.

: α1 := 70 ⋅ α2 := 12 ⋅

W

m ⋅K 2

W

m ⋅K 2

δ1 := 100 ⋅mm δ2 := 60 ⋅mm δ3 := 8 ⋅mm

λ1 := 0.81 ⋅ λ2 := 0.23 ⋅ λ3 := 45 ⋅

W

m⋅K W

m⋅K

W

m⋅K

T1 := ( 780 + 273) ⋅K

T2 := ( 20 + 273) ⋅K

F := 1

: k :=

⎛ 1 δ1 δ2 δ3 1 ⎞ + + + + ⎜ ⎟ ⎝ α1 λ1 λ2 λ3 α2 ⎠ 1

ΔT := T1 − T2

Q := k ⋅F ⋅ΔT

k = 2.074

W

m ⋅K 2

Q = 1.576 × 10

3

W 2

m

Tw1 := T1 − Q ⋅

1

α1

T1_2 := Tw1 − Q ⋅

δ1 λ1

T2_3 := T1_2 − Q ⋅ Tw2 := T2_3 − Q ⋅

⎛ ⎜ ⎜ ⎜ T := ⎜ ⎜ ⎜ ⎜ ⎝

δ2 λ2

δ3 λ3

T1 ⎞



Tw1 ⎟ T1_2 ⎟

i := 0 .. 5

⎟ T2_3 ⎟ Tw2 ⎟ ⎟ T2 ⎠

1000 Ti K Ti

⎛⎜ ⎜ ⎜ ⎜ T= ⎜ ⎜ ⎜ ⎜ ⎝

3 1.053 × 10 ⎟⎞

1.03 × 10 835.868 424.644 424.363 293

3

⎟ ⎟ ⎟K ⎟ ⎟ ⎟ ⎟ ⎠

800

K Ti K Ti

600

K Ti K

400

0

2

i, 1, 4, 2, 3

4

6

Tw2 Tw2_ := T2 + Q ⋅

t2-t1

q0 := 0

Tw2 = 424.363 K

1

α2

Tw2_ = 424.363 K

N := 100

t1 := 0

t2 := 100

D( t , q) := k ⋅F ⋅ΔT

Q_ := rkfixed( q , t1 , t2 , N , D) 0

2 .10

5

Q_

〈1〉 1 .105

Q_ = 0

0 Q_

(Q_ (Q_ 5.5.

〈1〉

〈1〉

50 〈0〉

100

)1 = 1.576 × 103 ) N = 1.576 × 105

1

85

85 .34·10 5

86

86356·10 5

87

87371·10 5

88

88387·10 5

89

89403·10 5

90

90419·10 5

91

91434·10 5

92

92 .45·10 5

93

93466·10 5

94

94482·10 5

95

95498·10 5

96

96513·10 5

97

97529·10 5

98

98545·10 5

99

99561·10 5

100

100576·10 5

. [16]. , – γ = 10 ,

L=3

, h = 0.16 –

,

/ 3, –

: (

.

.

. 5.2). ν = 1/6.

x h R

L

p

. 5.2.

5.5

R = 2 ,

,

, x. , x d 4ω dx k=4

3 ⋅ ( 1 − ν2 ) R2 ⋅ h2

ω– x– q– D–

4

+ 4⋅k ⋅ω =

q , D

(5.1)

, ,

, ,

D= ν– R– h–

E ⋅ h3

12 ⋅ ( 1 − ν2 )

,

, , . (5.1)

ω ( x ) = e k ⋅x ⋅ (C1 ⋅ sin(k ⋅ x ) + C 2 ⋅ cos(k ⋅ x )) + + e −k ⋅x ⋅ (C 3 ⋅ sin(k ⋅ x ) + C 4 ⋅ cos(k ⋅ x )) +

ϕ ( x) =

M ( x) = −D ⋅



Q( x ) = − D ⋅



d 2ω dx 2 d 3ω

dx 3 E ⋅h⋅ω N( x ) = , R

– E– γ–

dω , dx

γ ⋅ ( L − x) ⋅ R 2 E ⋅h

.

, ,

, ,

. 1 –

4

. ω(0)=0

ϕ(0)=0

,

, x=0.

,

M(x)= 0 ω(0)=0

, Q(x)= 0 ϕ(0)=0

(

x=L. 1 = 2 = 0.

x=0,

). R=2000

.

q, :

ϕ(x),

4

, L=3000

, h=160 . q = γ ⋅( L − x ) (

.

. 5.2).

, M(x), Q(x), N(x). ,

1 –

,

(5.1), MathCAD. MathCAD: «sbval», «load», «score»

«rkfixed».

MathCAD

E⋅h⋅ω ( x)

N ( x)

R

φ ( x)

d dx

ω ( x)

−D ⋅

M ( x)

−D ⋅

Q ( x)

d2 2

dx d3

3

dx

ω ( x)

ω ( x)

ω ( x)

e

k⋅ x

+ γ⋅

-

− k⋅ x

⋅( C1 ⋅sin( k ⋅x) + C2 ⋅cos ( k ⋅x) ) + e ( L − x) ⋅R E ⋅h

⋅( C3 ⋅sin( k ⋅x) + C4 ⋅cos ( k ⋅x) ) ...

2

h k⋅ x − k⋅ x E⋅ ⋅⎡ e ⋅( C1 ⋅sin( k ⋅x) + C2 ⋅cos ( k ⋅x) ) + e ⋅( C3 ⋅sin( k ⋅x) + C4 ⋅cos ( k ⋅x) ) ... ⎤ ⎥ R⎢ 2 ⎢ + γ ⋅ ( L − x) ⋅R ⎥ E ⋅h ⎣ ⎦

N ( x)

φ(x), M(x)

" -

φ ( x)

/ x:

Q(x)

/

"

k ⋅exp( k ⋅x) ⋅( C1 ⋅sin( k ⋅x) + C2 ⋅cos ( k ⋅x) ) ... + exp( k ⋅x) ⋅( C1 ⋅cos ( k ⋅x) ⋅k − C2 ⋅sin( k ⋅x) ⋅k) ... + −k ⋅exp( −k ⋅x) ⋅( C3 ⋅sin( k ⋅x) + C4 ⋅cos ( k ⋅x) ) ... + exp( −k ⋅x) ⋅( C3 ⋅cos ( k ⋅x) ⋅k − C4 ⋅sin( k ⋅x) ⋅k) −

γ ⋅R ( E⋅h) 2

M ( x)

−2 ⋅D⋅k ⋅exp( k ⋅x) ⋅C1 ⋅cos ( k ⋅x) + 2 ⋅D⋅k ⋅exp( k ⋅x) ⋅C2 ⋅sin( k ⋅x) ... 2

2

+ 2 ⋅D⋅k ⋅exp( −k ⋅x) ⋅C3 ⋅cos ( k ⋅x) − 2 ⋅D⋅k ⋅exp( −k ⋅x) ⋅C4 ⋅sin( k ⋅x) 2

2

-

2 ⋅D ⋅k ⋅exp ( k ⋅x) ⋅C1 ⋅sin( k ⋅x) − 2 ⋅D ⋅k ⋅exp ( k ⋅x) ⋅C1 ⋅cos ( k ⋅x) ... 3

Q ( x)

+ − 2 ⋅D ⋅k ⋅exp ( − k ⋅x) ⋅C3 ⋅cos ( k ⋅x) ...

3

3

+ 2 ⋅D ⋅k ⋅exp ( k ⋅x) ⋅C2 ⋅sin( k ⋅x) − 2 ⋅D ⋅k ⋅exp ( − k ⋅x) ⋅C3 ⋅sin( k ⋅x) ... 3

+ − 2 ⋅D ⋅k ⋅exp ( − k ⋅x) ⋅C4 ⋅cos ( k ⋅x) ...

3

3

+ 2 ⋅D ⋅k ⋅exp ( − k ⋅x) ⋅C4 ⋅sin( k ⋅x) + 2 ⋅D ⋅k ⋅exp ( k ⋅x) ⋅C2 ⋅cos ( k ⋅x) 3

3 ω ( 0)

3

4

0

φ ( 0)

0

C1

0

C2

0

Given L ⋅R

0

C4 + γ ⋅

0

− k ⋅C4 + C3 ⋅k −

2

E ⋅h

γ ⋅R

2

E ⋅h

⎡ − γ ⋅R2 ⋅ ( k ⋅L − 1 ) ⎢ ( k ⋅E ⋅h) Find ( C3 , C4 ) → ⎢ 2 ⎢ R ⎢ − γ ⋅L ⋅ ( E ⋅h) ⎣ ORIGIN ≡ 1 L := 3 ⋅m ν :=

1 6

h := 0.16 ⋅m

E := 2 ⋅10 ⋅MPa 3

R := 2 ⋅m

γ := 10 ⋅

kN 3

m

:

kN ≡ 10 ⋅newton 3

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

MPa ≡ 10 ⋅Pa 6

: D D := E⋅

(

3

h

12 ⋅ 1 − ν

)

k D = 702.171 kN ⋅m

2

1

4 ⎛ E ⋅h ⎞ ⎟ 2 ⎝ D⋅R ⋅4 ⎠

k := ⎜

k = 2.31 m

-1

1, 2, 3 C1 := 0 ⋅m C3 := −γ ⋅R ⋅ 2

( k ⋅L − 1) ( k ⋅E⋅h)

C3 = −3.209 × 10

−4

m

x := 0 ⋅mm, 10 ⋅mm.. L

4 C2 := 0 ⋅m C4 := −γ ⋅L⋅

2

R ( E⋅h)

C4 = −3.75 × 10

−4

m

k⋅ x − k⋅ x ⋅( C3 ⋅sin( k ⋅x) + C4 ⋅cos ( k ⋅x) ) ... ⎤ ω ( x) := ⎡ e ⋅( C1 ⋅sin( k ⋅x) + C2 ⋅cos ( k ⋅x) ) + e

⎢ 2 ⎢ + γ ⋅ ( L − x) ⋅R E ⋅h ⎣

⎥ ⎥ ⎦

h k⋅ x − k⋅ x N ( x) := E⋅ ⋅⎡ e ⋅( C1 ⋅sin( k ⋅x) + C2 ⋅cos ( k ⋅x) ) + e ⋅( C3 ⋅sin( k ⋅x) + C4 ⋅cos ( k ⋅x) ) ... ⎤ R⎢ ⎥ 2 ⎢ + γ ⋅ ( L − x) ⋅R ⎥ E⋅h ⎣ ⎦

⎤ ⎥ ⎥ 2 ⎥ ⎢ γ ⋅R ⎢ + exp( −k ⋅x) ⋅( C3 ⋅cos ( k ⋅x) ⋅k − C4 ⋅sin( k ⋅x) ⋅k) − ⎥ ( E⋅h) ⎦ ⎣

φ ( x) := ⎡ k ⋅exp( k ⋅x) ⋅( C1 ⋅sin( k ⋅x) + C2 ⋅cos ( k ⋅x) ) ... ⎢ + exp( k ⋅x) ⋅( C1 ⋅cos ( k ⋅x) ⋅k − C2 ⋅sin( k ⋅x) ⋅k) ... ⎢ + −k ⋅exp( −k ⋅x) ⋅( C3 ⋅sin( k ⋅x) + C4 ⋅cos ( k ⋅x) ) ...

2 2 M ( x) := ⎛ −2 ⋅D ⋅k ⋅exp( k ⋅x) ⋅C1 ⋅cos ( k ⋅x) + 2 ⋅D ⋅k ⋅exp( k ⋅x) ⋅C2 ⋅sin( k ⋅x) ... ⎞

⎜ ⎟ 2 2 ⎝ + 2 ⋅D⋅k ⋅exp( −k ⋅x) ⋅C3 ⋅cos ( k ⋅x) − 2 ⋅D⋅k ⋅exp( −k ⋅x) ⋅C4 ⋅sin( k ⋅x) ⎠

3 3 Q ( x) := ⎛ 2 ⋅D⋅k ⋅exp( k ⋅x) ⋅C1 ⋅sin( k ⋅x) − 2 ⋅D⋅k ⋅exp( k ⋅x) ⋅C1 ⋅cos ( k ⋅x) ...

⎞ ⎜ ⎟ 3 ⎜ + −2 ⋅D⋅k ⋅exp( −k ⋅x) ⋅C3 ⋅cos ( k ⋅x) ... ⎟ ⎜ + 2 ⋅D⋅k3⋅exp( k ⋅x) ⋅C2 ⋅sin( k ⋅x) − 2 ⋅D⋅k3⋅exp( −k ⋅x) ⋅C3 ⋅sin( k ⋅x) ... ⎟ ⎜ + −2 ⋅D⋅k3⋅exp( −k ⋅x) ⋅C4 ⋅cos ( k ⋅x) ... ⎟ ⎜ ⎟ 3 3 ⎝ + 2 ⋅D⋅k ⋅exp( −k ⋅x) ⋅C4 ⋅sin( k ⋅x) + 2 ⋅D⋅k ⋅exp( k ⋅x) ⋅C2 ⋅cos ( k ⋅x) ⎠

IV d

4 4

dx

ω + E⋅

h

D⋅R

2

⋅ω

(

d dx

Qx( 0) LL :=

−D ⋅ −D ⋅

d

D )

ω ( 0)

Mx( 0)

q

ω ( 0)

2 2

dx d3

3

dx

L m

−D ⋅

Mx( L)

0 φ

0

Qx( L)

−D ⋅

d

ω ( L)

0

ω ( L)

0

2 2

dx d

3 3

dx

ω ( 0)

ω ( 0) LL = 3

⎛ 0⎞ ⎟ ⎝ 0⎠

v := ⎜

⎛ ⎜ load( x1 , v) := ⎜ ⎜ ⎜ ⎝

0 ⎞

x=0



0 ⎟ v1 ⎟



v2 ⎠

ω2 ⎤ ⎡ ⎢ ⎥ ω3 ⎢ ⎥ ⎢ ⎥ DD(x , ω ) := ω4 ⎢ ⎥ ⎢ ( LL − x) ⎥ h ⎢ γ ⋅ D − E⋅ 2 ⋅ω 1 ⎥ D⋅R ⎣ ⎦

⎛ ω3 ⎞ ⎟ ⎝ ω4 ⎠

score(x2 , ω ) := ⎜

x=L

B := sbval( v , 0 , LL, DD , load, score)

⎛ 3.425 × 10− 3 ⎞ ⎟ ⎝ −0.017 ⎠

B=⎜

bnd := load( 0 , B)

IV

"rkfixed"

y := bnd

t1 := 0

y1 = 0 y2 = 0

t2 := LL

NN := 100 x=0

−3

y3 = 3.425 × 10 y4 = −0.017

Z := rkfixed( y , t1 , t2 , NN , DD)

(

〈〉 x_ := Z 1

(

〈〉 ω_ := Z 2

⎡ 〈2〉 Nθ_ := ⎢Z

(



〈〉 φ_ := Z 3

〈4〉 Mx_ := ⎡⎣Z

〈〉 Qx_ := ⎡⎣Z 5

Z

) ⋅m

) ⋅m

)

⎛ E⋅h ⎞⎤ ⋅m ⎟⎥ ⎝ R ⎠⎦

⋅⎜

⋅( −D)⎤⎦ ⋅m

−1

⋅( −D)⎤⎦ ⋅m

−2

, mm

, mm

3000

2000

1000

0.1

0

0.1

0.2

"MathCAD" ω ( 0 ⋅mm) = 0 mm

ω ( L) = −4.815 × 10

−4

L = 3 × 10 mm 3

max(ω_) = 0.252 mm

ω_1 = 0 mm

mm

ω_101 = −1.021 × 10

−3

mm

x_101 = 3 × 10 mm 3

ω_33 = 0.252 mm

ω ( 960 ⋅mm) = 0.252 mm

x_90 = 2.67 × 10 mm

ω ( x_90) = 0.041 mm

x_33 = 960 mm

3

ω_90 = 0.04 mm

, Q*m/kN

, mm

3000

2000

1000

10

0

10

20

30

40

50

"MathCAD" ,

, mm

3000

x_

2000

mm x mm 1000

2 .10

4

0

2 .10 4 .10 φ_ , φ( x) 4

4

"MathCAD"

, M*m/(kN*m)

, mm

3000

x_ 2000

mm x mm

1000

3

2 Mx_⋅

1 m

kN⋅ m

, M ( x) ⋅

1

m

kN⋅ m

"MathCAD"

, Q*m/kN

3000

, mm

0

2000

1000

5

0

5

10

15

MathCAD

5.6.

.

,

XX [8].

,

( . «0»

. , 1966),

«1»: 0 → 1 ( 0.1), «2»: 1 + 1 → 2 + 1 (103). , «2» «1»: 1 + 2 → 0 + 2 (102). , D(t, y).

) , [8].

«1» ,

(

: y0 := 1

0

y1 := 0

1

y2 := 0

−0.1 ⋅ y0 + 10 ⋅ y1 ⋅ y2

d y0 dt d dt

2 2

0

0.1 ⋅ y0 − 10 ⋅ y1 − 10 ⋅ y1 ⋅ y2 3

y1

2

1

10 ⋅ y1 + 10 ⋅ y1 ⋅ y2 − 10 ⋅ y1 ⋅ y2

d y2 dt

3

2

2

2

: 5.6.1.

t1 := 0

I

t2 := 100

Stiffb

N := 100

2 ⎛ −0.1 ⋅ y0 + 10 ⋅ y1 ⋅ y2 ⎜ D( t , y) := ⎜ 0.1 ⋅ y0 − 103 ⋅ y1 − 102 ⋅ y1 ⋅ y2 ⎜ ⎜ 103 ⋅ y + 102 ⋅ y ⋅ y − 102 ⋅ y ⋅ y 1 1 2 1 2 ⎝

Stif

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

⎡ d( d ( d ( d ( 2 2 2 2 − 0.1 ⋅ y0 + 10 ⋅ y1 ⋅ y2) − 0.1 ⋅ y0 + 10 ⋅ y1 ⋅ y2) −0.1 ⋅ y0 + 10 ⋅ y1 ⋅ y2) −0.1 ⋅ y0 + 10 ⋅ y1 ⋅ y2) ⎢ dt dy0 dy1 dy2 ⎢ ⎢ d d ( d ( d ( 3 2 3 2 3 2 3 2 0.1 ⋅ y0 − 10 ⋅ y1 − 10 ⋅ y1 ⋅ y2) 0.1 ⋅ y0 − 10 ⋅ y1 − 10 ⋅ y1 ⋅ y2) 0.1 ⋅ y0 − 10 ⋅ y1 − 10 ⋅ y1 ⋅ y2) ⎢ (0.1 ⋅ y0 − 10 ⋅ y1 − 10 ⋅ y1 ⋅ y2) dy0 dy1 dy2 ⎢ dt ⎢ d( 3 d ( 3 d ( 3 d ( 3 10 ⋅ y1) 10 ⋅ y1) 10 ⋅ y1) 10 ⋅ y1) ⎢ dt dy0 dy1 dy2 ⎣

"

/

/

"

0 − .1 100 ⋅ y2 100 ⋅ y1 ⎜⎛ ⎜ 0 .1 − 1000 − 100 ⋅ y2 − 100 ⋅ y1 ⎜ 0 0 1000 0 ⎝

100 ⋅ y1 100 ⋅ y2 ⎛ 0 − .1 ⎜ J ( t , y) := ⎜ 0 .1 − 1000 − 100 ⋅ y2 − 100 ⋅ y1 ⎜ 0 1000 ⎝ 0 0

Z := Stiffb( y , t1 , t2 , N , D , J)

⎟⎞ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎟ ⎠

Z1 := Stiffr( y , t1 , t2 , N , D , J)

1

Z Z Z

〈1〉 〈2〉 〈3〉

⋅5000 0.5

0 Z

50 〈0〉

100

0 1 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

5.6.2.

I stiffb

stiffr

acc := 0.000000001

kmax := 5

s := 10

ZZ := stiffb( y , t1 , t2 , acc , D , J , kmax , s)

ZZ1 := stiffr( y , t1 , t2 , acc , D , J , kmax , s)

1 ⎛ 0 ⎜ ⎜ 12.718 0.295 ⎜ ZZ = ⎜ 36.521 0.033 ⎜ −4 ⎜ 92.637 2.01 × 10 ⎜ −4 ⎝ 100 1.029 × 10

0

2.757 × 10

3.019 × 10

1.827 × 10

9.355 × 10

1 ⎛ 0 ⎜ ⎜ 10.107 0.377 ⎜ ZZ1 = ⎜ 20.295 0.146 ⎜ 0.058 ⎜ 30.327 ⎜ −4 ⎝ 100 1.029 × 10

(Z

(Z1 (Z

(Z1

〈0〉

〈0〉 〈2〉 〈2〉

−5 −6 −8 −9

0

3.549 × 10

1.347 × 10

5.326 × 10

9.355 × 10

)100 = 100

)100 = 100

(Z

〈1〉

(Z

〈3〉

(Z1

)100 = 9.355 × 10 − 9

)100 = 9.356 × 10 − 9

(Z1

⎞ ⎟ 0.705 ⎟ ⎟ 0.967 ⎟ ⎟ 1 ⎟ ⎟ 1 ⎠ 0

−5 −5 −6 −9

〈1〉

〈3〉

⎞ ⎟ 0.623 ⎟ ⎟ 0.854 ⎟ ⎟ 0.942 ⎟ ⎟ 1 ⎠ 0

)100 = 1.029 × 10 − 4

)100 = 1.029 × 10 − 4

)100 = 1

)100 = 1

6. 6.1.

( .

∂ u 2

∂t 2 x=0

L, F, x=L

= v2

. ∂ 2u ∂x 2

. ,

x=L. .

, F –S

E,

[11].

: E := 2.1 ⋅10

11

,

ρ := 7850

⎛ E⎞ ⎟ ⎝ρ⎠

v := ⎜

,

/ ^3

1 2

, /

v = 5.172 × 10

3

F := 1000

S := 0.0001

,

)

, , ^2

: N := 20

N1 := 40

i := 0 .. N

i1 := 1 .. N − 1 j := 0 .. N1

j1 := 1 .. N1 − 1

t_end := 0.00005 L_max := 1

t_begin := 0 L_min:= 0

L_max − L_min

h_x :=

h_t :=

N

h_x

N1 −6

h_x = 0.05 v⋅h_t

t_end − t_begin

h_t = 1.25 × 10

≤ 1 xi := 0 + h_x⋅i

u1( x) := F ⋅

u2( x) := 0

t j := 0 + h_t⋅ j

x

E ⋅S

Ui, 0 := u1( xi)

Ui, 1 := u1( xi) + h_t⋅u2( x)

U0 , j := 0 r :=

v⋅h_t h_x

(

)

Ui1, j1+ 1 := 2 − 2 ⋅r ⋅Ui1, j1 + r ⋅( Ui1+ 1 , j1 + Ui1−1 , j1) − Ui1, j1−1 2

2

x20 = 1

U

t20 = 2.5 × 10

−5

u( x , t)

∑ ∞

( −1) 8 ⋅F ⋅L_max ⎡ ( 2 ⋅n − 1) ⋅π ⋅v⋅t⎤ ⎡ ( 2 ⋅n − 1) ⋅π ⋅x⎤ ⋅cos ⎢ ⋅ ⎥ ⋅sin⎢ ⎥ 2 2 ⎣ 2 ⋅L_max ⎦ ⎣ 2 ⋅L_max ⎦ π ⋅E⋅S n = 0 ( 2 ⋅n − 1) n+ 1



Mathcad

8 ⋅F ⋅L_max ( −1) ⎡ ( 2 ⋅n − 1) ⋅π ⋅v⋅t⎤ ⎡ ( 2 ⋅n − 1) ⋅π ⋅x⎤ u1( x , t) := ⋅cos ⎢ ⋅ ⎥ ⋅sin⎢ ⎥ 2 2 ⎣ 2 ⋅L_max ⎦ ⎣ 2 ⋅L_max ⎦ π ⋅E⋅S n = 0 ( 2 ⋅n − 1) n+ 1

20

U1i, j := u1( xi , t j)

U1

6.2.

U

.

u( 0,t ) = u( L ,t ) = 0,

.

.

∂u ∂ 2u =k 2 , ∂t ∂x x ⋅( L − x ) [13]. u( x ,0 ) = L2 : λ := 45.4

, /( * )

Cp := 460

,

ρ := 7850

k :=

,

λ

Cp ⋅ρ

/ ^3

−5

k = 1.257 × 10

: N := 100

N1 := 20

i := 0 .. N

j := 0 .. N1

i1 := 1 .. N − 1 t_end := 4

t_begin := 0

L_max := 2 h_x :=

L_max − L_min N h_x = 0.02 h_t ≤

2

h_x

2 ⋅k

=1

L_min:= 0

h_t :=

t_end − t_begin N1

h_t = 0.2

/( * )

xi := 0 + h_x⋅i

t j := 0 + h_t⋅ j

x⋅( L_max − x)

u1( x) :=

2

Ui, 0 := u1( xi)

L_max

U0 , j := 0

UN , j := 0



Ui1, j+ 1 := ⎜ 1 −



⋅( Ui1−1 , j + Ui1+ 1 , j) ⎟ ⋅Ui1, j + 2 h_x h_x ⎠

2 ⋅k ⋅h_t ⎞

k ⋅h_t

2

0.26 0.24 0.22 0.2 Ui , 0 0.18 0.16 U N1 0.14 i, 0.12 2 0.1 0.08 Ui , N1 0.06 0.04 0.02 0 − 0.02 0.02

U

0 0

u( x , t)

20

40

60

80

i

100 100

2 2 ⎡⎢ ⎤⎥ − ( 2⋅ n + 1) ⋅ π ⋅ k⋅ t 2 ( 2 ⋅ n + 1 ) ⋅ π ⋅ x 8 1 ⎡ ⎤ L_max ⎢ ⎥ ⋅ ⋅e ⋅sin⎢ ⎥⎥ 3 3 ⎢ L_max ⎣ ⎦ π n = 0 ⎣ ( 2 ⋅n + 1 ) ⎦

∑ ∞

Mathcad

2 2 ⎡⎢ − ⎡⎣( 2⋅ n + 1) ⋅ π ⋅ k⋅ t⎤⎦ ⎥⎤ 2 1 8 ( 2 ⋅ n + 1 ) ⋅ π ⋅ x ⎡ ⎤ L_max ⎢ ⎥ u1( x , t) := ⋅ ⋅e ⋅sin⎢ ⎥⎥ 3 3 ⎢ L_max ⎣ ⎦ π n = 0 ⎣ ( 2 ⋅n + 1 ) ⎦

∑ 20

U1i, j := u1( xi , t j)

U1

6.3.

.

.

. -

∂u ∂ 2u =d 2 , ∂t ∂x u( 0,t ) = u( L ,t ) – u( x ,0 ) = 1

: 0 ≤ x ≤ 0.5

, [9].

1

0

,

^2/c

: d := 5.9 ⋅10

−5

: N1 := 10

N := 100

i := 0 .. N

j := 0 .. N1

i1 := 1 .. N − 1 t_end := 2

t_begin := 0

L_max := 0.5 h_x :=

L_min:= 0

L_max − L_min

h_x = 5 × 10 h_t ≤

h_t :=

N

−3

t_end − t_begin N1

h_t = 0.2

2

h_x

2 ⋅d

xi := 0 + h_x⋅i

t j := 0 + h_t⋅ j

u1 ( x) := 1

Ui , 0 := u1 ( xi)

U0 , j := 1 −

h_t ⋅j t_end

U N , j := 1 −

t_end



h_t

Ui1, j+ 1 := ⎜ 1 −



⋅j

⋅( Ui1−1 , j + Ui1+ 1 , j) ⎟ ⋅Ui1, j + 2 h_x ⎠ h_x

2 ⋅d ⋅h_t ⎞

d ⋅h_t

2

1.1 Ui, N1 U N1 i,

2

Ui, 0 0

U

1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

50

100

0

i

100

0≤t≤2

,

6.4.

.

.

.

∂x 2

u( x , y ) = x 2 − y 2

: -1 =x= 1, -1 =y= 1.

: u( x , y) := x − y 2

2

: 6.4.1.

. .

"relax" N := 10

i := 0 .. N

N1 := 10

j := 0 .. N1

i1 := 1 .. 9

j1 := 1 .. 9

L_max := 2

L_min:= 0

h_x :=

h_y :=

L_max − L_min N

h_x = 0.2

xi := −1 + h_x⋅i

∂ 2u

L_max − L_min N1

h_y = 0.2

yj := −1 + h_y⋅ j

+

∂ 2u ∂y 2

= 0,

yj

0

1

0

1

xi

U1i, j := u( xj , yi)

U0 , j := U10 , j

UN , j := U10 , j

Ui, 0 := U1i, 0

Ui, N := U1i, 0 "relax"

ai, j := 1 b := a c := a

d := a

e := −4 ⋅a

f i, j := 0

v0 , j := U10 , j

vN , j := U10 , j

vi, 0 := U1i, 0

vi, N := U1i, 0

U := relax( a , b , c , d , e , f , v , 0.95)

U

U − U1

6.4.2.

. .

N := 4

N1 := 4

i := 0 .. N

j := 0 .. N1

i1 := 1 .. N − 1

j1 := 1 .. N1 − 1

L_max := 2

L_min:= 0

h_x :=

h_y :=

L_max − L_min N

h_x = 0.5

h_y = 0.5

xi := −1 + h_x ⋅ i

yj

L_max − L_min N1

yj := −1 + h_y ⋅ j

0

1

0

1

xi

U1i, j := u( xj , yi)

U0 , j := U10 , j

UN , j := U10 , j

u1 := 0

u6 := 0

u2 := 0

u7 := 0

Ui, 0 := U1i, 0

Ui, N := U1i, 0

u3 := 0

u8 := 0

u4 := 0

u9 := 0

u5 := 0

Given 0.75 − 4 ⋅ u1 + u2 − 0.75 + u4

1 − 4 ⋅ u4 + u5 + u1 + u7

0

0

0.75 − 4 ⋅ u7 + u8 + u4 − 0.75 u1 − 4 ⋅ u2 + u3 − 1 + u5

0

0

u4 − 4 ⋅ u5 + u6 + u2 + u8

0

u7 − 4 ⋅ u8 + u9 + u5 − 1

0

u2 − 4 ⋅ u3 + 0.75 − 0.75 + u6 u5 − 4 ⋅ u6 + 1 + u3 + u9

0

0

u8 − 4 ⋅ u9 + 0.75 + u6 − 0.75

0

Z := Find( u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8 , u9) Ui1, j1 := Z3⋅( i1−1) + j1−1

⎛ ⎜ ⎜ U=⎜ ⎜ ⎜ ⎝

0

−0.75

0.75

0

−0.25

1

0.25

0.75

0

0

−0.75

−1

−0.75

0

0

0.75

0

0.25

1

−1

0

0.75

−0.25

−0.75

0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎛ ⎜ ⎜ U − U1 = ⎜ ⎜ ⎜ ⎝

0 0 0 0 0 ⎞

⎟ ⎟ 0 0 0 0 0 ⎟ ⎟ 0 0 0 0 0 ⎟ 0 0 0 0 0 ⎠ 0 0 0 0 0

U

6.5.

.

.

∂ u 2

∂x

u(x,y)

2

+

∂ u 2

∂y 2

= − f ( x, y ) ,

.

,

f(x,y) 8 8.

1000,

:

N := 8

i := 0 .. N

j := 0 .. N

Mi, j := 0

M4 , 4 := 1000

: A := multigrid( − M , 2 )

A

A

7.

(

7.1.

.

)

,

(

. 7.1) [7].

n y

p

Vi (vi)

E1 A1 xi

C α i

L

Ui (ui) yi

x . 7.1.

C

, E1 – , ui , vi –

p–

, A1 –

,L– , i, n –

, Ui , Vi –

, xi , yi –

, .

, .

. 7.2

, 1

n.

, .

,

Y4

{R4} p

4

3

X4

(c) y (a)

(d) (b) 5

2

1

6

x

V3

p y

3

(a)

U3

2 ы

1

x

ч ы э е е . 7.2.

,

, .

,

1–3

,

a, .

p , U, V

-

u, v

,

,

{F }ap

. .

(

),

a,

{F }a

⎧U1 ⎫ ⎪V ⎪ 1 ⎧ F1 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪⎪U 2 ⎪⎪ = ⎨ F2 ⎬ = ⎨ ⎬ , ⎪ F ⎪ ⎪V2 ⎪ ⎩ 3 ⎭ ⎪U ⎪ 3 ⎪ ⎪ ⎪⎩ V3 ⎪⎭

{δ}a

⎧ u1 ⎫ ⎪v ⎪ 1 ⎧ δ1 ⎫ ⎪ ⎪ ⎪ u ⎪ ⎪ ⎪ ⎪⎪ = ⎨δ 2 ⎬ = ⎨ 2 ⎬ . ⎪ δ ⎪ ⎪ v2 ⎪ ⎩ 3 ⎭ ⎪u ⎪ 3 ⎪ ⎪ ⎪⎩ v3 ⎪⎭

,

{F }a = [k ]a {δ}a + {F }ap + {F }εa , {F }aε – , 0



,

0

(7.1) ,

,

,

,

.

,

.

{σ}a ,

.

{σ}a = [S ]a {δ}a + {σ}ap + {σ}εa

0



[k ]a

,

[S ]a

, (7.2) .

(7.2)

,

.

(7.1)

,



. ,

,

.

b .

, ,

, ,

.

.

{F }a Fi

δi

⎧ F1 ⎫ ⎪ . ⎪ ⎪⎪ ⎪⎪ =⎨ . ⎬ ⎪ . ⎪ ⎪ ⎪ ⎪⎩ Fm ⎪⎭

⎧ δ1 ⎫ ⎪ . ⎪ ⎪⎪ ⎪⎪ =⎨ . ⎬, ⎪ . ⎪ ⎪ ⎪ ⎪⎩δm ⎪⎭

.

,

[k ]

e

kii

{δ}a

,

⎡ kii ⎢ . ⎢ =⎢ . ⎢ ⎢ . ⎢k mi ⎣

kij . . . kmj

l ×l , l –

. .–

kim ⎤ . ⎥⎥ . ⎥, ⎥ . ⎥ kmm ⎥⎦

. C

E (

A

. 7.1).

p

xi , y i

xn , y n ,

L=

{(x

ε 0 = αT .

n

}

− x i )2 + ( y n − y i )2 , ⎛ y n − yi ⎞ ⎟⎟ . ⎝ x n − xi ⎠

α = arctg ⎜⎜

. ,

,

,

{F }ap

ε0

⎧U i ⎫ ⎧− sin α ⎫ ⎪ ⎪ ⎪ cosα ⎪ ⎧F ⎫ ⎪V ⎪ ⎪ ⎪ pL =⎨ i⎬ =⎨ i ⎬ =⎨ . ⎬ F U sin α − ⎩ n ⎭p ⎪ n ⎪ ⎪ ⎪ 2 ⎪⎩Vn ⎪⎭ ⎪⎩ cosα ⎪⎭ p , EαTA ,

pL 2 .

{F }εa

0

⎧U i ⎫ ⎧− cosα ⎫ ⎪ ⎪ ⎪ − sin α ⎪ ⎛ F ⎞ ⎪V ⎪ ⎪ ⎪ = ⎜⎜ i ⎟⎟ = ⎨ i ⎬ = − ⎨ ⎬(EαTA). F U cos α ⎝ n⎠ ⎪ n⎪ ⎪ ⎪ ⎪⎩Vn ⎪⎭ ⎪⎩ sin α ⎪⎭ ε0

,

{δ}a (un − ui )cos α + (vn − vi ) sin α .

,

,

{F }δa

⎧ ui ⎫ ⎪ ⎪ ⎧δ ⎫ ⎪v ⎪ =⎨ i⎬=⎨ i⎬ ⎩δn ⎭ ⎪un ⎪ ⎪⎩vn ⎪⎭ EA L , .

EαTA

⎧U i ⎫ ⎪V ⎪ ⎧ Fi ⎫ ⎪ ⎪ =⎨ ⎬ =⎨ i ⎬ = F ⎩ n ⎭δ ⎪U n ⎪ ⎪⎩Vn ⎪⎭ δ

⎡ cos 2 α ⎢ EA ⎢ sin α cos α = L ⎢ − cos 2 α ⎢ ⎣⎢− sin α cos α

sin α cos α

− cos 2 α

sin α

− sin α cos α

− sin 2 α

sin α cos α

2

− sin α cos α

cos 2 α

− sin α cos α ⎤ ⎥ − sin 2 α ⎥ × sin α cos α ⎥ ⎥ sin 2 α ⎦⎥

,

⎧ ui ⎫ ⎪v ⎪ ⎪ ⎪ × ⎨ i ⎬ = [k ]a {δ }a . ⎪u n ⎪ ⎪⎩v n ⎪⎭

(7.1). .

,

(7.2)

, ,

, I–

− sin α − sin α

cos α cos α

sin α ⎤ a ⎧ 1 ⎫ pL2 d ⎧1⎫ {δ } + ⎨ ⎬ − ⎨ ⎬ E αT , sin α ⎥⎦ ⎩− 1⎭ 8 I ⎩1⎭

.

(7.2). ,

:

L := 1000 ⋅mm d := 20 ⋅mm h := 40 ⋅mm

α := 45 ⋅deg

E := 2.1 ⋅10 ⋅MPa 5

a := 1.1 ⋅10

−5

T := 30 ⋅K

p := 10 ⋅

⎛ ⎜ ⎜ δ := ⎜ ⎜ ⎝

,

C,

⎧σ 1 ⎫ E ⎡ − cos α ⎨ ⎬ = ⎢ ⎩σ 2 ⎭ C L ⎣ − cos α

d–

,



1 K

N 1 ⎞

mm



2 ⎟ ⋅mm 1.5 ⎟



0.5 ⎠

.

:

A := 2 ⋅d ⋅h

−sin(α ) ⎞

⎛ ⎜ Fp := ⎜ ⎜ ⎜ ⎝



cos (α ) ⎟ p ⋅L ⋅ −sin(α ) ⎟ 2

,



⎛ ⎜ ⎜ Fε0 := ⎜ ⎜ ⎝

cos (α ) ⎠

−cos (α ) ⎞



−sin(α ) ⎟ ⋅E⋅a ⋅T⋅A cos (α ) ⎟

,



sin( α ) ⎠

: 2 2 ⎛ −sin( α ) ⋅cos ( α ) −cos (α ) cos ( α ) sin( α ) ⋅cos ( α ) ⎜ 2 2 ⎜ sin(α ) ⋅cos (α ) sin(α ) −sin( α ) −sin(α ) ⋅cos ( α ) k := ⎜ 2 ⎜ −cos (α )2 −sin( α ) ⋅cos ( α ) sin( α ) ⋅cos (α ) cos (α ) ⎜ 2 2 sin(α ) ⋅cos ( α ) −sin( α ) sin(α ) ⎝ −sin(α ) ⋅cos (α )

Fδ := k ⋅δ

,

F := Fδ + Fp + Fε0

⎛ ⎜ F=⎜ ⎜ ⎜ ⎝

7.2.

,

86.06 ⎞



93.132 ⎟ kN −93.132 ⎟



−86.06 ⎠

.

(

. 7.3),

,

,

E– ,L– L = 2 , h = 2 , E = 2·1011

⎞ ⎟ ⎟ E⋅A ⎟⋅ ⎟ L ⎟ ⎠

p⋅L ( . 7.3), 2⋅E ,h– , p = 1·108 , = 0.3.

p

ΔL

. ΔL =

.

,μ–

p

L . 7.3.

, 100

[7]:

.

,

,

,p–

h

100

.

1)

(

)

,

,

. 2)

,

,

; 3)

. (

),

. e,

, . 7.4

i, j, m. .

i, j, m,

{δi } = ⎨

⎧ui ⎫ , ⎬ ⎩vi ⎭

⎧ δi ⎫ . {δ} = ⎪⎨ δ j ⎪⎬ ⎪δ ⎪ ⎩ m⎭ e

.

y

m

vi(Ui) i

ui(Ui)

xi yi

j x

. 7.4.

u = α1 + α2 x + α3 y ,

v = α4 + α5 x + α6 y .

αi

,

,

(7.3)

,

(7.3) ,

. ui = α1 + α2 xi + α3 yi ,

u j = α1 + α 2 x j + α3 y j ,

(7.4)

um = α1 + α2 xm + α3 ym ,

α1 ,α 2 ,α 3

{

(

ui ,u j ,um

u=

)

}

1 (ai + bi x + ci y )ui + a j + b j x + c j y u j + (am + bm x + cm y )um , (7.5 ) 2Δ ai = x j y m − x m y j ,

bi = y j − ym = y jm ,

(7.5 )

ci = xm − x j = xmj ;

i , j ,m ,

⎡1 xi 2Δ = det ⎢1 x j ⎢ ⎢⎣1 xm v

yi ⎤ y j ⎥ = 2⋅ ⎥ ym ⎥⎦



щ

ь

ь

ijm

.

(7.5 )

v=

(7.5 )

(

(7.6)



,

)

}

[

u⎫ e ⎬ = [N ]{δ} = IN i′ IN ′j v ⎩ ⎭

{ f } = ⎧⎨

2×2 ,

I – N i′

{

1 (ai + bi x + ci y )vi + a j + b j x + c j y v j + (am + bm x + cm y )vm . 2Δ

N i′ =

4)

ai + bi x + ci y 2Δ

]

IN m′ {δ}e ,

(7.6)

(7.7)

. .

(7.8).

. , :

(7.7)

⎧ ∂u ⎫ ⎪ ⎪ ⎧ ε x ⎫ ⎪ ∂x ⎪ . {ε} = ⎪⎨ ε y ⎪⎬ = ⎪⎨ ∂v ⎪⎬ ⎪ε ⎪ ⎪ ∂y ⎪ ⎩ xy ⎭ ⎪ ∂u ∂v ⎪ + ⎪ ∂y ∂x ⎪ ⎩ ⎭

(7.5 )

(7.6), ⎡ ∂N i′ ⎢ ⎢ ∂x {ε} = ⎢⎢ 0 ⎢ ⎢ ∂N i′ ⎢⎣ ∂y

[B] .

[B]

{ε0 }

⎡bi 1 ⎢ 0 = 2Δ ⎢ ⎢ci ⎣

0

∂N i′ ∂y ∂N i′ ∂x

,

я

∂x

0 cj

bm 0

bi

cj

bj

cm

,

в

∂y

∂y ∂N ′j

bj 0

0

∂N m′ ∂y

а

а

.

(7.9)

0⎤ ⎥ c m ⎥{δ}e , bm ⎥⎦

.

.

⎧ ε x0 ⎫ . {ε0 } = ⎪⎨ ε y 0 ⎪⎬ ⎪γ ⎪ ⎩ xy 0 ⎭

я

∂N ′j

∂N ′j

⎤ ⎧ ui ⎫ 0 ⎥⎪ v ⎪ i ⎥⎪ ⎪ ∂N m′ ⎥ ⎪⎪ u j ⎪⎪ ⎨ ⎬= ∂y ⎥ ⎪ v j ⎪ ⎥ ∂N m′ ⎥ ⎪u m ⎪ ∂x ⎥⎦ ⎪⎪v ⎪⎪ ⎩ m⎭

⎞ ⎛ ⎧ εx ⎫ ⎧ σx ⎫ ⎟, ⎜⎪ ⎪ ⎪ ⎪ {σ} = ⎨ σ y ⎬ = [D ]⎜ ⎨ ε y ⎬ − {ε 0 }⎟ ⎟⎟ ⎜⎜ ⎪ ⎪ ⎪σ ⎪ ε ⎩ xy ⎭ ⎠ ⎝ ⎩ xy ⎭ {σ0 }). (7.11)

,

а

0

∂N m′ ∂x

,

( –

∂x

0

0 ci

, ,

[D] ,

∂N ′j

(7.10)

σx ν ⋅ σ y + ε x0 , − E E ν ⋅ σx σ y εy = − + + ε y0 , E E 2 ⋅ ( 1 + ν ) ⋅ τ xy γ xy = + γ xy 0 . E

εx =

,

[D]

⎡ ⎤ 0 ⎥ ⎢1 ν , ⎢ν 1 [D ] = 0 ⎥ 2 1− ν ⎢ 1− ν⎥ ⎢0 0 ⎥ 2 ⎦ ⎣ E

E–

, ф

ν – ва

. я

σz.

,

в

εz = 0 = −

{δi }

.

⎡ ⎢ 1 ⎢ [D ] = E (1 − ν ) ⎢ ν ( 1 + ν )( 1 − 2ν ) ⎢ 1 − ν ⎢ ⎢ 0 ⎣

.

,

.

ν ⋅ σx ν ⋅ σ y σz − + + α ⋅ θe . E E E

[D]

(7.10)

{F }e

а

σx ν ⋅ σ y ν ⋅ σz − + α ⋅ θe , − E E E ν ⋅ σx σ y ν ⋅ σz εy = − + − + α ⋅ θe , E E E 2 ⋅ ( 1 + ν ) ⋅ τ xy . γ xy = E

σz ,

(

а

εx =

,

5)

ν 1− ν 1 0

⎤ ⎥ ⎥. 0 ⎥ ⎥ 1 − 2ν ⎥ 2( 1 − ν ) ⎥⎦ 0

,

{Fi }

,

,

.

)

, . -

e

{δ}e



{F }e = [k ]e {δ}e + {F }ep + {F }oe

-

[k ]e

(7.11)

,

, , –

[k ]e = ∫ [B]T [D] [B] dV .

ijm

[k ]e = ∫ [B]T [D] [B] t dx dy ,

,

t–

,

.

,

,

Δ–

,

y,

x

[k ]e = [B]T [D] [B] t Δ ,

[B] ,

.

,

,

[

(7.5 )]. (7.9),

[B ] = [Bi , B j ,B m ], ⎡ kii

[k ]e = ⎢⎢ k ji

⎢kmi ⎣

2×2

⎧bi [Bi ] = ⎪⎨ 0 ⎪c ⎩ i kij k jj k mj

0⎫ ⎪ ci ⎬ 2 Δ bi ⎪⎭

kim ⎤ ⎥, k jm ⎥ kmm ⎥⎦

. .

(7.12)

(7.13)

[k rs ] = [Br ] [D ] [Bs ] t Δ . :

T

6)

[K ]

7)

[Kij ] = ∑[kij ]

.

(7.14)

{F } = [K ]{δ} {δ} = [K ]−1{F }

.

,

,

x, y

.

{p} = ⎧⎨

X⎫ ⎬ ⎩Y ⎭

.

{F }ep = − ∫ [N ]T ⎧⎨

X⎫ , ⎬ dx dy ⎩Y ⎭

(7.7)

,

X⎫ ⎬∫ [N ]i dx dy ⎩Y ⎭

{Fi }p = − ⎧⎨ X

Y

.

,

Ni

.

∫ x dx dy = ∫ y dx dy = 0 , ,

,

(7.8),

{Fi }p = − ⎧⎨

X⎫ ⎧X ⎫ , ⎬ ∫ ai dx dy 2Δ = − ⎨ ⎬ ai 2 ⎩Y ⎭ ⎩Y ⎭

.

{ }p = {Fm }p .

{Fi }p = − ⎧⎨

X⎫ ⎬ Δ 3 = Fj Y ⎩ ⎭

{F }

e p

,

,

x

⎧X ⎫ ⎪Y ⎪ ⎪ ⎪ . ⎪X ⎪ = −⎨ ⎬ Δ 3 ⎪Y ⎪ ⎪X ⎪ ⎪ ⎪ ⎩Y ⎭

(7.15)

.

y,

8) ,

. Mathcad .

. :

μ := 0.3

E := 2 ⋅10

11

,

q_x := 10

8

X,

q_y := 0

Y, ,

a := 1

L := a t := 1

:

ORIGIN ≡ 1

. (7.11)

⎛ 1 μ 0 ⎜ E ⎜ μ 1 0 ⋅ D_N := 2 ⎜ 1−μ 1−μ ⎜ 0 0 2 ⎝

X

σ_x := q_x

ΔLmax :=

σ_x⋅L E

−4

ΔLmax = 5 × 10

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

1 - 4 (3

X (1 )

)

Y (2

)

0 a a 0 ⎞ ⎜⎛ ⎟ Nodes := ⎜ a a 0 0 ⎟ ⎜ 1 2 3 4⎟ ⎝ ⎠

( -

)

-

⎛ 1 3 2⎞ ⎟ ⎝ 1 4 3⎠

Top := ⎜

⎛ n_element ⎞ ⎛ rows( Top) ⎞ ⎜ ⎟ := ⎜ ⎟ ⎝ nn_point ⎠ ⎝ cols( Top) ⎠ ie := 1 .. n_element j := 1 .. nn_point

M_ie , 2⋅ j−1 := 2 ⋅Topie , j − 1

M_ie , 2⋅ j := 2 ⋅Topie , j

,

X, Y

⎛ Xnie , j ⎞ ⎡ Nodes1, (Top ie , j) ⎤ ⎜ ⎟ := ⎢ ⎥ ⎝ Ynie , j ⎠ ⎣ Nodes2, (Top ie , j) ⎦ j =

⎛⎜ 2 3 ⎞⎟ l := ⎜ 3 1 ⎟ ⎜ 1 2⎟ ⎝ ⎠

1 2 3

y1_2ie := ymedie :=

Ynie , 1 + Ynie , 2 2

Ynie , 3 + 2 ⋅y1_2ie 1+ 2

xmed1 = 0.667

ymed1 = 0.667

x1_2ie := xmedie :=

ymed2 = 0.333

bnie , j := Ynie , ( lj , 1) − Ynie , ( lj , 2) anie , j := Xnie , ( lj , 1) ⋅Ynie , ( lj , 2) − Xnie , ( lj , 2) ⋅Ynie , ( lj , 1)

2

Xnie , 3 + 2 ⋅x1_2ie

xmed2 = 0.333

(7.5 )

cnie , j := Xnie , ( lj , 2) − Xnie , ( lj , 1)

Xnie , 1 + Xnie , 2

. (7.5 )

1+ 2

1

0

0.5

0

0.5

1

1.5

. (7.5 )

⎛ 1 Xnie, 1 Ynie , 1 ⎜ Δ2nie := ⎜ 1 Xnie , 2 Ynie , 2 ⎜ 1 Xn ie , 3 Ynie , 3 ⎝

⎞ ⎟ ⎟ ⎟ ⎠ -

. (7.12)

⎛ bnie , j 0 ⎜ Bnie , j := ⋅⎜ 0 cnie , j Δ2nie ⎜ ⎝ cnie , j bnie , j 1

(

(

⎞ ⎟ ⎟ ⎟ ⎠

BBn( ie) := stack Bnie , 1 , Bnie , 2 , Bnie , 3 T

T

K_ie := BBn( ie) ⋅D_N ⋅BBn( ie) ⋅t⋅ T

ii := 1 .. 2 ⋅nn_point

T

))T

Δ2nie

-

. (7.13)

2

jj := 1 .. 2 ⋅nn_point

(7.14)

K( M_ ie , ii , M_ ie , jj) := K( M_ ie , ii , M_ ie , jj) + ( K_ie) ii, jj

Kii, jj := 0

A := K NN

⎛ 1 ⎞ ⎟ ⎝ 4 ⎠

NN := ⎜

n := 1 .. 2

k := 1 .. cols( Nodes ) ⋅2

Ak , 2⋅ NN n−1 := 0

A2⋅ NN n−1 , 2⋅ NN n−1 := ∞ Ak , 2⋅ NN n := 0

A NN n⋅ 2 , NN n⋅ 2 := ∞

⎛ ⎜ Uzl := ⎜ ⎜ ⎜ ⎝

2 ⎞



1 ⎟ 2 ⎟



1 ⎠

p := 1 .. cols( Nodes ) P2⋅ p −1 := P2⋅ p :=

-

. (7.15)

Δ2nie 2 − q_x ⋅ ⋅ ⋅Uzl( Top ie , j) cols( Top) 2 L

Δ2nie 2 − q_y ⋅ ⋅ ⋅Uzl( Top ie , j) cols( Top) 2 L

U := − A

−1

⋅P

U_xp := Up ⋅ 2−1

U_yp := Up ⋅ 2 uie_xie :=



⋅⎢

Δ2nie ⎢ 1



-

3

⎣ j =1 ⎡ 3 1 ⎢ uie_yie := ⋅ Δ2nie ⎢ ⎣ j =1



. (7.6)



( anie , j + bnie , j⋅xmedie + cnie , j⋅ymedie) ⋅U_x(Top ie , j) ⎥⎥

⎦ ⎤ ( anie , j + bnie , j⋅xmedie + cnie , j⋅ymedie) ⋅U_y(Top ie , j) ⎥⎥ ⎦ . mno := 1000

⎛ xdie , j ⎞ ⎢⎡ Nodes 1 , ( Top ie , j) + (mno⋅U_xTop ie , j) ⎜ ⎟ := ⎝ ydie , j ⎠ ⎢⎣ Nodes 2 , ( Top ie , j) + ⎡⎣( mno⋅U_y) Top ie , j⎤⎦

⎤ ⎥ ⎥ ⎦

1

0

1

0.5

0

ΔLmax = 5 × 10

0.5

−4

uie_x1 − uie_x2 = 2.266 × 10

1

−4

1.5

2

s := 1 .. max( Top)

⎡(

x_s := ⎣ Nodes

⎡(

xs := ⎣ Nodes

T

T

)

)

X δ_u2 := δ_u3 :=

⎡(

〈1〉 ⎤

⎦ s + U_xs

ys := ⎣ Nodes

⎡(

〈1〉 ⎤

⎦s

2

y_s := ⎣ Nodes

3

( x_2 − x2) − ΔLmax ΔLmax ( x_3 − x3) − ΔLmax ΔLmax

δ_u2 = 35.943 %

δ_u3 = 17.941 %

1

0

0

0.5

T

1

1.5

2

) T

)

〈2〉 ⎤

⎦s

〈2〉 ⎤

⎦ s + U_ys

σ

UUii, ie := U( M_ ie , ii )

εε

σσ

〈ie〉

ε

:= BBn( ie) ⋅UU

〈ie〉

:= D_N ⋅εε

〈ie〉

〈ie〉

1

σσ

εε

〈1〉

〈1〉

2

⎛ 1.371 × 108 ⎜ = ⎜ 3.728 × 106 ⎜ 6 ⎝ −3.728 × 10

⎞ ⎟ 〈2〉 ⎟ σσ ⎟ ⎠

⎛ 1.296 × 108 ⎜ = ⎜ 3.888 × 107 ⎜ 6 ⎝ 3.728 × 10

⎛ 6.797 × 10− 4 ⎜ = ⎜ −1.87 × 10− 4 ⎜ −5 ⎝ −4.847 × 10

⎞ ⎟ 〈2〉 ⎟ εε ⎟ ⎠

⎛ 5.897 × 10− 4 ⎜ 0 =⎜ ⎜ −5 ⎝ 4.847 × 10

σ_x = 1 × 10 ΔLmax L

1. . 2. , 1985. – 240 3.

. . ., . . .,

−4

. .,

.: .

. . . .

σy

σxy

⎞ ⎟ ⎟ ⎟ ⎠

εx

εy

εxy

εx

.– . .,

σx

σx

8

= 5 × 10

⎞ ⎟ ⎟ ⎟ ⎠

.

, 1954. – 288 . . . I: . . II:

.– .–

.:

.:

, 1985. –

496 . 4. ё . . . . – .: , 1994. – 268 . 5. . ., . . . – .: , 2001. – 267 . 6. . . MathCAD PLUS 6.0 PRO. – .: , – 1997 7. . . . . – .: , 1975. – 541 . 8. . MathCAD 2001. – .: , 2001. – 544 . 9. . ., . . . – .: . , – 1991 10. . . Excel: . – .: , 2002. – 336 . 11. . ., . . . – .: . . , 1996. – 368 . 12. . . . . 1. – .: , 1985. – 432 13. . . . . 2. – .: , 1985. – 560 14. . . .– : . ., 1988. – 367 . 15. . . Microsoft Excel .– .: , 2005. – 368 . 16. . . . – .: . ., 1982. – 264 . 17. . . . – .: . ., 1988. – 367 . 18. . . : . . – .: , 1987. – 320 . 19. . . . . – .: - , 1977. – 120 .

. . .

.