Svolosl Nonlinear

Duffing Van der Pol

: ...

2014

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Duffing

Van der Pol.

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Abstract

The seismic isolation is part of the broader scientific area of passive control of

seismic response of structures. After the use of seismic isolation for buildings and

bridges, it is applied to protect invaluable objects as museum artifacts. In this

diploma thesis a seismic isolator, which was designed for the previously mentioned

purpose and has nonlinear behavior due to mechanical and geometrical

characteristics, is examined. Formulating the equation of motion for the Single

Degree Of Freedom (SDOF) oscillator isolator, we observe that the non-linearity is

mainly due to Duffings type of stiffness and Van der Pols type of damping. In order

to solve the aforementioned nonlinear problems, the multiple time scales method

and the averaging method, which are analytical techniques for approximating

solutions, were presented. The latter methods were used in order to highlight the

effects of nonlinear systems which were ignored in the linear theory of vibrations

and were concealed by the use of numerical methods. An example of these

phenomena is the secondary resonances which are critical in design procedure. Their

criticality based on existing of jump phenomena in amplitude of response of

isolator. In this thesis, the region of frequencies where resonances may be

developed is examined using the frequency - response curves. The study reveals the

influence of the mechanical characteristics of isolator on the response to the

resonances (primary and secondary). As a result of all observations, we came to

useful conclusions concerning the design of the isolator in order for it to operate

both economically and safely.

1

1. ...................................................................................................... 1

1.1. ...................................................................................... 1

1.2. & ................................................... 2

1.3. ...................................................................... 3

1.3.1. ( ) ....................................................... 3

1.3.2. ................................................................. 5

1.4. .................................................. 6

1.4.1. ....................................................................... 6

1.4.2. ................................................. 6

1.4.3. () ......................................................... 9

1.4.4. ............................................................. 10

2

2. ......................... 11

2.1. ....................................................... 11

2.2. Duffing................................................................................... 12

2.2.1. & ........................................ 12

2.2.2. ........................................... 13

2.2.3. ......................................................................... 15

2.2.4. ............................................................ 17

2.2.5. ..................................................................... 18

2.3. Van der Pol ............................................................................ 21

2.3.1. & ........................................ 21

2.3.2. ........................................... 22

2.3.3. ......................................................................... 24

2.3.4. ............................................................ 26

2.3.5. ..................................................................... 27

3

3. ............................ 29

3.1. .......................................................................................... 29

3.2. ...................................................... 30

3.3. ........................................................................... 32

3.4. (The Straightforward Expansion) ......................... 34

3.5. (Multiple Time Scales Method) ...... 39

3.6. & (Averaging Method) ...... 50

4

4. ........................................................................................ 61

4.1. ............................................................................................. 61

4.1.1. ................... 61

4.1.2. ............................ 62

4.2. .................................................... 65

4.2.1. ................................................................... 65

4.2.2. ......................................... 69

4.3. .................................................... 72

4.3.1. ................................................ 72

4.3.2. ................................................................. 73

4.3.3. ................................................................. 76

4.3.4. .................................... 77

4.4. ......................................................... 78

5

5. ........................................... 87

5.1. ............................................................................................. 87

5.2. ............................. 88

5.2.1. ..................................... 88

5.2.2. ....................................................... 90

5.3. ..................................................... 93

5.3.1. ............................................................ 93

5.3.2. Duffing ............................................................................ 95

5.3.3. Duffing & Van der Pol ................................................................. 119

5.3.4. ....................................................... 141

5.3.5. ........................................ 146

6

6. ......................................................................................... 149

6.1. .......................................................... 149

6.2. ............................................................ 149

6.3. .................................... 150

6.3.1. .................................................................... 151

6.3.2. ....................................................... 153

6.4. ...................................... 158

6.5. ...................................................... 161

.................................................................................................. 163

..................................................................................................... 165

1

1.

1.1.

, ,

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, .

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(. 1.1).

. 1.1 :

1

2

(t)u

(t) . .

(t)u

(t) . m

,

( )u t ().

1.2. &

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() .

. :

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()

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() (,

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1

3

(. 1.2) .

. 1.2 :

1.3.

1.3.1. ( )

. (. 1.3)

, , .

,

( ) .

. 1.3 :

( )u t t

,

:

1. ( )p t

2. Sf

3. Df

4. If

1

4

Sf u

( )S Sf f u .

() Sf

, Hooke :

Sf ku (1.1)

k .

Sf ,

.

Df , .

.

, :

Df cu (1.2)

c .

.

, .

.

If . m

u . :

If mu (1.3)

,

() .

D Alembert.

: D Alembert

,

.

1

5

( )

:

( ) ( ) ( ) ( )I D Sf t f t f t p t (1.4)

1.3.2.

, (1.1) , (1.2) (1.3),

() :

( )mu cu ku p t (1.5)

. 1.4 :

(1.5)

. ,

.

.

. 1

( ) 0p t

0mu ku (1.6)

0

k

m

0mu cu ku (1.7)

- : 02crc m

- : crc c

- : crc c

cr

c

c

0( ) sin( )p t p t

0 sin( )mu ku p t (1.8)

0

1

D

1 : D

t

0 sin( )mu cu ku p t (1.9)

D

max2

1

2 1D

21 2

1

6

1.4.

1.4.1.

,

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(1.6) , (1.7) , (1.8) (1.9)

.

( ) ,

, .

, :

(-) .

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1.4.2.

,

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() .

.

1

7

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(1 cm) ;

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.

1 : ( (1.7) )

2

20

d u dum c kudt dt

(1.10)

0u

:

00

(0) , (0) 0t

duu u u

dt (1.11)

, u

t .

( ,U T ).

U

T , u

uU

tt

T , u t

. t :

1d d dt d

dt dt dt T dt (1.12)

2

2 2 2

1 1 1d d d d d d d dt d

dt dt dt dt T dt dt T dt dt T dt

(1.13)

(1.10) :

2 2

2 2 2 2

1 10 0

d Uu d Uu mU d u cU dum c k Uu kU uT dt T dt T dt T dt

1

8

2

2

20

d u c du kT T u

dt m dt m

(1.14)

2

2 2

020

d u c duT T u

dt m dt

(1.15)

T U 2 20 1T 0U u

(1.15) :

2

2

0

2 0 2 cr

d u du c cu

dt dt m c

(1.16)

:

0

0

(0) 1 , (0) 0t

u duu u

U dt (1.17)

2 : ( (1.9) )

2

02sin( )

d u dum c ku p tdt dt

(1.18)

, ,

:

u t

u tU T

(1.19)

U T .

(1.12) (1.13), (1.18) :

2

02 2sin( )

U d u U dum c kU u p TtT dt T dt

(1.20)

2

2 2 2 002

sin( )pd u c du

T T u T Ttdt m dt mU

(1.21)

,

0 . , ,

1 secT ,

.

1

9

(1.21) :

2

2 002

2 sin( ) 2

pd u du cu F t F

dt dt m mU (1.22)

, u t ( (1.17)).

u t

( (1.16) (1.22) ). u t

u t , .

:

u t

u tU T

(1.23)

2

2 2 2

1 1

d d d d

dt T dt dt T dt (1.24)

. 2

. (

) ( 1 2 ).

:

T

0

1T

(1.25)

1 secT (1.26)

2

20

d uu

dt (1.27)

22

02sin( )

d uu F t

dt (1.28)

.

0 u u (1.29)

2

0 sin( )u u F t (1.30)

. 3

1.4.3. ()

.

(1.29) (1.30)

.

.

1

10

,

( )

.

,

(, , )

c crc (. 1).

( )p t ,

Duhamel

Laplace. :

(t )0

(0) (0)( ) sin (0)cos

1 ( ) e sin

t

D D

D

t

D

D

u uu t t u t e

p t dm

(1.31)

(1.31) Duhamel

.

,

1.

1.4.4.

.

( )p t Duhamel

(1.31). .

,

.

. :

(Newmark)

(Runge-Kutta)

-

2

2.

2.1.

,

D Alembert,

(1.4).

, ,I D Sf f f

(1.1) , (1.2) (1.3). :

( )If m t u (2.1)

( , , )D Df f u u t (2.2)

( , , )S Sf f u u t (2.3)

( , , ) , ( , , )D Sf u u t f u u t

u u .

:

( ) ( , , ) ( , , ) ( )D Sm t u f u u t f u u t p t (2.4)

.

(2.4) , .

(. )

.

Df Sf u , u t ,

.

, ,

, ,

.

:

: , :

, : g g

x u x u

x x x x

2

12

2.2. Duffing

2.2.1. &

Georg Duffing (1861-1944) ,

.

, ,

.

( Hooke), Duffing

[5]

.

.

Duffing.

Duffing :

3 cosx x x x t (2.5)

( )x x t

t , , ,

. , :

(2.5) .

.

:

3( , , )S Sf f x x t x x (2.6)

/ .

, (

) (2.6)

. ,

.

2

13

2.2.2.

Duffing (2.5),

() .

, . 2.1, m

k .

.

.

d M

0d 0d d

k (Hooke)

. 4

F F t u u t

.

( sF ) (F ).

. 2.1 : &

: F ma (2.7)

x (. 2.1):

2x sxF F F 2

2x

d uma m mu

dt (2.8)

sxF sF x .

2

14

: 2 2u d (2.9)

, sF ( Hooke)

: sF k 0d

0( )sF k d (2.10)

, (. 2.1) :

00sin ( d ) 1

sxsx s

s

F du u uF F k ku

F

02 2

1sxd

F kuu d

(2.11)

(2.11)

02 2

2 2 1S sxd

f F kuu d

(2.12)

,

(2.6). ,

(2.12)

Duffing.

Maclaurin [] :

22

1 11

21x

x

(2.13)

2

20 0 0

32 2 2 2

1 1 1

2 2

d d du d ux u

d d d du d u d

(2.14)

(2.11) (2.14) sxF :

30 0

31

2sx

d dF k u k u

d d

(2.15)

(2.15) (2.11) 0, 4u d .

(2.8):

30 0

32 2 1x sx x

d dF F F F k u k u ma mu

d d

2

15

:

3

1 2 ( )mu k u k u F t (2.16)

01 2 1d

k kd

02 3

dk k

d 0 u 0 0u

, Duffing (2.16)

.

0F t

(2.16) :

2

3

1 220

d um k u k udt

(2.17)

( )

0( 0)u t u 0u( 0)t v .

(2.16) .

Duffing

.

2.2.3.

(. 1.4.2)

Duffing.

1:

23

1 220

d u dum c k u k udt dt

(2.18)

0( 0)u t u (2.19) 0u( 0)t v (2.20)

. 5

(2.18) (1.23) ,

(1.24) :

23

1 22 2

1 10

d Uu d Uum c k Uu k UuT dt T dt

2

16

2 2 22

31 2

20

T k T k Ud u Tc duu u

dt m dt m m

(2.21)

2

2 2 2 2 2 2 320 0 02

1 1

0kd u c du

T T u U udt k dt k

(2.22)

(1.25) :

2

3

22 0

d u duu u

dt dt (2.23)

22

1

kU

k 10 2

1 0 2 0

12

2 2

kc c c

k m k U m

(2.19) (2.20)

,u t (1.23) :

0( 0)

( 0)uu t

u tU U

(2.24)

0( )

( 0) ( 0)Tvdu dt d Uu T du T T

u u t u tdt dt dt U dt U U U

(2.25)

0U u . :

0

0 0

( 0) 1 ( 0)v

u t u tu

(2.26)

2:

23

1 22cos

d u dum c k u k u P tdt dt

(2.27)

0( 0)u t u (2.28) 0u( 0)t v (2.29)

. 6

(2.27) (1.23) ,

(1.24) :

23

1 22 2

1 1cos

d Uu d Uum c k Uu k Uu P TtT dt T dt

2 2 22 2

31 2

2cos

T k T k Ud u Tc du T Pu u Tt

dt m dt m m mU

(2.30)

2

17

2 2

2 2 2 2 2 2 320 0 02

1 1

coskd u c du T P

T T u U u Ttdt k dt k mU

(2.31)

(1.26) :

2

2 3

022 cos

d u duu u F t

dt dt (2.32)

2 2201

kU

k 20 2

1 2

22 2

c c c

k m k U

PF

mU

(2.28) (2.29)

,u t (1.23). (2.24) (2.25) :

0( 0)u

u tU

(2.33) 0 0( 0)Tv v

u tU U

(2.34)

2.2.4.

Duffing , .

23

20

d uu u

dt (2.35)

23

22 0

d u duu u

dt dt (2.36)

( )

2

2 3

02cos

d uu u F t

dt

(2.37)

2

2 3

022 cos

d u duu u F t

dt dt

(2.38)

P(t) :

2

3

22

d u duu u P t

dt dt (2.39)

. 7

: ,

.

. :

0( 0)u

u tU

(2.40) 0( 0)Tv

u tU

(2.41)

2

18

2.2.5.

Duffing,

. Runge-Kutta 4

(2.36).

:

32 0u u u u (2.42)

(2.42) , u v (2.43)

32

u v

v v u u

(2.44)

( ), ( )u t v t . ( )u t

( )v t .

( )u t ( ), ( )u t v t .

uv

(2.44)

. ,

uv. ,

, .

.

.

. 0

. ,

.

. . 2.2

( )u t .

0 , 0.1 , 2 0 .

2

19

0 , 0 0 00.2 , 0u v

0.1 , 0 0 3

2 , 0 0.2 , 1/ 3U T

. 2.2 : 1 Duffing

0

uv :

0 0cos( )u A t (2.45) 0 0 0sin( )v A t (2.46)

2 2

2 2 2

0

1u v

A A (2.47)

,

.

. 2.3 .

2 (2.36),

(. 2.4)

2 . ,

( )

.

2

20

0.1 , 0.1 0 00.2 , 0u v

0.1 , 1.0 0 3

0.2 , 1/ 3U T

. 2.3 : 2 Duffing

2 , 0.05 0 00.2 , 0u v

0.1 , 1.0 0 3

2 0.2 0.2 , 1/ 3U T

. 2.4 : 3 Duffing

2

21

2.3. Van der Pol

2.3.1. &

Balthasar van der Pol (1889-1959),

,

. ,

,

( )

( Hooke).

, , -

. Van der Pol,

.

Van der Pol :

2(1 ) 0x x x x (2.48)

( )x x t

t

.

(2.48)

.

. :

2( , , ) ( 1)D Df f x x t x x (2.49)

/ .

(2.49) 1x

1x . ,

( )

( ).

- ,

. .

,

.

2

22

(2.48) .

,

.

2.3.2.

Van der Pol (2.48)

,

.

, . 2.5, m

k .

0x

. 1c

.

s 2( )z x x . . 2.5 .

.

k (Hooke)

1c

( )z x

. 8

. 2.5 :

2

23

F F t cx

x x t .

( sF ) ,

(F ) ( cF )

: F ma (2.50)

x (. 2.1):

x s cF F F F 2

2x

d xma m mx

dt (2.51)

cF dF z .

1c dF :

1 1 1 1( ) ( ) 2d dd d dx

F c z x c z x c z x F c x xdt dx dt

(2.52)

o dF cF (. 2.5) :

2 21 1tan 2 2 4c

c d c

d

Fz F z F x c x x F c x x

F (2.53)

. sF

( Hooke) :

sF kx (2.54)

F c x

van der Pol (2.48)

.

(2.51):

2 2

14x s c xF F F F c x kx c x x ma mx

:

22 0mx c x c x kx (2.55) 22 14c c 0 0 0x x

2

24

, Van der Pol (2.55)

.

F .

:

22 ( )mx c x c x kx F t (2.56)

( )F t F .

1. 0F t

(2.56) :

2

2

220

d x dxm c x c kxdt dt

(2.57)

2. 0F t

(2.56) :

2

2

22( )

d x dxm c x c kx F tdt dt

(2.58)

( )

0( 0)x t x 0( 0)x t v .

(2.56) .

Van der Pol

.

2.3.3.

(. 1.4.2.)

Van der Pol.

1:

22

220

d u dum c u c kudt dt

(2.59)

0( 0)u t u (2.60) 0u( 0)t v (2.61)

. 9

2

25

(2.59) (1.23) ,

(1.24) :

2

2

22 2

1 1( ) 0

d Uu d Uum c Uu c k UuT dt T dt

22 2

22

20

Tc Ud u Tc du T ku u

dt m m dt m

(2.62)

2

2 2 2 2 2 220 0 02

0cd u c du

T U u T T udt k k dt

(2.63)

(1.25) :

2

2

21 0

d u duu u

dt dt (2.64)

0 1c

k (2.65) 22c c U (2.66).

(2.60) (2.61)

,u t (1.23) :

0( 0)

( 0)uu t

u tU U

(2.67)

0( )

( 0) ( 0)Tvdu dt d Uu T du T T

u u t u tdt dt dt U dt U U U

(2.68)

0U u . :

0

0 0

( 0) 1 ( 0)v

u t u tu

(2.69)

2:

22

22cos

d u dum c u c ku P tdt dt

(2.70)

0( 0)u t u (2.71) 0u( 0)t v (2.72)

. 10

(2.70) (1.23) ,

(1.24) :

2

2

22 2

1 1( ) cos

d Uu d Uum c Uu c k Uu P TtT dt T dt

2

26

22 2 2

22

2cos

Tc Ud u Tc du T k T Pu u Tt

dt m m dt m mU

(2.73)

2 2

2 2 2 2 2 220 0 02

coscd u c du T P

T U u T T u Ttdt k k dt mU

(2.74)

(1.26) :

2

2 2

021 cos

d u duu u F t

dt dt (2.75)

20 1c

k (2.76) 22c c U (2.77)

PF

mU (2.78)

(2.71) (2.72)

,u t (1.23). (2.67) (2.68) :

0( 0)u

u tU

(2.79) 0 0( 0)Tv v

u tU U

(2.80)

2.3.4.

Van der Pol , .

( )

2

2

21 0

d u duu u

dt dt (2.81)

22 2

021 cos

d u duu u F t

dt dt (2.82)

P(t) :

2

2

21 0

d u duu u P t

dt dt (2.83)

. 11

:

.

. :

0( 0)u

u tU

(2.84) 0( 0)Tv

u tU

(2.85)

2

27

2.3.5.

Van der Pol,

. Runge-Kutta 4

.

:

2 1 0u u u u (2.86)

(2.42) , u v (2.87)

2(1 )

u v

v u v u

(2.88)

( ), ( )u t v t . ( )u t

( )v t .

( )u t ( ), ( )u t v t .

.

.

. 0

. ,

.

. . 2.2

( )u t .

0 , 0.1 2 .

, .

, ( 0 0.05 mu 0 0.6 mu )

. 2.7.

.

2

28

0 0 00.2 , 0u v

0.1 0 3

2 0.2 , 1/ 3U T

. 2.6 : 1 Van der Pol

0.4

0 00.05 , 0u v 0 3

0 00.6 , 0u v 0.2 , 1/ 3U T

. 2.7 : 2 Van der Pol

3

3.

3.1.

.

,

, .

. ,

.

,

.

(

) .

,

( ).

(. )

(. )

.

3

30

3.2.

,

. .

Landaus Notations

( )f ( )g 0 .

1. : ( ) ( )f g 0

0

( )lim 0

( )

f

g

f g 0

2. : ( ) ( )f g 0

M

( ) g( )f M

0 .

f g 0

1. .

, ( )g

g .

2.

. , ( )g

g .

3. g - (gauge function).

( ) , ng n .

4. gf f

g 0 .

5. , 0

0 0 , 0 .

,f g .

6. n ( )f x

1nx 1nx 1x .

3

31

,

, ( , )f t

.

( , )h t 0 t

I . :

0

lim (t, ) 0h

I

0 0 0 t :

( , )h t t I 0

( , )h t ,

, t I .

Landau

( , )f t ( , )g t t I

0 . ( , ) ( , )f t g t

0 , I 0

( , )lim 0

( , )

f t

g t

I . , ( , ) ( , )f t g t

0 , I

( ) 0M t t I ( , ) ( ) ( , )f t M t g t t I

0 .

.

( , )au t

( , )u t I 0

( , ) ( , ) ( , )aE t u t u t 0 ,

t I .

( , )E t n

n n 0 ( ).

3

32

-

( , )ng t (gauge functions)

0 t I 1( , ) ( , )n ng t g t 0 n .

( , )u t 0 ,

( , )ng t , 0

( , )n nn

g t

:

0

( , ) ( , ) ( , )N

n n N

n

u t g t g t

0 N .

1. t I

.

2. : ( , ) ( ) ( ) ( ) nn n n ng t y t y t

( , )u t :

0 0 0

( , ) ( ) ( )n nn n n n nn n n

g t y t u t

(3.1)

3. : 0

( ) nnn

u u t

(3.2)

4. (3.1)

.

, ,

(

Taylor ).

5.

(Straightforward Expansion , . 3.4.) :

1

0

( , ) ( ) ( )m

n m

n

n

u t u t

, m (3.3)

3.3.

( )

.

3

33

.

.

.

( )

( ).

0 ,

.

()

(perturbation theory).

Duffing Van der Pol.

:

, , , ; 0F t u u u (3.4)

, 1

( , )au t (3.4)

t I 0 :

( , ) , ( , ), ( , ), ( , ), 0a a ar t F t u t u t u t (3.5)

I 0 .

:

1. (The Straightforward Expansion)

2. (Multiple Time Scales Method)

3. (Averaging Method)

:

1. Lindstedt-Poincar

2. (Boundary Layers)

3. (Matching)

3

34

3.4. (The Straightforward Expansion)

.

,

.

.

(3.4) :

20 1 2; ( ) ( ) ( ) ( )n

nu t u t u t u t u t (3.6)

( ) 1n nu t n .

.

( ; )u t (3.6)

, (3.4), .

(n ) .

,

, .

: 20 1 2( ) ( ) ( ) ( ) 0 ( ) 0 ,

n

i i i n i n if u f u f u f u f u n (3.7)

n :

iu : iu

.

n ( ) 0n if u , ( )iu t .

.

Duffing (2.35).

0t 0A

( 0 0(0)u x A U ). ( )u t

.

3

35

,

t (3.6).

.

( ) :

2

0 1( ; ) ( ) ( ) ( )u t u t u t (3.8)

(3.8) (2.35) :

32 2 2

0 1 0 1 0 1( ) ( ) ( ) 0u u u u u u (3.9)

:

3 2 32 3 2 2 2 2

0 1 0 0 1 0 1 1( ) 3 ( ) 3 ( ) ( )u u u u u u u u

3 2 20 0 13 ( )u u u (3.10)

(3.10) (3.9)

, :

3 20 0 1 1 0 ( ) 0u u u u u (3.11)

2 .

(3.7) :

0 03 20 0 1 1 0 31 1 0

0( ) 0

0

u uu u u u u

u u u

(3.12)

0 0 0u u (3.13) 3

1 1 0u u u (3.14)

(3.13) 0 0 0cosu t (3.15)

0 0, .

0u (3.15) (3.14) :

3 31 1 0 0cosu u t (3.16)

- []:

33 1

cos ( ) cos( ) cos(3 )4 4

(3.17)

3 0 0 03 1

cos ( ) cos(t ) cos(3 3 )4 4

t t (3.18)

3

36

(3.16) :

3 31 1 0 0 0 03 1

cos(t ) cos(3 3 )4 4

u u t (3.19)

- .

1,0u

- 1,pu , 1 1,0 1,pu u u (3.20).

( (3.13) ):

1,0 1 1cosu t (3.21) , 1 1,

: (3.19) , . .

31 1 0 03

cos(t )4

u u (3.22)

31 1 0 01

cos(3 3 )4

u u t (3.23)

(3.22) : 31, 1 0 03

sin( )8

pu t t (3.24)

(3.23) : 31, 2 0 01

cos(3 3 )32

pu t (3.25)

H , , :

3 31, 1, 1 1, 2 0 0 0 03 1

sin( ) cos(3 3 )8 32

p p pu u u t t t (3.26)

(3.19) :

3 31 1,0 1, 1 1 0 0 0 03 1

cos sin( ) cos(3 3 )8 32

pu u u t t t t (3.27)

0 1,u u (3.15) (3.27)

(3.8) :

30 0 1 1 0 03

( ; ) cos cos sin( )8

u t t t t t

3 2

0 0

1cos(3 3 ) ( )

32t

(3.28)

0 0 1 1, , , :

30 0 0 0 1 1 0 01

(0) cos( ) cos cos(3 )32

u x x

3

37

0 0 0

3

1 1 0 0

cos( )

1cos cos(3 ) 0

32

x

(3.29)

3 30 0 0 0 1 1 0 0 0 03 3

(0) 0 sin( ) sin sin( ) sin(3 )8 32

u x x

0 0 0

3 3

1 1 0 0 0 0

sin( ) 0

3 3sin sin( ) sin(3 ) 0

8 32

x

(3.30)

(3.29) (3.30) :

3

00 0 0 1 1 , 0 ,

32

xx (3.31)

:

3

200( ; ) cos cos 12 sin( ) cos(3 ) ( )

32

xu t x t t t t t (3.32)

Duffing

(2.17) . 2000k kN m

00,4 m , 0,36 md d (2.16)

01 2 1 400 d

k k kN md

302 3 11250

dk k kN m

d .

0 0,05 mA .

0,05 mU

00 1

Ax

U 22

1

0,07 1kU

k (3.32) :

7

( ) cos cos 12 sin( ) cos(3 )3200

u t t t t t t (3.33)

3

38

0 (t)u

1(t)u .

0 (t)u 0 1(t) ( )u u t .

t ,

. 0 (t)u ,

. ,

sin( )t t t .

sin( )t t (mixed-secular terms)

.

. ,

,

t . .

.

.

.

.

Duffing 0(0)u x :

203

2 18

T x

(3.34)

( ) ,

.

(secular terms)

I .

,

, Lindstedt-Poincare.

( ) .

3

39

3.5. (Multiple Time Scales Method)

(3.6)

.

.

iu . , , u t

(3.6) u 2 3 , t , , , , t t t . , :

2 3( ; ) ( , , , , ; )u t u t t t t (3.35)

0 1 2 3( ; ) (T ,T ,T , , ; )u t u T (3.36)

, nnT t n (3.37)

nT

. 0T t

, 1T

0T .

1T ,

. , 1T

0T .

nT . ,

.

u t

. u

, u 0 1 2 3T ,T ,T , ,T

(3.36).

:

20 1 2

0 1 2 0 1 2

dT dT dTd

dt T dt T dt T dt T T T

(3.38)

2 2 2 2 2

2

2 2 2

0 0 1 0 2 1

2 2d

dt T T T T T T

(3.39)

: i

i

DT

(3.38) (3.39) :

3

40

20 1 2d

D D Ddt

(3.40)

2

2 2 2

0 0 1 0 2 122 2

dD D D D D D

dt (3.41)

(3.40) (3.41) ,

u

0 1 2 3T ,T ,T , ,T .

,

.

.

,

.

:

0 0 1 2 1 0 1 2 (T ,T ,T , ) (T ,T ,T , )u u u (3.42)

u (3.42)

(3.7)

.

0u 1u .

()

( secular terms).

.

:

2

3

20

d uu u

dt (3.43)

Duffing (2.35).

0t 0A

( 0 0(0)u x A U ). ( )u t

(Multiple Scales).

3

41

,

0T 1T (3.42).

.

( ) :

2

0 0 1 1 0 1( , ) ( , ) ( )u u T T u T T (3.44)

(3.40) (3.41) :

0 1d

D Ddt

(3.45) 2

2

0 0 122

dD D D

dt (3.46)

(3.44),(3.45) (3.46) (3.43) :

2 2

0 0 1 0 1

32 2

0 1 0 1

2 ( )

( ) ( ) 0

D D D u u

u u u u

(3.47)

(3.10) :

2 2 2 3 20 0 0 1 0 1 0 0 1 02 ( ) ( ) 0D u D u D Du u u u (3.48)

2 2 3 20 0 0 0 1 1 0 0 1 02 ( ) 0D u u D u u u D Du (3.49)

(3.7) :

20 0 0 0D u u (3.50) 2 3

0 1 1 0 0 1 02D u u u D Du (3.51)

2

002

0

0u

uT

(3.52)

223 01

1 02

0 0 1

2uu

u uT T T

(3.53)

(3.50) (3.52) :

0 1 0 1( ) cos ( )u T T T (3.54)

1 :

0u (3.54) (3.51) (3.53) :

2 2

311 0 02

0 0 1

cos( ) 2 cos( )u

u T TT T T

(3.55)

1T .

2 3, ,T T

3

42

231

1 0 02

0 1 1

3

0

32 sin( ) 2 cos( )

4

1 cos(3 3 )

4

u d du T T

T dT dT

T

(3.56)

1

d

dT

1T

2 3, ,T T .

0sin( )T 0cos( )T (3.56),

, (secular terms).

t . :

1

0d

dT

(3.57) 3

1

32 0

4

d

dT

(3.58)

0 (3.59) 2

0 1 0

3

8T (3.60)

(3.56) (3.57) (3.58) :

2

311 02

0

1cos(3 3 )

4

uu T

T

(3.61)

: 31 01

cos(3 3 )32

u T (3.62)

(3.43) :

2 3 20 0 0 1 0 0 0 0 1 03 1 9

cos cos(3 )8 32 8

u T T T T

(3.63)

2 3 20 0 0 0 0 03 1 9

(t) cos cos(3 )8 32 8

u t t t t

(3.64)

0 0 .

(3.64) :

20 0 03

(t) cos8

u t t

(3.65)

3

43

2 :

. (3.54) :

:

1

cos2

i ie e (3.66)

: 0 0 0 0( ) ( )

0

1 1 1( )

2 2 2

i T i T iT iTi iu e e e e e e (3.67)

: 1 1 11

( ) ( )exp ( )2

A A T T i T 0 0exp( )e iT (3.68)

(3.54) :

0 0 0u Ae Ae (3.69)

cc - complex conjugate

z : z z z cc (3.70)

z , .

0u (3.69) (3.51) (3.53) :

2 2

31

1 0 0 0 02

0 0 1

2u

u Ae Ae Ae AeT T T

(3.71)

2

2 3 311 0 02

0 1

2 3u A

u i A A e A e ccT T

(3.72)

(3.44)

A 1T , (3.72)

1

dA

dT .

.

0e 0e , (3.72),

.

t ,

, . , , :

2

1

2 3 0A

i A AT

(3.73)

3

44

A (3.68) :

2

2

1 1

1 12 3 0

2 2 4 2

i i i ii e ie e eT T

(3.74)

3

1 1

30

8iT T

(3.75)

, (3.75) :

1

0T

(3.76) 3

1

30

8 T

(3.77)

0

2 3, ,T T :

2 3 0( , , )T T (3.78) 2 2

1 2 3 0 1 0

3 3( , , )

8 8T T T T (3.79)

(3.59) (3.60).

, (3.65).

( ) cc

.

, :

( )

.

- Duffing

(3.65) Duffing (3.43)

(Multiple Scales).

. :

20 0 03

(t) cos8

u t t

(3.80)

3

45

(1.23) (1.25) :

20 0 0 0 03

(t) cos8

u U t t

(3.81)

0 0u 0 0v 0 0 :

00u

U (3.82) 0 0 (3.83)

(3.81) , , :

2

0 00 0 2

3(t) cos

8

uu u t t

U

(3.84)

(3.84) Duffing. ,

Runge-Kutta

(3.84) (3.43).

0.2 , 1/ 3U T .

0 00.2 , 0u v

Runge-Kutta 0.1

Multiple Scales 0 3

. 3.1 : Duffing 1

3

46

0 00.2 , 0u v

Runge-Kutta 0.4

Multiple Scales 0 3

. 3.2 : Duffing 2

0 00.2 , 0u v

Runge-Kutta 2

Multiple Scales 0 3

. 3.3 : Duffing 3

3

47

Duffing (2.36) [ ] :

32 0u u u u (3.85)

.

(3.51) :

2 30 1 1 0 0 1 0 0 02 2D u u u D Du D u (3.86)

0u (3.69),

(3.73) :

22 3 2 0iA A A i A (3.87)

A (3.68)

:

1

d

dT

(3.88) 2

1

3

8

d

dT

(3.89)

u (3.54) :

11

Tc e (3.90)

1221 23

216

Tc c e

(3.91)

1 1221 0 1 23

cos 216

T Tu c e T c c e

(3.92)

u , t 1.4.2 :

0 0221 0 1 23

( ) cos 216

t tu t Uc e t c c e

(3.93)

0 0u 0 0v :

0 02

200 0 2

3(t) cos 1

16

t tuu u e t e

U

(3.94)

(3.94)

. : 0 3 , 0.2 , 1/ 3U T .

3

48

0 00.2 , 0u v

Runge-Kutta , 0.1 , 0.5

Multiple Scales 2 0.1

. 3.4 : Duffing 1

0 00.2 , 0u v

Runge-Kutta , 0.1 , 2


. 3.5 : Duffing 2

3

49

0 00.2 , 0u v

Runge-Kutta , 0.3 , 0.5


. 3.6 : Duffing 3

. 3.1 . 3.2 (3.84)

0.1

0.4 2 .

(3.94) Duffing

(. 3.5)

2 0.4 . ( 0.1 )

(. 3.4).

3

50

3.6. & (Averaging Method)

.

,

(Multiple Time Scales)

( ).

( ).

Verhulst [9].

,

,

.

(Variation of Parameters)

Duffing:

3 0u u u (3.95)

0 . :

cos( )u t (3.96) sin( )u t (3.97)

(3.96) (3.97) .

0 (3.95) :

( ) cos ( )u t t t (3.98) ( ) sin ( )u t t t (3.99)

( )u t (t) (t) .

(3.98), (t) (t) , :

( ) cos ( ) cos( ) (1 ) sin( )d

u t t t t tdt

(3.100)

(3.98) (3.99) :

cos( ) (1 ) sin( ) sin( )t t t (3.101)

cos( ) sin( ) 0t t (3.102)

3

51

(3.95) ( )u t :

3 0

( ) cos ( )

cos( ) sin( ) 0

u u u

u t t t

t t

(3.103)

( )u t , (t) (t) .

0

(3.95) . ,

(3.95).

.

( )u t (3.103),

(3.95) u u (3.98) (3.99) :

33 0 sin(t ) cos(t ) cos(t ) 0

du u u

dt (3.104)

3 3sin( ) cos( ) cos (t )t t (3.105)

(3.102) (3.105) , , (3.103) :

3 3

2 4

cos ( ) sin( )

cos ( )

t t

t

(3.106)

(3.106) (Standard Form)

- (3.98)

. ,

.

(3.106) (3.95)

. ,

t u ( t ).

( Averaging

Method) ( Runge-Kutta).

(3.106) , ,

(. 3.7).

t

( ).

3

52

. 3.7 : (t) , (t) ( )u t

- Averaged Equations

, ,

.

(3.106) :

( , t)x f x (3.107)

T

x (3.108) & 3 3 2 4( , ) cos ( ) sin( ) cos ( )T

f x t t t t (3.109)

(. 3.7)

, ( , )

(3.107).

(3.107) t T

. :

( )z f z (3.110)

0 0

1 1T Tdx d d

z ds xdsT dt dt T dt

(3.111)

0

1( ) ( , )

T

f z f z s dsT

(3.112)

T : ( , )f x t x .

(3.110)

(3.107)

.

(Averaging Method) .

(3.95) .

3

53

(3.107)

(3.108) (3.109).

(3.110) .

3 3 2 4

0

1( ) cos ( ) sin( ) cos ( )

TT

f z s s s dsT

(3.113)

T (3.113) :

23

( ) 08

T

f z

(3.114)

( )f z (3.110) :

2 2

0 0

( ) 3 3

8 8

z f z

(3.115)

:

0

2

0 0

(t)

3( )

8t t

(3.116)

(t) (t) :

0(t) (3.117) 2

0 0

3( )

8t t (3.118)

Duffing (3.95) (3.98) :

20 0 03

( ) cos8

u t t t

(3.119)

(Averaging Method)

(Multiple Scales) (3.65).

3

54

Van der Pol

(Averaging Method)

Van der Pol.

2(1 )u u u u (3.120)

- (3.98) :

2(1 ) 0

( ) cos ( )

cos( ) sin( ) 0

u u u u

u t t t

t t

(3.121)

( )u t :

2 2sin( ) cos( ) 1 cos ( ) sin( )

cos( ) sin( ) 0

t t t t

t t

(3.122)

:

2 3 2 2

2 3

sin ( ) sin ( )cos ( )

sin( )cos(t ) cos ( )sin( )

t t t

t t t

(3.123)

(3.107) : ( , t)x f x T

x

2 3 2 2

2 3

sin ( ) sin ( )cos ( )( , )

sin( )cos(t ) cos ( )sin( )

t t tf x t

t t t

(3.124)

(Averaged Equations),

(3.110): ( )z f z T

z (3.125) 0

1( ) ( , )

T

f z f z s dsT

(3.126)

T : ( , )f x t x .

(3.126) .

31 1

( ) 02 8

T

f z

(3.127)

:

21 112 4

0

(3.128)

3

55

Van

der Pol. 0 2 (3.128).

.

2 :

0( ) 2 cos( )u t t (3.129)

0 .

(3.128) :

1

20

2

0

11 ( 1)

4

t

t

e

e

(3.130) 0 (3.131)

(3.120) :

1

20

0

2

0

( ) cos( )1

1 ( 1)4

t

t

eu t t

e

(3.132)

(3.129) .

.

Van der Pol

(3.132) Van der Pol (3.120)

(Averaging Method).

.

:

1

20

0

2

0

( ) cos( )1

1 ( 1)4

t

t

eu t t

e

(3.133)

(1.23) (1.25) :

0

0

1

20

0 0

2

0

( ) cos( )1

1 ( 1)4

t

t

U eu t t

e

(3.134)

3

56

0 0u 0 0v 0 0 :

00u

U (3.135) 0 0 (3.136)

(3.81) , , :

0

0

1

20

02

0

2

( ) cos( )

1 ( 1)4

t

t

u eu t t

ue

U

(3.137)

(3.137) Van der Pol.

,

Runge-Kutta (3.137)

(3.120).

0.2 , 1/ 3U T .

0 00.2 , 0u v

Runge-Kutta 0.1

Averaging 0 3

. 3.8 : Van der Pol 1

3

57

0 00.2 , 0u v

Runge-Kutta 0.4

Averaging 0 3


0 00.2 , 0u v

Runge-Kutta 2

Averaging 0 3


3

58

0 00.05 , 0u v

Runge-Kutta 0.2

Averaging 0 3


0 00.6 , 0u v

Runge-Kutta 0.2

Averaging 0 3


3

59

Van der Pol

. 3.8, . 3.9 . 3.10.

. 3.8 (3.137)

.

. 3.9 2

(. 3.10). , ,

0 . (3.137)

(. 3.10) .

, Van der Pol

.

. 3.11 . 3.12

(3.137)

.

.

4

4.

4.1.

4.1.1.

, ,

.

,

.

.

. ,

. ,

.,

.

.

.

.

() ,

.

.

, ,

.

4

62

4.1.2.

,

.

.

,

.

:

. 4.1 :

. ,

, x

gx . 4.1.

x

.

,

g gx x x x gx x .

gmx mx .

4

63

,

x

.

. 4.1, x m

gx

gx x . ,

( 0g g gx x x x x )

( gx ).

0mx .

(. 4.2)

.

. 4.2 :

. 4.3 :

,

. ,

( )

.

.

4

64

,

,

.

:

S R (4.1)

.

(4.1).

.

,

, .

,

.

.

. 4.4 : () ()

4

65

:

1. ,

2. ,

3. ,

.

4.2.

.

.

4.2.1.

.

1.

,

(. 4.4).

:

.

.

,

, () .

4

66

2.

.

.

:

.

.

( )

, ,

.

. ,

.

,

.

.

, ,

,

.

4

67

3.

.

. ,

( )

.

,

, ,

,

, .

.

. 4.5 :

.

.

, ,

.

4

68

J.Paul Getty Malibu California. ,

.

.

.

(). :

&

4

69

4.2.2.

, , .

:

.

.

:

1. ,

().

2. , (PTFE)

.

.

.

1. (Rubber Bearings)

(ELB)

( 1-2 cm) (Steel Shims).

.

( Bulging).

.

, (LRB).

4

70

2. (Sliding Systems)

i.

,

.

, .

( W )

.

.

.

ii. (Friction Pendulum System , FPS)

(FPS)

.

. , ,

.

. 4.6 : FPS ( )

:

1.

, , .

2.

.

3.

.

4

71

.

.

(Lead Rubber Bearing , LRB)

ELB

. , LRB

,

.

.

. ,

( ).

. 4.7 : LRB

( )

.

.

.

( LRB) .

4

72

4.3.

4.3.1.

,

, .

( )

()

( ).

,

o ,

.

(. 4.8)

( ).

. 4.8 : (Shimizu Isolator)

,

.

, (sliding

bearings)

( ) ,

.

4

73

.

. ,

,

.

. 4.9 :

.

4.3.2.

. ,

,

( ).

4

74

.

. 4.10 : -

(. 4.9) ,

,

.

( ).

:

i.

ii. -

iii.

-.

- ( )

().

-

.

.

-

.

4

75

-

. ,

- ,

.

. :

2z x x x (4.2)

,

. 4.11 :

, ,

, ( )

. x,

x

z(x) .

()

-

z(x).

() .

. 4.11 : - Fr

4

76

4.3.3.

.

,

.

.

.

. 4.12 : Fc

4

77

4.3.4.

( , , ) .

(. 4.9 . 4.10)

0.364 0 .

.

,

0,625m (24,625 inches)

0,122m

(4,8 inches) (. 4.9) .

( )

0,235m (9,25 inches) .

( ) ,

,

.

33,57kg (74lbs),

74,84kg (165lbs), o 7,71kg (17lbs).

116,12kg (256lbs)

180,08kg (397lbs) .

,

.

.

.

,

.

, ,

.

,

.

( , , ) (. 4.11).

4

78

4.4.

, . ,

, xx yy.

( )

.

,

( x) .

,

x,

.

,

( )p t , ( )If t , ( )Df t

( )Sf t .

:

( ) ( ) ( ) ( )I D Sf t f t f t p t (4.3)

, , I D Sf f f .

,

x :

1. If

( )x t

D Alembert If mx .

2. p

( )

gx

.

4

79

, . 2.2,

( ) gp t mx , gx .

x ,

: gx x x .

,

x ,

If mx .

( ) ( ) (t) mI gf t mx p t p x (4.4)

3. Sf

. 4.13 :

xx

x

.

x

( )z x

. (4.2)

.

( ) ( )z x

(. 4.13).

4

80

xx ( ).

( )z x :

tan ( )dz

z xdx

(4.5)

RF ,

(. 4.14).

. 4.14 :

tanR spF F (4.6)

:

( )sp spF K z x (4.7)

spK : ( )z x : .

spF (4.7) (4.6) RF :

(x) tanR spF K z (4.8)

, (4.5), RF :

( ) ( )R spF K z x z x (4.9)

(4.2) RF :

4

81

2(x) z (x) K 2 ( )R sp spF K z x x ax sign x (4.10)

2 3 2 2( ) 2 3 ( ) ( 2 ) ( )R S spF f t K x a x sign x x sign x (4.11)

( )sign x : ( )x

sign xx

(0) 0sign .

:

3 23 2 1 0( ) ( )RF k x k x sign x k x k sign x (4.12)

:

23 2 spk K (4.13) 2 3 spk K (4.14)

21 2spk K (4.15) 0 spk K (4.16)

RF

( x ).

(4.12).

0 .

,

, x

.

, ,

( (4.3)) :

3 23 2 1 0( ) ( ) gmx k x k x sign x k x k sign x mx (4.17)

:

21

2 22sp

W WT

g k gK

(4.18)

g , W ,

,

sp .

4

82

0 (4.17)

Duffing.

33 1 gmx k x k x mx (4.19)

,

. ,

. ,

,

(4.20).

.

1 1 1 1 2

1 2 1 2 1 2

Fsp Fsp Fsp K KK

K K K K K K K K

(4.20)

4. Df

. 4.15 :

xx

x

.

1 2 d dC C . 4.15. , , x

2cF

. :

2 2c dF C x (4.21)

4

83

,

1dC .

spK , ,

.

(. 4.14)

N

.

1dC 1cF ,

,

( . 4.14).

1 tanc cpF F (4.22)

cpF

1dC . (

) , .

1 1( ) ( )cp d dF C z x C z x x (4.23)

( )z x : ( )z x ()

(4.23) (4.22)

(x)z (4.5) :

2

1 1 1( ) ( ) C 2 ( )c d dF C z x z x x x sign x x (4.24)

2 2 21 1 4 4 ( )c dF C x x sign x x (4.25)

0cF .

, ,

. (

) 0cF :

0 ( )cF mg sign x (4.26)

: m g :

. ( )sign x

x .

4

84

,

0 1 2C c c cF F F F ,

.

,

( )Df t (4.3)

.

(4.2) cF :

2 2 21 2( ) 4 4 ( ) ( )C D d dF f t C x x sign x x C x mg sign x (4.27)

2 2 21 1 1 2( ) 4 4 ( ) ( )C D d d d dF f t C x C x sign x C C x mg sign x (4.28)

( )

(4.28) :

22 1 0( ) ( )CF c x c x sign x c x c sign x (4.29)

2 1 0, , c c c c :

22 14 dc C (4.30) 1 14 dc C (4.31)

20 1 2d dc C C (4.32) c mg (4.33)

.

.

,

x x .

(4.3).

, (

x ) , :

22 1 0( ) ( ) gmx c x c x sign x c x c sign x kx mx (4.34)

:

2W

Tg k

(4.35)

4

85

g : , W :

k : .

0 0 (4.17)

Van der Pol.

22 0 gmx c x c x kx mx (4.36)

: ( ) ( )I C R gF F F P t m x t

:

IF mx

2 2 21 24 4C d dF C x x sign x x C x mg sign x

2 2 2 32 3 2R sp sp sp spF K x K sign x K x sign x K x

:

2

2 1 0 1 0

2 3

2 3 ( )g

mx c x c x sign x c x k x k sign x c sign x

k x sign x k x m x t

:

22 14 dc C 1 14 dc C

2

0 1 2d dc C C

21 2spk K 0 spk K c mg

2 3 spk K

2

3 2 spk K

5

5.

5.1.

.

Duffing Van der Pol

.

.

.

.

-

-

.

3.

.

(

) ( ).

.

5

88

5.2.

5.2.1.

, 4

, :

2

2 1 0 1 0

2 3

2 3 cos( )


k x sign x k x F t

(5.1)

, ,

1 2, , ,sp d dK C C

(5.1) .

m

. w .

(5.1)

F :

( )x f x (5.2)

:

v x (5.3) T

x x v (5.4)

(5.2) Runge-Kutta

gF w u , gu

( 0.2 ggu ).

(5.2)

D - / .

( )x t (5.2)

max_time Runge-Kutta.

(Steady State), I

0 .

:

0 max ( ) , x t t I (5.5)

5

89

.

0 0 ( ) . ,

-

().

0 0 ( )

D -

/ .

x . 0x :

3 23 2 1 0k x k x k x k F (5.6)

(5.6) x .

D :

0( )

Dx

(5.7)

: ( / )D D .

.

,

[3, 111].

1.8 kNw

0 0.364

0

8.08spK

1 0dC

2 0.088dC

( 10%)

0

2.601T

0.36F

kN m s

5

90

( / 1 ).

0 0 ( ) ( / )D D .

. 5.1 : -

. 5.2 : ( / )D

, / 1

.

.

5.2.2.

D

/ .

.

.

.

o

:

1.

.

2.

.

3. ,

.

5

91

1.8 kNw

0.8 0

0.04

16.72spK

1 0dC

2 0.176 dC

( 20%)

0

2.602T

0.36F

. 5.3

1.8 kNw

0.8 0.3

0

11.89spK

1 0dC

2 0.176dC

( 20%)

0

2.602T

0.36F

. 5.4

1.8 kNw

0.8 0.2

0.02

14.86spK

1 0dC

2 0.176dC

( 20%)

0.02

2.602T

0.36F

. 5.5

5

92

1.8 kNw

0.8 0

0.04

16.72spK

1 1.5dC

2 0.176dC

( 20%)

0

2.602T

0.36F

. 5.6

1.8 kNw

0.8 0.3

0

11.89spK

1 1.5dC

2 0.176dC

( 20%)

0

2.602T

0.36F

. 5.7

1.8 kNw

0.8 0.2

0.02

14.86spK

1 1.5dC

2 0.176dC

( 20%)

0

2.602T

0.36F

. 5.8

5

93

5.3.

5.3.1.

:

2

2 1 0 1 0

2 3

2 3 ( )g


k x sign x k x m x t

(5.8)

,

.

( )P t .

( ) cos( )gP t mx P t (5.9)

, :

2

2 1 0 1 0

2 3

2 3 cos( )


k x sign x k x P t

(5.10)

:

.

.

, ,

0 0 ( )sign .

:

2 32 0 1 3 ( )gmx c x c x k x k x m x t (5.11)

3.

( (1.26))

0 .

5

94

(5.11), (5.9),

1 :

2 2 30 2 0 cos( )x x c x c x kx F t (5.12)

:

2 102 spKk

m m

(5.13)

2 22 1 02

2 0

4 dC xcc xm m

(5.14)

0 20dc Cc

m m (5.15)

2 2

0230

2 spK xkk x

m m

(5.16)

0

PF

mx (5.17)

0x : ( )

(5.12)

,

.

1 : Duffing 2 0 0c k

2 30 0 cos( )x x c x kx F t (5.18)

2 : Duffing & Van der Pol 2 0 0c k

2 2 30 2 0 cos( )x x c x c x kx F t (5.19)

2 c k

. :

2 2c (5.20)

k k (5.21)

0c

0 02c (5.22) .

5

95

5.3.2. Duffing

, (5.18) :

2 30 02 cos( )x x x kx F t (5.23)

0 .

0 .

0 ,

.

,

,

.

,

( 0 ). :

0 (5.24)

,

0 .

,

,

. , 0 0 ( 0k

) ,

.

,

. ,

. :

F f (5.25)

5

96

(5.23) :

2 30 02 cos( )x x x kx f t (5.26)

(5.26)

.

.

:

0 0 1 1 0 1( ; ) ( , ) ( , )x t u T T u T T (5.27)

0T t 1T t

. ,

.

:

0 1 0 1 0 0 0 1 1 0x D D u u D u D u Du (5.28)

2 2 20 0 1 0 1 0 0 0 1 0 0 12 2x D D D u u D u D Du D u (5.29)

(5.28) (5.29) (5.26)

:

2 20 0 0 0 0D u u (5.30)

2 2 30 1 0 1 0 1 0 0 0 0 0 02 2 cos( )D u u D Du D u ku f T (5.31)

,

(5.31)

.

(5.24) :

0 0 0 0 0 0 1cos( ) cos( ) cos( )T T T T T (5.32)

(5.30) (5.31) :

2 20 0 0 0 0D u u (5.33)

2 2 30 1 0 1 0 1 0 0 0 0 0 0 0 12 2 cos( )D u u D Du D u ku f T T (5.34)

5

97

(5.33) :

0 0 1 1 0 0 1( , ) ( )cos ( )u T T a T T T (5.35)

(5.34), 0u

. , (7.9) , :

0 1 0( )u A T e cc (5.36)

0 00 :i T

e e (5.37)

1 1 11

( ) ( ) exp ( )2

A T a T i T (5.38)

(5.34) , (7.9), :

2 2 3

0 1 0 1 0 1 0 0 0 0 0

0 0 1 0 0 1

2 2

0.5 exp( ) exp( )

D u u D Du D u ku

f i T i T i T i T

(5.39)

MAPLE

0u (5.36) (5.39).

(RH ) (5.39) 1u .

2 3 30 0 0 1 0 02 2 3 0.5 exp(i )RH iA i A kA A f T e kA e cc (5.40)

(5.39)

0e , :

20 0 0 12 2 3 0.5 exp(i ) 0iA i A kA A f T (5.41)

A (5.38) :

30 10.5 cos 0.375 0a f T ka (5.42)

0 1 0 00.5 sina f T a (5.43)

, 1T :

30 0 0.5 cos 0.375 0a a f ka (5.44)

0 0 00.5 sina f a (5.45)

(Steady State) ,

( 0 0,a ), 0 0 0a

5

98

30 0 0 00.5 cos 0.375 0a f ka (5.46)

0 0 0 00.5 sin 0f a (5.47)

- 0 :

2 4 2 2 2 2 2 2 20 0 0 0 0 0 00.5625 3 4 4a a k a k f (5.48)

0 :

0 00 20 0

8tan

3 8a k

(5.49)

(5.26) :

1 0 0 1( ) ( )cos ( ) ( ) cos t ( )x t a T T T a (5.50)

1( )T 1( )T

(5.44) (5.45).

(5.50) 0 0,a (5.48) (5.49) :

0 0( ) cos t ( )x t a (5.51)

MATLAB 0 ( )a

(5.48).

0 3

20k

0 1

f

2

max 14f

. 5.9 : - - Duffing

5

99

- :

. 5.10 : -

-

.

(. )

f .

1 (. 5.10).

,

2 3.

3 4

.

.

5 ,

6.

.

6 2

(). .

3 6

.

f .

.

. 5.11 : f -

0a

5

100

, 0

Duffing ,

0 . 0 0 ( ) (5.48)

(5.49). 20

. MAPLE :

3 2 2 30 0 0 0 0 010.667 ( cos sin ) sinkf (5.52)

. 5.12

0 3

20k

0 1

f

2

max 14f

0 ( ) 0 ( )

. ,

.

.

.

5

101

. :

0 0

0 0

(5.53)

(5.48),

:

3 2 2 2 2 2 2 20 0 0 0 0 0 0 0( , ) 0.5625 3 4 4 0x x k x k x f (5.54)

20 0:x a (5.55)

, (5.53) :

00 0

0 0xx x

(5.56)

(5.54) :

2 2 2 2 2 20 0 0 0 0 01.6875 6 4 4 0x k x k (5.57)

(5.54)

- .

MATLAB

0 ( ) ( 0 020, 1, 3,maxf 14k ).

. 5.13 :

.

,

(Backbone Curve). .

5

102

Backbone Curve

(5.48) , :

2 2 2 4 6 2 2 2 2 20 0 0 0 0 0 0 0(4 ) ( 3 ) (0.5625 4 ) 0a a k a k a f (5.58)

:

4 2 2 2 2 2

0 0 0 0 0 0 0 2 2

0 02 2 2 2

0 0 0 0 0

3 4 4 3

8 8 4

a k a f a k fa

a a

(5.59)

0a . :

2

2

0 02 2

0 0 0 0

0 max4 2

f fa

a

(5.60)

(Backbone Curve) :

2

0 3 2

0 0 0 0

3, ,

32 2

kf fa

(5.61)

. 5.14

0 3

20k

0 1

f

0.2

max 14f

(5.60)

. , ,

(5.24). (5.61)

, . 5.14 .

5

103

. 5.15 :

(Backbone Curve)

.

f (. 5.15).

(. 5.16)

f

.

. 5.16

0 3

20k

0 3

f

5

max 40f

5

104

, ,

(.. 0, ,k )

. (5.48)

(5.49)

( 1 2, , , , K ,C ,sp d dm C ) 0, .

(5.13) (5.17).

,

0A .

(5.48) :

2 4 2 2 2 2 2 2 20 0 0 0 0 0 00.5625 3 4 4a a k a k f (5.62)

(5.21) (5.25) :

k

k

(5.63) F

f

(5.64) 002

c

(5.65)

0 , Fc k (5.15) , (5.16) (5.17) .

0a :

0

(5.66) 00

0

Aa

x (5.67)

(5.62)

, - :

2 4 2 4 2 2 2 2 2 2 2 20 0 0 0 0 0 0 2 02.25 6 ( ) 4 ( )sp sp dA A K mA K m C P (5.68)

0 :

2 00 2 20 0 0

4tan

6 8 ( )

d

sp

C

A K m

(5.69)

(5.68) (5.69) 0x ,

.

5

105

(5.68)

, - :

. 5.17 :

1.

.

2. , ,

.

3.

(.. 0 0.35 A m )

4. k , 0

(1) 1

1. (5.63) (5.66)

(5.68).

5

106

, .

2 30 02 cos( )x x x kx F t (5.70)

,

:

2 20 0 0 0 0cos( )D u u F T (5.71)

2 2 30 1 0 1 0 1 0 0 0 0 02 2D u u D Du D u ku (5.72)

(5.71) :

0 1 0( )u A T e e cc (5.73)

2 2

02( )

F

(5.74) 0: exp(i )e T (5.75)

(5.73) (5.72)

(

).

.

(5.72)

:

0 00 , , 33

(5.76)

,

. ,

. ,

,

.

. , ,

.

-

.

. 5.10 .

5

107

0 , :

03 (5.77)

(5.72) .

(5.73)

, MAPLE:

(5.72) , MAPLE :

3 2 21 0 0 0 0exp( ) 2 6 3 2RH k i T i A k A kAA iA e cc NST (5.78)

NST ,

(Non Secular Terms).

0e :

3 2 21 0 0 0exp( ) 2 6 3 2 0k i T i A k A kAA iA (5.79)

A (5.38)

( ,a )

(5.79) .

1( )A T

(5.70) :

0 1 0( )x u A T e e cc (5.80)

0( ) ( ) cos ( ) 2 cos( )x t a t t t t (5.81)

1( ) ( )a T a t 1( ) ( )T t

. 0a

( 0cos( )a t )

.

5

108

.

,a . :

1T (5.82)

( 0 0,a ) ( )x t .

( 0 0,a ) :

0 ,

0a .

4 2 2 2 2 4 2 2 2 6

0 0 0 0 0

4 2 2 2 2 2 2 6 2

0 0 0 0 0 0 0

9 2.25 6 0.1406

0.75

k k k k

k k

(5.83)

0sin 0cos :

0 00 2 20 0

8tan

24 3 8k ka

(5.84)

( 0 0,a )

:

0 0 2 20

( ) cos 3 cos ( )F

x t a t t

(5.85)

3 .

,

( ).

(5.85),

.

5

109

(5.85)

.

() () () ()

. 5.18

(5.83) 0a

. (1)

03

(5.77) 0a

. 5.18 ().

,

0a

.

.

0 3

4k

0 1

F

5

max 25F

. 5.19 :

5

110

. 5.19

(. 5.9).

. , .

(Backbone Curve) :

32 2 4

0 2 2

0 0 0 0 0

3( , ) (1 ),

8

kk ka

(5.86)

(5.83)

0a

,k 0 (Envelope Curve).

0 3

19F

0 1

k

2

max 8k

. 5.20 : k

k . k 0

( 0 , )

. .

. 5.21 k

5

111

0 3

4k

0 1

0.25

max 1.50

. 5.22 :

0 3

15F

4k

0

0.5

0max 1.5

. 5.23 : 0

5

112

,

.

0 3

4k

0 1

15

max 45

. 5.24 :

[8]

(. 5.11).

. 5.25 :

.

5

113

, F ( ),

0a ().

. 5.25. ,

. . 5.11

.

10

- (. 5.26).

k

.

, . 5.20,

0max a .

0 / 3 ,

.

. 5.26

0 3

10

0 1

k

5

max 20k

5

114

0 , :

03 (5.87)

(5.87)

-. 0e

( ). :

2 2 21 0 0 03 exp( ) 2 6 3 2 0k A iT i A Ak AA k iA (5.88)

.

,a

. :

0 0a (5.89)

-.

0 .

4 2 2 2 2 2 4 2

0 0 0

2 2 2 2 2

0 0 0 0 0

16 3 3.5556 0.25

0.4444 0.1975 1.7778 0

k a k k a k

a k

(5.90)

0 00 2 20 0

24tan

72 9 8k a k

(5.91)

( 0 0,a )

( ):

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Svolosl Nonlinear