svolosl nonlinear
DESCRIPTION
svolosTRANSCRIPT
-
Duffing Van der Pol
: ...
2014
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... ,
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,
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. ,
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,
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-
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,
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,
Duffing
Van der Pol.
,
. ,
. ,
,
.
,
.
( ).
, .
-
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
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1
1.
1.1.
, ,
.
. , ,
, .
.
,
,
.
. ,
,
.
(. 1.1).
. 1.1 :
-
1
2
(t)u
(t) . .
(t)u
(t) . m
,
( )u t ().
1.2. &
,
,
.
.
.
.
,
, .
,
, .
() .
. :
,
()
,
() (,
, ),
,
,
.
-
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.
,
. ,
(1.6) , (1.7) , (1.8) (1.9)
.
( ) ,
, .
, :
(-) .
.
,
, .
. ,
.
,
.
, .
,
. ,
, .
1.4.2.
,
.
,
.
() .
.
-
1
7
. ,
. ,
(1 cm) ;
.
.
. ,
.
.
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
Multiple Scales 2 0.4
. 3.5 : Duffing 2
-
3
49
0 00.2 , 0u v
Runge-Kutta , 0.3 , 0.5
Multiple Scales 2 0.3
. 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
. 3.9 : Van der Pol 2
0 00.2 , 0u v
Runge-Kutta 2
Averaging 0 3
. 3.10 : Van der Pol 3
-
3
58
0 00.05 , 0u v
Runge-Kutta 0.2
Averaging 0 3
. 3.11 : Van der Pol 4
0 00.6 , 0u v
Runge-Kutta 0.2
Averaging 0 3
. 3.12 : Van der Pol 5
-
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)
.
.
-
60
-
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
-
86
-
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( )
mx c x c x sign x c x k x k sign x c sign x
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
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
(5.8)
,
.
( )P t .
( ) cos( )gP t mx P t (5.9)
, :
2
2 1 0 1 0
2 3
2 3 cos( )
mx c x c x sign x c x k x k sign x c sign x
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 )
( ):