# 幻灯片 1 by dffhrtcv3

VIEWS: 10 PAGES: 40

• pg 1
```									Brief Survey of Nonlinear Oscillations
Li-Qun Chen
Department of Mechanics, Shanghai University, Shanghai 200444, China
Shanghai Institute of Applied Mathematics and Mechanics, 200072, China

lqchen@staff.shu.edu.cn
Nonlinearity

Nonlinear Phenomena in One DOF Systems

Nonlinear Phenomena in Multi-DOF Systems

Approximate Analytical Methods

Descriptions of Chaos
1 Nonlinearity
1.1 Linearity versus Nonlinearity

input-output possibilities for linear and nonlinear systems

1/36
1.2 Nonlinearity Everywhere
in mechanical systems

nonlinear elastic or spring elements

nonlinear damping, such as stick-slip friction

backlash, play, or bilinear springs

nonlinear boundary conditions

most systems with fluids

2/36
in electromagnetic systems

nonlinear resistive, inductive, or capacitive elements

hysteretic properties of ferromagnetic materials

nonlinear active elements such as vacuum tubes,
transistors, and lasers
moving media problems, for example vB voltages
electromagnetic forces, for example, J  B and MB
nonlinear feedback control forces in servosystems
3/36
physical sources of nonlinearity

nonlinear material or constitutive properties, for
example, stress-strain or voltage-current relations
geometric nonlinearities such as nonlinear stain-
displacement relations due to the large deformations

nonlinear body forces including gravitational, magnetic
or electric forces

nonlinear acceleration or kinematic terms such as
convective acceleration, centripetal or Coriolis
accelerations
4/36
1.3 Theories of Nonlinearity
classic theory of nonlinear oscillations
focus on periodic motions and equilibriums as well as
their stabilities, via approximate analytical approaches
including the method of multiple scales, the averaging
method, the Lindstedt-Poincaré method, the KBM
asymptotic method, the method of harmonic balance, etc

modern theory of nonlinear dynamics
focus on more complicated motions such as chaos and
the evolution of motion patterns such as bifurcation, via
more advanced mathematical techniques and numerical
experiments
5/36
2 Nonlinear Phenomena
in Single Degree-of-Freedom Systems
2.1 Free Oscillations
conservative systems without damping

u  f u   0

which is always integrable

u  h  F u 
1 2

2
where

F u    f u d u
6/36
phase plane for a conservative system with a single DOF

trajectory, equilibrium points, saddle points, a center,
separatrixes (homoclinc/heteroclinic orbits), static bifurcation
7/36
nonconservative systems with damping
      
    sin   0

phase plane for a simple pendulum with viscous damping
foci, attractors, domains of attraction
8/36
2.2 Self-Exciting Oscillations
nonconservative systems with nonlinear damping

   1 3
u   u   u  u 
       2
0
  
   3 
physical model

oscillator with dry friction       dry friction via relative speed

9/36
limit cycle

response of van der Pol oscillator

phase plane for van der Pol’s equation
10/36
relaxation oscillation

a physical model

responses of van der Pol oscillator
11/36
2.3 Forced Oscillations

u  0 u  f u , u   E
   2

ideal energy source E=E(t), nonideal energy source E  E u, u, u 
 
Duffing equation
u  0 u  2 u   u 3  K cos t
   2

away from any resonance
          1 2 
K cos t  tan
                    
2 
0   
                  
2
u      
 a e 2t cos 0  4 2  2 t  
2

         
0    4 2  2 2
2     2 2

12/36
primary (main) resonance
  0 
detuning parameter =O(1)
u  a cost     O 
frequency-response equation
2
    3a    2
     K2
2
 
                  2 2 2

     80         4 0 a

13/36
jump phenomenon resulted from the multivalueness

hardening characteristic   softening characteristic

14/36
domains of attraction

state plane for the Duffing
equation when three
exist: upper-branch stable
and the lower-branch
stable focus

15/36
superharmonic resonance of order 3
3  0 

u  a cos3t     2     cos t  O 
K
  0
2

superharmonic resonances   primary resonances
16/36
one-third subharmonic resonance
  3 0 
1       
cos t  O  
K
u  a cos t     2
3          0
2

17/36
multifrequency excitations
N
E   K i cos i t   i ,  i   i 1
i 1
primary, subharmonic, and superharmonic resonances
1
 0  i ,  0  i ,  0  3i
3
other resonances for N=2

 0   2  1 ,  0  2 2  1 ,  0   2  21
1
2
combination resonance

 0   3   2  1
18/36
2.4 Forced Self-sustainging Oscillations

forced van der Pol equation
    1 3
u   u    u  u   K cos t
      2
0
   
    3 
away from primary resonance, subharmonic resonance of
order 1/3 and superharmonic resonances of order 3

4
cos0t             cos t  O 
K
u
  a 4  a     e
2 2    t                      0
2   2        2                                   2   2
0   0        0   0

 2K 2
  1
where

2   2

2 2
0

Motion is aperiodic if the frequencies 0 and  are not
commensurable.                                                                19/36
quenching
definition: the process of increasing the amplitude of the
excitation until the free-oscillation term decays
condition: K large enough such that <0

unquenched response with K=0.9, 0=1 and =2

quenched response with K=1, 0=1 and =2
20/36
synchronization

steady-state response for small K such that >0

2                      K
u         cos0t     2     cos t  O 
0                     0
2

 0
u  a cost     O 
frequency-response equation
K2
4    1        2
2             2

4
where
1 2 2
  0 a ,     0
4                                   21/36
frequency-response curves for primary resonances of the
forced van der Pol oscillator

22/36
locking

pulling-out (beating phenomenon)

23/36
2.5 Parametric vibrations
stability in linear parametric vibrations
Mathier equation
u     cos 2t u  0

stable and unstable (shaded) regions in the parameter
plane for the Mathieu equation

24/36
effects of the damping on the stability

u  2u     cos 2t u  0
    

25/36
Steady-state response in nonlinear parametric vibrations

         
u   2u  2u cos 2t   2u  u 3  0
                          
8   4
a       1  4 2
3 3
1    

stability boundaries

26/36
3 Nonlinear Phenomena
in Multi-Degree-of-Freedom Systems
3.1 Free Oscillations
u1  12u1  21u1  1u1u1
               
u2  2 u2  2 2u2   2u12
    2

internal resonance     2  21  

27/36
3.2 Forced Oscillations
u1  12u1  21u1  1u1u1
               
u2  2 u2  22u2   2u12  F2 cost   2 
    2

primary resonance and internal resonance   1 , 2  21
saturation phenomenon

28/36
4 Approximate Analytical Methods

4.1 The method of Harmonic Balance

assume the periodic solution in the form
N
u   Ak coskt  k 0 
k 0

substitute the expression into the equation

equate the coefficient of each of the lowest N+1 harmonics
to zero

solve the resulting N+1 algebraic equations
29/36
4.2 The Lindstedt-Poincaré Method

assume the solution in the form

u t;    x1 t    2 x2 t    3 x3 t   
    0  1   22  
substitute the expression into the equation
equate the coefficient of each power of  to zero

solve the resulting nonhomogeneous linear differential
equations
eliminate the secular term in each solution by solving an
algebraic equations
30/36
4.3 The Method of Multiple Scales

assume the solution in the form
u t;    x1 T0 , T1 , T2 ,   2 x2 T0 , T1 , T2 ,   3 x3 T0 , T1 , T2 ,  
Tk   k t   k  0,1,2,
substitute the expression into the equation
equate the coefficient of each power of  to zero

solve the resulting nonhomogeneous linear differential
equations
eliminate the secular term in each solution by solving a
differential equation
31/36
4.4 The Method of Averaging

assume the solution in the form
ut    t  cos0t   t 
ut   0 t sin0t   t 


substitute the expression into the equation
express the derivatives of  and  as function of ,  and
=0t+ based on the resulting algebraic equations about
the derivatives

average the expressions over  from 0 to 2, assuming 
and  to be constants

32/36
4.5 The Krylov-Bogoliubov-Mitropolsky Method
assume the solution in the form
u t ;    a cos  x1 a,    2 x2 a,   
a  A1 a    2 A2 a   

  0  B1 a    2 B2 a   

substitute the expression into the equation
equate the coefficient of each power of  to zero
equate the coefficient of each of the harmonics to zero
eliminate the secular term
solve the resulting algebraic equations and differential
equations
33/36
5 Descriptions of Chaos
5.1 Sensitivity to Initial States
Duffing’s oscillator of Ueda type
  0.05x  x3  7.5 cost
x        

tiny differences in the initial conditions can be quickly
amplified to produce huge differences in the response
butterfly effect                                         34/36
5.2 Recurrent Aperiodicity
a bounded steady-state response that is not an equilibrium
state or a periodic motion, or a quasiperiodic motion

Poincaré map: sample a trajectory stroboscopically at
times that are integer multiples of the forcing period
35/36
5.3 Intrinsic Stochasticity
random-like motion in a deterministic system that is
seemingly without any random inputs (spontaneous
stochasticity )

36/36
Practical Example
axially tensioned nanobeam: bending vibration

N  P  A
 4w           2  w
6
2w
EI   4   e0 a         6 
 A 2
 x               x            t

    EA L  w 
2
2 EA  e0 a  L
2
  2 w 2 w  3 w    2 w

2 L 0  x                            0  x  x x3  x
 P                  dx                       2                dx  2  0
                                  L
                           2
                       
dW
 n2W  W 3  0
dt 2
  n4 4 4
 AL4
n  n P  n4 4 2  n2 2             n  n
EI
Thank you!

```
To top