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					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



     steady-state response
                                                              12/36
primary (main) resonance
                    0 
            detuning parameter =O(1)
  steady-state response
             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
steady-state responses
exist: upper-branch stable
focus, the saddle point,
and the lower-branch
stable focus




                              15/36
superharmonic resonance of order 3
                 3  0 
 steady-state response

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




superharmonic resonances   primary resonances
                                                   16/36
one-third subharmonic resonance
                   3 0 
 steady-state response
              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
                                    
nontrivial steady-state response
      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
  a system with quadratic nonlinearities
        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!

				
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posted:2/26/2013
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