Brief Survey of Nonlinear Oscillations

Li-Qun ChenDepartment of Mechanics, Shanghai University, Shanghai 200444, ChinaShanghai Institute of Applied Mathematics and Mechanics, 200072, China

lqchen@staff.shu.edu.cn

Nonlinear Phenomena in One DOF Systems

Nonlinearity

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

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1.2 Nonlinearity Everywhere

in mechanical systems

nonlinear damping, such as stick-slip friction

nonlinear elastic or spring elements

backlash, play, or bilinear springs

nonlinear boundary conditions

most systems with fluids

2/36

in electromagnetic systems

hysteretic properties of ferromagnetic materials

nonlinear resistive, inductive, or capacitive elements

nonlinear active elements such as vacuum tubes, transistors, and lasers

moving media problems, for example vB voltages

electromagnetic forces, for example, J B and MB

nonlinear feedback control forces in servosystems

3/36

physical sources of nonlinearity

geometric nonlinearities such as nonlinear stain-displacement relations due to the large deformations

nonlinear material or constitutive properties, for example, stress-strain or voltage-current relations

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

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2 Nonlinear Phenomena in Single Degree-of-Freedom Systems

2.1 Free Oscillations

which is always integrable

0 ufu

uFhu 2

2

1 where

uufuF d

conservative systems without damping

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phase plane for a conservative system with a single DOF

trajectory, equilibrium points, saddle points, a center, separatrixes (homoclinc/heteroclinic orbits), static bifurcation

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phase plane for a simple pendulum with viscous damping

0sin

foci, attractors, domains of attraction

nonconservative systems with damping

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2.2 Self-Exciting Oscillations

32

0 3

1uuuu

nonconservative systems with nonlinear damping

physical model

oscillator with dry friction dry friction via relative speed

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phase plane for van der Pol’s equation

limit cycle

response of van der Pol oscillator

10/36

relaxation oscillation

responses of van der Pol oscillator

a physical model

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2.3 Forced Oscillations

Euufuu ,20

ideal energy source E=E(t), nonideal energy source uuuEE ,,Duffing equation

tKuuuu cos2 320

away from any resonance

2222220

220

1

4

2tancos

tK

u

steady-state response

ta t 2220

2 4cose

12/36

0

primary (main) resonance

detuning parameter =O(1)

steady-state response

Otau cos

frequency-response equation

220

2

22

0

22

48

3

a

Ka

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

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03

superharmonic resonance of order 3

steady-state response

OtK

tau

cos3cos20

2

superharmonic resonances primary resonances16/36

03

one-third subharmonic resonance

steady-state response

OtK

tau

cos

3

1cos

20

2

17/36

11

,cos

ii

N

iiii tKE

multifrequency excitations

primary, subharmonic, and superharmonic resonances

iii 3,3

1, 000

other resonances for N=2

120120120 2,2,2

1

combination resonance

1230

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2.4 Forced Self-sustainging Oscillations

forced van der Pol equation

tKuuuu cos3

1 320

away from primary resonance, subharmonic resonance of order 1/3 and superharmonic resonances of order 3

taa

ut 022

020

20

20

cose4

4

where

tK

cos20

2 O

220

2

22

21

K

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

Otau cos

synchronization

steady-state response for small K such that >0

tu 00

cos2

tK

cos20

2 O

0

frequency-response equation

2

222

414

K

where

022

0 ,4

1 a21/36

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

02cos utu stable and unstable (shaded) regions in the parameter plane for the Mathieu equation

24/36

effects of the damping on the stability

02cos2 utuu

25/36

Steady-state response in nonlinear parametric vibrations

022cos2 32 uutuuu

nontrivial steady-state response

2413

4

3

8

a

1

stability boundaries

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3 Nonlinear Phenomena in Multi-Degree-of-Freedom Systems

3.1 Free Oscillations

212222

222

111111211

2

2

uuuu

uuuuu

a system with quadratic nonlinearities

internal resonance 12 2

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3.2 Forced Oscillations

22212222

222

111111211

cos2

2

tFuuuu

uuuuu

primary resonance and internal resonance 121 2, saturation phenomenon

28/36

4 Approximate Analytical Methods

4.1 The method of Harmonic Balance

N

kk ktkAu

00cos

assume the periodic solution in the form

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

22

10

33

22

1;

txtxtxtu

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

,2,1,0

,,,,,,,,,; 21033

21022

2101

ktT

TTTxTTTxTTTxtuk

k

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

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4.4 The Method of Averaging

assume the solution in the form

ttttu

ttttu

00

0

sin

cos

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 Methodassume the solution in the form

aBaB

aAaAa

axaxatu

22

10

22

1

22

1 ,,cos;

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

txxx cos5.705.0 3

Duffing’s oscillator of Ueda type

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

Poincaré map: sample a trajectory stroboscopically at times that are integer multiples of the forcing period

a bounded steady-state response that is not an equilibrium state or a periodic motion, or a quasiperiodic motion

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5.3 Intrinsic Stochasticity

random-like motion in a deterministic system that is seemingly without any random inputs (spontaneous stochasticity )

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22 3

20n

d WW W

dt

4 4 4n

4 4 2 2 2n n P n n

4

n n

AL

EI

4 6 22

04 6 2

222 2 3 20

2 3 20 0

20

2

L L

w w wEI e a A

x x t

EA e aEA w w w w wP dx dx

L x L x x x x

N P A

Practical Example

axially tensioned nanobeam: bending vibration

Thank you!