brief survey of nonlinear oscillations li-qun chen department of mechanics, shanghai university,...
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Brief Survey of Nonlinear Oscillations
Li-Qun ChenDepartment of Mechanics, Shanghai University, Shanghai 200444, ChinaShanghai Institute of Applied Mathematics and Mechanics, 200072, China
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
1/36
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
5/36
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
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
phase plane for a simple pendulum with viscous damping
0sin
foci, attractors, domains of attraction
nonconservative systems with damping
8/36
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
9/36
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
11/36
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
15/36
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
18/36
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
26/36
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
27/36
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 )
36/36
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!