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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 vB voltages electromagnetic forces, for example, J B and MB 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 2t 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 cost O frequency-response equation 2 3a 2 K2 2 2 2 2 80 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 cos3t 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 3i 3 other resonances for N=2 0 2 1 , 0 2 2 1 , 0 2 21 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 cos0t 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 cos0t 2 cos t O 0 0 2 0 u a cost 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 2u cos 2t u 0 25/36 Steady-state response in nonlinear parametric vibrations u 2u 2u cos 2t 2u 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 21u1 1u1u1 u2 2 u2 2 2u2 2u12 2 internal resonance 2 21 27/36 3.2 Forced Oscillations u1 12u1 21u1 1u1u1 u2 2 u2 22u2 2u12 F2 cost 2 2 primary resonance and internal resonance 1 , 2 21 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 coskt 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 22 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 ut t cos0t t ut 0 t sin0t 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!