Nonlinear spring-mass system

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Basic phenomenology of simple nonlinear vibraion (free and forced) Manoj Srinivasan (6) Nonlinear spring-mass sysem Damper Spring Mass Hardening Sofening O No damping graviy g lengh l A mass m

Do.8.6.. -. -. -.6 -.8 Nonlinear spring-mass sysem - - -.8 -.6 -. -....6.8 No damping Frequency (ime period) of free vibraion oscillaions depends on oscillaion ampliude unlike for linear spring-mass sysem 6.55 sofening cubic nonlineariy 6. hardening cubic nonlineariy 6.5 6.5 Time period ime period of oscillaion 6.5 6. 6.5 Sofening 6. Time period 6.5 ime period of oscillaion Hardening 6. 6. 6.5.....5.6.7.8.9 oscillaion ampliude Oscillaion ampliude 6.5.....5.6.7.8.9 oscillaion ampliude Oscillaion ampliude Damped free vibraions of Duffing equaion 8 6 how would you epec he frequency o change as he oscillaion ampliude decreases? Damped Duffing wih alpha> siffening spring guiar pich glide due o ampliude dependence of frequency? (assume siffening) - - -6-8 noice frequency change - 6 8 6 8 hps://www.youube.com/wach?v=vkkoflouqds

Linear spring-mass-damper sysem Damper Spring Mass Eernal forcing Damping raio Naural frequency Normalized force ampliude = (unis of lengh) ( Saic deflecion o saic force F ) Frequency response of linear spring-mass-damper sysem figure source: Wikipedia

Duffing equaion wih forcing Damper Spring Mass Eernal forcing Hardening cubic nonlineariy Sofening O graviy g lengh l A mass m Cubic nonlineariy wih or wihou quadraic nonlineariy When we do Taylor series of an odd funcion abou an equilibrium for wih spring force = Duffing s equaion cubic When we do Taylor series of odd funcion abou an equilibrium wih spring force no = (Recall HW for illusraive eample) cubic quadraic (or) jus Taylor series of a no-odd funcion

Frequency response of Duffing equaion (cubic nonlineariy) Primary resonance. Big response ampliude when forcing frequency ωo ~ linear naural frequency ωn Secondary resonance Super-harmonic resonance: Big response ampliude when forcing frequency ωo ~ ωn / ineger (eample: ωo ~ ωn/) Sub-harmonic resonance: Big response ampliude when forcing frequency ωo ~ ωn ineger. (eample: ωo ~ ωn) Primary resonance and secondary resonances Muliple peaks due o nonlineariy even hough i is a single DOF sysem ( Recall ha an N DOF linear sysem (N>) will have muliple peaks due o here being N modes and corresponding naural frequencies)

Linear vs Hardening vs Sofening How he primary resonance s frequency response changes Response ampliude (y ais) vs forcing frequency ( ais) for cases Nayfeh and Mook Linear sysem response Hardening spring Sofening spring forcing frequency arrows indicae response obained in forward and backward sweeps forcing frequency Hardening vs Sofening Ampliude response by simulaion unil seady sae see MATLAB program Forward sweep (magena) and Backward sweep (blue) shown Hardening Sofening Seady response ampliude vs frq. wih he poins conneced cubic nonlineariy alpha =.5 Seady response ampliude vs frq. wih he poins conneced cubic nonlineariy alpha = - 6 forward frequency sweep backward frequency sweep 9 forward frequency sweep backward frequency sweep 5 7 Seady Oscillaion Ampliude Seady Oscillaion Ampliude 8 6 5.5.5 Forcing frequency.5.5.5.5 Forcing frequency.5.5

Ampliude response obained by finding fied poins of Poincare maps (so we can find boh sable and unsable moions) response ampliude Frequency response for forced Duffing wih hardening spring see MATLAB program response ampliude response ampliude.5.5.5.5 Frequency response for forced Duffing wih sofening spring forcing frequency blue = sable periodic response red = unsable periodic response.5.5.5.5 forcing frequency forcing frequency Ueda shows ha he fully nonlinear forced Duffing (linear siffness = ) has many parameer regimes wih many differen behavior Damping coeff c See paper uploaded, Ueda 99. Forcing ampliude A Following slides show some MATLAB illusraions of his compleiy

Five differen co-eising periodic moions For he same parameer values: k =, A =., c =.8, omega =.5.5 -.5 - -.5 5 6 7.8.6. do. Do -. -. -.6 -.8 - - -.5.5 Chaoic moion (unique) For he same parameer values: k =, A =, c =., omega = Chaoic - - - - 6 8 6 8 6 do - - -6-8 -6 - - 6 8

Co-eising periodic moion and chaoic moion For he same parameer values: k =, A =.5, c =., omega = periodic moion chaoic moion - - - - - - - 5 6 7 8 9-5 6 7 8 9 8 6 do - - -6-8 - -8-6 - - 6 8 Frequency conen of response Fourier ampliude specrum Primary resonance (forcing ω = ) Fourier Specrum of Eernal Forcing funcion 5 6 7 8 9 Omega (Hz) -- in angular frequency.5.5 Inpu Fourier Specrum of he oscillaion Response Response 5 6 7 8 9 in angular frequency (Hz) frequency c =. m =, k = epsilon =. omega = A = 5 Even when he response has he same frequency as he forcing, here can be oher harmonics in response due o nonlineariy (unlike linear sysems)

Frequency conen of response Fourier ampliude specrum Fourier Specrum of Eernal Forcing funcion 6 8 6 8 Omega (Hz) -- in angular frequency.5.5.5.5 Forcing freq ω = Inpu Fourier Specrum of he oscillaion Response Response 6 8 6 8 in angular frequency (Hz) frequency c =. m =, k = epsilon =. omega = A = 5 far from resonance we don see much higher harmonics in response Chaos = broad band frequency conen Fourier ampliude specrum 8 6 Fourier Specrum of Eernal Forcing funcion 5 6 7 8 9 Omega (Hz) -- in angular frequency..8.6.. Fourier Specrum of he oscillaion Response 5 6 7 8 9 in angular frequency (Hz) frequency - - - - Phase space in D chaoic moion 6 8 6

....8.6.. Fourier Specrum of Eernal Forcing funcion 5 6 7 8 9 Omega (Hz) -- in angular frequency.5..5..5..5 Fourier Specrum of he oscillaion Response period moion 5 6 7 8 9 in angular frequency (Hz) do.5.... Do -. -. -. -. -. -....6 Ueda (Manoj noaion) c =.8 k = epsilon = ; omega = A =.;....8.6.. Fourier Specrum of Eernal Forcing funcion 5 6 7 8 9 Omega (Hz) -- in angular frequency..5..5..5 Fourier Specrum of he oscillaion Response period moion 5 6 7 8 9 in angular frequency (Hz)... Do -. -. -. -. -. -. -..... do Beware: sofening cubic nonlineariy and purely quadraic nonlineariy have regimes where he siffness is negaive for some large ampliudes => sysem can go o infiniy if he forcing is no small enough Fi: A quadraic nonlineariy could be accompanied by a siffening cubic nonlineariy which keeps he moion bounded A sofening cubic nonlineariy could be accompanied by a siffening ^5 nonlineariy, which keeps he moion bounded

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