Harmonic response of a class of finite extensibility nonlinear oscillators

Abstract

Finite extensibility oscillators are widely used to simulate those systems which can not be extended to infinity. For example, they are used when modelling the bonds between molecules in a polymer or DNA molecule or when simulating filaments of non-Newtonian liquids. In this work, the dynamic behavior of a harmonically driven finite extensibility oscillator is presented and studied. To this end, the harmonic balance method is applied to determine the amplitude-frequency and the amplitude-phase equations. The distinguishable feature in this case is the bending of the amplitude-frequency curve to the frequency axis, making it to approach asymptotically to the limit of maximum elongation of the oscillator, which physically represents the impossibility for the system<br />to reach this limit. Also, the stability condition which defines stable and unstable steady-states solutions is derived. The study of the effect of the system parameters in the response reveals that a decreasing value of damping coefficient or an increasing value of excitation amplitude leads to the appearance of a multi-valued response and to the existence of a jump phenomenon. In this sense, the critical amplitude of the excitation, which refers to here as a certain value of external excitation that results in the occurrence of jump phenomena, is also derived. Numerical experiments to observe the effects of the system parameters on the frequency-amplitude response are performed to compare them to analytical calculations. For a low value of damping coefficient or a high value of excitation amplitude the agreement is poor for low frequencies but good for high frequencies.<br />It is demonstrated that the disagreement is caused by neglecting the higher-order harmonics in the analytical formulation. These higher-order harmonics, which appear as distinguishable peaks at certain values in the frequency response curves, are possible to calculate considering not the linearized frequency of the oscillator but its actual frequency which is strongly amplitude-dependent. On the other side, for a high value of damping coefficient or a low value of excitation amplitude, the agreement between numerical and analytical calculations is excellent. For these cases, the system is prevented to explore large amplitudes of vibration and, therefore, the<br />nonlinearity is not too much manifested.