We study the dynamical response of an asymmetric forced, damped Helmholtz-Duffing oscillator by using Jacobi elliptic functions, the method of elliptic balance, and Fourier series. By assuming that the modulus of the elliptic functions is slowly varying as a function of time and by considering the primary resonance response of the Helmholtz-Duffing oscillator, we derived an approximate solution that provides the time-dependent amplitude-frequency response curves. The accuracy of the derived approximate solution is evaluated by studying the evolution of the response curves of an asymmetric Duffing oscillator that describes the motion of a damped, forced system supported symmetrically by simple shear springs on a smooth inclined bearing surface. We also use the percentage overshoot value to study the influence of damping and nonlinearity on the transient and steady-state oscillatory amplitudes.

Since most of the nonlinear differential equations that characterized the motion of several physical systems do not have closed-form solution, we have to use numerical or perturbation techniques to study the dynamical response of these systems. However, most of the perturbation techniques such as multiple scales, averaging, and harmonic balance, to say a few, focus on only the determination of steady-state approximate solutions because of the complexity involved in finding transient solutions of nonlinear differential equations [

The aim of this paper is to investigate the influence on the dynamical behavior of the transient and steady-state solutions of the system

Recently, Kovacic and coworkers studied the primary resonance response of (

Here the approximate solution of (

In the next section, the approximate general solution of (

In order to obtain the general solution of (

Notice that (

Next, the average of (

To determine the unknown parameters

We then follow the harmonic balance procedure and ignore higher harmonics terms in (

We will next discuss the accuracy of our derived approximate solution by studying the dynamical response of an asymmetric Duffing oscillator by plotting the amplitude-frequency and the percentage overshoot response curves and show how the evolution of time influences the systems behavior.

In order to verify the accuracy of our proposed solution described in (

To find the amplitude-frequency response curves of (

Evolutive amplitude-frequency response curves of a rigid body supported symmetrically by a simple shear spring and sliding over a smooth inclined bearing surface with moderate static shear deflection

Phase portraits and amplitude-time response curves for parameter values of

We next investigate the influence of the transient solution on the dynamics response of (

Percentage overshoot versus the driving frequency for parameter values of

Transient and steady-state amplitude ratio

To examine damping effects on the system overshoot, we next use the parameter values of

System nonlinear dynamic behavior for parameter values of

In this paper, we have used elliptic functions to obtain the transient and the steady-state solutions of an asymmetric Duffing oscillator and studied its influence on the region of the primary resonance response. The theoretical predictions of our derived solution given by (

Based on these findings, we can conclude that if one really wants to have a better understanding of the system dynamical behavior, transient oscillations amplitudes must be estimated to avoid undesirable system effects such as peak transient amplitude values that could violate design constraints or unstable system behavior when the excitation is suddenly switched on. In this case, the percentage overshoot charts could be used to identify the frequency bandwidth region at which the system can be tuned to have a reliable dynamical response.

The terms

The terms

This work was partially funded by the Tecnológico de Monterrey, Campus Monterrey, through the Research Chairs in Nanotechnology and Intelligent Machines. Additional support was provided by the project FOMIX NL-2010-C30-145429.