On the transient response of forced nonlinear oscillators

We consider the transient response of a prototypical nonlinear oscillator modeled by the Duffing equation subjected to near resonant harmonic excitation. Of interest here is the overshoot problem that arises when the system is undergoing free motion and is suddenly subjected to harmonic excitation with a near resonant frequency, which leads to a beating type of transient response during the transition to steady state. In some design applications, it is valuable to know the peak value of this response and the manner in which it depends on system parameters, input parameters, and initial conditions. This nonlinear overshoot problem is addressed by considering the well-known averaged equations that describe the slowly varying amplitude and phase for both transient and steady state responses. For the undamped system, we show how the problem can be reduced to a single parameter χ that combines the frequency detuning, force amplitude, and strength of nonlinearity. We derive an explicit expression for the overshoot in terms of χ, describe how one can estimate corrections for light damping, and verify the results by simulations. For zero damping, the overshoot approximation is given by a root of a quartic equation that depends solely on χ, yielding a simple bound for the overshoot of lightly damped systems.