A non-linear oscillator with history dependent force

We study the dynamics of a non-linear damped one-degree-of-freedom oscillator whose response force is dependent on the history of the motion and is of the relaxing type. Such an oscillator may serve as a simple model for mechanical applications where viscoelastic materials are used for the purpose of vibration damping and/or isolation. In the simplest cases history dependent forces are characterized by two parameters: the relaxation modulus Φ_O and the relaxation time γ. Through the use of a version of Melnikov's method we show how the two parameters govern the existence of non-linear subharmonic orbits. We suggest an experimental method for determining both parameters.