The dynamically shifted oscillator.

The dynamically shifted oscillator is a nonlinear system described by a differential equation of motion which includes a Hooke's law restoring force plus a stiffness force which depends only upon the sign of the displacement. The natural frequency of the oscillator is a function of the displacement amplitude; for positive stiffness it increases as the amplitude decreases. Because of the special nature of the nonlinearity it is possible to find an exact expression for the displacement waveform as a function of time. In the limits of zero Hookian force or zero stiffness, the waveform becomes, respectively, a cosine wave or a parabola wave. Conservation of energy leads to an integral equation which can be solved to find an exact expression for the amplitude-dependent frequency. If the model equation is extended to include damping then the model predicts the time dependence of the frequency. The latter is compared with the measured time-dependent frequencies of two systems to which the dynamically shifted oscillator theory can be applied, the spring doorstop and the two-point librator. [ABSTRACT FROM AUTHOR]