Valid initial conditions for the Duffing model of the controlled, driven clamped–clamped beam

Determine which initial conditions are valid for the Duffing equation employed to model the controlled, driven clamped–clamped beam so that the resulting solution behavior is appropriate for the system and consistent with existence and uniqueness guarantees.

Background

The report models transverse vibrations of a clamped–clamped beam with axial tension nonlinearity, damping, external harmonic drive, and saturated feedback control. Through separation of variables and reduction, the temporal dynamics are expressed as a forced, damped Duffing equation, whose behavior is sensitive to initial conditions and can exhibit chaos.

In discussing the well-posedness of the problem, the authors note that although existence and uniqueness can be ensured under appropriate conditions (by Cauchy–Kowalevski), they do not know which specific initial conditions are valid for the case studied, due to the Duffing system’s sensitivity and multiple applicable choices. Identifying these initial conditions is necessary to guarantee physically meaningful and deterministic behavior of the modeled oscillator.