Exact solution of stable periodic one contact per N cycles motion of a damped linear oscillator contacting a unilateral elastic stop

An exact closed-form solution is developed for a damped single-degree-of-freedom oscillator contacting a unilateral stop once per N cycles of external sinusoidal force. It also includes the effect of a bias force. This solution was obtained by using two different displacement and velocity expressions for a two-region piecewise linear oscillator, continuity of displacement and velocity at the boundary of the two regions, and periodicity conditions. Contact duration is assumed as known and four simultaneous equations in four unknowns are solved exactly. The four unknowns are the amplitude of the sinusoidal force, phase angle of the assumed first contact, entry velocity, and exit velocity of mass. The stability of motion is checked using the closed-form expressions of elements of a 2×2 stability governing matrix. These elements are obtained by considering only first-order perturbations in periodic motion. Theoretical predictions based on exact solution agree with previous results and with results obtained using a numerical simulation approach. The relationships are studied in detail between input parameters (such as amplitude of external force, frequency of external force, bias force, gap, spring stiffness and damping constants) and output parameters (such as contact duration, phase angle at contact, entry velocity, exit velocity, maximum displacement and minimum displacement).