Response of linear parametric amplifiers with arbitrary direct and parametric excitations

Abstract

The effect of parametric amplification, where parametric excitation is added to direct excitation, on the performance of a dynamical system is investigated. To model the system, the linear forced Mathieu equation is considered. In order to obtain the response of the system without restrictions such as requiring the system parameters to be small or the parametric excitation frequency to be exactly twice the direct excitation frequency, the method of varying amplitudes (MVA) is used. Employing the MVA, the response of the system and its maximum value for arbitrary values of the system parameters including the excitation frequencies are derived. It is shown that when the ratio between the excitation frequencies is rational, the maximum response of the system is highly dependent on the phase angle between the excitations. However, when the frequency ratio is irrational, the maximum response of the system is not dependent on the phase angle. Consequently, large values of response can be attained by using an irrational frequency ratio regardless of the phase angle. MVA results for peak responses show good agreement with numerical results obtained by direct integration of the equation of motion.