EXACT SHORT-TIME IDENTIFICATION OF MATHIEU EQUATION OF PARAMETRIC RESONANCE

Mathieu equation is a 2nd order differential equation with a periodically modulated coefficient relevant to the squared natural frequency. It describes the parametric excitation of oscillation, 2D vibration in elliptic coordinates, wave propagation in periodically varying medium, etc. Depending on the parameters, those systems have a highly non-stationary nature such as exponential growth of amplitude, vibrato, tremolo, and beat (periodic modulation of frequency and amplitude), etc. In this presentation, we will propose an exact mathematical technique for estimating the parameters and excitation conditions of the Mathieu equation systems from a short-time observation of waveform. Applying the weighted integral method [Ando and Nara, IEEE Trans. SP, 2009] to the Mathieu equation, we obtain a set of algebraic equations for solving the coefficients from the Fourier coefficients of the waveform. When the excitation frequency of the parameter is known, the algebraic equations becomes simple linear equations with three Fourier coefficients of equally spaced orders, hence the solution process is very easy to implement. Terminal wave values and differentials at the window edges are also estimated which provide a good indicator to judge non-stationary and non-periodic natures of the excitation. Some actual estimation results using an experimental setup will be shown.