A Stability Criterion for the Mathieu-Hill Equation

Abstract

Abstract A criterion for the stability of the solutions of the Mathieu-Hill equation containing even periodic potential is developed. The criterion is based on a result derived by Hill relating the infinite determinant, the Floquet exponent, and the constant term of the even Fourier series expansion of the potential function. It is shown that the criterion is easy to apply in practical problems to obtain stability maps in a suitable parametric plane. In the special case of the Mathieu equation, the criterion is applied to generate the stability maps that compare fairly well with those obtained by other well-known, but more tedious methods. A qualitative relation between the stable and unstable regions of the stability map and the infinite determinant is also provided. In another example, the problem of undamped, finite uniaxial vibrations coupled with a superimposed small twisting oscillation of a rigid disk attached to a hyperelastic neo-Hookean shaft is briefly discussed to bring out the usefulness of the criterion when applied to problems of engineering interest. It is found that the stability of small twisting motion is governed by the Mathieu-Hill equation containing an even periodic potential. Use of the criterion is made to generate the stability maps in a relevant parametric plane. This stability chart can be directly used in designing such a shaft-disc assembly to avoid undue high amplitude torsional vibration.