Dynamics of a circular membrane with an eccentric circular areal constraint: Analysis and accurate simulations

Abstract

The dynamics of a circular membrane with an eccentric circular areal constraint is studied under arbitrary initial conditions. The symmetric and antisymmetric modes of vibrations are investigated and the results are compared with those in the literature. The accuracy of the eigenfrequencies and mode shapes are studied for different sizes and locations of the constraint and it is shown that proper choice of the number of angular modes is critical to accurate computation of the mode shapes; fewer or more number of angular modes can provide relatively accurate eigenfrequencies but completely inaccurate mode shapes. The orthogonality of distinct modes of constrained membranes is mathematically established in this work. The orthogonality property of the modes is used to compute the modal coefficients for simulation of the dynamics. In comparison to prior work, where the objective has been limited to computing the first few eigenfrequencies, this work presents a method for accurately computing the eigenfrequencies, mode shapes, and modal coefficients needed for dynamics simulation. Numerical simulations and video animations of vibration of constrained membranes are presented for arbitrary initial conditions.