Complex modes of vibration due to small-scale damping in a guitar topplate

Abstract

Modal analysis is one of the preeminent methods used by scientists and engineers to study vibrating structures. The frequency response functions obtained through this method, are, in general, complex-valued. There is, however, no agreed-upon interpretation given to the real and imaginary parts of these functions, even though it is acknowledged that their relative magnitude for different frequencies is related to the behaviour of the corresponding modes. A simple model is deduced to describe the shape of the spectrum associated with a finite-length time-signal. There is very good agreement between results obtained using this model and numerical results obtained for, in this case, the vibration of a guitar top-plate using finite element methods. One interpretation of the relative magnitudes of the real and imaginary parts of the frequency response functions is advanced. It is found that stationary-wave behaviour is associated with the dominance of the real or imaginary part; traveling-wave behaviour, on the other hand, occurs when the real and imaginary parts are of the same order of magnitude, as long as the scale of damping is large enough and resonance peaks in the spectrum are close enough.

Solving the Partial Differential Problems Using Maple

“…[25] considered two differential equations whose independent variables involve the partial derivatives. [26] discussed the distance functions whose expressions contain the partial derivatives, and [27] found the solutions of some type of partial differential equation. In this article, some examples are used to demonstrate the proposed calculations, and the manual calculations are verified using Maple.…”