A method of calculating the reflection and transmission coefficients required for controlling the nodal positions of a mode shape as desired in a three-dimensional beam structure

Abstract

This paper addresses a method of controlling the nodal positions of mode shapes as desired. The method is developed to calculate reflection and transmission coefficients at discontinuities and ends required for shaping mode shapes. The method is based on the one-dimensional Ray-Tracing method, which is a method that has been used to obtain eigenfrequencies, mode shapes, frequency response functions of a structure, based on the analysis of wave propagation on three-dimensional beam structures. Natural vibration in an undamped system refers to a state in which waves propagating on a structure are superimposed and become standing waves everywhere on the structure. Therefore, the mode shape indicates the shape of this standing wave. In the paper, a method is derived to estimate the reflection and transmission coefficients at discontinuities and ends, which is necessary to change the nodal points of the mode shape of interest. This method requires a matrix operation that changes only the values of the reflection and transmission coefficients at discontinuities and ends without changing the shape of the structure. Therefore, a rank-one reduction method via similarity transformation of matrices is developed. Finally, the effectiveness of the proposed method is confirmed through numerical examples.