Design of piezoelectric modal sensor for non-uniform Euler–Bernoulli beams with rectangular cross-section by using differential transformation method

Abstract

In this paper, a solution to the problem of finding the shape of piezoelectric modal sensors for non-uniform Euler–Bernoulli beams with rectangular cross-sections is proposed by using the differential transformation method (DTM). A general expression for designing the shape of a piezoelectric modal sensor is presented, in which the output signal of the designed sensor is proportional to the response of the target mode. The modal sensor shape is expressed as a linear function of the second spatial derivative of the structural mode shape function as well as the beam width and thickness function. Based on the DTM and employing some simple mathematical operations, the closed-form series solution of the second spatial derivative of the mode shapes for beams consisting of an arbitrary number of steps can be determined through a recursive way. The solution is obtained by solving a set of algebraic equations with only five unknown parameters. Furthermore, the method can be extended to obtain an approximate solution of the second spatial derivative of the mode shapes for any type of non-uniform beams. Then the shapes of the piezoelectric modal sensors for arbitrary non-uniform Euler–Bernoulli beam with rectangular cross-section are obtained. Finally, several numerical examples are given to demonstrate the feasibility of the proposed modal sensors with various boundary conditions.