Dynamic Analysis of a Beam with Additional Auxiliary Mass Spatial Via Spectral Element Method

Abstract

This paper aims to propose a new spectral element with additional mass. Methodologies for structural health monitoring are used to include additional auxiliary mass in the structure to change of natural frequencies. Therefore, the additional auxiliary mass can enhance the effects of discontinuities in the structural dynamics response, which could improve the identification and location of the discontinuities. The proposed approach deals with the wave propagation in structures regarding the spectral analysis method. The change in the natural frequencies due the mass is examined by comparing the differences between the dynamic responses of the beam with and without additional auxiliary mass. Similar analyses also performed with the Galerkin assumed modes technique to validate the new spectral element. The proposed technique is validated with numerical simulation and then compared to experimental data.

Appendix

According to Galerkin’s method [15], the solution of Eq. (1) can be expanded as function of:

$$\begin{aligned} w (x,t) \cong w_h (x,t) = \sum ^N_{j=1} {{\psi }_j (x) {\ q}_j (t)} \end{aligned}$$

(15)

where ${\psi }_j (x)$ are test functions in domain $\mathbb {D} = [0,L]$ that satisfying boundary conditions of problem, $q_j (t)$ are generalized coordinates of discretized system. By substituting (15) into (1) to obtain residual function $\mathscr {R}$. As base functions are not exact solutions of problem (1), the residual function$\ \ \mathscr {R}$ results a non-null function. But, according to Garlerkin’s method, a kind of weighted residual method [6, 10], we search minimize the residual function $\mathscr {R}$ in domain $\mathscr {D} \in [0,L]$,

$$\begin{aligned} \int _{\mathscr {D}} {\mathscr {R} \left[ w_h (x,t) \right] \cdot {\psi }_i (x) dx} = 0, \ \ \ \ \ \ i=1,2,\dots ,N \end{aligned}$$

(16)

This problem can be solved by a numerical time integrator, e.g., Newmark method, Hilbert-Hughes-Taylor method, or a simple Runge-Kutta.