Wavelet-Based Model for One-Dimensional Periodic Structure

Abstract

In this work, a wavelet based method is developed for wave propagation analysis of a generic multi-coupled one-dimensional periodic structure (PS). The formulation is based on periodicity condition and use dynamic stifiness matrix of the periodic cell obtained from flnite element. The proposed numerical scheme uses Daubechies scaling function and enables both time and frequency domain analysis under arbitrary loading conditions. This is unlike the Fourier transform based analysis which are restricted only to frequency domain analysis. Here, in frequency domain, the dispersion characteristics of the periodic structures, especially the band gap features are studied. Next, the method is implemented to simulate time domain wave response under impulse loading condition. The proposed method is able to accurately simulate the wave responses with substantially reduced computational cost. I. Introduction This paper aims at developing a generalized numerical method based on Floquet theorem 1 to simulate the wave response of multi-coupled linear one-dimensional periodic structure (PS) in both time and frequency domains. These simulations will help to understand the dynamic characteristics and explore the functionality of difierent engineering PS like periodically stifiened fuselage and wing of an aircraft, periodic layered media, bridge with repeated truss-like structures. In addition, their band gap characteristics are utilized in applications like vibration attenuation, 2 design of piezoelectric transducers. 3 With the recent advent of nano-structured materials, for example, carbon nanotubes and their composites, study of their periodic properties will help in exploring their functional properties. The proposed scheme is based on the use of Daubechies scaling function 4 as approximation bases. These basis functions are bounded in both time and frequency domains. The localized nature of the functions in time allows accurate simulation of time domain responses unlike the Fourier transform based methods which are restricted only to frequency domain analysis. However, the localization in time domain causes reduced accuracy in frequency domain. As a result, in this wavelet based method, the frequency dependent wave characteristics can be obtained only up to a certain fraction of the Nyquist frequency. 5 The formulation starts with obtaining the dynamic stifiness matrix KD of a periodic cell. Here, this matrix is obtained in the wavelet domain in contrast to the conventional dynamic stifiness matrix which is obtained in frequency domain through Fourier transform. KD can be calculated from the mass and stifiness matrices obtained from flnite element (FE) followed by wavelet transform. Next, the periodicity of the structure is considered using Floquet theorem to develop the numerical scheme for wave propagation analysis. The basis of the scheme is to solve a polynomial eigenvalue problem (PEP) in the transformed wavelet domain. The solution of PEP gives the dispersion constants as eigenvalues and the wave amplitudes as eigenvectors. The particular solution is obtained using the boundary condition of the flnite periodic structure. The method developed is then implemented flrst to obtain the dispersion relation i.e frequency dependence of the propagation constants for two examples of periodic structures. The examples considered are the periodically simply-supported Euler-Bernoulli and trusses. Next, the time domain wave responses of these structures due to broad-band impulse loading are simulated. These responses obtained through the periodicity assumption are compared with the response simulated using FE method without considering periodicity. The response obtained using the proposed method is referred as the periodic solution in the remaining parts of the paper.