Nonlinear Spectral Methods for the Analysis of Wave Propagation

Abstract

The in∞uence of stress-strain nonlinearity on waves propagating in structures is investigated through a numerical model formulated by combining Spectral Finite Element Method (SFEM) and Multiple Scales Perturbation Techniques (MSPT). The considered approach is illustrated with the help of a simple case of one dimensional wave propagation in a nonlinear rod. However the proposed approach can be extended to complex structures such as beams and plates in order to guide the application of the nonlinear analysis as a damage assessment tool to a wide range of structures. The results show that the flrst order perturbation corresponds to a higher harmonic, equal to twice the excitation frequency. Correspondingly, the spatial variation shows a wavenumber which is double that corresponding to the excitation frequency. In addition, the flrst order solution has amplitude increasing in space as predicted by the analytical solution. Following early attempts to detect changes in vibration characteristics of the structure, current Structural Health Monitoring (SHM) approaches explore the interaction of elastic waves with damages. Detection of the initiation crack zone is one of the major challenges of SHM. The initiation of fatigue cracks is attributed to loading cycles, creation of dislocations or movement of the material along slip planes. Detection of smaller cracks and investigation of damage precursors need interrogation at higher frequencies, and the support of suitable numerical tools which allow establishing relations between measured quantities and microstructural features related to the state of health of the structure. These models are expressed by nonlinear difierential equations and a simple tool is needed to solve them. This paper investigates the application of SFEM for the analysis of wave propagation in nonlinear media. The change from the linear wave equation, due to the damage and its precursors is considered small. Perturbation techniques 1 are used to separate the equation into coe‐cients of a small parameter. The perturbed equations are solved through the SFEM 2 which formulates dynamic problems in the frequency domain through the application of suitable transforms. The basic steps in the development of SFEM can be formulated as follows. First, the governing partial difierential equations are transformed into ordinary difierential equations using a transform function. Doing this, the dimensionality of the problem is reduced by one. From the transform function point of view, two types of SFEM have been developed: one is based on fast Fourier transform (FFT) 2 and the second one is based on Wavelet transform. 3 FFT based SFEM has problems handling flnite structures (small dimensions) and is valid only when the initial conditions (displacements and velocities) are zero. The more recent application of the Wavelet transform allows overcoming the above two restrictions and avoids signal processing errors due to wraparound or aliasing. The ordinary difierential equations can be then solved in the frequency domain to obtain dynamic shape functions, which exactly reproduce the dynamic response of the structure up to the desired frequency. This allows reductions