Excitation of waves in elastic waveguides by piezoelectric patch actuators

Abstract

Piezoelectric patch actuators are employed for monitoring the health of structures and for performing active control of vibration. When the structure is long, slender and has constant cross-section, such as a pipe, rail or rod, it can be considered to be an infinite waveguide. The excitation of waves in waveguides may be analysed in the time domain using conventional finite element methods. This analysis is computationally very demanding as the model must be a number of wavelengths long to avoid the influence of reflections from the ends of the model. In addition, separating the contributions of the individual modes of propagation can be difficult. This paper presents a highly efficient analysis method for computing the waves excited as a function of frequency for waveguides excited by piezoelectric patch actuators. The waveguide is modelled using specially developed waveguide finite elements. These elements are formulated using a complex exponential to describe the wave propagation along the structure and finite element interpolation over the area of the element. Therefore only a two-dimensional finite element mesh covering the cross-section of the waveguide is required. The response of the waveguide to harmonic forces at the nodes which are in contact with the piezoelectric patch can be computed to produce a stiffness matrix (complex and frequency dependent) representing the waveguide. This matrix is included in a model of the piezoelectric patch developed using conventional three-dimensional piezoelectric finite elements. The response of the piezoelectric patch actuator (attached to the waveguide) to electrical excitation is computed and the displacements at the interface degrees of freedom are used to compute the forces applied to the waveguide and hence the waves excited in the waveguide. The method is applied to a patch actuator on an infinite beam of rectangular cross-section, which has been studied analytically, to verify the method. A model of a rail, excited by piezoelectric patches, is used to demonstrate the ability of the method to analyse the excitation of high frequency waves in waveguides with complex cross-sections. This modelling technique would be useful during the design of piezoelectric patches and arrays of patches for excitation of a specific mode of wave propagation