Wave motion analysis in arch structures via wavelet finite element method

• Wave motion in arch and crack is simulated by the wavelet finite element method.
• The influence of crack for wave propagation in arch structure is presented.
• Comparisons of different impulses show the narrow-band input is better.
• The dispersion and mode split are presented.
• Reciprocal length is introduced in wave motion analysis.

The application of B-spline wavelet on interval (BSWI) finite element method for wave motion analysis in arch structures is presented in this paper. Instead of traditional polynomial interpolation, scaling functions at certain scales have been adopted to form the shape functions and construct wavelet-based elements. Different from other wavelet numerical methods adding wavelets directly, the element displacement field represented by the coefficients of wavelets expansions is transformed from wavelet space to physical space via the corresponding transformation matrix. The energy functional of the arch is obtained by the generalized shell theory, and the finite element model for wave motion analysis is constructed according to Hamilton's principle and the central difference method in time domain. Taking the practical application into account, damaged arch waveguides are also investigated. Proper analysis of the responses from structure damages allows one to indicate the location very precisely. This paper mainly focuses on the crack in structures. Based on Castigliano's theorem and the Pairs equation, the local flexibility of crack is formulated for BSWI element. Numerical experiments are performed to study the effect of wave propagations in arch waveguides, that is, frequency dispersion and mode spilt in the arch. The responses of the arch with cracks are simulated under the broad-band, narrow-band and chirp excitations. In order to estimate the spatial, time and frequency concentrations of responses, the reciprocal length, time-frequency transform and correlation coefficient are introduced in this investigation.