Lee S., Mace B., Brennan M.

A generalised approach based on reflection, transmission and propagation of waves is applied for the analysis of non-uniform Euler-Bernoulli beams whose properties vary rapidly but deterministically. The variation of the properties with position is such that no wave reflection occurs. Examples are given that include an Euler-Bernoulli beam with geometric variation that can be described by a polynomial. The state vector in the physical domain is transformed to the wave domain using the displacement and internal force matrices. The wave amplitudes at one point are then related to those at another point by the propagation matrix, which is diagonal for the cases considered. By normalising the elements of the propagation matrix with respect to energy, their magnitudes are less than or equal to unity, so that the problem is always well-posed. The energy transport velocity, at which energy is transported by the waves, is derived using the relationship between power and energy. It is shown that this energy velocity decreases as waves move toward the vertex. A numerical example for the wave transmission through a rectangular connector with linearly tapered thickness and constant width is presented. This well-conditioned approach can be used to predict the transmission of vibration through the connector without any approximation errors and at a low computational cost, irrespective of the frequency.