Modelling of longitudinal dynamics of beams with variable cross-sectional area

This paper presents a new method of modelling the longitudinal dynamic behaviour of beams with variable cross-sectional area. The beam is considered as an one-dimensional distributed-parameter system, whose dynamics is described by a PDE with boundary conditions. The solution is expanded in a Fourier series, where the eigenfunctions of the eigenvalue problem for the PDE are used as basis functions. The eigenvalue problem can be rewritten as an homogeneous linear spatial-varying state space model. Its solution, represented by the transition matrix, is derived via using the unified spectral theory for LTV-systems. By defining spatial-transfer-functions, it is possible to derive the eigenfunctions and eigenfrequencies of beams with piecewise defined cross-sectional area. An example demonstrates the practicability of the new method of modelling.