Static and dynamic analysis of repetitive structures

Abstract

In this thesis, a one-dimensional (beam-like) repetitive pin-jointed structure with point masses located at nodal cross-sections is analysed with the aim to provide a simple physical explanation for the existence of frequency-propagation zones and decay zones. Two forms of dynamic transfer matrix are derived for the model structure: the displacement-force transfer matrix G, and the displacement-displacement transfer matrix H; the focus is on the relationships between the two, including their respective (dis)advantages. Similarity matrices are introduced to relate G and H, together with their respective metrics. Symplectic orthogonality relationships for the eigenvectors of both G and H are derived, together with relationships between their respective sets of eigenvectors. New expressions for the group velocity are derived. For repetitive structures of finite length, natural frequency equations are derived employing both G and H, including phase-closure and the direct application of boundary (end) conditions. The wave propagation and decay characteristics of the model structure are described in depth. The existence of propagation zones is explained in terms of phase-closure (implying a natural frequency) over the cross-section at the cut-on frequency, and phase-closure over the smallest axial unit – the repeating cell – at the cut-off frequency; these zones can therefore be regarded as extended resonances. Wave interaction between branches displaying normal and anomalous dispersion is explained in terms of the Krein signature, and leads to attenuating waves. At frequencies below cut-on, waves are generally monotonic evanescent, while above cut-off they are generally oscillatory evanescent. Energetics of different wave types under the new taxonomy is investigated. Equations for energy and power are derived in terms of eigenvectors of G and H. Numerical results for axial phase velocity and group velocity of the different modes show some familiar and some peculiar phenomena. Condition for maximum group velocity is proposed. Numerical study on anomalous dispersion reveals that pin-jointed structure which allows negative eigenvalue to occur under static self-equilibrating load will presage anomalous dispersion under dynamic condition. The solution of a two point boundary value problem is numerically stable when one employs the Riccati transfer matrix method. An alternative numerically stable transfer matrix method, which is more direct and transparent, is developed for a repetitive structure fixed at one end, and subject to point-wise distributed loading, with and without an intermediate support. This is achieved by constructing mixed column vectors of participation coefficients, so that spatial evolution involves multiplication by powers of the eigenvalues which are less than or equal to unity.