The Method Of Reverberation-Ray Matrix And Its Applications

Since its first publication in 1998, the method of reverberation-ray matrix (MRRM) has been successfully applied to analyzing transient responses of planar framed structures as well as transient waves in isotropic and transversely isotropic layered media. To have a thorough understanding of MRRM and extend its applicability, the mathematical foundation of MRRM is explored and the applications to some complex dynamic problems are investigated.Firstly, based on the solution of matrix differential equations, a uniform procedure for formulating MRRM is established in a strict mathematical sense along with the utility of physical essentials of the problem. The similarities and differences between MRRM and various other methods, including the spectral element method (SEM), the exact dynamic stiffness method (EDSM), the finite element method (FEM), the method of transfer matrix (MTM) and the traveling wave approach (TWA), are explained from the viewpoint of mathematical formulation as well as physical interpretation. Numerical algorithms are proposed when adopting MRRM to perform dynamic analysis of complex structures via PCs.Secondly, uniform formulations of MRRM for complex planar and space framed structures are derived in a way suitable for programming. Dynamic analyses, including modal analysis, steady-state response analysis and transient response analysis, of complex structures, are then carried out by further combining with the appropriate numerical algorithms. Numerical examples are given to validate the derivation and study the influence of various parameters of the complex framed structures on its dynamic behavior.Thirdly, based on the continuum model, the natural modes of undamped complex framed structures are proved to be orthogonal with each other in a generalized sense. This leads to an alternative method for transient response analysis of undamped framed structures via the expansion of normal modes.Finally, the method of reverberation-ray matrix is further modified for layered media based on state space formulism. Exponentially growing functions are excluded from the phase relation and matrix inversion operation is avoided in the scattering relation, guaranteeing the numerical stability in all cases. As examples, the modified method of reverberation-ray matrix (MMRRM) is applied to study the propagation of guided waves in functionally graded anisotropic elastic and piezoelectric layered media. The effect of material gradient on the characteristic of the guided waves is also shown graphically.It is indicated that MRRM (or MMRRM) bears a solid mathematical basis, a clear physical background and a uniform formulation procedure. Numerical examples show that MRRM (or MMRRM) has a high accuracy and less computational cost when applied to dynamics of complex framed structures and propagation of guided waves in FGM and FGPM layered media.