Method Of Reverberation-ray Matrix Research Articles

Transient wave propagation and early short time transient responses of laminated composite cylindrical shells

The generalized ray method (GRM) and the method of reverberation ray matrix (MRRM) have been successfully used to study the transient elastic wave transmission in the beams, planar trusses, space frames and infinite layered media. In this paper, the GRM and MRRM are extended to investigate the early short time transient responses of laminated composite cylindrical shells with finite size under impact load. Using the Laplace transformation, the ray groups transmitting in the laminated cylindrical shells under the shock load are yielded by means of the boundary conditions. The reverberation-ray matrix representing the multi-reflected and scattered waves in the laminated cylindrical shells with finite structural size is deduced. Using FFT algorithm, the transient response corresponding to each ray group can be derived. Through the numerical simulations, it is seen that the early short time transient accelerations of the laminated composite cylindrical shells under impact loads are very large, but the early short time transient shear strain and displacement are very small. Furthermore, the effects of the stacking sequence, thickness of the composite cylindrical shells and different shock loads on the early short time transient responses of the laminated composite cylindrical shells with finite size are also analyzed.

Formation of longitudinal wave band structures in one-dimensional phononic crystals

Existing research on the mechanism of frequency bands in phononic crystals were without exception concerned with the band gaps (stop-bands). This paper reports, for the first time, the formation mechanism of the band structures, including both the phase constant and the attenuation constant spectra, of longitudinal waves in one-dimensional phononic crystals based on the periodic ternary rod model. Closed-form dispersion relation is obtained by our proposed method of reverberation-ray matrix. The band structures of characteristic longitudinal wave in the infinite one-dimensional phononic crystal are found to be generated from the dispersion curves of equivalent longitudinal wave in the unit cell due to zone folding effect and wave interference phenomenon. The characteristic times of longitudinal wave traversing the individual components of the unit cell as well as traversing the unit cell itself and the contrast of characteristic impedances of components are revealed to be the essential parameters determining the band structures. The deep understanding of band-structure formation provides direct scheme for adjusting frequency bands and propagation characteristics of elastic waves in one-dimensional phononic crystals. It will facilitate the design and optimization of phononic crystals to satisfy specific requirements for wave guiding/filtering and vibration control/isolation applications.

Transient responses of Timoshenko beams subject to a moving mass

The dynamic behavior of beam structures subject to a moving mass is a topic of practical importance in many research fields. In this study, the method of reverberation-ray matrix is presented to investigate the dynamic responses of an undamped Timoshenko beam subject to a moving mass. Based on Inglis’s assumption, the moving mass is simplified into a moving force and a concentrated mass fixed at the mid-span of the beam. Two dual local coordinates are introduced. Based on the theory of elastodynamics, the general wave solutions with two sets of unknown amplitude coefficients are derived in the transformed domain by the dual integral transform. From continuity conditions of forces and displacements at each joint and the compatibility conditions with respect to the dual coordinates, the unknown amplitude coefficients can be determined exactly. The transient dynamic motions for a Timoshenko beam under a moving mass are then determined numerically by inverse integral transform in which the Neumann series expansion is employed to avoid the integral singularities. Two simple numerical examples are presented and the results so obtained are compared with both the experimental and theoretical ones. It is shown that the present method can be a simple alternative for determining dynamic responses of bridges subject to a moving vehicle.

Early short time transient response of finite L-shaped Mindlin plate

The early short time transient response of an L-shaped plate is investigated by the method of reverberation ray matrix (MRRM) and the generalized ray method (GRM). The MRRM and GRM have been successfully used to study the transient elastic wave transmitting in the beams, planar trusses, space frames and infinite layered media. In the present paper, these two methods are extended to investigate the flexural wave propagation in a finite L-shaped Mindlin plate. The reverberation-ray matrix representing the multi-reflected and scattered waves in the L-shaped plate is presented. The rays in the GRM propagating in the L-shaped plate are determined by the shock source and the boundary conditions. From the numerical results, it indicates that the early short time transient responses of L-shaped plate calculated by the MRRM are the same as those by the GRM. The early short time transient acceleration responses are very large, while the early short time transient displacement responses are very small for the L-shaped plate subjected to the unit impulse. Furthermore, the effects of the different shock signals and the plate thickness on the early short time transient responses of the finite L-shaped Mindlin plates are also analyzed.

Dynamics of a prestressed Timoshenko beam subject to arbitrary external load

The free vibration and transient wave in a prestressed Rayleigh-Timoshenko beam subject to arbitrary transverse forces are analyzed by the newly developed method of reverberation-ray matrix (MRRM). The effects of shear deformation and rotational inertia are taken into consideration. With a Fourier transform technique, the general wave solutions with two sets of unknown amplitude coefficients are obtained in the transformed domain for an unbonded prestressed beam under the action of arbitrary external excitations. From the coupling at joints and the compatibility of displacements in each member, the free and forced vibration responses of a beam with various boundary conditions are finally evaluated through certain numerical algorithms. Results are presented for a simply-supported beam subject to either a point fixed load or moving load. Good agreement with the finite element method (FEM) is obtained. The present work is instructive for high-speed railway bridge design and structural health monitoring.

An electro-mechanical impedance model of a cracked composite beam with adhesively bonded piezoelectric patches

A model of a laminated composite beam including multiple non-propagating part-through surface cracks as well as installed PZT transducers is presented based on the method of reverberation-ray matrix (MRRM) in this paper. Toward determining the local flexibility characteristics induced by the individual cracks, the concept of the massless rotational spring is applied. A Timoshenko beam theory is then used to simulate the behavior of the composite beam with open cracks. As a result, transverse shear and rotatory inertia effects are included in the model. Only one-dimensional axial vibration of the PZT wafer is considered and the imperfect interfacial bonding between PZT patches and the host beam is further investigated based on a Kelvin-type viscoelastic model. Then, an accurate electro-mechanical impedance (EMI) model can be established for crack detection in laminated beams. In this model, the effects of various parameters such as the ply-angle, fibre volume fraction, crack depth and position on the EMI signatures are highlighted. Furthermore, comparison with existent numerical results is presented to validate the present analysis.

Recursive formulae for wave propagation analysis of FGM elastic plates via reverberation-ray matrix method

Through a laminated plate model, a recursive formulation for the method of reverberation-ray matrix (MRRM) is proposed for the analysis of free wave propagation in an elastic plate with material properties varying along the thickness direction. In contrast to the traditional MRRM, the present new formulation behaves well for high frequency computation, while the heavy computational cost of storage and memory can also be cut down. Numerical examples are given to analyze dispersion characteristics of waves in FGM plates, and the comparison among traditional MRRM and the recursive formulation demonstrates the significant efficiency of the proposed recursive formula.

Structural Health Monitoring Using High-Frequency Electromechanical Impedance Signatures

An overview of recent advances in electromechanical impedance- (EMI-) based structural health monitoring is presented in this paper. The basic principle of the EMI method is to use high-frequency excitation to sense the local area of a structure. Changes in impedance indicate changes in the structure, which in turn indicate that damages appear. An accurate EMI model based on the method of reverberation-ray matrix is introduced to correlate changes in the signatures to physical parameters of structures for damage detection. Comparison with other numerical results and experimental data validates the present model. A brief remark of the feasibility of implementing the EMI method is considered and the effects of some physical parameters on EMI technique are also discussed.

Generalized Reverberation Matrix Formulation for Wave Propagation in Multilayered Medium

Generalized reverberation matrix (GRM) formulation is presented to investigate elastic wave propagation in a complex multilayered solid by the combination of reverberation-ray matrix (RRM) method and stiffness matrix (SM) method. RRM method formulates a reverberation matrix, which reflects the reflection or refraction of the elastic waves in the multilayered solid. However, the dimension of RRM increases as the sublayer number increases, which may result in lower calculation efficiency of the generalized rays. SM formulation yields a system matrix of the constant dimension to promise higher calculation efficiency, but it is difficult to identify the generalized rays. In order to calculate the generalized rays in the complex multi-layered solid efficiently, the RRM formulation is applied to the interested sublayer for the evaluation of the generalized rays and SM formulation to the other sublayers, to construct a generalized reverberation matrix of the constant dimension, which is independent of the sublayer number. Numerical examples show that GRM formulation has higher calculation efficiency for the generalized rays in the complex multilayered-solid configuration compared with RRM formulation.

Vibration Mode Analysis of Frames by the Method of Reverberation Ray Matrix

The method of reverberation ray matrix (MRRM) has been developed by (Pao et al. 1999, “Dynamic Response and Wave Propagation in Plane Trusses and Frames,” AIAA J., 37(5), pp. 594–603) recently based on the theory of wave propagation for transient analysis of truss or frame structures. In this study, the MRRM is employed to obtain the frequency response function (FRF) of displacement of a frame under the action of a unit impulse load. The natural frequencies of the frame are determined from the FRF, since the curve of FRF has peak when a resonant frequency is approached. The vibration mode is retrieved from the adjoint matrix of the coefficient matrix of the governing equations of MRRM. The MRRM has advantage over numerical methods, such as finite element method (FEM), since in MRRM the frame is treated as an assembly of multiconnected beams, and exact solutions to the beam differential equations are employed to yield the system matrix of the frame. The vibration mode obtained is therefore exact. A planar frame made of 17 aluminum bars is analyzed. The vibration modes, as well as natural frequencies obtained from MRRM, coincide accurately with those obtained from FEM of ANSYS for the first a few modes; however, the difference of the frequencies between the two methods becomes a bit obvious when high order modes are examined.

Reverberation-ray matrix analysis of free vibration of piezoelectric laminates

Based on the theory of first-order ordinary differential equations, a dual relation between two solutions in the dual local coordinates for a single layer in the laminate is derived, which is further arranged in a manner that can avoid the numerical instability usually encountered in the state space method. Joint coupling relation can be established by the consideration of equilibrium and compatibility conditions at the interfaces/surfaces or nodes. The two relations correspond to the phase relation and scattering relation in the method of reverberation-ray matrix (MRRM), as demonstrated by considering the free vibration of orthotropic (or cross-ply) piezoelectric laminates in cylindrical bending. Another contribution of the paper is that the case of repeated eigenvalues of the coefficient matrix of the state equation is discussed, which has never been tackled before. The discussion completes the mathematical formulation of MRRM. The approach is first verified by comparing the results with those obtained from the state space method for laminates with simple supports. Calculations are then performed to show the dominating modes at some given frequencies and particular discussions are presented.

Recursive formulations for the method of reverberation-ray matrix and the application

A recursive formulation is proposed for the method of reverberation-ray matrix (MRRM) to exactly analyze the free vibration of a multi-span continuous rectangular Kirchhoff plate, which has two opposite simply-supported edges. In contrast to the traditional MRRM, numerical stability is achieved by using the present new formulation for high-order frequencies or/and for plates with large span-to-width ratios. The heavy computational cost of storage and memory are also cut down. An improved recursive formulation is further proposed by modifying the iterative formula to reduce the matrix inversion operations. Numerical examples are finally given to demonstrate the effectiveness and efficiency of the proposed recursive formulae.

On free wave propagation in anisotropic layered media

The method of reverberation-ray matrix (MRRM) is extended and modified for the analysis of free wave propagation in anisotropic layered elastic media. A general, numerically stable formulation is established within the state space framework. The compatibility of physical variables in local dual coordinates gives the phase relation, from which exponentially growing functions are excluded. The interface and boundary conditions lead to the scattering relation, which avoids matrix inversion operation. Numerical examples are given to show the high accuracy of the present MRRM.

Elastodynamic theory of framed structures and reverberation-ray matrix analysis

Based on one-dimensional theory of elastodynamics, the vibration and wave propagation in members of a framed structure are analyzed by the recently developed method of reverberation-ray matrix. Unidirectional traveling wave solutions for axial, torsional, and two flexural waves in six modes that are reverberated in structural members through repeated reflection and multiple scattering by joints of the structure are expressed in matrix form with two sets of unknown amplitude coefficients. From joint coupling equations and from compatibility conditions of displacements in dual coordinates of each member, the two sets of unknowns are determined in terms of a reverberation-ray matrix for given source excitations at joints. Free and forced wave motion in steady-state response can all be evaluated from the matrix solutions, and transient response is determined in one more step of Fourier inverse transform. The method is particularly effective to determine the early time transient response through Neumann series expansion of the inverse transformed solution with a special numerical algorithm.

Reverberation-ray analysis of moving or distributive loads on a non-uniform elastic bar

The method of reverberation-ray matrix has been developed and successfully applied to analyze the wave propagation in a multibranched framed structure or in a layered medium. However, the formulation is confined to the case of external concentrated loads applied at the junctions. This paper aims to extend the formulation of reverberation-ray matrix to cases of continuously distributed loads and point moving loads. To this end, a non-uniform bar subjected to these new types of loads is considered for illustration. The difference lies largely in the exact solutions, which include the particular parts due to the loads considered in this paper. The compatibility between displacements in the dual coordinates for a single member is utilized to derive the phase relations. For several types of loadings, numerical results are given and compared with the exact solutions or those obtained by other available method. Exact agreement is observed, thereby validating the present approach. The commonly adopted method that transforms distributed load to equivalent nodal forces is also discussed.

Dynamic analysis of space frames: The method of reverberation-ray matrix and the orthogonality of normal modes

The formulation of reverberation-ray matrix analysis has been proposed to study wave propagation in planar frames. It is applied here to modal analysis of complex three-dimensional framed structures, optionally with lumped masses and/or elastic supports. Furthermore, by means of Betti's reciprocity theorem, orthogonal conditions are established for different natural modes, and hence transient response analysis based on mode superposition is developed. Both the reverberation-ray matrix analysis for free vibration and the mode superposition method for transient response are illustrated by numerical examples.

Transient elastic waves in a transversely isotropic laminate impacted by axisymmetric load

A method of reverberation-ray matrix has been extended to the investigation of the field of wave propagation in a transversely isotropic laminate. By using the decomposition in a local coordinate system, any complicated waves can be separated into a departing part and an arriving part, which are expressed in the local scattering matrix at structural interface. Together with the local phase matrix, we obtain a certain wave transmitting from a layer to the neighboring one. Thus, the wave propagation in the whole laminate can be described when assembling the local information with global phase and global permutation matrices. This method is perfectly suitable for evaluating the transient waves involving a large number of generalized-rays. In this paper, the method is applied to laminate made of transversely isotropic material. Numerical results show the influence of the change of thickness and elastic constants of the layers on the wave propagation in laminate.

Effects of Causality and Joint Conditions on Method of Reverberation-Ray Matrix

We reexamine the approximations for amplitude vectors of steady-state waves in framed structures as reported previously by Pao et al. (Pao, Y. H., Howard, S. M., and Keh, D. C., Dynamic Response and Wave Propagation in Plane Trusses and Frames, AIAA Journal, Vol. 37, No. 5, 1999, pp. 594-603). The arrival and departure wave vectors have been determined exactly in the reverberation-ray matrix from the joint condition of the structure and the phase relation between them within each member. In the approximations the departure vector is obtained by truncating an infinite power series of reverberation-ray matrix to a polynomial of N degree. The arrival vector is given by another polynomial of N degree if it is calculated from the phase relation as in a previous paper, but by polynomials of (N - 1) degree if calculated from the joint conditions as in this study. The newly found set of approximation is preferred to the previous one as it satisfies the causal condition as well as the joint conditions in successive orders of approximation. The differences between these two solutions are, however, very small numerically for large orders of N.

Dynamic Bending Strains in Planar Trusses with Pinned or Rigid Joints

The method of reverberation ray matrix is applied to analyze the dynamic behavior of structural members in trusses with pinned joints subjected to suddenly applied force. The results are compared with those in planar trusses with rigid joints. Detailed calculations are made with two types of trusses, a small laboratory model made of slender aluminum bars, and a hypothetical real size Pratt truss made of structural steel. It is found that the maximum dynamic bending strains in the members of both types of trusses are very large, comparable with the dynamic axial strains in the same member. The magnitudes of bending strains in the pinned truss differs little from those in the truss with rigid joints for both types of trusses. Such a large dynamic bending strain in a structure member, which is contradictory to the static behavior of the same truss, could be caused by the inertial force of the mass in structural members of the truss.

Application of the reverberation-ray matrix to the propagation of elastic waves in a layered solid

This paper extends the newly developed method of reverberation-ray matrix (J. Sound Vibrat. 230(4) (2000) 743) to the propagation of elastic waves in a layered solid. The steady state waves generated by point source (axisymmetric problem) or a line source (plane strain problem) are expressed by the Sommerfield-weyl integrals of wave numbers. The waves radiated from the source are reflected or refracted at the interface of two adjacent layers, and the process of transmission and reflection is represented by a local scattering matrix; and the process of wave transmitting from one interface to the neighbouring one is represented by a local phase matrix. The local matrices of all layers are then stacked to form the global scattering matrix and global phase matrix of the layered medium separately. The product of these two matrices together with a global permutation matrix gives rise to the reverberation-ray matrix R, which represents the multi-reflected and transmitted steady state waves within the entire medium. The transient waves are then determined by another integration over the frequency, and the integrand of the double integral in frequency and wave number, known as the ray-integrals, contains a power series of R. The ray-integrals so formulated are particularly suitable for evaluating the transient waves involving a large number of generalized-rays by calculating the double integrals numerically as illustrated by the example of a laminated plate in this paper.