Method Of Reverberation-ray Matrix Research Articles

Flexural wave propagation and attenuation through Timoshenko beam coupled with periodic resonators by the method of reverberation-ray matrix

A new perspective of physical understanding is presented in this paper for the propagation and attenuation behaviours of the beam coupled with periodic resonators. Wavenumber spectral relation of the beam-resonators coupling system is established based on the Timoshenko beam theory and its wave form solution, the formulation of the method of reverberation-ray matrix (MRRM) and the Bloch theorem of periodic structures. The complex wavenumbers of the coupling system are calculated by numerical techniques. The validity and accuracy of the MRRM in analysing the propagation and attenuation characteristics of the beam-resonators coupling system are verified by the analytical solution of the homogeneous beam and the numerical results of the beam coupled with periodic resonators reported in the published literature. Numerical examples are analysed to demonstrate the general wavenumber spectral characteristics of the beam-resonators coupling system, together with those of the homogeneous beam and the beam on elastic foundation. The effects of the parameters of the beam and the resonators on the wavenumber spectral characteristics are respectively evaluated. Numerical results show that the coupling between the beam and the resonators results in a local resonant attenuation band and multiple non-local resonant attenuation bands. In all these attenuation bands, the real wavenumber of the flexural wave is an integer multiple of π, and it does not vary with frequency within each of the attenuation bands. There are two resonant frequencies respectively corresponding to the translational and rotational coupling between the resonators and the beam. The effect of the parameters of the beam on the wavenumber spectral characteristics of the coupling system is mainly reflected in shifting all the attenuation bands to the higher or lower frequency ranges. The parameters of the resonators have a more significant effect on the local resonant attenuation band with respect to the non-local resonant ones. With the increase of the spacing of two adjacent resonators, the bandwidth and the attenuation factor of all the attenuation bands decrease, the number of the non-local resonant attenuation bands increases linearly, and the real wavenumber spectrum curve of the beam coupled with periodic resonators gradually converges to the real wavenumber spectrum curve of the homogeneous beam. Mergence of the local and non-local resonant attenuation bands enlarges the bandwidth of the local resonant attenuation band significantly. The occurrence of mergence of the local and non-local resonant attenuation bands is tunable by selecting appropriate structural parameters such as the mass-related parameters of the beam, the translational coupling stiffness and the spacing of two adjacent resonators. The innovative findings and practical suggestions could provide potential references for the researchers and engineers in vibration reduction design of engineering structures.

A procedure of the method of reverberation ray matrix for the buckling analysis of a thin multi-span plate

A procedure of the method of reverberation ray matrix (MRRM) is developed to perform the buckling analysis of thin multi-span rectangular plates having internal line supports or stiffeners. A computation algorithm for the reverberation ray matrix in the MRRM is derived to determine the buckling loading. Specifically, the analytical solutions are presented for the buckling of the structure having two opposite simply-supported or clamped-supported edges with spans, while the constraint condition of two remaining edges may be in any combination of free, simply-supported, and clamped boundary conditions. Furthermore, based on the analysis of matrices relating to the unknown coefficients in the solution form for the deflection in terms of buckling modal functions, some recursive equations (REs) for the MRRM are introduced to generate a reduced reverberation ray matrix with unchanged dimension when the number of spans increases, which promotes the computation efficiency. Several numerical examples are given, and the present results are compared with the known solutions to illustrate the validity and accurateness of the MRRM for the buckling analysis.

Thickness-twist waves in the nanoplates with flexoelectricity

The effects of flexoelectricity on thickness-twist waves propagating in the nanoplates are analytically investigated. Detailed calculations are performed for nonpiezoelectric material using the method of reverberation-ray matrix (MRRM), by which the numerical instability usually encountered in the conventional methods can be successfully avoided. Based on the derived characteristic equation, the dispersion relations, phase velocity and group velocity of the structures with different geometry and material parameters are subsequently plotted and compared. The numerical results show that the flexoelectric effect is really essential and cannot be neglected to predict wave propagation behavior especially in the nanoscale structures.

Dynamic analysis of laminated piezoelectric cylindrical shells

The method of reverberation-ray matrix (MRRM) has been successfully utilized to study the transient waves in beams, plates and laminated solids. In this work, the MRRM is adopted to study the transient responses and wave propagation of piezoelectric cylindrical shells with finite size. Based on the Donnell shell theory, the reverberation matrix formulation in the cylindrical shell is derived. Using Laplace transformation, the transient responses under imposed impact load can be predicted. Through the numerical simulations, the early short time transient responses can be further elucidated. Furthermore, the effects of the geometric parameters of the composite shells on the wave propagating in the laminated piezoelectric shells are also analyzed.

Free In-Plane Vibrations of Orthotropic Rectangular Plates by Using an Accurate Solution

Many numerical methods have been developed for in-plane vibration of orthotropic rectangular plates with various boundary conditions; however, the exact results for such structures with elastic boundary conditions are very scarce. Therefore, the object of this paper is to present an accurate solution for free in-plane vibration of orthotropic rectangular plates with various boundary conditions by the method of reverberation ray matrix (MRRM) and improved golden section search (IGSS) algorithm. The boundary condition studied in this paper is defined as that a set of opposite edges is with one kind of simply supported boundary conditions, while the other set is with any kind of classical and general elastic boundary conditions or their combination. Its accuracy, reliability, and efficiency are verified by some numerical examples where the results are compared with other exact solutions in the published literature and the FEA results based on the ABAQUS software. Finally, some new accurate results for free in-plane vibration of orthotropic rectangular plates with elastic boundary conditions are examined and further can be treated as the reference data for other approximate methods or accurate solutions.

Extracting torsional band gaps and transient waves in phononic crystal beams: Method and validation

Due to an absence of proper measuring techniques, torsional waves/vibrations in one-dimensional phononic crystals (PCs) or elastic metamaterials are usually theoretically or numerically studied without experimental validations. This work proposes a measurement method capable of extracting torsional band gaps and transient torsional waves from the total responses of PC and metamaterial beams. Specifically, a pair of point-wise self-demodulated fiber Bragg grating (SFBG) displacement sensors, which can be arranged at points very close to each other on a thin PC beam, is utilized along with the proposed extraction method. The SFBGs are symmetrically set up at two locations about the central line on the surface of the PC beam. The torsional band gaps and the transient torsional waves are extracted from the differential responses of the two SFBGs, in which the common bending waves are cancelled out. In addition to the torsional behaviors, the bending waves can also be extracted from the summation responses of the two SFBGs. We first examine the dynamic linearity of the SFBGs. Then, steady-state detection of the torsional and bending band gaps of a PC beam is performed. Finally, a cantilever PC beam subjected to an eccentric steel ball impact, whose loading history is recorded by a pair of polyvinylidene fluoride (PVDF) films, is investigated to demonstrate the extraction of the torsional/bending bandgaps and waves from the early short time transient responses. The feasibility of the proposed torsional (or bending) extraction method is justified by the method of reverberation-ray matrix (MRRM), which is capable of providing “pure” torsional (or bending) behaviors of the PC beam. With the proposed extraction method, we also observe surface localized modes in the transmission of the PC beam, which are strongly related to the arrangement of the constituent components with different impedances in the unit cells. We believe that the proposed extraction method for torsional wave, along with the SFBG sensing technique and the MRRM, can provide future researchers with measurement and analysis method to validate their designs of torsional phononic crystals or metamaterials.

Investigation on dynamic performances of a set of composite laminated plate system under the influences of boundary and coupling conditions

• The MRRM method is adopted to analyze the combination of composite laminated plate system. • A simple first-order shear deformation plate theory (S-FSDPT) is proposed. • The dynamic performance of the composite laminated plate system is discussed. • Parametric studies of the structural parameters are carried out. The method of reverberation-ray matrix (MRRM) is adopted to analyze the dynamic performance including the free vibration and power-low transfer characteristics of the combination of composite laminated plate system with arbitrary boundary and coupling boundary conditions. The classical plate theory (CPT) and a simple order shear deformation plate theory (S-FSDPT) are adopted. The artificial virtual boundary condition technology is taken to achieve arbitrary boundary conditions. At the coupling edges of the system, the generalized forces and displacements in local coordinates are transformed into the global coordinate of each component according to the coordinate transformation relationship. By using the artificial virtual coupling technology, the support springs at the coupling boundary are evenly arranged in the generalized displacement direction under the global coordinates. By changing each spring stiffness value to satisfy the generalized equilibrium relationship and the generalized displacement coordination conditions, thereby arbitrary coupling conditions of the coupled plate system are achieved. Through the numerical calculation of the common “L” type, “T” type and “ □ ” type coupled plate structure, the superior calculation performance of the method is proved, and it is not limited by the number of coupling plates and the coupling angle. On this basis, the effects of material parameters and coupling parameters on power flow and its transfer efficiency are discussed separately.

Vibration isolation of saturated foundations by functionally graded wave impeding block under a moving load

Based on the Biot’s model for saturated porous media and functionally graded materials (FGMs), the isolation effect of saturated porous FG wave impeding block (WIB) on reducing vibration induced by a moving load in a saturated foundation is studied. The dynamic governing equations for saturated foundations under a moving load are obtained by using the coordinate transformation, and the expression of displacement and stress is derived based on the Fourier transform and the reverberation-ray matrix method. Via numerical examples, the isolation effect of saturated porous FG WIB under a moving load is analyzed. The results show that the saturated porous FG WIB can effectively reduce the vibration amplitude and has more advantages and effective under a moving load.

Gauge sensitivity analysis and optimization of the modular automotive body with different loadings

The structural optimization method based on sensitivity analysis is an effective way to reduce the mass of automotive body structure. In this paper, an effective structural optimization method is proposed to facilitate the lightweight design of modular automotive body, where gauge sensitivity analysis values are used to determine the optimized direction to increase or decrease the thickness of each beam element in the body model. An object-oriented MATLAB toolbox constructed in our previous study is adopted as a black box for the structural analysis and optimization of the body-in-white (BIW) model, in which the beam element sensitivity values of BIW structure under different loading conditions are fast calculated by the method of reverberation ray matrix (MRRM) and the finite difference method (FDM). Then, the optimized direction of each beam element is identified, and a structural optimization model is formulated and solved by the genetic algorithm (GA). In order to verify the effectiveness of this method, a simplified modular automotive body model is constructed to implement the performance indexes comparison between the initial body model and optimized body model. The analysis results show that this method is feasible and effective for the optimal design of modular automotive body structure.

An exact solution for free vibration of cross-ply laminated composite cylindrical shells with elastic restraint ends

In this paper, a new exact solution based on the classical shell theory (CST) for free vibration of cross-ply laminated composite cylindrical shells with elastic restraint ends is proposed. The present exact solution can be summed up in the following steps: Firstly, the displacement functions are constructed by the governing differential equations with the exact closed form solutions; Then, the artificial spring technology is introduced to simulate the general boundary conditions of the two end edges of shell; Thirdly, the equation for natural frequencies is obtained by means of the method of reverberation-ray matrix (MRRM); Lastly, the vibration results are presented by the modified golden section search (MGSS) algorithm. By comparing the present method with published papers, the accuracy of present method is verified. On the basis of that, some new exact nature frequencies and mode shapes of the cross-ply laminated composite cylindrical shells with various classical boundary conditions and elastic restraints are performed and they can be served as the benchmark data for the future.

A closed form solution for free vibration of orthotropic circular cylindrical shells with general boundary conditions

In the past decades, the exact closed form solutions for the free vibration of thin orthotropic circular cylindrical shells have been merely restricted to some classical boundary conditions. Therefore, the target of the current paper is to present a new exact closed form solution for free vibration of orthotropic circular cylindrical shells with general boundary conditions by means of the method of reverberation-ray matrix (MRRM). Based on the Donnell–Mushtari shell theory, the wave solutions are constructed by the exact closed form solutions of the governing differential equations. The artificial spring technology is introduced to achieve the general boundary conditions of two end edges of shell. Hereby, the reverberation ray matrix can be easily obtained by using the MRRM together with the wave solutions, boundary conditions and dual coordinates of the orthotropic circular cylindrical shells. Then, the vibration results are obtained from the extrapolation method and golden section search (GSS) algorithm. By the comparison with other published methods and the finite element method, the accuracy of the present method is verified. On the basis of that, some new exact nature frequencies of the orthotropic circular cylindrical shells with general elastic restraints are shown which can serve as the benchmark data for the future computing method.

Analysis of Isolation Ground Vibration by Graded Wave Impeding Block under a Moving Load

Combined with the coordinate transformation, the dynamic governing equations for elastic foundation under a moving load are established by using the reverberation ray matrix method, and the isolation effect of a graded wave impeding block is studied. The displacement and the stress are obtained in the time domain by using numerical inverse Fourier transformation. Via numerical examples, the effectiveness of vibration isolation of graded wave impeding block is compared to conventional single phase solid homogenous wave impeding block. The results show that compared with the single phase solid homogenous wave impeding block isolation system, the graded wave impeding block can effectively reduce the vibration amplitude and has more advantages and is effective under a moving load.

A self-demodulated fiber Bragg grating for investigating impact-induced transient responses of phononic crystal beams

This work presents a fiber Bragg grating (FBG) displacement sensing system to experimentally investigate the dynamic behaviors of phononic crystal (PC) beams through impact-induced early short time or long time transient responses. Based on the couple-mode theory and the optical transfer matrix (T-matrix) formulation, we first show that it is feasible to achieve linear displacement sensing using a single FBG without extra demodulators such as matching gratings. To validate its effectiveness, the proposed self-demodulated FBG system is applied to measure the point-wise transient displacement responses of a cantilever PC beam subjected to steel ball impacts. To demonstrate the transient sensing performance, the measured early short time transient responses are compared with the corresponding ones predicted by the method of reverberation-ray matrix (MRRM) theoretically and the finite element method (FEM) numerically. The excellent agreements verify together the experimental, theoretical and numerical results. Finally, conducting Fast-Fourier transform (FFT) to the early short time or long time transient responses gives the frequency responses that clearly show the existence of the band gaps. In addition to providing a new displacement sensing method using the self-demodulated FBGs, this work also offers another route to investigate the band structures of PC beams through the frequency responses transformed from the early short time transient responses, long before the structural damping becomes dominant.

Reverberation-Ray Matrix Analysis and Interpretation of Bending Waves in Bi-Coupled Periodic Multi-Component Beams

Most existing research on periodic beams concerns bending waves in mono-coupled and bi-coupled periodic mono-component beams with the unit cell containing only one beam segment, and very few works on bi-coupled periodic multi-component beams with the unit cell containing more than one beam segments study the bending waves in structures with only binary unit cells. This paper presents the method of reverberation-ray matrix (MRRM) as an alternative theoretical method for analyzing the dispersion characteristics of bending waves with the wavelength greater than the size of the cross-sections of all components in bi-coupled periodic multi-component beams. The formulation of MRRM is proposed in detail with its numerically well-conditioned property being emphasized, which is validated through comparison of the results obtained with the counterpart results by other methods for exemplified bi-coupled periodic beams. Numerical examples are also provided to illustrate the comprehensive dispersion curves represented as the relations between any two among three in frequency, wavenumber (wavelength) and phase-velocity for summarizing the general features of the dispersion characteristics of bending waves in bi-coupled periodic multi-component beams. The effects of the geometrical and material parameters of constituent beams and the unit-cell configuration on the band structures are also demonstrated by numerical examples. The most innovative finding indicated from the dispersion curves is that the frequencies corresponding to the Brillouin zone boundary may not be the demarcation between the pass-band and stop-band for bending waves in bi-coupled periodic multi-component beams.

An efficient structural optimization approach for the modular automotive body conceptual design

Appropriate structural analysis and optimization methods are of great significance for automotive body in conceptual design stage. This paper proposes a mathematical simulation approach to promote the conceptual design of modular body-in-white (BIW) model fast and effectively, which is based on the method of reverberation ray matrix (MRRM) and the genetic algorithm (GA). A simplified modular BIW conceptual model is used to predict early-stage static and dynamic characteristics, and a minimum mass of the BIW model is set as an optimization objective. Then the static and dynamic behaviors of auto-body are analyzed by the MRRM, and a cross-sectional size optimization model for the BIW structure is formulated and solved by the GA. Afterward, an object-oriented MATLAB toolbox is constructed to achieve the mathematical simulation method for the fast modeling analysis and the structural optimization processes of the modular BIW conceptual structure. Lastly, the validity of this structural optimization approach is demonstrated by a modular automotive body conceptual model.

Analysis of Bending Waves in Phononic Crystal Beams with Defects

Existing investigations on imperfect phononic crystal beams mainly concern periodic multi-span beams carrying either one or two channel waves with random or deterministic disorder in span-length. This paper studies the two channel bending waves in phononic crystal beams consisting of many phases of materials with defects introduced as one structural segment having different cross-sectional dimensions or material parameters. The method of reverberation-ray matrix (MRRM) based on the Timoshenko beam theory, which can conduct high-frequency analysis, is extended for the theoretical analysis of dispersion and transmission of bending waves. The supercell technique and the Floquet–Bloch theorem are adopted for modeling the dispersion characteristics, and the whole finite structural model is used to calculate the transmission spectra. Experimental measurements and numerical calculations are provided to validate the displacement transmission obtained by the proposed MRRM, with the effect of damping on transmission spectra being concerned. The high-frequency calculation applicability of the proposed MRRM is also confirmed by comparing the present results with the corresponding ones either using the transfer matrix method (TMM) or MRRM based on Euler—Bernoulli beam theory. The influences of defect size, defect form, and unit-cell number on the transmission spectra and the band structures are discussed. The drawn conclusions may be useful for designing or evaluating the defected phononic crystal beams in bending wave control. In addition, our conclusions are especially potential for identifying the defect location through bending wave signals.

Dispersion curves of magneto-electro-elastic imperfect cylinders filled with fluid

Imperfect interfaces due to manufacturing errors can affect the dynamic performance of multiferroic composite cylinders, especially in the high frequency range. However, the traditional transfer matrix method will encounter numerical instability when high frequency is involved. In this paper, an alternative approach called the reverberation-ray matrix method is presented for the wave propagation of a fluid-filled multiferroic composite cylinder with imperfect interfaces, which is unconditionally numerically stable for high frequency analysis. Based on the three-dimensional magneto-electro-elastic theory and the Hamiltonian state equation, the reverberation-ray formulations are established to analyze the dispersion relations of laminate composite cylinders. The effects of interfacial imperfection as well as different material and geometric parameters on the dispersion curves of multiferroic composite cylinders are investigated. The present method could be useful for the analysis and damage detection of layered composite cylinders.

A Hybrid Method for Transverse Vibration of Multi-Span Functionally Graded Material Pipes Conveying Fluid with Various Volume Fraction Laws

Both functionally graded materials (FGMs) and fluid-conveying pipes have wide applications in engineering communities. In this paper, the transverse vibration and stability of multi-span viscoelastic FGM pipes conveying fluid are investigated. Volume fraction laws including power law, sigmoid law and exponential law are introduced to describe the variations of material properties in FGM pipes. A hybrid method which combines reverberation-ray matrix method and wave propagation method is developed to calculate the natural frequencies, and the results determined by present method are compared with the existing results in literature. Then, a comparative study is performed to investigate the effects of fluid velocity, volume fraction laws and internal damping on transverse vibration and stability of the FGM pipes conveying fluid. The results demonstrate that the present method has high precision in dynamic analysis of multi-span pipes conveying fluid. It is also found that natural frequencies of FGM pipes can be adjusted by devising the volume fractions laws. This particular feature can be tailored to fulfill the special applications in engineering.

A simple first-order shear deformation shell theory for vibration analysis of composite laminated open cylindrical shells with general boundary conditions

This paper presents, for the first time, a simple first-order shear deformation shell theory (S-FSDST) for free and transient vibration analysis of composite laminated open cylindrical shells with general boundary conditions. By partitioning the radial displacement into bending and shear components, the present theory contains only four unknowns and can be regarded as an enhanced classical shell theory with the consideration of the effects of shear deformation and rotary inertia terms. The governing equations and appropriate boundary conditions are derived from Hamilton’s principle. To obtain natural frequencies and transient responses accurately, the method of reverberation ray matrix (MRRM) is employed based on the obtained exact closed-form solutions. The artificial spring technology is adopted to achieve the general boundary conditions. Accordingly, the scattering matrix is redefined in MRRM to make it suitable for different boundary cases. The excellent accuracy, reliability and efficiency of the present theory and approach are verified by examining the free and transient vibrations of composite laminated open cylindrical shells under various combinations of classical and non-classical boundary conditions. Meanwhile, a variety of new parameter studies regarding the influence of the boundary conditions, geometry parameters, lamina number, material properties and loading forms are performed in detail.

Vibration Analysis and Energy Transmission of Finite Coupled Mindlin Plates

Power flow analysis of finite coupled Mindlin plates and energy transmission through the structure are investigated by employing the method of reverberation-ray matrix (MRRM). The rectangular Mindlin plates are connected at an arbitrary angle. Both in-plane and out-of-plane waves propagation solutions are considered by establishing the dual local coordinates in each plate. The boundary conditions at the plate edges, continuous conditions at the driving force locations, and coupling conditions at the line junction between several rectangular plates are established and solved simultaneously. Then the flexural and in-plane vibrations of the finite coupled Mindlin plate are obtained by using the MRRM, which are verified by comparing the results obtained with those by the finite element method (FEM). The vibration behaviors of coupled plates such as L-shaped structure, T-shaped structure and box-shaped structure are calculated and verified.