Rayleigh-Ritz Research Articles

Local Stability Analysis of a Composite Corrugated Steel Plate Pipe-Arch in Soil

The straight part of the corrugated steel plate (CSP) pipe-arch structure in soil may cause local buckling instability due to insufficient load-bearing capacity. Recently, composite CSP pipe-arch has been widely utilized to enhance structural stability, and their stability needs to be thoroughly investigated. This paper studies the local buckling stability problem of the straight part of composite CSP pipe-arch in soil by simplifying the soil support and introducing the inter-layer bonding effect. Based on elastic stability theory, a theoretical mechanical model of composite CSP pipe-arch was proposed. The Rayleigh–Ritz method and the semi-combined composite structure stiffness approximation were used to derive the critical buckling conditions for the straight part of the composite CSP pipe-arch. Through numerical calculation and influencing factors analysis, it is concluded that the critical buckling load of the straight part of the composite CSP pipe-arch structure is affected by the elastic modulus, thickness, Poisson’s ratio, rotational restraint stiffness and side length of the straight part of the material. In particular, it is found that as the inter-layer bonding coefficient increases, the critical buckling load is improved, while the critical buckling wave number is mainly influenced by the width of the straight part, elastic modulus, and inter-layer bonding coefficient. Additionally, we discussed the coupling effect of several key parameters on the stability of the structure. The results of this study offer theoretical foundations and guidance for the application of composite CSP pipe-arch in soil engineering, such as culverts, tunnels, and pipeline transportation.

A piezoelectric acoustic black hole absorber for rail vibration reduction and energy harvesting: Semi-analytical model

Theoretically, vibration energy can be converted and harvested during the process of vibration reduction, thus enabling a self-powered supply for sensor nodes. However, little research was reported on rail absorbers that combine vibration reduction and energy harvesting. Accordingly, this study aims to design an absorber based on the acoustic black hole (ABH) effect, termed the P-ABH absorber, that can simultaneously achieve broadband vibration reduction and energy harvesting. Firstly, the P-ABH absorber was preliminarily designed, and the semi-analytical model of the track with the P-ABH absorber was established by the Rayleigh-Ritz method. Secondly, the semi-analytical model was verified by the 3D finite element method, and the vibration reduction and energy harvesting performance were initially evaluated. The dimensions of the ABH structure, along with the sizes of the piezoelectric patches, were then determined. Finally, the vibration reduction and energy harvesting performances of the P-ABH absorbers fully deployed between each fastener were evaluated. The results demonstrated that the P-ABH absorber exhibited a wider vibration reduction bandwidth than the uniform piezoelectric absorber and the piezoelectric patches, with an increase in energy harvesting power by 27.47 % and 115.26 %, respectively. The vibration reduction and energy harvesting performance were significantly affected by the number of piezoelectric patches but little affected by the thickness and total length. When the P-ABH absorbers were fully attached to the rail, the overall mean square velocities at the three resonant frequencies, which required emphatic control, were reduced by 36.77 %, 98.37 %, and 3.52 %, respectively. Meanwhile, the energy harvesting performance was excellent. The P-ABH absorber presents a promising solution for improving the sustainability and cost-effectiveness of track systems.

Uncertainty sensitivity analysis for vibration properties of composite doubly-curved shallow shells using Kriging method

This paper presents a Kriging based global sensitivity analysis (GSA) method for the frequency response of displacements of composite doubly-curved shallow shells. A unified solution is utilized to develop the dynamic vibration formulation using the First-order Shear Deformation Theory (FSDT) and the Rayleigh-Ritz method. Kriging surrogate model is employed to substitute the frequency response function (FRF) of displacements. Ten parameters including materials and geometrical dimension are considered as input uncertain variables. A variance-based GSA method for dynamic model is employed to quantify the influence of each uncertain parameter. In addition, to avoid the computational burden of Monte Carlo simulation method (MCS), the presented sensitivity indices are computed analytically based on the Kriging mode, which further improves computational efficiency. Based on the convergence studies and comparison with traditional methods, the accuracy and efficiency of the present method are validated. The results shows that the frequency response of displacements exhibits greater sensitivity to changes in width, and thickness is more influential than others in the example from this article. Finally, the presented numerical results demonstrate vibration characteristics of different types of shells and observation points, which can also serve as a reference for further study on uncertainty-propagation analysis.

T-matrix of piezoelectric shunt inclusions on a thin plate

Developing piezoelectric-based plate-like metamaterials necessitates an effective modeling method to elucidate the omnidirectional wave properties of piezoelectric coupled inclusions on a thin plate. The commonly used methods, such as the transfer-matrix method and finite element method, are inadequate for analyzing the transmission and reflection of omnidirectional waves in a two-dimensional elastic medium. Unlike the existing methods, the multiple scattering method employs Bessel functions as the displacement-basis methods which can accurately describe the propagation and scattering characteristics of omnidirectional waves. This paper develops a novel T-matrix formulation to support the multiple scattering method, representing the input-output relationship between incident and reflected waves from a piezoelectric shunt inclusion on a host thin plate. The piezoelectric shunt inclusion comprises a varying-thickness substrate bonded with piezoelectric shunting patches. The T-matrix of the piezoelectric shunt inclusion is formulated by integrating the wave-based method with Rayleigh-Ritz method. The derived T-matrix is then used to semi-analytically analyze the far-field scattering and reflection properties of a single inclusion. Numerical results capture the scattering properties resulting from trapped mode resonances of the piezoelectric shunt inclusion. Additionally, the capability of the piezoelectric shunt damping to design and tune multiple critical coupling conditions for axisymmetric modes of thin plates is parametrically investigated by varying the values of inductors and resistors.

Analytical Solution to Estimate the Effect of Underlying Tunnel Excavations on the Existing Tunnel

ABSTRACTThe tunneling underlying inevitably leads to the displacement of adjacent soil, greatly influencing the deformation of the tunnel above. Most theoretical studies primarily concentrate on analyzing the mechanical equilibrium of individual tunnel sections, ignoring the energy generated by the system during tunnel deformation. Based on this, in view of the energy relationship, the overlying tunnel's deformation is simulated by using the Rayleigh–Ritz method. Further, its potential energy equation can be established based on the Vlasov foundation. The corresponding energy variational solution is solved according to the minimum potential energy principle, leading to an analytical solution for the shield tunneling‐induced overlying tunnel's response. The efficacy of the suggested method is verified by contrasting it with centrifuge experiments and field case studies derived from prior studies. Relative to the Winkler foundation model, which deviated from the suggested approach, the results derived from the suggested method show a closer correlation with the collected measurement data. Further parameter studies show that the vertical clearance and skew angle between two tunnels, the volume loss rate, and elastic modulus are significant factors affecting the tunnel behaviors due to tunneling underneath. The suggested theoretical model can be applied to forecast potential risks that an existing tunnel may encounter during the excavation of a new tunnel underlying in similar engineering projects.

Study on stable configurations and snap through of the rectangular VS bi-stable laminates with curvilinear fiber alignment

With advances in fiber placement technology, variable stiffness bistable laminates (VSBL) composited by curvilinear fiber format become a subject of intense interest for morphing structures due to their lower weight, adjustable stiffness and a richer range of stable configurations compared to straight fiber bistable laminate. Merely conducting research on square laminates is not enough allowing for practical application scenarios. This article investigates the influence of geometrical parameters of VSBL on the stable state configuration and static snap through actuated by MFC systematically. A quadratic variable curvature polynomial is used to fit the transverse displacement according to classical laminate theory. By means of von-Karman geometrical relationship and the Rayleigh-Ritz method, stable configurations of the VS bistable laminate are solved. By comparing with finite element results, the reliability and precision of the current results are verified. It shows that changing fiber orientation leads to significant effects on the steady state configuration and critical bifurcation points. Theoretical analysis of VS bistable laminates containing the smart piezoelectric material MFC is followed by analytical modeling. The voltage threshold required for actuating snap through of two stable states is acquired by smart material layer MFC. The specific reasons for the differences in the configurations of VS bistable laminates are illustrated by analyzing distribution of the stiffness of VSBL in the plane.

Vibration of composite open shell of hydrogen-electric fuselage with rectangular cutout in hygrothermal circumstances: theoretical and experimental research

An effective formulation is performed to approximate calculate the vibration of the typical composite fuselage battery compartment structures of the hydrogen-electric aircraft, which are highly sensitive to hygrothermal circumstances. The strain energy, kinetic energy and hygrothermal potential energy of CLOCSRC are deduced in frame of Donnell's shell assumption and a series of hypothetical elastic springs operating on the edges of divided components are introduced to simulate coupling connection and the elastic supports of the open shell, which contribute to total system energy in the form of elastic spring potential energy. The equations of motion for CLOCSRC subjected to distinct hygrothermal circumstances and arbitrary boundaries are eventually obtained in frame of the Rayleigh-Ritz method by means of the modified orthogonal polynomial as the displacement admissible function. After confirming the convergence and accuracy of the presented formulation by experiment and finite element method, several numerical examples are presented to investigative the effects of geometric parameters, elastic boundaries and hygrothermal circumstances on the vibration of CLOCSRC. Differ from traditional investigation, the present paper offers an effective approach to insight the hygrothermal mechanism of open shell with rectangular cutouts creatively. The relevant investigation demonstrated that the current strategy is beneficial for further research on the vibration characteristics of CLOCSRC.

HIGH-ORDER ZIG-ZAG THEORIES TO STATIC LAMINATED COMPOSITE BEAM ANALYSIS BY THE RAYLEIGH-RITZ METHOD

Technological advancements across various engineering fields have grown demand for highly efficient materials for specific applications. Laminated composite materials have emerged as a promising solution, offering a combination of favorable mechanical properties by layering different materials. Theories, such as zig-zag theories, have been developed to predict the mechanical behavior of these materials under external stresses. This study investigates the outcomes provided by the Rayleigh-Ritz variational method, employing different approximation functions, particularly focusing on unified higher-order zig-zag kinematics. All proposed shape functions agree to the reference values, but the model with polynomial shape function shows more applications possibilities. Keywords: composite materials, composite laminated beams, zig-zag theory, Rayleigh-Ritz method.

Nonlinear effect on stable state and snap-through bistability of square composite laminate

Bistable structures with the elastic instability caused by buckling are confirmed to perform significantly in micro-electromechanical systems, metamaterials, energy harvester, vibration isolation and morphing structures. The target of this paper is to explore the dynamic responses of cross-ply bistable composite laminate, focusing on the nonlinear effect of the single- and double-well vibration. Both theoretical and finite element (FE) methods are employed to simulate the nonlinear vibration and dynamic snap-through of bistable composite laminate under the foundation excitation at the center. In the theoretical model, the governing equations are established via Lagrange's equation based on the first-order shear deformation theory, von Karman nonlinear strain-displacement relation and Rayleigh-Ritz method. The fourth-order Runge-Kutta method is adopted to solve the governing equations, and the numerical results are validated by FE model. Subsequently, the details of the dynamic responses are analyzed to identify the nonlinear effects in the form of bifurcation diagram, phase portrait, time history, Poincare maps and amplitude spectrum. The dynamic responses are examined for a series of excitation parameters in both time and frequency domain. Through fixed frequency and frequency sweep tests, the nonlinear phenomenon of the single- and double-well vibrations are analyzed including superharmonic resonance, stiffness softening, hysteresis phenomenon, various periodic and chaotic vibration. The diverse responses to external inputs contribute significantly to efficiently predicting mechanical behaviors in real-world conditions, thereby offering indispensable theoretical support for structural design applications.

Dynamic characteristics and vibration control of composite laminate wall panels in electric aircraft using NiTi shape memory alloys

This paper investigates the dynamic characteristics and vibration control of a new-energy electric aircraft cockpit wall panel. An innovative boundary-splitting Rayleigh-Ritz method is introduced for modal analyses of wall panel structures in thermal environments, verified with the hammering and finite element methods, showing a maximum error of 5.42 %. NiTi shape memory alloy (SMA) wires are embedded in composite structures for vibration control, with the first theoretical-experimental validation performed. A parametric fitting-computational model for NiTi-SMA is developed to extract mechanical parameters from hysteresis data. Theoretical models are decoupled into dynamical equations using the Galerkin truncation method. The damping capability of NiTi-SMA is demonstrated by solving the amplitude using the harmonic balance and Runge-Kutta methods. Room-temperature and variable-temperature vibration tests show over 30 % vibration control capability with optimized embedding methods. These experimental results confirm the theoretical findings, providing support for the analysis and vibration control of new-energy aircraft.

Elastic Local Buckling Analysis of a Sandwich Corrugated Steel Plate Pipe-Arch in Underground Space

In underground spaces, corrugated steel plate (CSP) pipe-arches may experience local buckling instability, which can subsequently lead to the failure of the entire structure. Recently, sandwich CSP pipe-arches have been used to enhance the stability of embedded engineering outcomes, and their buckling behaviors require in-depth research. In this paper, we establish a theoretical model by simplifying soil support and using Hoff sandwich plate theory to focus on the local buckling stability of the straight segment in embedded sandwich CSP pipe-arches using the Rayleigh–Ritz method. Through stability analysis, the instability criteria for embedded sandwich CSP pipe-arches are analytically determined. Numerical calculations reveal that the critical buckling load of a sandwich CSP pipe-arch is affected by several factors, including the elastic modulus, thickness, Poisson’s ratio, rotational constraint stiffness, and the length of the straight segment. Specifically, increasing the thickness of the sandwich CSP pipe-arch can substantially enhance the critical buckling load. Meanwhile, the wavenumber is affected by the elastic modulus and the length of the straight segment. The analytical results are in agreement with those obtained from finite element analysis. These findings provide a theoretical basis and guidance for the application of sandwich CSP pipe-arches in fields such as subway stations, tunnel construction, underground passages, and underground parking facilities.

The Vibration Characteristics Analysis of Fiber-Reinforced Thin-Walled Conical–Cylindrical Composite Shells with Artificial Spring Technique

In this paper, the vibration characteristics (free vibration and forced vibration) of fiber-reinforced thin-walled conical–cylindrical composite shells (FTCCS) are analyzed by combining theory and experiment. Based on the classical shell theory and Kirchhoff–Love assumption, the overall structure of the FTCCS is theoretically modeled. The artificial spring technology is used to simulate the arbitrary boundary conditions at the joint, which is divided into a main spring and a secondary spring. At the same time, the displacement functions of the two sub-structure shells of the FTCCS are established by the orthogonal polynomial method, and the energy function of the FTCCS is constructed according to the continuity condition of the bolt joint. Then, the vibration characteristics of the FTCCS are solved using the Rayleigh–Ritz method and the modal superposition method. Finally, the TC300/epoxy resin-based fiber-reinforced thin-walled conical–cylindrical composite shells are selected as the research object, and the rationality of the mathematical model is verified by pulse, frequency sweeping and resonance excitation tests. The experimental results show that the errors between the natural frequency and the resonance displacement calculated by the theory and the experimental results are 1.75–4.76% and 3.79–9.28%, which confirms the rationality of the proposed model. By changing the length of the shell, the lamination angle and the semi-cone angle of the conical shell, the vibration characteristics parameters under different conditions are calculated to evaluate the influence of three physical parameters on the vibration characteristics of the FTCCS.

Traveling wave vibration characteristic analysis of FRC sandwich cylindrical shells with FGP-GPLRC core under discontinuous elastic boundary conditions

In this study, a fiber-reinforced composite (FRC) sandwich cylindrical shell with a functionally graded porous graphene platelet reinforced composite (FGP-GPLRC) core is investigated, and the traveling wave vibration characteristics of this new sandwich cylindrical shell under discontinuous elastic boundary conditions are analyzed for the first time. The influences of centrifugal and Coriolis forces are introduced into the model. By employing the first-order shear deformation theory (FSDT) and the continuity assumption of the layerwise theory, the governing equations of the rotating sandwich shell are obtained. The artificial springs are implemented at the ends of the sandwich shell to represent the discontinuous elastic boundary conditions. The natural frequency of the shell is solved via the Rayleigh-Ritz method and the state space method. Then, the approach proposed is validated by comparing the present results with those reported in the literature and the finite element method. Finally, the effects of various parameters on the traveling wave frequency of the shell are evaluated. It was found that in practical engineering, the appropriate thickness of the FGP-GPLRC core should be selected according to the boundary conditions and other factors, and the ply angle should be adjusted to obtain more stability of vibration.

Nonlinear dynamic analysis and vibration reduction of two sandwich beams connected by a joint with clearance

The dynamics and vibration reduction characteristics of the clamped–clamped two sandwich beams jointed with clearance is studied theoretically and experimentally. A transverse and torsional spring system with clearance is used to equivalent the joint model. The homogenization method is used to equivalent the core layer and Rayleigh-Ritz method is utilized to derive the mode function of the interconnected sandwich beam by using a sequence of orthogonal polynomials. The nonlinear motion equation of the two jointed sandwich beam structure with clearance is derived by the application of the Hamilton principle and then solved using an improved Newmark integration approach. In order to validate the accuracy of the natural frequency and vibration mode, the finite element model is established. This paper examines the impact of clearance on the amplitude frequency response and vibration transmission of the two jointed sandwich beam structure, it is found the jointed sandwich beams show obvious nonlinear characteristics and intermittent vibration transmission phenomenon due to the clearance. Moreover, the vibration transmission analysis reveals that the existed clearance demonstrates significant vibration reduction effect, for which an experiment is conducted to validate the results. In general, this work proposes a novel approach for modeling sandwich structures with clearance with improved vibration reduction performance.

Vibro-acoustic modeling of the turbulent excited cavity-plate-exterior space coupled system

To mitigate flow-induced noise in sonar self-noise, this study introduces a vibro-acoustic modeling technique for sonar cabins. This technique couples a turbulent-flow-excited plate with the interior cavity and the exterior field within a unified model. Employing the Rayleigh-Ritz method, the Chebyshev spectral method, and the Rayleigh integral, the model accounts for general boundary restraints, wall impedances, and reinforcement configurations. Flow-induced responses are determined through the Monte Carlo quadruple integration of frequency response functions and the cross-spectral density model, which characterizes turbulent pressure fluctuations. The convergence and reliability of the present method are verified, achieving excellent agreement with numerical methods and previous results. The study reveals that periodic impedance arrangements in the cavity significantly attenuate sound wave propagation, especially when periodic arrangements are added at low frequencies. The reinforcement configuration perpendicular to the flow direction minimizes low-frequency vibrations. However, the influence of reinforcement on acoustic energy within the cavity is marginal and could potentially lead to increases under simply supported boundary conditions. This method effectively addresses vibro-acoustic challenges in sonar cabins, offering substantial practical value for the development of low-noise sonar systems.

Coupled axial-torsional free vibration analysis of a rotating thin-walled composite beam in hygrothermal environments

ABSTRACT The coupled axial-torsional vibrations of a composite beam in hygrothermal environments are studied. In the existing literature, most studies investigated the torsional/axial-bending vibrations of rotating composite beams. Few works considered the axial-torsional vibrations in hygrothermal environments. By considering this, the coupled axial-torsional vibration characteristics of composite thin-walled beams in hygrothermal environments are studied. Vibration characteristics are solved using the Rayleigh-Ritz method. The accuracy of the calculation results is verified using the finite element method. The following conclusions are obtained: (1) The frequencies decrease with the hygrothermal factors and mounting angle. (2) Compared to temperature, moisture has a smaller impact on frequencies. (3) The influence of rotating speed and mounting angle is minimal and can be ignored. (4) The influences of ply angle on frequencies are more significant than other parameters. (5) The axial inertia force decreases torsional frequencies when the ply angle is 15°~75°. (6) The influences of temperature, moisture, rotating speed, and mounting angle on the first two modes are very small and can be ignored.

Vibration analysis of Ti-SiC composite airfoil blade based on machine learning

In this study, machine learning (ML) methods are integrated with Rayleigh-Ritz method and first-order shear deformation theory (FSDT) to predict the vibration properties of Ti-SiC fiber-reinforced composite airfoil blade. The natural vibration characteristics of airfoil blade are largely determined by various geometric and material parameters, which leads to the high computational cost of numerical methods. Therefore, the low-cost ML models in conjunction with Ti-SiC fiber-reinforced composite material is developed to replace traditional numerical methods in order to predict the vibration characteristics of airfoil blade. Random Forest (RF), Gradient Boosting Decision Tree (GBDT) and Back Propagation (BP) neural network models are utilized to compare the predicted results with existing data. Among these models, the BP neural network demonstrates superior performance. Additionally, the SHapley Additive exPlanation (SHAP) method is utilized to elucidate BP neural network model, facilitating the prioritization of input features. This approach offers a feasible auxiliary solution for investigating the vibration characteristics of airfoil blade.

Instability of single polymer chain in an electroelastic problem

Under electric loading produced by compliant electrodes, a dielectric elastomer is prone to material instabilities which, in a microstructural level, may connect to single polymer chain instability. To reveal if such connection exists, we aim to use an electroelastic energy model for single polymer chain to study the chain instability. We approximate curvature shape of the chain under the electric loading by using trigonometry series in a spatial coordinate. The Rayleigh–Ritz method is then applied to solve the energy equation formed by the trigonometry series. We demonstrate in the numerical examples that the instability of the chain may occur at the value of electric field with corresponding configuration of the chain close to its full length.

A shell theory approach for the analysis of metal-FRP hybrid toroidal pressure vessels

This study focuses on the analysis of metal-FRP hybrid toroidal pressure vessels (TPV) using shell theory. The analytical approaches for the design of metal-FRP TPVs considering bending effects are currently not available. Invariably the designs are done based on the most simplified approach like linear membrane theory. In this work, a potential energy functional for the metal-FRP TPV considering the membrane and bending deformations is developed using Love's shell theory. The classical laminate theory is employed to determine the stresses/strains throughout the thickness of the FRP laminate. The Rayleigh-Ritz method is adopted to solve the proposed potential energy functional. A finite element analysis (FEA) is conducted using ABAQUS to validate the proposed analytical solution. In addition, a parametric analysis is carried out to understand the influence of various parameters viz. thickness of base metal, thickness of FRP, and ratio between radii of toroid and cross-section on bending deformations in the hybrid TPV. The results from the proposed solution are in good agreement with that from FEA. Therefore, the proposed solution can be used in the analysis and design of the metal-FRP TPV irrespective of any variations in the geometric parameters. It is found that the inclusion of bending deformations can improve the accuracy of solution and the bending deformations in the base metal decreases or become negligibly small with the increase in R/r ratio and thickness of FRP. The orientation of fibers can change the direction of failure in the base metal. The important design parameters that influence the yield pressure are thickness of base metal and FRP, orientation of fibers and R/r ratio.

Free vibration analysis and multi-objective robust optimization of three-dimensional pyramidal truss core sandwich plates with interval uncertain parameters

In this study, the free vibration analysis and multi-objective robust optimization of three-dimensional pyramidal truss core sandwich plates with interval uncertain parameters are fulfilled. The numerical model for free vibration of the plate is derived by combining the three-dimensional elasticity theory and Rayleigh-Ritz method, and the validity of the model is illustrated by numerical results. On this basis, considering various uncertainties within the plate, a new uncertainty-propagation analysis method is constructed by integrating the numerical model, interval-analysis model, kriging model and optimization. The one-dimensional and multi-source uncertainty-propagation analysis of the fundamental frequency is finished using this method, and the accuracy is proved by comparing with Monte Carlo simulation results. Meanwhile, to minimize the effects of uncertainties on the performance of plates at the design stage, an objective multi-objective robust optimization model with the objectives of maximizing the fundamental frequency and minimizing the robustness factor is established. Finally, an improved pelican optimization algorithm is proposed by introducing the improved opposition-based learning strategy, tracking strategy with step control and convergence strategy. And the Pareto front and corresponding design schemes applicable to different working environments are obtained without repeating the optimization.