Rayleigh-Ritz Approach Research Articles

An analytical model for wrinkle-free forming of composite laminates

Double Diaphragm Forming (DDF) of 3D shapes significantly increases production rates for laminated composite parts when compared with Automated Fibre Placement. However, it is prone to the occurrence of wrinkling defects, which can lead either to concession of part strength or expensive and wasteful part scrappage. Finite element simulation of the process is computationally costly and not suited to preliminary design for manufacture. In this work a novel analytical model is presented, based on obtaining the wrinkle profile via energy minimisation using a Rayleigh–Ritz approach, which enables rapid mechanics-based assessment of formability for the design of wrinkle-free composite products. The new modelling approach is validated against eight experimental DDF forming trials performed over C-shaped tools with double curvature using both dry Non-Crimped Fabric and unidirectional prepreg material. The model successfully predicts the three cases for which wrinkling is absent and the five cases where it occurs.

Nonlinear vibration control of interconnected functionally graded fluid-conveying pipeline

In this study, an exploratory vibration experiment is established and an interconnected functionally graded material (FGM) pipeline system is installed. This experiment represents the first time that NiTiNOL-steel wire ropes (NiTi-ST) are applied to FGM pipeline system. And the experimental results demonstrate that the NiTi-ST has a practical vibration damping effect. Based on the experiment, a further theoretical pipeline system model is created. The effects of fluid velocity, temperature difference, and clamp stiffness on the modal and response of the pipeline system are added. The modal analysis part is solved through the implementation of the Rayleigh-Ritz approach using a set of modified orthogonal polynomials. In order to verify the accuracy of the Raylei-Ritz calculations, the finite element method (FEM) is also used. The response analysis part is solved through the Galerkin truncation method (GTM) and harmonic balance method (HBM). To match the harmonic balance solutions, the Runge-Kutta method (RK) are adopted. The results of the response analysis reveal that NiTi-ST demonstrates significant nonlinear vibration control capabilities to the interconnected FGM pipeline system under various conditions.

Dynamic behavior of pin-free beam with torsional spring and hub at pinned end

This paper described the free vibration of a Euler-Bernoulli beam and discovered the first five dimensionless natural frequencies and mode shapes of motion for the pin-free beam model with a helical spring and rotating mass on the pinned side of the beam. The effect of changing the dimensionless torsional stiffness ratios was investigated, as was the effect of changing the dimensionless rotational inertia torque ratios Concerning the vibration properties of the beam. The mathematical equations that regulate the pin-free beam model were constructed using the exact analytical solution, establishing the Admissible Function and the natural frequency equation for the basic pin-free beam system. The approximate Rayleigh-Ritz approach was then used to determine the first five dimensionless natural frequencies, as well as the movement pattern for the pin-free beam model with a helical spring and a rotating mass on the side of the hinge. The results demonstrated that modifying the beam's natural frequencies is dependent on the ratios of the additional torsional spring parameters as well as the ratios rotational moment of inertia. The threshold system frequencies increase when the spring parameters are the major factor, and decrease when the inertia moment parameters are the main factor. Both play a role in influencing natural frequency to varied degrees. Unifying the final distribution of parameters might lead to optimizing the impact of parameters on the natural frequencies of the system.

Free vibration characteristics of elastic foundation-supported porous functionally graded nanoplates using Rayleigh-Ritz approach

PurposeThe purpose is to explore the free vibration behaviour of elastic foundation-supported porous functionally graded nanoplates using the Rayleigh-Ritz approach. The goal of this study is to gain a better knowledge of the dynamic response of nanoscale structures made of functionally graded materials and porous features. The Rayleigh-Ritz approach is used in this study to generate realistic mathematical models that take elastic foundation support into account. This research can contribute to the design and optimization of advanced nanomaterials with potential applications in engineering and technology by providing insights into the influence of material composition, porosity and foundation support on the vibrational properties of nanoplates.Design/methodology/approachA systematic methodology is proposed to evaluate the free vibration characteristics of elastic foundation-supported porous functionally graded nanoplates using the Rayleigh-Ritz approach. The study began by developing the mathematical model, adding material properties and establishing governing equations using the Rayleigh-Ritz approach. Numerical approaches to solve the problem are used, using finite element methods. The results are compared to current solutions or experimental data to validate the process. The results are also analysed, keeping the influence of factors on vibration characteristics in mind. The findings are summarized and avenues for future research are suggested, ensuring a robust investigation within the constraints.FindingsThe Rayleigh-Ritz technique is used to investigate the free vibration properties of elastic foundation-supported porous functionally graded nanoplates. The findings show that differences in material composition, porosity and foundation support have a significant impact on the vibrational behaviour of nanoplates. The Rayleigh-Ritz approach is good at modelling and predicting these properties. Furthermore, the study emphasizes the possibility of customizing nanoplate qualities to optimize certain vibrational responses, providing useful insights for engineering applications. These findings expand understanding of dynamic behaviours in nanoscale structures, making it easier to build innovative materials with specific features for a wide range of industrial applications.Originality/valueThe novel aspect of this research is the incorporation of elastic foundation support, porous structures and functionally graded materials into the setting of nanoplate free vibrations, utilizing the Rayleigh-Ritz technique. Few research have looked into this complex combo. By tackling complicated interactions, the research pushes boundaries, providing a unique insight into the dynamic behaviour of nanoscale objects. This novel approach allows for a better understanding of the interconnected effects of material composition, porosity and foundation support on free vibrations, paving the way for the development of tailored nanomaterials with specific vibrational properties for advanced engineering and technology applications.

Bending analysis of functionally graded plates resting on elastic foundation: A Rayleigh-Ritz approach and ANN method

This study presents a comprehensive investigation into the static bending characteristics of rectangular plates constructed from functionally graded materials resting on an elastic foundation. The research examines numerical factors that include central deflection, bending, and normal stresses, in functionally graded rectangular plates (FGRP) subjected to mixed edge constraints under various loading conditions. Classical plate theory is employed, and a computationally efficient mathematical expression for bending analysis is proposed, utilizing the energy principle. The Rayleigh-Ritz method with an algebraic function is applied to solve the governing equation, and the computation is performed using MATLAB. The study investigates the impact of aspect ratios, material property exponents, and the moduli of the Winkler and Pasternak foundations on the numerical factors of embedded FGRPs. The findings reveal that material property exponents and aspect ratios significantly affect numerical factors. The Pasternak foundation moduli demonstrate a more pronounced influence compared to the Winkler foundation moduli. This research provides valuable insights about the static analysis of FGRPs on Winkler-Pasternak elastic foundations, offering practical applications in the design and analysis of structures with similar configurations. Additionally, a salient aspect of this research is in the implementation of an artificial neural network (ANN) for the purpose of predicting the central deformation of FGRP while manipulating the input parameters. This approach is especially advantageous in engineering applications as it facilitates precise predictions for any set of input parameters without using complex computational work.

Advanced Solutions to Boundary Value Problems: A Comparative Study of Rayleigh-Ritz and the Finite Element Method

This study investigates the solution of a boundary value problem using two numerical methods: The Rayleigh-Ritz method and the Finite Element Method (FEM). The aim is to compare the performance of these methods and assess the reliability of FEM as a generalization of the Rayleigh-Ritz approach for more complex problems. The Rayleigh-Ritz method and the linear element formulation of the finite element method were employed to solve the boundary value problem. A detailed comparison of the results obtained from both methods was performed. Graphical illustrations were used to present the solutions, and potential sources of error were analyzed, including element and domain approximation errors, round-off errors, and the impact of using linear rather than quadratic elements in FEM. The solutions generated by both methods were found to be in close agreement, demonstrating that FEM is a viable alternative to the Rayleigh-Ritz method for solving boundary value problems. The minor discrepancies observed can be attributed to approximation errors and the choice of linear elements in the finite element analysis. This work highlights the applicability and effectiveness of both the Rayleigh-Ritz method and FEM in solving boundary value problems. It underscores the finite element method’s flexibility, especially in handling more complex boundary conditions and geometries, and contributes to the understanding of the factors influencing the accuracy of numerical methods in structural analysis.

Modeling and dynamic analysis of functionally graded porous spherical shell based on Chebyshev–Ritz approach

Abstract This study proposes a unified modeling method to investigate the dynamic behaviors of the functionally graded porous (FGP) spherical shell with elastic boundary conditions. First, three kinds of FGP distributed patterns are defined. Then, the first-order shear deformation theory is selected to build the governing equations of the spherical shell with elastic boundary conditions, which can be solved by the Rayleigh–Ritz approach. Moreover, Chebyshev polynomials of the third kind are selected as an admissible function to express the motion equation. With the constructed model, the correctness is verified by comparing the natural frequency and forced response obtained from both open literature and finite element method. Ultimately, the parameter study is conducted to conclude the effect of the design parameter on the dynamic characteristics of the spherical shell.

An Open Database to Evaluate the Fundamental Frequency of Historical Masonry Towers through Empirical and Physics-Based Formulations

The fundamental frequency plays a primary role in the dynamic assessment of Cultural Heritage towers. Local and global features may impact its value: geometric, material features, interaction with the soil and adjacent buildings, aging, the construction phase, and repairs. A database is assembled to study the relationship between the fundamental frequency and the slender masonry structure features. Empirical and physics-based approaches were developed to assess the fundamental frequency from different sources of information. A Rayleigh–Ritz approach is proposed and compared with a 3D finite element model. A sensitivity analysis is then performed to quantify the contribution of each feature. As expected, it is shown that the height of the tower contributes the most to the fundamental frequency. The other tower features have a second-order impact on both the fundamental frequency and the mode shape. A comparison between the different approaches shows that the Rayleigh–Ritz drastically minimizes the difference between numerical and experimental frequencies when all information is available. Empirical relations are a good compromise when less information is available.

Vibration evolution of laminated composite conical shell with arbitrary foundation in hygrothermal environment: experimental and theoretical investigation

This contribution addresses on a systematic experimental and theoretical investigations on the vibration of laminated composite conical shell with hygrothermal environment and arbitrary supports. The Donnell’s thin shell assumption is employed to derive the elastic potential energy, and the hygrothermal potential energy due to the thermal and humid environment is considered at the same time. A series of linear artificial springs are imported on the free boundaries of the conical shell to approximate the possible arbitrary support. The governing equations of motion of laminated composite conical shell in hygrothermal environment with arbitrary foundation have been derived in frame of Rayleigh-Ritz approach by utilizing a series of improved orthogonal polynomials as the admissible displacement functions. Particularly, the mechanical parameters of composite material were determined via the thermomechanical and mechanical parameter analysis instruments. Then, the convergence of the present analytical formulation was confirmed via comparing the experiment and finite element results. Finally, the influence of hygrothermal environment and boundaries stiffness on the vibration of laminated composite conical shell was investigated in detail and several interesting phenomena have been confirmed by theoretical analysis, experiment and finite element method.

Vibration behaviours of foam-filled grille composite sandwich cylindrical shells

In this study, a novel lightweight composite sandwich cylindrical shell, namely an all-composite sandwich cylindrical shell (ACSCS) with a hollow grille core (HGC) filled with polyurethane foam (PF), is designed and prepared, and its free and forced vibration behaviours are investigated for the first time. First, an analytical model of the PFHGC-ACSCSs subjected to both base excitation and single-point pulse excitation loads is developed by employing the shear deformation theory enriched by the layerwise method, virtual artificial spring technique, Rayleigh-Ritz approach, modal superposition principle, Newmark-Beta approach, and improved cross-fill equivalent theory. Furthermore, the solution procedures for the natural frequencies, mode shapes, and vibration responses in the time and frequency domains of the PFHGC-ACSCSs are defined. Convergence investigations are performed to determine the appropriate truncation numbers and stiffness values for the virtual artificial springs used in the prediction model. Both literature and experimental validations are conducted on the current model, the results of which indicate that it is reliable for predicting the concerned dynamic parameters. Finally, the effects of the critical parameters on the free and forced vibrations of the PFHGC-ACSCSs are studied. The results suggest that the vibration suppression capability could be significantly strengthened by adopting a large core-to-skin thickness ratio, high ratio of the circumferential length of the hollow grille ribs to the whole thickness, large axial and circumferential number of the hollow grille ribs, high ratio of the wall thickness to the circumferential length of the hollow grille ribs, and type of foam with a high elastic modulus. The modelling and solving techniques, preparation processes, testing methods, and conclusions of this study pave the way for the application of such shell structures.

Buckling Analysis of Laminated Stiffened Plates with Material Anisotropy Using the Rayleigh–Ritz Approach

An energy-based solution for calculating the buckling loads of partially anisotropic stiffened plates is presented, such as antisymmetric cross-ply and angle-ply laminations. A discrete approach, for the mathematical modelling and formulations of the stiffened plates, is followed. The developed formulations extend the Rayleigh–Ritz method and explore the available anisotropic unstiffened plate buckling solutions to the interesting cases of stiffened plates with some degree of material anisotropy. The examined cases consider simply supported unstiffened and stiffened plates under uniform and linearly varying compressive loading. Additionally, a reference finite element (FE) model is developed to compare the calculated buckling loads and validate the modelling approach for its accuracy. The results of the developed method are also compared with the respective experimental results for the cases where they were available in the literature. Finally, an extended discussion regarding the assumptions and restrictions of the applied Rayleigh–Ritz method is made, so that the limitations of the developed method are identified and documented.

Block-partitioned Rayleigh–Ritz method for efficient eigenpair reanalysis of large-scale finite element models

Abstract In this manuscript, we propose a new effective method for eigenpair reanalysis of large-scale finite element (FE) models. Our method utilizes the matrix block-partitioning algorithm in the Rayleigh–Ritz approach and expresses the Ritz basis matrix using thousands of block matrices of very small size. To avoid significant computational costs from the projection procedure, we derive a new formulation that uses tiny block computations instead of global matrix computations. Additionally, we present an algorithm that recognizes which blocks are changed in the modified FE model to achieve computational cost savings when computing new eigenpairs. Through selective updating for the recognized blocks, we can effectively construct the new Ritz basis matrix and the new reduced mass and stiffness matrices corresponding to the modified FE model. To demonstrate the performance of our proposed method, we solve several practical engineering problems and compare the results with those of the combined approximation method, the most well-known eigenpair reanalysis method, and ARPACK, an eigenvalue solver embedded in many numerical programs.

Vibration Analysis of Bistable Unsymmetric Laminates with Curvilinear Fiber Paths

Multistable laminates have been actively studied in recent years due to its potential applications in morphing and energy harvesting devices. Variable stiffness (VS) bistable laminates provide opportunities for further improvements in design space in comparison with constant stiffness bistable laminates. The snap-through process involving shape transition between the stable configurations is highly nonlinear in nature and exhibits rich dynamics. Exploiting the dynamic characteristics during the snap-through transition is of considerable interest in designing the morphing structural components. In this paper, we present a semi-analytical model based on Rayleigh–Ritz approach in conjunction with Hamilton’s principle to predict the natural frequencies of bistable VS laminates. The obtained results are compared with the results from the full geometrically nonlinear finite element (FE) model. The proposed FE model is further extended to study the dynamics of VS laminates subjected to external forces with different amplitudes. Subsequently, a parametric study is performed to explore the effect of different curvilinear fiber alignments on natural frequencies, mode shapes, free vibration characteristics and forced vibration characteristics (single-well and cross-well vibrations).

On the nonlinear post-critical responses of VAT sandwich beams with variable stiffness composite skins under axial compression

In this article, the nonlinear post-critical problem of Variable Angle Tow (VAT) sandwich beams with variable stiffness composite skins under axial compression is studied for the first time. The present sandwich beam model is constructed from a modified version of extended high-order sandwich panel theory (EHSAPT), in which the first-order shear deformation theory with von Kármán’s strain–displacement relation is utilized for the skin layer, whereas the high-order theory with linear strain–displacement is applied to characterize the deformation of the core layer. Based on the principal of virtual work, a system of differential equations with variable coefficients is firstly formulated for the postbuckling problem of VAT sandwich beams with variable stiffness composite skins under plane-strain state. Meanwhile, a more efficient Rayleigh–Ritz approach combined with variational formulas is employed to derive the nonlinear algebraic equations in weak forms, and thereafter a sophisticated path-tracing technique is applied to determine the postbuckling equilibrium path. Both super-critical and sub-critical postbuckling states of VAT sandwich beams can be readily captured under the present analytical model framework. The accuracy and effectiveness of the developed Rayleigh–Ritz procedure are validated by comparing against the results available in literature. Effects of variation in skin stiffness on both the critical instability and post-critical behaviours of VAT sandwich beams are discussed in numerical examples. It is found that the change in skin stiffness plays a determinative role in the formation of nonperiodic wrinkling patterns of VAT sandwich beams. In addition, the change in skin stiffness may cause global and local instability loads to be close to each other, and accordingly, the possibility of encountering interactive buckling is increased. The results presented herein may be beneficial for the design of VAT sandwich structures under compression.

Nonlinear dynamic modeling of bistable variable stiffness composite laminates

The room-temperature equilibrium stable states of cured unsymmetric composite laminates have been the focus of recent research, with a particular emphasis on shape morphing applications. It has been shown that changing the fiber orientation of unsymmetrical laminates using curvilinear fiber path description can results in a plethora of bistable configurations with an enriched design space. In bistable structures, snap-through involves transition from one stable shape to another, which is a non-linear phenomenon exhibiting rich dynamics during the shape transition. Past works involving such dynamic characteristics show encouraging potential in designing efficient morphing strategies. In this work, a novel semi-analytical model using Föppl von Kármán kinematics has been formulated to predict the non-linear dynamic characteristic of bistable variable stiffness (VS) laminates. An efficient energy formulation is adopted where the membrane and bending energies are decoupled using the semi-inverse constitutive equation. The in-plane stress resultants and the energy components are expressed in terms of curvatures using the in-plane equilibrium equations and compatibility conditions. Using Hamilton’s principle in conjunction with the Rayleigh–Ritz approach, a set of non-linear equations are generated, which is solved to obtain the dynamics of the snap-through process. The accuracy of the predicted non-linear vibration results of bistable plates from the semi-analytical model is verified using a fully non-linear finite element framework and validated exemplarily by tests on a straight fiber laminate configuration. Finally, a parametric study is performed by tailoring the VS parameters to identify the effect of different curvilinear fiber alignments on the dynamic characteristics of bistable VS laminates.

Numerical study of the accuracy and reproducibility of sound absorption measurements in reverberation rooms at low frequencies

Standardised absorption measurements in reverberation rooms suffer from relatively low reproducibility, especially at low frequencies and for highly absorptive samples. In these conditions, complete sound field diffusivity cannot be achieved. However, recent research has demonstrated that the theoretical diffuse absorption emerges as the ensemble average across a wide range of rooms with different geometries, even at very low frequencies. The present study aims to investigate the influence of the room geometry, sound source positioning, and presence of panel diffusers on the sound absorption values obtained in a specific reverberation room, as well as on the difference between those values and the theoretical diffuse values. The focus is on the lowest frequency bands.A numerical simulation approach is proposed, in which the room without sample is modelled in full detail using the finite element method and coupled to the sample with a Rayleigh–Ritz approach. The measurement of diffuse sound absorption is simulated in an efficient way using a stationary power balance approach. This approach is validated against measured data and against a detailed simulation of the impulse response. From the parametric study conducted with the model, it can be concluded that, for highly absorptive samples, good agreement with the theoretical diffuse absorption values can be obtained for certain room designs. In these rooms, the measured absorption is also less sensitive to the number and positioning of sources and diffusing elements.

Investigations on the multistability of series-connected unsymmetric laminates

Multistable composite shells have received great attention in shape-changing morphing applications due to their ability to attain more than one stable shape when cooled down from curing to room temperature. However, an individual bistable shell may not completely fulfil all the requirements of a morphing structure as they may require more than two stable states during the morphing action. Also, on several occasions, cylindrical bistable shapes can be found to be limiting for morphing applications. Therefore, studies on highly multistable non-cylindrical shapes are a subject of interest. In this study, we explore how unsymmetric laminates connected in series can result in an increased number of stable shapes. To analyze series-connected laminates, a semi-analytical model is developed using the Rayleigh-Ritz approach. The shapes predicted by the semi-analytical model have been validated using the results from a full geometrically non-linear finite element model and corresponding experimental results. A further enhancement in the design space of series-connected laminates is proposed by replacing the conventional cross-ply laminate with curvilinear fiber variable stiffness (VS) laminates as they allow the designer to tailor stiffness properties. Although VS laminates can be used to generate the similar stable configurations as that of conventional cross-ply laminates, they may result in different curvature values leading to tailored snap-through requirements. The developed finite element model is extended to account for unsymmetric variable stiffness laminates, where effect of curvilinear fiber alignment on the multistable design space is investigated.

Nonlinear aeroelastic analysis of a damped elastica-aerofoil system

This work formulates a comprehensive model of a nonlinear aeroelastic system developed for the analysis of complex aeroelastic phenomena related to structural and aerodynamic nonlinearities. The system is formulated as a two-dimensional cantilevered elastica with a rigid airfoil section firmly attached at its tip undergoing large displacements in the crosswind conditions. The system can demonstrate a wide range of domain specific as well as coupled nonlinear phenomena. The structural model is developed by means of the Rayleigh–Ritz approach, with shape functions discretizing both vertical and horizontal displacements and Lagrangian multipliers enforcing inextensibility. Damping is modeled based on a non-local strain-based mechanism in the Kelvin–Voigt arrangement. The resulting structural model is examined through studying the behavior under a follower load and with a tip-attached tendon under tension to study the shape convergence properties and the alignment of the results with known characteristics in the literature. The ONERA dynamic stall model is used to model the aerodynamics of the problem to accurately capture post-stall behavior at large deformations. The LCO responses of the aeroelastic problem are evaluated through time-marched simulations, and the combined airspeed–damping interactions are studied in this manner.

Global buckling and wrinkling of variable angle tow composite sandwich plates by a modified extended high-order sandwich plate theory

The advanced Automated Fibre Placement (AFP) manufacturing technologies make the synthesis of Variable Angle Tow (VAT) composites, which enable the design of lightweight sandwich structures to possess variable stiffness facesheets. However, both global and local instability phenomena of VAT sandwich plates under in-plane compressive loads are rarely explored till now. The objective of this article is to fill this gap by developing a Rayleigh–Ritz procedure based on a modified extended high-order sandwich plate theory (EHSAPT). The original two-dimensional EHSAPT proposed by Phan et al. (2012) is extended to a three-dimensional case with a minor modification that the first-order shear deformation theory instead of the classical Kirchhoff–Love hypothesis is adopted for each facesheet, which brings several merits such as the conciseness in derivation process and the easiness in program implementation. The Rayleigh–Ritz approach combined with the principle of minimum potential energy is employed to derive the eigenvalue equation that governs the instability problem of VAT sandwich plates under in-plane compressive loads. Both global buckling and wrinkling patterns of VAT sandwich plates can be captured under this proposed analytical model framework. Before instability analysis, the nonuniform prebuckling stresses over the entire sandwich plate are determined under the assumption of membrane prebuckling state. The usage of Lagrange multiplier method in the prebuckling regime removes the restrictions inherent in conventional Rayleigh–Ritz formulation and thus provides a general way to model in-plane boundary conditions. The accuracy and effectiveness of the developed Rayleigh–Ritz procedure are validated by comparing with previously published results and FE solutions given by ABAQUS. Effects of core thickness, core orthotropy, and fibre orientation angle of the facesheets on the instability behaviour of VAT sandwich plates are investigated. Finally, the mechanism of steering the fibre trajectory over the facesheets to improve the buckling resistance for the sandwich plate is studied and discussed.

Vibration Suppression Performance of FRP Spherical-Cylindrical Shells with Porous Graphene Platelet Coating in a Thermal Environment

In this study, the vibration suppression effect of a fiber reinforced polymer (FRP) spherical–cylindrical shell coated with the porous graphene platelet (PGP) in a thermal environment is investigated. The dynamic equilibrium equation is derived by combining the first-order shear deformation theory and the Rayleigh–Ritz approach, together with the virtual spring technology and the multi-segment partition technique. After the free and forced vibration responses of this FRP combined shell with PGP coating are solved. The model is validated by the convergence analysis and comparison of the present results and literature or finite element results for different shell structures with and without coating under various boundary conditions. Using the present model, the parametric study is conducted to explore the effects of porosity distribution type, dispersion pattern of volume fraction porosity, nanofiller weight fraction, and thickness ratio of PGP coating on the transient response. This study provides a useful model and some suggestions for better improving the vibration suppression capability of coated shell structures in a thermal environment.