Rayleigh-Ritz Technique Research Articles

Non-homogeneity effect on the vibration of the rectangular visco-elastic plate subjected to the linear temperature effect with quadratic thickness variation in both directions

The present work investigates the effect of non-homogeneity on the vibration of a rectangular visco-elastic plate. The plate is subjected to linear temperature variation in the x-direction with quadratic thickness variation in both directions. The quadratic variation has been considered in the material density of the plate only along the x-axis, and it is assumed that non-homogeneity transpires because of this variation. The governing differential equation is solved using the Rayleigh-Ritz technique. All four edges of the plate should be clamped to drive the frequency equation. Deflection and time period have been evaluated for several combinations of values of the thermal constant, constant of nonhomogeneity, taper constants, and length-to-width ratio (aspect ratio) for the first two vibrationmodes of the clamped plate. The results presented are compared with those found in the literature.

A comprehensive study of elastically constrained beams under compression and tensile forces employing analytical and numerical approaches

This research investigates elastically constrained beams, including Timoshenko, shear, Rayleigh, and Euler–Bernoulli beams, positioned on diverse foundations such as Pasternak, Hetényi, and Filonenko–Borodich, and subjected to compression and tensile forces. The significance of this study lies in providing a critical dynamics response to the behavior of various beam types under varying foundation conditions, elucidating the influences of important factors such as shear modulus, flexural rigidity, foundation stiffness, rotatory effect, and shear deformation effects. The determination of the dynamical behavior of the problem is particularly challenging, dealing simultaneously with four engineering theories of beams positioned at different foundations under elastic constraints. The problem has been analytically solved by extending the separation of variables approach, involving coupled eigenvalues in the dispersion relation and leading to the development of the spatial matrix for slopes and displacements, along with their derivatives. The complexity of the problem necessitates a rigorous procedure for coupling eigenvalues, deviating from the standard approach. Consequently, analytical handling proves challenging and relatively sensitive to variations. Therefore, a numerical scheme using Galerkin finite element method (GFEM) has been obtained, recognized for its accuracy compared to other techniques like Differential Transform Technique and Rayleigh Ritz technique in finite element methods. The GFEM approach employed here offers a relatively simple and convenient solution, demonstrating effective convergence to exact solutions without shear locking. This further opens a pathway for delving into more intricate and challenging problems in beam dynamics within the scope of investigation. The obtained results indicate that beam eigenfrequencies of different models are higher under tensile forces and decrease with reduced compression forces. The study provides valuable insights for ideal design under specific conditions, highlighting the importance of incorporating foundation characteristics, compression, and tensile forces in beam dynamics.

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.

Vibrational analysis of orthotropic rectangular plate having combination of circular thickness and parabolic temperature

The objective of this study is to investigate the natural vibration of a rectangular plate made of orthotropic material with circular thickness (two dimensions) and temperature variation on the plate is parabolic (two dimensions) in nature. The solution to the problem is obtained by utilizing the Rayleigh-Ritz technique and the first four frequency modes are obtained under clamped edge conditions. The study aims to provide numerical data that demonstrate how circular variation in tapering parameters of plate can effectively control and optimized vibrational frequencies of the plate. Orthotropic rectangular plate, thermal gradient, circular tapering, aspect ratio.

Vibrational frequency of triangular plate having circular thickness

In the current research, modes of frequency of isotropic tapered triangular plate having 1-D (one dimensional) circular thickness and 1-D (one dimensional) linear temperature profile for clamped boundary conditions are discussed. Authors implemented Rayleigh Ritz technique to solve the frequency equation of isotropic triangular plate and computed the first four modes with a distinct combination of plate parameters. Authors have performed the convergence study of modes of frequency of the isotropic triangular plate. Also, conducted comparative analysis of modes of frequency of the current study with available published papers and the results presented in tabular form. The aim of the present study is to show the impact of a one dimensional circular thickness and one dimensional linear temperature on modes of frequency of vibration of an isotropic tapered triangular plate.

Effect of Graphene Reinforcement on Free Vibration and Material Properties of the FG-GPLRC Porous Cantilever Torsional Plate

This paper researches the free vibrations and corresponding material parameters of a functional gradient graphene platelets-reinforced composite (FG-GPLRC) cantilever torsional plates with both pore and graphene variations in the transverse direction. Utilizing the feature of closed-cell cellular solids, the mixture rule, and the modified Halpin–Tsai model, the material parameters of the composite are determined for different volume fractions of component materials. Then, using the classical plate theory (CLPT), the Rayleigh–Ritz technique and polynomials, the dynamic equation that can be used to obtain the free vibration mode shapes and frequencies of the rotating cantilever torsional plate is given. Comparison studies with the previous calculation results from available literature and the finite element (FE) models of cantilever plates are conducted, and the correctness of the present theoretical formulation and numerical calculation is verified. Finally, the effects of graphene platelet (GPL) distribution, porosity distribution (PD), GPL content, rotational speed, and average geometric size of GPL on free vibrations of the system are studied in depth.

The vibration of the double-plated structure containing the inner fluid and elastic connections

Analytical studies of the vibration characteristics of double-plated structure containing the inner fluid and elastic connections have been performed. An analytical model is first derived to predict the vibration characteristics of the double-plated structure. The transverse displacements of plate and the sound pressure of inner fluid are described by the improved Fourier series. Elastic connections are regarded as springs. In order to simulate arbitrary boundary conditions, the virtual spring technology is applied along each side of the plates. The vibration and sound pressure can be obtained by the Fourier coefficients which calculated by the Rayleigh-Ritz technique. Results from the finite element method (FEM) are well consistent with ones calculated by the analytical method. The effects of parameters on vibration are studied by analytical analyses. It is found that the inner air shows well vibration isolation, while inner water acts as a short circuit in vibration transmission. The stiffness of the elastic connection has significant effects only when the inner fluid is air. When the inner fluid is water, the effect of the spring will be greatly weakened.

Analytical and experimental studies for the vibration transmission of the double-plated structure containing the inner fluid and elastic connections

The vibration transmission of the double-plated structure containing the inner fluid and elastic connections are researched by the analytical and experimental techniques. An analytical method is firstly proposed to predict vibration responses of the double-plated structure. The improved Fourier series are used to describe the transverse displacements of the plates and the sound pressure of the acoustic cavity. The elastic connections are regarded as springs. Motion equations are derived by the energy method and vibration results are gained via the Rayleigh–Ritz technique. Meanwhile, a test sample of the double-plated structure is manufactured. Four vibration isolators are utilized as the elastic connections. The inner fluid is air or water. Forced vibration tests are conducted with a random white-noise force. The results from both experiments and the finite element method (FEM) are well consistent with ones calculated by the analytical method. Then the effects of important parameters on the vibration transmission are studied by the analytical and experimental analyses, including the stiffness of elastic connections, the height of the inner fluid, the boundary condition of plates, the spring position, the inner fluid loading and the coating damping layer. It is found that the inner water acts as a short circuit in the vibration transmission, and the inner air leads to the less vibration transmission from the upper plate to the lower one. Some other new phenomena are also observed and analyzed in this paper.

Time Period Analysis of Orthotropic Skew Plate with 2-D Circular Thickness and 1-D Circular Density

In the present study, the time period of vibration of an orthotropic parallelogram plate with 2 − D (two-dimensional) circular thickness under the effect of 2 − D parabolic temperature is investigated for the first time. The different edge conditions are C C C C , C C C F , C F C F , C S C F , and S F S F boundary conditions, where C, S, and F stands for clamped, simply supported, and free edges of the plate, respectively. The variation in density of plate material is considered to be 1 − D (one-dimensional) circular. The Rayleigh–Ritz technique is used to solve the differential equation and evaluate the time period for the first two modes of vibration. A convergence study of an orthotropic parallelogram, rectangle plate, and square plate for modes of frequency at various edge conditions is also carried out. The authors performed a comparative analysis of the time period and modes of frequency of orthotropic parallelogram, rectangle plate, and isotropic square plate with the available published results at various edge conditions. The main conclusion which we made from this study is that by choosing the above-mentioned plate parameters, we obtained fewer modes of frequency in comparison to other variations mentioned in the literature. Also, the study suggests that the variation in modes of frequency is less in comparison to other variations.

Thermal Vibration of Rectangle Plate with Two-Dimensional Circular Thickness Effect

Frequencies comprising two-dimensional circular thickness impacts are computed for the first time, for clamped rectangle plate under nonhomogeneity and temperature field. The study also comprises the two-dimensional temperature and one-dimensional density effect, on frequencies computation, utilizing the Rayleigh–Ritz technique. The objective of the study is to reduce the frequencies of the plate as well as to provide the effect of various plate parameters on frequencies. Numerical illustration and comparative study of the frequencies with other available results well authenticate the objective of the study. A comparison of frequency modes done for the clamped rectangle and square plate corresponding to tapering and aspect ratio and nonhomogeneity concludes that frequency obtained for the present study is very less in comparison with other studies done. A convergence study of the obtained results is also presented with the help of figures. The convergence study done concludes that rate of convergence in case of circular variation is faster than the other variations.

Behavior of frequencies of orthotropic rectangular plate with circular variations in thickness and density

In this article, authors dealt with general solution of differential equation of orthotropic rectangular plate with clamped boundary conditions under bi-parabolic temperature variations. Rayleigh Ritz technique is adopted to solve the resultant equation and evaluate the frequencies for first four modes of vibration. The effect of circular thickness and density on frequencies of orthotropic plate are analyzed which is not done yet. The objective of the study is to optimize the frequency modes by choosing the appropriate variation in plate parameters. Th findings of the study complete the objective of the article. All the results are provided in tabular form.

Morphing of bistable variable stiffness composites using distributed MFC actuators

The use of piezoelectrically controlled bistable composite laminates for morphing applications has received increased attention in recent years. So far, most existing investigations have explored the possibility to trigger snap-through using large-sized piezoelectric Macro Fibre Composite (MFC) actuators bonded at the centre of the bistable laminate. However, bonding large-sized MFCs at the centre of the plate leads to flattening of the midsection of the laminate, which can result in the loss of bistability. This study presents an alternative approach of bonding distributed smaller MFCs over the entire surface of the laminate and investigate the resulting stable shapes and the change in snap-through voltages. Self-resetting piezoelectrically controlled active laminates where the MFC patches are distributed over the laminate surface have been investigated. A semi-analytical model using the Rayleigh–Ritz technique is developed to account for the distributed actuation system. Results from the proposed semi-analytical framework are verified using a corresponding finite element model. The bistable shapes, as well as the snap-through and snap-back voltages, are calculated for different distributed MFC configurations and are compared with a single MFC laminate system. Snap-through voltage predictions from the proposed semi-analytical formulation and the finite element model are compared with the experimental results for a single MFC patch available from the literature. Further, the possibility to tailor the snap-through voltages of the proposed laminate-MFC configuration is explored by replacing conventional cross-ply laminates with variable stiffness (VS) laminates generated from curvilinear fibre alignments. A finite element parametric study is performed by tailoring the VS fibre orientation parameters to achieve a bistable laminate-MFC configuration with lower snap-through requirements where the designed laminate is actuated with distributed MFC patches.

Bending analysis of nanobeams based on the integral form of nonlocal elasticity using the numerical Rayleigh-Ritz technique

The nonlocal theory is commonly applied for nanomaterials due to its capability in considering size influences. Available studies have shown that the differential version of this theory is not suitable for some problems such as bending of cantilever nanobeams, and the integral version must be used to avoid obtaining inconsistent results. Therefore, an attempt is made in this paper to propose an efficient variational formulation based on the integral nonlocal model for the analysis of nanobeams. The formulation is developed in a general form so that it can be used for arbitrary kernel functions. The nanobeams are modeled using the Bernoulli-Euler beam theory, and their bending behavior is analyzed. Derivation of governing equations is performed according to an energy-based approach. Also, a numerical approach based on the Rayleigh-Ritz method is developed for the solution of problem. Moreover, the results of integral and differential models are compared. It is revealed that by the proposed numerical solution, the paradox in the behavior of nanocantilever is resolved.

Frequency Analysis for Functionally Graded Material Cylindrical Shells: A Significant Case Study

Cylindrical shells play an important role for the construction of functionally graded materials (FGMs). Functionally graded materials are valuable in order to develop durable materials. They are made of two or more materials such as nickel, stainless steel, zirconia, and alumina. They are extremely beneficial for the manufacturing of structural elements. Functionally graded materials are broadly used in several fields such as chemistry, biomedicine, optics, and electronics. In the present research, vibrations of natural frequencies are investigated for different layered cylindrical shells, those constructed from FGMs. The behavior of shell vibration is based on different parameters of geometrical material. The problem of the shell is expressed from the constitutive relations of strain and stress with displacement, as well as it is adopted from Love’s shell theory. Vibrations of natural frequencies (NFs) are calculated for simply supported-simply supported (SS-SS) and clamped-free (C-F) edge conditions. The Rayleigh–Ritz technique is employed to obtain the shell frequency equation. The shell equation is solved by MATLAB software.

Performance of FGM bilayered cylindrical shell placed on cantilever edge

Functionally graded materials (FGMs) are designed for specific purpose and applications. Functionally graded materials for bi-layered cylindrical shell was discussed for different boundary conditions. Functionally graded materials (FGMs) are that kind of material in which function and formation may deviate continuously. Cylindrical shells are mainly significant in various fields of science as well as advanced technology of engineering like aerospace engineering, mechanical engineering and civil engineering. Wide applications of cylindrical shell in different fields like aircraft, aerospace and pressure vessels etc. Bi-layered cylindrical shells consist of two layers and in this work, one layer is of FGM material whose constituents are nickel (Ni) and zirconia (Zr) and other is of isotropic material whose constituent is stainless steel. In this work, effect of trigonometric volume fraction law on cantilever FGM bi-layered cylindrical shell with internal pressure has analyzed by using Rayleigh-Ritz technique and Love's shell theory. Present results of FGM bi-layered cylindrical shell are compared with FGM cylindrical shell. Validity of present technique has verified by way of comparisons with current conclusions and those obtained in the past studies.

Nonlinear vibration analysis of horizontal axis wind turbine blades using a modified pseudo arc-length continuation method

The horizontal axis wind turbine (HAWT) blades are exposed to parametric and direct excitations of harmonic nature due to the gravity and aerodynamic loads. Hence, the study of the periodic response of the HAWT blades under these operating loads is of great importance in structural design. In this paper, a new computationally efficient dynamic model for the blade is presented including the pre-twist effects. The Rayleigh-Ritz technique is applied to the derived dynamic model and a reduced order model (ROM) is devised in a systematic manner. An efficient method based on the pseudo arc-length continuation method is introduced to find out the edge-wise and flap-wise amplitude-frequency response (AFR) curves of the blade under harmonic wind inflow. Furthermore, the forced vibration of the blade under different operating loads including gravity, centrifugal and aerodynamic loads (due to the uniform and shear wind inflows) is studied separately. Applying the proposed method, the effects of structural damping, nonlinear terms, pre-twist ratio, and cyclic excitation due to the gravity force as well as the centrifugal force on the AFR curves are investigated. Eventually, the dynamic model and the presented method for the frequency analysis are validated with the available data.

Study on Acoustic Characteristics of a Flexible Plate Strongly Coupled with Rectangular Cavity

This paper presents a method to predict the acoustic characteristics and steady-state responses of a flexible plate strongly coupled with rectangular cavity based on energy principle theory and Legendre polynomial series. First, the displacement of the plate and the sound pressure in the cavity are constructed in the form of two-dimensional and three-dimensional Legendre polynomial series, respectively. The unknown expansion coefficients are obtained using the Rayleigh–Ritz technique based on the energy expressions for the strongly coupled plate-cavity system. The accuracy, convergence, and efficiency of the present method are verified by comparing with the results available in the FEM and literature. Finally, the effects of the structural boundary conditions, cavity depth, and structural length-width ratio on the coupling natural frequency and the steady-state responses under three excitation conditions are analyzed.

Динамічна втрата стійкості складеною нанокомпозитною оболонкою

Free oscillations and dynamic instability due to supersonic airflow pressure are investigated in a functional-gradient compound composite conical-cylindrical shell made of a carbon nanotubes-reinforced material. Nanocomposite materials with a linear distribution of the volumetric fraction of nanotubes over the thickness are considered. Extended mixture rule is used to estimate nanocomposite’s mechanical characteristics. A high-order shear deformation theory is used to represent the shell deformation. The assumed-mode technique, along with a Rayleigh-Ritz method, is applied to obtain the equations of the structure motion. To analyze the compound structure dynamics, a new system of piecewise basic functions is suggested. The pressure of a supersonic flow on the shell is obtained by using the piston theory. An example of the dynamic analysis of a nanocomposite conical-cylindrical shell in the supersonic gas flow is considered. The results of its modal analysis using the Rayleigh-Ritz technique are close to the natural frequencies of the shell obtained by finite element analysis. In this case, finite element analysis can only be used for shells made of material with a uniform distribution of nanotubes over the thickness. The dependence of the natural frequencies of a compound shell on the ratio of the lengths of the conical and cylindrical parts is studied. The dependence of the critical pressure of a supersonic flow on the Mach numbers and the type of carbon nanotubes reinforcement is investigated. Shells with a concentration of nanotubes predominantly near the outer and inner surfaces are characterized by higher values of natural frequencies and critical pressure than the shells with a uniform distribution of nanotubes or with a predominant concentration of nanotubes inside the shell.

APPLICATION OF RAYLEIGH-RITZ TECHNIQUE IN ANALYSIS OF NATURAL VIBRATION

The paper is the first step towards the aerodynamic analysis of slender structures to wind actions. By Rayleigh-Ritz technique and Lagrange equation, the paper describes the vibrationof a continuous structure though a system of ordinary differential equations in a way more familiar with structural analysts. This vibration method will be applied to antenna pole for dynamic analysis.

Dynamics and power flow control of irregular elastic coupled plate systems: Precise modeling and experimental validation

The vibration characteristics of irregular elastic coupled plate systems are studied by Chebyshev-Ritz method in this paper, including dynamics and power flow control. Firstly, the accurate geometric model is established, and the coupling conditions and boundary conditions are simulated by artificial virtual springs. In order to facilitate integral operations, the irregular integral area is mapped into the regular integral area by a special coordinate transformation. Then all energy equations of the coupled system are established according to the first order shear deformation theory. The Chebyshev polynomial is used as the allowable displacement functions. Finally, the unknown coefficients in the allowable displacement functions are determined by the Rayleigh-Ritz technique. On the basis of the free vibration study, the dynamic behavior is studied by applying the external load to the system surface, including the forced response, the admittance and the power flow. While deriving the method, some modal verification experiments about coupled plate systems are designed. The innovation of the study is the establishment of the vibration analysis model of irregular elastic coupled plate systems based on the elastic theory, the coordinate transformation criterion and the two-dimensional Chebyshev-Ritz principle. The mechanism and control of energy transfers of the coupled plate system are studied by the power flow intensity analysis. The numerical results show that both the coupling condition and the boundary condition are important parameters in the vibration. This study provides some new insights into the vibration characteristics and energy transfer of the irregular elastic coupled plate system.