Rayleigh-Ritz Research Articles

A General Approach for the Analysis of Transverse Vibration Characteristics of a Rectangular Plate with Arbitrary-shaped Openings

Based on the Rayleigh-Ritz method, the transverse vibration characteristics of a rectangular plate with an arbitrarily shaped inner opening are investigated. A modified two-dimensional Fourier series was chosen as the trial function and springs of different stiffnesses were used to equate the arbitrary boundary conditions of the rectangular plate. The Lagrange generalization of the transverse vibration of the open rectangular plate is obtained by equating the open area to a special film with physical properties of zero according to the energy superposition principle. The variational method is used to derive the characteristic equations of the thin plate and to find the intrinsic frequencies and corresponding vibration modes. The convergence of this theoretical method is analyzed by means of an example, and the accuracy of the method is verified by comparison with the results of finite element software calculations, which provides a reference for practical engineering problems.

An investigation into the static configurations and snapthrough response of clamped hybrid bistable symmetric laminates

This paper presents a numerical analysis of the room-temperature shapes and snapthrough response of thermally induced hybrid bistable symmetric laminates. The bistable laminate is clamped at one end and free at the other end to extend the applications of bistable laminates to non-free-free boundary conditions. The hybrid layup resolves the issue of losing the bistability of the laminate when attached to a larger structure or clamped. Bidirectional (BD) glass-epoxy layers are symmetrically imbedded in the laminate’s layup to trigger the bistability. An approximate analytical model that is based on the Rayleigh-Ritz method along with the ABAQUS FE package are used in the analysis. The model is validated against the results available in the literature and a good agreement is obtained. The significance of the BD layer’s parameters on the thermally induced room-temperature shapes and the snapthrough response is examined. Three parameters are considered: the BD layers’ width, thickness, and location from the laminate’s center. It is found out that the three parameters greatly affect the static equilibrium shapes and the snapthrough/snapback response. This analysis complements the ongoing research on the bistable laminates for morphing applications.

Vibration analysis of sandwich cylindrical shells made of graphene platelet polymer–viscoelastic–ceramic/metal FG layers

In this paper, natural frequencies and loss factors of cylindrical sandwich shells composed of the viscoelastic core layer, surrounded by functionally graded graphene-platelet reinforced polymer composite (FG-GPLRPC) and ceramic/metal (FG-ceramic/metal) are investigated. The viscoelastic layer is modeled via the fourth parameter fractional viscoelastic pattern, and the functionally graded ceramic/metal layer is theoretically modeled using a power-law function. The uniform, symmetric and un-symmetric patterns are considered for simulating the graphene platelet (GPL) nanofillers distributions along with the thickness direction. The classical shell theory is used for functionally graded layers and properties of the effective materials of GPLRPC multilayers are determined by using a modified Halpin–Tsai micromechanics model and the rule of mixture. The governing equations of motion are extracted by applying the Lagrange equation and the Rayleigh-Ritz method. The determinant of the coefficient matrix of the characteristic equation is calculated, and the natural frequencies and loss factors of the system are extracted. A study of the interactions of materials and geometrical factors such as the ratio of radius to length, the properties of functionally graded materials, and GPL weight fractions for patterns of proposed distributions are presented and some conclusions have been formed.

Studying The Necessary Optimality Conditions and Approximates a Class of Sum Two Caputo–Katugampola Derivatives for FOCPs

In this paper, the necessary optimality conditions are studied and derived for a new class of the sum of two Caputo–Katugampola fractional derivatives of orders (α, ρ) and( β,ρ) with fixed the final boundary conditions. In the second study, the approximation of the left Caputo-Katugampola fractional derivative was obtained by using the shifted Chebyshev polynomials. We also use the Clenshaw and Curtis formula to approximate the integral from -1 to 1. Further, we find the critical points using the Rayleigh–Ritz method. The obtained approximation of the left fractional Caputo-Katugampola derivatives was added to the algorithm applied to the illustrative example so that we obtained the approximate results for the state variable x(t) and the control variable u(t) by assumed α, β ∈ (0,1) with different values for two periods of ρ > 0 (ρ∈ (0,1) , ρ∈ (1,2)). In both cases, the algorithm steps show the accuracy and efficiency of the approximate results of the proposed system. An approximation to the fractal derivative was obtained and then added to the algorithm applied to the illustrative example so that we obtained the approximate results for the state variable x (t) and the control variable u (t) by imposing different values for two periods of ρ > 0. In the first case we take ρ∈ (

Vibration analysis of the combined conical–cylindrical​ shells coupled with annular plates in thermal environment

In this paper, a unified method is proposed for the first time to predict the free and forced vibration behavior of the combined conical–cylindrical shells coupled with annular plates in the steady-state thermal environment. The energy equations of each substructure are obtained based on the first-order shear deformation theory (FSDT), and the artificial spring technique is introduced to realize the coupling between each substructure, the circumferential closure of each substructure, and the arbitrary boundary conditions. The displacements of the midface in each sub-structure are expanded using the spectral-geometry method (SGM), and the energy expression is solved by using the Rayleigh–Ritz method to obtain the free and forced vibration behavior of the combined structure. The numerical examples derived from the calculations are compared with the results obtained by the finite element method (FEM) to verify the accuracy of the present method. The effects of boundary conditions, temperature and geometric parameters on the free and forced vibration behaviors of the combined structure are investigated. The method in this paper is a meshless method, which is faster and more efficient as it is less dimensional and less computationally expensive than the FEM. This paper provides a compelling reference for vibration control of the combined conical–cylindrical shells coupled with annular plates in practical applications.

Resonant Actuation Based on Dynamic Characteristics of Bistable Laminates

Bistable or multi-stable structures have found broad applications in the fields of adaptive structures, flow control, and energy harvesting devices due to their unique nonlinear characteristics and strong local stability behavior. In this paper, a theoretical model based on the principle of minimum potential energy and the Rayleigh–Ritz method is established to study the dynamic characteristics of a bistable unsymmetric laminate with a fixed center. Numerical results of this theoretical model were obtained and verified by an FEA model using ABAQUS. The nonlinear dynamic characteristics and the structural response under different levels of external excitation were investigated and verified by experiments. The realization conditions of single-well vibration and cross-well vibration of bistable laminates were determined, with which the actuation strategies can be optimized for targeting modal frequencies of bistable laminates.

Properties of local buckling of spherical shell under external pressure

An important property of localization in buckling of spherical shells under external pressure is discussed. It is shown that the localization is possible for structures with nonlinear softening. An analytical model of local buckling of spherical shell is developed. Rayleigh–Ritz method is used at small and moderate deflections. At large deflections corresponding to relatively small pressure (less than 20% of classical bucking load) it is based on asymptotic method. The asymptotic model is then expanded to the practically important range of the load. The response of the structure to local perturbations of different types (including radial probing force, prescribed deflection at the shell pole, and energy barrier) is studied. Special attention is paid to the energy barrier which is required for structure transition from initial equilibrium state to the post-buckling dimple-like state. Energy barrier criterion is used as a measure of metastability of the structure and applied for estimation of load level separating high and low sensitivity of the shell to local perturbations. Based on this pressure value, formulae for design buckling load are proposed and deliberated. Similarities and differences of local buckling of spherical shells under external pressure and axially compressed cylindrical shells are discussed.

Hygro-thermo-mechanical vibration analysis of functionally graded porous curved nanobeams resting on elastic foundations

This paper presents an analytical solution using Chebyshev polynomials based on Rayleigh–Ritz method and Navier's solution to analyze the free vibration response of functionally graded porous (FGP) curved nanobeams embedded resting on an elastic medium under hygro-thermo-mechanical load. Applying Hamilton's principle based on the quasi-3D higher-order shear deformation beam theory and the nonlocal elasticity theory, the governing equation of FGP curved nanobeam is derived. Material properties of nanobeam continuously change through the thickness via a power-law distribution and porosity distribution is described by two laws including even porosity distribution and uneven porosity distribution, respectively. The impact of thermal and moisture on structures are assumed to cause tension load in the plane and do not change the material properties. The elastic foundation used in this work is the Winkler–Pasternak type. The accuracy of the proposed method is verified by comparing the obtained numerical results with those of the published works in the literature. In addition, the influence of curvature radius, nonlocal coefficient, porosity coefficient, stiffness of foundation, and boundary conditions on the free vibration of the nanobeam are examined in detail.

Peeling behaviour of a hard magnetic soft sheet driven by the magnetic field: Geometric nonlinearity analyses

Magneto-responsive smart structures can produce large deformation and realize the remote non-contact control driven by the magnetic field, which are widely applied in a lot of engineering areas. In the present study, the peeling behavior of a hard-magnetic soft sheet driven by the magnetic field is comprehensively investigated both experimentally and theoretically. Firstly, the peeling experiments of two identical magnetic sheets under the control of a cylindrical magnet are performed, where one is adhered by the liquid film and the other takes a cantilever configuration. In addition, we build the expression of the total energy functional of the sheet-liquid-magnet system, and then derive the approximate solutions on the beam deflection and the adhered length during the peeling process based on the Rayleigh-Ritz method. The numerical result is in excellent agreement with the experimental data. The effects of the external magnetic field and work of adhesion on the maximum deflection and adhered length are fully explored, and more results are predicted via calculation. These findings can provide some inspirations on engineering of many new devices, such as smart structures, micro-sensors and intelligent robots, etc.

Thermoelastic damping in cylindrical shells with arbitrary boundaries

An effective method is proposed for the dynamic modelling and thermoelastic damping (TED) evaluation of the cylindrical shells with arbitrary boundaries. The arbitrary boundary conditions of the cylindrical shell are simulated by a set of springs, and the degree of the boundary condition is achieved by adjusting the spring stiffnesses. Hamilton's principle with the Rayleigh-Ritz method is employed to establish the equation of motion of the cylindrical shell, and the natural frequencies of the cylindrical shell with arbitrary boundaries are obtained by solving the eigenvalue problem of the equation of motion. An analytical model for the TED in the cylindrical shell is obtained by computing the dissipated energy and the maximum elastic potential energy in the cylindrical shell. The correctness and feasibility of the present theoretical model are verified by comparing the natural frequencies and TED obtained by this analytical method with those from the published literature and the finite element method (FEM). The convergence analysis of the analytical expression of TED is carried out. The influences of geometrical parameters and different types of spring stiffnesses on the TED are studied. In addition, the effects of the boundary conditions and the vibration modes of the structure on the TED characteristics of the cylindrical shells with arbitrary boundaries are analyzed. The present model provides theoretical reference for optimizing the cylindrical shell resonators with high quality factors.

Evanescent waves in a metabeam attached with lossy acoustic black hole pillars

In this paper, we suggest a metabeam having periodically distributed integrated acoustic black holes (IABHs). Compared to conventional embedded ABHs, the thickness of the base beam has not been modified, so it adapts well to application occasions with load-bearing requirements. The ABH pillars behave as dynamic vibration absorbers (DVAs) that not only provide abundant modes with high damping performance but also generate local resonance and Bragg scattering bandgaps to stop wave propagation. To characterize the evanescent waves, a wave and Rayleigh–Ritz method (WRRM) is proposed, based on the nullspace method (NSM) that assumes the response is constituted by the linear combination of the nullspace basis of the constraint matrix. The proposed WRRM is validated using numerical partial differential solutions. Analysis of the complex dispersion curves shows that multiple local resonance and Bragg scattering bandgaps take place in the whole frequency range of interest, and the damping effect can merge these bandgaps so as to form very wide attenuation bands. Not only that, remarkable evanescent waves are also observed in the passbands. The lattice constant and ABH length are studied parametrically. Results show that the larger the lattice constant, the more attenuation bands appear, and the longer ABH length can enhance the attenuation capability. Finally, the transmissibility and mean square velocity for a finite metabeam with 5 cells are investigated, indicating substantial wave damping in a broadband frequency range.

Do we need to satisfy natural boundary conditions in energy approach to nonlinear vibrations of rectangular plates?

Despite a large literature, it seems still unclear if it is necessary to satisfy the natural boundary conditions in an energy approach to the study of nonlinear vibrations of rectangular plates. In the present study, which is based on Lagrange equations, the case of simply supported movable edges is considered. In the past, special nonlinear terms were added to the in-plane displacements to satisfy the natural boundary conditions. However, it was unclear if this was just helping the fast convergence of the series, reducing the number of degrees of freedom in the model, or it was actually necessary. In fact, it is well-known that for linear vibrations only geometric (essential) boundary conditions must be satisfied in variational energy approaches like the Rayleigh-Ritz method. Therefore, this doubt should be clarified. In the present study, accurate expansions of the plate displacements are introduced with additional in-plane trigonometric terms. They allow to satisfy the natural boundary conditions by energy minimization without the introduction of additional nonlinear terms. Nonlinear damping is also introduced in order to obtain results which agree with experiments.

Vibro-acoustic coupling characteristics of the microperforated panel with local resonators

This paper provides a unified approach to evaluate the sound absorption performance of the metamaterial microperforated panel (MMPP) with additional local resonators under elastic boundaries. The elastic boundary conditions can be realized by setting the stiffness of restrained spring at the edges as a specific value. The vibration equations of the metamaterial panel (MP) are derived by combining the Spectral Geometry method with the Rayleigh Ritz method, and the acoustic electrical analogy method is used to solve the absorption coefficient considering the vibro-acoustic coupling. The program developed by these methods can be used to study the vibro-acoustic coupling model of the MMPP with various parameters under various boundaries. To validate the accuracy of the present method, a structure-acoustic simulation model is established, and a good agreement is achieved between the theoretical method and the finite element method. A comprehensive parametric analysis is performed to obtain the optimum suppression performance of sound absorption and the variation of the absorption coefficients has been explained justifiably in detail. The results demonstrate that additional local resonators can improve the absorption performance of the microperforated panel, and the absorption band can be appropriately broadened by reducing the spring stiffness or increasing mass ratio. This paper provides a better sound absorption prediction for the practical application of MMPP under complex conditions.

Thermal vibration analysis of a composite sandwich plate in the space environment

This article proposes the method for conducting thermal vibration analyses of the composite sandwich plate of a satellite in the space environment. Based on the continuous model, the method is derived using the Rayleigh-Ritz method and includes thermo-elastic coupling vibration by the space environment. In addition, the validity of the method is verified by the results of the finite element analysis and experimental analysis. Finally, parametric analysis is performed to investigate thermo-elastic coupling vibration characteristics of a satellite structure. This result allows a database to be design guidelines of composite sandwich structures for aerospace applications. We then concluded that the method is well-suited to thermally induced vibration analysis of a composite sandwich plate for aerospace applications due to their relative simplicity and computational efficiency.

Improving climate resilience: Reduced-order mechanical modelling of a temperature-adaptable sun protection facade system

Analytical models to describe the mechanical behaviour of an adaptive sun protection facade system (Adaptex) are presented. First, a one-dimensional (1D) rigid link model is developed. Stretching and bending deformation of textile bands are represented with the aid of extensional and rotational springs respectively. The model is described by two generalized coordinates only. The second analytical model employs a continuous description based on the Kirchhoff plate theory. The Rayleigh–Ritz method is applied to analyse the deformation behaviour of the textile bands. In comparison with the experimental results, it can be found that the continuous model simulates the real deformation of the facade system, thus its morphological behaviour, very effectively. The results of a parametric study indicate that the stiffness of the textile bands, the geometric dimensions and the position of load application have significant effects on the deformation behaviour.

Free Vibration Analysis of Rectangular Plate with Cutouts under Elastic Boundary Conditions in Independent Coordinate Coupling Method

Based on Kirchhoff plate theory and the Rayleigh-Ritz method, the model for free vibration of rectangular plate with rectangular cutouts under arbitrary elastic boundary conditions is established by using the improved Fourier series in combination with the independent coordinate coupling method (ICCM). The effect of the cutout is taken into account by subtracting the energies of the cutouts from the total energies of the whole plate. The vibration displacement function of the hole domain is based on the coordinate system of the hole domain in this method. From the continuity condition of the vibration displacement function at the cutout, the transition matrix between the two coordinate systems is constructed, and the mass and stiffness matrices are completely obtained. As a result, the calculation is simplified and the computational efficiency of the solution is improved. In this paper, numerical examples and modal experiments are presented to validate the effectiveness of the modeling methods, and parameters related to influencing factors of the rectangular plate are analyzed to study the vibration characteristics.

Complicated nonlinear oscillations caused by maneuvering of a flexible spacecraft equipped with hinged solar arrays

We present a general methodology for the nonlinear dynamical modeling of a three-axis stabilized spacecraft equipped with flexible solar arrays. The large-span multi-panel solar arrays are modeled as flexible thin plates that are connected to the rigid central body of the spacecraft by means of nonlinear flexible hinges. We construct a low-dimensional yet accurate dynamical model by using a Galerkin expansion in terms of global modes of the system that we compute first by using the Rayleigh–Ritz method. The hinged connections between the rigid and flexible parts of the system are imposed employing Lagrange multipliers. The model is used to study the spacecraft response triggered by various maneuvering scenarios. We in particular focus on the coupling between vibrations of the flexible components and the rigid motion of the spacecraft induced by hinge nonlinearities during these orbital and attitude maneuvering operations. In all cases considered, it is found that four global modes are sufficient to accurately compute the system’s response. We also observe other complicated nonlinear dynamical phenomena, such as hysteresis and superharmonic resonance that may be of concern in spacecraft design. Our modeling approach can be applied to other multibody systems directly.

Nonlinear analysis of Euler beams resting on a tensionless soil with arbitrary configurations

BackgroundThe nonlinear interaction between an elastic Euler beam and a tensionless soil foundation is studied. The exact analytical solutions of the nonlinear problem are rather complicated. The main difficulty is imposing compatibility conditions at lift-off points. These points are determined as a part of the solution, although being needed to get the solution itself. In the current work, semi-analytical solutions are derived using the Rayleigh–Ritz method. The principle of vanishing variation of potential energy is adopted. The solution is approximated using a set of suitable trial functions. Accurate high-order approximate analytical solutions are obtained using MAXIMA symbolic manipulator. Lift-off points are identified through an iterative procedure and compatibility conditions are satisfied automatically. The methodology is designed to accommodate arbitrary configurations for the load distribution and the beam properties.ResultsExact solutions are revised briefly to verify the semi-analytical solutions in terms of deflection, bending moment, and shear. Semi-analytical solutions for constant beam properties including various support conditions and load distributions are verified. Convergence of high-order semi-analytical solutions is illustrated for cases including one and two contact points. A parametric study is provided to illustrate the effect of soil stiffness on the contact length. The case of a finite beam with free ends is considered. The semi-analytical solutions for variable beam moment of inertia are provided and verified.ConclusionsHighly accurate semi-analytical solutions can be obtained for the problem considered using the Rayleigh–Ritz method along with a symbolic manipulator. Arbitrary load and support configurations can be modeled, and the locations of lift-off points are well predicted. The semi-analytical solutions are extremely valuable for cases of variable moment inertia since exact solutions are rather rare.

LTB analysis of pre-stressed mono-symmetric thin-walled beams with deviators under non-uniform bending moments using a simplified FEM

A FE formulation has been newly presented for the lateral-torsional buckling (LTB) analysis of an externally pre-stressed (PS) steel beam, possessing deviators under non-uniform bending moments. Firstly, the total potential energies of the PS mono-symmetric thin-walled beams in possession of un-bonded/bonded deviators are derived depending on the number of deviators. Afterwards, the in-plane solutions of the PS mono-symmetric simple/cantilever beams under the lateral load can be analytically derived through the use of discontinuous functions. A simplified FE procedure is then developed based on the total potential energy, taking into consideration the effects of double tendons. Finally, the critical buckling loads of simple and cantilever PS mono-symmetric beams which were subjected to the non-uniform moments with the increasing of deviators are evaluated using the FEM and Rayleigh-Ritz method. As a result, the FE solutions were found to be very well matched to solutions by Ritz method and Abaqus. Additionally, the effects of double tendons, eccentricity, and the initial tension on LTB of the PS beam system were investigated with the increase of un-bonded/bonded deviators.

Natural Characteristics Analysis for the Spacecraft Equipped with Constructed Cantilever Solar Panels

The power series polynomial constraining method is proposed in this paper. The dynamical model of the cantilever plate can be established by applying the constraint, which is different from the traditional polynomial. Firstly, the characteristic orthogonal polynomial was used to describe the displacement field of the rectangular plate of which all edges are free. Then the four-sided free plate was equivalent to cantilever plate by power series multiplier constraint method. The characteristic equation of the constructed cantilever plate was obtained by the Rayleigh–Ritz method. Natural frequencies and modal shapes of the plate were obtained by solving the characteristic equation. Next, the proposed method was adopted to establish dynamical model of a pair of solar panels clamped on the central platform symmetrically. The convergence of the proposed method was verified by comparing the calculated results of the cantilever plate with that of the finite element software ANSYS 15.0. The optimum order the power series polynomial was obtained by comparing different results. The analysis of the dynamical characteristics of the cantilever plate and the spacecraft demonstrates the validation of the proposed method. This method can provide a new idea for the plate with local edge constrained.