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The dynamics of nonlinear polar orthotropic circular plates with simply supported boundary condition are investigated. Kirchhoff strain displacement relations for thin plates plus next higher-order nonlinear terms (von Karman type geometric nonlinearity) are considered. Lagrangian density function and Hamilton's principle are utilized to derive Lagrange's equations, from which the equations of motion and associated boundary conditions are derived. Analytical solution is obtained by the perturbation techniques and numerical solution by the Runge–Kutta method. Phase diagrams, discrete Fast Fourier Transform (FFT diagrams) and time history responses are presented for studying the forced vibration behavior. The sub-harmonic and primary resonances are studied as well as the effect of adding damping foil layers. The quadratic term in the governing equation plays a softening role on the overall behavior of the plate due to its relatively large coefficient. The increase of damping tends to smooth out the unstable region (i.e. jump phenomenon) in the system.

Based on the Maxwell equations, the electromagnetic constitutive relations and boundary condition, the electrodynamic equation and the electromagnetic force expressions in an electromagnetic field are derived. Using the principle of virtual work, the basic set of equations for nonlinear electromagnetic elasticity vibration expressed by the displacement of a thin plate in a longitudinal and a transverse magnetic field is obtained, respectively. In addition, we study the nonlinear principal resonance and the solution stability of a thin plate with two opposite sides simply supported and subjected to a mechanical live load and in a constant transverse magnetic field. By the method of multiple scales, the amplitude frequency response equation and the approximate analytic solution in steady motion are also derived. According to the characteristic of singularity and the Lyapunov stability theory, the stability of the solution is analyzed and the critical condition of stability is determined. Finally, by means of numerical calculations, the amplitude frequency response curves, time history response plots and phase charts of the magnetoelasticity vibration are obtained.

This paper presents the discrete singular convolution (DSC) method for solving buckling and vibration problems of rectangular plates with all edges transversely supported and restrained by uniform elastic rotational springs. The opposite plate edges are subjected to a linearly varying uni-axial in-plane loading. The rationale for using DSC method stems from its numerical stability and flexible implementation for structural analysis. To verify the present approach, convergence and comparison studies for rectangular plates with different combinations of elastically restrained and classical edges are carried out. Accurate buckling and vibration solutions of plates having two opposite edges elastically restrained and the other two sides clamped, or all edges elastically restrained are presented.

Fatigue life, stability and performance of majority of the structures and systems depend significantly on dynamic loadings applied on them. In many engineering cases, the dynamic loading is random vibration and the structure is a plate-like system. Examples could be printed circuit boards or jet impingement cooling systems subjected to random vibrations in harsh military environments. In this study, the response of thin rectangular plates to random boundary excitation is analytically formulated and analyzed. In the presented method, closed-form mode shapes are used and some of the assumptions in previous studies are eliminated; hence it is simpler and reduces the computational load. In addition, the effects of different boundary conditions, modal damping and excitation frequency range on dynamic random response of the system are studied. The results show that increasing both the modal damping ratio and the excitation frequency range will decrease the root mean square acceleration and the maximum deflection of the plate.

The Hp-Cloud meshless method was developed to study the dynamic analysis of arbitrarily shaped thin plates with intermediate point supports. By proposing a special pattern for the influence radius of nodes and a polynomial type of enrichment function, the Hp-Cloud shape functions with Kronecker delta property were constructed. They can satisfy the zero deflection conditions for the field nodes at the point supports. The results obtained from these shape functions agree well with the previous ones, showing good accuracy and convergence. For plates with sharp corners, it is not possible to construct the Hp-Cloud shape function with Kronecker delta property. To this end, the Lagrange multiplier method was used for enforcing the boundary conditions. The computations were carried out by the Ritz method, and the cell structure method is refined to improve the speed and accuracy of numerical integration on the subscription surface of clouds intersecting with the plate boundaries. Using the algorithm developed, the natural frequencies of plates of various shapes and support patterns were computed. By increasing the number of point supports on the plate edges, the natural frequencies computed of the plate tend to those of the simply supported plate. Appropriate pattern of point supports distribution was presented for modeling the simply supported plates of various shapes by comparing the corresponding natural frequencies.

Geometrically nonlinear free vibrations of thin rectangular plates are studied using the recently developed weak form quadrature element method (QEM). The nonlinear von Karman plate theory is employed to express the strain-displacement relations. The weak form description of the plate is formulated on the basis of the variational principle. The integrals involved in the variational description are evaluated by an efficient numerical integration scheme, and the partial derivatives at the integration points are approximated by differential quadrature analogs. A system of algebraic equations is eventually derived, and the nonlinear frequencies and mode shapes are extracted from solving the equations. The efficiency of the method is demonstrated by a convergence study. The accuracy of the method is illustrated by comparing the computed nonlinear to linear frequency ratios with those available in the literature. The influences of the nonlinearity on higher order frequencies and mode shapes are exhibited as well.

The free and forced vibration characteristics of three-layered sandwich plates with thin isotropic faces and Leptadenia pyrotechnica rheological elastomer (LPRE) core are studied in this investigation. The LPRE core is fabricated and experimented to determine its shear storage modulus and loss modulus. It is observed that the stiffness and damping characteristics of the LPRE core is significantly higher than those of the room-temperature vulcanized silicone rubber elastomer (RTVE) core. The governing equation of motion for the sandwich plate is derived by the Lagrange principle and given in finite element form. The natural frequencies and loss factors of the three-layered sandwich plate are studied by varying the thicknesses of the core and the constraining isotropic layer, and material of the constraining layer with different boundary conditions. The results are compared with those of similar structures with different core materials and boundary conditions. In addition, a LPRE-based sandwich plate is fabricated and its fundamental frequency is determined experimentally and compared with the finite element result. The forced vibration response of the three-layered sandwich plate is also explored under a harmonic excitation force. This study provides supports for the application of the LPRE-based sandwich plates potentially to the passive vibration suppression of structures.

In this paper, principal parametric resonance of an axially moving piezoelectric rectangular thin plate under thermal and electric field is investigated. Based on Kirchhoff–Love plate theory and Von Karman theory, the transverse vibration differential equation of a piezoelectric rectangular thin plate under thermal and electric field is derived by using Hamilton’s principle. The dimensionless vibration equation of piezoelectric rectangular thin plate with parametric excitations is discretized by Galerkin’s method. Then, the multiple scales method is applied to derive amplitude-frequency response equation and the stability conditions of the steady-state solution are obtained by Lyapunov stability theory. Numerical method is used to find the influences of specific parameters on the vibration performance and stability of the system. Based on the global bifurcation diagram and corresponding response diagram, the influences of bifurcation control parameters on the nonlinear dynamic characteristics of the system are discussed. Numerical results illustrate that the system amplitude frequency characteristic curve presents soft spring characteristics. There are periodic and chaotic motions with the increase of velocity and the central temperature difference, and the decrease of plate thickness and velocity will result in the decrease of chaotic threshold. The results also show that increase the velocity perturbation amplitude can prolong the chaotic motion.

This paper presents a thin plate formulation with nodal integration for bending analysis using three-node triangular cells and linear interpolation functions. The formulation was based on the classic thin plate theory, in which only deflection field was required and dealt with as the field variables. They were assumed to be piecewisely linear and expressed using a set of three-node triangular cells. Based on each node, the integration domain has been further derived, where the curvature in the domain was computed using a gradient smoothing technique (GST). As a result, the curvature in each integration domain is a constant whereby the deflection is compatible in the whole problem domain. The generalized smoothed Galerkin weak form is then used to create the discretized system equations where the system stiffness is obtained using simple summation operation. The essential rotational boundary conditions are imposed in the process of constructing the curvature field in conjunction with imposing the translational boundary conditions in the same way as undertaken in the standard FEM. A number of numerical examples were studied using the present formulation, including both static and free vibration analyses. The numerical results were compared with the reference ones together with those shown in the state-of-art literatures published. Very good accuracy has been achieved using the present method.

This paper presents a curvature-constructed method (CCM) for bending analysis of thin plates using three-node triangular cells and assumed piecewisely linear deflection field. In the present CCM, the formulation is based on the classic thin plate theory, and only deflection field is treated as the field variable that is assumed piecewisely linear using a set of three-node triangular background cells. The slopes at nodes and/or the mid-edge points of the triangular cells are first obtained using the gradient smoothing techniques (GST) over different smoothing domains. Three schemes are devised to construct the curvature in each cell using these slopes at nodes and/or the mid-edge points. The generalized smoothed Galerkin weak form is then used to create the discretized system equations. The essential rotational boundary conditions are imposed in the process of constructing the curvature field, and the translational boundary conditions are imposed in the same way as in the standard FEM. A number of numerical examples, including both static and free vibration analyses, are studied using the present CCM and the numerical results are compared with the analytical ones and those in the published literatures. The results show that outstanding schemes can obtain very accurate solutions.

Load identification has long been a difficult issue for distributed load acting on structures. In this paper, the dynamic load identification technology based on the modal coordinate transformation theory is developed for dealing with identification problem of the two-dimensional thin plate structure. For the distributed dynamic load acting on a plate, we decompose it with the mode functions in the modal coordinate space and establish the liner relationship between the time function coefficients of the distributed load and the modal excitations which are solved out based on the known response data of the measuring points. Then the distributed dynamic load is rebuilt based on orthogonal decomposition and inverse Fourier Transform method. The simulation examples and elastic thin plate structure tests show that the proposed method has a good accuracy with the allowable error range and is reliable and practical. The proposed method can be also used for load identification of complicated structures in a wide range of engineering applications.

A double finite sine integral transform method is employed to analyze the buckling problem of rectangular thin plate with rotationally-restrained boundary condition. The method provides more reasonable and theoretical procedure than conventional inverse/semi-inverse methods through eliminating the need to preselect the deflection function. Unlike the methods based on Fourier series, the finite integral transform directly solves the governing equation, which automatically involves the boundary conditions. In the solution procedure, after performing integral transformation the title problem is converted to solve a fully regular infinite system of linear algebraic equations with the unknowns determined by satisfying associated boundary conditions. Then, through some mathematical manipulation the analytical buckling solution is elegantly achieved in a straightforward procedure. Various edge flexibilities are investigated through selecting the rotational fixity factor, including simply supported and clamped edges as limiting situations. Finally, comprehensive analytical results obtained in this paper illuminate the validity of the proposed method by comparing with the existing literature as well as the finite element method using (ABAQUS) software.

In this paper, the improved complex variable element-free Galerkin (ICVEFG) method is proposed for solving the bending problem of thin plate on elastic foundations. In the ICVEFG method, the approximation function regarding the deflection of thin plate is formed with the improved complex moving least-squares (ICVMLS) approximation, the discrete equation is obtained from Galerkin weak form of bending problem of thin plate on different elastic foundations, and essential boundary conditions are considered based on penalty method. As the ICVMLS approximation is based on the complex variable theory, it can obtain the shape function quickly with high precision. Three sample problems are used to discuss the advantages of the ICVEFG method, and the numerical results show that the ICVEFG method presented in this paper has a fast convergence speed and great computational accuracy.

A three-dimensional model of the hydro-elastic waves in the mammalian cochlea is presented along with numerical simulations. The cochlear fluid is treated as linear, incompressible, and inviscid. The cochlear partition is treated as a massless thin plate loaded by the fluid. This model is then reformulated by analytically removing the fluid variable with the use of a Dirichlet-to-Neumann operator. The resulting fifth-order nonlocal PDE for the motion of the partition is simulated using a novel implicit numerical scheme. Simulations demonstrate that this model exhibits traveling wave characteristics and a clear place principle. Asymptotic analysis in the small aspect ratio of the cochlea is performed on the given model equations with energetic concerns in mind. The results of simulations along with these asymptotic arguments suggest a relationship between the form and function of the cochlea which we compare to physiological data.

The paper is devoted to simulating the impact of a thermal shock on a thin homogeneous plate in the ANSYS package. The assessment of the stress–strain state is carried out and the dynamics of changes in the temperature field of the plate is determined. The obtained results were compared with the data of other authors and can be used when taking into account the thermal shock of large elastic elements of spacecraft.

This paper makes the first attempt to propose a novel hybrid collocation solver based on the generalized finite difference method (GFDM) and singular boundary method (SBM) to analyze underwater acoustic radiation and propagation around the thin plate structures excited by simple harmonic force under shallow sea environment. In the proposed hybrid solver, the meshless GFDM is employed to obtain the fluid–structure coupling vibration response of thin plate structure, and then the SBM with Pekeris waveguide Green function is used to calculate the external acoustic field of thin plate structure. The simplified Price–Wu boundary condition is adopted on the surface of plate structure to connect the computational domains between the GFDM and SBM. Three benchmark examples are carried out to demonstrate the accuracy and efficiency of the proposed hybrid GFDM–SBM solver in comparison with the COMSOL simulation for computing underwater acoustic wave fields around the thin plate structures under shallow sea environment.

The paper is devoted to simulating the impact of a thermal shock on a thin homogeneous plate in the ANSYS package. The assessment of the stress–strain state is carried out and the dynamics of changes in the temperature field of the plate is determined. The obtained results were compared with the data of other authors and can be used when taking into account the thermal shock of large elastic elements of spacecraft.