Free Vibration Of Thin Plates Research Articles

Analysis of Size-Dependent Linear Static Bending, Buckling, and Free Vibration Based on a Modified Couple Stress Theory

The purposes of this paper are to study bending, buckling, and vibration by considering micro-scale effects using the Kirchhoff thin-plate theory and to consider small deflections, neglecting higher-order nonlinear terms. The governing equations for the bending, buckling, and vibration of the system are obtained using the equilibrium method coupled with the Kirchhoff thin-plate theory and a modified couple stress theory (MCST). The concept of the equivalent bending stiffness (EBS) of micro-thin plates is proposed to describe the scale effect. The Navier method is used to obtain analytical solutions for the bending, buckling, and free vibration of thin plates under simply supported boundary conditions with scale effects. The numerical results are presented to investigate the influence of scale effects on deflection, critical buckling load, buckling topography, and thin-plate natural frequency. The results show that the scale effect increases the equivalent stiffness of the thin plate, which leads to a decrease in deflection, a larger critical buckling load, and an increase in natural frequency, but does not affect the buckling topography. The MSCT is invalid when the thickness is greater than 10 times the scale effect parameter, thus defining the scope of application of the scale effect. This research study may contribute to the design of micro-scale devices such as MEMSs/NEMSs.

The Effect of the Material Periodic Structure on Free Vibrations of Thin Plates with Different Boundary Conditions.

Thin elastic periodic plates are considered in this paper. Since the plates have a microstructure, the effect of its size on behaviour of the plates can play a crucial role. To take into account this effect, the tolerance modelling method is applied. This method allows us to obtain model equations with constant coefficients, which involve terms dependent of the microstructure size. Using the model equations, not only can formulas of fundamental lower-order vibration frequencies be obtained, but also formulas of higher-order vibration frequencies related to the microstructure. In this paper, the effect of the material periodic microstructure on free vibration frequencies for various boundary conditions of the plates was analysed. To obtain proper formulas of frequencies, the Ritz method is applied. Moreover, some results are compared to the results calculated using the FEM.

A Unified Solution Method for Free Vibration of Arbitrarily Shaped Plates without or with Cracks

This paper presents a new approach for analyzing the free vibration of thin plates with arbitrary piecewise smooth curvilinear contour under various boundary conditions. It can also be applied to plates with cracks. This approach is based on the transformation of energy integral expressions and the domain decomposition technique. Furthermore, boundary conditions are modeled by using linear springs to restrain the plate edges. For obtaining the expressions of energy integral, the arbitrarily shaped domain of integration is divided into several trapezoid domains with curved sides by a set of parallel lines passing through the intersection points of contour curve segments. Then, Jacobi orthogonal polynomials are introduced as the admissible functions, so that the repeated integrals in the energy expressions are reduced to definite integrals analytically. At this point, the calculation method of the energy functional is determined by the equation forms of the curve segments of the plate contour. When the equations of the curve segments allow the integrands to have analytic primitive functions, the energy functional has an analytical solution. Otherwise, the Gauss–Legendre method is used to obtain the numerical solution. Accordingly, the arbitrarily shaped plate with cracks is decomposed into several arbitrarily shaped subdomains based on the cracks. Each subdomain is handled according to the above procedure. The continuity conditions at the interconnecting interfaces of the subdomains are realized by linear springs. The plate without or with crack is modeled by setting the spring stiffness to infinity or 0. The accuracy of the proposed method is verified by comparing the obtained results with the published results. Furthermore, the vibration characteristics of plates with various shapes, such as astroid-shaped plates and cracked elliptical plates, are investigated. These new results can serve as benchmarks for further studies on the vibration of plates.

Exact Solutions of Fully Nonstationary Random Vibration for Rectangular Kirchhoff Plates Using Discrete Analytical Method

This paper proposes the discrete analytical method (DAM) to determine exactly and efficiently the fully nonstationary random responses of rectangular Kirchhoff plates under temporally and spectrally nonstationary acceleration excitation of earthquake ground motions. First, the fully nonstationary power spectral density (PSD) model is suggested by replacing the filtered frequency and damping of Gaussian filtered white-noise model with the time-variant ones. The exact solutions of free vibration of thin plates with two opposite edges simply supported boundary conditions are introduced. Then, the full analytical procedure for random vibration analysis of the plate is established by using a pseudo excitation method (PEM) that can consider all modal auto-correlation and cross-correlation terms. Owing to involving a series of Duhamel time integrals of single degree of freedom systems, it is difficult to fully analytically evaluate the PSD of time-variant responses such as the transverse deflection, velocity, acceleration and stress components. Thus, DAM that combines the PEM with precise integration technique is developed to enhance the computational efficiency. Finally, comparison of the results by the DAM with Monte Carlo simulations and the analytical stationary random vibration analysis demonstrates the high efficiency and accuracy of DAM. Moreover, the fully nonstationary excitation imposes a remarkable effect on the response PSD of rectangular Kirchhoff plates.

Vibration Analysis of Plate with Irregular Cracks by Differential Quadrature Finite Element Method

A universal method combining the differential quadrature finite element method with the virtual spring technique for analyzing the free vibration of thin plate with irregular cracks is proposed. Translational and rotational springs are introduced to restrain the vertical displacement and orientation of the plate. The mass matrix and stiffness matrix for each element are deduced involving the effects of the virtual springs. The connection relationships between elements can be modified by setting the stiffness of the virtual springs. The vibration of two rectangular plates with three irregular shaped cracks and different boundary conditions are presented. The results are compared with those obtained by ANSYS, where the good agreement between the results validates the accuracy and efficiency of the present method.

Hermite type radial basis function-based differential quadrature method for higher order equations

In the paper, the radial basis function-based differential quadrature method (RBF-DQM) that uses Hermite type interpolation is developed. The method is an extension of the known RBF-DQM, which is a meshless numerical technique for solving differential equations. According to this technique, derivatives in a governing equation are approximated by a linear weighted sum of the sought function values defined at scattered nodes. To allow the method to be applied in higher order equations, where more than one boundary condition is imposed at an edge, Hermite type interpolation for the radial basis functions is used and appropriate weighting coefficients for differential quadrature method are determined in the paper. As a numerical test the method is used to discretize the governing equation for the free vibration of thin plates with various boundary conditions. Different shaped plates with various boundary conditions are analyzed. The convergence tests carried out in the work confirm usefulness of the method as a truly meshless technique.

Free vibration analysis of variable thickness thin plates by two-dimensional differential transform method

This paper aims at extending the application of two-dimensional differential transform method (2D-DTM) to study the free vibration of thin plates with arbitrarily varying thickness. First, the differential equation of motion governing thin plates with varying thickness is derived using Hamilton’s principle. Afterward, the 2D-DTM, a numerical method which is capable of reducing the size of computational work and can be applied to various types of differential equations, has been applied to derive the natural frequencies of variable thickness thin plates with different boundary conditions. Several numerical examples have been carried out to demonstrate the applicability and accuracy of the present method in free vibration analysis of both uniform plates and plates with variable thickness.

Simplified approach of free vibration analysis of plates supported in vicinity of the corners by BEM

Free vibration analysis of Kirchhoff plate by the Boundary Element Method is presented in the paper. The boundary integral equation are derived according to the Bettie theorem. The collocation version of BEM with non-singular approach with one and double collocation points is used. The constant type of element is introduced. Boundary support at selected point is modelled as support in vicinity of point along single boundary element. Introduction Plates supported at points, in vicinity of selected points, for example by the columns, are often used in building structures. The bending problem of plates supported on boundary or rested on internal supports is the classic boundary issue in theory of plates and shells. The columns are often located on plate edges. The first well known work which included strict solutions of these problems is a publication of Nadai [1]. Iguchi [2] considered free vibrations of thin plates supported at the corners. Kączkowski [3] applied double Fourier range of sines to derive a general solution of boundary problem of orthotropic rectangular plates with optional boundary conditions including internal supports located at the plate corners. Timoshenko and Woinowsky-Krieger [4] analysed a static problem of a square plate rested on supports located at the corners by. The solutions as polynominals and range of cosines were assumed. Woźnica [5] solved analytically similar static and dynamic problem using Fourier transformation. A lot of papers present numerical approach, especially the Finite Difference Method (FDM), the Finite Element Method (FEM) and the Finite Strip Method (FSM), as a useful tool in plate bending analysis. Wide review of the literature is included and cited by Guminiak and Sygulski [6]. The Boundary Element Method (BEM) was proposed as independent numerical tool for applying for engineering applications [7, 8]. Bezine [9] and Stern [10] introduced BEM for analysis of plate bending according to Kirchhoff theory. Vander Weeen [11] proposed fundamental solutions for Reissner plate and used BEM to numerical analysis. Thick plates according to Reissner theory were analysed by Litewka and Sygulski [12] by the BEM. Authors applied fundamental solution derived by Ganowicz [13]. Myślecki [14, 15], Myślecki and Olenkiewicz [16] and Olenkiewicz [17] applied BEM tech-

Vibration Analysis of Thin Plates Resting on Pasternak Foundations by Element free Galerkin Method

The element free Galerkin method is used to analyze free vibration of thin plates resting on Pasternak elastic foundations with all possible types of classical boundary conditions. Convergence of solution is studied by increasing number of nodes for different boundary conditions and foundation parameters. Upon comparison with available results in literature, it was found that the method converges very fast and has very good accuracy even with small number of nodes. Applicability of the method was shown by solving numerical examples with all possible combinations of boundary conditions and different values of foundation parameters.

Vibration Analysis of Thin Plates Resting on Pasternak Foundations by Element free Galerkin Method

The element free Galerkin method is used to analyze free vibration of thin plates resting on Pasternak elastic foundations with all possible types of classical boundary conditions. Convergence of solution is studied by increasing number of nodes for different boundary conditions and foundation parameters. Upon comparison with available results in literature, it was found that the method converges very fast and has very good accuracy even with small number of nodes. Applicability of the method was shown by solving numerical examples with all possible combinations of boundary conditions and different values of foundation parameters.

New exact solutions for free vibrations of rectangular thin plates by symplectic dual method

The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simply-supported (S) and clamped (C) boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.

Active Control of Vibration

Introduction to Mechanical Vibrations: Terminology. Single-degree-of-freedom (SDOF) Systems. Free Motion of SDOF Systems. Damped Motion of SDOF Systems. Forced Response of SDOF Systems. Transient Response of SDOF Systems. Multi-degree-of-freedom (MDOF) Systems. Free Motion of MDOF Systems. Forced Response of MDOF Systems. Damped Motion of MDOF Systems. Finite Element Analysis of Vibrating Mechanical Systems. Introduction to Waves in Structures: Longitudinal Waves. Flexural Waves. Flexural Response of an Infinite Beam to an Oscillating Point Force. Flexural Wave Power Flow. Flexural Response of an Infinite Thin Beam to an Oscillating Line Moment. Free Flexural Motion of Finite Thin Beams. Response of a Finite Thin Beam to an Arbitrary Oscillating Force Distribution. Vibration of Thin Plates. Free Vibration of Thin Plates. Response of a Thin Rectangular Simply Supported Plate to an Arbitrary Oscillating Force Distribution. Vibration of Infinite Thin Cylinders. Free Vibration of Finite Thin Cylinders. Harmonic Forced Vibration of Infinite Thin Cylinders. Feedback Control: Single-channel Feedback Control. Stability of a Single-Channel System. Modification of the Response of an SDOF System. The Effect of Delays in the Feedback Loop. The State Variable Approach. Example of a Two-degree-of-freedom System. Output Feedback and State Feedback. State Estimation and Observers. Optimal Control. Modal Control. Feedforward Control: Single Channel Feedforward Control. The Effect of Measurement Noise. Adaptive Digital Controllers. Multichannel Feedforward Control. Adaptive Frequency Domain Controllers. Adaptive Time Domain Controllers. Equivalent Feedback Controller Interpretation. Distributed Transducers for Active Control of Vibration. Active Control of Vibration in Structures: Feedforward Control of Finite Structures. Feedback Control of Finite Structures. Feedforward Control of Wave Transmission. Actuator Arrays for Control of Flexural Waves. Sensor Arrays for Control of Flexural Waves. Feedforward Control of Flexural Waves. Feedback Control of Flexural Waves. Active Isolation of Vibrations: Isolation of Periodic Vibrations of an SDOF System. Vibration Isolation From a Flexible Receiver the Effects of Secondary Force Location. Active Isolation of Periodic Vibrations Using Multiple Secondary Force Inputs. Finite Element Analysis of an Active System for the Isolation of Periodic Vibrations. Practical Examples of Multi-Channel Feedforward Control for the Isolation of Periodic Vibrations. Isolation of Unpredictable Vibrations from a Receiving Structure. Isolation of Vibrating Systems from Random External Excitation the Possibilities for Feedforward Control. Isolation of Vibrating Systems from Random External Excitation Analysis of Feedback Control Strategies. Isolation of Vibrating Systems from Random External Excitation Formulation in Terms of Modern Control Theory. Active Isolation of Vehicle Vibrations from Road and Track Irregularities. Active Structural Acoustic Control, I. Plate Systems: Sound Radiation by Planar Vibrating Surfaces the Rayleigh Integral. The Calculation of Radiated Sound Fields by Using Wavenumber Fourier Transforms. Sound Power Radiation From Structures in Terms of Their Multi-Modal Response. General Analysis of Active Structural Acoustic Control (ASAC) for Plate Systems. Active Control of Sound Transmission Through a Rectangular Plate Using Point Force Actuators. Active Control of Structurally Radiated Sound Using Multiple Piezoelectric Actuator Interpretation of Behaviour in Terms of the Spatial Wavenumber Spectrum. The Use of Piezoelectric Distributed Structural Error Sensors in ASAC. An Example of the Implementation of Feedforward ASAC. Feeback Control of Sound Radiation From a Vibrating Baffled Piston. Feedback Control of Sound Radiation From Distributed Elastic Structures. Active Structural Acoustic Control, II. Cylinder Systems: Coupled Cylinder Acoustic Fields. Response of an Infinite Cylinder to a Harmonic Forcing Function. Active Control of Cylinder Interior Acoustic Fields Using Point Forces. Active Control of Vibration and Acoustic Transmission in Fluid-Filled Piping Systems. Active Control of Sound Radiation From Vibrating Cylinders. Active Control of Sound in Finite Cylinder Systems. Control of Interior Noise in a Full Scale Jet Aircraft Fuselage. Appendix. References. Index.

A boundary integral equation formulation for large amplitude nonlinear vibration of thin elastic plates

An integral equation formulation for large amplitude free vibration of thin plates is presented. The Von Karman nonlinear plate bending theory is adopted. By using the harmonic balance method and discretizing both on the boundary and in the domain, the amplitude dependent eigenvalue equations are obtained. An appropriate iterative procedure is proposed to solve the nonlinear free vibration problems. A number of numerical examples are given to demonstrate its feasibility and efficiency.

Free vibrations of thin plates with inhomogeneities

Free vibrations of thin plates with inhomogeneities