An approximate solution for the bending vibrations of a combination of rectangular thin plates

The appropriate method often used for calculating the bending vibration of a single rectangular plate is extended to calculate the bending vibrations of a global system of combinations of rectangular plates with elastically supported and damped non-coupled edges. Two examples, a series of T-combinations and an L-combination of rectangular thin isotropic plates, are considered and the input and transfer mobilities due to point excitation derived. Numerical results are presented for the case of combinations of concrete plates and the effects of varying the material damping of the plates and edge damping are investigated.The eigenfrequencies of an L-combination of plates with one plate of very high bending stiffness are calculated and results compare well with the eigenfrequencies of a single plate calculated by means of the classical Ritz-Rayleigh method.     

Nonlinear free transverse vibrations of in-plane moving plates: Without and with internal resonances

In this paper, nonlinear free transverse vibrations of in-plane moving plates subjected to plane stresses are investigated. The Hamilton principle is applied to derive the governing equation and the associated boundary conditions. The method of multiple scales is employed to analyze the nonlinear partial differential equation. The solvability conditions are established in the cases without internal resonance and with 3:1 or 1:1 internal resonances. Some numerical examples are presented to demonstrate the effects of in-plane moving speeds on the frequencies. The nonlinear frequencies of the in-plane moving plate without internal resonances are numerically calculated. The relationship between the nonlinear frequencies and the initial amplitudes is showed at different in-plane moving speeds and the nonlinear coefficients, respectively. It is feasible to investigate resonances without the modes not involved in the resonances. The effects of the related parameters are demonstrated for the case of 3:1 and 1:1 internal resonances, respectively. The differential quadrature scheme is developed to solve numerically the governing equation and confirm results via the method of multiple scales.     

Transverse vibrations of thin,elastic plates with concentrated masses and internal elastic supports

The fundamental frequency of vibration of a plate carrying concentrated masses and with internal elastic supports is determined. The case of an orthotropic, rectangular plate elastically restrained against rotation along the four edges is tackled first by using simple polynomial approximations and the Galerkin method. Then, vibrations of clamped and simply supported isotropic plates of regular polygonal shape are studied by using the conformal mapping technique coupled with the variational method. Finally the case of a circular plate elastically restrained against translation and rotation is considered.     

Localized bending vibrations of piezoelectric plates

This paper investigates the existence and propagation of electro-elastic bending waves localized at the free edge of a piezoelectric plate. The problem is considered within the framework of the high-order refined plate theory introduced by Ambartsumian. The condition for existence of a localized bending wave is obtained, and the dispersion equation solved with respect to a dimensionless frequency. It is shown that the piezoelectric effect can increase the attenuation coefficient for a localized wave by up to 70% compared with that for a purely elastic plate, thus significantly decreasing the depth of penetration. The problem is also solved within the classical Kirchhoff theory. A comparison of results is carried out between two theories.     

Piezoelastic vibrations of composite Reissner-Mindlin-type plates

Linear vibrations of Reissner-Mindlin-type composite plates in the presence of piezoelectric eigenstrains are studied. Piezoelectric eigenstrains are produced by applying electrical loads to piezoelectric layers embedded in or attached to substrate layers. The influence of the mechanical field upon the electric field is taken into account in the modelling, ending up with electro-mechanically coupled field equations and boundary conditions, which describe the mechanical and the electrical dynamic response of the plate.The mechanical displacements are approximated by means of the kinematic hypothesis of Hencky. The electric potential distribution is assumed to be composed of a superposition of a linear and a parabolic distribution in the thickness direction. The linear part accounts for the electric potential difference between the electrodes of the totally electroded piezoelectric layers. The parabolic part is considered in order to take into account the influence of the mechanical field upon the electric potential by means of the direct piezoelectric effect. A weak two-dimensional formulation of the three-dimensional field equations is obtained by utilizing mechanical and electrical variational principles. This formulation is characterized by resultants of stress and electric displacement. The electro-mechanically coupled behaviour comes into play by means of the constitutive relations. In case the electric potential difference is not prescribed, it can be calculated from a relation, which connects the total electric charge and the electric potential difference to each other. Because this relation is obtained from the Gauss law of electrostatics, requiring integration with respect to the area of the electrode, non-local constitutive relations for the plate are found. The non-local constitutive relations bring a new aspect into the theory of plates. An analysis for the practically interesting one-dimensional case of composite, piezoelectric plates in cylindrical motion completes the paper.     

Transverse vibrations of hybrid laminated plates

In this paper, an attempt is made to obtain the free vibration response of hybrid, laminated rectangular and skew plates. The Galerkin technique is employed to obtain an approximate solution of the governing differential equations. It is found that this technique is well suited for the study of such problems. Results are presented in a graphical form for plates with one pair of opposite edges simply supported and the other two edges clamped. The method is quite general and can be applied to any other boundary conditions.     

Free vibrations of randomly inhomogeneous plates

Consider a large collection of elastic rectangular plates with random inhomogeneities, but otherwise indistinguishable in any overall sense. An expression is obtained for the natural frequency, μ, of such plates, vibrating freely under simply supported boundary conditions, in the form μ = μ⁽⁰⁾ + ?μ⁽¹⁾ + ?²μ⁽²⁾ + … where μ⁽⁰⁾ is the natural frequency of a homogeneous comparison plate, ? is a small real parameter measuring the degree of inhomogeneity, and the coefficients μ⁽¹⁾, μ⁽²⁾, …, are given explicitly. By constructing geometrically a correlation function for a special type of composite plate, μ⁽¹⁾ is computed and hence, to first order in ?, the variance of μ. The paper concludes with a theorem linking the mean and variance of μ to the volume concentration and geometry of the inclusions in the plate.     

On the effect of different edge flexibility coefficients on transverse vibrations of thin,rectangular plates

To solve problems of transverse vibration of thin, rectangular plates with different edge flexibility coefficients polynomial co-ordinate functions are used which identically satisfy the boundary conditions. It is shown that by a proper combination of the polynomials several modes of vibrations can be analyzed with a minimum amount of labour. A variational formulation is used to generate the frequency equation. Eigenvalues calculated by using a two-term approximation seem to possess extremely good accuracy, at least from an engineering viewpoint. It is also shown that the effect of in-plane forces or the case where the plate is supported on a Winkler-type foundation can be studied without any formal difficulties.     

Nonlinear vibrations of buried landmines

The seismo-acoustic method is one of the most promising emerging techniques for the detection of landmines. Numerous field tests have demonstrated that buried landmines manifest themselves at the surface through linear and nonlinear responses to acoustic/seismic excitation. The present paper describes modeling of the nonlinear response in the framework of the mass-spring model of the soil-mine system. The perturbation method used in the model allows for the derivation of an analytical solution describing both quadratic and cubic acoustic interactions at the soil-mine interface. This solution has been compared with actual field measurements to obtain nonlinear parameters of the buried mines. These parameters have been analyzed with respect to mine types and burial depths. It was found that the cubic nonlinearity could be a significant contributor to the nonlinear response. This effect has led to the development of a new intermodulation detection algorithm based on dual-frequency excitation. Both quadratic and intermodulation nonlinear algorithms were evaluated at the U.S. Army outdoor testing facilities. The algorithms appear to complement each other in improving the overall detection performance.     

Nonlinear vibrations of loudspeaker-like structures

During high amplitudes of vibration, nonlinearities affect the electroacoustical behavior of electrodynamic transducers and are responsible for audible distortions. We distinguish two types of nonlinearities: electrical and mechanical. In this study, attention is paid to the mechanical and geometrical properties of loudspeaker-like structures. The loudspeaker is viewed as a combination of an annular plate with a circular plate. Nonlinear vibrations of such a structure are investigated, using the dynamic analog of the Von-Kármán equations. Furthermore, the influence of both material properties and geometrical parameters is studied. It is shown that nonlinear effects can be substantially reduced by choosing appropriate material and geometrical parameters.     

On the non-linear vibrations of rectangular plates

A solution, based on a one-term mode shape, for the large amplitude vibrations of a rectangular orthotropic plate, simply supported on all edges or clamped on all edges for movable and immovable in-plane conditions, is found by using an averaging technique that helps to satisfy the in-plane boundary conditions. This averaging technique for satisfying the immovable in-plane conditions can be used to resolve many anisotropic and skew plate problems where otherwise, when a stress function is used, the integration of the u and v equations becomes difficult, if not impossible. The results obtained herein are compared with those available in the literature for the isotropic case and excellent agreement is found. Results available for the one-term mode shape solutions of these problems are compared and the non-linear effect is presented as functions of aspect ratio and of the orthotropic elastic constants of the plate. The results are further compared with those based on the Berger method and the detailed comparative studies show that the use of the Berger approximation for large deflection static and dynamic problems and its extension to anisotropic plates, skew plates, etc., can lead to quite inaccurate results.     

Non-linear flexural vibrations of orthotropic rectangular plates

This study is an analytical investigation of free flexural large amplitude vibrations of orthotropic rectangular plates with all-clamped and all-simply supported stress-free edges. The dynamic von Karman-type equations of the plate are used in the analysis. A solution satisfying the prescribed boundary conditions is expressed in the form of double series with coefficients being functions of time. The model equations are solved by expanding the time-dependent deflection coefficients into Fourier cosine series. As obtained by taking the first sixteen terms in the double series and the first two terms in the time series, numerical results are presented for non-linear frequencies of various modes of glass-epoxy, boron-epoxy and graphite-epoxy plates. The analysis shows that, for large values of the amplitude, the effect of coupling of vibrating modes on the non-linear frequency of the fundamental mode is significant for orthotropic plates, especially for high-modulus composite plates.