Free vibration of a circular plate elastically restrained along some radial segments

An analysis is presented for the free vibration of a circular plate restrained against deflection along radial segments. With the reaction forces acting on the segments regarded as unknown harmonic loads, the stationary response of the plate to these loads is expressed by the use of the Green function. The force distributions along the segments are expanded into Fourier series with unknown coefficients, and the homogeneous equations for the coefficients are derived by restraint conditions on the supports. The natural frequencies and the mode shapes of the plate are determined by calculating the eigenvalues and eigenvectors of the equations. The method is applied to circular plates supported along several radial segments located at equal angular intervals, the natural frequencies and the mode shapes of the plates are calculated numerically and the effect of the supports is discussed.     

Free vibration of two elastically coupled rectangular plates with uniform elastic boundary restraints

An analytical method is derived for determining the vibrations of two plates which are generally supported along the boundary edges, and elastically coupled together at an arbitrary angle. The interactions of all four wave groups (bending waves, out-of-plane shearing waves, in-plane longitudinal waves, and in-plane shearing waves) have been taken into account at the junction via four types of coupling springs of arbitrary stiffnesses. Each of the transverse and in-plane displacement functions is expressed as the superposition of a two-dimensional (2-D) Fourier cosine series and several supplementary functions which are introduced to ensure and improve the convergence of the series representation by removing the discontinuities that the original displacement and its derivatives will potentially exhibit at the edges when they are periodically expanded onto the entire x-y plane as mathematically implied by a 2-D Fourier series. The unknown expansions coefficients are calculated using the Rayleigh-Ritz procedure which is actually equivalent to solving the governing equation and the boundary and coupling conditions directly when the assumed solutions are sufficiently smooth over the solution domains. Numerical examples are presented for several different coupling configurations. A good comparison is observed between the current results and the FEA models. Although this study is specifically focused on the coupling of two plates, the proposed method can be directly extended to structures consisting of any number of plates.     

Free axisymmetric vibration of a circular plate elastically supported along two concentric circles

This letter deals with the vibration of homogeneous circular plates resting on linear and torsion spring supports along the outside periphery and along a circle of arbitrary radius. The solution includes two special cases. In the first, the intermediate elastic support becomes rigid and in the second the elastic coefficient goes to zero thereby removing the intermediate support. Results are shown in the form of graphs. These graphs also include results for classical conditions at the outer periphery which are found to be in excellent agreement with the available results.     

Nonlocal-integro-differential modeling of vibration of elastically supported nanorods

In the previously established nonlocal continuum-based models, small characteristic length was commonly incorporated into the mass matrix and the driving force vector which is a bit in contradiction with our sense regarding these factors. Herein, a nonlocal-integro-differential version of the constitutive relations is employed for the bulk and the surface layer of the nanorod. By adopting Hamilton's principle, integro-partial differential equations of motion of elastically supported nanorods are established accounting for both nonlocality and surface energy effects. Then, these are solved by an efficient meshless methodology. For fixed–fixed and fixed–free nanorods, modal analysis of the problem is also performed and the explicit expressions of the mass and stiffness matrices are derived. For these special cases, the obtained results by the meshless technique are successfully verified with those of the modal solution. In the newly developed numerical model, the small-scale parameter is only incorporated into the stiffness matrix which gives us a more realistic sense about the nonlocality effect. Subsequently, the roles of the surface energy, small-scale parameter, elastic supports, and kernel function on natural frequencies of the nanostructure are discussed and explained. This work can be considered as a pivotal step towards a more reasonable nonlocal modeling of vibration of nanoscale structures.     

On the non-linear vibration of circular plates of variable thickness elastically restrained along the edges

Berger's technique of neglecting the second strain invariant in the middle plane of the plate is used in an investigation of the effect of non-linearity on the frequency of a vibrating circular plate of variable thickness which is elastically restrained along its circumference. A simple polynomial expression for the deflection function and a Galerkin method are used in the analysis.     

Non-linear axisymmetric vibration of orthotropic thin circular plates on elastic foundations

This study deals with the large amplitude axisymmetric free vibrations of cylindrically orthotropic thin circular plates resting on elastic foundations. Geometric non-linearity due to moderately large deflections has been included. Movable and immovable simply supported plates and immovable clamped plates resting on Winkler, Pasternak and non-linear Winkler foundations have been considered. The von Kármán type governing equations have been employed. Harmonic vibrations are assumed and the time t is eliminated by the Kantorovich averaging method. An orthogonal point collocation method is used for spatial discretization. Numerical results are presented for the linear natural frequency of the first axisymmetric mode and for the ratio of the non-linear period to the linear period of natural vibration. The effects of foundation parameters, the orthotropic parameter and the edge conditions on the non-linear vibration behaviour have been investigated.     

Vibration of plates elastically supported on a non-homogeneous foundation

This study deals with the determination of the fundamental frequency of vibration of circular and regular polygonal plates elastically supported over a non-homogeneous foundation. Both cases are tackled in a unified fashion by adopting polynomial co-ordinate functions and making use of the Ritz method to obtain the frequency equation.     

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.     

Non-linear vibration and postbuckling of isotrophic thin circular plates on elastic foundations

An approximate solution for the large deflection axisymmetric responses of isotropic thin circular plates resting on Winkler, Pasternak and non-linear Winkler foundations is presented. Plates with edges elastically restrained against rotation and -in-plane displacement are considered. Von Kármán type equations in terms of transverse deflection and stress function are employed. A one term mode shape is used to approximate the transverse deflection and Galerkin's method is used to obtain an equation for the central deflection which has the form of a Duffing's equation. Non-linear frequencies, postbuckling response to radial load at the edge and the maximum transient response to transverse step load have been obtained. It is shown that sufficiently accurate results are obtained by this method. Numerical results are presented to illustrate the effect of various parameters.     

Fundamental frequency of vibrating rectangular,non-homogeneous plates

An approximate solution is obtained in the present paper using the classical Ritz method. The use of polynomial coordinate functions allows for the treatment of edges elastically restrained against rotation.     

Experiments on elastic cloaking in thin plates

Following a theoretical proposal [M. Farhat et al., Phys. Rev. Lett. 103, 024301 (2009)], we design, fabricate, and characterize a cloaking structure for elastic waves in 1?mm thin structured polymer plates. The cloak consists of 20 concentric rings of 16 different metamaterials, each being a tailored composite of polyvinyl chloride and polydimethylsiloxane. By using stroboscopic imaging with a camera from the direction normal to the plate, we record movies of the elastic waves for monochromatic plane-wave excitation. We observe good cloaking behavior for carrier frequencies in the range from 200 to 400?Hz (one octave), in good agreement with a complete continuum-mechanics numerical treatment. This system is thus ideally suited for demonstration experiments conveying the ideas of transformation optics.