Flexural Vibrations Of Thin Plates Research Articles

Wave based method for flexural vibration of thin plate with general elastically restrained edges

Wave based method is a numerical technique to predict steady-state vibration and solve vibroacoustic problems. It is a deterministic approach that is acknowledged to be more efficient than the element based methods in the mid-frequency range. The wave based method for plates with prescribed boundary conditions has been proposed and verified for the ideal constraint cases such as clamped, free and simply supported. To deal with different kind of or more realistic boundary conditions, this paper proposes a modified wave based method. The method is aimed at analyzing the flexural vibration of thin plates with general elastically restrained edges, and it can also consider the damping effect associated with the edge restraint. A number of numerical validation examples are shown, where the plate edges are uniform, non-uniform or partially supported. Within the examples, the responses of the plate under different edge conditions are compared, including elastic supports with different translational and rotational stiffness and damping parameters, rigid supports like clamped and simply supported, and free edges. For all the considered cases, the proposed method is proved to be effective, though the convergence performance is influenced by the boundary conditions. More wave functions may be required for simply supported, non-uniform or partially supported cases.

Non-uniform rational Lagrange functions and its applications to isogeometric analysis of in-plane and flexural vibration of thin plates

New basis functions were developed for isogeometric analysis (IGA) to overcome the difficulties of IGA using NURBS (Non-Uniform Rational B-Splines) on coping with Dirichlet boundary conditions. The new basis functions were constructed through nesting rational local Lagrange interpolations like the T-spline and were evaluated in similar procedure as the finite difference method. Explicit expressions for the new basis functions were presented. Due to their equivalence to the NURBS, the new basis functions were named as non-uniform rational Lagrange (NURL) basis. IGA using NURL includes the finite element method as a special case but the geometry in IGA using NURL is exact. IGA using NURL can carry out p-refinement that has the nesting feature of the k-refinement of NURBS. Dirichlet boundary conditions can be directly imposed in IGA using NURL because the NURL are interpolation basis functions. A method of directly transforming the tensor product basis of triangular patches to area coordinates was presented and the singularity problem at the edge degenerated to a single point was solved. The methods developed in this work were applied to in-plane and flexural vibration of thin plates. Comparisons with available results in literatures showed the fast convergence and high accuracy of IGA using NURL and the transformation method for triangular patches. Introduction of a MATLAB toolbox of the NURL was appended.

Thermoelastic relaxation in elastic structures, with applications to thin plates

A new result enables direct calculation of thermoelastic damping in vibrating elastic solids. The mechanism for energy loss is thermal diffusion caused by inhomogeneous deformation, flexure in thin plates. The general result is combined with the Kirchhoff assumption to obtain a new equation for the flexural vibration of thin plates incorporating thermoelastic loss as a damping term. The thermal relaxation loss is inhomogeneous and depends upon the local state of vibrating flexure, specifically, the principal curvatures at a given point on the plate. Thermal loss is zero at points where the principal curvatures are equal and opposite, that is, saddle shaped or pure anticlastic deformation. Conversely, loss is maximal at points where the curvatures are equal, that is, synclastic or spherical flexure. The influence of modal curvature on the thermoelastic damping is described through a modal participation factor. The effect of transverse thermal diffusion on plane wave propagation is also examined. It is shown that transverse diffusion effects are always small provided the plate thickness is far greater than the thermal phonon mean free path, a requirement for the validity of the classical theory of heat transport. These results generalize Zener's theory of thermoelastic loss in beams and are useful in predicting mode widths in micro- and nano-electromechanics systems oscillators.

Application of the R-function method to nonlinear vibrations of thin plates of arbitrary shape

In this paper free large-amplitude flexural vibrations of thin plates with various planforms and boundary conditions are studied by the R-function method. This method is based on the joint application of the R-function theory and variational methods. The main feature of the R-function theory is the possibility to present all geometric information given in the boundary value problem in analytical form, which allows one to seek a solution in the form of some formula called the solution structure. A method of constructing the solution structures for the given nonlinear vibration plate bending problems is developed. Numerical examples of large-amplitude flexural vibrations of thin plates with arbitrary shapes and boundary conditions for illustrating the aforementioned R-function method and comparison against the other methods are made to demonstrate its merits and advantages.

A theory for flexural vibration of thin plates with thermoelastic damping

Thermoelastic damping in thin plates can be the dominant loss mechanism under certain circumstances, e.g., as demonstrated by recent measurements for MEMS paddle oscillators. However, modeling of thermoelastic damping in flexural vibration of plates has lagged, despite the classical theory of Zener for beams. This talk shows how the thinness of the plate permits a very accurate approximation to the fully coupled system of stress and temperature. In this asymptotically valid theory, the flexural plate equations are modified by damping terms of two types. The first are viscosity-like terms in the plate equation of motion. These depend on the local curvature, being zero at saddles. The damping also modifies the boundary moment and shear force. Both the bulk and boundary effects are important in estimating the Q for resonant modes. The theory is derived from a generalized form of Hamilton’s principle, using Kirchhoff’s kinematic assumption and asymptotic approximations for small thermal coupling valid for all relevant materials. Modal damping can be estimated simply for certain boundary conditions, including fixed edges. Plates with free or partially free edges present a more interesting challenge, and various alternative estimates of Q are presented and illustrated numerically for rectangular plates and MEMS structures.

Large amplitude free flexural vibrations of thin plates of arbitrary shape

A finite element formulation is developed for analyzing large amplitude free flexural vibrations of elastic plates of arbitrary shape. Stress distributions in the plates, deflection shape and nonlinear frequencies are determined from the analysis. Linearized stiffness equations of motion governing large amplitude oscillations of plates, quasi-linear geometrical stiffness matrix, solution procedures, and convergence characteristics are presented. The linearized geometrical stiffness matrix for an eighteen degrees-of-freedom conforming triangular plate element is evaluated by using a seven-point numerical integration. Nonlinear frequencies for square, rectangular, circular, rhombic, and isosceles triangular plates, with edges simply supported or clamped, are obtained and compared with available approximate continuum solutions. It demonstrates that the present formulation gives results entirely adequate for many engineering purposes.