Free vibration of circular plates with elastic edge supports using the receptance method

Flexural vibrations of free circular plates elastically constrained along parts of the edge

This paper deals with the in-plane vibration of circular annular disks under combinations of different boundary conditions at the inner and outer edges. The in-plane free vibration of an elastic and isotropic disk is studied on the basis of the two-dimensional linear plane stress theory of elasticity. The exact solution of the in-plane equation of equilibrium of annular disk is attainable, in terms of Bessel functions, for uniform boundary conditions. The frequency equations for different modes can be obtained from the general solutions by applying the appropriate boundary conditions at the inner and outer edges. The presented frequency equations provide the frequency parameters for the required number of modes for a wide range of radius ratios and Poisson's ratios of annular disks under clamped, free, or flexible boundary conditions. Simplified forms of frequency equations are presented for solid disks and axisymmetric modes of annular disks. Frequency parameters are computed and compared with those available in literature. The frequency equations can be used as a reference to assess the accuracy of approximate methods.

https://downloads.hindawi.com/archive/2010/501902.pdf

Frequency Equations for the In-Plane Vibration of Circular Annular Disks

Finite element analysis of vibrations of initially stressed thin shells of revolution

In-plane freevibrations of an elasticand isotropic annular disk withelasticconstraints at theinner and outer boundaries, whichareappliedeither alongtheentireperipheryof thediskorat apoint areinvestigated. Theboundary characteristicorthogonal polynomials are employed in the Rayleigh-Ritz method to obtain the frequency parameters and the associated mode shapes. Boundary characteristic orthogonal polynomials are generated for the free boundary conditions of the disk while artificial springs are used to account for different boundary conditions. The frequency parameters for different boundary conditions of the outer edge are evaluated and compared with those available in the published studies and computed from a finite element model. The computed mode shapes are presented for a disk clamped at the inner edge and point supported at the outer edge to illustrate the free in-plane vibration behavior of the disk. Results show that addition of point clamped support causes some of the higher modes to split into two different frequencies with different mode shapes.

/pdf/in-plane-free-vibration-analysis-of-an-annular-disk-with-3dfsu1ebu8.pdf

In-Plane free Vibration Analysis of an Annular Disk with Point Elastic Support

= radius of the plate = £/i/[12(l — v)], flexural rigidity of the plate = Young's modulus of material = thickness of plate = rotational spring constant = radial load per unit length of circumference = buckling load per unit length of circumference = [_ppha/D']' characteristic value of frequency = characteristic value of frequency without radial load = deflection of plate = polar coordinates = nondimensional coordinate defined by r/a = mass density of plate material = frequency of oscillation, rad/sec = Poisson's ratio = nondimensional boundary restraint factor defined by ka/D = nondimensional radial load defined by Ra/D

C. L. Amba-Rao

Papers

Lateral vibration and stability relationship of elastically restrained circular plates.