A unified solution for free vibration of orthotropic circular, annular and sector plates with general boundary conditions

Abstract

This paper adopts a modified Fourier-Ritz approach to study the free vibration characteristics of orthotropic circular, annular and sector thin plates subjected to general boundary conditions. For the arbitrary plate forms and the boundary conditions, the displacements can be written in the form of a standard Fourier cosine series supplemented with several auxiliary functions. The auxiliary functions, which are closed-form and introduced to remove all the potential discontinuities of the original displacement function and its derivatives in the whole domain, can be usefully employed in improving the convergence of the results. The artificial boundary spring technique and artificial coupling spring technique are adopted to simulate the arbitrary boundary conditions and to ensure appropriate continuity conditions along the radial edges, respectively. Because the displacement field is sufficiently smooth in the whole solution domain, the accurate solution can be obtained by using the Ritz procedure on the basis of the energy functions. The accuracy, reliability and versatility of the current method are fully demonstrated and verified through numerical examples involving plates with various shapes and boundary conditions.

A generalized solution procedure for in-plane free vibration of rectangular plates and annular sectorial plates

“… B.C. method 1 2 3 4 5 6 7 8 CCCC present 3.3677 4.4791 5.8203 5.9742 6.7084 7.2168 7.7834 8.6131 FE AB [ 25 ] 3.3692 4.4807 5.8240 5.9810 6.7152 7.2240 7.7894 8.626 S 1 S 1 S 1 S 1 present 1.5882 3.063 3.6308 4.4535 5.8097 6.0961 4.9339 6.4738 FE AN [ 25 ] 1.5884 3.0641 3.6320 4.4572 4.9362 5.8163 6.1036 6.4793 S 2 S 2 S 2 S 2 present 1.3469 2.3639 2.8909 3.4244 ...…”

“…In table 7 , the results of an annular plate with different classical boundary conditions are compared to data from the literature. The results by ABAQUS-V6.12 and ANASYS-V14.5 are from [ 25 ]. Table 7 shows excellent agreement between the current model and existing data.…”

“…The presented method improves the efficiency for the sectorial plate in two aspects. First, the stiffness matrix and the mass matrix in equation (2.34) has the explicit form, which does not need the tedious numerical integration process [ 25 ], so it is helpful in generating the global matrices. Second, in equation (2.25), the terms in the truncated series of this paper are M = N = 10, while the chosen number of M and N are 15 in [ 25 ].…”

“…First, the stiffness matrix and the mass matrix in equation (2.34) has the explicit form, which does not need the tedious numerical integration process [ 25 ], so it is helpful in generating the global matrices. Second, in equation (2.25), the terms in the truncated series of this paper are M = N = 10, while the chosen number of M and N are 15 in [ 25 ]. The present method significantly reduces the computation cost of the eigen-problems for the matrix dimension in equation (2.34) is 4/9 of that in [ 25 ], which does not use the logarithmic radial variable.…”

“…Wang et al . [ 25 ] used a modified Fourier–Ritz approach [ 26 , 27 ] to solve the free in-plane vibration of orthotropic circular, annular and sectorial plates subjected to general boundary conditions. Recently, the modified Fourier series technique has been extended to study the in-plane vibration of plate and shell structures with general boundary conditions by the modified Ritz method [ 25 , 28 – 31 ].…”