Applying the modified Fourier-Ritz approach, the admissible functions of the structure elements are expanded into the improved Fourier series which consist of two-dimensional (2D) Fourier cosine series and auxiliary functions to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions and then solving the natural frequencies by means of the Ritz method. In order to obtain the unified computational model of the functionally graded cylindrical, conical, spherical panels and shells of revolution, as one merit of this paper, the coupling spring technology is introduced to ensure the kinematic and physical compatibility at the common meridian, if a complete shell of revolution needs considering.
The presented method consists of two main steps: firstly, the admissible functions of the structure elements are expanded into the modified Fourier series which consist of the two-dimensional (2D) Fourier cosine series and auxiliary functions to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions; secondly, the natural frequencies of the structure elements are obtained by using the variational operation (Ritz method) in terms of the unknown the expanded coefficient.
Fazilati, "Bending buckling and vibration problems of nonlocal Euler beams using Ritz method
," Composite Structures, vol.
Leissa and Zhang  employed the Ritz method to study the three-dimensional problem of determining the free vibration frequencies and mode shapes for a rectangular parallelepiped which is completely fixed on one face and free on other five faces.
The modified Fourier series are chosen as the basis functions of the admissible function of the thick plates in the Ritz method. The arbitrary boundary conditions are obtained by laying out three types of liner springs on all edges.
Cheng, "The improved moving least-square ritz method
for the one-dimensional sine-gordon equation," Mathematical Problems in Engineering, vol.
The problem was solved employing a combination of the Ritz method
and the Lagrange multiplier method.
Liew and Lim  proposed a pb-2 Ritz method
to study the free vibration of the doubly curved shallow shells in form of a rectangular plane.
Lam and Loy  studied the effects of boundary conditions on the free vibration characteristic for a multilayered cylindrical shell based on Love's first approximation theory using Ritz method
. The displacements were represented as combination beam functions with trigonometric functions.
It was left as an open question how to generalize the harmonic Rayleigh-Ritz approach for the MER This paper addresses this issue, and also introduces a refined Ritz method
In case both p and q are polynomials of degree one, the rational harmonic Ritz approach is related to the standard harmonic Ritz method
In the Ritz method
, a dependent unknown (e.g., the displacement) w(x,t) is approximated by a finite linear combination as the form