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DEEM, designed to calculate high eigenfrequencies, also gives enough accurate results for lower vibration modes at kinematic boundary conditions. For static conditions, the accuracy of determining the lowest natural frequencies decreases. Attempts to apply the method to stability problems have shown that the error of determining the buckling load is quite high.

One of the promising ways to improve the DEEM accuracy is its combination with variational approaches. The first works in this direction were the papers (Vijaykumar and Ramaiah 1978a, 1978b), where the Rayleigh–Ritz method (RRM) was applied and the asymptotic expressions for natural modes were used as basis functions (the Rayleigh–Ritz–Bolotin method, RRBM). According to the comparative estimates, this modification grants a much more accurate determination of natural frequencies (see also Krizhevskii 1988, 1989).

As an example, we use RRBM for natural oscillations of a square plate (0 ≤ x, y ≤ a) with free contour. The governing equation is

[1.42]

Here, , h is the plate thickness and ν is Poisson’s ratio.

Boundary conditions have the form

[1.43]

[1.44]

According to the principle of virtual work,

[1.45]

where U and V are, respectively, the potential and kinetic energy, defined as follows:

[1.46]

[1.47]

Using the ansatz

we obtain from equations [1.45]–[1.47]

[1.48]

The expression for the eigenfunction W(x, y) obtained using DEEM has the form

[1.49]

where

[1.50]

[1.51]

[1.52]

On satisfying the boundary conditions [1.43] and [1.44] to determine the wave numbers, we obtain a system of transcendental equations

[1.53]

where , , ; .

For constants Cij, we obtain

[1.54]

Using the DEEM solution [1.49]–[1.54], we can determine the desired frequency from expression [1.48].

The square of dimensionless frequencies λ for ν = 0.225 and various m, obtained by RRM (Gontkevich 1964), RRBM and DEEM are shown in Table 1.1. Wave forms along a cylindrical surface are not considered since in this case an exact solution can be obtained. The numbers corresponding to the indicated modes of vibration are omitted in Table 1.1.

Table 1.1. Comparison of frequencies obtained using various approximation methods

m	λ, RRM (Gontkevich 1964)	λ, RRBM	Discrepancy with Gontkevich (1964), %	λ, DEEM	Discrepancy with Gontkevich (1964), %
1	14.10	14.48	2.7	12.41	13.6
3	35.96	36.68	2.0	34.60	3.9
5	65.24	66.33	1.7	63.44	2.8
6	74.45	75.28	1.1	73.59	2.5
7	109.30	109.10	0.2	106.30	2.8

RRBM gives more accurate results than DEEM for the first natural frequencies. When m increases, both solutions asymptotically approach the exact one, namely, from above in the case of applying RRMB and from below in the case of using DEEM.

RRBM can also be used for stability problems of plates and shells with complicated boundary conditions. This method was applied to plates of complicated form (skew, circle, sector (Andrianov and Krizhevskiy 1988, 1989, 1991)) and structures (Andrianov and Krizhevskiy 1987, 1993).

An interesting modification of DEEM for determining natural frequencies and mode shapes of isotropic and orthotropic rectangular plates with various types of boundary conditions was given in Pevzner et al. (2000). This approach does not postulate the formula for the eigenfrequency, but rather it is based on the condition that the frequency obtained from the governing differential equations has to be equal to that given by the Rayleigh method. The paper by Pevzner et al. (2000) claims that this modification is more straightforward and computationally faster, and the mode shapes derived are valid on a larger part of the plate.