The governing equation is
[1.42]
Here,
Boundary conditions have the form
According to the principle of virtual work,
where U and V are, respectively, the potential and kinetic energy, defined as follows:
[1.46]
Using the ansatz
we obtain from equations [1.45]–[1.47]
The expression for the eigenfunction W(x, y) obtained using DEEM has the form
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
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
