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 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.
1.6. Quasi-separation of variables and normal modes of nonlinear oscillations of continuous systems
Bolotin did not give an exact definition of the concept of quasi-separation of variables. Intuitively, this means that the difference between solutions of boundary value problems with separated and quasi-separated variables is sufficient only near the boundaries. In other words, the energy accumulated in the EE zone is small compared to that accumulated in the inner zone. This allows us to not take into account DEE when expanding the natural mode of vibration during the calculation of forced oscillations. Bolotin’s conception of quasi-separation of variables (Bolotin 1961c, 1984) can be used in the theory of normal modes of nonlinear oscillations for continuous systems.
When studying linear oscillatory systems with a finite number of DOF, normal oscillation modes play a key role. Kauderer (1958) indicated the existence of solutions in a nonlinear system, which were, in a sense, similar to the normal modes of linear systems. He called these solutions the principal ones and showed how to construct their trajectories in the configuration space. Rosenberg (1962) defined normal vibrations of nonlinear systems with a finite number of DOF, formulated the problem in the configuration space and found several classes of nonlinear systems that allowed solutions with straight-line trajectories (for details, see Mikhlin and Avramov 2011; Avramov and Mikhlin 2013). Generalizations of this concept to continuous systems are related to the exact separation of spatial and time variables (Wah 1964; Avramov and Mikhlin 2013), i.e. to the possibility of representing the sought solutions in the form
The restriction of this approach is clear since the separation of variables only works for some boundary conditions. Based on Bolotin’s conception of the quasi-separation of variables, we can propose the following definition (Andrianov 2008): a function U(x,t) is called the normal mode of