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Substituting function w0 into equation [1.29], we obtain a PDE for function wee :
where
It is important that PDE [1.30] is linear. The spatial and time variables are not separated exactly; therefore, we apply the Kantorovich variational method (Kantorovich and Krylov 1958) to solve equation [1.30], presenting wee in the form
On substituting ansatz [1.31] into PDE [1.30] and applying the Kantorovich method (Kantorovich and Krylov 1958), the following ODE is obtained:
with
[1.33]
Hereinafter, we use the principal value of the arcsin(…) function.
Among the four roots of the characteristic equation for ODE [1.32], two purely imaginary ones correspond to the generating solution W0. To construct DEE, we should use real roots of the characteristic equation. Then, the DEE solution is
Let us construct DEE near the edge x = 0 . For a sufficiently long beam, we can suppose
[1.35]
Then, at x = 0, we have from the boundary conditions
Using expressions [1.34]–[1.36], we obtain
Note that when c* → 0 and c* → ∞, formulas [1.34]–[1.38] yield solutions for simply supported and clamped ends of the beam, respectively.
Similarly, we can construct DEE localized at the edge x = L .
The modes of natural nonlinear oscillations of the beam can be divided into groups according to the types of symmetry. For the modes that are symmetric relative to the point x = L/2, from the condition
we obtain
For antisymmetric modes, from the condition
we have
Equations [1.39] and [1.40] can be reduced to the following form:
in which even values of m correspond to antisymmetric modes, and odd values of m to symmetric modes relative to the point x = L/2 .
Thus, the system of equations [1.37], [1.38] and [1.41] can be applied to determine the constants λ and x0.
The described technique was used to study nonlinear oscillations of isotropic (Andrianov et al. 1979; Zhinzher and Denisov 1983; Awrejcewicz et al. 1998; Andrianov et al. 2004) and orthotropic (Zhinzher and Khromatov 1984) plates, circular cylindrical and shallow shells (Zhinzher and Denisov 1983; Andrianov and Kholod 1985; Zhinzher and Khromatov 1990; Andrianov and Kholod 1993a, 1993b, 1995).
1.5. DEEM and variational approaches
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
