Modern Trends in Structural and Solid Mechanics 2

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 W 0. To construct DEE, we should use real roots of the characteristic equation. Then, the DEE solution is

[1.34]

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

[1.36]

Using expressions [1.34]– [1.36], we obtain

[1.37]

[1.38]

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

[1.39]

For antisymmetric modes, from the condition

we have

[1.40]

Equations [1.39]and [1.40]can be reduced to the following form:

[1.41]

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 x 0.

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).

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]