Modern Trends in Structural and Solid Mechanics 2. Группа авторов

Modern Trends in Structural and Solid Mechanics 2 - Группа авторов


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      [1.33] image

      Hereinafter, we use the principal value of the arcsin(…) function.

      Let us construct DEE near the edge x = 0 . For a sufficiently long beam, we can suppose

      [1.35] image

      Then, at x = 0, we have from the boundary conditions

      Similarly, we can construct DEE localized at the edge x = L .

image

      we obtain

      For antisymmetric modes, from the condition

image

      we have

      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 .

      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).

      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, ya) with free contour. The governing equation is

      [1.42] image

      Here, in13-1.gif, h is the plate thickness and ν is Poisson’s ratio.

      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] image

image

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