Читать книгу Modern Trends in Structural and Solid Mechanics 2 - Группа авторов - Страница 18

To describe the generalization of DEEM to the nonlinear case, we use the nonlinear Kirchhoff beam equation (Kauderer 1958):

[1.19]

Let the beam be elastically supported:

[1.20]

where c^*= c/EI, c is the coefficient characterizing elastic support.

We search a generating solution in the form

[1.21]

Substituting ansatz [1.21] into PDE [1.19], we obtain an ODE for determining the time function ξ (t) :

[1.22]

where

ODE [1.22] with initial conditions

[1.23]

has the solution

[1.24]

where cn(σt, k) is the Jacobi cosine elliptic functions with period T = 4K, is the complete elliptic integral of the first kind with modulus (Abramowitz and Stegun 1965).

The solution to the problem far from the edges is

[1.25]

where .

Solution [1.25] satisfies the original equation [1.19], but does not satisfy the boundary conditions [1.20]. To construct the states localized near the edges, we represent the solution of the original problem in the form

[1.26]

Substituting ansatz [1.26] into ODE [1.19], we obtain

[1.27]

In contrast to the previously considered linear case, the equations for functions w₀ and wee are coupled due to the nonlinearity of the problem. At the same time, the solution in the inner domain and EEs differ energetically since the EEs are localized in a small vicinity of the beam ends (Andrianov et al. 1979; Awrejcewicz et al. 1998; Andrianov et al. 2004, 2014). Let us estimate the orders of the integrand terms in equation [1.27] with respect to L/λ ≫ 1:

[1.28]

Restricting ourselves to the term of order (π/λ)² ≫ 1 in equation [1.27], we reduce it to the form:

[1.29]

Substituting function w₀ into equation [1.29], we obtain a PDE for function wee :

[1.30]

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

[1.31]

On substituting ansatz [1.31] into PDE [1.30] and applying the Kantorovich method (Kantorovich and Krylov 1958), the following ODE is obtained:

[1.32]

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

[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 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).