Numerical study of convergence of nonlinear models of the theory of shells with thickness decrease

Abstract

The article is devoted to numerical study of convergence of calculation results obtained on the basis of two nonlinear models of the theory of shells with thickness decrease. As models are considered nonlinear theory of thin shells, based on the hypotheses of the Kirchhoff-Chernykh and hypotheses type Tymoshenko, modified K.F. Chernykh for the case of hyperelastic rubber-like material. As an example, we consider the problem of axisymmetric conical compression and spherical shell by axial force. The convergence of results with decreasing thickness is disturbed in areas stability loss(buckling). Also happens when in the deformation process is violated the basic assumption of the theory of shells -the thickness is much smaller than radius of curvature (h < < R).

On the equation of fourth order with quadratic nonlinearity

“…The comparison with the numerical solution of equation (1) shows the accuracy of the analytical solution in the form (10)(11). Figure 1 shows the graph of the solution of the modified equation with quadratic nonlinearity with the following parameters:…”

“…For the construction of solutions of nonlinear differential equations in partial derivatives [6][7][8][9][10][11][12] is used different analytical and numerical methods: the perturbation methods, the small parameter method, the separation of variables method, the linearization method, the averaging method, the method of the stretched coordinates, the method of composite expansions, grid methods -the method of finite differences and the finite element method [13][14][15][16][17][18][19].…”

Mathematical modeling of the guidance system of satellite antenna

“…The solution of nonlinear differential equations [18][19][20][21][22][23][24][25] can be carried out various approximate analytical methods [26][27][28][29][30][31][32][33][34][35]: the method of Van der Pol, the harmonic balance method, the averaging method, the small parameter method, the method of Krylov-Bogolyubov, method of harmonic linearization, the method of Poincare. We obtained an approximate analytical solution of the modified method of harmonic linearization with Chebyshev polynomials [36][37][38][39][40][41][42] 2 2 4 2 4 2 2 2 2 3 3 3 3 11 11 11 22 3 15 3 1 2 3 3 3 3 3 Figure 15 shows graphs of the vertical oscillations of mobile satellite antenna obtained by analytical method (blue), a numerical method (yellow) and the graph the oscillation without vibration protection devices (green).…”

Non-linear vibrations of rubber membrane

“…For large deformations of shells and membranes to calculate such characteristics are not simply [22][23][24][25][26]. This is due to not only to the difficulties of constructing solutions of nonlinear boundary value problems, but also to the fact that they can have more than one solution [27][28][29][30][31][32][33][34][35]. If the problem has several solutions, there are difficulties in the numerical solution of boundary value problems [36][37][38][39].…”