A Kirchhoff type equation in a nonlinear model of shell vibration

Abstract

The paper deals with the study of the initial boundary value problem for the Donnell‐Mushtari‐Vlasov nonlinear system of differential equations, which describes large deflections of a rectangular shallow shell. The sought functions and are respectively longitudinal and transverse displacements of points of the shell midsurface. It is assumed that Dirichlet and Neumann type homogeneous conditions are fulfilled for functions u1 and u2. After expressing u1 and u2 through w, i.e. after constructing the relations with a help of Fourier series method, we come to a nonlinear integro‐differential hyperbolic equation for the transverse displacement function w, which by its structure is analogous to Kirchhoff string, Woinowsky‐Krieger beam and Berger plate equations. These equations are united under the common name – Kirchhoff type equations. The advantage of the described technique consists in splitting the initial boundary value problem at the end into two subproblems. From the integro‐differential equation we first find w and using the explicit relations we construct u1 and u2. The issue of the use of this approach in the general case not restricted by the rectangular domain and Dirichlet and Neumann conditions on the boundary is discussed. It is shown that in order to obtain formulas we have to solve a linear plane problem of elasticity with respect to the functions u1 and u2.