EQUATIONS OF MOTION OF AN ISOTROPIC SPHERICAL MOMENT ELASTIC SHELL*2

Abstract

Using the previously obtained equations of motion of thin moment elastic shells of constant thickness with an arbitrary middle surface, the equations of motion of an isotropic moment spherical shell in terms of forces and “displacements” (kinematic parameters) are constructed. In this case, the metric of the middle spherical surface is taken into account, as the curvilinear coordinates of which two angular coordinates of the standard spherical coordinate system with the origin at the center of the middle surface are used. First, a closed system is written down, which includes the equations of motion in the physical components of the tensors of internal forces and moments, as well as additional similar characteristics corresponding to the moment properties of the model, and physical relationships. Then, by excluding physical relations, it is reduced to twelve equations of motion in kinematic parameters, written in operator form. In this case, the coefficients of partial derivative operators are simplified by neglecting terms that have a higher order of smallness relative to the thickness of the shell. Despite the bulkiness of the system, it is written in a compact form. Boundary conditions are not written, since the shell is considered closed. By introducing additional hypotheses similar to those used in the classical theory of shells (neglecting the compression of a normal fiber, the Kirchhoff – Love hypothesis about the connection between the tangential components of the rotation angle vector of a normal fiber and normal displacement, as well as the hypothesis about the connection between the normal to the middle surface part of the coordinate of the rotation angle vector and its tangential components) the number of equations and unknowns decreases. To carry out this procedure, a variational Hamilton equation is constructed, which takes into account the connections between kinematic parameters imposed by the hypotheses, and then the corresponding one is transformed using the generalized Ostrogradsky – Gauss theorem.