Vibrations of a rotating circular membrane in the field of inertial and gravitational forces

Abstract

We study the vibrations of a circular membrane rotating with a constant angular velocity about the symmetry axis perpendicular to the plane of the membrane.The center of mass of the membrane moves along a circular trajectory in a central force field.The partial differential equations describing the small periodic motion of the membrane about the state of steady rotation are constructed.Gravitational forces and inertial forces (corresponding to the translational acceleration, relative accelerations, and Coriolis acceleration) are taken into account.The vibrational modes of the membranes for different rotational angular velocities are found. This problem arises in the development of space characterized by large size and small mass, and where the required shape of the structure is maintained by the rotation.i. We consider the vibrations of a rotating circular membrane with a free boundary whose center of mass moves along a circular trajectory in a central force field.The solution is constructed in two steps.In the first step we consider the motion relative to the center of mass of an equivalent rotating rigid body. The conditions corresponding to steady rotation in the orbital coordinate system are found. The results are used in the second step in considering the equilibrium and vibrations of a rotating membrane, treated as a continuum.We introduce the following right-handed rectangular coordinate systems (Fig. i): OaX=YaZa is the absolute coordinate system with its origin at the attraction center; OXYZ is the orbi--> tal system, where the Z axis is the continuation of the radius vector P(0) drawn from the attraction center through the center of mass of the membrane.The X axis lies in the plane passing through the radius vector and the velocity vector of the center of mass. We will use the coordinate system Oxyz whose axes are along the principal central axes of inertia of the membrane, and the related cylindrical coordinates rn, 8n, z n. The displacements of the membrane along the coordinate curves r n, 8 n, z n will be denoted as u, v, w, respectively (Fig. i).The equations of motion of an axisymmetric rigid body rotating about its center of mass, which itself moves along a circular trajectory in a central force field, have the formA--jr