Small vibrations of mechanical systems in the case of degenerate eigenvalues

Abstract

Abstract Free vibrations of mechanical and hydrodynamic systems are considered. Some well-known works in this field are summarized and some aspects of computational nature are discussed. The eigenvalue boundary value problem is solved in the case of multiple roots for a system with n degrees of freedom. The case of the zero root of a semidefinite system is considered. The question of orthogonality of forms of natural oscillations is considered. It is shown that in some cases, the proximity of eigenvalues occurs in problems of dynamics of an aircraft containing elastic containers with a liquid. On the example of a mechanical analog of transverse vibrations of a cylindrical tank with a liquid, the possibility of rotation of the free surface of the liquid is shown.