Parametric oscillations and the stability of a mechanical system with considerable dissipation

A detailed investigation is carried out into the problem of parametric oscillations when there is linear dissipation. Using constructive numerical-analytical methods, the boundaries of the domains of stability are constructed for a wide range of variation of the parameters, that is, the modulation factor and the friction coefficient. By solving non-self-adjoint eigenvalue and eigenfunction problems, the phase vectors of the three lower oscillation modes are determined and the principal features of the behaviour of the boundaries when the linear friction coefficient is varied are established. The eigenvalues and eigenfunctions of the adjoint boundary value problem are found. A complete biorthogonal system is constructed and its functional properties are determined. Modified expressions are obtained for scalar products and the squares of the norms of the characteristic phase vectors.