Investigation of the stability of rotation of a rigid body on a flexible rod

The stability of rotation of a symmetrical rigid body on a vertical flexible rod (shaft) is investigated. Both ends of the shaft are secured from offsets. The lower end of the shaft is supported by a bearing, and the bearing of the upper end is attached to an elastic ring plate, pinched along the outer diameter. This is one of the designs of a real-life centrifuge. The shaft material is considered inherently viscoelastic. The equations of motion of the flexible shaft are constructed within the framework of the linear Euler-Bernoulli model. The mathematical model of the mechanical system under consideration is an initial boundary value problem for partial differential equations and an infinite (integral) delay of the argument. The boundary conditions contain the highest time derivatives and the time lag. The concept of a generalized solution of an initial boundary value problem is introduced as a solution to some variational problem, functional spaces for initial conditions and a generalized solution are defined. The existence of a generalized solution, its uniqueness and continuous dependence on the initial conditions and parameters of the initial boundary value problem are proved. The stability of solutions of the initial boundary value problem is investigated depending on the main parameters of the mathematical model - the rotation speed and the coefficient of external friction. In the plane of the main parameters, the stability domains of solutions to the initial boundary value problem are constructed by the D-partitioning method. Mechanisms of stability loss are investigated. It is shown that the loss of stability of solutions can be caused by the passage of one or two pairs of complex conjugate points of the spectrum of the characteristic beam of operators through the imaginary axis of the complex plane.