Ziegler paradox in viscoelastic system stability problems

This work is devoted to the study of the motion stability of a viscoelastic rod whose material obeys the Kelvin–Voigt law. Solutions to two problems of determining the rectilinear shape stability of a thin homogeneous rod are given. In problem I, the rod is restrained at one end, and a tracking force is applied to the other end such that it is always tangent to the rod line when the rod is bent. Such a force can be realized, for example, by installing a gunpowder rocket engine at the end of the rod. In problem II, the rod executes a uniformly accelerated motion under the action of a tracking force. This problem can be considered as a simplified model of a rocket moved by a reactive force. Mathematically, the problem reduces to a partial differential equation of the fifth order with specified boundary conditions at the rod ends. The solution of the problem is presented as a series expansion in beam functions. To determine the critical load, the Rouss–Hurwitz criterion is applied. The number of expansion terms is justified. The critical load is found as a function of the internal viscosity coefficient. The paradox of destabilization is confirmed: any infinitesimal viscosity coefficient significantly reduces the critical load as compared to the elastic model. The critical force is found as a function of the internal viscosity coefficient is given. To confirm the obtained analytical results, a numerical solution of the corresponding Cauchy problem using the Runge–Kutta method of the 8th order is given. An agreement between the numerical and the analytical results indicates the validity of the latter. The practical application of research in the field of the motion stability of viscoelastic rods is extremely wide because viscoelastic rods are the basic element of various engineering structures. Determining the critical load allows one to ensure their reliability and efficiency.