Stability of a non-potentially loaded cantilever beam restrained by an elastic spring

One of the characteristic features of mechanical systems loaded by non-conservative positional (follower) forces is that under certain conditions various instability types can occur [1, 2]. If all solutions of the equations of motion are monotonic near the instability boundary in the parameter space, the instability is referred to as divergence. If some of the solutions are oscillatory, the instability is referred to as flutter. For systems with lumped parameters, a classical example of such a system is the Ziegler pendulum [3], while an example for distributed elastic systems is a cantilever beam loaded by the potential ("dead") and follower forces [4]. When studying such systems, the main aim is to construct the critical surface separating in the parameter space the stable equilibrium region from the instability region. This critical surface (the stability boundary) consists of different parts, crossing of which corresponds to different instability modes. The bifurcation types of non-conservative systems depend both on the external loading parameters and on the internal parameters. In this study, a systematic stability analysis is carried out for a beam clamped at one end, restrained by an elastic spring at the other end, and loaded by a potential and follower forces. The influence of the elastic spring stiffness on the position of the critical surface and on the instability type occurring upon crossing this surface is analyzed. The stability boundary is constructed by two methods, using the continuum mechanics approach, i.e., by solving the boundary-value problem, or by means of the series expansion with respect to the natural vibration modes (the method of normal coordinates). For the Ziegler pendulum restrained by an elastic spring, the divergence-type bifurcation problem has been considered in [5].