The Use of Simulation Modeling to Study the Stability of Ziegler’s Pendulum

The stability and post-critical behavior of Ziegler's pendulum under the effect of potential and follower forces are studied by simulation modeling in the Simulink software environment. The mathematical formulation of the problem is given; the equations of motion are derived, and the simulation modeling scheme is presented. The trivial equilibrium state is analyzed using the Routh–Hurwitz criterion. By using the developed model and under the control of numerical integration, the stability domain boundaries are determined depending on the values of external load parameters. The effect of energy dissipation on the position of these boundaries is investigated; the time histories of generalized coordinates after loss of stability are obtained; the chaotic motion of the considered deterministic nonlinear system in the post-critical domain is illustrated, and the phase portraits and the Poincare cross section are given. Using the developed scheme, the pendulum behavior under various loads can be studied, and corresponding types of the nonlinear non-conservative system dynamic behavior can be observed: the process of settling equilibrium conditions in the stability domain, loss of stability when crossing the boundary, for example, hard loss of stability corresponding to divergence and Andronov–Hopf bifurcation, corresponding to a soft loss of stability. These can be periodic motions (limit cycles), quasi-periodic and chaotic motions, a so-called deterministic chaos.