Conditions for paradoxical response of Ziegler’s 3-DOF system under partially follower load prior to divergence

The Ziegler’s inverted pendulum system has traditionally been regarded as a fundamental reference model for investigating nonconservative instabilities in mechanical systems subjected to follower-type loading. The 2-D.O.F. corresponding model under either fully or partially follower loading has been the subject of a large number of works, as far as its static and dynamic stability is concerned. Contrary, the 3-DOF model has attracted less attention, which also may be associated with paradoxical dynamics, the study of which is the subject of the present investigation, focusing on regimes before divergence instability. Analytical conditions are derived through linearized equations of motion, employing the Lienard-Chipart criteria for asymptotic stability. Parametric studies identify specific mass and stiffness distributions that amplify the paradoxical behavior, offering new insights into the sensitivity of nonconservative systems to load partitioning. A key contribution of this study is the recognition that dynamic instability is not solely a consequence of large external loads, as often assumed in seismic or structural engineering, but can be an intrinsic feature of particular combinations of nonconservative and dissipative parameters. This outcome is of considerable importance, as it advances theoretical comprehension while informing engineering practice, influencing preliminary design models and the assessment of existing structures.