Flutter instability and Ziegler destabilization paradox for elastic rods subject to non-holonomic constraints

TL;DR

This paper investigates the flutter and destabilization phenomena in elastic rods with non-holonomic constraints, revealing that conservative systems can exhibit instability and paradoxical destabilization when dissipation is introduced.

Contribution

It demonstrates that models of elastic rods with non-holonomic constraints can exhibit flutter and Ziegler destabilization, challenging the notion that these are only due to non-conservative loads.

Findings

Conservative elastic systems can exhibit flutter and divergence instability.

Introduction of dissipation leads to Ziegler's destabilization paradox.

Models align with classical structures like Beck and Reut columns.

Abstract

Two types of non-holonomic constraints (imposing a prescription on velocity) are analyzed, connected to an end of a (visco)elastic rod, straight in its undeformed configuration. The equations governing the nonlinear dynamics are obtained and then linearized near the trivial equilibrium configuration. The two constraints are shown to lead to the same equations governing the linearized dynamics of the Beck (or Pfluger) column in one case and of the Reut column in the other. Therefore, although the structural systems are fully conservative (when viscosity is set to zero), they exhibit flutter and divergence instability. In addition, the Ziegler's destabilization paradox is found when dissipation sources are introduced. It follows that these features are proven to be not only a consequence of 'unrealistic non-conservative loads' (as often stated in the literature), rather, the models proposed by Beck, Reut, and Ziegler can exactly describe the linearized dynamics of structures subject to non-holonomic constraints, which are made now fully accessible to experiments.