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

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.

arXiv comment10 pages, 4 figures

References in corpus (7)

• Paradoxes of dissipation-induced destabilization or who opened Whitney's umbrella?
• Flutter and divergence instability in the Pfluger column: experimental evidence of the Ziegler destabilization paradox
• Snapping of elastic strips with controlled ends
• The destabilizing effect of external damping: Singular flutter boundary for the Pfluger column with vanishing external dissipation
• Non-holonomic constraints inducing flutter insability in structures under conservative loadings
• Detecting singular weak-dissipation limit for flutter onset in reversible systems
• Structures loaded with a force acting along a fixed straight line, or the "Reut's column problem"

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