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 Pflüger) 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 and the Ziegler destabilization paradox. 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.

Flutter instability in solids and structures, with a view on biomechanics and metamaterials

“…More specifically, the linearized equations of motion for this conservative system subject to a non-holonomic skate constraint coincide with those for the ‘Ziegler’s double pendulum’, equation (1.2). Analogous examples have been provided for the Reut’s column, equipped now with a non-holonomic constraint [170,171]. It can be concluded that non-conservative forces are not necessary so that a plethora of dynamic instabilities can simply be obtained using non-holonomic constraints, within a framework of conservative forces, easily achievable in a laboratory .…”

“…'conservative system can not become dynamically unstable since, by definition, it has no energy source from which to supply the extra kinetic energy involved in the instability'. Disproving all the above claims and beliefs, a variant of the device invented by Bigoni & Noselli [129], in which the freely rotating wheel becomes a non-holonomic constraint, impeding the velocity to be orthogonal to the wheel movement, shows that dynamic instabilities and effects related to the dissipation are possible in structures only subject to conservative forces [170,171]. Figure 7 shows an example of a structure equipped with a non-holonomic constraint, which, though different in the nonlinear dynamic response, has the same critical loads for flutter and divergence as the 'Ziegler's double pendulum' and exhibit dissipation-induced instability and Ziegler-Bottema paradox.…”