Dynamic Instability of Certain Conservative and Non-Conservative Systems

Abstract

This paper is concerned primarily with the analysis and resolution of the problems and contradictions that arise in the classification as conservative or non-conservative of a certain type of dynamical system. The systems concerned have equations of motion of gyroscopic type when the deflections are expressed with reference to particular co-ordinate axes, and they exhibit dynamic instability. Three examples are considered which possess their own special characteristics and subtleties; these are the rotating flexible asymmetric shaft, the helicopter ground resonance system and the clamped clamped flexible tube conveying fluid. For each of these cases, the energy input mechanism is examined and the problems of classification resolved.

Some unresolved issues in fluid-structure interactions

“…Thus, allowing axial extension at the downstream end evidently permits energy to flow from the fluid to the shell, thereby inducing flutter. This is clearly reminiscent of the Done and Simpson (1977) findings referred to in Section 2.1. The reader is reminded here of the conjecture made concerning the influence of mathematically explicitly absent (but implied) physical mechanisms influencing the dynamics.…”

“…An ingenious attempt to resolve the paradox was made by Done and Simpson (1977), by considering the downstream end to be free to slide axially ( Fig. 1), though prevented from moving laterally.…”

A Stability Criterion for Parameter-Dependent Gyroscopic Systems

“…When F~ = F2 = -1, (14) and (15) can be used to model shaft whirl, the motion of a discrete point particle of unit mass attached to a flexible massless shaft with principal lateral stiffnesses kl and k2 (both positive numbers) that rotates at angular speed a (Huseyin andPlaut, 1974-1975;Done and Simpson, 1977). This system exhibits divergence instability at certain speeds when k~ ¢ k2, no divergence instability when kj = k> and does not exhibit flutter instability.…”

Nonlinear dynamics of extensible fluid-conveying pipes, supported at both ends

“…On the other hand, the work done by the fluid on the pipe in a presumed cycle of oscillation is zero. 2 This, then, constitutes a paradox: for how is it possible for the pipe to flutter if the system is conservative and no energy is supplied to sustain the oscillation (Done and Simpson, 1977;Païdoussis, 1998? To resolve this paradox, the system must be analysed by a nonlinear model to ascertain the reliability of the predicted post-divergence behaviour, as linear theory is valid only up to the onset of the first instability, in this case divergence.…”