Non-holonomic constraints inducing flutter instability in structures under conservative loadings

Abstract

Non-conservative loads of the follower type are usually believed to be the source of dynamic instabilities such as flutter and divergence. It is shown that these instabilities (including Hopf bifurcation, flutter, divergence, and destabilizing effects connected to dissipation phenomena) can be obtained in structural systems loaded by conservative forces, as a consequence of the application of non-holonomic constraints. These constraints may be realized through a 'perfect skate' (or a non-sliding wheel), or, more in general, through the slipless contact between two circular rigid cylinders, one of which is free of rotating about its axis. The motion of the structure produced by these dynamic instabilities may reach a limit cycle, a feature that can be exploited for soft robotics applications, especially for the realization of limbless locomotion. . 1 Bolotin [9] writes 'The Euler method is [only] applicable if the external forces have a potential (i.e. if they are conservative forces), and in general is not applicable if they do not.' 2 The attempt by Willems [33] of producing a tangential follower load was indicated as misleading by Huang et al. [13]. Elishakoff [10] reports that 'Bolotin felt -if my memory serves me well!-that it should be impossible to produce Becks column experiment via a conservative system of forces.' Anderson and Done [2] write 'Sometimes, the creation of a force like [a follower force] in the laboratory presents awkward practical problems, and the simulation of this force wherever possible by a conservative force would be very convenient. However, because of the differing nature of the fundamental properties of the conservative and non-conservative systems, the simulation could only work in a situation where the two systems behave in similar ways; that is when the conservative system is not operating in a regime of oscillatory instability. (The conservative system can not become dynamically instable since, by definition, it has no energy source from which to supply the extra kinetic energy involved in the instability).' Koiter [19] states that '[...] it appears impossible to achieve any non-conservative loading conditions in the laboratory by purely mechanical means', because 'non-conservative external loads always require an external energy source, much as a fluid flow or an interaction with electro-dynamic phenomenon'. Koiter was strongly convinced that follower forces were a sort of 'physical non-sense' (Koiter,[20], [21]), so that Singer et al. [31] write 'An example in the field of elastic stability of what Drucker referred to as playing useless games was presented by Koiter, in his 1985 Prandtl lecture, where he discussed the physical significance of instability due to non-conservative, purely configurationdependent, external loads.' Several years after these negative views, Bigoni and Noselli [5] and Bigoni and Misseroni [8] respectively showed how to realize a tangentially follower force (Ziegler, 1952 [34]) and a fixed-line force (Reut, 1939 [30]) with devices involving Coulomb fr...

Flutter Instability and Ziegler Destabilization Paradox for Elastic Rods Subject to Non-Holonomic Constraints

“…Because the characteristic equation (58) applies to the elastic rod connected to both the 'skate' and 'violin bow', flutter and divergence loads are the same for both structures and in turn coincide with the Beck and Reut columns. This feature corresponds to an analogous property found for discrete columns made up of of rigid bars connected by visco-elastic hinges [8].…”

“…The behaviour of the elastic rod analyzed in this paper is also confirmed using a discretized rod model[8] at increasing number of its constituent rigid bars, so that the continuous and discrete models show coincidence of the instability loads in the limit.…”

“…Research on structures subject to nonconservative loads is a timely topic in view of several, different applications [18,19,20], nevertheless the fundamental models proposed by Reut, Ziegler, and Beck have Published in Journal of Applied Mechanics (2021), 88(3): 031003 doi: https://doi.org/10.1115/1.4047132 often been considered unrealistic and therefore harshly criticized [21,22]. The results presented here (and those relative to discrete systems [8]) demonstrate that the Reut, Ziegler, and Beck models represent the exact linearized behaviour of corresponding structures subject to non-holonomic constraints.…”

“…The flutter and divergence loads arẽ QS flu ≈ 7.920 and̃ QS div ≈ 40.646, respectively. (Right) Flutter load of the rod obtained as limit value at increasing number of elements for a chain of rigid bars connected through viscoelastic hinges [8]. such limit value of the flutter load is numerically found to be never higher than that evaluated for the same structure, but with null dissipation 'from the beginning'.…”

“…The flutter load in the limit of vanishing viscosities confirms the Ziegler's destabilization paradox for a rod subject to non-holonomic constraints. Indeed, 3 Using equations available in [8] as Supplementary Material and following [28], the flutter load for the discrete counterpart of the continuous system can be evaluated by considering the following parameters of internal rotational damping and of rotational stiffness ( = 1, ..., ) for a visco-elastic chain of rigid bars…”