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

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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.

Introduction

Flutter and divergence (in a word, ‘dynamic’) instabilities of elastic structures are connected to a number of counterintuitive and surprising features: (i.) they may occur in the absence of quasi-static bifurcations, and may or may not degenerate into a limit cycle, determining a so-called ‘Hopf bifurcation’ in the former case; (ii.) they are facilitated by a sufficiently small viscosity. Moreover, (iii.) a vanishing viscosity leads to a discontinuity in the critical load value, an oddity called ‘Ziegler paradox’, so that (iv.) this value may depend on the direction of the vanishing viscosity limit, when more than one dissipative source is present. All these features and the presence itself of flutter and divergence instabilities in structures are believed to be strictly related to the action of non-conservative loads,1 which are considered of difficult realization.2 Indeed, quoting Anderson and Done (Anderson and Done, 1971), a ‘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’.

We show in the present article that the last sentence is wrong, as a supply of ‘extra kinetic energy’ can be provided by means of a mass falling within a gravitational field, or through a release of the elastic energy initially stored in a spring and therefore:

We introduce two ways to induce dynamical instablities (such as flutter and divergence), Hopf bifurcation, destabilizing effect of dissipation, Ziegler paradox and its directionality dependence in visco-elastic mechanical structures under purely conservative loading conditions. The key to this behaviour is the use of non-holonomic constraints.

Recently, flutter and divergence instabilities have been shown to occur in structures loaded through non-conservative forces produced with frictional devices by Bigoni and Noselli (2011), realizing the tangentially follower load postulated by Ziegler (1952), and by Bigoni and Misseroni, 2020, obtaining the fixed-line load introduced by Reut (1939). The same type of instabilities is disclosed here in structural systems subject to conservative loadings when non-holonomic constraints are applied to the structure. More specifically, similarly to Jarzebowska and McClamroch (2000), Kuleshov and Rybin (2013), Neimark and Fufaev (1972), the non-holonomic constraint is realized through two rigid (massless) cylinders in slipless contact, where one of these cylinders can freely rotate about its axis while the other can not. The non-existence of a potential (Lanczos, 1952, Neimark, Fufaev, 1972) characterizes both a frictional device and a non-holonomic constraint, but only the latter is conservative. Indeed, the non-holonomic constraint introduces a kinematic condition prescribed in terms of velocity, thus realizing a (reaction) force acting on the structure and having a modulus varying in time during motion (Meijaard et al., 2007). In particular, the velocity r˙C of the instantaneous contact point C between the two cylinders is constrained to have a null component along the axis of the freely rotating cylinder (Fig. 1)r˙C·e=0,where a superimposed dot represents the derivative in the time variable t and e(t) is the unit vector aligned parallel to the rotating cylinder’s axis. The reaction p(t) transmitted by the constraint is parallel to e, so that Eq. (1) makes clear that this reaction does not work for every possible displacement. This means that every mechanical system composed of elastic elements, dead loads, and (time-independent) non-holonomic constraints is conservative.

When properly connected to the end of a structure, for instance a visco-elastic double pendulum, the two cylinders imposing condition (1) may be exploited to realize either a ‘skate’, or a ‘violin bow’ constraint, the former transmitting to the structure a tangential follower reaction similar to the Ziegler’s load and the latter a reaction acting on a fixed line similar to the Reut’s load, but now varying in their modulus during motion. In Fig. 2, the two proposed structures with non-holonomic constraints (which will be subject to conservative loads) are shown on the left, while the non-conservative counterparts are shown on the right.

In particular, the ‘skate’ constraint can be realized (i.) with a freely-rotating, but non-sliding wheel, or (ii.) with a perfect skate, or (iii.) by connecting the freely-rotating cylinder to the end of the structure, with the other cylinder initially orthogonal to it and fixed in space. On the other hand, the ‘violin bow’ constraint can be obtained by connecting the fixed cylinder to the end of the structure, with the freely-rotating cylinder initially orthogonal to it and fixed in space, Fig. 2. Note that the structural systems subject to non-holonomic constraints are conservative, when loaded with dead forces or springs and if the viscosity is set to be zero.

In the present article, discrete structural systems made up of N rigid bars and subject to one of the two proposed types of non-holonomic constraints (Fig. 2, left) are investigated. Such structures are loaded in a conservative way, namely, either with a dead force F, or with a deformed linear spring (see Figs. 5 and 6 and details presented in the next Sections).

The non-holonomic constraint acting on these structures permits the existence of infinite equilibrium configurations (Neimark and Fufaev, 1972) and of Hopf bifurcations connected to stable limit cycles in the presence of dissipation, as partially anticipated by the motion of the non-holonomic Chaplygin’s sleigh on an inclined plane (Neimark and Fufaev, 1972) and by the problem of shimmy instability (Beregi, Takacs, Gyebroszki, Stepan, 2019a, Beregi, Takacs, Stepan, 2019b, Facchini, Sekimoto, du Pont, 2017, Ziegler, 1977).

With reference to a column made up of N rigid segments and subject to a perfectly aligned non-holonomic constraint (realized for β0=0 in Figs. 5 and 6), the trivial configuration becomes the unique quasi-static solution. It is shown that:

A visco-elastic column subject to the ‘skate’ or the ‘violin bow’ constraint and the same structure subject to non-conservative load, respectively of the Ziegler or Reut type, evidences exactly the same critical loads for flutter (Hopf bifurcation) and divergence, and the same Ziegler paradox, with the same directionality dependence occurring when multiple sources of dissipation are considered.

More specifically, the directionality dependence in the limit of null viscosity has been analyzed by Bolotin (1963) and Kirillov (2005) for non-conservative systems subject to follower forces, showing that the ideal critical load for the undamped system may be recovered only for special ratios between different viscosities.

In addition, a ‘viscosity-independent Ziegler paradox’ is found, in which the flutter load becomes independent of the viscosity, but cannot become higher than that evaluated for the corresponding system assumed without viscosity ‘from the beginning’. In particular, such a surprising behaviour is shown to be related to the presence of two specific damping parameters acting on the proposed structures when subject to non-holonomic constraints.

The stability of Hopf bifurcation, and therefore the achievement of limit cycles in the neighbourhood of the critical point (Agostinelli, Lucantonio, Noselli, DeSimone, 2020, Bigoni et al., 2018, Bigoni, Misseroni, Tommasini, Kirillov, Noselli, 2018b, Jenkins, 2013), is influenced by the considered mechanical system, so that a perfect match between the mechanical behaviour of the proposed non-holonomic systems and of their non-conservative counterparts is lost, particularly when non-linearities dominate.

A typical dynamic evolution of a double visco-elastic column connected to a non-holonomic ‘skate’ constraint at its final end (represented as a non-sliding wheel) is shown in Fig. 3. The structure is loaded on the left end through a dead force of constant magnitude (so that the system is conservative when the viscosity is set to zero), selected within the flutter region. The occurrence of flutter is shown, through evaluation of the first Lyapunov coefficient (Kuznetsov, 2004, Marsden and McCracken, 1976), to correspond to a supercritical Hopf bifurcation, so that the complex motion taking place after bifurcation [sketched in parts (a)–(f) of Fig. 3] reaches a periodic orbit in the neighbourhood of the bifurcation point (lowest part of Fig. 3). It may be interesting to note that the structure shown in Fig. 3 is subject to a constant force, so that it would suffer a varying acceleration in the absence of viscosity. However, the presence of viscosity (even the rotational viscosity at the hinges is enough) is sufficient to allow the mechanical system to reach, after a transient phase, a steady motion with a constant mean velocity, as detailed in Section 6.

The presented results provide a new key to theoretically interpret and experimentally realize dynamic instabilities until now believed to be possible only as connected to non-conservative loads. Several applications in energy harvesting and soft robotics can be envisaged, but two in particular merit a special attention, namely, frictional contact and locomotion. In particular, our results show how a micromechanism might act at a sliding surface between two solids to reduce friction through instability. Moreover, it will be shown that imposing rotations (instead than applying forces) to the structures that will be analyzed, a motion is induced, which means that our results have implications in the problem of limbless locomotion.

Section snippets

A first insight on flutter instability of a conservative system from a double pendulum with non-holonomic constraint and dead load

With the purpose to provide a first insight on flutter instabilities in conservative non-holonomic systems, an elastic double pendulum (two rigid bars of equal length l, each equipped with a centred mass m and elastic hinges of stiffness k) is considered as sketched in Fig. 4. The structure is loaded on its left end with a dead force F, while at its right end is subject to a ‘skate’ non-holonomic constraint, which can be realized for instance with a freely rotating and massless wheel.

The planar

The column and its loadings

The motion in the xy plane is analyzed for an elastic column of total length L, discretized as a chain of N rigid bars of length li (i=1,,N, so that L=i=1Nli), connected to each other with visco-elastic hinges of elastic stiffness ki and viscous parameter ci, Fig. 5. The first hinge is connected to a rigid block which may only slide along the x-direction and is loaded in one of the following conservative ways :

  • A - elastic device: the rigid block is loaded by imposing a compressive

Linearized dynamics and stability of the column with non-holonomic systems

A first-order expansion of Eq. (23) about a generic quasi-static solution in both the generalized coordinates qQS and the Lagrangian multiplier pQS, therefore satisfying Eq. (24), can be performed by assumingq(t)=qQS+ϵq^(t),p(t)=pQS+ϵp^(t),where ϵ is an arbitrarily small real parameter and {q^(t),p^(t)} is the set denoting the perturbation fields. From the quasi-static nature of the configuration qQS, it follows thatq˙(t)=ϵq^˙(t),q¨(t)=ϵq^¨(t).

A Taylor series expansion of the Eq. (23) about ϵ=0

The double pendulum subject to the ‘skate’ and ‘violin bow’ constraints

The geometric and inertial properties are considered for simplicity coincident for the two bars, so that {li,mi,di}={l,m,d} (i=1,2) and L=2l, with a total mass M=2m. By introducing the characteristic time T=LM/k of the structure and the stiffness k=k2 for the rotational springs, the following dimensionless quantities can be introducedχ=XL,Δ˜=ΔL,d˜=dl,l˜=lL=12,τ=tT,p˜=pLk,F˜=FLk,M˜X=MXM,M˜L=MLM,m˜=mM=12,I˜r,L=Ir,LL2M,K˜=KL2k,k˜1=k1k,c˜e=ceL2kM,c˜t,L=ct,LLkM,c˜r,L=cr,LLkM,c˜i=ciLkM,andΞ˜={12K˜[χ(τ

Post-critical behaviour and limit cycles

The linearized eigenvalue analysis so far developed shows that the straight configuration of the visco-elastic double pendulum is always stable (unstable) for loads smaller (higher) than the critical value for flutter, but nothing can be said about stability at the critical load and also on the stability of the post-critical dynamics involving large displacements.

The post-critical behaviour has been analyzed numerically to show that, in the presence of dissipation, limit cycles can be attained

Locomotion and friction

Limbless locomotion. The periodic solutions so far obtained for the analyzed structures can be exploited in the design of limbless locomotion devices. In fact, the time evolution of the rigid bars’ rotations exhibiting a generic limit cycle can directly be imposed to the same structures which are left only with the horizontal motion X of the end of the structure as ‘free coordinate’. With this set-up, the non-holonomic constraint is capable of converting the imposed oscillations of the rigid

Conclusions

It has been shown that flutter instability and Hopf bifurcations can be displayed by visco-elastic structures under conservative loads, when subject to non-holonomic constraints. This finding opens a new perspective in structures suffering instabilities and provides a proof of the mechanical equivalence between non-holonomic conditions and polygenic forces, which cannot be derived from any scalar functional (for instance follower loads) (Lanczos, 1952). In this context, the reaction force

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

Financial support is acknowledged from: ERC Advanced Grant ERC-2013-ADG-340561-INSTABILITIES (AC), PRIN 2015 2015LYYXA8-006, PITN-GA-2019-813424-INSPIRE, and ARS01-01384-PROSCAN (DB and FDC). The authors also acknowledge support from the Italian Ministry of Education, University and Research (MIUR) in the frame of the ‘Departments of Excellence’ grant L. 232/2016.

References (35)

  • V.V. Bolotin

    Nonconservative Problems of the Theory of Elastic Stability

    (1963)
  • I. Elishakoff

    Controversy associated with the so-called “follower force”: critical overview

    Appl. Mech. Rev.

    (2005)
  • G. Facchini et al.

    The rolling suitcase instability: a coupling between translation and rotation

    Proc. R. Soc. A

    (2017)
  • M. Golubitsky et al.

    Singularities and Groups in Bifurcation Theory - Volume I

    (1985)
  • N.C. Huang et al.

    Willems’ experimental verification of the critical load on becks problem

    J. Appl. Mech.

    (1967)
  • E. Jarzebowska et al.

    On nonlinear control of the Ishlinsky system as an example of a non-holonomic non-chaplygin system

    Proceedings of the 2000 American Control Conference

    (2000)
  • O.N. Kirillov

    A theory of the destabilization paradox in non-conservative systems

    Acta Mech.

    (2005)
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