## Abstract

One edge of an elastic rod is inserted into a friction-less and fitting socket head, whereas the other edge is subjected to a torque, generating a uniform twisting moment. It is theoretically shown and experimentally proved that, although perfectly smooth, the constraint realizes an expulsive axial force on the elastic rod, which amount is independent of the shape of the socket head. The axial force explains why screwdrivers at high torque have the tendency to disengage from screw heads and demonstrates torsional locomotion along a perfectly smooth channel. This new type of locomotion finds direct evidence in the realization of a ‘torsional gun’, capable of transforming torque into propulsive force.

## 1. Introduction

Motion, based on self-propulsion or locomotion, is a research topic currently attracting strong attention in mechanics, robotics and biology. Since pioneering studies by Gray on serpentine propulsion [1–3], elastic bending of a rod has been shown to produce an axial tractive force, whereas torsion has never been linked to locomotion. In mechanics, torsion of elastic rods is an old, but still ongoing and important research topic [4–11], which is linked in this article to locomotion through the following model problem.

A rectilinear inextensible elastic rod is subjected to an applied torque at one end, whereas the other edge is inserted into a perfectly smooth and fitting female constraint, able to react to the applied moment (figure 1*a*). For instance, the elastic rod can be realized as a blade of thin rectangular cross section inserted in a flathead screw, or as a cylindrical rod of hexagonal cross section inserted in a hex socket. In these conditions, if *l* is the length of the rod between the application point of the torque *M* and the end of the female constraint, *D* the torsional rigidity (product of the elastic shear modulus *G* and the torsion constant *J*_{t}) of the rod, the total potential energy of the system at equilibrium is
*l* of the rod be fixed, nothing special follows, but, because this length is a free parameter, an ‘Eshelby-like’ or ‘configurational’ force^{1} *P* is obtained as negative of the derivative of the potential energy with respect to the configurational parameter, namely the length *l*
*M*, was never previously noted. It is, at a first glance, unexpected because of the smoothness of the female constraint, and simply explains why a screwdriver tends to disengage from a screw head. Even more interestingly, this axial force (1.2) can be understood as a propulsive force opening new possibilities for locomotion, while previously Lavrentiev & Lavrentiev [14] and Kuznetsov *et al.* [15] related locomotion of snakes and fish to the possibility of a system of releasing elastic flexural energy. The analytical expression, equation (1.2), for the propulsive force *P* is rederived and confirmed in §2 through two different methodologies, namely the variational and the perturbative approaches, whereas, in §3, the experimental evidence of this force is provided through its measure for different settings.

Torsional locomotion is finally proved in §4 through realization of a prototype, which generates a propulsive force from a release of torsional energy.

## 2. The existence of the torsionally induced axial force

The existence of the propulsive force *P*, equation (1.2), can be proved with a variational argument and recurring to a perturbation technique.

### (a) Variational approach

The total potential energy *S* and on the right end to a torque *M* is (figure 1*b*)
*z* is the coordinate along the rod's axis, *θ*(*z*) is the cross-section rotation in its plane, *θ*(*l*_{in})=0 and the statical boundary condition

Considering the rotation field *θ*(*z*) and the length *l*_{in} as the sum of the equilibrium configuration {*θ*_{eq}(*z*);*l*_{eq}} and the respective variations {*ϵθ*_{var}(*z*);*ϵl*_{var}} through a small parameter *ϵ*, the boundary conditions define as compatibility equations

Equilibrium can be obtained by imposing the stationarity of the functional *θ*_{var}(*z*) and in the length *l*_{var}. The first variation

the latter providing the axial equilibrium and showing the Eshelby-like or configurational force *P*, equation (1.2), once the former is solved taking into account the statical boundary condition

### (b) Perturbative approach

The Eshelby-like force (1.2) can be obtained by introducing the assumption that the female constraint, though perfectly frictionless, has some geometrical imperfection. In particular, (i) there is a gap between the rod's cross section and the female, and (ii) the profile of the female is not sharply cut, but has a curvature (sketched for the sake of simplicity as circular in figure 1*b*,*e*). This imperfection will be shown to lead to the configurational force *P*=*M*^{2}/2*D* (independently of the misfit gap and of the female's profile) and therefore to remain unchanged in the limit when the imperfection tends to zero (differently from the propulsive forces generated by bending [13]). This approach was introduced by Balabukh *et al.* [16] for a system subjected to bending, and is extended now to torsion where its results are complicated by the three-dimensional nature of the problem.

The elastic rod (with a polygonal cross section) of *z*-axis is assumed to be constrained by *N* (equal to 3 in figure 1) smooth cylindrical rigid profiles having a plane normal to their axes containing the *z*-axis. The shape of the cross-section boundary of each rigid profile (assumed circular for the sake of simplicity in figure 1*b*,*c*,*d*) is described by *g*_{i}=*h*_{i}(*z*), with *i*=1,…,*N*. The contact points may vary along *z*, so that the contact points are defined by the set *c*,*d*,*e*), expressed by the line force *q*_{i}(*z*), with *t*_{i}(*z*) and axial component *p*_{i}(*z*) given by
*z*. The cross section of the elastic rod (triangular in figure 1), considered rigid in its plane, is subjected to an internal twisting moment *m*(*z*) varying along the elastic rod in the zone of contact and in equilibrium in its plane with the contact forces *t*_{i}(*z*), so that the principle of virtual work written for an incremental torsion angle d*θ* and corresponding incremental displacements d*g*_{i}=*h*_{i}′(*z*)d*z* can be written as
*θ*=*m*(*z*)/*Ddz* and the definition (2.4), becomes
*P* is generated that can be obtained as
*P*, because *m*(0)=0, and *P* is independent of the shape of the female's profile and of the amount of the initial gap, present between the rod and the smooth profiles, meaning that the amount of propulsive force, equation (1.2), is not affected by imperfections of the female constraints.

## 3. Experimental proof of the torsionally induced axial force

The system sketched in figure 1*b* has been realized to provide a direct experimental measure of the axial thrust *P*, equation (1.2). In particular, the torsional apparatus (figure 2*a*) has been designed and manufactured at the Instabilities Lab (http://ssmg.unitn.it/) of the University of Trento. The torque *M* is provided through a pulley (180 mm diameter) loaded at a constant rate with a simple hydraulic device in which water is poured into a container at 10 gr s^{−1} (the applied load is measured with a miniaturized cell from Leane, type XFTC301, R.C. 500 N). The elastic rod under twist is constrained against rotation by employing roller bearings from Misumi Europe (press-fit straight type, 20 mm diameter and 25 mm length), modified to reduce friction. Where the torque is applied, the elastic rod has been left free to slide axially through a double system, consisting of a linear bushing (LHGS 16-30 from Misumi Europe) mounted over a linear bearing (type easy rail SN22-80-500-610, from Rollon), so that longitudinal friction has been practically eliminated.^{2}

Experimental results, presented in figure 2 for different cross section, length, elastic modulus and constraint condition of the elastic rod subjected to torsion, fully confirm the theoretical predictions. In particular, results obtained with rods of different lengths *l* and different misfit gaps Δ between the rod's cross section and the female constraint (*c*) show unequivocally the indifference of the Eshelby-like force from these parameters. Moreover, tests have been conducted with different elastic moduli for the rod employing high-density polyethylene (HDPE) and polycarbonate (PC) and different (thin rectangular, square, triangular and trapezoidal, corresponding to *D*={31.29;36.37;156.97;638.86} Nm^{2}, respectively) cross sections (*b*). In all cases, the theoretical predictions have been found to be extremely close to experimental results (see the movie available as the electronic supplementary material for a sample of the test).

## 4. Torsional locomotion and torsional guns

Gray [1–3] has been the first to point out that a release of flexural elastic energy of a rod free of sliding in a frictionless channel can produce a locomotion force and Gray employed this force to explain fish and snake movement, so that a snake can propel itself producing bending by the backbone and its muscles. Within the terminology introduced in this article, the axial thrust produced during flexural deformation is the Eshelby-like force related to the release of elastic energy associated with curvature changes [17].^{3} It is therefore obvious to conclude that the configurational force *P*, equation (1.2), can be interpreted as a propulsive force capable of producing longitudinal motion through the application of a torque *M*.

To definitely prove that a torsional deformation can generate a longitudinal propulsion, a proof-of-concept device has been developed as shown in figure 3*a*,*b*. In particular, an elastic strip (19.5 mm wide and made in PC, weight 0.62 N) has been used, realized with two pieces with different rectangular cross section (one is 1.8 mm and the other 5.3 mm thick), so that one half of the strip, called ‘soft’ in the following, has *D*_{1}=3.02 Nm^{2}, whereas the other, called ‘stiff’, has *D*_{2}=67.36 Nm^{2}. The elastic strip is constrained with two pairs of roller bearings (at a distance *M* or a relative rotation *Θ*. Initially, the elastic strip is inserted within the rollers, so that the soft part of the strip has a length *l*_{1}, and the stiff one has a length *Θ* or a constant torque *M* is imposed between the two roller pairs, the total potential energy is, respectively,
*l*_{1}
*M* is imposed while it is a decreasing function of *l*_{1} when *Θ* is fixed. The elastic properties of the rod affect the amount of the propulsive force *P*. For instance, for a material with low shear modulus *G*, the torsional rigidities *D*_{1} and *D*_{2} of the projectile would decrease, whereas the propulsive force *P* would increase (decrease) for a given twisting moment *M* (for an imposed angle *Θ*).^{4} With the employed materials and geometrical set-up (*l*_{1}=215 mm and *l*_{2}=320 mm) and for an imposed angle *Θ*=*π*/2, the device realizes an initial propulsive force *P*=0.68 N, enough to overcome gravity when the device is held in a vertical configuration.

During manual use of the torsional gun, neither *Θ* nor *M* is precisely imposed, but a quick hand torsion of the device originates a propulsive longitudinal force able to eject the rod, even against gravity, see figure 3*c* and the movie available as the electronic supplementary material.

Note that, different from a bow or a catapult, in the ‘torsional gun’ the elastic deformation is stored in the projectile. The prototype of a torsional gun proves in an indisputable way that an axial motion can be produced via torsion, even in the absence of friction, so that a ‘flat animal’ can climb a frictionless narrow channel by employing a muscular torque.

## 5. Conclusion

Locomotion associated with torsional deformation of an elastic rod in a frictionless system has been introduced and substantiated both theoretically and experimentally, opening a new perspective in animal propulsion and in the mechanical design of deformable systems. The proof-of-principle ‘elastic gun’ shows how a torque can be transformed into a longitudinal thrust (or vice versa) without employing any mechanism, thus proving the realization of torsional locomotion.

## Funding statement

Financial support from the ERC advanced grant ‘Instabilities and nonlocal multiscale modelling of materials’ FP7-PEOPLE-IDEAS-ERC-2013-AdG (2014-2019) is gratefully acknowledged.

## Footnotes

↵1 In the context of the virtual work principle, the Eshelby force is the generalized force associated with the axial kinematical (Lagrangean) variable,

*P*−*S*in the following. The nomenclature ‘Eshelby-like’ force, used here to denote the propulsive force*P*, has been introduced by Bigoni*et al.*[12,13] for a different system subjected to bending. Results presented in this article confirm and extend their results to structures undergoing torsional deformation.↵2 The Eshelby-like force has been measured using a Gefran OC-K2D-C3 (R.C. 50 N) load cell and all data have been acquired with a NI CompactDAQ system, interfaced with Labview v. 8.5.1 (National Instruments). The torsional device has been mounted on an optical table (from TMC, equipped with four Gimbal piston air isolators) to prevent spurious vibrations, which have been checked to remain negligible employing two IEPE accelerometers (PCB Piezotronics Inc., model 333B50).

↵3 Change in curvature is essential to produce energy release, so that (in the words of Gray [3]) ‘a snake cannot glide round the arc of a circle or along a perfectly straight line’.

↵4 The effects of a longitudinal extensibility of the rod could affect the propulsion process, although this aspect is not explored in this article (the rod is assumed inextensible). Furthermore, the propulsion could also be affected by buckling.

- Received August 5, 2014.
- Accepted August 26, 2014.

© 2014 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.