Abstract

The concept of a ‘deformable arm scale’ (completely different from a traditional rigid arm balance) is theoretically introduced and experimentally validated. The idea is not intuitive, but is the result of nonlinear equilibrium kinematics of rods inducing configurational forces, so that deflection of the arms becomes necessary for equilibrium, which would be impossible for a rigid system. In particular, the rigid arms of usual scales are replaced by a flexible elastic lamina, free to slide in a frictionless and inclined sliding sleeve, which can reach a unique equilibrium configuration when two vertical dead loads are applied. Prototypes designed to demonstrate the feasibility of the system show a high accuracy in the measurement of load within a certain range of use. Finally, we show that the presented results are strongly related to snaking of confined beams, with implications for locomotion of serpents, plumbing and smart oil drilling.

1. Introduction

For millennia, the (equal and unequal) arm balance scales have been used, and still are used (see the overview in [1]), to measure weight by exploiting the equilibrium of a rigid lever, so that a deformation of the arms would merely represent an undesired effect. On the other hand, equilibrium is always satisfied for a spring balance, where the weight measure is directly linked to deformation and a counterweight is not needed. A new paradigm, based on the exploitation of nonlinear kinematics and configurational mechanics of elastic rods, is proposed here for a scale with deformable arms, where an inflected equilibrium configuration can be exploited to measure weight. In a sense, the proposed balance is a sort of combination between a rigid arm and a spring balance, because both equilibrium and deformation are simultaneously exploited. Therefore, the concept introduced here differs completely from that underlying traditional scale design, so that the proposed device can work with or without a counterweight.

The ‘elastically deformable arm scale’ is shown infigure 1a (photo of prototype 1) as a realization of the scheme shown infigure 1b, where an elastic rod (inclined at an angleα∈[0,π/2] with respect to the two vertical dead loads applied at its edges) is free to slide in a frictionless sleeve of lengthl*. For given loads (P₁ andP₂), the scale admits an equilibrium configuration, which is possible by virtue of the flexural deformation of the arms (if these were rigid, the equilibrium would be trivially violated). This equilibrium configuration is inherently nonlinear, as it necessarily involves the presence of configurational or ‘Eshelby-like’ forces [2], but can be derived from the stationarity of the total potential energy in a form suitable for direct calculations. Therefore, the nonlinear equilibrium equations (§2) can be exploited to determine a load from the measurement of a configurational parameter (the lengtha_eq). Considerations of the second variation of the total potential energy (§3) show that the equilibrium configurations of the scale are unstable, a feature that may enhance the precision of the load measure and that does not prevent the feasibility of the scale, as demonstrated by two proof-of-concept prototypes (§4). Furthermore, a sensitivity analysis as well as the experiments performed on the prototypes indicate that the deformable arm balance works correctly and that it can perform better than traditional balances in certain load ranges.

Finally, as the elastic lamina in the sliding sleeve (forming the deformable arm scale) can be viewed as a ‘snake in a frictionless and tight channel’, our results demonstrate that Eshelby-like forces play a fundamental role in the problem of snaking of a confined elastic rod, with implications for smart oil drilling [3], plumbing, and locomotion of reptiles and fish [4–6].

2. Flexural equilibrium through Eshelby-like forces

The system shown infigure 1b attains equilibrium because two forces exist, tangential to the sliding sleeve, which can be interpreted as ‘configurational’ (or ‘Eshelby-like’ [2]), in the sense that they depend on the configuration assumed by the system at equilibrium. These forces and the equilibrium conditions of the system can be obtained for an inextensible elastic lamina of bending stiffnessB and total length$\overline{l}+l^{\ast }$ from the first variation of the total potential energy of the system [7]

where$s\in [0;\overline{l}+l^{\ast }]$ is a curvilinear coordinate,θ(s) is the rotation of the rod's axis,a anda+l* are the curvilinear coordinates at which, respectively, the left arm terminates and the right one initiates, so thatθ(s)=0 fors∈[a;a+l*]. The parametera, defining the position of the rod with respect to the sliding sleeve, is variable, to be adjusted until the equilibrium configuration is reached.

Considering a small parameterϵ and taking variations (subscript ‘var’) of an equilibrium configuration (subscript ‘eq’) in the formθ=θ_eq(s)+ϵθ_var(s) anda=a_eq+ϵa_var, four compatibility equations are obtained from a Taylor series expansion of the rotation field$\theta (\hat{s})$ for$\hat{s}=a$ and$\hat{s}=a+l^{\ast }$, namely (see [8] for details)

Through integration by parts and consideration of the first two compatibility conditions (2.2) and the static conditions at the free edges,${\theta }_{\mathrm{eq}}^{\mathrm{\prime }}(0)={\theta }_{\mathrm{eq}}^{\mathrm{\prime }}(\hspace{0.17em}\overline{l}+l^{\ast })=0$, the first variation of the total potential energy (2.1) can be obtained as

and must vanish (for every variation in the rotation fieldθ_var(s) and in the lengtha_var) to obtain the equilibrium configuration. This is governed by the following criteria:

• (i) the elastica [9] for the two arms of the lamina

  $$B{\theta }_{\mathrm{eq}}^{\mathrm{\prime }\mathrm{\prime }}(s)-P_j\sin [{\theta }_{\mathrm{eq}}(s)-{(-1)}^j\alpha ]=0,$$

  2.4

  wherej=1 for the left arm (s∈[0,a_eq]) andj=2 for the right one ($s\in [a_{\mathrm{eq}}+l^{\ast },\overline{l}+l^{\ast }]$), and
• (ii) the rigid-body equilibrium condition along the sliding direction of the sleeve

  $$(P_1+P_2)\cos \alpha +\underset{\mathrm{Eshelby}\text{-}\mathrm{like}\hspace{0.17em}\mathrm{forces}}{\underbrace{\frac{M_1^2-M_2^2}{2B}}}=0,$$

  2.5

  where$M_1=B{\theta }_{\mathrm{eq}}^{\mathrm{\prime }}(a_{\mathrm{eq}})$ andM₂=Bθ^′_eq(a_eq+l*).

  Condition (2.5) reveals the presence of two so-called ‘Eshelby-like forces’ [2], provided by the sliding sleeve at its left and right ends and generated by the flexural deformation of the left and right arms, respectively, which define the equilibrium condition of the system and are the key concept of the deformable arm scale.

The rotations at the free ends at equilibrium,θ₀=θ_eq(0) and${\theta }_{\hspace{0.17em}\overline{l}+l^{\ast }}={\theta }_{\mathrm{eq}}(\hspace{0.17em}\overline{l}+l^{\ast })$, can be obtained by double integration of the elastica (2.4), leading to the following conditions:

where$\mathcal{K}({\kappa }_j)$ and$\mathcal{K}(m_j,{\kappa }_j)$ are, respectively, the complete and incomplete elliptic integral of the first kind

and

Further integration of the elastica (2.4) leads to the solution for the rotation field at equilibrium

where sn is the Jacobi sine amplitude function. Since solution (2.9) implies

the equilibrium along the sliding direction of the sleeve (2.5) can be expressed as a ‘geometrical condition’ of equilibrium, which relates the angles at the free edges to the two applied vertical dead loads as

and represents the balance of axial thrust of the deformable scale (0≤α+θ₀≤α and$\pi /2\le \alpha +{\theta }_{\hspace{0.17em}\overline{l}+l^{\ast }}\le \pi$).

When$\alpha +{\theta }_{\hspace{0.17em}\overline{l}+l^{\ast }}=\pi /2$ the equilibrium equation (2.11) impliesP₁=0, so thata counterweight is not needed (figure 2).

As a conclusion, the following modes of use of the elastic scale can be envisaged:

• — The easiest way to use the elastic scale is by referring to equation (2.11) and measuring the two anglesθ₀ and${\theta }_{\hspace{0.17em}\overline{l}+l^{\ast }}$. Assuming thatP₁ andα are known,P₂ can be calculated. Note thatB is not needed in this mode of use.

• — Another mode of use of the elastic scale is through measurement of the lengtha_eq. By knowingP₁,B andα,P₂ can be determined as follows: (i) equation (2.6)₁ givesθ₀; (ii) equation (2.11) gives${\theta }_{\hspace{0.17em}\overline{l}+l^{\ast }}$ as a function of the unknownP₂; (iii) equation (2.6)₂ provides an equation for the unknownP₂, to be numerically solved.

Note that equations (2.6) definea_eq as a one-to-one function, respectively, ofθ₀, equation (2.6)₁, and${\theta }_{\hspace{0.17em}\overline{l}+l^{\ast }}$, equation (2.6)₂, while equation (2.11) defines a unique relationship betweenθ₀ and${\theta }_{\hspace{0.17em}\overline{l}+l^{\ast }}$ (within the limits of variability of these two angles). Therefore, if all the possible deformations of the elastica which are unstable even for the clamped end are not considered, the equilibrium solution of equations (2.6) and (2.11), when it exists, is unique.

3. Stability

Equilibrium configurations of the proposed mechanical system are expected to be unstable by observing that a perturbation of the system at an equilibrium position,a=a_eq, through an increase (or decrease) ina, yields a leftward (or rightward) unbalanced Eshelby force, which tends to increase the perturbation itself. However, instability of the equilibrium configuration exploited in the deformable arm scale does not necessarily represent a drawback, as it could increase precision in the measurement.

In a rigorous way, the instability of a deformed configuration can be detected by investigating the sign of the second variation of the total potential energy, which can be written as

The second variation, equation (3.1), becomes

when the two auxiliary functionsΓ₁(s) andΓ₂(s) are introduced as the solutions of the following boundary value problems:

Considering the necessary condition of stability [10], an equilibrium configuration can be stable if

Introducing the Jacobi transformation

which leads to the following Jacobi boundary value problems:

the auxiliary functionsΓ_j(s) withj=1,2 have been numerically evaluated for all configurations considered in the experiments and, although conjugate points are not present, the unstable character of the configurations follows from Δ<0.

4. Prototypes of the deformable scale and experiments

To test the possibility of realizing a deformable scale, two prototypes (called ‘prototype 0’ and ‘prototype 1’) have been designed, produced and tested (at the Instabilities Lab of the University of Trento, Italy).

In prototype 0 (shown infigure 3a, and in the movie available as the electronic supplementary material) the sliding sleeve is 296 mm in length and is made up of 27 roller pairs (each roller is a Teflon cylinder 10 mm in diameter and 15 mm in length, containing two roller bearings). In prototype 1 (shown infigures 1a,2 and3b) the sliding sleeve, 148 mm in length, is realized with eight roller (press-fit straight type, 20 mm in diameter and 25 mm in length) pairs from Misumi Europe. The tolerance between the elastic strip and the rollers inside the sliding sleeve can be calibrated with four micrometric screws. Two elastic laminas were made from solid polycarbonate (white 2099 Makrolon UV from Bayer, elastic modulus 2250 MPa), one with dimensions 980×40.0×3.0 mm and the other 487×24.5×1.9 mm; the latter has been used for the experiments reported infigures 2,4 and5, while the former is shown infigure 1a. The sliding sleeve is mounted on a system (made from PMMA) that may be inclined at different anglesα. The two vertical dead loads applied at the edges of the elastic lamina were imposed manually. The tests were performed on an optical table (1HT-NM from Standa) in a controlled temperature 20±0.2° and humidity 48±0.5% room.

Experimental results (presented infigure 4 in terms of the measured values of the lengtha_eq, for different weightsP₂) are in excellent agreement with the theory.

The sensitivity of the scale$\mathcal{S}=\mathrm{\partial }{a}_{\mathrm{eq}}/\mathrm{\partial }{P}_2$ is reported infigure 5, together with the maximum absolute error ‘err’ found in the experimental determination of the loadP₂. The figure correctly shows that errors decrease at high sensitivity. Moreover, the sensitivity is so high for smallP₂ that the scale could in a certain range of use become more accurate than a traditional balance.

The prototypes represent proof-of-concept devices, demonstrating the feasibility of the elastic scale, with an accuracy which can be highly improved in a more sophisticated design. A movie with experiments on the prototypes is available in the electronic supplementary material (and athttp://ssmg.unitn.it).

5. A link to snaking and locomotion

The sliding sleeve employed in the realization of the elastic arm scale and considered also in [2,8] can be viewed as a perfectly frictionless and tight channel in which an elastic rod can move.¹ Our results show that a motion along the channel can be induced even when the applied forces are orthogonal to it. Moreover, it has been shown that the Eshelby-like forces can have a magnitude comparable to the applied loads. These forces are the essence of snake and fish locomotion [4–6] and must play an important role in the problem of beam snaking occurring during smart drilling of oil wells and in plumbing [3]. Investigation of these and related problems is deferred to another study.

6. Conclusion

A new concept for a deformable scale has been introduced in which equilibrium is reached through nonlinear flexural deformation of the arms and generation of configurational forces. This equilibrium configuration is unstable, a feature which can increase the precision of the measure and which does not prevent the practical realization of prototypes, showing that real balances can be designed and can be effectively used to measure weights. The reported findings represent a first step towards applications to deformable systems, in which non-trivial equilibrium configurations at high flexure can be exploited for actuators or to realize locomotion.

Footnotes

Funding statement

The work reported is supported by the ERC Advanced Grant ‘Instabilities and nonlocal multiscale modelling of materials’ (ERC-2013-ADG-340561-INSTABILITIES).

References