An elastica arm scale

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.

Some perspectives on Eshelby-like forces in the elastica arm scale

“…The balance law includes the jump condition (see (2.7) 1 ) that is obtained from variational principles and, additionally, a local form (see (2.2) 1 ) of the material momentum balance law. With the help of (2.2) 1 and (2.7) 1 , we find that (1.1) is simply a consequence of conservation of a contact material force C and are easily able to generalize (1.1) to the cases where the self-weight of the rod is considered (see (4.2)) or applied moments act at the ends of the rod (see (6.4)). The conservations we discuss also illuminate connections between (1.1), (4.2) and (6.4) and the conservation law for a terminally loaded elastica discussed in Love's classic text [5, eqn.…”

“…In a recent paper, Bosi et al [1] present a novel measuring scale featuring an elastic rod of length that is free to move inside a frictionless sleeve which is inclined at an angle α to the vertical. Weights P 1 and P 2 are attached to the respective ends of the lamella and, assuming that one of the weights and the slope of the tangent at the ends of the rod are known, the second weight can be determined from the relation P 1 cos(θ(0) + α) + P 2 cos(θ( ) + α) = 0.…”

“…The discussion in [1] on the mechanics of the scale features a detailed variational formulation based on Euler's elastica. Here, we find it illuminating to consider a complementary (and equivalent) formulation using the balance laws for the elastica supplemented by a material momentum balance law from O'Reilly [4].…”

Chain paradoxes

“…15 I believe that they were also at the basis of the explanation given by Biggins & Warner [35,36] [40], rejected this assumption and based instead the solution of the problem of the folded chain on the conservation of the total mechanical energy. 17 This latter implies that the velocity of the falling arm increases indefinitely as the chain approaches the fully stretched configuration. This can be easily understood by realizing that conservation of energy requires a finite amount of potential energy to be transferred to a moving chain fragment whose mass decreases to nought as the chain straightens to its full length.…”

“…where we made use of the condition 17) which limits the variation of the motion to the portion of string under consideration. Correspondingly, Requiring the variational principle in (2.3) to be identically valid for arbitrary variations δv and δṡ 0 , we arrive at the equations…”