Dynamics of Elastic Rods in Perfect Friction Contact

Abstract

One of the most challenging and basic problems in elastic rod dynamics is a description of rods in contact that prevents any unphysical self-intersections. Most previous works addressed this issue through the introduction of short-range potentials. We study the dynamics of elastic rods with perfect rolling contact which is physically relevant for rods with rough surface, and cannot be described by any kind of potential. We derive the equations of motion and show that the system is essentially non-linear due to the moving contact position, resulting in a surprisingly complex behavior of the system. Introduction Take two rubber strings, stretch them a bit and cross them so they remain in contact, as shown on Figure 1. As long as the deformations are not too large, the strings will roll at the contact without sliding. Of course, that simple experiment is dominated by the energy loss from the internal deformation of strings at the contact point; however, improving the quality of strings will allow them to oscillate for a reasonable time before the energy loss takes over. Interestingly, this simple and familiar experiment has deep mathematical and physical implications that go beyond a toy problem.

A study of snake-like locomotion through the analysis of a flexible robot model

“…We deduce the equations of motion through the Lagrange-d'Alembert principle, similarly to what is done in [35] and [36]. The principle states that, in the presence of the dissipative force F , a solution (r, θ) that satisfies constraint (28) must solve…”

“…but k = k(s 0 (t) + s) is no longer predetermined. On the other hand, the equations are closed through the boundary conditions obtained by setting the three generalized edge loads equal to zero N (0, t) + EJ k(s 0 (t)) − α(0, t) k(s 0 (t) = 0 , N (L, t) = 0 , EJ k(s 0 (t) + L) − α(L, t) = 0 (35) together with the initial curvature k(s 0 (t 0 ) + s) = K(s), with s ∈ [0, L], and the initial values for s 0 andṡ 0 at t = t 0 . Such values must satisfy the compatibility relationṡ…”

“…A key role is played by the third boundary condition in (35), coming from the vanishing of the bending moment at the leading edge. This latter condition, namely, k(s 0 (t) + L) = α(L, t)…”

“…Suppose we have a solution of the free-path locomotion problem (32)-(33)-(34) satisfying the boundary conditions (35) and the extra requirements (31) and (36) (43) The general case in which s 0 (t) > L may also occur (i.e., the trailing edge is no longer contained in the image of the initial configuration, see Fig.5B) can be handled by applying the following simple, yet technical, argument. As we will show, a local solution s 0 for (43) exists and is unique once a positive initial velocityṡ 0 (t 0 ) is given, by requiring that s 0 (t) ≤ L. If a local solution s 0 of (43) exists and is unique then, for every given initial conditions, there exists only one solution with a maximal interval of definition.…”

“…In order to solve the locomotion problem, we need to find the unknown functions N (s, t), H(s, t), f (s, t), together with s 0 (t) and k(s 0 (t) + s). The three equations of motion and the three boundary conditions (35) are sufficient to solve this problem uniquely. This leads to a unique solution also for r and θ, once the initial position r(0, t 0 ) and orientation θ(0, t 0 ) of the first end are prescribed, by integrating the equations θ s = k and r s = Γ as done, e.g., in [15,16].…”