Loading paths for an elastic rod in contact with a flat inclined surface

This paper computes stationary profiles of an isotropic, homogeneous, linearly elastic rod with its endpoint locations and tangents specified. One end of the rod is clamped and the other end makes contact with a flat, rigid, impenetrable surface, which is displaced towards the clamped end. This boundary value problem has applications to biomechanical sensory devices such as mammal whiskers. The paper gives exact analytical solutions to the boundary value problem, embracing the planar equilibrium configurations for both point contact and line contact with the wall. Plots of loading paths for different inclinations of the wall provide an insight into the force-displacement relationship appertaining to real world slender rods under this type of loading. This report is complemented by data obtained from corresponding experimental studies which shed light on the differences between the model, which is based on the mathematical theory of elasticity, and the mechanics of real world long slender bodies, such as mammalian vibrissal systems

Critical Points of the Clamped-Pinned Elastica

We investigate equilibrium configurations of the clamped-pinned elastica where the pinned end can be displaced towards, and past, the clamped end. Solving the non-linear ordinary differential equation for the clamped-pinned elastica for any mode in terms of elliptic integrals, we find sets of equations which govern the equilibrium configurations for given displacements. Equilibrium configurations for various displacements of the pinned end and any mode are obtained by numerically solving those sets of equations. A physical quantity, such as the force that arises in the elastica due to displacement of the pinned end, is taken to be a function of displacement. Although, it is generally not possible to define a physical quantity as a function of displacement explicitly, an equation for the rate of change of this physical quantity with respect to displacement can be found by partial differentiation of the sets of equations which govern the equilibrium configurations. Setting that rate of change to zero provides a constraint equation for locating the critical points of that physical quantity. That constraint equation and the sets of equations which govern the equilibrium configurations are solved numerically to obtain the critical points of the physical quantity. Our procedure is demonstrated by finding local extrema on force-displacement plots (useful when analysing stability of equilibrium configurations) and the maximum deflection of the elastica. Finally, we suggest how our procedure has scope for wider application

Critical Points of the Clamped-Pinned Elastica

We investigate equilibrium configurations of the clamped-pinned elastica where the pinned end can be displaced towards, and past, the clamped end. Solving the non-linear ordinary differential equation for the clamped-pinned elastica for any mode in terms of elliptic integrals, we find sets of equations which govern the equilibrium configurations for given displacements. Equilibrium configurations for various displacements of the pinned end and any mode are obtained by numerically solving those sets of equations. A physical quantity, such as the force that arises in the elastica due to displacement of the pinned end, is taken to be a function of displacement. Although, it is generally not possible to define a physical quantity as a function of displacement explicitly, an equation for the rate of change of this physical quantity with respect to displacement can be found by partial differentiation of the sets of equations which govern the equilibrium configurations. Setting that rate of change to zero provides a constraint equation for locating the critical points of that physical quantity. That constraint equation and the sets of equations which govern the equilibrium configurations are solved numerically to obtain the critical points of the physical quantity. Our procedure is demonstrated by finding local extrema on force-displacement plots (useful when analysing stability of equilibrium configurations) and the maximum deflection of the elastica. Finally, we suggest how our procedure has scope for wider application

Asymptotic analysis of the clamped-pinned elastica

Asymptotic solutions for the clamped-pinned elastica when the displacement of the pinned end is small (immediately after buckling) and when it approaches its limiting displacement (when the force in the rod tends to infinity) are presented. Simple leading order relationships describing the force as a function of the pinned end’s displacement are derived. Those approximate the force-displacement behaviour of the clamped-pinned elastica for small and limiting end displacements. All our results are valid for the clamped-pinned elastica in any mode

Large deflections of a clamped-slider-pinned rod with uniform intrinsic curvature

© 2017. This paper examines large planar deflections of a slender elastic rod with uniform intrinsic curvature, which is clamped at one end and the other end is pinned to a slider allowing it to move freely in the vertical direction as the ends are displaced horizontally in a straight line. The analysis encompasses force-displacement loading paths and corresponding configurations, with a particular focus on the formation of loops. The analysis is accompanied with data obtained from experiments on nickel titanium alloy strips, demonstrating a good match with theoretical predictions

Ecomorphology reveals Euler spiral of mammalian whiskers

Whiskers are present in many species of mammals. They are specialised vibrotactile sensors that sit within strongly innervated follicles. Whisker size and shape will affect the mechanical signals that reach the follicle, and hence the information that reaches the brain. However, whisker size and shape have not been quantified across mammals before. Using a novel method for describing whisker curvature, this study quantifies whisker size and shape across 19 mammalian species. We find that gross two‐dimensional whisker shape is relatively conserved across mammals. Indeed, whiskers are all curved, tapered rods that can be summarised by Euler spiral models of curvature and linear models of taper, which has implications for whisker growth and function. We also observe that aquatic and semi‐aquatic mammals have relatively thicker, stiffer, and more highly tapered whiskers than arboreal and terrestrial species. In addition, smaller mammals tend to have relatively long, slender, flexible whiskers compared to larger species. Therefore, we propose that whisker morphology varies between larger aquatic species, and smaller scansorial species. These two whisker morphotypes are likely to induce quite different mechanical signals in the follicle, which has implications for follicle anatomy as well as whisker function

Describing whisker morphology of the Carnivora

One of the largest ecological transitions in carnivoran evolution was the shift from terrestrial to aquatic lifestyles, which has driven morphological diversity in skulls and other skeletal structures. In this paper, we investigate the association between those lifestyles and whisker morphology. However, comparing whisker morphology over a range of species is challenging since the number of whiskers and their positions on the mystacial pads vary between species. Also, each whisker will be at a different stage of growth and may have incurred damage due to wear and tear. Identifying a way to easily capture whisker morphology in a small number of whisker samples would be beneficial. Here, we describe individual and species variation in whisker morphology from two-dimensional scans in red fox, European otter and grey seal. A comparison of long, caudal whiskers shows inter-species differences most clearly. We go on to describe global whisker shape in 24 species of carnivorans, using linear approximations of curvature and taper, as well as traditional morphometric methods. We also qualitatively examine surface texture, or the presence of scales, using scanning electron micrographs. We show that gross whisker shape is highly conserved, with whisker curvature and taper obeying simple linear relationships with length. However, measures of whisker base radius, length, and maybe even curvature, can vary between species and substrate preferences. Specifically, the aquatic species in our sample have thicker, shorter whiskers that are smoother, with less scales present than those of terrestrial species. We suggest that these thicker whiskers may be stiffer and able to maintain their shape and position during underwater sensing, but being stiffer may also increase wear

Large Llamas with Silver Shoes

Drawing upon contemporary accounts, this paper analyzes conquistadors’ and Incas’ perceptions of each other’s ungulates—that is, camelids and horses—from the first encounters in 1532 until 1536. The paper traces the evolution of those perceptions within the wider context of human-nonhuman animal relations, which differed between Spaniards and Andeans. Those differences are reflected in the respective languages. The paper finds a tension between a sense of familiarity and a sense of otherness. That tension manifested in a supernatural realm. The paper argues that nonhuman animal relations, particularly with respect to horses, played a central role in the invasion, but as the conflict unfolded the meanings of “human” and “animal,” as understood by the protagonists, were perturbed. The paper presents a critique of Diamond’s theory of nonhuman animal domestication

Whisker Sensing by Force and Moment Measurements at the Whisker Base

We address the theoretical question which forces and moments measured at the base of a whisker (tactile sensor) allow for the prediction of the location in space of the point at which a whisker makes contact with an object. We deal with the general case of three-dimensional deformations as well as with the special case of planar configurations. All deformations are treated as quasi-static, and contact is assumed to be frictionless. We show that the minimum number of independent forces or moments required is three but that conserved quantities of the governing elastic equilibrium equations prevent certain triples from giving a unique solution in the case of contact at any point along the whisker except the tip. The existence of these conserved quantities depends on the material and geometrical properties of the whisker. For whiskers that are tapered and intrinsically curved, there is no obstruction to the prediction of the contact point. We show that the choice of coordinate system (Cartesian or cylindrical) affects the number of suitable triples. Tip and multiple point contact are also briefly discussed. Our results explain recent numerical observations in the literature and offer guidance for the design of robotic tactile sensory devices