Reach of an Inclined Cantilever with a Tip Load

We investigate the problem of determining the reach of an inclined cantilever for a given point load suspended from its tip. Two situations are considered. Firstly, we find the maximum reach of the cantilever by varying its angle of inclination. Secondly, we find the reach of the cantilever subject to the condition that its tip is at some specified height, above or below, the level of the clamped end. In the second case, the reach of the cantilever is maximised by shortening its physical length whilst keeping the physical load and physical height of load deployment constant

The Clamped-Free Rod Under Inclined End Forces and Transitions Between Equilibrium Configurations

We investigate the problem of the straight, inextensible and unshearable clamped-free elastic rod subjected to an inclined end force. Exact analytic solutions representing all equilibrium configurations of the deformed rod are presented in elliptic integral form. Those exact solutions, for a given angle of inclination of the end force and number of inflection points, are characterised by two quantities; the end force and the elliptic modulus. Critical points are discussed and analytic conditions for determining their location are presented. Certain critical points where transitions between two equilibrium configurations whose numbers of inflection points differs by one are pointed out. Simple formulae for the total number of equilibrium configurations for a given end force are given. Applying arguments based on the elastic strain energy of the rod, we discuss scenarios where highly inflectional equilibrium configurations can transition to equilibrium configurations with fewer inflection points

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

Snap buckling, writhing and loop formation in twisted rods

A plethora of literature has appeared over the last twenty years with respect to large deflection rod theory. Some of this has been purely theoretical, and some has been applied. However there is a marked absence of experimental research. One of the aims of this thesis is to help fill that gap. A suitable rigid loading device was designed for this purpose. The rig facilitates the input of end rotation and end shortening (slack), and the measurements of torque and axial force at the ends of the rod. The rods were generally fixed in chucks. The thesis begins with an extensive review of the history of rod theory. This is followed by an outline of the mathematical model, in which we formulate a well defined boundary value problem for computing spatial (non-planar) solutions for finite length uniformly isotropic rods. The experimental apparatus and procedure, including details of the rods used in the experiments are outlined. Classical solutions (the helix and the localised homoclinic) are reviewed. Bent rods buckling into and out of planar configurations are investigated. We derive analytical solutions to four qualitatively different types of planar elastica with aligned welded ends. Our work includes some new analytical results. Data obtained from experiments on bent rods are compared with planar elastica theory. We find a secondary bifurcation not previously reported upon in the literature. Data obtained from experiments on rods under combined compression and twisting are compared to numerical results, exploring commonly encountered phenomena such as snap-buckling, hysteresis and the formation of loops. We include in our report, experimental data for which we have no numerical results to compare with. These experiments involve highly twisted rubber rods investigating such phenomena as snarling and pop-out. The isotropic case is rather special because the equations are integrable. However the theory for flat rods, which cannot bend about one axis, (i.e. "strips") is not well-established. We include some results from experiments on strips which could be useful for future research. We end the thesis with an industrial application of rod theory. Combining a version of our mathematical model with some simple experiments and some analysis, we study the out of plane deflections which arise during pipe-laying operations in deep seas