Notes for Practical Kinetic Shape Application

(4.112) This force output F_{OU T}(θ1, θ2) is shown in the three dimensional plot in Figure 4.26. This plot shows a percent force transfer, that is, given 100% input force, this kinetic shape system reacts with a cumulative output % force transfered. Figure 4.26 also shows two sample kinetic shape orientations, θ₁and θ₂.

4.4 Notes for Practical Kinetic Shape Application

As previously mentioned, my kinetic shape equations are able to exactly produce a shape that will exert kinetic properties. In this section, I will review how kinetic shapes can be used and what things need to be considered when integrating a kinetic shape into practical designs.

The most critical application consideration for the kinetic shape is the contact surface between the shape and a level plane/ground. Theoretically the plane/ground onto which the shape is pressed on is infinitely rigid and perfectly flat and the kinetic shape is also infinitely rigid and makes contact with the ground perfectly along the shape definition, and the friction between ground and the kinetic shape is assumed to be infinitely high. However, in practice these conditions are often times never the case and I am going to outline some work-around techniques which may help enhance the theoretically predicated values obtained by my kinetic shape equations.

Note for clarity I am only describing these practical techniques for the 2D kinetic shape, however these practicalities outlined in this section are also applicable for 3D kinetic shapes but may carry additional practical design problems.

4.4.1 Ground-Shape Contact

The contact between the kinetic shape and the flat surface onto which it is pressed should be as close to a point as possible. For the two dimensional kinetic shape, this point can also be the orthogonal line between the kinetic shape faces. If the 2D kinetic shape is cut from a flat material with finite thickness, the cut could be made with an angle (or curvature) to force the

Figure 4.27: Practical 2D kinetic shape application. Practical suggestion on how to create a point contact between the two dimensional kinetic shape and flat ground surface. Note that when creating a point contact or increasing friction it is possible to alter the kinetic shape or the ground onto which it is placed.

kinetic shape to roll on a edge instead of a surface, which may be irregular depending on the cut. Such edge modification can be seen in Figure 4.27. Note that edge modification may not be necessary, however may improve results. If however the angled edge is cut sideways, it may cause an unwanted moment at the shape axle.

The ground contact point may be improved by scaling the kinetic shape up. A small kinetic shape may be more affected by contact point imperfections such as ground contact deformation or imperfections in surfaces. To clarify, this shape alteration can be viewed in Figure 4.27.

4.4.2 Ground-Shape Contact Friction

For the kinetic shape equation to work the friction between the shape and the flat surface has to be as high as possible. After some experimentation and implementation of the kinetic shape, I have found that there are potentially three different ways of increasing friction (Figure 4.27). This can be achieved by layering the flat surface (or the kinetic shape rolling surface) with sand paper,

rubber, or even gearing the kinetic shape, in a sense creating a rack-and-pinion. If one is to use a rubber surface, it should be noted that the ground-shape contact point may be blunted yielding inacurate results. As I’ve previously mentioned, this may be alleviated to some degree using a larger shape. If one is to use a geared kinetic shape, it is important that the gear teeth are relatively small compared to the kinetic shape. A rough approximation would be using gear teeth which are under 5% of smallest shape radius. Relatively large gear teeth create large angled edges which can affect the force redirection of the kinetic shape.

4.4.3 Force Application

Naturally, the application of the applied vertical force at the shape axle, F_v, can be done by pushing or pulling. It is only important that this force is applied orthogonal to the flat rolling surface. Depending on the application of the kinetic shape one means of force application may be better then the other.

If one is to push onto a kinetic shape that is in static equilibrium, the force applicator experiences a bending moment as shown in Figure 4.28(a). This bending moment is directly proportional to the radial ground reaction force. In the two dimensional kinetic shape this bending moment, M, can be defined by Equation 4.113 and 4.114.

M = F_v(θ )[R(θ ) cos(ϕ(θ ))] = Fr(θ )[R(θ ) sin(ϕ(θ ))]. (4.113)

ψ (θ ) = tan⁻¹ F_v(θ ) F_r(θ )



. (4.114)

Pulling on the shape axle may be more beneficial in design situations if one can pull perpendicular to the flat surface. In some situations this may be easier since one could hang a weight from the shape axle. Naturally due to gravitational forces, the weight will produce a perfectly orthogonal force to the flat surface if the flat surface is perfectly horizontal. This method can be seen in Figure 4.28(b). However, note that if the shape is rolling in this setup, depending

Figure 4.28: Vertical force application. Via (a) pushing, (b) hanging a weight, (c) or a spring A swaying weight during shape rolling could be compensated for by defining the applied force, F_v in terms of the swinging weight dynamics.

A third possibility of applying a force to the kinetic shape axle is by a spring/rubber band/elastic member pulling the shape toward the ground (Figure 4.28(c)). This setup applies a variable force to the kinetic shape axle. As the shape changes orientation, so does the distance from the shape axle to the ground contact point and in turn the force applied to the shape changes.

The force applied to the shape axle in this setup can be described as Equation 4.115.

F_v(θ ) = k [R(θ )sin[ψ(θ )] + (xpre− x₀)] (4.115)

Here, k is defined as the stiffness of the spring, x₀ is the nominal/free/unstreched spring length, and xpre is the distance the spring is already stretched from its nominal/free length (pre-tension).

Note that the force application has to come from below the kinetic shape. If the the spring does not move and pulls directly below the shape axle, proper geometric force decomposition needs to be applied. This misalignment will result in a force pulling the shape along the ground plane, adding or subtracting from horizontal ground reaction force, F_r.

4.4.4 Kinetic Shape Radius Change with Time

One thing that I thought of as I was developing the kinetic shape is that a kinetic shape radial function can be dependent with time, i.e. R(θ ,t) for the 2D kinetic shape. This means that it is possible to implement the kinetic shape such that it physically changes its shape to produce other radial force functions over time. This dynamic morphing property opens up many more possibilities for the kinetic shape application.

This dynamic shape change could be implemented with internal spokes that are actuated with some lead screw and a electric motor or electric servos. One could also change the shape by using Bowden cables like used in bicycle brakes and gear shifter. However, for a changing kinetic shape it to be rigid enough to transfer the applied force while also flexible enough to change its shape. If the rim material is too loose, one may consider a larger shape to offset the deformation of the rim.