Examples

To demonstrate a few examples of two dimensional kinetic shapes defined using Equa-tion 4.10, I have chosen three different desired radial ground reacEqua-tion force (RGRF) funcEqua-tions with constant applied vertical weight. The three radial force functions are constant, sinusoidal with an offset, and Fourier series expanded non-smooth. Each derived kinetic shape assumed a constant vertical force of 800 N. The magnitude of all these force functions were chosen for the convenience of experimentation. However, as explained in the previous Section 4.1.1, it is possible to apply a variable weight with respect to the angle around the kinetic shape. Although the analysis can be expanded to kinetic shapes that revolve more than once, I will focus on shapes that range from zero to 2π rads.

I verified all three 2D kinetic shapes with the setup shown in Figure 4.2. The weight was applied to the shape axle and the reaction forces exerted by the shape axle are simultaneously measured with a pair of load cell sensors (Omega LC703) placed in line with the forces. The load cell that measured the radial ground reaction force was supported by a small platform to minimize any cable tension force due to the sensor weight. This platform was adjustable in height. The signal from both force sensors was read by a Phidgets^® force sensor interface board. The interface board was connected to a desktop computer, acquiring data with a C++ program. To prevent the kinetic shape from slipping, I placed two-sided course grade sandpaper at the point the kinetic shape was touching the ground. As the applied weight was gradually loaded, the RGRF increased as well.

The tested kinetic shapes were loaded at π/6 rads intervals from zero to 2π rads. Some perimeter points, such as the lowest radii on a spiral shape, were omitted because the ground contact could not physically reach that particular perimeter point due to other parts of the kinetic shape touching the ground, however this usually was only one kinetic shape orientation angle.

The reaction load for each perimeter point was recorded with a mass of 7.9 kg to 18.0 kg at four even intervals applied to the shape axle. The mean and standard deviation for each point

Figure 4.2: Schematic of test structure for 2D kinetic shape examples

was calculated in terms of percent force transfer (100 ∗ F_r(θ )/Fr(θ )), which was then multiplied by 800 N.

The three 2D kinetic shapes examples chosen for verification were laser cut from tough 0.25 in (0.64 cm) thick Acetal Resin (Delrin^®) plastic. The laser cutter used to cut test shapes was a 60 Watt Universal Laser System^® VLS4.60.

4.1.1.1.1 2D Shape Example 1: Constant RGRF. To introduce the KS design concept, I will start with a shape defined by a constant force function and a constant applied weight function.

Equation 4.13 and Equation 4.14 describe the input force functions used to derive the first 2D kinetic shape. The kinetic shape was started with an initial shape radius of 2.5 in (6.35 cm) and ends with a 5.46 in (13.86 cm) radius.

F_v(θ ) = 800 N (4.13)

F_r(θ ) = 100 N (4.14)

Plugging in these force functions and the value for initial radius, Equation 4.10 becomes the following Equation 4.15.

Figure 4.3: 2D kinetic shape example 1. 2D example kinetic shape 1 forms a spiral with a monotonically increasing radius. Theoretical and measured RGRF are in good agreement.

R(θ ) = 2.5 exp 100 800 θ

θ =2π θ =0

(4.15) As an 800 N force is applied at the shape axle, the shape will react with a 100 N force regardless of the rotation angle. As seen in Figure 4.3, the gradual and slight exponential increase in shape radius, dR/dθ , statically produces a constant force at any perimeter point around the shape, creating a spiral KS. Note that the units, and thus the scaled size, are irrelevant and this KS would behave the same if scaled up or down, the shape radius change is established by the ratio between the the applied weight and the RGRF.

As seen in Figure 4.3, the physical measurements are in good agreement with theoretical values. Although the force profile standard deviation is not always within predicted theoretical range, the trend is relatively constant. There are some variations; however, these can be accounted for by shape surface and test setup imperfections, hence an overall larger kinetic shape may be more resistant to surface imperfections.

Figure 4.4: 2D kinetic shape example 2. 2D Shape 2 forms a monotonically increasing radius spiral, however when a constant weight is applied, it reacts with a sinusoidal RGRF around its perimeter.

4.1.1.1.2 2D Shape Example 2: Sinusoidal RGRF. A kinetic shape can also be derived using a more complicated sinusoidal force function with a constant offset. Equations 4.16 and 4.17 describe the input functions that define this 2D kinetic shape.

F_v(θ ) = 800 N (4.16)

F_r(θ ) = 100 sin (θ ) + 100 N (4.17)

With these forces, Equation 4.10 then becomes Equation 4.18.

R(θ ) = 1.75 exp 1

8[θ − cos(θ )]

θ =2π θ =0

. (4.18)

Unlike in the previous example that produces a constant RGRF, this shape creates a varying sinusoidal force throughout the rotation. In this example it is clear that the reaction force is dependent on dR/dθ of the shape. As the sinusoidal force reaches a maximum at θ = π/2 radians, radius change, dR/dθ , is steepest and so produces the highest RGRF. Likewise, as the

Figure 4.5: 2D kinetic shape example 3. The shape forms a continuous shape because, when a constant weight input is applied, it initially reacts with a positive reaction force and then switches directions to form a negative RGRF. All physical measurements are in good agreement.

input force reaches a minimum of zero at θ = 3π/2, dR/dθ is zero as well. At θ = 3π/2 the KS instantaneously behaves as a circular wheel would, and, like a circular wheel, it does not produce a RGRF when vertically loaded at its axle.

This kinetic shape assumes a spiral shape with a starting radius of 1.75 in (4.44 cm) and a final radius of 3.82 in (9.70 cm). The shape again resembles a spiral due to the fact that the sum of force around the shape perimeter is non-zero as defined by Equation 4.12. The physically measured force profile for this 2D kinetic shape, shown in Figure 4.4, was slightly higher than predicted, however the sinusoidal trend was in good agreement with theoretical.

4.1.1.1.3 2D Shape Example 3: Fourier Expanded Piecewise Force. It is clear now that a kinetic shape can be designed with any input force function. I will now expand my analysis to a piecewise force function that has been expanded using a ten term Fourier series to demonstrate that nearly any force profile can be created. This piecewise force function is defined by Equations 4.19

F_v(θ ) = 800 N (4.19)

Note that this time the RGRF function crosses zero at θ = 4.1 radians (Figure 4.5). This point is an unstable maxima of the shape radius function. At exactly this point, the shape produces no force, while any slight deviation off this point will cause RGRF. At exactly θ = 0/2π rads the shape radius is at its global minima with a RGRF equal to zero. This point is a stable point of the kinetic shape. This shape does not form a spiral, but is continuous around its perimeter, starting and ending at the same radius, hence Equation 4.12 is satisfied.

Measurements on the physical shape verified the predicted values. As seen in Figure 4.5, physical data falls well within theoretical values. Note that the standard deviation of measurements increases where the force profile fluctuates the most. Note that no measurements are possible at exactly θ = 0/2π rads