Geared Jitterbugs

Year: 2019 Authors: Elisabetta A. Matsumoto; Henry Segerman

Core claim

Adding bevel or acircular gears to jitterbug linkages can stabilize motion, preserve dihedral relationships, and enforce desired rotation-rate constraints.

Topics

jitterbug mechanisms, bevel gears, acircular gears, kinetic sculpture

Domains

kinematics, geometry, numerical methods, gear theory, kinetic sculpture, 3D printing, mechanical design, product fabrication

Methods

bevel gear construction, numerical solution, integral equation, rack-based tooth cutting

Media

3D printed parts, involute gear profiles, rack profile, bevel gears

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2019 Conference Proceedings

Geared Jitterbugs

Elisabetta A. Matsumoto¹ and Henry Segerman²

¹School of Physics, Georgia Institute of Technology, Atlanta, USA; sabetta@gatech.edu

²Department of Mathematics, Oklahoma State University, Stillwater, USA; henry@segerman.org

Abstract

We describe geared versions of Buckminster Fuller’s jitterbug mechanism, and a variant cuboctahedral mechanism. In the cuboctahedral case, differences between the triangular and square faces led us to a numerical method to construct acircular gears, allowing for rotation rates that vary with angle.

Introduction

In 1948, Buckminster Fuller discovered the jitterbug mechanism, a linkage consisting of eight rigid triangular faces joined at their corners by point hinges. The jitterbug expands from an octahedral to a cuboctahedral shape by twisting neighbouring triangles in opposite directions. See Figure 1 (a-c). However, this original version of the jitterbug has more than a single degree of freedom; other modes of movement are possible. In 1974, Dennis Dreher discovered that one could restrict the motion to only the expanding mode by replacing the point hinges between triangles with pairs of linear hinges attached to an intermediate part. See Figure 1 (d-f). Following Kiper and Söylemez [2], we refer to these intermediate parts as $DAP$ parts, where “DAP” stands for “dihedral angle preserving”, as they enforce the angle between the normals of adjacent triangular parts. See Schwabe [4] for a brief history of the jitterbug mechanism.

Our first contribution is to add bevel gear teeth onto the two triangular parts that mesh in the vicinity of a DAP part. Mathematically, the DAP parts reduce the possible motion of the jitterbug to a single degree of freedom. However, the real world does not behave: play in the hinges can result in the mechanism departing from this path to the extent that it can jam. The added gear teeth work in concert with the DAP parts to enforce preservation of the dihedral angles. Furthermore, the teeth constrain the relative rotation rates of the triangular parts. Both of these effects help to restrict the motion of the physical model to the desired single degree of freedom, resulting in a smoother mechanism.

Verheyen [5], Kiper and Söylemez [2], and others have investigated generalizations

of the jitterbug mechanism to other polyhedra. We apply our gearing technique to another example: a jitterbug mechanism based on a cuboctahedron which expands to form a rhombicuboctahedron. Here the gear design is complicated by the fact that the triangular and square faces of the mechanism must rotate at different, varying rates as the mechanism expands.

(a)

(b)

(c)

(d)

(e)

(f) Figure 1: (a-c) The jitterbug mechanism with point hinges. (d-f) The jitterbug mechanism with DAP parts.

Matsumoto and Segerman

Gearing the Jitterbug

To design the bevel gears for our octahedral jitterbug mechanism, we used a two-dimensional profile for involute gears [1], coned to a point. Standard circular gears have parallel axes, while bevel gears have axes that intersect at a point. To construct a pair of bevel gears, take a pair of gears with parallel axes and rotate one of them about the common tangent vector to the two midcurves of the gears – curves of common tangency without teeth (here drawn dashed in Figure 2). The axes of the gears now meet in a single point. Coning the gear faces to this intersection point creates a set of bevel gears. The faces of the gears are always perpendicular to their axes, so this rotation angle determines the cone angles for both gears. The cone point for these bevel gears is the intersection of the two hinge axes on the DAP part, which is also a vertex of the underlying jitterbug mechanism with point hinges. The resulting geared mechanism is shown in Figure 3.

Figure 2: Making conical gears.

Figure 3: 3D printed geared jitterbug.

Cuboctahedral Jitterbug

Figure 4

In the octahedral jitterbug, neighbouring triangles rotate in opposite direction. In the cuboctahedral jitterbug, the triangles all rotate in the same direction while the squares rotate in the other. When the mechanism expands from a cuboctahedron to a rhombicuboctahedron, the triangles rotate by $2\pi /3$, as they do in the octahedral design. However, the squares rotate by only $\pi /2$. Moreover, the angle of rotation of the triangles, $\theta$ say, is not linearly related to the angle of rotation of the squares, $\varphi$ say. This makes gearing considerably more complicated than the octahedral case.

To see how to calculate the relationship between $\theta$ and $\varphi$, consider Figure 4. Here a red square and a blue triangle are joined at a point hinge and start as faces of a cuboctahedron. We rotate the triangle by an angle $\theta$ about an axis perpendicular to its face, while allowing it to also move outwards along that axis. We also allow the square to rotate and translate along its perpendicular axis. With these constraints, there is a line perpendicular to the triangle’s axis on which the point hinge must lie. The point hinge must also lie on a cylinder centered on the square’s axis. This line and cylinder intersect twice. One of these two intersections is the position of the hinge, and from this we derive the following relation between the rotation angles $\theta$ of the triangle and $\varphi$ of the square:

Geared Jitterbugs

Since the relation between $\theta$ and $\varphi$ is not linear, the gears for this mechanism cannot be circular.

Acircular Gears

Consider two gears with distance one between their axles. Let the rotation angles of the driving and driven gears be $\theta$ and $\varphi =\varphi (\theta )$ respectively. Initially we think of these gears as being toothless but rolling against each other without slipping. The radii $a(\theta )$ and $b(\varphi (\theta ))$ thus add to one. Moreover, as the gears roll, no slipping means that the arc length traversed by the driving gear between angles 0 and $\theta$ is the same as the arc length traversed by the driven gear between angles 0 and $\varphi$ , so

For our cuboctahedron jitterbug, we have the relationship $\varphi =\varphi (\theta )$ given in equation (1), and we must solve an integral equation to find the radii $a(\theta )$ and $b(\varphi )$ that will achieve this relationship. This is difficult to solve analytically, so we move to a numeric solution. We will use the set up shown in Figure 5.

Here, the gear centered at $A$ is the driving gear, and the gear centered at $B$ is the driven gear. The driving gear has radius $a_i$ when rotated by ${\theta }_i$ and the driven gear has complementary radius $b_i=1-a_i$ . After the driving

Figure 5

gear has been rotated by an additional $\Delta {\theta }_i$ , it has radius $a_{i+1}$ , and the additional arc length is $s_i$ , where $s_i^2=a_i^2+a_{i+1}^2-2a_i{a}_{i+1}\cos (\Delta {\theta }_i)$ by the law of cosines. The no slipping condition means that the driven gear must also have arclength $s_i$ , so by the law of cosines again, we get that $s_i^2=b_i^2+b_{i+1}^2-2b_i{b}_{i+1}\cos (\Delta {\varphi }_i)$ . We also know that the new radius of the driven gear must satisfy $b_{i+1}=1-a_{i+1}$ . Putting all of this information together, we can solve:

Figure 6: Profiles of the triangular and square gear midcurves.

Using the derivative ${\varphi }^{\prime }(\theta )$ , we can work out the initial relative speeds of the gears, and therefore the initial value $a_0$ . We chose a constant driving step size $\Delta {\theta }_i=\delta =\pi /300$ . From this we calculate the driven step size $\Delta {\varphi }_i=\varphi ((i+1)\delta )-\varphi (i\delta )$ . We can then generate all of the radii $a_i$ and $b_i$ . The gear profiles we obtain by applying this technique with equation (1) are shown in Figure 6.

Note that this numerical method does not suffer much from “drift”: so long as the gears we generate keep matching up at the corners of

subsequent triangles (as in Figure 5), then the relative rotation rates will be correct.

Adding Teeth

The midcurve of the acircular gears prescribe the required rotation rate of the square gear with respect to the triangular gear due to the no-slip condition. However, as these just barely touch, there would be nothing except friction to transfer torque from the triangular gear to the square gear to induce rotation. Circular gears use involute teeth to more efficiently transfer torque. To add teeth to our acircular gears, we carve out teeth using a linear rack (black in Figure 7) with the same total length as the arc length of the midcurve of the acircular gears. This is similar to a method described by Laczic [3]. As the gears rotate, the midcurve of the rack is oriented

Matsumoto and Segerman

to be tangent to the midcurves of the acircular gears at the point of common tangency. The rack also rolls without slipping along the midcurve of the acircular gears so that as the triangular gear rolls through an angle, all three midcurves will have moved by the same arc length. The tooth profile is the envelope of the rack profile as this motion is executed. See Figure 8. The benefit of this design is that the pressure angle of the teeth can be set easily by the linear rack. This ensures that a constant torque can be transferred between gears.

Figure 7: Cutting teeth.

(a) Gear teeth cut by the envelope of a rack profile.

(b) Gear tooth profile.

Figure 8

Result

The final kinetic sculpture (Figure 9) consists of three type of parts: square faced parts $(\times 6)$ which sit at the square faces of the cuboctahedron, triangular faced parts $(\times 8)$ which sit at the triangular faces of the cuboctahedron, and connectors $(\times 24)$ which align each pair of triangular/square acircular bevel gears.

Figure 9: 3D printed geared cuboctahedral jitterbug.

References

[1] Leonhard Euler. Supplementum de figura dentium rotarum. Novi Commentarii academiae scientiarum imperialis Petropolitanae, 11:207-231, 1767. [2] Gökhan Kiper and Eres Söylemez. Homothetic Jitterbug-like linkages. Mechanism and Machine Theory, 51:145-158, May 2012. [3] B. Laczik. Design and manufacturing of non-circular gears by given transfer function. 2007. [4] Caspar Schwabe. Eureka and serendipity: The rudolf von laban icosahedron and buckminster fuller’s jitterbug. In George W. Hart and Reza Sarhangi, editors, Proceedings of Bridges 2010: Mathematics, Music, Art, Architecture, Culture, pages 271-278, Phoenix, Arizona, 2010. Tessellations Publishing. [5] H F Verheyen. The complete set of Jitterbug transformers and the analysis of their motion. Computers & Mathematics with Applications, 17(1-3):203-250, 1989.