Aerospace Engineering 2220: Dynamics

Prof. Eric FeronHW 4

Due June 16, 2014

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Location of instantaneous center of rotationIn this problem, we study the location of instantaneous center of rotations

for a variety of rolling objects.Consider a circle, a square, an ellipse and a ’lens’ (made of two arcs of a

circle), all shown in fig. 1.

Figure 1: Four rolling shapes

We assume each object has the same perimeter, equal to 1 meter. Eachobject is shown in its initial position. Assume each object rolls withoutslipping on the ground at constant angular velocity ω = 1 rad/sec. Findan expression for the location (on the ground) of the instantaneous centerof rotation for each rolling object as a function of time t, from t = 0 untilt = 8π. Plot all four curves on top of each other and discuss your findings.

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More instantaneous centers of rotationConsider the mechanical assembly in Fig. 2. A disc is able to rotate about

its center O. The tip of an arm is attached to the disc at the point M , withthe other tip P sliding along a rod. The rod is horizontal and it is alsoattached to the center of the disk O. Plot the locus of the instantaneouscenter of rotation of the arm as the disc rotates.

Figure 2: Disk-Rod arrangement

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A rolling Reuleaux triangleI am not sure if you knew that, but circles are not the only objects to

have a constant diameter! Consider the geometrical construct in Fig. 3:Three circles are mounted as an equilateral triangle as shown: Each circlegoes through the center of the other two circles. The object in bold is named

Figure 3: Left: Releaux triangle. Right: The triangle has a constant diame-ter.

a Reuleaux triangle (look at Wikipedia to learn more about it).We define the diameter of an object as the minimum distance between

two parallel lines, the area within which contains the object. (See diagramin Fig. 3).

1. Show that the diameter of the Reuleaux triangle is constant, that is,no matter how the triangle is oriented, the measured diameter remainsthe same. (According to Richard Feynman, famous Cal Tech educatorand Nobel prize in physics, NASA Engineers used to check that theThiokol space shuttle solid rocket boosters were round by measuringthree diameters, and concluding ‘it’s round’ if the three diameters werethe same. Then the 1986 Challenger accident happened. Thiokol’sname is now ATK.)

2. Consider two parallel lines as shown in Fig. 4. The bottom line is fixedand lies on the ground. The top line is free to move. A Reuleauxtriangle is sandwiched between the two lines and rolls without slippingon both (like a fancy gear). We assume the Reuleaux triangle rotates

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at constant angular speed ω. What is the longitudinal speed V of thetop plate as a function of time? Is it constant?

3. Numerical application: The perimeter of the Reuleaux triangle is1m. The angular speed ω is 1 rad/sec. The initial position of theReuleaux triangle is as shown in Fig. 4. Compute and plot V fromt = 0 to t = 20 sec.

Figure 4: Experimental setup

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Fancy gearsGears need not be round. There are several cases when non-circular gears

are useful. For example, Fig. 5 shows a nearly square gear set. Believe it ornot, this gearset works. In the following exercise, we want to build a similargearset

Figure 5: Non-round gearset

In this problem, we have a perfectly square gear (we assume the teeth ofthe gear are infinitely small), as shown in Fig. 6.

You are asked to find what the shape of the other gear should be. Doesthis shape depend on the distance between O1 and O2, the centers of rotationof the two gears? Are there many shapes possible? Explain your work.

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Figure 6: The left gear is a perfect square. What is the right gear like?

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