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Appendix C C-59<br />
PROJECT<br />
The Motion of a Piston<br />
To see the way a machine works, you can take the covers off and look inside.<br />
But to understand what goes on, you need to get to know the principles<br />
that govern its actions. —David Macaulay, The Way Things Work (Boston: Houghton<br />
Mifflin Company, 1988)<br />
In this project we examine a simple linkage that exhibits an oscillatory motion<br />
that is somewhat more complicated than simple harmonic motion. The tools that<br />
we’ve developed in the last two chapters will enable us to derive a function<br />
describing this motion. This function provides a very good mathematical<br />
model for the motion of a piston in a conventional internal combustion car<br />
engine. Although you won’t need to be familiar with car engines to follow the<br />
mathematics, your appreciation of its applicability would be greatly enhanced<br />
if someone in your group could explain how a crankshaft piston and cylinder<br />
move in a car engine. The Internet or a reference such as the one quoted above<br />
might be useful.<br />
Figure A shows a line segment OC of length R rotating counterclockwise<br />
about the fixed origin O and another line segment CP of length L, with L<br />
greater than R. The line segments are thought of as being linked together at<br />
point C as if they were flat rods with a pin through them so that they can rotate<br />
freely about C, their common endpoint. The point P is constrained to move<br />
along the x-axis. As the line segment OC rotates about the origin, the point P<br />
moves back and forth along the x-axis. The two figures illustrate the configuration<br />
at two different rotation positions. In the language of car engines the origin<br />
would be the center of a cross-section of the crankshaft, the segment OC<br />
would be a crank arm, the segment CP would be a piston rod, and point P<br />
would be the center of a cross-section of a wrist pin.<br />
y<br />
y<br />
R<br />
O<br />
¨<br />
C<br />
L<br />
P<br />
x<br />
C<br />
R<br />
¨<br />
O<br />
L<br />
P<br />
x<br />
(i)<br />
(ii)<br />
Figure A<br />
Exercise 1 Let f(u) equal the x-coordinate of the point P when the line segment<br />
OC is at an angle u (measured in radians) from the positive x-axis. What are the<br />
maximum and minimum values of f and at what values of u do they occur?<br />
Exercise 2 Figure B can be used to derive a formula for f(u). Express the<br />
lengths of OA and CA in terms of R and u. Now use the Pythagorean theorem
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