STATEMENT OF PROBLEM

The reciprocating engine plays an important part in modern life. You may be interested in studying the mathematics behind this device, and this problem provides an opportunity for you to do that. You are to begin by describing the motion of key parts of the engine.

As the crankshaft rotates, the piston moves back and forth along the x-axis. Your job is to find the position, velocity and acceleration of the piston as a function of the angle q that the crank makes with the x-axis. Since the crankshaft is assumed to be rotating at a constant angular velocity w, (measured in radians/sec), this is equivalent to finding the quantities as functions of time. (q = w t) You are also required to find the angular velocity and angular acceleration of the connecting rod. These will be the first and second derivatives with respect to time of the angle f that the connecting rod makes with the x-axis.

The quantities
x,
dx/dt,
d2x/dt2,
f,
df/dt
d2f/dt2
should be plotted as either functions of time, or as functions of q. Find the maximum and minimum values of each quantity. The maximum and minimum values are of interests to engineers calculating the stresses in the engine.

Data

Here is some realistic data that you can use. Let the crank length, r, be 8 cm, and the connecting rod length be 24.5 cm. Assume that the crank is turning at a constant 1000 rpm, (revolutions per minute). We must convert this to radians per second. Let T be the time in seconds it takes to do one rotation.

Input := 

data = { r -> .08, L -> .245, w -> 1000 (2 Pi / 60)} //N;
T = N[60/1000];