Another way to measure how fast the circle is revolving is the frequency or number of revolutions per unit time. Typical units are revolutions per minute or revolutions per second. The equation below shows how the frequency is related to and .

As described in Prof. Norton's text *Design of Machinery*,
the slider crank mechanism is used in most internal combustion
engines. A schematic diagram of this mechanism appears below. The
circle represents the crankshaft and the line from point to point
is the connecting rod which links the crankshaft to the
piston. Burning of fuel in the combustion chamber above the piston
forces the piston down, and the vertical movement of the connecting
rod is converted to rotational motion of the crankshaft.

In class, we derived an equation that relates the coordinate of
the point to the angle that a radius through the point
makes with the positive axis. This equation appears below.

If the crank is rotating at an angular speed then this equation gives the coordinate of the point as

The first and second derivatives of this function then give the velocity and acceleration of the piston.

In engine design, the speed and acceleration of the pistons are two of the factors which determine the stress on certain parts of the engine. For example, a larger crankshaft radius can increase the torque the engine produces, but can also lead to more stress on the crankshaft, piston and the connecting rod. In the exercises, you will use Maple to help you investigate how changes in the parameters , , and affect the piston velocity and acceleration.

The command below shows how to define the function in Maple.

> y := t -> r/12*sin(omega*t) + sqrt(L^2-r^2*cos(omega*t)^2)/12;Here the units of and are in inches and is in seconds. The mysterious factor of just converts the units for into feet. We will want to plot this function and its derivatives for several parameter values and the following commands show a convenient way to do this, by defining sets of parameter values, and using the

> par_set1 := {r=1,omega=80*Pi,L=5}; > plot(subs(par_set1,y(t)),t=0..0.1); > plot(subs(par_set1,diff(y(t),t)),t=0..0.1); > plot(subs(par_set1,diff(y(t),t,t)),t=0..0.1);From the plots, it is easy to approximate the minimum and maximum values of the position, velocity, and acceleration by using the mouse. If you don't know how to do this, ask your IA to explain.

- Suppose that a circle of radius 6 inches is revolving about its center at a frequency of revolutions per minute. Find the value in seconds of the period and the angular frequency in radians per second. Then plot graphs of the position in feet, velocity in feet per second, and acceleration in feet per second squared of the coordinate of a point on the circle that starts at at .
- In the example in the background, radians per second. Convert this value to revolutions per minute. Is this a reasonable value for an internal combustion engine?
- Find approximate values for the maximum and minimum values for the position, velocity, and acceleration of point for the example in the background. The easiest way to do this is to use the mouse.
- The maximum and minimum values of the velocity of the point on the crankshaft are and the maximum and minimum values of the acceleration of the point are . Compute these values for the example in units of feet per second for the velocity and feet per second squared for the acceleration. Compare them to the approximate values you obtained in the previous exercise.
- Plot the position, velocity, and acceleration of the piston for
the parameter set ,
and . Compare the
maximum and minimum values for position, velocity, and acceleration
to the ones in the example in the background. Is the motion of the
piston simple
harmonic motion? Why or why not?

2003-12-02