Example 1.5-1: Piston Motion

1.   Understand the purpose of the problem.

¨     Verify that the design above will convert the angular motion of the crankshaft into up and down motion of the piston (and vise versa) and visualize this relationship to give us insight into how this conversion takes place.  We want to know if the piston motion will be smooth or whether it will experience abrupt changes in direction, which can cause significant wear and tear.

 

¨     Determine how tall the cylinder needs to be so that the piston will not come out of the bottom or bust through the top (given the dimensions L1 and L2).

 

¨     This means we must calculate and plot the distance traveled by the piston (d) as a function of the angle of rotation or crankshaft angle (A) given the connecting rod length L1 and the crankshaft offset L2.

 

2.   Collect the known information. Realize that some of it might be discovered later to be unnecessary.

¨     L1=1.0 ft

¨     L2=0.5 ft

¨     Motion is symmetrical about A=0 therefore we need only consider the following range of angles: 0° £ A £180°

¨     We assume that the height of the piston is known (although not specifically provided in the book problem).  Let this height be h (in ft).

 

3.   Determine what information you must find.

¨     Need to determine d (the distance traveled – in ft) as a function of A (degrees).

¨     By plotting d vs. A, we will know the full vertical range of motion of the piston.  We can then specify the cylinder height required.

 

4.   Simplify the problem only enough to allow the required information to be obtained. State any assumptions you make.

¨     The problem requires no further simplification. The relationship between d and A can be deduced using trigonometric analysis.

 

5.   Draw a sketch and label any necessary variables.

6.   Determine which of the fundamental principles are applicable.

 

¨     Using Cosine Definition (cos(angle)=adjacent/hypotenuse):

             

                (Remember SOH CAH TOA)

¨     Using Law of Sines (sides proportional to the sine of the opposite angle):

                               

 

7.   Think about your proposed solution approach in general and consider other approaches before proceeding with the details.

¨     Is this the best way to analyze the geometry?

¨     Am I missing anything? Is the motion of the piston really symmetric?

 

8.   Label each step of the solution process.

a.     Pick a range of angles A

b.    Solve for angle B for each A

c.     Solve for d for each A and corresponding B

d.    Plot d vs. A and see what the range of displacements is

e.     Make the cylinder tall enough to contain the piston throughout its entire displacement.

 

9.   Solve the problem.  Assuming you are using a program:

 

·       State the problem concisely.

¨     Determine d using

where the angle B is calculated as

 

·       Specify the data to be used by the program. This is the “input”.

¨     L1=1.0, L2=0.5, 0° £ A £ 180°.

 

·       Specify the information to be generated by the program. This is the “output”.

¨     d (ft) will be calculated for a range of values of A (degrees) and the results will be shown in an annotated plot

 

·       Work through the solution steps by hand with a simple data set.

¨     Here is a partial solution that will help us reality check our program output.

¨     A=0° Þ d=1.5 ft

¨     A=180° Þ d=0.5 ft

By inspection

·       Write and run the program.

 

L1=1.0;

L2=0.5;

Ad=[0:.05:180];

Ar=Ad*pi/180;

Br=asin(L2*sin(Ar)/L1);

d=L1*cos(Br)+L2*cos(Ar);

plot(Ad,d);

title('Locus of piston in Example 1.5-1 (Pg. 19)')

xlabel('Crankshaft Angle, A (degrees)')

ylabel('Piston Travel, d (ft)')

grid

 

·       Check the output of the program with your hand solution.

q       All calculated points check out!

q       A=0°

q       A=180°

 

·       Run the program with your input data (if different from the simple data set) and perform a reality check on the output.

Output distance ranges from 1.5 to .5 as expected. Looks good!

 

·       If the program will be used as a general tool in the future, test it by running it for a range of reasonable data values; perform a reality check on the results.

¨     It would make sense to test the program for other values of the input variables, L1 and L2.

¨     This is not really convenient at the moment for several reasons:

v   We'd need to retype the code!

v   We'd probably like to have the other results on the same plot!

v   We'd probably like to have a real customer who wants to pay for it!

 

See Problem 1.5-2 in Text p. 25

 

 

10.         Present the results.

 

Do not state the answer with any greater precision than is justified by any of the following:

·       The precision of the given information.

·       The simplifying assumptions.

·       The requirements of the problem.

 

 

From the plot it appears that the angular motion of the crankshaft does translate into up and down motion of the piston.  This relationship appears to be sinusoidal.  Thus, you now know that the pistons in your car are tracing out sine functions as they move up and down.  Note that the change in direction occurs smoothly, which is important to minimize wear and tear.

 

The center of the piston moves from .5 feet above the center of the crankshaft to 1.5 feet, and never goes higher or lower.  Thus the cylinder must start .5 – h/2 feet and must extend to at least 1.5 + h/2 feet above the crankshaft center.

 

Our precision is only limited by our knowledge of L1 and L2.

 

 

This document was originally created by Dr. Malcolm Daniels of the Department of Electrical and Computer Engineering and has been modified by Dr. Russell Hardie.