Introduction
Calculus can be used to determine the acceleration of an engine piston, which is important to design engineers. Your task is to use the calculus to develop a model that can be used to determine the acceleration equation of a given engine piston, and to explore the effect of different engine configurations on piston acceleration.
Guiding Question
How does the design of an engine affect its performance? Why might it be important to know the acceleration equation for an engine piston?
Questions
1. Obtain a diagram of the crankshaft of a car engine with an inline configuration. Conduct research to determine how fast this car engine's crankshaft rotates in revolutions per minute (rpm) when it idles, when it is driving down the freeway, when it is at redline, etc.
2. As the crankshaft turns in the xy-plane, the piston moves up and down inside the cylinder. What is the radius of your car engine's crankshaft? How long is the connecting rod? What is the distance from the top of the piston to the point at which the connecting rod fastens?
3. Let y be the distance from the top of the piston to the center of the crankshaft. Let [theta] be an angle in standard position measured from the positive x-axis and increasing as the crankshaft rotates counterclockwise. Write an equation expressing y in terms of [theta]. The law of cosines is helpful here.
4. Find an equation for the rate of change of y with respect to t in terms of [theta] and d[theta]/dt.
5. Find an equation for the acceleration of the piston when the engine is at idle. Treat d[theta]/dt as a constant. Then find an equation for the acceleration of the piston when the engine is driving down the freeway. This time the value for d[theta]/dt will be a different constant. Finally, find an equation for the acceleration of the piston when the engine is at redline. Again, treat d[theta]/dt as a constant.
6. Between what two values of [theta] is the piston turning down with an acceleration greater than that of gravity (980 centimeters per second squared)?
7. Repeat this project with an engine with a different configuration (V, narrow-V, box, radial, etc.). What conclusions can you draw?
8. Why might it be important to know the acceleration of a piston?
Internet Resources
Crankshaft, Piston, and Engine Flywheel Animation
Velocity, Acceleration, and Calculus
Law of Cosines
Honda 2006 Accord Sedan Specifications
Honda 2006 Civic Sedan Specifications
Honda 2006 Pilot Specifications
Honda 2006 S2000 Roadster Specifications
Calculus and Math Definitions
Calculus Help
Visualize Derivatives
Understanding Derivatives through Linear Approximation
The Physical Concept of a Derivative
More Information on Derivatives
Derivative in the Encyclopedia
Using Data To Graph A Sine Curve
Ti-83 Graphing Using Data Points Tutorial (PDF)
Activities
Your write-up must include:
1. Objective: Develop the main question you are trying to answer with this project.
2. Bibliography: List of all resources used.
3. Data: Organize and record data about the engine and its dimensions.
4. Procedures: Describe what you did in this project and methods used to answer the questions posed.
5. Visuals: Create a diagram of the crankshaft and graphs of the functions developed in this project, complete with labels and captions.
6. Calculations: Show work for all calculations and include written explanations for each.
7. Conclusions: Write up all of your conclusions, including a discussion of how the design of an engine affects its performance, and why might it be important to know the acceleration equation for an engine piston.
Jean Nanjo, Teacher, and Jerry Lee, AP student, collaborated to develop this web hunt.
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