Homework Help: Cam Lift as a Periodic Forcing Function

1. Feb 18, 2013

PatrickRay007

1. The problem statement, all variables and given/known data
Background: Given a cam and follower system in the valvetrain of an internal combustion engine and a table of values relating follower lift with respect to angular displacement, I would like to model the input delivered to the valvetrain by the cam lobe as a periodic forcing function for dynamic analysis of the response. I have an old vibrations textbook from college, wherein I found an example very similar to what I'm trying to do (in fact, it's labeled as a function one might see in a cam, though it would be a particularly poorly designed cam if it were). In this example, the author has jumped straight to a Fourier series representing the motion of the cam, plots this, and compares it to a graph of the motion created by the cam, all the while assuming I can jump straight to the Fourier Series (his approximation was very close, however!). The problem for me is that I have a table of lift values, which I can graph, but have been grappling with my rusty mathematical knowledge on just how I can develop an equation from this table of values.

Things I know:
- Angular displacement
- Lift (distance from the contact point to the center of the cam shaft minus base circle (zero lift) radius
- Mass of the system components
- Spring constants of the system

For the moment, I'd like to focus on obtaining the equation of motion as a function of angular displacement and understanding that before relating the motion to time (should be simple) and then developing the dynamic model, which I expect will be much more challenging. Additionally, I will have the ability to compare my model to an actual camshaft in operation. The idea here is to predict undesirable valvetrain motion (such as flutter and float) and compare the predictions to the test results and further refine the model to use in future valvetrain development. Once I have a dynamic model, I hope to be able to test the effect of changing the properties of the system without having to reconfigure an actual valvetrain. I found a wonderful dissertation by a South African graduate student that has helped, but his development of the mathematical model is, frankly, over my head.

2. Relevant equations

3. The attempt at a solution
My first attempt at a solution has been to overlay a sinusoidal curve (EQ 1) to the data, matching by eye, then plot the difference between the two. I then fitted a sinusoid to the difference by eye and subtracted it from EQ1. Plotted the difference, fitted a sinusoid, and so on three or four times. This produced mediocre results, as the lift profile is apparently complex, so this is why I am now leaning toward a fourier series representing the data. I have posted a truncated excel file with the lift values for one revolution of the cam shaft for your consideration.

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