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Let me begin by saying that this will be a long post and will involve a few homework questions. I'm not looking for answers for per-say. Instead, I am looking to deepen my vague understanding of the Rayleigh-Ritz Method and how one can use it to find the mode shapes and natural frequencies of distributed and lumped parameter systems. Also note that I have already attempted to utilize my professor and TA as resources, but I've had other, more pressing issues to take up with my professor and my TA is just atrocious.

Anyway, the homework in question is as follows:

The very first question (1a) I believe I can solve without problem. I've never had to find the natural frequencies and mode shapes in transverse vibration for a bean with a point mass at the end, but I have done so with longitudinal vibrations. In a like manner, I'm assuming that the Euler-Bernoulli beam equation can be used with an adjustment to the boundary conditions. 1a) For the figure given, cantilevered beam with a tip find the exact eigenvalues and eigenfunctions for the system. (graduate student required, extra credit for undergraduates).

1b) Using the Rayleigh-Ritz method, compare and approximate solution of the following beam problem to a close form beam solution (part 1a) (undergraduates choosing not to do the extra credit, could compare to a cantilevered beam, with no tip mass). Compare natural frequencies (in Hertz) and mode shapes for each system.

Use the mode shape of a cantilevered beam in your Ritz method.

2) Using the Rayleaigh-Ritz method predict the eigenvalues and mode shapes for the following system.

Compare the eigenvalues of the system for kL/EI = .1, 1, 10; assume a four function expansion. Discuss how the solution changes for the system with increasing spring constant. Tabulate the natural frequencies for the beam alone, and for the beam/spring system.

1b and 2 are the questions I cannot solve. I'm not explicitly looking for solutions, as I'd like to do and learn them myself. I would, though, really appreciate a thorough explanation of the following example problem (see the next section), or a similar example problem lifted from some other text.

2. Relevant equations

The example problem I have been given is attached.

3. The attempt at a solution

I have a basic understanding of the Rayleigh-Ritz Method, but I do not understand how to apply it to the attached questions, nor do I understand the attached example. Again, all I'm looking for is an explanation of the example, or similar problem, so that I can finish the last 2 homework questions. I've read a few technical writings concerning the Rayleigh-Ritz method, but none deal explicitly wit this subject. There are about 250 different books in my schools library that deals with this variational method, in one way or another, but I don't have time to inspect them all. I figured this website would hasten my search.

Thanks guys.

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# Homework Help: Rayleigh-Ritz Method and Distributed Parameter Systems

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