CA-AA242B-Hw5

CA-AA242B-Hw5 - 4. Assume that the bar has a length L = 1...

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AA 242B: Mechanical Vibrations (Spring 2010) Homework #5 Due May 28, 2010 1 Problem 1 1. Using Hamilton’s principle, write the equation of dynamic equilibrium of a bar in extension. For this purpose, assume that the cross section A , Young modulus E , and mass per unit length m of the bar are constant. 2. Assume that the bar is clamped at one end and free at the other end. Derive in this case all its eigenfrequencies ω k . 3. Write a MATLAB program for building the ±nite element sti²ness and mass matrices of a clamped-free bar associated with a uniform decompo- sition of the bar in N elements. For this purpose, use a discretization based on linear shape functions.
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Unformatted text preview: 4. Assume that the bar has a length L = 1 m, a circular cross section of radius R = 0 . 02 m, a Young modulus E = 60 10 9 Pa, and a mass per unit length m = 3 , 391 . 2 Kg/m 3 . Using the above program and any other MATLAB tool, compute the lowest three eigenfrequencies of the bar and plot them as functions of the number of elements used in the nite element discretization. 5. Using the above program and any other MATLAB tool, compute the highest eigenfrequency of the bar and plot it as a function of the number of elements used in the nite element discretization. 6. Formulate some interesting conclusions. 1...
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