CA-AA242B-Hw5

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

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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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