TAM310hw02 - TAM 310 Advanced Engineering Analysis I Spring...

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Unformatted text preview: TAM 310 Advanced Engineering Analysis I Spring 2008 Prof.R.Rand Homework No.2 (Due Tuesday Feb. 5) 1. Using separation of variables on the following PDE's, obtain 2 ODE's for each. Underline in red all variable coefficients which occur in the ODE's. DO NOT SOLVE the ODE's. a. b. c. u 2u 1 u = 2 + t r r r 1 u 1 2 + 2 sin 1 sin u sin =0 u 1 2u sin + =0 sin2 2 2. Find the general solution of the following ODE's: a. b. y + 8y + 25y = 0 y - 8y + 16y = 0 3. The transverse vibrations of a simply-supported elastic bar (also known as a beam) are governed by the PDE: 2u 4u =- 4 t2 x with the boundary conditions: u = 0 at x = 0 2u = 0 at x = 0 x2 u = 0 at x = 1 2u = 0 at x = 1 x2 and the initial conditions: u = x(1 - x) when t = 0 u = 0 when t = 0 t a. Use separation of variables to obtain the solution in the form of an infinite series. Be sure to evaluate all integrals. b. Write out the first 5 non-zero terms of the series. c. What is the period of vibration? That is, how long before the bar returns to its original position? 1 d. Using your result from part b above, draw the deformed shape of the bar at time t = . 4 Compare your result with the initial shape u = x(1 - x). ...
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