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GSW Problem Set 4

# GSW Problem Set 4 - t → ∞ 4 Consider a spring/mass...

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Engineering Analysis 4 Workshop questions - week of 10/26/08 1. Consider the forced oscillator x 00 + x = cos 2 t. (1) (a) Find the general solution. (b) Suppose you are interested in the initial conditions x (0) = 1 , x 0 (0) = 0 . Explain the fallacy in the following argument. Incorrect argument - determine α and β to match the initial conditions so that 1 = α cos 0 + β sin 0 , 0 = - α sin 0 + β cos 0 . or α = 1 , β = 0 . Thus, the solution to the initial value problem is x ( t ) = cos t - 1 3 cos 2 t, which clearly does not satisfy the initial condition. (c) Determine the correct solution to the initial value problem. 2. Consider a forced, undamped oscillator with m = k = 1 and forcing sin 2 t . The equation is x 00 + x = sin 2 t. Show that for all initial conditions the displacement is bounded as t → ∞ . 3. Consider a forced, undamped oscillator with m = k = 1 and forcing sin t . The equation is x 00 + x = sin t. Show that for all initial conditions the displacement is unbounded as

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Unformatted text preview: t → ∞ . 4. Consider a spring/mass system held vertically, so that the mass also experiences a force due to gravity. Assume that the rest of the system, e.g., the spring and the damper, are essentially weightless. Since the spring is held vertically you might want to call the displacement y . (a) Determine the equation satis±ed by the displacement. (b) Determine a particular solution to the equation in terms of an undetermined coef±cient A . (c) Verify the units of the particular solution. 1 (d) Explain the physical signifcance oF the particular solution. (e) Make the transFormation z = y-A and determine the equation satisfed by the new variable z . (F) Explain the physical signifcance oF z and the equation that you Found. (g) Assuming c > so that there is damping, determine lim t →∞ y ( t ) . 2...
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GSW Problem Set 4 - t → ∞ 4 Consider a spring/mass...

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