home5_09F - = mg (with g = 9 . 8 m/s 2 ) of the machine...

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Department of Mathematical Sciences University of Delaware Prof. T. Angell October 21, 2009 Mathematics 351 Exercise Sheet 5 Exercise 21 : For the forced damped oscillator ¨ x +2˙ x +26 x = 82 cos (4 t ) (a) Find the transient solution. (b) Find the steady state solution. (c) Find the “zero-state response”. Exercise 22 : For the forced damped oscillator ¨ x +6˙ x +45 x = 50 cos ( ωt ) , ±nd the frequency that gives the maximum amplitude response. ( HINT: Look at problem #43, p. 165 of your textbook.) Exercise 23 : Assume that the mass-spring-dashpot system m ¨ x + γ ˙ x + kx = 0 is either critically damped ( γ =2 km ) or over damped ( γ> 2 km ). Show that the mass can pass through the equilibrium position at most once regardless of the initial conditions. HINT: Determine all possible values of t for which x ( t )=0 . Exercise 24 : A front-loading washing machine is mounted on a thick rubber pad that acts like a spring; the weight
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Unformatted text preview: = mg (with g = 9 . 8 m/s 2 ) of the machine depresses the pad exactly 0 . 5 cm. When its rotor spins at radians per second, the rotor exerts a vertical force F o cos ( t ) newtons on the machine. At what speed (in revolutions per minute) will resonance vibrations occur? Exercise 25 : Show that t and t 2 are linearly independent on 1 < t < 1; indeed, they are linearly independent on every interval. Show also that W ( t, t 2 ) is zero at t = 0. What can you conclude from this about the possibility that t and t 2 are solutions of the equation y + p ( t ) y + q ( t ) y = 0? Verify that t and t 2 are solutions of t 2 y 2 ty +2 y = 0. Does this contridict your conclusion? Does the behavior of the Wronskian of t and t 2 contradict Abels theorem? Due Date: Wednesday, October 28 in class...
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