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Unformatted text preview: Answer Key to Third Exam (3021) • It is quite possible I did some mistake typing this Answer Key, so if you think you spotted one, please let me know. (I) (20 pts). If an undamped springmass system with a mass that weighs 2 lb and a spring constant 4 / 3 lb/in is suddenly set in motion at t = 0 by an external force of 4 cos t lb , determine the position of the mass at any time and write the solution so that it would convenient to graph. The first thing to do is to put all the constant under consistent units. We have the weigh, we need the mass: m = w g = 2 32 = 1 16 slugs There is no damping so γ = 0. The sprong constant should be expressed in lb/ft rather than lb/in to be consistent with the mass unit. That is k = 16 lb/ft . The question says ’suddenly set in motion’, which you should interpret as: at initial time ( t = 0), the mass is at equilibrium ( u (0) = 0) and has no velocity ( u prime (0) =)). Thus the IVP to solve is 1 16 u primeprime + 16 u = 4 cos t ; u (0) = 0 ft, u prime (0) = 0 ft/sec or equivalently u primeprime + 256 u = 64 cos t ; u (0) = 0 , u prime (0) = 0 The characteritic equation of the corresponding homogeneoux ODE is r 2 + 256 = 0 ⇔ r = ± 16 i ⇒ u c ( t ) = c 1 cos16 t + c 2 sin 16 t The frequency of the forcing function ( ω = 1) not being the same as the natural frequency of the system ( ω = 16), a particular solution of the ODE is of the form: Y ( t ) = A cos t + B sin t This has to satisfy the ODE, so Y primeprime + 256 Y = 255 A cos t + 255 B sin t = 64 cos t That is A = 64 / 255 and B = 0. Now, considering the initial condition, we have: u (0) = c 1 + 64 255 = 0 ⇒ c 1 = 64 255 1 u prime (0) = 16 c 2 = 0 ⇒ c 2 = 0 So the solution of the problem is given by: u ( t ) = 64 255 (cos t cos16 t ) = 128 255 sin parenleftbigg 17 t 2 parenrightbigg sin parenleftbigg 15 t 2 parenrightbigg Note that the last expression of the solution as a product of sin is the one that is convenient to use in order to graph the solution....
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