Question

The displacement of an object in a spring-mass system in free damped oscillation is:

2y''+12y'+26y=0 and has solution: y1 =−5e^(−3t)*cos(2t−0.5π)

if the motion is under-damped.

If we apply an impulse of the form f(t) = αδ(t−τ) then the differential equation becomes :

2y''+12y'+26y=αδ(t−τ) and has solution y =−5e^(−3t)cos(2t−0.5π)+αu(t−τ)w(t−τ) where w(t) = L(^−1)*(1/(2s^2+12s+26))

a. When should the impulse be applied? In other words what is the value of τ so that y1(τ) =0 . Pick the positive time closest to t = 0

b. What intensity should the impulse be so that the object is in equilibrium for t > τ (i.e. what should the value of α be so that y(t)=0 for t>τ ).

c. Write up your work for this problem including the inverse transform needed to write out w(t) and the completed form of the impulse function f(t)   

Answer 1

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