A mass weighing 8 lb is attached to a spring hanging from the ceiling and comes

Question

A mass weighing 8 lb is attached to a spring hanging from the ceiling and comes to rest at its equilibrium position. At t=0, an external force F(t)=2cos(2t) lb is applied to the system. If the spring constant is 10 lb/ft and the damping constant is 1 lb-sec/ft, find the steady-state solution for the system. What is the resonance force for the system?

So I found A to be 18/85 and B to be 4/85 for the steady-state solution (although I'm not entirely sure this is the final answer for the first part). I don't have any idea of how to go about finding the resonance force! Please help :) A mass weighing 8 lb is attached to a spring hanging from the ceiling and comes to rest at its equilibrium position. At t=0, an external force F(t)=2cos(2t) lb is applied to the system. If the spring constant is 10 lb/ft and the damping constant is 1 lb-sec/ft, find the steady-state solution for the system. What is the resonance force for the system?

So I found A to be 18/85 and B to be 4/85 for the steady-state solution (although I'm not entirely sure this is the final answer for the first part). I don't have any idea of how to go about finding the resonance force! Please help :)

Explanation / Answer

mass = 8lb/(32 ft/s^2) = 1/4
damping = 1
k = 10

1/4*u'' + u' +10u = 2cos(2t)
u'' + 4u' + 40u = 8cos(2t)

solution to homogenous is u=e^(-2t)[a*cos(6t) + b*sin(6t)]
solution to particular equation is 18/85cost(2t) + 4/85sin(2t) using undetermined coefficients.

u(t) = e^(-2t)[a*cos(6t) + b*sin(6t)] + 18/85cost(2t) + 4/85sin(2t)

Resonance frequency has to do with the solution to the homogenous part. cos(4t) & sin(4t)
frequency is cycles per second, and cos & sin repeat every 2*pi. So set 6t=2*pi and you get the resonance frequency at pi/3 sec.
If the applied force had been something with sin(6t) or cos(6t), the system would have been at resonance

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