Express cos 3t - cos t in the form A sin alpha t sin beta t Express sin 3t - sin

Question

Express cos 3t - cos t in the form A sin alpha t sin beta t Express sin 3t - sin t in the form A sin alpha t cos beta t For Problems 3-5, find the steady state solution having, h, form U_ss = A cos (omega t - delta). u + 4u + 4u = cos t u + 2u + 2u = 2 cos t y+ u + = 4 cos 3t Resonance A mass of one slug is hanging at rest on a spring whose constant is 12 lb/ft. At time t = 0 an external force of f(t) = 16 cos omega t lb is applied to the system. What is the frequency of the forcing function that is in resonance with the system? Find the equation of motion of the mass with resonance. Good and Bad Resonance Sometimes resonance in a physical system is desirable, and sometimes it is undesirable. Discuss whether the resulting resonance would be helpful or destructive in the following systems. Soldiers marching on a bridge with the same frequency as the natural frequency of the bridge. A person rocking a car stuck in the snow with e s frequency as the natural frequency of the same

Explanation / Answer

d2u/dt2 + 4 du/dt + 4u = cost

complimentary function

m2 +4m +4 =0

(m+2)(m+2)=0

(c1+ c2t)e-2t

to find particular integral

let u= Acost + B sint

du/st= -Asint + Bcost

d2u/dt2= -Acost -B sint

d2u/dt2 + 4 du/dt + 4u = cost

-Acost -B sint + 4( -Asint + Bcost ) +4( Acost + B sint)=cost

-A+4B+4A=1 => 3A+4B=1

-B-4A+4B=0   => -4A+3B=0

12A +16B=4

-12A + 9B =0

25B=4, B=4/25

4A=3B

A= 3/4 B= 3/4 4/25=3/25

solution is

(c1+ c2t)e-2t + 3/25 cost + 4/25 sint

3/25 cost + 4/25 sint

=1/5( 3/5 cost + 4/5 sint)

=1/5 cos( t + x) where x=arc tan 4/3

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