Find the forced response of Qt 4 Qt 20 Qt 16 e2t With ini

Find the forced response of: Q\"(t) + 4 Q\'(t) + 20 Q(t) = 16 e^-2t With initial conditions: Q(0) = 2 and Q\'(0)= 0 What type of response does this system have: over damped, critically damped or under damped. Why?

Solution

First we solve the associated homogeneous ode

Q\'\'+4Q+20Q=0

THis is a linear homogeneous ode with constant coefficients so solution is fo teh form:y=exp(kx)

Substituting gives

k^2+4k+20=0

k=-2+4i,-2-4i

SO general solution to homogeneous ode is

y=exp(-2t)(A sin(4t)+B cos(4t))

Let particular solution be :yp=C e^{-2t}

SUbstituting gives

-4yp-8yp+20yp=16e^{-2t}

8yp=16e^{-2t}

yp=2e^{-2t}

SO, C=2

y=exp(-2t)(A sin(4t)+B cos(4t))+2 e^{-2t}

y(0)=B+2=2

B=0

y=Aexp(-2t) sin(4t)+2e^{-2t}

y\'=-2Aexp(-2t) sin(4t)+4A exp(-2t) cos(4t)-2 e^{-2t}

y\'(0)=4A-2=0

A=1/2

y=exp(-2t) sin(4t)/2+2e^{-2t}

(2)

Underdamped system as small oscillations with some oscillations