There is another damped spring-mass model which can be used for further comparis

Question

There is another damped spring-mass model which can be used for further comparison and contrast, namely the automobile suspension system given in pset 2. It is equivalent to a spring-mass system which is driven through both the spring and dashpot As before, we will take the input f(t) = B cos(wt). a) Derive the formula for the amplitude response g(w). As before, give the formal answer in terms of P(iw) and Arg(P(iw)) and the detailed answer in terms of m, b, k, w. (b) Derive the formula for the practical resonant frequency. Does practical resonance always occur in this case?

Explanation / Answer

given for the spring damped model

mx" + bx' + kx = kf(t) + bf'(t)

f(t) = Bcos(wt)

f'(t) = -Bwsin(wt)

hence

mz" + bx' + kx = B(kcos(wt) - bwsin(wt)) = M(cos(a)cos(wt) - sin(a)sin(wt))

comparing

M^2 = B^2*(k^2 + b^2w^2)

M = B*sqrt(k^2 + b^2*w^2)

and

Bk/M = cos(a)

cos(a) = k/sqrt(k^2 + b^2*w^2)

sin(a) = bw/sqrt(k^2 + b^2w^2)

tan(a) = bw/k

hence

mx" + bx' + kx = B*sqrt(k^2 + b^2w^2)*cos(wt + arctan(bw/k))

a. the response steady state, g = Acos(wt - phi) is given by

g = Acos(wt - pphi)

st

A = B*sqrt(k^2 + b^2w^2)/m*sqrt((k/m - w^2)^2 + c^2*w^2/m^2)

phi = arctan(cw/(k - mw^2)) - arctan(bw/k) = arctan((cwk - bwk + bmw^3)/(k^2 - mkw^2 + cw*bw)))

b. practical resonant freq

w' = sqrt(k/m - c^2/4m^2)

hecne practical resonance does not always occusr but depends on the relativer values of m, b and k

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