Need help with this questions. Need details, like free body diagram. Thank you s

Question

Need help with this questions. Need details, like free body diagram. Thank you so much!!

For the electromechanical system shown in figure 4, determine the transfer function X(s) / F(s) using the free-body diagram, where f(t) is the applied force, x(t) is the displacement of the mass, k is the spring constant, and c, is the damping constant of the dashpot and the friction coefficient. Using the results of PLA-1, write the transfer function C(s) / R(s) of the system shown in figure 4. Assume KEMS = KsKenc. Given the step response of a system shown in figure 5, write the transfer function of the prototype second-order system in the form of equation (1). and find zeta and omegan n of the response using equations (2) and (3). Using the results of PLA-2 and PLA-3, find the values of KEMS, c, and k in terms of m. Note that for the ECP system, m is mass of the carriage and the additional known mass of the brass weights, c is the damping constant of the dashpot plus the friction coefficient and k is the stiffness of the spring.

Explanation / Answer

1.

from the figure we can write the governig equation as

f(t) - kx -cx' = mx'' where x' and x'' represents the first and second derivative of x w.r.t t.

taking laplace transform we get

F(s) = X(S)*(mS^2 + cS + k)

so

X(S)/F(S) = 1/((mS^2 + cS + k))

2.

we have

X(S)/F(S) = 1/((mS^2 + cS + k))

now

from the figure

C(S) = X(S)*(Kenc*Khw)

or

C(S) = F(S)*1/((mS^2 + cS + k))*(Kenc*Khw)

now

F(S) = R(S)*K(S)

so

C(S)/R(S) = (K(S)*(Kenc*Khw))/((mS^2 + cS + k))

3.

equating both the equations we get

Wn = sqrt(K/m)

and

gi = c/(2*sqrt(mK))

4.

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