Question

s) Given the following rotational mechanical system, hot relates the input variable T (applied torque) to the output a) Write the differential equation that re variable angular displacement) b) Convert the differential equatio c) Write the Transfer function of the system (I. w ent the differential equation to Laplace domain assuming initial conditions Zero Consider the following values for the parameters: J - 2 kg-m? (moment of inertial of the mass) D = 0.5 N-m-s/rad (coefficient of friction) K-1 N-m/rad (spring constant)

Answer 1

Question 1 (a)

Writing the Torque Balance Equation

det) T(t) - R.Ot) - D. dt d?(t) . dt2

Substituting the values

T(t) - 1.0(t) -0.5. de (t) dt de(t) = 2.-

So the differential equation that relates applied Torque and the angular displacement is

,d Ꮎ t , ᏧᎾ (t , , + Ꮎ (t = Ꭲ(t t . dt2 +0.5.4

Question 1(b)

,d Ꮎ t , ᏧᎾ (t , , + Ꮎ (t = Ꭲ(t t . dt2 +0.5.4

Taking Laplace Transform on both sides, assuming zero initial conditions

2.s? (s) + 0.5.s. (s) + 0(s) = T(s)

We have used the differentiation property of Laplace transform under zero initial conditions which states that

If

(5)X = {(1)x}7

Then

, (dx(t)} = s.X(5)

(6)x";s={,3}

So the differential equation in Laplace domain is

2.s? (s) + 0.5.s. (s) + 0(s) = T(s)

Question 1(c)

2.s? (s) + 0.5.s. (s) + 0(s) = T(s)

(2.s2 + 0.5.s + 1)(s) = T(S)

e(s) T(S) (2.+0.5.s +1)

So the transfer function is

e(s) T(S) (2.+0.5.s +1)