Question

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Question 1 Figure Q1 shows a mechanical system. The system input is T) and output is supposed to be 0. Please find the transfer function from T to θ 3, and discuss the stability of the system if the input is a unit impulse signal. (30 marks) To 01(t) 01t) I kg-m2 N 10 030) N2 100 100 kg-m2 100 N-m/rad 100 N-m-s/rad Figure Q1

Answer 1

Body 1:

$J_1\ddot \theta_1 = T(t)$

$\frac{\dot \theta_1}{\dot\theta_2} = \frac{100}{10} = 10$

$J_1*10\ddot \theta_2 = T(t)$

$10\ddot \theta_2 = T(t)$

Laplace transform:

$10s^2 \theta_2(s) = T(s)$

Body 2:

$J_2(\ddot \theta_3 )+k*(\theta_3 -\theta_2)+c*\dot \theta_3= 0$

$100\ddot \theta_3+100*(\theta_3 -\theta_2)+100*\dot \theta_3= 0$

$\ddot \theta_3+\theta_3 -\theta_2+\dot \theta_3= 0$

Laplace transform:

$s^2\theta_3(s)+\theta_3 (s)-\theta_2(s)+s \theta_3(s)= 0$

$(s^2+s+1) \theta_3(s)-\theta_2 (s)= 0$

$\theta_2 (s)= (s^2+s+1) \theta_3(s)$

But,

$10s^2 \theta_2(s) = T(s)$

$10s^2 *( (s^2+s+1) \theta_3(s)) = T(s)$

Transfer function:

$\frac{\theta_3(s)} {T(s)} = \frac{1}{10s^2 *( (s^2+s+1)}$

Given a unit impulse signal for T.

Laplace transform of a unit impulse signal is 1.

$\theta_3(s)= \frac{1}{10s^2 *(s^2+s+1)}*1$

This system has 2 poles at the origin (due to s²).

Which indicates a type 2 system.

$ss \;error= \lim_{s\rightarrow 0}\frac{1}{10s^2 *(s^2+s+1)}s = \infty$

The system is unstable.