Find the differential equations relating input force ft and

Question

Physical system modeling, time response and stability

For a mass-spring-damper mechanical systems shown below,

1. Find the differential equations relating input force f(t) and output displacement x₁(t) and x₂(t) in the system.

(Hint: K, f, and M are spring constant, friction coefficient and mass respectively)

2. Determine the transfer function G(s)= X₁(s)/F(s).

For the following questions, suppose G(s) = 3 x (s +1.3)/(s +1.4)(s² + 2s + 1.49)

3. Determine the system impulse response in time domain.

4. Draw its root locus and comment on system stability.

5. If the system can be simplified using the system dominant pole technique into G1(s) below, obtain the system response for a unit step input, and estimate the overshot and settling time.

G1 = 3 x (0 +1.3)/(0 +1.4)(s² + 2s + 1.49) = 3.9/(1.4(s² + 2s + 1.49))

(Hint: use the second order system formulas)