An electromechanical system has the following equations of motion:

aα¨ + b sin( ˙α) + c tan(α) = ki + α 0.1 e^ −0.05α 0.15 (1)

di/dt = αi˙ (d1 + d2α) + d3 − i d4 (2)

where α indicates its rotation angle and i stands for the actuator’s current. Set a = 1, b = 0.54, c = 1, k = 0.07, d1 = 140, d2 = 320, d3 = 24, and d4 = 1000.

1. Develop its nonlinear code in MATLAB using ode45 and plot α and i for t = 0 : 0.01 : 6500.

2. Find the system’s equilibrium points and discuss them.

3. Then using the equilibrium points linearize the equations of motion by knowing that the actuator has slow dynamics.

4. Find the transfer function of Output Input = α(s) I(s) assuming α(0) = ˙α(0) = 0.

5. Use the transfer function and make its unity feedback block diagram through Simulink.

6. Now plot the system response for the inputs of impulse, ramp, and parabolic along with their error signals. Validate the error signals by examining the final value theorem.

7. Apply a harmonic input of sin(0.1t) and plot its time-domain response. Explain your observations for both the amplitude and phase angle. Now apply sin(7.5t), plot and discuss it. Also analytically plot Bode for both the cases, as I taught in class, and then validate those with MATLAB coding.

8. Now repeat Section 7 for b = 9. What are your observations? Discuss them.

**Mechanical Engineering Question**

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