Questions: Figure 2.81: Electromechanical system.

Solution Steps

The given figure represents an electromechanical system. Let's define the variables and parameters:

• \(R\): Resistance of the electrical circuit
• \(L\): Inductance of the electrical circuit
• \(K_e\): Back emf constant of the motor
• \(v\): Input voltage
• \(i\): Current flowing through the circuit
• \(T\): Torque generated by the motor
• \(\omega\): Angular velocity of the motor shaft
• \(J_1\): Moment of inertia of the first rotating mass
• \(J_2\): Moment of inertia of the second rotating mass
• \(k\): Spring constant
• \(c\): Damping coefficient
• \(\theta_1\): Angular displacement of the first rotating mass
• \(\theta_2\): Angular displacement of the second rotating mass

The electrical subsystem can be described by Kirchhoff's voltage law:

\(v = Ri + L \frac{di}{dt} + K_e \omega\)

Also, the torque generated by the motor is proportional to the current:

\(T = K_t i\)

where \(K_t\) is the torque constant. In many cases, \(K_t = K_e\). Therefore, we'll assume \(K_t = K_e\). So:

\(T = K_e i\)

The mechanical subsystem can be described by Newton's second law for rotational motion. For the first rotating mass:

\(J_1 \ddot{\theta}_1 = T - k(\theta_1 - \theta_2)\)

And for the second rotating mass:

\(J_2 \ddot{\theta}_2 = k(\theta_1 - \theta_2) - c\dot{\theta}_2\)

Also, the angular velocity of the motor shaft is equal to the angular velocity of the first rotating mass:

\(\omega = \dot{\theta}_1\)

Final Answer: