Questions: Figure 2.81: Electromechanical system.

Transcript text: Figure 2.81: Electromechanical system.

Solution

Step 1: Define the variables and parameters

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

Step 2: Write the equations for the electrical subsystem

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:

\(T = K_e i\)

Step 3: Write the equations for the mechanical subsystem

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:

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

The governing equations for the electromechanical system are:

Electrical subsystem: \(v = Ri + L \frac{di}{dt} + K_e \dot{\theta}_1\)
Mechanical subsystem: \(J_1 \ddot{\theta}_1 = K_e i - k(\theta_1 - \theta_2)\) \(J_2 \ddot{\theta}_2 = k(\theta_1 - \theta_2) - c\dot{\theta}_2\) \(\omega = \dot{\theta}_1\)

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