Two Machine Dynamics

First and in my opinion, the most crucial part of this problem is the obtainment of the expressions for $P_{E1}$ and $P_{E2}$ .

Consider the equation of the circuit shown in the diagram. Consider the current to be flowing clockwise at an arbitrary instant of time. The equation is:

$L\frac{dI}{dt} = V_{t1} - V_{t2}$

Multiplying both sides by $Idt$ and integrating from $0$ to $t$ gives us:

$\frac{1}{2}LI^2 = \int_{0}^{t} V_{t1} Idt - \int_{0}^{t} V_{t2} Idt =\int_{0}^{t} P_{E1}dt+ \int_{0}^{t} P_{E2}dt$

The way to interpret the above equation is that the electrical energy outputs of the generators are stored in the inductor. In other words,

$P_{E1} = V_{t1} I$

and

$P_{E2} = -V_{t2} I$

Having established this, the next step is to derive the governing differential equations of this electromechanical system. After simplifying and rearranging the given equations, the following are obtained:

$\ddot{\theta}_1 = \frac{V_m}{\omega_oJ} I\sin{\theta_1} \ \dots (1)$ $\ddot{\theta}_2 = -\frac{V_m}{\omega_oJ} I\sin{\theta_2}\ \dots (2)$

Replacing terms in the circuit equation gives:

$\frac{dI}{dt} = -\frac{V_m}{\omega_oL} \left(\sin{\theta_1}\dot{\theta}_1 - \sin{\theta_2}\dot{\theta}_2\right) \ \dots (3)$

Now, this given system of equations is converted into a nonlinear state-space form as such:

Let: $x_1 = \theta_1;\ x_2 = \theta_2 ; \ x_3 = \dot{\theta}_1 ;\ x_4 = \dot{\theta}_2 ;\ x_5 = I$

Doing so gives us the following system description:

$\left[\begin{matrix}\dot{x}_1\\\dot{x}_2\\\dot{x}_3\\\dot{x}_4\\\dot{x}_5\end{matrix}\right] = \left[\begin{matrix}x_3\\x_4\\\left(\frac{V_m}{\omega_oJ}\right)x_5\sin{x_1}\\-\left(\frac{V_m}{\omega_oJ}\right)x_5\sin{x_2}\\\left(\frac{V_m}{\omega_oL}\right)\left(x_4\sin{x_2}-x_3\sin{x_1}\right)\end{matrix}\right] \implies \dot{x} = f(x)$

Initial conditions are:

$x(0) = \left[\begin{matrix}\pi/9\\0\\\omega_o\\\omega_o\\0\end{matrix}\right]$

Integrating the above using the explicit Euler method and evaluating the required expression using a computer program gives:

$\boxed{\dot{\theta}_{1M} - \omega_o = 16.5667}$

As for the bonus question:

The following plots should provide some insight:

It is seen that initially flows anticlockwise. Looking back at the angular acceleration equations, one can see that the value of the angular acceleration of the first machine is negative and that of the second machine is positive. Therefore, the speed of the first machine initially reduces while that of the second machine increases. If $\theta_1(0) = -\pi/9$ and $\theta_2(0) = 0$ , the opposite would happen. This can again be understood by looking at the structure of the equations of the system. We see an oscillatory behavior as the flux associated with the two machines is of a periodic nature. As a result of that, the output voltages are oscillatory in nature and consequently, from the energy conservation principle, the rotational rate of both the machines are periodic. If there is more to this than meets the eye, please share your comments.