Machine Dynamics

Consider a hypothetical rotating machine, whose angular position $\theta$ varies as follows:

$\large{\tau_{in} - \tau_{out} - \alpha \, (\dot{\theta} - \dot{\theta}_{ref}) = I \, \ddot{\theta}}$

In the above equation, $\tau_{in}$ and $\tau_{out}$ are the input and output torques, $\alpha$ is a damping parameter, $\dot{\theta}_{ref}$ is the machine's reference speed, and $I$ is the machine's moment of inertia.

Initially, the input and output torques are at their nominal values, and the machine rotates at the reference speed.

Then, the output torque increases to $150 \%$ of its nominal value, and the machine approaches a new steady-state speed.

How many $\text{rad/s}$ below the reference speed is the new steady-state speed?

Nominal Parameter Values:
$\tau_{in} = \tau_{out} = 10 \, \text{N m} \\ \alpha = 3 \, \text{N m s} \\ I = 4 \, \text{N m s}^2 \\ \dot{\theta}_{ref} = 120 \pi \, \text{rad/s}$