How to calculate the max torque due to angular momentum of a beam given motor specs

Question

I have non-rigid beam attached to a gearbox and motor assembly. The gearbox outputs $1000 N \cdot m$ at the shaft. The beam is rotated at $3000 RPM$ around it's center. If the motor is switched off immediately and acts as a brake, is it possible that the torque generated from the momentum of the beam could be greater than the $1000N \cdot m$ that the motor originally produced, and if so how?

I am trying to work out the max torque a beam could create at the gearbox mount when the power is cut suddenly, however we don't know any of the beam characteristics such as where the point loads are, how long it takes to stop etc. All I have is the gearbox and motor specifications.

Edit: This question is different from a similar one I asked here. In the other question I mentioned that the beam was uniformly distributed, however in this example I have no beam information. All I have to go off is the motor and gearbox specifications.

Answer 1

is it possible that the torque generated from the momentum of the beam

I'm going to stop you right there. Momentum doesn't generate torque. Momentum is moment of inertia times speed. Torque is moment of inertia times acceleration. Just like speed and acceleration aren't the same thing, momentum and force aren't the same thing.

If you apply brakes, the torque that results is whatever the braking torque is. The load doesn't dictate what the braking torque is. The load will determine how quickly it decelerates for a given braking torque, but the brake is what sets the torque.

Again, momentum is $p = mv$, or $L = I\omega$. Torque affects acceleration. It changes the speed, so it changes momentum $\tau = dL/dt$, but that's just because again $L = I\omega$, so really what that's saying is that $\tau = (I)d\omega/dt$, or the more common expression for torque, $\tau = I\alpha$.

So again, to reiterate, the load doesn't set the torque. The brakes set the torque. The load reacts to torque.

I am trying to work out the max torque a beam could create at the gearbox mount when the power is cut suddenly, however we don't know any of the beam characteristics such as where the point loads are, how long it takes to stop etc.

If you want to calculate the load, but you don't know anything about the load, you're pretty well out of luck.

Answer 2

Depending on the elasticity of your beam and its mass, it will vibrate in a complex harmonic pattern advancing the 3000rpm ad retarding it, in a pattern similar to that of the winded spring of mechanical hand watches.

In the absence of more detail on the beam we assume this simplified harmonic motion of the beam rotation above and below 3000rpm, while its center is attached to motor at constant 3000 rpm. Lets call the difference in rotation, or swinging like a pendulum about center, delta rpm, DN .

$$ DN_{(t)} = Acos(\omega t)$$ and $$\space \omega = \sqrt {k/m} $$

So the torque follows the same harmonic pattern, it can be more or less than 1000N.m.

There may also be higher frequency modes on the beam where it can vibrate on 2nd harmonic mode, meaning the is vibration along the length of the beam, while it advances and retards.

These all have impact on the torque!