Doubt on a shaft-turbine-generator system

Question

Consider a shaft which connects a turbine to a generator. Initially, none of the parts are moving. Now the turbine starts to rotate and the torques act on the shaft from both the turbine and generator side. I'm interested in knowing, what happens to the torques acting on the shaft from both ends and the angular speed of rotation of the shaft, from the start of the turbine until some angular speed $\omega$ is reached.

It makes sense to me that as the turbine starts rotating it applies a torque on the shaft from the turbine side, and because of some net unbalanced torque the speed of the shaft starts to increase. It is my guess that the shaft starts to rotate at some constant angular speed when an equal and opposite torque is developed from the generator side. I do not understand how the generator applies this torque? why the torque from the generator side increases along with the torque from the turbine side?

Answer 1

I am not entirely certain about your question but I think, it basically boils down to this equation

$$M = I \alpha $$

where:

• M is the torque applied
• I is the mass moment inertia of all rotating masses
• $\alpha$ is the angular acceleration $(\frac{d \omega}{dt})$

In another form this is written as:

$$\Delta M = I \frac{d \omega}{dt}$$

I.e. the turbine has a load which can be translated to a torque $M_{turb}$ which is resisting motion. The generator is producing $M_{gen}$.

If:

• $M_{gen}>M_{turb} \rightarrow M_{gen}-M_{turb}=\Delta M>0$ then the system accelerates
• $M_{gen}=M_{turb} \rightarrow M_{gen}-M_{turb}=\Delta M=0$ then the system does not change its kinetic state.
• $M_{gen}<M_{turb} \rightarrow M_{gen}-M_{turb}=\Delta M<0$ then the system decelerates

Answer 2

The generator start-up torque demand is the sum of the load on it, if any, its angular momentum, frictions in bearings, cogging of magnets. etc.

When a torque by the turbine is applied to the The generator starts to accelerate until the RPM of the generator is at the systems design RPM. At design RPPM the generator power is $P= 2πNT/60$

• N is rpm
• T= torque in Nm
• P =power in watts

So the torque of the shaft initially goes to accelerate the generator and later to have it produce power turning at a constant speed, $\omega$.

At all times the torque at the two ends of the shaft is equal and opposite.

Edit

After OP's comment

Shaft unbalanced torque is a major subject in power generation plants.

At the most basic level, one can compare it to SDF torsional vibration. In reality, it is a continuous mass multi-node force function with tapering or stepping shaft diameter.

In the SDF form with damping the equation simplifies to

$$ I \frac{d^2\theta}{d^2t} +C\frac{d\theta}{dt} + k\theta = \tau(t)$$

At low damping ratios frequency of the system is very near the natural frequency

$f_n= \frac{1}{2\pi}\sqrt{\frac{k}{I}}$.

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