Dear all,
The rotor of a synchronous machine (motor or alternator) connected to the grid may show up torsional vibrations. There must be a resonance frequency basically determined by rotor moment of inertia and the armature (or whatever) reactance. (And also some damping from the squirrel cage bars in the rotor poles).
Can someone help me to find the basic formulas to calculate this type of torsional resonance frequency? Or same, how to transform electrical reactance of a synchronous machine into mechanical torsional stiffness [kNm/radians] ?
Regards from NL,
Arie

It is actually a nonlinear problem. If the oscillation (around the synchronous speed) does occur, the frequency of this oscillation depends on the load of the machine. The basic formula for the circular frequency (ignoring the damping) is:

w = sqrt(C.p/J) (1)

w …. circular frequency of oscillation
p….. number of pole pairs
J……moment of inertia of the rotor
C….. synchronization moment (torque)
In (1) the …… C= Mmax. * cos(beta) (2)
This is valid only for small angles of oscillation (basically for angles of oscillation x where sin(x) = x).

(beta) is the angle between the rotor field and the stator field. It grows with the load (and here is the non-linearity). Equation for the Mmax:

Mmax = 3U*Ui/(ws*Xd) (3)

3 is for three phases
U …line to neutral voltage
Ui…induced voltage (it is a voltage that would be induced to the stator by rotor field if there were not saturation)
ws….circular synchronous speed
Xd…. Synchronous reactance
The above is for a turbo-generator (simple round rotor). The expressions for the machine with salient poles would be more complicated.

Hence, for the calculations, a detailed information about the machine is required. Far more that one usually finds on the nameplate.
jank

It certainly would be interesting to solve it as characteristics of the machine with the grid assumed to hold a constant frequency/phase.

For large generators (I'm not sure to what extent it applies to smaller), the frequencies of such vibration depend heavily on the grid characteristics and the control characteristics. Google generator subsynchronous oscillation

Thanks for your responses.
The 10 kV grid can be considered as an infinite bus bar system with a very large MVA short-circuit capability (power plant next door).
Jan, regarding your formulas:
a) Can you direct me to an article where these formulas are derived?
b) A quick analysis of your formulas tells me the torsional resonance frequency (or subsynchronous oscillation) should be well below 10 hz, right?
Regards,
Arie Mol

I have got the equations from the textbook from my school years. It is called (like many others): Theory of electric machinery, actually in Check “Teorie elektrickych stroju”. I am suspicious that it would sound all Greek to you. (I do not doubt your technical abilities, I just assume, you do not speek Check).
But basically they are starting with a simple differential equation:

J/p * d^2 (x)/dt^2 + Cx=0

In the steady state the load angle (beta) is constant. If some disturbance appears, there is a moment (or torque) C that tries to return everything to a steady state. Similarly as a mechanical pendulum, the excursions from steady angle beta must be small. It means the x has to be small. Then the differential equation is a linear one. For large disturbances you would have to solve a nonlinear differential equation.
The other equations are equations from the theory of the synchronous machine. And it is just my opinion, that the synchronous machines are much less understood than the asynchronous ones. The main reason is because they are so much less common.
I have never encountered any need to use this theory, up to now. But you are obviously right estimating the frequency below 10 Hz. My book talks about 1 to 3 Hz.
jank

Here is my attempt at a derivation, although it ends up with slightly different expression:

A familiar starting point is that power is given by:

P = U1 * U2 * sin(delta) / XL
where U1 is generator induced votlage, U2 is system voltage, sin(delta) is angle between those two voltages,m and XL is inductive reactance between them (would have to include any transformers in there too).

Then Torque = P / w = U1 * U2 * sin(delta) / (XL*w)

Rotational form of F = MA:
Torque = - J d^2 (delta)/dt
- J d^2 (delta)/dt^2 = U1 * U2 * sin(delta) / (XL*w)

For small angles sin(delta) ~ delta
- J d^2 (delta)/dt^2 = U1 * U2 * (delta) / (XL*w)

0 = J d^2 (delta)/dt^2 + U1 * U2 * (delta) / (XL*w)

If we assume the variation in w is small and treat w as a constant (not strictly true), then it looks just like familiar M / K system:

0 = M * dx^2/dt^2 + kx
we know the solution form wnat = sqrt(k/m)
wnat = sqrt(U1 * U2 / (XL*w*J)

It bears some resemblance to Jan's and I'm sure if we looked closely we could understand the differences. How to actually figure out those constants would be a challenge for me.

Thanks guys for your responses. It cleared the sky a lot.
Jan,
More familiar than Greek! Believe me this is on my bookshelf in my office: cesko-anglicky technicky slovnik
I bought this dictionary on one of my trips to the Czech Republic. In Western Europe a lot of electrical machinery OEM's have closed facilities and today your father's homeland CZ has become an important player.
In the meantime, on my bookshelf I have also found some textbooks that deal with this posting. Funny, today I tend to consult discussion boards like this one first and tend to forget there are excellent textbooks still at hand.
BTW, I guess it will be of interest to you, my problem is this:
A recurrent shear pin collapse in a spacer coupling of a 10 MW turbine - gearbox - alternator drive. I have found a torsional resonance frequency near to 1*RPM. However on one hand, to me this frequency seems too high to point towards a subsynchronous oscillation problem or a steam turbine governor problem. On the other hand, too low to stick it to a mechanical torsional resonant condition (provided submitted coupling stiffness data is correct ?!).
Greetings from NL

Arie,

I suggest measuring torsional vibrations. Perhaps there is a higher torsional mode near 2x line frequency that is causing high dynamic stress. There may be a operator or control system issue that causes high transient torque when generator goes synchronous.

Walt

I see second order DE's. My eyes are tired, or maybe it is just hard to read.

I have measured a low frequency torsional resonance on synchronous electrical machinery. The torsional theory without a torsional spring to ground gives a rigid body resonance for the first torsional, 0 Hz.

The coupling should be computed as a first order Taylor approximation (for linear analysis). d(torque)/d(theta) - torsional spring to ground - torque applied to rotor from stator through electrical stuff. Units like N-m/radian

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In the above I assumed the rest of the story. The (lumped parameter) torsional model consists of shaft inertias (polar mass moment of inertias) and shaft connection torsional stiffnesses (see strength of materials reference or rotordynamic book).

Usually, the model has no connection to earth, unlike radial rotordynamic models. The rotor is free to rotate with no constraints, and this results in the rigid body mode (0 eigenvalue - modeshape = 1,1,1,1,...,1) at 0 Hz.

The electrical stuff connects the rotational degrees of freedom to earth, eliminating the eigenvalue at 0 (The mode is not purely rigid any more.)