Example of rotor critical speed calculation

Determine the critical speed of turbine rotor taking into account gyroscopic bending moment influence.

Input data: disc weight m_d = 1,48 kg; disc polar moment of inertia J_р=0,512 kgm²; disc equatorial moment of inertia J_d =0,325 kgm²; operational rotor rotational speed n=12300 r/min; rotor rotational speed at an idle rating n_i =10400 r/min.

The design scheme and the indispensable sizes are shown in Fig. 8.30. To simplify, without large inaccuracy we obtain l₂=1.

The rotor critical angular velocity is determined with the help of an equation (8.12):

.

Influence coefficients are determined, using Vereshchagin’s rule. For this purpose we segment the shaft and construct the diagrams of bending momenta caused by force Р=1 (at М_G=0) and bending moment М_G =1 (at Р=0).

1. The compiling of bending momenta equations and construction of diagrams.

The bending moment under a support, caused by concentrated singular force P=1 N, is equal to:

The bending moment, caused by a singular gyroscopic moment, on a console shaft segment is constant and equals М_G = 1 Nm.

a b

Fig. 8.30. Example of rotor critical rotational speed determination:

a – scheme of a rotor; b – design scheme and diagrams

2. Determination of influence coefficients by Vereshchagin’s rule.

The sag, caused by a singular moment action, and the turn angle of shaft section, caused by a singular force action, are equal to:

where

The sag, caused by a singular force action, is equal to:

where

The turn angle of shaft section caused by a singular moment action is equal to:

where

3. Determination of critical angular velocity taking into account gyroscopic moment influence.

From equation (8.12) we have:

For direct synchronous precession the disc moment of inertia is equal to:

Then,

Hence

or

As we can see, rotor critical rotational speed is less than rotor operational rotational speed (n=12300 r/min) and less than rotor rotational speed at an idle rating (n_i =10400 r/min). Therefore, the shaft of the given rotor is flexible.

The rotor operational rotational speed margin is equal to:

The rotational speed margin at an idle is equal to:

The rotor critical rotational speed taking into account support elasticity (according to experimental data) n_cr=3750 r/min, i. е. it is 2,18 times smaller, than for a rotor on absolutely rigid supports.

The rotor critical rotational speed disregarding influence of gyroscopic moment and in the conjecture of shaft weightlessness is equal to (by the formula (8.5а)):

where

Thus, gyroscopic moment has caused increase of rotor critical rotational speed by

which has reduced a critical rotational speed margin.

8.8. Measures taken to reduce intensity of rotor oscillation connected with critical rotational speeds

GTE rotating rotor can be affected by various forces, damping oscillations. They include, first of all, friction forces in bearings, friction forces of rotor elements about operating environment, internal friction forces in the shaft material and other. These forces create a moment of resistance to rotation. The redundant turbine torque overcomes it, but it does not influence a shaft sag value. Besides, these forces hinder the shaft precession motion, and, therefore, promote its sag decreasing.

The shaft sag decreasing at operating ratings is also promoted by initial eccentricity decreasing. It is reached by good rotor balancing.

However, high accuracy of rotor initial balancing, i.e. balancing during manufacturing and repair does not solve all problems, since at long operation the unbalance value increases, which affects shaft sags at critical rotational speeds.

Therefore, the various design measures are taken at a design stage of the GTE. They must reduce the shaft sag at a critical rotational speed, and also change the value of critical rotational speed.

So, the critical rotational speed increases with increase of support casings and shaft rigidity at bending at the expense of the shaft diameter and cross-section area increase or intermediate support addition.

The reduction of a critical rotational speed is reached at the expense of shaft rigidity decreasing at bending, and also support elastic yielding increase, for example, by use of elastic rings placed between support casing and fixed bearing race, or at the expense of application of elastic bearing bushings like a “squirrel cage” in a support design (they will be considered in Chapter 11).

The shaft sag reduction at a critical rotational speed is provided with oil or band-oil dampers. The oil dampers in a combination with elastic elements promote safe transition through a critical rotational speed at engine start-up and cut-off.

Such complex of design and technological measures at the GTE rotors and their supports development provides reliable and safe engine operation within the whole range of operational ratings.