4. Factors influencing the disc normal mode frequency

Disc temperature. In case of evenly heated disc the frequency reduces with temperature increase, i.e. with elasticity module Е_T decrease. The computational value of disc normal mode frequency at temperature T is determined by the formula

where f_{(S-N) 20}, Е₂₀  are the disc normal mode frequency and elasticity module of a material at the temperature of 20 С, respectively.  

For the unevenly heated disc, the normal mode frequency is difficult to determine analytically. In this case Rayleigh’s method is applied.

So, if the temperatures along disc radius r are distributed under the cubic law

where T₀ is the temperature difference on the external disc outline (on the rim, where r=r_r) and in the centre, then it is possible to evaluate the normal mode frequency of the disc with blades for the vibration form with two nodal diameters (N=2) by the formula:

The lowering of normal mode frequencies due to high temperatures and disc heating unevenness reaches 20…25 %. This influence is more essential for thin large diameter discs at high squeezing stresses in a rim part.

E ngine rotor rotational speed. The rotor wheel angular speed significantly influences the disc normal mode frequency. The radial inertial forces prevent disc bending and increase a normal mode frequency (Fig. 7.5). For highly loaded discs the so-called dynamic normal mode frequency can be doubled. The relation, connecting static and dynamic normal mode frequencies, can be shown by the formula:

where n_s is a rotor rotational speed, r/s; B is a factor dependent on geometry, design, conditions of fastening and disc vibration form.

Fig. 7.5. Influence of rotor rotational speed on disc normal mode frequency

Factor B is determined either experimentally or theoretically by Rayleigh’s energy method. The values of factor B for the flat disc without blades fixed in the centre are shown in Tab. 7.3.

Table 7.3.

Factor B values for the flat disc without blades fixed in the centre

Number of nodal circles, S	Number of nodal diameters, N
	0	1	2	3
0	0	1	2,35	4,05
1	3,30	5,95	8,95	12,30

For turbine discs in operation conditions the angular speed and temperature have the simultaneous effect on normal mode frequency. Therefore at small angular speed the blade radial forces have the main influence, while at higher engine ratings temperature factor does because of disc rim high temperature (Fig. 7.6). As a result, at lower vibration forms the normal mode frequency increases, and at higher forms it decreases.

Influence of disc design features. The normal mode frequency increases if the disc is fixed on the rim and has centre hole strengthened by bead (the disc of disc-drum-type compressor).

Fig. 7.6. Change of disc normal mode frequency  with temperature

The increase in conicity or hyperbolicity of disc body section makes normal mode frequency of disc 1,5…2 times bigger. Broad and massive rim increases normal mode frequency by 10…15 %.

The presence of rotor blades on the disc essentially reduces the normal mode frequency. The normal mod frequency is reduced by 15 % if blades weight makes 25…30 % of disc weight. Moreover, the disc normal mode frequency can not exceed blade normal mode frequency if the blades are fixed on the disc (Fig. 7.7).