RESULTADOS

The distribution of weight and supports along the length of a rotor estab-lishes the static deflections, plus the static bearing loads. For example, consider a constant diameter shaft that is simply supported between two points as shown in Fig. 3-1. This shaft will have a maximum deflection at the midspan, and each support location will carry one half of the shaft weight.

If one support is moved to the rotor midspan, the condition described in Fig.

3-2 will occur. In this case, the maximum deflection occurs at the unsupported end of the shaft. The load applied to each support will now be dependent upon the support characteristics. Specifically, if the left support is a free support, it will not have a static load. Under this condition, the center support will carry the entire weight, and the shaft will be balanced on this center pivot. On the other hand, if the left support is connected to the rotor (e.g., a bearing), it will produce a vertical reaction if the middle support is not perfectly centered. For this reason,

68 Chapter-3

overhung machines such as power turbine rotors for dual shaft gas turbines, or overhung blowers must be carefully examined to determine the static and dynamic bearing loads, and directions.

Next consider the addition of a concentrated load (e.g., an impeller) at the middle of the rotor. The diagram presented in Fig. 3-3 represents the deflection associated with the additional force applied at the midspan of the simply sup-ported shaft. Clearly the center deflection must increase when the additional load is applied. In addition, with the supports located at the shaft ends, it is rea-sonable to conclude that the total weight (shaft plus midspan load) will be equally shared between the two supports.

Fig. 3-4 illustrates the condition of an overhung rotor with the addition of a center weight. In this configuration, one support is located directly below the concentrated midspan load. In this case, the mode shape, and the maximum deflection are identical to Fig. 3-2 with zero external load. However, the force bal-ance has been altered, and the center support must now carry the shaft weight plus the center load.

Fig. 3–1 Simply Supported Shaft With Static Deflection Due To Beam Weight

Fig. 3–2 Overhung Shaft With Static Deflection Due To Beam Weight

Fig. 3–3 Simply Supported Shaft Deflected By Beam Weight And Center Load

Fig. 3–4 Overhung Shaft Deflected By Beam Weight And Center Load

1/2 Weight 1/2 Weight

Uniform Shaft

Maximum Deflection

20

Maximum Deflection Uniform Shaft

20

Maximum Deflection Concentrated

Load

35

Concentrated Load

Maximum Deflection

20

Mass and Support Distribution 69

Next, consider the static mode shapes displayed in Figs. 3-5 and 3-6 that describe the influence of moving the concentrated load from the midspan to the end of the rotor. For the shaft simply supported between bearings (Fig. 3-5), the mode shape returns back to the initial condition (Fig. 3-1). The additional load is directly transmitted to the right hand support. Under this configuration, the deflected mode shape is dependent only on the shaft weight, but the support loads are clearly different.

Finally, for the overhung case of Fig. 3-6, the cantilevered load at the end of the shaft results in an increase in the maximum deflection. This type of behavior certainly makes intuitive sense, and it is representative of real overhung machines. It should also be recognized that the application of this load to the free end of the rotor will result in a downward vertical restraining force at the left end support. Again, this is consistent with the forces and moments encountered in machines such as power turbines and overhung blowers.

Overall, it is recognized that the static support forces (i.e., at the bearings), plus the location and magnitude of the maximum deflection are dependent upon the support characteristics. It is apparent that the addition of elements such as impellers, couplings, balance pistons, and spacers will directly influence the resultant support forces, and the associated maximum static deflection.

Simple mechanical systems can often be modeled as a uniform weight dis-tribution for the shaft, combined with concentrated loads for the impellers. The static deflections can be calculated with beam theory, and the static bearing loads determined by summation of moments. Reference books such as Roark’s Formulas for Stress and Strain¹provide characteristic equations for many typi-cal mechanitypi-cal systems without resorting to detailed beam typi-calculations.

For more complex rotors, it is necessary to divide the rotor into discrete and Fig. 3–5 Simply Supported Shaft

Deflected By Beam Weight And End Load

Fig. 3–6 Overhung Shaft Deflected By Beam Weight And End Load

1 Warren C. Young, Roark’s Formulas for Stress & Strain, Sixth Edition, (New York: McGraw-Hill Book Company, 1989).

Concentrated Load

Maximum Deflection

20

Maximum Deflection Concentrated

Load

50

70 Chapter-3

definable segments. Based upon the dimensions and material density, it is possi-ble to calculate the weight for each station. From this weight distribution, the static loads at both bearings can be computed. This approach identifies the mass distribution along the rotor, plus the resultant bearing forces in a static position.

The weight of each section or portion of a rotor is dependent upon the phys-ical dimensions, plus the density of the shaft material. For example, the hollow circular cylinder depicted in Fig. 3-7 is dimensionally specified by an outer diam-eter D_o, an inner diameter D_i, and an overall length L. The shaft radius to the outer diameter is R_o, and the internal bore radius is identified as R_i. Based on these dimensions, a variety of necessary calculations may be performed. For instance, from plane geometry, the cross sectional area of this annulus is com-puted by subtracting circular areas in the following manner:

(3-1)