INTRODUCTION

An overhung rigid rotor is one that has its balance correction planes located outside the supporting bearings, as shown in Figure 11, and whose rotational speed is well below the first critical speed3.

Cross effect, also known as the correction plane interference , can be defined as the change in the unbalance indication at one correction plane caused by a change in the unbalance at the other correction plane. The overhung rotors can be difficult to balance because of high levels of cross effect. To minimize cross effect, two trial weights, which generate a couple of forces, are added for the third run1. Under identical conditions, the correction plane interference ratios for the new static/couple solution are comparable with the correction plane interference ratios for the effective static/couple solution2.

ANALYSIS

Figure 1 illustrates an overhung rigid rotor as a shaft with a disk of width l and radius r for attaching trial and correction weights rotating in a clockwise direction with an angular speed ω as seen from the right side. The rotor is supported on the near bearing N close to the rotor and on the far bearing F far from the rotor, which are separated by a distance h. The center of gravity G of the rotor is farther away from the right correction plane R and closer to the left correction plane L, which is localized at a distance c from the near bearing N1. This is considered a perfectly balanced rotor.

pair2

Figure 1: Overhung rotor supported on bearings

A condition of dynamic unbalance is created by adding the weights WL and WR to the left and right correction planes, respectively. These two weights generate the two corresponding centrifugal forces, which are proportional to the factor S = (r/g)ω2, where g is the acceleration due to gravity1. Assuming the radii to attach trial and correction weights are equal, that is rL=rR=r, then the factors SL=SR=S2. The unbalance weight in the left correction plane, WL=(WL,aL), as well as the unbalance weight in the right correction plane, WR=(WR,aR), expressed in polar coordinates, are complex vectors that are known as phasors, whose magnitudes WL and WR are the amounts of unbalance weight in the left and right correction planes, respectively, and whose directions are the corresponding angles, aL and aR of the unbalance weights1,2.

pair2

If the two additional unbalance weights, WR and – WR, are added to the left correction plane, the state of unbalance is not altered. The vector combination of the left unbalance weight WL with the right unbalance weight WR in the left correction plane gives the quasi-static unbalance weight WE=(WE,aE) in the left correction plane, whose magnitude WE is the amount of unbalance weight in the left correction plane and whose direction is the angle aE of the unbalance weight in the left correction plane. This can be written as follows1 in Equation 1:

part2

Therefore, the dynamic unbalance produced by the unbalance weights in the left and right correction planes is equivalent to the dynamic unbalance produced by the quasi-static unbalance weight in the left correction plane plus the couple unbalance created by the set of opposite unbalance weights in the left and right correction planes1.

From the definition of the mobilities for the near and far bearings, MN and MF, respectively2, the influence factors are given by the following expressions1 in Equation 2:

part2

The general equations that relate the unbalance weights in the left and right correction planes of the rotor with the vibrations in the near and far bearings are the ones that follow at continuation1 in Equation 3:

part2

Within the common balancing procedure, for the second run, the addition of a trial weight WTL in the left correction plane according to Equation 3 leads to Equation 41:

pair2

And, within the effective balancing procedure, for the third run, the addition of a trial weight –WTR in the left correction plane and of the opposite trial weight WTR in the right correction plane to form a couple according to Equation 3 leads to Equation 51:

part2

The indicated couple unbalance weight in the right correction plane WIR caused by a trial weight WTL in the left correction plane is deduced from Equation 6:

part2

The correction plane interference ratio, using Equation 2, is calculated as a percentage, as follows in Formula 7:

part2

Therefore, the correction plane interference ratios for the effective static/couple solution1 are exactly the same as the correction plane interference ratios for the new static/couple solution2.

It is observed in the second run of the balancing procedure that instead of the addition of the left trial weight WTL in the left correction plane to obtain the effective static/couple solution1, that the quasi-static trial weight WTE is added directly in the left correction plane to obtain the new static/couple solution2.

CONCLUSION

The correction plane interference ratios for the effective static/couple solution and the new static/couple solution are equal. The new solution is more direct than the effective solution, which is advantageous for practical applications.

REFERENCES

  1. Méndez-Adriani, José A., “Considerations on the Field Balancing of the Overhung Rigid Rotors”, The Shock and Vibration Digest, Vol. 37, No. 3, 2005, pp. 179-187.
  2. Méndez-Adriani, José A., “A Detailed Equation Formulation for the Analysis of the Unbalance of an Overhung Rigid Rotor”, Proceedings of the Vibration Institute Training Conference, Jacksonville, Florida, June 19-21, 2013.
  3. Vance, John M., Rotordynamics of Turbomachinery, John Wiley & Sons, New York, 1988, pp.116-120.
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