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An overhung rigid rotor is one that has its balance correction planes located outside the supporting bearings, as shown in Figure 1^{1}, and whose rotational speed is well below the first critical speed^{3}.

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 run^{1}. 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 solution^{2}.

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 *N*^{1}. This is considered a perfectly balanced rotor.

A condition of dynamic unbalance is created by adding the weights *W _{L}* and

If the two additional unbalance weights, *W _{R}* and –

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 planes^{1}.

From the definition of the mobilities for the near and far bearings, M^{N} and M^{F}, respectively^{2}, the influence factors are given by the following expressions^{1} in Equation 2:

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 continuation^{1} in Equation 3:

Within the common balancing procedure, for the second run, the addition of a trial weight *W _{TL}* in the left correction plane according to Equation 3 leads to Equation 4

And, within the effective balancing procedure, for the third run, the addition of a trial weight *–W _{TR}* in the left correction plane and of the opposite trial weight

The indicated couple unbalance weight in the right correction plane *W _{IR}* caused by a trial weight

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

Therefore, the correction plane interference ratios for the effective static/couple solution^{1} are exactly the same as the correction plane interference ratios for the new static/couple solution^{2}.

It is observed in the second run of the balancing procedure that instead of the addition of the left trial weight *W _{TL}* in the left correction plane to obtain the effective static/couple solution

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.

- 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. - 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. - Vance, John M.,
*Rotordynamics of Turbomachinery*, John Wiley & Sons, New York, 1988, pp.116-120.

*Jose Alberto Mendez-Adriani, Ing Mec (UCV, 1970), MSE Mech Eng (U of Michigan, 1973), FASP (MIT, 1980), ScD Mech Eng (City U of Los Angeles, California, 1989), QVA (IRD Mechanalysis, 1981), CMDB (Entek IRD, 1997), CTDPB (DSS, 2011); Industrial Practice: in the assembly plant of General Motors of Venezuela, 1976; Professor Emeritus of Mechanical Engineering and Postgraduate Tutor in Theoretical and Applied Mechanics at the Central University of Venezuela.*

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