In particular, the deduction of the direct two-plane static/couple solution is presented according to the convention for phase measurement of the rotating angular scale with the fixed reference mark [3]1. The development of the standard solution for the two-plane balancing of a symmetric rigid rotor using complex vectors and based on the modern method of influence coefficients was done for the phase convention of the fixed angular scale with the rotating reference mark [4]. For the verification, it was necessary to express the direct static/couple solution for the two-plane field balancing of an overhung rigid rotor according to the convention for phase measurement of the fixed angular scale with the rotating reference mark [6].

**Demonstration**

An overhung rotor has its balance correction planes located outside of the supporting bearings as shown in Figure 1.

Figure 1(a) illustrates an overhung rigid rotor as a disk with equal radii r to attach trial and correction weights in the left balancing plane *L* and in the right balancing plane *R*, whose shaft is rotating in the clockwise direction and supported on the near bearing *N* and on the far bearing *F*, which are localized toward the left side as seen by a frontal observer. Figure 1(b) shows the same overhung rigid rotor rotating in the counterclockwise direction, with an interchange of the relative position of planes and bearings now localized toward the right side, as seen by a rear observer.

With reference to Figure 1(a), in the coordinate system associated with the convention of the rotating scale, whose angles are measured in a sense opposite to the rotation of the rotor, the vibration of the near bearing is represented by the vector *N *and the vibration of the far bearing is represented by the vector *F*.

For the second run of the balancing procedure, *N _{2} *is the vibration of the near bearing and

*F*is the vibration of the far bearing when a quasi-static trial weight

_{2}*W*is added to the left plane in the direct solution to produce the effects

_{TEL}*Q*and

For the third run, *N _{3} *is the vibration of the near bearing and

*F*is the vibration of the far bearing when a pair of trial weights,

_{3}*W*& -

_{TR}*W*, forming a couple are added to produce the effects,

_{TR}*vV*and

*V*, given by the following expressions [3, 6]:

The direct static/couple solution, using the convention for phase measurement of the rotating angular scale with the fixed reference mark, gives the quasi-static correction weight in the left balancing plane and the right correction weight to form a couple in the right and left balancing planes, respectively as [6]:

The phasor, *N*=*N ^{eia}_{x}*=(

*N,a*), which is also expressed in polar coordinates, where

_{N}*N*=|

*N*| is the modulus and

*a*is the angle, being

_{N}*e*the base of the natural logarithms and

*i*= √-1 the imaginary unit, as a tensor of rank one over the complex field, is a mathematical object whose magnitude is invariant under coordinate transformations [1].

In the coordinate system associated with the convention of the fixed scale, whose angles are measured in the same sense of the rotation of the rotor, the angle of the complex vector representing the vibration of the near bearing is calculated employing the following conversion formula [1]:

Therefore, in the coordinate system associated with the convention of the fixed scale, the vibration of the near bearing is represented by the complex vector:

Taking the complex conjugate of the preceding equation in this coordinate system, the conjugate complex vector representing the vibration of the near bearing is found for the phase convention of the fixed angular scale with the rotating reference mark as [6]:

Following a similar development for each one of the terms in equation (3), or by expressing equation (3) in this coordinate system and taking the complex conjugate, you have that:

Taking again the complex conjugate of the previous equation, in the new and direct solution, the quasi-static correction weight in the left balancing plane (*L*) and the right correction weight forming a couple in the right and left balancing planes (*R* and *L*), respectively, for the fixed scale convention, are given by [6]:

In the associated coordinate system, *W' _{TEL}* is the quasi-static trial weight in the left plane and

*Wʹ*is the right trial weight.

_{TR}The conjugate of the quasi-static vector factor *V' _{E}* and the conjugate of the couple vector factor

*V'*in this coordinate system are defined as [6]:

_{C}With reference to Figure 1(b) in the coordinate system associated with the convention of the fixed scale, whose angles are measured in the same sense of the rotation of the rotor, the vibration of the near bearing is represented by the vector *N*′ and the vibration of the far bearing is represented by the vector *F*'.

For the second run of the balancing procedure, *N' _{2}* is the vibration of the near bearing and

*F'*is the vibration of the far bearing when a quasi-static trial weight,

_{2}*W'*, is added to the right plane in the direct solution to produce the effects

_{TER}*Q*' and

*q'Q'*, which is expressed as:

At this point, it is necessary to explain that the quasi-static trial weight has to be added to the right plane to avoid a high level of cross effect [2, 5].

For the third run, *N' _{3}* is the vibration of the near bearing and

*F'*is the vibration of the far bearing when a pair of trial weights,

_{3}*W*&

_{TL}*-W*, to form a couple are added to produce the effects,

_{TL}*v'V'*and

*V'*, given by the following expressions:

However, the values would be entered in equations (9), as follows:

The results demonstrate the fact that a different orientation of the overhung rotor does not change the values of the vector factors.

Nevertheless, in the new and direct solution, the quasi-static correction weight in the right balancing plane (*R*) and the left correction weight to form a couple in the left and right balancing planes (*L* and *R*), respectively, for the fixed scale convention are given by:

With reference to Figure 1(a), a comparison of the standard solution, which was obtained by the addition of a left trial weight, *W' _{TL'}*, for the second run and the addition of a right trial weight,

*W'*, for the third run [4], with the static/couple direct solution, which was obtained by the addition of a left quasi-static trial weight,

_{TR'}*W'*, for the second run and the addition of a pair of trial weights,

_{TEL'}*W'*&

_{TR}*-W'*, in the right and left planes (

_{TR'}*R*and

*L*), respectively, to form a couple for the third run [6]; it can be seen that the corresponding effects in such formulas come from the same differences at the same points of measurement

*N*and

*F*. Then, it is possible to utilize the standard solution to find the static/couple direct solution, employing a pair of trial weights to form a couple during the third run, assuming that the left correction weight,

*-W'*, is interpreted as the left quasi-static correction weight,

_{L'}*-W'*, and that in addition to the right correction weight,

_{EL'}*-W'*, an opposite left correction weight of equal magnitude,

_{R'}*W'*, also must be added.

_{R'}**Conclusion**

The invariance of the vector factors in the direct solution for the two-plane field balancing with the orientation of the overhung rigid rotor has been demonstrated for the fixed scale convention.

The correct application of the formulas involved to a practical case also has been established.

The standard solution for the two-plane field balancing of a general rigid rotor developed for the fixed scale convention could be used as the direct static/couple solution for the fixed scale convention to balance an overhung rigid rotor since the effects come from the same points of measurement.

**References**

- Méndez-Adriani, J. A. Dynamic Balancing of Rotative Machinery. EdIT, 2000: 117-118, 273-281 (in Spanish).
- Méndez-Adriani, J. A. "Considerations on the Field Balancing of the Overhung Rigid Rotors." The Shock and Vibration Digest, Vol. 37, No. 3, May 2005: 179-187.
- Méndez-Adriani, J. A. "A Detailed Equation Formulation for the Analysis of the Unbalance of an Overhung Rigid Rotor." Conference Proceedings of the Vibration Institute, Jacksonville, Florida, June 19-21, 2013.
- Méndez-Adriani, J. A. "Rigid Rotor Two-Plane Balancing Solution for the Fixed Scale Convention." Conference Proceedings of the Vibration Institute, Jacksonville, Florida, June 19-21, 2013.
- Méndez-Adriani, J. A. "Comparison between Cross Effects in New and Effective Static/Couple Solutions for Two-Plane Field Balancing of an Overhung Rigid Rotor." Uptime Magazine, February/March, 2014: 50-51.
- Méndez-Adriani, J. A. "Verification of the New Two-Plane Static/Couple Solution for the Field Balancing of the Overhung Rigid Rotor Using the Fixed Scale Convention." ASME Verification and Validation Symposium, Las Vegas, Nevada, May 7-9, 2014: 69 in program, presentation V&V2014-7035.

**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.