## Basics

Before the subject of balancing can be properly discussed, unbalance should be described. An unbalance exists when the mass center is displaced from the rotating center. Figure 9-la shows a massless disk with a finite mass, m, located at radius, r, from the center of rotation. If the disk rotates at an angular velocity of (0, the force, F, exerted by the finite mass, m, is

where

Equation 9.1 is the basic equation for unbalance. For such a simple arrangement, balancing (referred to as static balance) could be done by placing the shaft on knife edges. Initially, the location of the mass would rotate the disk gravitationally until the mass was on the bottom. If

370 Compressors: Selection and Sizing m a

Figure 9-1. A one mass (a) and a two mass (b) model of an unbalance.

weights were added opposite the location of m, until the mass of the weights equaled the mass, m, and were located at the same radius, r, the disk would be motionless, regardless of the position of the angular placement of the disk on the edges.

In Figure 9-lb, two masses are located exactly 180° apart, but on two different weightless disks. The two disks are separated by a distance x. If this rotor were placed on the knife edges, it could be placed in any angular position desired, and it would not rotate. The rotor is now in static balance. The diagram also shows that the forces, when viewed from the end of the rotors, including the summation of forces, appear to be zero. However, if the figure is viewed longitudinally, the vectors, F, are separated by the distance, x. If a summation of moments is made, a moment unbalance is revealed. This type of balance is referred to as a dynamic unbalance. Correcting this type of unbalance requires spinning the rotor as in a balance machine. This method takes the name of dynamic balancing.

The balancing of a rotor certainly has changed from a "black art" with the introduction of more sophisticated equipment. While it is now much more scientific, it certainly is not a job for the novice. For several obvious

Figure 9-1. A one mass (a) and a two mass (b) model of an unbalance.

reasons, the various balance methods will not be detailed. Rather, the effects of balance results are of more interest to the average person working with the compressor.

### Unbalance

Figure 9-2 is a chart showing the residual unbalance and the resulting force in pounds. It can be generated from Equation 9.1 using the unbalance in inch-ounces at several different speeds. The real problem is that there has not been any practical method of relating unbalance to rotor

Figure 9-2. Residual unbalance and the resulting force in pounds. (Courtesy of A-C Compressor Corporation

Figure 9-2. Residual unbalance and the resulting force in pounds. (Courtesy of A-C Compressor Corporation vibration. The more sophisticated damped unbalance response computer programs tend to close the gap, but to someone at a balance machine, these programs are somewhat remote. Also, the accuracy is dependent on the program and the skill of the user. Like the balance machine, it is not simply a matter of reading an answer on a printout.

Since compressors had to be balanced, and though acceptable vibration levels had been determined, the two still elude each other. Another method was devised by the field personnel in charge of keeping the compressors running. By trial and error, the 0.1 g criterion evolved. This sim ply says that if the force due to the residual unbalance is less than 1096 of the rotor weight, the rotor will no longer respond to the unbalance. It' Figure 9-2 is used, the value of 55 inch-ounce yields 1,550 pounds of force at 4,000 rpm. Such a rotor might weigh 15,000 to 20,000 lbs. This would be a very large compressor. To exactly meet the 10% criteria, the rotor would have to weigh 15,500 lbs. To the end that a balancer did not want to use a curve and back-calculate, a more straightforward melhod was developed. Equation 9.2 was used in API Standard 617, fourth edition and earlier standards

For a comparison, the example can be evaluated

This value falls near the .lg limit, which just happens to be the basis for the equation.

Balance machines rate their sensitivity in eccentricity, e, where e is the apparent center of gravity shift due to the unbalance. To obtain e, simply divide the inch-ounce of unbalance by the rotor weight in ounces, in this case, e -- 54.6/(15,500 x 16)

This balance level can be readily achieved.

In the API Standard 617, fifth edition, the allowable value for residual unbalance was changed to

where

U = allowable unbalance, in.-oz.

W = rotor overall weight or journal static weight (if unbalance taken for each end of rotor)

N = operating speed, alternatively maximum continuous speed, rpm

The equation is based on Mil-Std-167 (SHIPS) dated 1954, a Navy specification [2]. The basis for the change was that the allowable residual unbalance for rotors operating under 14,000 rpm was lowered, while it permits higher residual unbalance values for rotors above 14,000 rpm. The balance is more realistically achievable on the balance machine.

If the previous example is applied in Equation 9.3,

The value is lower than the .lg would call for, but most users would prefer to install a compressor rotor balanced with some residual unbalance margin.

For comparison, the eccentricity is calculated for the value from Equation 9.3

Balance machines are limited to the 25 to 50 juin sensitivity level, indicating that Equation 9.3 for the example will take the balance close to the limit.

The international standard ISO 1940 is based on a linear equation similar to Equation 9.3 [3]. The specification calls for balance quality level by Grade Numbers with the lower the number the lower the permissible unbalance value. Equation 9.3 when restated in the ISO terms would require a Grade level of 0.66 or G 0.66 in ISO terms.

Figure 9-3 is a plot of the two equations for residual unbalance and includes two common ISO grades for comparison.

When the sensitivity in microinches in considered, it can be seen why, when a component is mandrel balanced, the required runouts will be very small. If the runout is larger than the balance machine sensitivity, the component could actually be unbalanced. Since mandrels do have runout, the alignment of the balanced component with shaft runout is desirable; however, even here, the differential between the mandrel runout and shaft may be high enough to miss the desired level—a reason for trim balance. It is also probably the best argument for the progressive stacking and balance as recommended by API.

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