1. Introduction
Balancing usually is performed in two ways  shop balancing and field balancing. Shop balancing mounts the mechanical parts on a balancing machine and balances each part separately. Field balancing balances the machine in normal operations with the rotor mounted in the bearings. Large machines, such as steam turbines, electric motors, and generator armatures, often require field balancing. Field balancing times average 4 hours and rates range from $60 to $200 per hour. Personal digital assistant (PDA) devices and CompactFlash data acquisition (CF DAQ) cards are ideal for field balancing because you can carry them easily. The PDA device can be used to measure the vibration and make a track record for the process instead of paper works. These features would benefit a lot because the balancing field might be narrow and disordered.
This twoplane balancing demo is built with LabVIEW PDA Module and NI CF6004 CompactFlash Data Acquisition for PDAs. This demo shows how to use PDAs to measure the vibrations of rotating machinery and reduce the vibrations through balancing. You can use the demo VIs in realworld applications with some simple modifications to the code.
Needs to be recreated with 9234
2. Fundamentals of twoplane balancing
There are usually two types of unbalance: static unbalance and dynamic unbalance.
Figure 1
As shown in Fig.1, static unbalance happens when the gravity centre of a rotor shift from the geometrical center of the thin plane. In some cases, you can place the rotor on knife edges, and allow gravity to pull the heavy spot down to the bottom of the assembly to detect static unbalance. The thin planes in real cases can be fans, grinding discs, pulleys, flywheels, gears and so on. They share a common characteristic that the rotors are quite narrow and have no axial swash motion. You can remove static unbalance by adding/removing balancing screws (or sliding weights) or drilling/grinding on the rotor elements.
Figure 2
As shown in Fig.2, the next level of unbalance complexity consists of weight maldistribution in two separate geometric planes. You need to establish twoplane balancing when the rotors are elongated and rigid, such as electric motors, generator armatures, machine spindles, grinding rolls and so on.
Correction of unbalance situations involves characterizing the heavy spot. The heavy spot is the radial location at which the excessive radial mass distribution exists. This heavy spot is always a location that is opposite the location where weight needs to be added. Unfortunately, unless previous balancing information is available, the location of the heavy spot cannot be identified directly. In this case, you can use the Influence Coefficient method for calculation of correction weights as many field balancing solutions do.
The influence coefficient is used to describe how the rotor system responds to the unbalanced weight changes. Once a rotor installs a trial weight to produce a change in vibration amplitude or phase, the influence caused by the trial weight, or the influence coefficients can be calculated. A singleplane balance procedure will produce one balance response coefficient. Multipleplane balancing will produce a number of coefficients depending upon the number of balance planes. Since the initial vibration can be viewed as the response of the rotor to the initial mass unbalance, you can calculate the initial mass unbalance with the initial vibration and the influence coefficients.
The system shown in Fig.3 is a typical twoplane balancing system. The whole system consists of a rotor, two bearings, two accelerometers and one optical tachometer. The accelerometers are mounted on the bearings and used to acquire vibration signals. The tachometer signal is used to calculate the rotating speed and synchronize the vibration signals in each revolution.
Figure 3
Assuming the rotor system is a linear system, the response at each measurement point (the bearing point) is equal to the vector summation of the unbalance response at each plane. This means the weight at plane 1 can contribute to the vibration acquired at bearing 1 as well as the vibration acquired at bearing 2. So does the weight at plane 2. The basic relationship between the vibration response and the unbalance weight at the planes can commonly be expressed by a group of sensitivity vectors. By adding a known calibration weight at a known angular location on the planes, and measuring the vibration response vectors at both bearings, you can determine the sensitivity vectors experimentally. Once you figure out the sensitivity vectors, you can calculate the initial unbalance weight at each plane and make corrections. The details of this method are introduced in the appendix. You can refer to the appendix if you want to know more about this method.
3. Hardware setup
Fig.4 shows the mechanical system of this demo provided by SpectraQuest, Inc. The mechanical system is called Machine Fault Simulator (MFS). It is constructed with special kinds of bearings, rotors with split collar ends, a split bracket bearing housing, multipurpose belt tensioning and gearbox mounting mechanism, and reciprocating system. More Details are shown in Fig.5. Two planes mounted on the shaft are made of aluminum, with two rows of tapped holes at every 20 degrees. You can install steel screws as calibration weight at these holes easily. The small piece of reflective tape on the shaft reflects the laser from the tachometer while the rest part of the shaft does not. When the shaft rotates, the reflected light would form an on/off signal corresponding to each cycle. Two pieces of magnets mount the accelerometers on both bearing housings. These connections are firm enough and easy to perform.
As shown in Fig.5, the accelerometers are mounted on the top of bearing housings. The accelerometers are powered by the SCCACC01 Accelerometer Input Modules, which are installed into a SC2345 SCC Carrier. The signals are buffered by the SCCACC01 module and connected to the Analog Input Pins (AI1 & AI2) of CF6004 DAQ card. The output of the tachometer is connected to AI0 directly, because the tachometer is an active device. All of the three signals are connected to the DAQ card in singleend mode with the GND signal connected to AGND of the SCC Carrier. Refer to Fig.6 for the hardware connections.
The process of this demo can be separated into 5 steps as shown in Fug.7. Before the measurement, you must establish the hardware connections and check if all components, including the tachometer, the accelerometers, and the CF6004 card, work well. You can confirm if everything goes well on the Waveform page of the demo program.
Figure 8
Fig.8 shows the Waveform page. This page consists of three waveforms that are corresponding to the signals from the tachometer, the 1st accelerometer, and the 2nd accelerometer. Tap the Acquire button to start data acquisition. The waveforms are displayed after the software acquires the data. The Speed indicator under the Tachometer graph shows the rotational speed of the rotor. You can use the waveforms to verify if all the signals are connected properly and the rotor is running at a constant speed.
Figure 9
The second step is to calculate the initial vibrations. The Initial page of the program displays the initial vibration of the system under testing as shown in Fig.9. You need to set the average time for the acquiring process in this page. Once the value of average times is set, this value would be used through the whole balancing measurement. Increasing the value of average times improves the accuracy but takes more time to complete the measurement. Tap the Acquire button to start the measuring process. The number beside the Acquire button counts from zero to the specified value of average times. Both the magnitudes and angles of the vibration vectors change every time the counter increases until the value of the counter reaches the specified value of average times.
Figure 10 Figure 11
The third step is to measure the vibrations with a trial weight added to the first plane. The trial weight is added at a known angular position of plane 1 with a known value. The 1st Trial page shown in Fig.10 is slightly different from the Initial page in Fig.9. Upon the vibration vectors, there is a trial calibration weight vector on plane 1. You must fill this vector (Grams at Degree) according to the weight you add to the first plane and where you add it. You also can find a Save button and a Load button on this page. You can use these two buttons to save and load all the environment variables so that you do not have to repeat the measurement if you stop the VI by mistake or any error occurs. You also can use the Save button to save a whole set of measurement results instead of marking them down manually.
The fourth step is to measure the vibrations with a trial weight added to the second plane. The 2nd Trial page of the program is almost the same as the 1st Trial page except that the calibration weight vector and the vibration vectors are assigned to the second plane. You do not need to set the trial weight if you use the same value as filled in the 1st Trial page. The program copies it from the 1st Trial page as the default value. But you still have to specify the phase of the trial weight.
Figure 12
The last two steps are on the Solution page. Once you have finished the measurement in the previous pages, you can switch to this page and tap the Solution button. The solution of balancing appears on the screen. You can add the balancing weights to the rotor system according to the solution. Tap the Acquire button on this page to acquire the vibration vectors after balancing and compare the vibration results before and after balancing.
5. Result
The following tables illustrate the processes and results of several groups of testings. They are all calculated under the rotational speed of 1470 RPM:
Group 1
Step  Operation  Result (Measurement or Calculation) 
Simulate an Unbalance  Add initial weight: 11.5Grams@340º (Plane 1) 
0.01442Grs@95.172º (Plane 1) 0.01437Grs@93.503º (Plane 2) 
Trail on Plane 1  Add trial weight: 4.9Grams@280º (Plane 1) 
0.01463Grs@68.809º (Plane 1) 0.01454Grs@66.551º (Plane 2) 
Trail on Plane 2  Remove trial weight: 4.9Grams@280º (Plane 1) Add trial Weight: 4.9Grams@80º (Plane 2) 
0.01802Grs@108.858º (Plane 1) 0.01875Grs@110.570º (Plane 2) 
Get the solution  Remove trial weight: 4.9Grams@80º (Plane 2) 
Weight should be add: 11.34Grams@197.783º (Plane 1) 1.25Grams@338.316º (Plane 2) 
Balance  Add correction weight: 11.4Grams@200º (Plane 1) 1.6Grams@340º (Plane 2) 
0.00184Grs@85.085º (Plane 1) 0.00137Grs@65.840º (Plane 2) 
Group 2:
Step  Operation  Result (Measurement or Calculation) 
Simulate an Unbalance  Add initial weight: 9.1Grams@280º (Plane 2) 
0.00953Grs@348.473º (Plane 1) 0.01250Grs@344.621º (Plane 2) 
Trail on Plane 1  Add trial weight: 6.5Grams@0.0º (Plane 1) 
0.01343Grs@25.572º (Plane 1) 0.01609Grs@19.127º (Plane 2) 
Trail on Plane 2  Remove trial weight: 6.5Grams@0.0º (Plane 1) Add trial Weight: 6.5Grams@120º (Plane 2) 
0.00361Grs@308.116º (Plane 1) 0.00591Grs@300.910º (Plane 2) 
Get the solution  Remove trial weight: 6.5Grams@120º (Plane 2) 
Weight should be add: 7.02Grams@170.800º (Plane 1) 9.86Grams@50.200º (Plane 2) 
Balance  Add correction weight: 7.1Grams@170º (Plane 1) 9.9Grams@50º (Plane 2) 
0.00131Grs@357.884º (Plane 1) 0.00225Grs@2.975º (Plane 2) 
Group 3:
Step  Operation  Result (Measurement or Calculation) 
Simulate an Unbalance  Add initial weight: 9.9Grams@340º (Plane 1) 9.5Grams@30º (Plane 2) 
0.01940Grs@73.552º (Plane 1) 0.02152Grs@75.120º (Plane 2) 
Trail on Plane 1  Add trial weight: 6.3Grams@160º (Plane 1) 
0.01152Grs@86.596º (Plane 1) 0.01305Grs@88.327º (Plane 2) 
Trail on Plane 2  Remove trial weight: 6.3Grams@160º (Plane 1) Add trial Weight: 6.3Grams@290º (Plane 2) 
0.02183Grs@61.915º (Plane 1) 0.02347Grs@60.733º (Plane 2) 
Get the solution  Remove trial weight: 6.3Grams@290º (Plane 2) 
Weight should be add: 14.69Grams@171.504º (Plane 1) 2.86Grams@278.291º (Plane 2) 
Balance  Add correction weight: 14.8Grams@170º (Plane 1) 2.9Grams@280º (Plane 2) 
0.00343Grs@73.533º (Plane 1) 0.00429Grs@73.709º (Plane 2) 
Comparison:
Group Number

Before Balancing

After Balancing

Group 1

0.01442Grs@95.172º (Plane 1) 0.01437Grs@93.503º (Plane 2) 
0.00184Grs@85.085º (Plane 1) 0.00137Grs@65.840º (Plane 2) 
Group 2

0.00953Grs@348.473º (Plane 1) 0.01250Grs@344.621º (Plane 2) 
0.00131Grs@357.884º (Plane 1) 0.00225Grs@2.975º (Plane 2) 
Group 3

0.01940Grs@73.552º (Plane 1) 0.02152Grs@75.120º (Plane 2) 
0.00343Grs@73.533º (Plane 1) 0.00429Grs@73.709º (Plane 2) 
From the comparison of the initial vibrations and the vibrations after balancing, you can notice that the vibration amplitudes are significantly reduced. That shows the effects of twoplane balancing with PDA
6. Some useful hints
A. If you want to compile the VI yourself instead of using the executable file directly, you need to configure the DAQmx Base Configuration Utility as follows:
Figure 13
Note: The name of the task that you set in the utility must be the same as the one used in the demo VI. The value of Scan Rate must be set to 10000 if you want to compile the VI without modifying it.
B. It is safe to add the initial balance weight at any angular location of the demo machine, because the demo machine has a split bracket bearing housing to prevent components from flying out. But it could become really dangerous doing the same thing in real cases. You must recognize that it is an extremely dangerous practice that can result in serious mechanical damage. In virtually all cases, the weight should be installed to reduce the residual unbalance, and lower the associated vibration amplitude.
7. APPENDIX: TwoPlane Balancing Theory
Assume some kind of weight unbalances exist in the twoplane system, you can use the following traditional twoplane vector equations for the initial unbalance response of a linear mechanical system:
(1)
(2)
where:
Initial Vibration Vector at Bearing 1 (Grs,pp at Degrees)
Initial Vibration Vector at Bearing 2 (Grs,pp at Degrees)
Sensitivity Vector at Bearing 1 to Weight at Plane 1 (Grams/Grs,pp at Degrees)
Sensitivity Vector at Bearing 1 to Weight at Plane 2 (Grams/Grs,pp at Degrees)
Sensitivity Vector at Bearing 2 to Weight at Plane 1 (Grams/Grs,pp at Degrees)
Sensitivity Vector at Bearing 2 to Weight at Plane 2 (Grams/Grs,pp at Degrees)
Mass Unbalance Vector at Plane 1 (Grams at Degrees)
Mass Unbalance Vector at Plane 2 (Grams at Degrees)
In order to calculate the influence coefficients of the system, we must add some trial weight at both planes to acquire vibration vectors under different conditions. For a linear system, the addition (or removal) of a calibration weight W1 at plane 1 should vectorially sum with the existing unbalance U1 to produce the following new pair of vector equations:
(3)
(4)
where:
Vibration Vector at Bearing 1 with Weight W1 at Plane 1 (Grs,pp at Degrees)
Vibration Vector at Bearing 2 with Weight W1 at Plane 1 (Grs,pp at Degrees)
Calibration Weight Vector at Plane 1 (Grams at Degrees)
Removal of the calibration weight at balance plane 1, plus another calibration weight W2 at balance plane 2 produces the following pair of vector equations:
(5)
(6)
where:
Vibration Vector at Bearing 1 with Weight W2 at Plane 2 (Grs,pp at Degrees)
Vibration Vector at Bearing 2 with Weight W2 at Plane 2 (Grs,pp at Degrees)
Calibration Weight Vector at Plane 2 (Grams at Degrees)
The six equations above contain eight known vector quantities: six vibration vectors and two calibration weights. The calculation procedure initially solves for the four unknown balance sensitivity vectors, and finally the two mass unbalance vectors are calculated. The following expression provides a general solution for balance sensitivity vectors for the solution:
(7)
In equation (7), the subscript m specifies the measurement plane, and the subscript p identifies the weight correction plane. Combining the solutions for the four balance sensitivity vectors within the initial equations (1) and (2) yields the following result for mass unbalance at both correction planes:
(8)
(9)
The calculated mass unbalance vectors ( and ) represent the amount of weight that should be used at each balance correction plane. The angles associated with these unbalance vectors represent the angular location of the mass unbalance. Hence, weight can be removed at the calculated angles, or an equivalent weight may be added at the opposite side of the rotor. That is, if weight must be added, the weight addition angle would be equal to the calculated mass unbalance vector angle plus or minus 180º. Although it is generally desirable to remove weight from a rotor during balancing, we only can add balance correction weights to the planes using in this demo. And, calibration weight W2 must be removed from Plane 2 before adding new balance correction weight.
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