Here is another paper which very specifically discusses the relationship between unbalanced voltages and twice line frequency vibration
EFFECT OF MOTOR VOLTAGE UNBALANCE ON MOTOR VIBRATION: TEST AND EVALUATION by Toshiba authors made available on the Toshiba website here:
Toshiba paper said:
This particular motor happened to be tested with a relatively large voltage unbalance, leading to an elevated 7200 RPM (2 times line frequency) vibration level. Once a set of suspects is identified, additional information can be incorporated into the analysis to either eliminate suspects or confirm the inclusion of the suspected cause. Analysis of 2 times line frequency (2f) will traditionally point to a short list of suspects [6]. Most of these potential causes indicate imbalances in the magnetic field of the motor.
Of the remaining suspects [causes for 2LF vib], eccentric air gap, soft foot conditions, loose stator core, interphase and ground faults, unbalanced voltages and loose top covers each have a set of specific tests that can be performed on the motor to either confirm or dismiss a suspected cause….
Motor voltage unbalance is difficult to diagnose without data of the motor voltage condition during running conditions. Data must be collected with the motor in operation and in its installation. Without having the vibration data collected concurrently with data regarding the voltage on each leg of the motor, analysis of vibration caused by voltage unbalance will be difficult to accomplish. By observing and reviewing the data collected it is easier to evaluate the unbalance of the voltages supplied, and make a determination about whether the power supply is a contributing factor. Removal of the motor from service and sending to a repair shop may, in the instance of voltage unbalance, mask the issue. Moving the motor to a different mounting base on a different power supply may produce a vibration unlike that of the end-user instillation, further complicating the analysis.
Section III goes on to explain that unbalanced voltages cause a negative sequence (reverse rotating) component of the field which leads to twice line frequency vibration.
“As the two sequences are opposing one another, this interaction leads to the two times line frequency vibration phenomenon associated with unbalanced voltage operation.”
Section IV provides results of testing motors with various controlled levels of voltage unbalance. Tables 2 through 7 provide results for individual motors. The column headings for each table are: % voltage unbalance, average radial vibration (ips), overall vibration (pu), 2x vibration (pu) where in this case pu is the vibration normalized so that the vibration with no unbalance is 1.0.
You and I know that by today’s typical interpretation of notation “2x” would mean 2x running speed. However I contend that is not the case for these tables, for these tables the heading 2X refers to 2x line frequency. The reason I say this is:
1. Nowhere do they talk about twice running speed in the entire paper.
2. All over the paper they talk about twice line frequency vibration
3. They don’t even break out 1x running speed as a column header, so why on earth would they break out 2x running speed.
With all that said, the results of the testing vary widely among motors. The Table 2 motor shows a strong trend of increasing 2LF with voltage unbalance but some of the others show decreasing trend and yet others show non-monotonic trends. The test data seems very inconclusive to me due to the wide variability, but clearly the authors who prepared the paper believed that unbalanced voltage leads to increased 2LF vibration. There may (must?) be other unknown factors associated with the specific motors or test conditions that explain why the individual motors acted differently.
New subject - axial vibration. I did some thinking about whether unbalanced currents (from unbalanced voltage) could result in axial 2*LF, and I say yes it can. But we have to assume first that the rotor iron is offset axially from the stator iron slightly. That is a condition that we wouldn't notice under balanced conditions because it results in no vibration (see below) but we would notice under unbalanced condition because it results in 2*LF axial force (see further below).
Why does the axially-offset rotor cause no vibration when currents are balanced? The 12 pole stator field It is like having 12 equally spaced magnets (N S N S N S N S N S N S) circulating around the circumference. Each one exerts a vector force on the rotor. The force includes radial and axial components. The radial forces cancel out by symmetry (I'm assuming perfectly centered airgap throughout my discussion). The axial forces (due to rotor off-center axially) do not cancel out. As the poles rotate, the radial components still cancel out but nothing changes in terms of axial force. There is a constant dc axial pull on the rotor, but it doesn’t change over time as the field rotates, so no vibration.
Now what happens when you add unbalanced currents. As mentioned in the Toshiba paper directly above, you can represent the field from the unbalanced currents as a positive sequence (forward rotating) and negative sequence (reverse rotating) set of fields. (it is similar to the principle of sequence analysis of three phase, except we disallow zero sequence because the homopolar flux path is assumed to be high reluctance). So now you have to imagine two sets of twelve magnets each, one set rotating in the forward direction and another smaller set rotating in the reverse direction. When the N magnets of the forward rotating field line up with the N magnets of the reverse rotating field, the fields add and the axial pull is the strongest. When the N magnets of the forward rotating field line with with the S magnets of the reverse rotating field, the fields subtract and the axial pull is the weakest. So there is a varying axial pull which can result in vibration.
What is the frequency? First we need to figure out how fast the poles move. Let's say at t=0 the snapshot of the forward rotating field is located as follows: N pole at 12:00, S pole at 1:00, N pole at 2:00… (12 pole motor). We know from the fundamental relationship among poles, frequency and speed that in one electrical cycle (0.02 seconds) the forward pole that was at 12:00 at time t-0 will move to 2:00 at time =0.02 sec. And (halving that) it means it would only take 0.01 sec to travel from 12:00 to 1:00 position and (halving that again) 0.005 sec to travel from the 12:00 to the 12:30 position.
So let's assume that at t=0 the poles of the forward and reverse sets of fields are lined up for maximum force.
Now let’s start the clock and watch where the poles go and what effect it has on force (12 pole 50hz motor):
[tt]=== Time 0 sec =====
Forward 12:00N / 1:00S / 2:00N / 3:00S/ 4:00N
Reverse 12:00N / 1:00S / 2:00N / 3:00S/ 4:00N
Forward and reverse poles are lined up and therefore add together for MAXIMUM AXIAL FORCE.
==== Time 0.005 sec =====
Forward ******** 12:30N / 1:30S / 2:30N / 3:30S/ 4:30N
Reverse 11:30N / 12:30S / 1:30N / 2:30S / 3:30N
Forward and reverse pole are opposing and therefore subtract for MINIMUM AXIAL FORCE.
=== Time 0.01 sec =====
Forward ***************** 1:00N / 2:00S / 3:00N/ 4:00S / 5:00N
Reverse 11:00N / 12:00S / 1:00N / 2:00S / 3:00N
Forward and reverse poles are lined up and therefore add together for MAXIMUM AXIAL FORCE.
==== ===== =====
[/tt]
The above shows that the period of the axial force (time between maximums) is 0.01 sec which corresponds to a frequency of 100hz or 2*LF (under assumption that rotor iron is not axially centered on stator iron).
I feel pretty good about my above theoretical discussion of axial force...meaning I don't think there's anything controversial/questionable in it. But whether it really explains what's going on in your particular motor, who knows. The Toshiba paper illustrates how widely different motors can act.
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(2B)+(2B)' ?
