Vibration Analysis -23 "Field Balance-2"

Following the previous procedure, we experimented with balancing measurements and corrections using a field balancing kit and an FFT analyzer. This time, we will discuss the process of plotting field balance vectors and correcting the balance using this data. Please also refer to the calculation method using trigonometric functions.

(1) Configuration of the experimental system

The experimental setup is shown in Figure 1.

Figure 1

(2) Operating conditions

The rotor rotation speed is 2480 r/min, and we will correct the imbalance at this speed. The rotor has threads cut every 22.5 degrees so that correction weights can be attached.

For the experiment, a screw was attached to the rotor at position 0 degrees to create an imbalance. This is shown in Figure 2.

Figure 2

(3) Measurement conditions

From the rotor rotation speed of 2480 r/min, the unbalance frequency f (first rotation frequency) is f = 2480 ÷ 60 = 41.3 (Hz).

In the FFT analyzer settings, if we assume a frequency range of 1kHz and 2048 samples, the sample frequency sf and frequency resolution Δf are:

　　　sf＝1000×2.56　　　　（Hz） 　　　

　　　⊿f＝1000÷800＝1.25　（Hz）

The phase resolution Δθ is

⊿θ = 360 (degrees) ÷ (number of samples per rotation) = 360 ÷ (1000 × 2.56 ÷ 41.3) ≈ 6 (degrees)

To improve the accuracy of balance correction, the phase resolution should be within 5 degrees. Based on this guideline, the frequency range would be 2kHz, but since the mounting angle of the correction weights is every 22.5 degrees, we set the range to 1kHz.

(4) Trigger

The reference signal was obtained using an HT-5200 tachometer, and the pulse output from the detected reflection mark was input to channel 1 of the CF-3600 FFT analyzer.

Adjust the trigger level while observing the waveform of Ch1, and set the position to 0. The waveform of Accelerometer captured after triggering will look like Figure 3. The X-axis has been enlarged for easier viewing.

Figure 3

When using field balancing software, the reference signal is input to an external sample terminal, and order analysis is performed, focusing on the first-order rotation component. However, in this experiment, we will trigger using the reference signal on channel 1 and measure the amplitude and phase from the Fourier spectrum of the acceleration waveform on channel 2. Note that triggering is a necessary operation to determine the reference position.

(5) Average of the data

In this experiment, the acceleration waveform is clean, so there is no problem measuring the unbalance without averaging. However, if the waveform contains noise due to a mixture of vibration waveforms from various elements, the measurement will vary. To remove such noise, we perform time-axis averaging. Time-axis averaging is a process that involves overlaying the waveforms and averaging them. The unbalance signal, which is synchronized with rotation, has its waveforms overlapped with each measurement due to the trigger function, but the noise component is asynchronous with rotation and therefore does not overlap, and is thus removed when averaged.

The average number of repetitions is determined based on when the waveform stabilizes. In this case, we used 50 repetitions.

Figure 4 shows the time-axis averaged waveform. Similar to Figure 3, the X-axis is magnified for display.

Figure 4 Average over time axis

(6) Initial Test

We measure the current imbalance. The power spectrum and phase spectrum, averaged over time, are shown in Figure 5 after operation at 2480 r/min.

Figure 5

From this measurement, we can read an acceleration voltage of 1.666mV and a phase difference of -91 degrees at a primary rotation frequency of 41.25Hz. The phase difference represents a waveform that is 91 degrees behind the reference waveform of cos(2π × 41.25Hz). The relationship between polarity and rotation direction is particularly important; a negative phase difference means that the phase is shifted in the opposite direction to the rotation direction (lag), while a positive phase difference means that the phase is shifted in the same direction as the rotation direction (lead). Although amplitude is generally expressed as displacement, it can be converted to displacement by converting the voltage to acceleration from the sensitivity of Accelerometer and then dividing by 2πf squared, and the correction weight can be calculated proportionally, so the conversion to displacement is omitted here.

(7) Test weight

A 2g test weight was attached to the shaft at a 90-degree angle from the reference position. Figure 6 shows the results measured under the same operating conditions as in (6).

Figure 6

From this measurement, we can read an acceleration voltage of 1.388mV and a phase difference of -70 degrees at a rotational primary frequency of 41.25Hz.

(8) Vector construction

The measurement data is summarized in Table 1.

If we convert the table to X and Y coordinates,

Initial test (F ̅)
X: 1.666cos(-91) = -0.0278
Y: 1.666sin(-91) = -1.670

Test weight test (F ̅+T ̅)
X: 1.338cos(-71) = 0.453
Y: 1.338sin(-71) = -1.314

As a side note, the calculations for X and Y above are the same as those for the real and imaginary numbers in a Fourier spectrum, and can be read by displaying the measurement as real and imaginary numbers, as shown in Figure 7.

Figure 7

Figure8showsthe result of plotting this as a vector.

Figure8

In Figure8, the phase difference polarity is left unchanged, so it is reversed from the actual direction of rotation. Note that the direction of rotation is reversed depending on whether you view the rotor from the right or left side.

Now,asshown in Figure 8,T and θcan be measured with a measuring tape and protractor, but they can also be calculated as follows.

  　　　　　T ̅=(F ̅+T ̅ )-F ̅

Therefore,calculating by separating the components into the XandY axes, respectively.

X：　0.453−（-0.0278）＝0.481　　　

Y：（-1.314）−（-1.670）＝0.356

The angle θbetween the test weight and the correction positionis...Since it is a parallelogram, the sum of the interior angles of a parallelogram is 360degrees.

θ= (360−2β)÷2= {360−2×(36.5+91)}÷2=52.5(degrees)

The size of the corrected weightWuis

Wu= (weight of test weight)×(size)÷(size) =2 × 1.670 ÷ 0.598=5.58(g)

Therefore, the imbalance can be corrected by removingthe 2gtest weight and attaching the 5.58g correction weight at a position 52degreesin the direction of rotation from where the correction weight was attached, which corresponds to a 90 + 52=142degree position on the axis.

A problem arose here.As shown in Figure 2, the correction weight cannot be attached at the142-degree position. The attachment positions are either135degrees or157.5 degrees. When we tried attaching 5.58g at the 135-degree position and taking a measurement, the vibration became too strong and dangerous, making operation impossible. Figure 9showsthe positional relationship between the unbalance, the test weight, and the correction weight.

Figure9shows the position of the test weight and the position of the correction weight (angles indicated on the axes).

(9)Calculationof component forces

Let's calculate the force by treating5.58g as a component force of 142degrees, divided into 135degrees and157.5 degrees.

C ̅ = A ̅ + B ̅twist

X：　5.58cos142＝Bcos135＋Ccos157.5 　　

Y：　5.58sin142＝Bcos135＋Ccos157.5

Solving this system of equations

B＝3.90　(g)　　　

C＝1.78　(g)

By addinga 3.9g correction weight at135 degrees on the axisandanother1.77g correction weight at157.5 degrees, it becomes equivalent to adding a5.58g weight at142 degrees. Please refer to Figure 9, which shows the positions A and B of these weights.

(10) ConfirmationTest

Since a 3.9gweight was unavailable, a 3.5gadjustment weight was attached at the 135-degree position, and the verification test was conducted. The datais shown in Figure 10. Note that the test was conducted at a rotational speed of2540, so the primary rotational frequencyis42.5Hz.

Figure10

In this instance, we did not have a suitable weight on hand, sowe only attached a correction weight atone point at 135degrees. However, the amplitude was 0.571mVandthe phase difference was +149, confirming the vibration reduction effect. The position of the imbalance also shifted. When the mounting position of correction weights is fixed, such as for fans, this method is used to calculate the component forces andcorrect the balance by attaching appropriate correction weights totwo locations. Table 2summarizes the measured and calculated values.

Table2

The DS-0227balancing softwareconveniently generates diagramslike the one shown in Figure 9 automatically.

(Excerpt from the email newsletter issuedonDecember15,2005)