Frequency Analysis from the Basics (37) - "Balancing Measurement"

Last time, we talked about "tracking analysis," a commonly used method for measuring the sound and vibration of rotating bodies. This time, we will discuss "balancing measurement," which is important for vibration control of rotating bodies.

It is generally said that approximately 40% of malfunctions in rotating machinery equipment such as power plants are caused by vibration, and about 30% of those vibrations are due to unbalancing. Therefore, balancing correction is an extremely important vibration control method during the installation and maintenance of rotating machinery. Here, we will briefly explain the principles and usage of field balancing techniques that can be performed on-site for rotating bodies.

F=Meω²

Here, F: centrifugal force (N)
M: Rotor mass (kg) e: Center of gravity
ω: angular frequency (rad/s) Let n be the rotational speed (r/min)

             f:振動数(Hz)

Figure 1: Imbalance due to centrifugal force of rotor rotation

As shown in Figure 1, imbalance occurs when the rotor's center of rotation (O in the figure) and center of gravity (G in the figure) are misaligned, generating centrifugal force and causing abnormal vibrations. The frequency of these vibrations is the frequency of the first-order component of rotation. As can be seen from equation (1) in Figure 1, the centrifugal force increases in proportion to the square of the angular frequency (or rotational speed), so balancing correction is very important at high rotational speeds.
The following describes the basic procedure for balancing with a single rotor.

1. Determine the amplitude and phase in the initial state, and let them be F.
   F = (A ₀, θ ₀)
2. Next, a test weight of mass mt is attached to an arbitrary point on the rotor (with the angle set to 0°), and the amplitude and phase of the state at that time are determined and defined as F+T.
   F + T = (A ₁, θ ₁)
3. Next, we will use vector calculations to determine the amplitude and phase of the test weight.
   T = (A ₂, θ ₂)
4. The angle of the correction point is calculated from the initial vector F and the vector T at the test weight, the test weight is removed, and the correction weight mc is attached.
   Here,The correction point angle is calculated using the point where the test weight was attached (angle 0°) as the reference point.
   (θ ₀ −0 ₂)±180°
5. Check the effect of adding the corrective weights.
   Here, the amplitude and phase are determined from those of the sine wave corresponding to the frequency of the first rotation component.

Thus, balancing measurements require accurately determining the amplitude and phase (from the reference point) of the rotational first-order component vibration waveform simultaneously. An FFT analyzer is used to measure the amplitude and phase of the rotational first-order component sine wave. Specifically, a "Fourier spectrum order ratio analysis" is performed using the reference position signal as a trigger signal.

In an FFT analyzer, the phase of the Fourier spectrum is determined with respect to the cosine wave. If we define the oscillation waveform of the first-order rotation component as x(t)=Acos(ωt+φ), then in Figure 4, although the amplitude A is the same, the phase of the cosine wave is 0° and the phase of the sine wave is -90°.

Order ratio analysis is an analytical technique in which the horizontal axis (frequency axis) of the Fourier spectrum is not the usual frequency, but rather the order based on the first rotation. By performing sampling synchronized with the reference signal (one pulse per rotation), it is possible to reduce errors due to the time window of the FFT and fix the first-order component to a constant number of spectral lines regardless of the rotation speed.

Now, let's try performing a specific balancing measurement of a rotating body using the DS-0227A field balancing software for the DS-3000 series.
For simplicity, we will perform the test with one surface, one condition, and one speed. The connection between the rotating body and the DS-3000 measurement system is as shown in Figure 5. For the sensors, a contact-type accelerometer mounted on the bearing and an optical rotation detector are used as reference position sensors.

Figure 6 shows the settings screen. The following measurement conditions will be used.

1. 1 surface 1 condition Ch1: Acceleration PU
2. Maximum analysis order: 50th order Phase resolution: Approx. 2.8°
3. Number of divisions 16/360° 22.5° division
4. Remove the test weight.
5. Rotation speed mode 1 speed (here, 2400 r/min)
6. Average number of times: 32

We will perform the measurements according to the procedure described above.

1. We measure the initial unbalance. ==> Initial test (Figure 7)
   Measurement result: Rotation speed 2400 r/min
   Acceleration amplitude 1.662 m/s ²
   Phase -74.7°
   
2. Next, a test weight with a mass of 2.1g was attached to the position at an angle of 0°, and the state at that time was
   Amplitude and phase are measured. ==> Trial test (Figure 8)
   Measurement result: Rotation speed 2400 r/min
   Acceleration amplitude 1.46 m/s ²
   Phase -89.6°Figure 8 Trial Test
3. Based on the above measurement results, we calculate the position of the balancing correction point and the correction weight.
   Results: Figure 9
   Taking the point where the test weight is attached as 0°, attach a 7.7g correction weight at a point 56.2° in the opposite direction of rotation.
   However, since 56.2° is not an angle division point, corrective weights are added to the following two points to match the angle division point.
   4.0 g for No. 2 (45°)
   3.9 g for No. 3 (67.5°)
   Figure 9 Balancing correction points
4. Remove the test weight, attach the corrected weight, take measurements, and check the balancing results. ==> Verification test (Figure 10)
   Here, we used approximately 5 g for No. 2 and approximately 2 g for No. 3.
   Measurement result: Rotational speed 2400 r/min
   Acceleration amplitude 0.180 m/s ²
   Phase -108°
   The amount of imbalance (acceleration amplitude) ^decreased from the initial value ^of 1.66 m/s² to 0.18 m/s², indicating an improvement of approximately 1/10.
   Figure 10 Confirmation Test

In conclusion, here's a summary.

1. Balancing correction is a very important vibration control method when installing or maintaining rotating machinery.
2. Unbalance is a phenomenon in which the rotor's center of rotation and center of gravity are misaligned, resulting in centrifugal force and abnormal vibrations. The frequency of these vibrations is the frequency of the first-order component of the rotation.
3. The field balancing method involves first conducting an initial test, then a trial test with test weights attached, and calculating the location of the correction points and the value of the correction weights based on the results. After that, a confirmation test is performed.
4. Balancing measurements require accurately determining both the amplitude and phase (relative to a reference point) of the rotational first-order vibration waveform simultaneously.
5. To measure the amplitude and phase of a first-order rotating sine wave, an FFT analyzer is used, and a "Fourier spectrum order ratio analysis" is performed using a reference position signal as a trigger signal.

【keyword】

Balancing, unbalancing, imbalance, field balancing, centrifugal force, angular frequency, rotational speed, vibration frequency, rotational first-order component, test weight, correction weight, reference position sensor, trigger signal, Fourier spectrum, order ratio analysis, time window, initial test, trial test, confirmation test

【reference】

Written by Shuichi Maki
"Practical Aspects of Equipment Diagnosis Using Vibration Methods," Japan Plant Maintenance Association (1985)
Edited by Kenichi Kido
"FFT Analyzer User Manual," Japan Plant Maintenance Association (1984)
(Hima)

(Excerpt from the email newsletter issued on March 22, 2018)