Manual Statistical Processing of Oscillograms to Obtain Vibration Spectra
Abstract Generated abstract
This paper presents a manual statistical method for deriving vibration spectra from oscillograms of pneumatic hammers when dedicated spectrum analyzers are unavailable. The method divides a vibrogram into repeated curve segments, uses a calibrated template to estimate half-period frequencies and vibration-velocity amplitudes, and tabulates occurrences by frequency and amplitude to compute duration, probability, mathematical expectation, variance, and root-mean-square deviation. An example for a BS3-135V pneumatic hammer shows how the resulting data produce a discrete vibration-velocity spectrum and provide probabilistic measures characterizing the vibration process. The authors report that the obtained spectrum in the frequency range up to 2 kHz corresponds to the relevant 1966 sanitary standards.
Full Text
Proceedings of the Tomsk Order of the October Revolution and Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov
Volume 188 1974
Manual Statistical Processing of Oscillograms to Obtain Vibration Spectra
V. F. Gorbunov, A. I. Pashnin
Instrumental analysis of the vibration spectra of pneumatic hammers can be performed using SICH spectrum analyzers, Bruel & Kjaer instruments, and the FS\(_p\)-10a. However, many production and research organizations do not have these expensive instruments. It is therefore proposed that, for small-scale studies of the vibration of pneumatic hammers, a method of manual statistical processing be used. In addition to making it possible to obtain vibration spectra without the use of scarce and bulky equipment, this method also has other positive qualities. Namely, it makes it possible to obtain criteria for a probabilistic evaluation of spectra: variance, root-mean-square deviation, probability of occurrence of amplitudes, etc.
Let us consider the method of obtaining spectra using vibration velocity as an example. The spectra of displacement or acceleration are obtained analogously, but for these purposes it is necessary to take, respectively, vibrograms of displacement or acceleration.
Figure labels visible in the oscillogram: \(Q = 20\,\mu\), \(P = 50\,\text{mm}\); BCS-135B; \(T_{\text{per}}\).
Fig. 1. Oscillogram of the vibration velocity of a BCS-135B chipping hammer.
The method of manual statistical processing consists of the following. The curve being processed is divided into periods characterized by identical changes in the shape of the curve (Fig. 1); then, using a special template, the frequency of each period and the amplitudes of the vibration velocity are determined. When one component is superimposed on another (period 2), the parameters of both components are determined (positions of template 3 and 4 in Fig. 1). Thus, from the complex form of the curve, there are isolated
principal harmonics. Along the vertical axis of the template, values of velocity amplitudes up to 7 cm/sec are plotted, depending on the recording scale on the vibrogram. Along the horizontal axis, frequency values are marked. The scale of the frequency axis is determined by the selected speed of motion of the photographic film, i.e., by the sweep speed. And since the measurement is made not of a whole period, but of a half-period, when calculating the scale the frequency values must be doubled. Thus, the scale of the frequency axis is determined by the formula
\[ x_i=\frac{V}{2f_i}, \]
where \(V\) is the sweep speed, cm/sec;
\(f_i\) is the frequency, Hz;
\(x_i\) is the distance from the starting point to the mark, cm. The scale of the vibration-velocity amplitude axis is determined entirely by the sensitivity of the transducer and is constructed from its calibration characteristic.
Decoding of the vibration-velocity curve is carried out as follows. The template is positioned so that its vertical line passes through the maximum of the curve perpendicular to the zero line (template position 1 in Fig. 1), and the horizontal line passes through the minimum of the curve for this period parallel to the zero line.
On the frequency scale, at the point of tangency with the curve, we read off the frequency value and obtain in this case \(f_i=45\text{--}47\) Hz. The amplitude is determined from the vertical scale. In this case it is equal to 3.2–3.3 cm/sec. The data obtained are entered into a table, which has a number of rows determined by the frequency range under study, while the number of columns depends on the accuracy with which the velocity amplitudes are obtained. After the vibration velocity has been read from the template, the reading data are entered into the corresponding cell of the table. Next, the half-period of the following oscillation is measured (Fig. 1), template position 2. We obtain a frequency of 145–150 Hz and an amplitude (velocity) of 0.7–0.6 cm/sec, etc. After completion of the measurements of the processed section of the pneumatic-hammer vibrogram, the number of marks in each cell of the table is counted, i.e., the number of oscillation cycles in a narrow frequency range at a constant amplitude. Knowing the number of oscillations \(N_i\) and the frequency \(f_i\), we determine the duration of action of the given frequency
\[ \tau_i=\frac{N_i}{f_i},\ \text{sec}. \]
Next, the probability of occurrence of oscillations of some frequency is determined:
\[ p_i=\frac{\tau_i}{\tau_{\text{cut}}}, \]
where \(\tau_{\text{cut}}\) is the total duration of the processed section of the vibrogram, calculated from the recording scale and the length of the diagram,
\[ \tau_{\text{cut}}=m_t\,l,\ \text{sec}. \]
After determining the probability of occurrence of the frequency, we calculate the mathematical expectation of the velocity amplitude at the given frequency
\[ M(A_{v_i})=\frac{\displaystyle\sum_{i=1}^{n} A_{v_i}p_i}{\displaystyle\sum_{i=1}^{n} p_i}\cdot \text{cm/sec}, \]
Results of manual statistical processing of oscillograms
Table
| \(f_i,\ \mathrm{Hz}\) / \(A_{\nu_i}\) | \multicolumn{2}{c}{0,2—0,4} | \multicolumn{2}{c}{0,5—0,7} | \multicolumn{2}{c}{0,7—0,9} | \multicolumn{2}{c}{1,0—1,2} | \multicolumn{2}{c}{1,3—1,5} | \multicolumn{2}{c}{1,6—1,8} | \multicolumn{2}{c}{1,9—2,2} | \multicolumn{2}{c}{2,3—2,5} | \multicolumn{2}{c}{2,6—3,5} | \(\sum\limits_1^9 P_i\) | \(M(A_{\nu_i})\) |
|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| | \(N_1\) | \(P_1\) | \(N_2\) | \(P_2\) | \(N_3\) | \(P_3\) | \(N_4\) | \(P_4\) | \(N_5\) | \(P_5\) | \(N_6\) | \(P_6\) | \(N_7\) | \(P_7\) | \(N_8\) | \(P_8\) | \(N_9\) | \(P_9\) | | |
| 30—33 | | | | | | | | | | | | | | | | | | | 0,30 | 3,32 |
| 37—42 | | | | | | | | | | | | | | | | | | | 0,41 | 3,48 |
| 46—47 | | | | | | | | | | | 1 | 0,075 | 1 | 0,085 | 1 | 0,085 | 1 | 0,085 | 0,34 | 2,61 |
| 53—65 | | | | | | | | | | | 1 | 0,055 | 1 | 0,055 | 1 | 0,055 | | | 0,165 | 2,30 |
| 90—120 | | | | | 1 | 0,030 | 2 | 0,061 | 1 | 0,031 | | | | | 1 | 0,030 | 1 | 0,030 | 0,182 | 1,66 |
| 180—210 | 1 | 0,014 | | | 3 | 0,043 | | | 1 | 0,014 | | | | | | | | | 0,072 | 0,87 |
| 240—250 | 1 | 0,01 | 2 | 0,01 | 2 | 0,02 | | | | | | | | | | | | | 0,04 | 0,70 |
| 275—300 | 5 | 0,056 | 2 | 0,022 | 1 | 0,011 | | | | | | | | | | | | | 0,089 | 0,67 |
| 370—400 | 8 | 0,08 | 4 | 0,04 | 3 | 0,03 | 1 | 0,01 | | | | | | | | | | | 0,16 | 0,53 |
| 920—930 | 12 | 0,045 | 3 | 0,01 | | | | | | | | | | | | | | | 0,055 | 0,68 |
| 1850—1950 | 29 | 0,003 | | | | | | | | | | | | | | | | | 0,003 | 0,32 |
where \(M(A_{v_i})\) is the mathematical expectation of the vibration-velocity amplitude, cm/sec;
\(A_{v_i}\) are specific (particular) values of the vibration-velocity amplitudes, cm/sec.
Knowing the mathematical expectations of the vibration velocity for the corresponding frequencies, we construct the discrete vibration-velocity spectrum.
To assess the correspondence of the obtained vibration-velocity spectrum to the real process, it is necessary to know the variance and the root-mean-square deviation. The variance (a characteristic of the scattering of the velocity amplitude around the mathematical expectation) is determined by the formula
\[ D(A_{v_i})= \frac{\displaystyle\sum_{i=1}^{n}\left[A_{v_i}-M(A_{v_i})\right]^2 p_i} {\displaystyle\sum_{i=1}^{n} p_i}. \]
The root-mean-square deviation is
\[ \sigma(A_{v_i})=\sqrt{D(A_{v_i})}. \]
Fig. 2. Spectrum of the pneumatic hammer BS3-135V at \(Q=20\) kg, \(p_0=5\) atm
The probability of the occurrence of a frequency, the value of the mathematical expectation, and the root-mean-square deviation fully characterize the acting vibration. For example, in studying the curve (Fig. 1), it turned out that at frequencies of 46–47 Hz there occurred 4 components (harmonics) of oscillation with amplitudes of 1.7 mm, 1.9 mm, 2.5 mm, and 2.8 mm. First we calculate the probability of occurrence of this frequency, if the duration of the regime is \(\tau_{\text{reg}}=0.3\) sec.
\[ \sum_{i=1}^{n} p_i = \frac{\displaystyle\sum_{i=1}^{n}\tau_i}{\tau_{\text{reg}}} = \frac{\left(\frac{1}{46.5}+\frac{1}{46.5}+\frac{1}{46.5}+\frac{1}{46.5}\right)}{0.3} =0.34. \]
Next we find the mathematical expectation of the vibration-velocity amplitude
\[ M(A_{v_i})= \frac{1.9\cdot0.085+1.7\cdot0.085+2.5\cdot0.085+2.8\cdot0.085}{0.34} =2.61\ \text{cm/sec}. \]
The variance is determined by the formula
\[ D(A_{v_i})= \frac{\left[(1.9-2.61)^2\cdot0.085+(2.5-2.61)^2\cdot0.085\right]}{0.34} + \]
\[ +\frac{(1.7-2.61)^2\cdot 0.085+(2.8-2.61)^2\cdot 0.085}{0.34} =0.317\ \text{cm}^2/\text{sec}^2. \]
The value of the root-mean-square deviation is
\[ \sigma(A_{v_i})=\sqrt{0.317}=0.56\ \text{cm}/\text{sec}. \]
The method described above is used in analyzing the vibration of pneumatic hammers, both production models and experimental ones.
The results of processing the oscillogram (Fig. 1) are given in the table. On the basis of the data in the table, the vibration-velocity spectrum of the BS3-135V hammer, shown in Fig. 2, was constructed.
As follows from Fig. 3 and the table, the vibration-velocity spectrum of the BS3-135V hammer in the frequency range up to 2 kHz corresponds to the 1966 sanitary standards.