Mathematical model of vibrating air pump unit

Based on the laws of classical mechanics, in particular, the law of conservation of momentum, the paper describes the developed mathematical model of signal propagation during vibration diagnostics. At the beginning, the problem of signal propagation was investigated, which was reduced to solving the problem of wave propagation. According to the analysis of experimental results investigated, that the attenuated nature of the signals must be taken into account. For this purpose, a mathematical model has been developed, which allows to solve the problem of the propagation of damped signals. Comparative analysis allows to conclude that the constructed model is adequate.


Introduction
Currently in the Republic of Uzbekistan about 70% of the land is irrigated using centrifugal pumps, the smooth operation of these pumps is necessary for the successful development of agriculture. In the Action Strategy for the further development of the Republic of Uzbekistan for 2017-2021, special attention is paid to the development of land reclamation and irrigation facilities to increase the level of the national economy". The implementation of this task, aimed at improving the accuracy of diagnosing the state of pumping units, becomes important.

Methods
As a rule, when vibrodiagnostics using an accelerometer installed at certain points of the machinery, a signal is recorded in the form of vibration acceleration. This signal, either in the device, which is called the vibrator, or integrated in the computer, is converted into vibration velocity or vibration displacement (Fig. 1). All these three types of signals in vibration diagnostics are considered periodic polyharmonic processes.
The leading direction in vibration diagnostics is the analysis of the spectrum of the vibration signal.
When constructing a mathematical model, we will proceed from the laws of theoretical mechanics, in particular, from the law of conservation of momentum, which leads to the equation of signal propagation in the form of vibration accelerations in integral form [1]. In order to pass from an integral equation to a differential one, suppose that the desired function has second derivatives.
The task of signal propagation is reduced to solving a differential equation of the form:  (2) And initial conditions: To solve the problem using the method of separation of variables. According to this method, equation (1) is represented as: Y (t) -function of variable t only. As a result of substituting equation (4) into equation (1), we obtain the following expression: x X (5) where: λ is a constant, which, for the convenience of subsequent calculations, is taken with a minus sign, without assuming anything about its sign.
From the expression (5) we obtain the differential equations for determining.
Moreover, the boundary and initial conditions are taken as zero. It is determined [2] that when the value of λ1 is equal to where: flourier coefficient for the k-th harmonic Equations (10) can also be represented as where:  -initial phase of the k-th harmonic of the spectrum.
When k = 1 we have Т f Further, by virtue of linearity and homogeneity, we represent equation (1) In particular, from equation (12) one can get equation (1), however, due to the fading nature of signal propagation, the third term of equation (12) cannot be neglected.
In this case, the problem is reduced to solving an ordinary differential equation: To find a unique solution of the differential equation (14) in partial derivatives, it is necessary to determine the initial and boundary conditions. Initial and boundary conditions are called the conditions specified at the initial time t. The boundary conditions are specified for different values of spatial variables.
Equation (14) is solved numerically under initial conditions.
To test the adequacy, we compare the results of solving equation (14) with the data of experimental studies and present them in the form of an accelerogram (Fig. 1.)

Fig. 1. Accelogram amplitude spectrum of vibration on the camera impeller pumps first Karshi
Machine channel.
Using the "Spline interpolation" method, we replace the accelelogram with broken curves (Fig. 2). Further, according to the well-known method [2], we average the values and as a result we obtain the averaged curve (Fig. 3). Based on the analysis and processing of experimental data with a given degree of accuracy, it is possible to assert that equation (14) is valid.
In case of replacement by a differential equation of the first order.
The pattern of signal change is shown in Fig. 4. The calculation of the spectrum module is performed by the formula: where is the frequency of the k-th harmonic ; , In complex form, this value is In the future, the amplitude of the k-th harmonic, calculated by the formula (6), we will denote Y(k).
If the signal is not periodic, then its spectrum is continuous and is determined by the direct Fourier transform: The values of the function are determined by the inverse Fourier transform.

At discrete frequencies
; where: c k is the complex Fourier coefficients (the complex coefficient is equal to the area of the rectangle with base f 1 and height s (k / T)).
Based on the Kotelnikov theorem, if the signal has a frequency-limited spectrum (f≤F b ), then to restore the signal it is enough to know the spectrum at discrete points f k.
Given the discreteness of the representation of the original signal y (i) the desire to represent the spectral density in discrete form, use the discrete Fourier transform (DFT) [3] (with the assumption that the values of y i ) If the cutoff frequency ) 2 /( t N Fc   is significant, then the number of ordinates N at a constant discretization step must also be large, which increases the amount of DFT calculations (the number of calculations is proportional to N2).
In this regard, in practice the methods of fast Fourier transform (FFT) are widely used (4) For FFT ); We introduce such notation:
Actually, formula (28) is valid in the case, if the constant component а0/2 equals zero. In vibration diagnostics vibration, removed from machinery, Filtered at low and high frequencies and set in the range (FH, Fb) FH˃0, so the formula (28) is true for the time function (wave) and coincides in meaning with the formula (19), fair to the spectrum.
In the case when the vibration signal breaks up in time into parts, either when integrating or defining an RMS envelope, you must first remove the constant component The number of terms in the formula (29) is equal to L. In addition to the VHC, in vibration diagnostics is used: а) peak value -the largest absolute value of the maximum deviations of the oscillating quantity. There is a positive peak value and a negative peak value; b) span -the difference between the highest and lowest values of the oscillating value. RMS is the most important indicator, since it takes into account the temporal development of the studied fluctuations, and it directly displays the value associated with the signal energy and, therefore, the destructive power of these oscillations.

Conclusions
Comparison of the obtained results with experimental data allows us to conclude that, that the proposed method adequately determines the process under consideration, signal propagation.