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Features of Pipeline Pump Unit Vibrodiagnostics at Cavitation Phenomena Repin A.I., Bosamykin V.A., Kovalsky V.N., Bogatenkov J.A. Contact

The paper discusses peculiarities of pipeline pump unit incipient fault detection under cavitation conditions. The problem is that a spektrum of cavitation chattering occupies a range of 0-10000 Hz and hence it masks off vibration frequency spectral lines, characteristic (as example) for bearing faults [1]. Methods of the Markovian optimal non-linear filtering theory are utilized for detecting informative vibration signals (from defects) on an additive noise background (due to the cavitation)

The diagnostic model establishes correlation of an informative set of the signs with collection of state parameters describing fault of a pipeline pump unit (PPU). So long as useful argument of signal changes during observation, the problem of optimal filtering of a phase-shift keyed pseudo-random signal is put with provision for fluctuating phase j(t), delay t(t) and amplitude A(t).

Such a signal with manipulation of angle (at this angle value, the dispersion of delay estimation has its minimum) is possible to write as it follows [2]:




where t (t) - delay of an accepted pulse train of duration t_u; X (t) - state vector which optimal estimation is to be received;



Taking into account, that the law of phase-shift keying is given by following expression:

we receive

Let's synthesize a system for a pulse train. Then its input is oscillation:

(1)

where n (t) - stochastic process of white noise type with zero mean value and known correlation function. The function f [X(t),t] allows for features of useful signal variations under cavitation condition of a centrifugal pump and accepts two values at keeping track of an argument t [3]:




where < T (t) > = T_o - period of envelope . f [X(t),t]
As the searching of solutions of equations for precise optimal estimation of the state vector and for precise covariance matrix of errors is rather complicated problem, because of abtragung coefficient and function, being defined by the useful signal model, are non-linear functions of X(t), we shall receive approximated algorithms of non-linear filtering at Gauss` approximation of a posteriori probability density.

The prior stochastic differential equations for a phase angle, amplitude and time delay are as follows:


where

The equations of optimal filtering in this case will be written as follows:

where is an estimation of delay; t₀(t) is error of an estimation of delay.
Taking into account, that the input process looks like (1) and argument A (t) has power format, function Q [X (t), t)] is possible to find using the formula:

if   and it becomes equal zero when f [t(t),t] = 0.
Whence

(2)


Bearing in mind that estimations are unbiased, a derivative of Q (X, t), taken in the point X(t)=X(t), where as an estimation of process X (t) it is accepted X (t), is possible to write (for small values of a posteriori dispersions), expression:

Operation of taking a derivative of a discretely - code chain (in particular, the M - series) with respect to delay is substituted by more simple calculations of finite differences. As the mean value of the members of the finite differences is close to zero point due to the properties of M - series, we get


Further , where p = {j, t,A}; q = {j, t,A}
When we receive, with using time averaging, accordingly,




The solution of the last system is




where s²_A,s²_t are the dispersions of errors of filtering at the beginning of a signal pulse.

Utilizing the found values K_{j j}, Kt t, K_AA and taking into account (2), we receive a following differential equations system:

(3)

If time delay is Wiener process (c=0), the system is simplified unsignificantly [2].
For a vector component of the transmitted messages we have

.
At We shall have for -10 and 100 mcs, accordingly , When and gets the same values, the dispersions .
As it is seen from accounts, the quality of filtering of intermittent signal is getting worse when the value T₀.h is getting more.
The intermittent signal mean power reduction and the accumulation of errors in spaces of a signal result in the deterioration of filtering. To avoid it, it is enough to increase the signal power in an impulse at number of times equal the pulse period-to-pulse duration ratio, if the impulse recurrence rate is high; and it is necessary to increase mean power yet, if the impulse recurrence rate is small.
A possible way to implement the diagnostic system is shown at Fig. 1.

Fig 1


During spaces of a signal for the time period the system should store information of the signal parameters, which value estimations were obtained at the moment of the last impulse termination. Time of memory for the delay should be not less than magnitude of . Increasing of duration of the spaces can require system memory enlarging. The Markov's property of input process is otherwise disturbed.
The shift register (SR) with a feedback is introduced at the schema as the code generator. To control a code chain delay it is possible to utilize a command element (CE) and controlled clock signal generator (CSG), which substitute both a booster and an integrator.
The set of equations (3) represents a mathematical model of PPU, supplying vibrodiagnostics at cavitation phenomena.

The References (in Russian):

1. Popkov V.I. and oth . Vibrodiagnostics in Shipbuilding .- St.-P. Sudostroenie. 1989.
2. Tuzov G.I. The Statistical Theory of Recieving of composite signals. - M.: Sov. Radio,1977.
3. Davydov I. T., Repin A.I. Fundamentals of Optimization and Integration of Board Intelligence Systems - M. MAI, 1996.

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