KARACHUN V.V.

NATIONAL TECHNICAL UNIVERSITY OF UKRAINE "IGOR SIKORSKY KYIV POLYTECHNIC INSTITUTE"

ZERO DISPLACEMENT OF THE MOVING PART OF THE FLOAT GYROSCOPE IN THE STOCHASTIC SPACE OF PERTURBING FACTORS

External inciting - kinematic and acoustic – are considred as the processes determined in time. Analytical providing of accordance of the of gyroscope parameters with the nameplate requirements in operating terms was satisfied by two differential equations of the first and second approximation:

(1)

. (2)

Thus, the right part of the equation (1) contained the parameters of the fuselage angular motion and acoustic vibration of the float surface in an obvious form, in other words, acoustic vibration of the float surface [1] -

, (3)

or more precisely -

where is a derivative in time from the turning angle about the sensitive axis of the device.

In its turn, the right part of equations (2) contains not only the functions of the aircraft tossing and acoustic vibration of the float surface, but also the equation solution (1). Using the expression, the right part can be written down as follows -

(4)

At the determined character of external inciting factors, task was traced to determination of the stable constituent of the right part of the expression (4), in other words

, 1,2 is a number of approximation and the systematic error of ARS was calculated by the formula -

. (5)

On the contrary, when the aircraft tossing and acoustic vibration of the float surface have a stochastic structure, it should be considered as the expected value of instrumental errors. Thus

. (6)

Assume matematical expectation of values and their derivatives in time equals zero. Mathematical expectation of products of these values, at the same time, can contain stable constituents.

For denotation of mathematical expectation of values ,,,, and so on, the symbols ,,,,, . From now we can use formula (5) and calculate the instrumental errors in the moment of time t. Mathematical expectation of the products can have stable constituents. We will denote them by symbols etc.

It is known from the theory of chances, if , then mathematical expectation of the products of two these functions, calculated in various moments of time, is called the “correlation function of connection” :

If , then this function is called the autocorrelation function of the stochastic process -

At we receive dispersion of the stochastic process :

On condition of stationarity of the process we have -

; .

Dispersion of the stationary process is a stable value. Further on stationarity of process is not obligatory.

So we have:

Then

(7)

Averaging of function can be obtained without efforts, if the given correlation functions of connection, which are present in expression (7) of the combinations,,, і .

We will use the obtained correlation for the calculation of "shift of zero" in the first approximation. For this purpose we take and. In other words, we assume the absence of systematic rotation of the base relatively to the ingoing axis of the device. From here the obvious equalities follow -

(8)

Averaging in time of mathematical expectation of "zero shift" is determined by the formula

(9)

If we use the known correlations

expression (9) can be written down a little different:

(10)

Thus, for determination of "zero shift" in the first approximation, it is enough to know the correlation functions of connection between the pitch angle and resilient radial moving of the float side surface , and also between the yaw angle and radial and tangential moving of the float side surface. In addition, it must be known the correlation function of connection between the yaw angle and bend vibrations of the float’s end under the action of acoustic wave. These functions can be determined experimentally.

If we take up the angles of the aircraft tossing and resilient moving of the float under the action of penetrating acoustic radiation as casual and stationary constrained functions, in other words, as those, for which the mutual correlation functions depend only on the difference of moments of time , then it is possible to write down the following correlations -:

(11)

The "zero shift" will be calculated from (10), using the correlation (11) :

(12)

We will illustrate the proven data. We will calculate the "zero shift", for example, at the following type of correlation functions of the connection -

Then

After a substitution in the formula (12) we have:

. (13)

We will pass to the second approximation. The right part of the equation (11) contains two types of elements – those, which do not contain the decision of equation of the first approximation and, opposite, those, which contain this decision. The function will be written down as -