ON DETERMINING THE PROBABILISTIC CHARACTERISTICS OF OUTPUT COORDINATES OF AIRCRAFT STABILIZATION SYSTEMS

A. A. Chapkovich

(Presented by the scientific and technical seminar of the Department of Gyroscopic Instruments and Devices.)

The accuracy requirements for aircraft stabilization systems, especially at the most complex and critical stages of flight, are constantly increasing. Under external random disturbances acting on the stabilization system, it is necessary to determine the probabilistic characteristics of the system’s output coordinates, which characterize the accuracy of its operation. For example, when there is a constraint on the value of a certain output coordinate of the stabilization system, it is necessary to determine the probability that this coordinate will not exceed the permissible value. In solving such questions, it is necessary to know the probabilistic characteristics of the external random actions and the characteristics of the stabilization system itself. In this connection, experimental data obtained during flights are widely used: records of aircraft load factors, flight altitude, values of the aircraft pitch angle, etc.

We shall further assume that a stabilization system of a certain coordinate of an aircraft is considered (load factor, flight altitude, vertical velocity, pitch angle, trajectory inclination angle, etc.) in longitudinal motion under the action of vertical wind gusts caused by atmospheric turbulence. We adopt the following simplifying hypotheses [1]:

a) atmospheric turbulence is a locally stationary and homogeneous random process;

b) the form of the spectral function of turbulence is invariant in shape and changes only in intensity. Then for the spectral density of the velocity of vertical wind gusts we have the following expression:

\[ S_w(\omega)=\frac{\sigma_w^2 L}{\pi V}\cdot \frac{1+3\left(\frac{\omega L}{V}\right)^2} {\left[1+\left(\frac{\omega L}{V}\right)^2\right]^2}, \tag{1} \]

where \(\sigma_w^2\) is the variance of the velocity of vertical wind gusts;
\(L=300\ \text{m}\) is the turbulence scale;
\(V\) is the flight speed of the aircraft;

\(\omega\) is the frequency in \(1/\mathrm{s}\).

The probability distribution of the root-mean-square velocity of vertical wind gusts, based on acceleration records obtained during a large number of flights of an aircraft, is determined by the expression [1]:

\[ f(\sigma_w)=P_1\sqrt{\frac{2}{\pi}}\,\frac{1}{b_1}\exp\left(-\frac{\sigma_w^2}{2b_1^2}\right) +P_2\sqrt{\frac{2}{\pi}}\,\frac{1}{b_2}\exp\left(-\frac{\sigma_w^2}{2b_2^2}\right), \tag{2} \]

where \(P_1, P_2, b_1, b_2\) are coefficients whose values, depending on the flight altitude of the aircraft, are given in Table 1 [1].

The equation describing the mean number of peak values of the coordinates of the stabilization system per unit flight length that reach a specified value has the form [1, 2]:

\[ N(y)=N_0\int_{0}^{\infty} f(\sigma_w)\exp\left\{-\frac{y^2}{2\sigma_w^2A^2}\right\}\,d\sigma_w, \tag{3} \]

Table 1

where \(\sigma_w\) is the root-mean-square value of the velocity of vertical wind gusts;

\(y\) is the magnitude of the coordinate of the stabilization system that is of interest to us;

\(A\) is the root-mean-square value obtained from the spectral function for the quantity \(y\) under the action of a unit value of the root-mean-square velocity of wind gusts;

\(N_0\) is the number of peaks in the characteristic for \(y\) per unit flight length in a turbulent atmosphere.

Substituting expression (2) into (3) and taking the integral [3], we then obtain:

\[ \begin{aligned} N(y)&=N_0\int_{0}^{\infty}\left[ P_1\sqrt{\frac{2}{\pi}}\,\frac{1}{b_1}\exp\left(-\frac{\sigma_w^2}{2b_1^2}\right) +P_2\sqrt{\frac{2}{\pi}}\,\frac{1}{b_2}\exp\left(-\frac{\sigma_w^2}{2b_2^2}\right) \right]\times \\ &\quad \times \exp\left\{-\frac{y^2}{2\sigma_w^2A^2}\right\}\,d\sigma_w = N_0\left\{P_1\exp\left(-\frac{y}{b_1A}\right)+P_2\exp\left(-\frac{y}{b_2A}\right)\right\}. \end{aligned} \tag{4} \]

Using expression (4), one can determine the probability distribution of the clustering of the values of the output coordinates of the stabilization system. Equation (4) should be applied separately to each part of the aircraft flight trajectory, since the parameters \(P_1, P_2, b_1, b_2\) change depending on the flight altitude. Depending on the flight conditions and the characteristics of the stabilization system, the values of \(N_0\) and \(A\) change. For example, in the presence of nonlinearity of the elements of the stabilization system, with a change in the level of the external random disturbance \(\sigma_w^2\), they chan-

the quantity \(y\) and the quantity \(A\) are determined. It should be noted that, with the method considered for determining the probabilistic characteristics of the output coordinates of the stabilization system, it is necessary to obtain experimentally the number of peaks \(N_0\) in the characteristic for \(y\) per unit flight distance in a turbulent atmosphere.

References

1. Progress in Aeronautical Sciences. Volume 3. Edited by A. Ferri, D. Küchemann and L. H. G. Sterne. Pergamon Press, 1962.

2. Lappe U. Oscar. “Low-altitude turbulence model for estimatind dust loads on aircraft,” Journal of Aircraft, vol. 3, No. 11, 1966.

3. I. S. Gradshteyn, I. M. Ryzhik. Tables of Integrals, Sums, Series, and Products. Moscow, Fizmatgiz, 1963.