PhD. Sedelnikov A.V.

Institute for Energy and Transport, Samara State Aerospace University named after Academician SP The Queen (National Research University), Russian Federation

PROBABILISTIC ASSESSMENT OF THE SUCCESSFUL IMPLEMENTATION OF GRAVITATIONAL-SENSITIVITY OF THE EXPERIMENT

In solving the evaluation of the implementation of favorable conditions for certain sensitive processes in space, a simple mechanical analysis of the physical factors that affect these conditions, is often not enough. This can be attributed to a number of random events that contribute to one or other developments in the real world. In practice, the acceptance of the decision on whether the experiments analyzed the most dangerous situation in terms of adverse outcome of the experiment. However, the point estimate is clearly insufficient for understanding the picture of all possible outcomes, because not even the estimated probability of occurrence of this very dangerous situation, not to mention the analysis of other possible outcomes. The aim of this work is the probabilistic analysis of the implementation of favorable conditions.

Without going into the essence of the ongoing processes, a detailed description of which is contained in [1], formalize the task. In the motion space laboratory in orbit periodically included orientation engines that violate favorable conditions, with probability 1, so all experiments are performed in the intervals between the engine. After turning off the engines are excited by the natural oscillations of large elastic elements of the laboratory, which generate a factor that could affect the enabling environment. The intensity of the vibrations depends on what exactly the engine work. In this laboratory disorientation is random, so random and is triggering a specific engine. Factor, which violates the favorable conditions, we represent the stochastic process W(t) with states of continuous and discrete time [2]. Perform an enabling environment. For the i-th realization of this process w_i(t) conditions are considered satisfied if " " t_j w(t_j) £ w_кр (w(t_j)Î w_i(t)). Otherwise, it is considered that the conditions are violated.

In [3] suggested that the conditions present a Markov chain with four states, two of which are absorbing (Fig. 1).

Fig. 1. Markov chain that characterizes the conditions

In Fig.1: s1 - absorbing state with an infinite time disorientation; s2 - a condition in which the conditions are met; s3 - a condition where the conditions are violated; s4 - absorbing state with zero misorientation.

Estimates show that . To evaluate and concretize the problem. Suppose there are n engines, which after the: . A_i event will be what works the i-th engine. In normal mode , because there is no reason to believe that any of the engines is included more often or, conversely, less the remaining [3]. For the problem under abnormal situation has no physical meaning due to the fact that all experiments will fail with probability 1. The situation where both included more than one engine, is excluded.

Then it is obvious:

.

From the normalization condition:

.

Similarly:

,

.

Form the matrix of transition probabilities for the k-th step:

,

considering that the . Next, you should use the recurrence relation for the inhomogeneous Markov chain [4] to determine the probabilities of the states on the m-th step:

,

where – the vector of initial probability distribution, as well . In the present problem can be (0, 1, 0, 0) or (0, 0, 1, 0). For practical purposes it is important to estimate the time spent in s₂ using the formula [5]:

,

where – estimate the residence time in s₂ and – the interval between two successive engagements engines. Often, in practice it is sufficient to consider the case of a homogeneous chain, ignoring the random scatter of the vector engine thrust. Then (1) can be simplified [6]:

.

In this simplification can be used during early design, when the demands for more thrust spread have not been formulated. In this case, possible to consider the circuit of Fig. 1 without absorption, referring to the fact that in practice there have been no actual cases of states s₁ and s₄. Then the Markov chain consisting of s2 and s3, will have the ergodic property. Therefore, to analyze it applies Markov's theorem [6]:

.

Final probabilities p_j can serve as estimates of p_ij starting with m = 5, when the chain a stationary regime.

Reference.

1. Sedelnikov A.V. The problem of microgravity: from awareness to the fractal model. – Moscow: Academy of Sciences. Selected works of the Russian School, 2010.

2. Sedelnikov A.V. Accelerations as a Markov random process / / Review of Appl. and indus. Math., 2011, v.18, №. 1, pp. 142-143.

3. Sedelnikov A.V. Estimating the probability of spacecraft orientation of the "Nika-T" in a passive mode / / Herald ISTU, 2011, № 3 (51), pp. 178-181.

4. Wentzel E.S., Ovcharov L.A. The theory of stochastic processes and its engineering applications. - Moscow: Higher School, 2007.

5. Rozanov, Yu.A. Random processes. - Moscow: Nauka, 1971.

6. Khrushchev, IV, VI Shcherbakov, Levanova DS Fundamentals of mathematical statistics and stochastic processes. - Moscow: Lan, 2009.