Operation process of training aircraft Diamond DA 20-C1

probability distribution of a random variable X(tn) depends solely on the probability distribu- tion of one of the random variables X(tn-1). Prop- erties of the Markov process at the moment of tn do not depend on the values that the process as- sumed at moments t1, t2 n-2. The Markov process is therefore fully characterized by the conditional distribution (MIL-STD-882D, 2000): ( , , , ) ( ) | ( ) , F s t x y P X t x X s y s t (2) or the total distribution of random vector (X(s), X(t)) with initial distribution F(s, y) = P[X(s)<y]. called the probability of transition is essential, which is defined for any moment t, state s and for any number of real y and any Borel set B, in the following way: ( , , B, ) ( ) | ( ) P s t y P X t X s y B (3) In the case of the Markov process, the probabil- ity distributions of time in the states must be ex- ponential. The exception is the calculation of asymptotic reliability indicates. In some cases it Operation process of training aircraft Diamond DA 20-C1 207 is also possible to transform the state space in such a way that non-explanatory probability dis- tributions will be replaced by a sequence of ex- ponential distributions (Leski et al., 2009). 2.2. Smoluchowski-Chapman-Kolmogorov equation In practical applications, especially in reliability considerations, the most important part is per- the range 0, ) with the state space S = 0, process are functions of fixed intervals and their graphs are stair line. For a point process Markov probability of transi- tion (Grabski, 2015): ], ) ( | ) ( [ ) , ( i s X j t X P t s pij (4) ,... 2 , 1 , 0 , , j i s t satisfy the relationships: 1 1 1 0 ( , ) ( , ) ( , ), ( ) ij ik kj k p s t p s t p t t s t t (5) known as Smoluchowski-Chapman-Kolmogorov equation. Moreover, for each i (i there is an equation 2009): 0 ( , ) 1 ij j p s t . (6) Introducing functions ij(t) known as process transition intensities or transition rates: ), , ( 1 lim ) ( 0 t t t p t t ij t ij (7) j i j i ,..., 2 , 1 , 0 , , the system of differential equations with variable coefficients is obtained (Niemiro, 2013): i j S j j ij i ij i t P t t P t dt t dP S i ) ( ) ( ) ( ) ( ) ( : i j S j j ij ij t P t t ) ( ) ( ) ( (8) where: Pi(t) the unconditional probability of the pro- cess remaining at time t at state i, ij(t) the transition rate of the process at t from state i to state j. When the Markov process is homogeneous, the transition rate is independent of the time ij(t) = ij = const., j and a system of differen- tial equations with constant coefficients is ob- tained: ( ) : ( ) ( ) i ii i ji j j S i j dd t i S d t d t dt (9) where: di(t) the unconditional probability of the process being in the state i at moment t, di(t) =P {Xt = i} for i S, t T, for which knowledge of initial probabilities for which knowledge of initial probabilities Pi(0), i S is needed. The above system can be written in vector form, as: ( ) ( )T d t t dt D (10) where: D(t) = [d1(t), d2(t) m(t)]mx1 column vector of probabilities of process presence in particular states, 1 12 1 2 21 2 2 1 2 1 1 2 1 m j m j m j m j j m m m mj j M M O M transition rate matrix, m number of sets S (number of process states). The elements of the = [ ij], i, j S transition rate matrix the following probabilistic interpreta- 208 tion may be given: · ( ) ( ) ij ij t o t p t (11) 1 · ( ) ( ) i ii t o t p t (12) where 0 ( ) 0 t o t t . The expected value of the random variable Tij can be interpreted as the average time of stay in the state Si before the state Sj. From equation (11) the estimator of parameter from the sample can be calculated from the formula (Paska, 2005): ( ) 1 1 · [ ] ij i ij ij N k i i i k n n p E T n t (13) ij ij i n p n (14) where: nij number of transitions from the state Si to the state Sj, ni number of outputs from the Si state, ti(k) time of the object being in the Si state for observation number k from the sample. 2.3. Markov process stationary distribution In many practical applications, only the asymp- totic probability values, i.e. D(t) values at t are relevant. If these values are assumed to exist, i.e. the process is ergodic, the differential equa- tions are transformed into algebraic equations. Therefore, for processes with a finite number of states and a non-zero transition intensity matrix, there are stationary (boundary) distribution of states ( : lim ( ) t t D (15) For continuous time, linear equations are solved from the stationary probabilities pi : T T 0 (16) where = [p1 , p2 , r ]1xm stationary dis- tribution (or invariant measure) vector (Kow- alenko et al., 1989). 2.4. Embedded Markov chain At the same time as a given Markov process, the corresponding Markov chain can be considered, called the embedded Markov chain, with the same phase space S and the moments of testing occurring at the moments of state change by the process. If the Markov process is taken into account only at those moments upon which the state of the system changes, then the probability matrix of chain transitions between individual operating states is P = [pij], where: ij ij ii p (17) with ij being the intensity of the Markov pro- cess. The elements of the P matrix should be treated as the probability that a certain transition to state j will occur at the moment of the immediate state change, exit of the process from state i. The most important characteristics of the chain are its unconditional distributions. The uncondi- tional distribution of the Markov chain at mo- ment n can be described as a vector (Taylor & Karlin, 1998): 1 2 1 , , , n n n nm m d d d D (18) where: 1 : 0 1 m nj n nj nj j d P X j j S d d . Using the formula for total probability, it is pos- sible to obtain: } ; | { } { } { 1 1 1 i X j X P i X P j X P n n m i n n m i ij i n j n p d d 1 , ,1 (19) which in the matrix notation leads to the follow- ing relationship between the unconditional dis- tributions of the variables Xn and Xn+1 - Operation process of training aircraft Diamond DA 20-C1 209 Rilola, 2002): 1 n n D D P (20) The stationary distribution of the finite, homoge- neous Markov chain with the transition matrix P is the boundary lim n c n D . It is a vector that * * 1 : 0 1 r i i i i S p p and (Decewicz, 2011): c c . (21) The stationary distribution is the only non-zero solution (Cappe et al., 2005): T T c P I . (22) 3. Operational state model The operation process is the transition of aircraft from one operating state to another. The transi- tion from state to state of the aircraft under oper- ation can be illustrated, by a direct graph or as a zero-one matrix (Rabiner, 1989). An aircraft may be in one of the states in the op- eration process: S1 standby state, S2 pre-flight service state, S3 flight state, S4 interstate service state, S5 after-flight service state, S6 hangar service state. The states S1, S2, S3 and S4 are classified as read- iness states. 3.1. Standby state The operating state in which the aircraft is air- worthy and completely available for the flight task. 3.2. Pre-flight service state The operating state during which pre-flight ser- vice is carried out. There are two types of pre- flight service: pre-flight inspection and thru- flight inspection. Pre-flight inspection is per- formed immediately prior to the first take-off (flight) of the aircraft from its base on a certain day, and is valid for 24 hours. Pre-flight inspec- tion is performed prior to each flight from the base, except for the first flight on that day. Pre- flight service shall be performed on an aircraft that is airworthy and all periodic services, rec- ommendations, bulletins, etc. have been com- pleted on it. 3.3. Flight state The performance of an aeronautical task is the flight state whereby the aircraft performs the scheduled flight. There are performed such flights as flights with passengers on board, train- ing, technical, etc. For the aircraft to be in this condition, it must be airworthy, and pre-flight service must be performed immediately before the flight. After the completion of the flight task, there is an after-flight service on the aircraft. 3.4. Interstate service state The intermittent service is the operating state, when periodic service is carried out, the level of which depends on the operating period (e.g. ACheck) or the number of flight hours (e.g. 2CCheck). Depending on the scope of the periodic service, after its completion: the aircraft may be ready for the flight task, the aircraft can be directed to make a bulletin, recommendations, etc., a failure may be detected, resulting in a diag- nosis or repair, the aircraft may be referred for technical veri- fication for a flight. 3.5. After-flight service state The operating status during which service is per- formed after the last flight of the day if the air- craft lands at home base. After the post-flight service: the aircraft may be ready for the flight task, the aircraft may be directed to periodic service or to the performance of a bulletin, recom- mendations, etc. a detected failure may be resulting in a diag- nosis or repair. 210 3.6. Hangar service state The operating state, when diagnostic technical activities, such as the identification and analysis of failures or eventual repair, are performed on the aircraft. The performance of these operations depends on the technical condition of the aircraft, and the transition to this state may occur when- ever the aircraft is maintained. Depending on the scope of technical activities performed during diagnostics and repair, it may be necessary to re- evaluate the technical condition of the aircraft. 4. Exploitation elements Nowadays, an aircraft structure must be designed in accordance with the assumed flight parame- ters, flight performance, but also in accordance with the operating principles. The concept of exploitation of an aircraft, plane, glider or heli- copter covers a very wide range of issues related to its use, maintenance, supply of spare parts and consumables (fuel, lubricants), repairs, storage and disposal. According to maintenance, repair, operational and diagnostic susceptibility, measures are determined which define the sus- ceptibility of an aircraft. Such vulnerabilities are determined, among others, on the basis of the following theories: flight safety, readiness and reliability. The essential elements of exploitation are the use of the aircraft, the maintenance of the aircraft and the operational investigation of the aircraft. Maintenance concerns not only the technical object itself, which in aviation is the aircraft, but also takes into account the influence of the envi- ronment in which the technical objects are ex- ploited, as well as the humans in the aircraft op- erational subsystem. The maintenance process of technical objects, including aircraft, is influenced by: type, category, class, age of aircraft, environment, operations to maintain the aircraft in readi- ness, the conditions under which the flight is per- formed. Aviation is a transport industry where safety is more important than anywhere else, so a lot of emphasis is given to information flow. Pilots need to be in effective contact with mechanics and avionics (who carry out work in the aircraft maintenance subsystem). Smooth transmission of information about the behaviour of the aircraft in the air, its failures, difficulties in piloting, vari- ances from the accepted standards, is necessary to maintain safety at the appropriate level re- quired. Environmental conditions often require unsteady states which deviate from the accepted aircraft construction load standards. Aircraft maintenance stations, with appropriately quali- fied personnel and a basic set of equipment, are needed for direct pre-flight and post-flight maintenance. This enables the aircraft to be maintained in a constant state of readiness for flight and ensures an adequate level of safety and reliability. 5. Analysis of readiness for Diamond DA 20-C1 aircraft to perform training task The operational process is the transition of air- craft from one operational state to another. At the same time it is a set of events that occur inside the operational state when the aircraft is in this state, and it is a set of physical and chemical events occurring in the aircraft itself, which are independent of human actions. They are partly influenced, e.g. by lubrication to reduce friction, proper maintenance to delay the development and destructive effects of corrosion (Hlinka, 2007). In addition, it is possible that an aircraft could be simultaneously in two or more operational states, for example: a resupply process would take place in parallel with an aircraft maintenance process. The possibility of transition between different states of an exploited aircraft can be represented, for example, by a directed graph or in the form of a zero-one matrix (Ross, 1996). In this chap- ter, the model of operating states will be visual- ised in the form of a graph. Graphs can illustrate the structure and relation- ships of states. An aircraft or an exploitation sys- tem, as mentioned earlier, may simultaneously be in several different exploitation states. The verti- ces of the graph represent the operational states, and the arrows mark possible transitions between states (Kowalski, 2005). Data for the analysis was collected during explo- ration process. On the basis of the collected data, the exploitation graph presented in Figure 1 was developed. The number of flights performed at intervals assumed are presented in Table 1. Operation process of training aircraft Diamond DA 20-C1 211 Table 1. Number of flights performed at intervals assumed Aircraft type Day Week Month Year DA 20-C1 4 27 108 1148 The average residence time of the aircraft in each state are presented in Table 2. Table 2. Average residence time of aircraft in each state No. State Time 1 Standby 6 h 2 Pre-flight service 55 min 3 Flight 90 min 4 Interstate service 22 min 5 After-flight service 35 min 6 Hangar service 8 24 h Figure 1. Directed graph of operating states of Diamond DA 20-C1 aircraft. The system shown in Figure 1 can be described by a system of differential equations: ) ( ) ( ) ( ) ( ) ( 6 61 5 51 1 16 12 1 t P t P t P dt t dP ) ( ) ( ) ( ) ( 1 12 2 26 23 2 t P t P dt t dP ) ( ) ( ) ( ) ( ) ( 4 43 2 23 3 35 34 3 t P t P t P dt t dP ) ( ) ( ) ( ) ( 3 34 4 46 43 4 t P t P dt t dP ) ( ) ( ) ( ) ( 3 35 5 56 51 5 t P t P dt t dP ) ( ) ( ) ( ) ( 2 26 1 16 6 61 6 t P t P t P dt t dP ) ( ) ( 5 56 4 46 t P t P (23) where: P1(t) the probability that the system is in a standby state; P2(t) the probability that the system is in a pre- flight service state; P3(t) the probability that the system is in a flight state; P4(t) the probability that the system is in a in- terstate service state; P5(t) the probability that the system is in an after-flight service state; P6(t) the probability that the system is in a hangar service state. These probabilities can be determined using the program Wolfram Mathematica ( Table 3 presents the data related to the probabil- ity of transitions between particular exploitation states. Table 3. Probability of transition between individual operating states of Diamond DA 20-C1 pij S1 S2 S3 S4 S5 S6 S1 0 0.982 0 0 0 0.018 S2 0 0 0.394 0.602 0 0 S3 0 0 0 0.989 0 0 S4 0 0 0 0 1 0 S5 0.263 0.732 0 0 0 0.005 S6 0.282 0.718 0 0 0 0 Table 4 summarizes the transition rates between particular operating states for the real exploita- tion process. Probability graphs P were generated from the time t in a year perspective. At the initial stage, the tested technical object may be in one of the 212 Table 4. Transition rates between individual operating states of Diamond DA 20-C1 ij S1 S2 S3 S4 S5 S6 S1 -0.075 0.074 0 0 0 0.001 S2 0 -0.346 0.137 0.209 0 0 S3 0 0 -1.823 1.823 0 0 S4 0 0 0 -0.212 0.212 0 S5 0.143 0.399 0 0 -0.545 0.003 S6 0.358 0.910 0 0 0 -1.468 assumed operational states. The probability of staying in a given state as a function of time has been calculated. The stationary distribution c of the finite, ho- mogeneous Markov chain was as follows: 1 6 0.0726; .2731; .1073; 0.2 0 0 0 0.004 7 9 08; .2708; c The stationary distribution of the process was as follows: 1 6 0.2676; .2190; .0165; 0.35 0 0 0 0.0005 74; .1390; Figure 2. Probabilities of being in one of analysed operating states as a function of time, assuming that the initial state was the standby state for Diamond DA 20-C1. Figure 2 shows the probabilities of being in one of the analysed operating states as a function of time, assuming that the initial state was the standby state. On this basis (Mattrand et al., 2011). Analysis of Fatigue Crack Growth under Random Load Sequences Derived from Military In-flight Load, it can be seen that the probability of an aircraft remaining in a standby state at the initial phase is 100% and decreases over time. After about 30 days, this probability reaches a constant level called the limit probability, which is approximately 24%. The probability of an air- craft being in between-flight service state in- creases until an estimated 30 days and has since remained stable at approximately 38%. Proba- bilities for states with lower values are shown in Figure 3. Figure 3. Probabilities of being in one of analysed operating states as a function of time, assuming that the initial state was the standby state for Diamond DA 20-C1 enlarged drawing. The probability of the aircraft being in flight condition is approximately 1.5% and the proba- bility of the aircraft being in hangar service state is negligible. The aircraft is likely to be in the state of maintenance before flight at 21% and the aircraft is likely to be in the state of maintenance after flight at 16%. For the process under study and the initial distri- bution 0 1 6 1,0,0,0,0,0 D , by calculating suc- cessive powers of the P matrix, the calculated dn values can be shown in Figure 4 Figure 4. Probabilities of state observation as a function of chain step. Operation process of training aircraft Diamond DA 20-C1 213 6. Conclusion The Markov's processes, while satisfying appro- priate assumptions, allow to determine the prob- ability in which the analysed technical object is standing. Concluding on the basis of the per- formed analysis of probabilities of staying of the studied technical objects in one of the exploita- tion states, approximate values of probabilities of staying of the aircraft of type Diamond DA-20 in the assumed exploitation states were determined. After about 30 days, the probabilities shown in the figures reach constant levels called limit probabilities. It has been assumed that hangar service operations are performed regularly in accordance with the guidelines described by the manufacturer in the operating instructions. The probability for both aircrafts of being in a hangar service state is negligible. The probability of being in a state of flight result from the more expensive operation of the Diamond aircraft, which is mainly used for night flights. In addi- tion, it is assumed that operations are carried out correctly by appropriately trained personnel. It can be concluded that Diamond aircrafts rarely fail. References Ayyub, B.M. 2011. Vulnerability, Uncertainty, and Risk Analysis, Modeling, and Manage- ment. American Society of Civil Engineers (ASCE). Callus, P. 2003. DEF STAN 00-970 Require- ments for the Design and Airworthiness of Composite Aircraft Structure. Defence, Scienc- es & Technology. Cappe, O., Moulines, E. & Ryden, T. 2005. In- terference in Hidden Markov Models. 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