PROBLEMS OF ASSESSMENT
THE OPERATIONAL SAFETY OF A COMBAT
AIRCRAFT USING SAFETY SYSTEMS
Szczepanik R., Tomaszek H., Żurek J.
Air Force Institute of Technology, Warszawa, Poland
Abstract: In this study we present an outline of a method of assessment of the aircraft pilot
safety when an ejection seat is used. Two methods are presented. The first method concerns a case when a seat must be used due to failures during flight, which do not occur too often.
The second one applies to cases when a pilot must be ejected as a result of his opponent’s/ enemy’s action (armed conflict) with combat means and the structure of an aircraft is destroyed.
1. Introduction
The assessment of the safety of an aircraft during flight is a complex task. In this study we will discuss only some related problems.
A combat aircraft pilot is rescued in emergency with an emergency feature known as an eject seat. Such a seat allows a pilot to get off an aircraft in a short time and to land with a parachute.
Therefore factors decisive for the pilot safety are as follows:
Reliability of the air craft;
Reliability of the eject seat;
Pilot health and his composure in emergency situations
In addition, in case of his opponent’s action the pilots safety is affected by:
Aircraft sensitivity to destruction by the opponent;
Self-defense capability;
Opponent’s action in order to destroy the aircraft in a combat;
Quality of the air traffic organization (operational dynamics) and crew skills. Let’s examine the pilot’s safety provided the following conditions are fulfilled:
A possibility of destroying an aircraft on an airfield – airbase (therefore an eject seat as well) may be passed over;
An aircraft is destroyed by an enemy during its combat mission;
The probability of an aircraft destruction during one mission is equal to
Q
and the eject seat operational capability is preserved (a seat can be used); The use of a seat is determined by the moment when a severe failure of the aircraft occurs or by the aircraft destruction due to enemy action;
The seat durability is equal to that of the aircraft.
The leading role in the assessment of the pilot’s safety is that of the life of an aircraft determined by the number of flights or the flight time up to the destruction of an aircraft during peace or warfare. The life of an eject seat can be measured on the basis of the number of successful flight missions of the aircraft up to the moment of its destruction. This study deals with a variant where failures enforcing the use of an eject seat are a rare occurrence and a second variant (applying to combat missions) when such a limitation can be passed over.
2. Outline of the method of determining the risk of grave failures
necessitating the use of an eject seat
Since in the process of operation of aircrafts a failure is a rare occurrence, in order to be able to describe such failures quantitatively, some assumptions are required. We will assume that the probability of a failure event within a given time interval is in direct proportion to this time interval and the number of operated aircrafts where a failure can occur. In other words, one can assume the risk of a failure is in direct proportion to the joint flight time of a fleet of aircrafts over this time interval.
Having stated as above we may formulate as follows: ) ( 0 ) ( ) , ( 1 t t t N t t t P (1) where: ) , ( 1 t t t
P - Probability of failure event within a time interval of t )
(t
N - Number of operated aircrafts where a failure under discussion can occur;
- Factor of proportionality;
t
- Assumed time interval of aircraft operation (or number of flight hours). Let’s designate the probability of n failures within the (0, t) time interval as Pn(t). On the basis of the Poisson process postulates, we can formulate a system of equations as follows: P0'(t)N(t)P0(t) ) ( ) ( ) ( ) ( ) ( 1 ' t N t P t N t P t Pn n n (2)
Initial conditions for the system of equations (2) are the following:
1 ) 0 ( 0 P 0 ) 0 ( n P for n > 0 (3)
The equations (2) are linear differential equations and are solved recurrently. First we have to find theP0(t). Once we know the P0(t), we determine next theP1(t), and so on and so forth.
The solution of the system of equations (2) will take a form:
dt t N t e t P ) ( 0 0( )
(4) dt L N t n t ne
dt t N n t P ) ( 0 0 ) ( ! 1 ) ( The probability of occurrence of n failures over the (0,t) time interval requiring catapulting is described by the Poisson distribution, where the role of the expression „t”
due to the assumed small number of failures during operation of aircrafts is replaced with the t N(t)dt
0
value.
For the (4) relationships the t N(t)dt
0
integral can be replaced with the following sum:
i N i t t dt t N 1 0 ) ( where:
N - Number of aircrafts operated over in a given year;
i
t
- Flight time of an aircraft in a given year.We are often interested not in the probability itself that during a given number of flight hours n failures will occur, but in the value of the coefficient which characterizes the intensity of failure occurrences.
For a single aircraft, the probability of a failure over one year of aircraft operation is:
t
e
q1 (5)
where
:
q - probability of a failure occurrence for one aircraft;
Since the value for failures of that type is small, the etcan be developed into a progressive series.
Therefore:
t
et 1 (6)
By substitution of (6) into (5) we obtain:
q t (7)
Consequently with the aid of the relation (7) one is able to assess the probability of a failure occurrence for a single aircraft when a pilot must catapult himself.
Assuming that the duration of a single flight of an aircraft is about one hour, the risk of a failure during such a single flight can be assessed by the parameter .
To determine the estimator of the parameter, we will use the method of greatest credibility.
Let’s assume we have observed and recorded failure generation / occurrences (when catapulting was necessary) over several disjointed time intervals for which the flight time of aircrafts was t1, t2, ..., ti.
As a result of this observation we have obtained:
over the (0, t1) interval there were n1 failures;
over the (t1, t2) interval there were n2 failures;
over the (ti-1, ti) interval there were n1 failures.
The probability of an occurrence of the mentioned number of failures during aircraft operation i.e. n1+ n2+...+in at the intensity of their generation equal to can be expressed
by the following relation:
1 2 2 2 2 1 1 1 ! ) ( ! ) ( ! ) ( T i i n i T n T i n e n T e n T e n T L ) ... 2 1 ( 2 1 2 2 1 1 ... 2 1 ! ... ! ! ... i T T T i i n i n n i n n n e n n n T T T
(8) where: 1 i i i t t T
The above-noted probability discussed as a function of the variable at fixed
i
i T T T
n n
n1, 2,..., , 1, 2,... is called credibility. Now we will find such a value for for which the credibility L has the greatest value. To achieve that, we will find the logarithm of the relation (8), and calculate the derivative in relation to and next compare it to zero. Thus, having solved the obtained equation, we find the relation for . Therefore:
i i T T T n n n ... ... ˆ 2 1 2 1 (9)
With the relation (9) we determine the estimator by the method of greatest credibility. The relation (7) takes then the form:
t
qˆˆ (10)
where:
t - number of aircraft flight hours in a year.
The relation (10) makes it possible to estimate the probability of a failure occurrence for a single aircraft over a year of its operation.
The probability of a non-occurrence of a failure for a single aircraft will be:
t
R1ˆ (11)
To estimate the average number of failures for the operated fleet of aircrafts within a year we may use the relation:
i N i n n t t P n n E 1 1 ˆ ) ( (12) where :ti - flight time of the “i” quantity of aircrafts N - number of flown aircrafts.
3. Outline of the method for assessment of the eject seat life by using
differential equations
This problem consists in determining a distribution of successful flights till the time point an eject seat was used within an assumed time interval or within the entire period of aircraft life under warfare conditions.
Having determined the distribution of successful flights, we will try to determine the factors of interest. It is assumed that the probability of aircraft destruction during one flight (with a possibility of catapulting preserved) is equal to
Q
. The process of the increase in the number of successive flights will be considered as a function of the flight time of an aircraft or the number of flights of an aircraft to perform a combat mission.Let Uz,t be a denomination of the probability that for the t flight time the number of
successful flights is z. Then, with these adopted designations, we can formulate the following differential equation:
t z t t z Q U U , (1 ) 1, (13) where: 1 ) 1 ( Q Q
z - random variable of the number of successful flights of an aircraft;
t - flight time of an aircraft.
The equation (13) as a function notation takes the following form: ) , 1 ( ) 1 ( ) , (z t t Q u z t u (14) where:
u(z,t) - density function of the number of successful flights at the moment t.
We will transform the differential equation (14) into a partial differential equation. From the equation (14) after applying the Taylor formula for (n = 2) and (n = 1) we can obtain a partial differential equation.
2 2 ) , ( 2 1 ) , ( ) , ( ) , ( z t z u a z t z u b t z u c t t z u (15) where: t Q c (16) t Q b(1 )1 (17) t Q a 2 1 ) 1 ( (18)
The solution of the equation (15) is the function:
u(z,t)ce-ctu(z,t) (19) where: at bt z e t a t z u 2 2 ) ( 2 1 ) , (
The expected value of the number of successful flights of an aircraft over the entire period of its life will be:
Q Q t Q t Q c b dt dz t z u z z E[ ]t ( , ) 1 (1 ) 1 (1 ) 0 (20)
The expected value of the number of successful flights of an aircraft over a finite time interval: ] 1 [ 1 ) , ( ] [ 1 0 Qk Qk t t e Qke Q Q dt dz t z u z z E (21) where: t t
k1 - number of flights of an aircraft.
If k, the formula (21) takes a form of the formula (20).
The probability of the number of successful flights being lower or equal to z1 at a time
point with the flight time equal to t1 , and with the possibility of aircraft destruction, is
expressed as: dt dz t z u t P z t z ( ) ( , ) 1 1 0 1 ) 1 ( 1 (22)
The probability of the number of successful flights being greater than z1 at a time point
when the flight time is t1 with the possibility of aircraft destruction is expressed as:
dt dz t z u t P z t z ( ) ( , ) 1 1 0 1 ) 2 ( 1 (23)
The probability that within the flight time of (0,t1) an aircraft will not be destroyed will
amount to: 1 1 0 1 ) 2 ( 1 1 ) 1 ( 1 3 ) 1 ( 1 ( ( ) ( ( )) 1 ct ct t z z t P t P t ce dt c P
We may formulate the dependence (31) as:
Qk t t Q e e k P(3)( ) 1 (24) where :
k - number of flights of an aircraft.
The probability that within the flight time of (0, k) an aircraft will be destroyed is therefore now:
Qk
e k
Q( )1 (25)
4. Estimation of the pilot’s safety
In the event of a possible loss of an aircraft, a pilot has to save his life by catapulting himself, i.e. using an eject seat.
The following factors are decisive for a successful use of an eject seat:
catapulting operation / process;
conditions under which a catapulting took place;
aircraft and eject seat type;
landing;
behavior and pilots skill during catapulting.
In real situations, the time to make a decision about catapulting is mostly very short. In addition, under such circumstances a pilot considers and attempts to avert a danger. Making a decision about catapulting is affected by the “human factor” and other factors determining the psychical state of mind of the catapulting pilots.
In case a decision about catapulting is made, there is a need to take several actions affecting the process of catapulting. These actions are performed more or less automatically. Their unfailing performance is essential for a successful catapulting. The reliability of catapulting may be expressed as follows:
i n i P R 1 (26) where:
Pi - probability of a correct performance of actions and n is the number of actions
required for a positive result.
The reliability of catapulting depends on the type of an aircraft and eject seat. Catapulting is not always successful; to a great extent it depends on conditions under which it was performed. The probabilistic notation of the pilot safety can vary; it can be more precise or less precise.
In this paper, we will confine ourselves to a notation taking into consideration the reliability of an aircraft and eject seat.
The reliability of an aircraft and eject seat may be regarded as separate sets of events, i.e.:
1 ) ( ) (t Q t R (27) 1 ) ( ) (
Q
R (28) where: ) (tR - aircraft reliability within the flight time (0, t);
) (t
Q - unreliability, i.e. the probability of an occurrence of at threat to pilot life over the flight time interval of (0, t);
) (
R - reliability of an eject seat during its use; )
(
Q - unreliability of an eject seat during its use;
- duration of catapulting.
The dependence (27) applies to an aircraft and (28) to an eject seat. With the dependences (27) and (28) we may determine the pilot’s safety within the flight time range (0, t) using the relation:
) ( ) ( ) (t Q t R R Rt (29)
On the other hand, the probability of an occurrence of a threat to pilot’s life during flights is: ) ( ) (t Q Q Qt (30)
The relations (27) and (28) can take different forms. In our case (taking into consideration the results of the points 2 and 3), these relations will take the following form:
in the event failures during operation of aircrafts are a rare occurrence:
i n i t t t P R 1 ˆ ) ˆ 1 ( (31) ) 1 ( ˆ 1 i n i t t P Q (32)
in the event of an enemy action (during armed conflict):
i n i Qk Qk k e e P R 1 ) 1 ( (33) (1 )(1 ) 1 i n i Qk k e P Q (34)
The description presented under “2” shows that for a case when during flights failures occur very seldom, the risk of an occurrence of such a failure (necessitating the use of an eject seat) during a single flight will be at a level of the
ˆ
parameter.5. Examples in figures and final conclusions
If there is a need to calculate the pilot’s safety during a single flight with available figures tabulated in the Table 1.
Table 1. Year of
operation Data
1 2 3 4
Aircraft flight time
[hours] 12.000 9.000 10.000 11.000 Qty. of failures during flight
which require catapulting 2 0 1 1
With the formulation (10) we calculate:
000 . 42 4 000 . 11 000 . 10 000 . 9 000 . 12 1 1 0 2 ˆ
00009523 , 0 10 523 , 9 5
In order to assess the pilot’s safety during a single flight we use the formulation (39):
n i i t t t P R 1 ˆ ) ˆ 1 (
for t = 1 [where] and 0,9 1 i n i P we obtain: R(10,00009523)0,0000950,9 99999903 , 0 0000855 , 0 9999048 , 0
From the presented outline for the assessment of the crew’s (pilot’s) safety, it is evident that the problem is a complex one. This thesis does not describe all the problems related to the aircraft crew’s safety. A further study is necessary to solve this problem. Is seems also advisable to verify the presented method which bases on real data obtained from operation of aircrafts.
As measure of the safety during operation of aircrafts, also indicators determined with the aid of the relations (12, 20, 21) can serve.
References
1. Borgoń J.: Reliability and Safety of the System Pilot – Aircraft, Informator ITWL, 1987, 269/87.
2. Skomra A., Tomaszek H.: The Method of Effectiveness Assessment of a Combat
Aircraft - ZEM, Zeszyt (Brochure) 3 1996 s. 355 – 367.
3. Szajnar S., Tomaszek H.: The Life of Eject Seat and Crew Safety of Combat Aircraft
under Conditions of Enemy Action, ZEM, 1999.
4. Szajnar S., Tomaszek H.: Problems of determining indicators of Catapulted Pilot
Seat and Combat Aircraft Crew Safety under Warfare Conditions, ZEM. Zeszyt
(Brochure) 3. 2001.
5. Szajnar W., Wojtkowiak H.: Problems of Aircraft Crew Safety in Emergency
Situation,- Warszawa 1999. Wydawca (Publisher): BIL-GRAF.
