VIEWS: 2 PAGES: 28 POSTED ON: 12/26/2012 Public Domain
Aviation Session 5. Aviation EFFECT OF LITHUANIA’S ENTRY TO THE EUROPEAN UNION ON THE NATIONAL AIR TRANSPORT Jonas Butkevičius Vilnius Gediminas Technical University Plytinės g. 27, Vilnius, LT-10105, Lithuania Phone: (850)2745099. E-mail:Jovas.Butkevicius@ti.vgtu.lt The entry of Lithuania into European Union has changed the situation in macroeconomics. It has influenced the changing situation of competition and trade including air transport. In this article the liberation of air transport market, development of air fleet, increasing coverage of flight, enlargement the geography of routes, development of charter flight, change of airports and air navigation system‘s work, consolidation in deference of unsure aviacompanies, change in opinion of specialists, influence to attract new aviacompanies, and consolidation competition is analysed. Keywords: air transport, market, liberation, flight park, passenger transportation 1. INTRODUCTION After May 1, 2004, when Lithuania became the Member State of the European Union (EU), the macroeconomic environment of the country has changed [1]. This caused changes in competition and business situation of the entire national economy, including transport sector as well [2]. Therefore it is important to identify the effect of joining the EU on Lithuanian transport system. This article analyses how Lithuania‘s entry to the EU has affected the national air transport. 2. LIBERALISATION OF AVIATION MARKET On April 1, 1997 all restrictions on entering the local market (cabotage) were repealed in the European Union. From that time the air transport markets of EU Member States are considered to be totally liberalised. Principal features of the liberalisation are as follows: − open market enabling flights between any airport within the European Union; − common licensing standards for carriers in all EU countries; − free rating of tariffs; − equal rules applied for regulation of operation of regular and charter airlines. After liberalisation of air transport market airlines received equal conditions for fair competition in the EU market, without restrictions of frequency of flights, costs and route network. Consequently, in the struggle for attracting the clients it is not only necessary to be competent in the field of tariffs, but also to regard such parameters as the frequency of flights, convenience of arrival and departure slots, quality of services supplied to passengers and the network of routes. After joining the EU, Lithuanian airlines are also directly influenced by the liberalisation of aviation market. The impact has produced positive and negative effects. Among positive effects the fact that national airlines can open their flights to any airport of the EU without bilateral agreements, without parity evidence, and without other bureaucratic restrictions can be mentioned. The geography of flights is constantly expanding thus influencing the growth of passenger flows and the increase of profits. Among other important effects of market liberalisation on national airlines are the following: – EU carriers are also able to choose airports of our country without limits, therefore new companies, such as “Air Baltic”, entered our market, and this boosted the competition for Lithuanian companies; – the increased competition reduced the costs of services, particularly because of the entry of “cheap flights” airlines, thus causing the decrease of airlines‘ profits. 210 The 7th International Conference “RELIABILITY and STATISTICS in TRANSPORTATION and COMMUNICATION - 2007” Liberalisation of airline companies has a positive influence on Lithuanian citizens’ travelling by air, as the ticket prices go down, and the geography of flights from national airports has expanded, and consequently airlines are forced to improve the quality of services, etc. Liberalisation of aviation market also has a positive influence on national airports because the handling of the airports and the increasing flows of passengers bring additional profits to these airports. 3. DEVELOPMENT OF AIRCRAFT FLEET A positive result of Lithuania’s entry to the EU is the fact that airlines has to renew their aircraft according to EU standards, and this influences directly the quality of services delivered to passengers, improves the environmental and flight safety. For airlines the purchase (or leasing) of new aircraft is related to considerable additional expenses. Lithuanian aircraft fleet has most intensely changed by the airline companies “Lietuvos avialinijos” (“Lithuanian Airlines”, LAL, present name: AB “flyLAL-Lithuanian Airlines”) and “Aurela”. “Lithuanian Airlines” have disposed of 2 airplanes Boeing 737-200, which do not meet the EU standards of noise emission. “Aurela” has disposed of 2 airplanes JAK-42 because of the same reasons. In 2006 “Lithuanian Airlines ” operated 5 airplanes of Boeing 737-500 type and 4 airplanes of SAAB-2000 type (in 2003 it owned 3 of the type) the noise emission of which meets the EU standards of noise emission. “Aurela” disposed of Soviet-type airplanes JAK-42 and purchased 1 airplane of Boeing-737-300 and 1 airplane of HS 125-700 B type. The airplane Boeing 737-700 owned by “Aurela” makes charter flights to non-EU countries. The airline “Aviavilsa” will have to change 2 airplanes of AN-26B type for the shipment of DHL because of the lack of certificate in line with EASA PART 145. 4. GROWTH OF PASSENGER AMOUNT The amount of serviced passengers in national airports have been increasing constantly – if in 2000 services are delivered to 580.043 thousand passengers, so in 2001 the number grows up to 649.968 thousand passengers (growth rate 12.1%), and in 2002 – 700.853 thousand passengers (growth rate 7.8%), in 2003 – 788.248 thousand passengers (growth rate 12.5%). Particularly extensive growth starts after Lithuania’s entry to the EU: in the year 2004 the amount of passengers reached 1112.916 thousand passengers (growth rate 41.2%), and in 2005 the amount even reached 1453.222 thousand (growth rate 30.6%). If compared with the years 2003 and 2005, the amounts of passengers increase actually by 84.4%. The main reason of such a growth of passenger amounts is the fact that Lithuania joined the EU, and this enabled travelling to EU countries without visas, also the extended geography and amount of flights, etc. In 2005 the largest amount of passengers is as follows to and fro the United Kingdom – 15.3%, Germany – 12.8%, Denmark – 12.3% and Ireland – 6.6%. 5. INCREASE OF FLIGHTS In 2005 from Lithuanian airports 39282 flights has been made by 16% more than in 2004. In 2005 from Vilnius airport 29193 flights has been made (i. e. by 23% more than in 2004), from Kaunas airport 4611 flights has been made (decreased by 5%) and from Palanga airport 5478 flights has been made (growth rate 4%). The growth rate of flights is caused by the fact that Lithuania has joined the EU. From 2003 to 2005 total quantity of flights from Lithuanian airports increases by 41.1%, of which from Vilnius airport grew by 59.2%, from Kaunas airport increases by 13.1%, from Palanga airport increases by 3.7 %. During the year 2005 Lithuanian airlines have carried 674.3 thousand passengers – it is by 14% more than during the year 2004. This is also the result of Lithuania‘s entry to the European Union. 6. DEVELOPMENT OF THE GEOGRAPHY OF FLIGHTS The development of new flight routes in air transport is an important fact of Lithuania‘s entry to the EU. 211 Session 5. Aviation When Lithuania joined the EU, the market became liberalised, and airlines received an opportunity to choose flight routes by themselves. In 2004, from Vilnius airport new destination flights have been opened to six towns: Milan (LAL), Dublin (LAL and “Air Baltic”), Cologne (“Air Baltic”), Hamburg (“Air Baltic”), Munich (“Air Baltic”) and Oslo (“Air Baltic”). Competing with other airlines, the company “Lietuvos avialinijos” (LAL, present name: AB “flyLAL-Lithuanian Airlines”) is prepared to open 11 new flight routes through the airport of Palanga: to Kaliningrad, Latvia, Germany and Scandinavia. Furthermore, the LAL airlines are prepared to open new flight routes from Vilnius to Dubrovnik in Croatia, Thessaloniki in Greece, Istanbul and Antalya in Turkey. “Air Baltic” is also planning the development of flights from Vilnius airport. 7. DEVELOPMENT OF CHARTER FLIGHTS Former system of bilateral agreements prevents airlines from free development of transportation for meeting the growing demands for travelling by air transport. It is particularly obvious in the market of charter flights where the demand is rapidly growing, and the capacities of foreign airlines to make flights from Lithuania are narrow. The market of charter flights is liberalised by Lithuania joining the EU. This stimulated the opening of new charter routes and entry of new airlines to the market of charter flights. In 1995 the passenger flows by charter flights make 11.2 % of the total amount of air transport passengers. In 2000 passengers of charter flights make up to 12.2%, and in 2004 – 16.6%. In 2004 Lithuanian airlines, as well as those from abroad (Egypt, Spain, Tunisia, Ukraine), make 2998 charter flights and carry 166983 passengers. Besides, 1986 passengers are carried by local flights. In 2004, from Vilnius airport two new airlines started their charter flights – “Joanos avialinijos” and “Gintarinės avialinijos”. New direct charter flights to Spanish resorts in Gran Canaria, Tenerife and Palma de Mallorca were started in 2004. 8. “CHEAP FLIGHTS” AIRLINES ENTERING THE NATIONAL MARKET Lithuania‘s entry to the EU and liberalisation of aviation market make favourable conditions for airlines with cheap flights to come into the national market. Thus it causes a strong competition to national airlines. In 2005 “cheap flights” airlines make 24% of inland regular flights within Europe. They deliver services for 900 flight routes linking more than 200 towns. They own over 250 airplanes and carriy almost 60 million passengers per year (i.e. approximately 16% of the inland passengers of Europe). As mentioned before, the entry of “cheap flights” airlines into the national market caused a strong competition for national airline companies. In 2004 “Air Baltic” carried 96817 passengers that made 9.7% of passengers of Vilnius airport, and LAL carried 430417 passengers that made 46.3%. However forecasts of Vilnius International Airport show that in 2008 “Air Baltic” will already carry 535.2 thousand passengers, and LAL (present name: AB “flyLAL-Lithuanian Airlines”) will carry 696.6 thousand passengers. Thus “Air Baltic” will obtain even bigger share of the market and will pose even stronger competition to LAL. 9. CHANGES IN THE AIR NAVIGATION SYSTEM Air navigation services in Lithuania are delivered by State Enterprise “Oro navigacija”. It should be noted that after the entry to the EU the amount of air navigation services grew considerably in Lithuania: if in 2003 – 86805 flight services were delivered, so in 2004 their amounts grew by 25.4% and made 108875 flights. Furthermore, in 2005 the amount of flight services delivered increased yet by 12.2% and made 122209 deliveries of flights. In the period of 2003-2005 the amount of serviced flights grew even by 40.8%. In 2005 – 68% airplanes flew by transit, during the year the amount of transit flights grew by 9.2%. Regarding the SE “Oro navigacija”, principal positive effects of Lithuania‘s entry to the EU are the following: – disappearance of customs duties in purchasing of equipment from EU countries; 212 The 7th International Conference “RELIABILITY and STATISTICS in TRANSPORTATION and COMMUNICATION - 2007” – rise of factors enabling lower prices for the delivery of services; – emerging opportunities for using financial support from EU Structural funds. An important change concerning the SE “Oro navigacija” is the fact that the implementation of the “Single European Sky” concept foresees the implementation of functional air space components regarding the procedural airplane demands, regardless of national borders of countries. 10. CHANGES IN THE ATTITUDE TOWARDS THE STAFF An important result of Lithuania‘s entry to the EU is the factor that business stakeholders have to consider the level of labour force – this is underlined in the questionnaires and interviews of all specialists employed in civil aviation. The problem of pilots is one of the most important subjects. In Lithuania pilots are qualified in line with EU standards. Licences issued by CAA are recognised in all EU countries. Before the Lithuania‘s EU membership here was a surplus in pilots. However, after joining the EU, a certain proportion of pilots have been lured away by foreign (especially by Latvian) airlines, where the wages are more considerable. So at present a lack of pilots is already palpable. The same can be said about the lack of aviation engineers. On January 1, 2006 there are 2302 specialists of civil aviation in Lithuania, they are holders of 2991 issued licences, from which: 201 licences of airlines’ transport pilot, 423 licences – commercial aviation pilot, 70 licences – flight engineer, 20 licences – navigator, 134 licences – air traffic manager. Concerning the migration of pilots national airlines are forced to raise the wages for pilots and high rank specialists. This, in turn, raises the costs of services of airlines and reduces their profits. From the viewpoint of the staff this is a positive result as the wages are growing, thus corresponding to the professional standards of the staff. 11. IMPROVEMENT OF LITHUANIA‘S NATIONAL IMAGE Great advantage of EU membership for Lithuania is a changed attitude to Lithuania as a country, whereas the EU membership is a factor guaranteeing that each EU Member State is a reliable country. Therefore the fact of joining the EU and the improvement of the national image brings the following positive results to the air transport sector: – rating of national airlines has improved, thus making easier negotiations on slots, aircraft handling in airports of other countries, supply of spare parts for aircraft, etc. – pilots trained in Lithuania have more opportunities of working in airlines of other countries; – EU membership guarantees a higher standard of civil aviation, therefore the flows of tourists increase in the national airports, and foreign airlines are more encouraged to select flights through Lithuanian airports, etc.; – different contracts with foreign countries in the field of civil aviation are easier to make; – no customs duties required for purchasing equipment in the EU countries; – other EU countries get more information about Lithuania, our country has become more attractive and more reliable to other EU countries, etc. All the above mentioned factors are basing on the improved national image of Lithuania. This applies not only to the air transport, but to other modes of transport as well. CONCLUSIONS 1. EU membership enables Lithuanian airlines to open flights without any restrictions to all eligible airports of the EU. On the other hand, new airline companies have entered Lithuanian market, thus boosting the competition for national airlines. 2. A positive factor is that national airlines has to renovate their aircraft fleet in line with EU standards. 3. EU membership enabled non-visa entrance to EU countries, therefore the amount of passengers in Lithuanian airports has grown considerably; if compared with the period of 2003 and 2005, the growth rate reached even 84.4%. 213 Session 5. Aviation 4. The proportions of flights from Lithuanian airports have increased significantly, and the geography of flights has expanded distinctly. 5. Proportions of charter flights from Lithuanian airports have increased as well. 6. After Lithuania’s entry to the EU “cheap-flights” airlines come to the national market and cause strong competition to Lithuanian airline companies. 7. Joining the EU resulted in the increased proportions of air navigation services in Lithuania. 8. An important effect of joining the EU is the necessity for business stakeholders to consider the level of labour force. 9. EU membership improves the positive image of Lithuania as a State. References 1. Butkevičius, J. The effect of Lithuania‘s entry into the European Union on the national transport system and the transport system development: manuscript of monograph. Vilnius: Technika, pp. 35-118. 2. Butkevičius, J. Development of passenger transportation by railway from Lithuania to European States, Transport – 2007, Vol. XXII, No 2, 2007, pp. 73-79. 214 The 7th International Conference “RELIABILITY and STATISTICS in TRANSPORTATION and COMMUNICATION - 2007” MANAGING AND CONTROL OF AIRCRAFT POWER PLANT USING ARTIFICIAL NEURAL NETWORKS Eugene Kopytov, Vladimir Labendik, Sergey Yunusov, Alexey Tarasov Transport and Telecommunication Institute Lomonosova 1, Riga, LV-1019, Latvia Phone: (+371)7100590. Fax: (+371)7100660. E-mail: tsi@tsi.lv The tasks of neural networks’ application in the airplane power plants automatic diagnostic systems are considered as well as their peculiarities and advantages. Keywords: aircraft power plant, technical maintenance, status condition, diagnostics system, neural network 1. INTRODUCTION The modern air engine is a complicated technical object that embodies the most progressive technologies of science and engineering. The provision of needed economic characteristics of the air engine power plant is possible by the rising the level of thermodynamic cycle operation parameters, that mostly have an influence on the air engine’s resource and reliability. During the gas turbine engines operation the various tasks of aircraft power plant vital activities guaranteeing and airplane systems and energy plants functioning are solved in the onboard computer. Opportune and qualitative aircraft engine parameters’ monitoring and diagnostics make it possible to realize the exploitation of the engine based on its status condition. In spite of the big number of various gas turbine engines diagnostics and monitoring methods, none of them is completely universal one. It can be explained mostly by a very high complication of the air engine: there are many parameters, links, the processes are non-linear, there are many modes of operation, etc. All it assumes the need of complex methods’ application for the solving the air-engine’s parameters’ control and diagnostics tasks. The promising method of air engine managing and conditions’ control is the neural networks’ application [1-3] that grants the solving of the wide spectrum of tasks. 2. THE NEURAL NETWORKS APPLICATIONS The main tasks that can be solved by neural networks application are the following: the identification of gas turbine engine mathematical models; the classification of aircraft power plant modes of operation; the control and diagnostics of aircraft power-plant status condition; the analysis of trends of aircraft power-plant status condition parameters, measured onboard, aimed for forecast of these parameters variation depending on the life length. The usage of gas turbine engines mathematical models is needed for many research and practical tasks solving, for example, for the synthesis and analysis of the power-plant managing during the various regimes of flight. The ideal mathematical model (MM) must satisfy the following mutually contradictory requirements: to describe adequately the connections between parameters and the processes in gas turbine engine; to provide the given precision of the parameters’ calculation; to be convenient for usage in calculations and modelling; to be adaptable (learnable) for the individual copy of the engine, etc. In practice usually the set of mathematical models of different complexity is used. Each model satisfies the part of requirements given above and has different areas of application. 215 Session 5. Aviation The most precise and complicated models are non-linear gas turbine engine components’ models, but they are used for the solving of research tasks and optimization of the characteristics of power-plants control system. Simplified non-linear models are widely spread, for example, regression models as well as the models with variable coefficients. But even using this approach it’s not always possible to receive the universal models because of their different representation forms and the need of models’ coefficients correction. Classical multidimensional functions approximation methods do not allow to realize simple mechanisms of the mathematical models structure choosing. The realization of classical interpolation methods on the base of spline-functions requires the considerable computing resources. In such a situation as a rule the providing of the calculations in the real-time is problematic. For the air engine throttle control model the altitude-velocity and throttle performances are described by the authors using the similar modes [4]. Also in the work [4] the simple and precious mathematical model is developed. Today in the works about the power plants management systems designing the approach of neural network usage for the mathematical model identification is proposed. The main idea of such approach usage is in the process of network learning that means the adjustment of the big number of coefficients (synaptic weights between neurons). The building of the neural network models is based on the standard procedures of the neural network structure and their learning methods. Multilayered neural network organization makes it possible to carry out the parallel calculations that support the solving of the characteristics approximation task in the real time [3]. Thermo-gas parameters of the aircraft engine (temperature, pressure, air consumption, etc.) in the different sections of the gas-air flow duct as well as mode and exit parameters (the rotor rotating frequencies, throttle, fuel consumption, etc.) are the carriers of information about the aircraft power plant condition. That’s why they can be used for the definite classes of the aircraft engine conditions (properly functioning, non-properly functioning) recognizing with the aid of different mathematical models of gas turbine engines. Using the power plant parameters trends the classes of the air-engine can be recognized by the uncovering the correlations between measured and calculated using mathematical model parameters. Statistical characteristics of the controlled parameters registration results (also caused by the appearing and developing of the defects in the aircraft power plant) make it possible to forecast the changes of the aircraft power plant condition changes during the exploitation process [5]. In the case when the number of the measurable parameters is not sufficient for the linear mathematical model development the neural network can be used as the model for the standard and defected engine’s model. The analysis of the given parameters’ deviations in the time is carried out by the calculation of the metrical distance between standard engine’s data basing on the neural network and data received during the exploitation of the engine. The results of the quantitative modelling are evidence of complex monitoring and exploitation management possibility of the aircraft power plant using new neural network technologies. Such methods extend and expand “classical” methods that can raise the reliability of management and trustworthiness of the parameters control and aircraft engine diagnostics as well as the decision making processes efficiency on-board while “critical” conditions are detected. 3. ON THE NEURAL NETWORK’S TYPE CHOOSING The mathematical model of the aircraft engine power plant is the model that generalizes many local models, for example, models of the flow path of the air engine. The relationship between elementary models of the physical processes can be described by the graph that makes possible to formalize the model integrity and coherence research. Direct application of the neural network models, for example, for the diagnostic model of the aircraft power plant’s flow path can be represented as multilayered perceptron with two hidden layers. Perceptron’s inputs are the controlled parameters of the gas turbine engine (y1, ..., yn), its outputs are the controlled parameters’ deviations from their nominal values (Δy1 ,.., Δyn ) . The distinctive feature of the neural network as the diagnostic model of the aircraft power plant is that it’s weight matrix is formed in the process of two-staged learning procedure basing on the models that are linear on the local aircraft engine’s parameters. 216 The 7th International Conference “RELIABILITY and STATISTICS in TRANSPORTATION and COMMUNICATION - 2007” The most simple example of the aircraft power plant mathematical model is the throttle characteristic of the aircraft engine in two coordinates (GF_coer, nc_coer), where GF_coer is an adjusted value of fuel consumption in the combustion chamber, nc_coer is an adjusted value of the gas turbine engine turbo compressor’s rotation frequency that is represented in the relative (non-dimensional) units. As the rule the powerful deviation on the throttle characteristics’ relations can be observed because of some disturbing factors (changes of the air engine’s flow path geometrical parameters, errors of the main engine’s parameters’ and external conditions used for engine’s parameters’ conversion to standard atmospheric conditions measurements errors, etc.). Recent investigations show, that in the case of modes operation near maximum the measured fuel consumption’s deviation from the nominal value can be up to 4,5% in the case when such an error can be caused by uniformly distributed errors of the direct measurements of the temperature and pressure as well as fuel characteristics’ changes. If tolerance to the parameters’ deviations is less than range explainable by random factors, then while engine parameters are regulated systematic deviations must be compensated at the expense of engine parameters’ deviations from the their standard values. Investigations carried out show that the most appropriate neural network architecture that can satisfy prescribed requirements for the aircraft power plant control quality is ensemble neural network. Such architecture in the given case is represented as the following chain: radial basis function networks (RBF) → perceptron → Kohonen’s neural network (KN) (see Figure 1). RBF1 P A R AM E RBF2 PERCEPTRON T E R S KN ...... RBF5 Figure 1. Ensemble neural network architecture In ensemble architecture the first (input) layer filters the measured data, the second (intermediate) layer represents the thickener of the aircraft power plant status characteristics, and the output layer is the status conditions classifier. The modelling of the given task using PC in the case of aircraft power plant’s work in different modes and with different combinations of parameters let to denote 3-5 most informative parameters that maximally affect the normal functioning process of the aircraft engine. In this case the usage of five RBF networks each with five input parameters (state vector) and two output parameters (state “1” 217 Session 5. Aviation – normal, state “0” – failure) is considered as the optimal one. Perceptron in the ensemble is the field that concentrates five RBF’s outputs. Kohonen’s network (classifier) has two inputs and one output. This network with a high degree of precision carries out the classification (recognition) of the aircraft power plant status condition, also in the case of partial or full uncertainty about its parameters. Computer experiment has shown that ensemble neural network recovered the absent data steady and determined the condition of the power plant. The estimation of the power plant according to its exploitation time based on neural network is carried out in the following way. The standard model of the power plant (parameters of engine) received in the process of factory development testing, which is stored in the neural network basis as the individual informational “portrait”, is compared to the power plant parameters during the exploitation [6]. In the process of the comparative analysis the special metrics is calculated. Its value can be used to estimate air engine status condition and to build the separating plane. For such task solution the 3-layered perceptron (2 inputs, 14 hidden neurons, 4 outputs) is used. It is trained using the error back-propagation algorithm. CONCLUSION The application of the neural network apparatus is effective in the solution of many tasks: aircraft engine power plant’s “image” identification, control, mode classification, diagnostics, trend analysis, forecasting, etc. The problem of aircraft power plant and its subsystems control and diagnostics is a complicated task that is concerned with the need of taking into account many factors also the uncertainty factors. The application of the artificial intelligence methods based on the neural networks makes it possible to find new ways of this problem solving that are based on the using the knowledge and experience of the experts. These are image recognition theory, learning theory, theory of adaptation to the changing external conditions, theory of the decision making in the case of the information deficiency, etc. Investigations considered in [7, 8] show that the neural networks using for analysis of the aircraft power plants is effective and promising. References 1. Yefimov, V.V., Yakovkin, V.A. Method of technical diagnostics based on Neural Networks / Priborostroenie, Vol. 9, 1999. (In Russian) 2. Vasilyev, V.I., Zhernakov, S.V., Urazbakhtina, L.B. Neural Network Monitoring of Gas Turbine Engine. Neurocomputers: design and application, Radiotehnika, Vol. 1, 2001, pp. 37-43. (In Russian) 3. Neurocomputers in Aviation / Ed. by Vasilyev V.I., Ilyasov B.G., Kusimov. In: Book 14: study book for high school establishments. Moscow: Radiotehnika, 2004. 496 p. (In Russian) 4. Labendik, V., Ozolinsh, I., Zvanchuk, P. Method for turbofan thrust control in flight using similar engine mode of operation work. In: Abstract collection of II International Scientific and Technical Conference «Aero Engines of XXI Century», Vol. II, 06-09.12.2005, Moscow, Russia. Moscow: CIAM, 2005, pp. 244-246. 5. Zhernakov, S.V. Definition of aviation engine parameters using active expert system, Aviacionnaja promishlennost, Vol. 4, 2001, pp. 24-28. (In Russian) 6. Artificial tools of aviation engines reliability diagnostics and forecasting / V.I. Dubrovin, S.A. Subbotin, A.V. Boguslaev and others. Zaporozhje: OAO “Motor-Sich, 2003. 279 p. (In Russian) 7. Kopytov, E., Labendik, V., Osis, A., Tarasov. A. Method of aviation engine diagnostics in the case of partial loss of information. In: Abstract collection of II International Scientific and Technical Conference «Aero Engines of XXI Century», Vol. II, 06-09.12.2005, Moscow, Russia. Moscow: CIAM, 2005, pp. 246-247. 8. Kopytov, E., Labendik, V., Osis, A., Tarasov, A. Neural Networks Application for Analysis of Flight Information in Aircraft Engine Diagnostic System. In: Transport and Telecommunication, 2006, vol. 7(2). Riga: TTI, 2006, pp. 287-294. 218 The 7th International Conference “RELIABILITY and STATISTICS in TRANSPORTATION and COMMUNICATION - 2007” RECOGNITION OF VIBRATION SIGNAL CHANGES VIA CFAR TEST FOR EARLY IDENTIFICATION OF FATIGUE CRACKS Konstantin N. Nechval1, Nicholas A. Nechval2, Gundars Berzins2, Maris Purgailis2, Juris Krasts2, Uldis Rozevskis2, Kristine Rozite2, Vladimir Strelchonok3, Natalie Zolova2 1 Mathematical Methods and Modelling Department, Transport and Telecommunication Institute Lomonosova 1, Riga, LV-1019, Latvia E-mail: konstan@tsi.lv 2 Mathematical Statistics Department, University of Latvia Raina Blvd 19, Riga, LV-1050, Latvia E-mail: nechval@junik.lv 3 Informatics Department, Baltic International Academy Lomonosova 1, Riga, LV-1019, Latvia E-mail: str@apollo.lv This paper introduces a new technique for early identification of fatigue cracks, namely the constant false alarm rate (CFAR) test. This test works on the null hypotheses that a target vibration signal is statistically similar to a reference vibration signal. In effect, this is a time-domain signal processing technique that compares two signals, and returns the likelihood whether the two signals are similar or not. The system monitors the vibration signal of the rotor as it cycles, and compares that vibration signal with, say, the original vibration signal. The difference vector reflects the change in vibration over time. By developing of a crack, the vector changes in a characteristic way. Thus, it is possible, during CFAR test, to determine whether the two signals are similar or not. Therefore, by comparing a given vibration signal to a number of reference vibration signals (for several crack scenarios) it is possible to state which is the most likely condition of the rotor under analysis. The CFAR test not only successfully identifies the presence of the fatigue cracks but also gives an indication related to the advancement of the crack. This test, despite its simplicity, is an extremely powerful method that effectively classifies different vibration signals, allowing for its safe use as another condition monitoring technique. Keywords: vibration signal, changes, recognition, CFAR test 1. INTRODUCTION The machines and structural components require continuous monitoring for the detection of cracks and crack growth for ensuring an uninterrupted service. Non-destructive testing methods like ultrasonic testing, X-ray, etc., are generally useful for the purpose. These methods are costly and time consuming for long components, e.g., railway tracks, long pipelines, etc. Vibration-based methods can offer advantages in such cases [1]. This is because measurement of vibration parameters like natural frequencies is easy. Further, this type of data can be easily collected from a single point of the component. This factor lends some advantages for components, which are not fully accessible. This also helps to do away with the collection of experimental data from a number of data points on a component, which is involved in a prediction based on, for example, mode shapes. Nondestructive evaluation (NDE) of structures using vibration for early detection of cracks has gained popularity over the years and, in the last decade in particular, substantial progress has been made in that direction. Almost all crack diagnosis algorithms based on dynamic behaviour call for a reference signature. The latter is measured on an identical un-cracked structure or on the same structure at an earlier stage. Dynamics of cracked rotors has been a subject of great interest for the last three decades and detection and monitoring have gained increasing importance, recently. Failures of any high speed rotating components (jet engine rotors, centrifuges, high speed fans, etc.) can be very dangerous to surrounding equipment and personnel (see Figure 1), and must always be avoided. Jet engine disks operate under high centrifugal and thermal stresses. These stresses cause microscopic damage as a result of each flight cycle as the engine starts from the cold state, accelerates to maximum speed for 219 Session 5. Aviation take-off, remains at speed for cruise, then spools down after landing and taxi. The cumulative effect of this damage over time creates a crack at a location where high stress and a minor defect combine to create a failure initiation point. As each flight operation occurs, the crack is enlarged by an incremental distance. If allowed to continue to a critical dimension, the crack would eventually cause the burst of the disk and lead to catastrophic failure (burst) of the engine. Engine burst in flight is rarely survivable. Figure 1. Jet engine fan section failure In Figure 2 is given micrograph showing one of the cracks detected in the bladed disk assembly of the High Pressure Turbine for over 17,000 cycles. Figure 2. Micrograph showing one of the cracks detected in the High Pressure Turbine (bladed disk assembly) Schematic showing the orientation of the columnar grains at the leading and trailing edges in the blade root region of the failed turbine rotor blisk is presented in Figure 3. 220 The 7th International Conference “RELIABILITY and STATISTICS in TRANSPORTATION and COMMUNICATION - 2007” Figure 3. Schematic showing the orientation of the columnar grains at the leading and trailing edges in the blade root region of the failed turbine rotor blisk 2. PROBLEM STATEMENT Suppose that we desire to compare a target vibration signal and a kth reference vibration signal, which have p response variables. Let xij(k) and yij be the ith observation of the jth response variable of the kth reference signal and the target signal, respectively. It is assumed that all observation vectors, xi(k)=(xi1(k), ..., xip(k))′, yi=(yi1, ..., yip)′, i=1(1)n, are independent of each other, where n is a number of paired observations. Let zi(k) = xi(k)−yi, i=1(1)n, be paired comparisons leading to a series of vector differences. Thus, in order to compare the above signals, and return the likelihood whether the two signals are similar or not, it can be obtained and used a sample of n independent observation vectors Z(k)=(z1(k), ..., zn(k)). Each sample Z(k), k∈{1, …, m}, is declared to be realization of a specific stochastic process with unknown parameters. It is assumed here that zi(k), i=1(1)n, are independent p-multivariate normal random variables (n≥p+2) with common mean a(k) and covariance matrix (positive definite) Q(k). A goodness-of-fit testing for the multivariate normality is based on the following theorem. Theorem 1 (Characterization of the multivariate normality). Let zi(k), i=1(1)n, be n independent p-multivariate random variables (n≥p+2) with common mean a(k) and covariance matrix (positive definite) Q(k). Let wr(k), r = p+2, …, n, be defined by r − ( p + 1) r − 1 ⎛ S (k ) ⎞ wr (k ) = (z r (k ) − z r −1 (k ) )′ S r−11 (k )(z r (k ) − z r −1 (k )) = r − ( p + 1) ⎜ r − ⎜ S (k ) − 1⎟, ⎟ p r p ⎝ r −1 ⎠ r= p+2, …, n, (1) where r −1 z r −1 (k ) = ∑ z (k ) /(r − 1), i =1 i (2) r −1 S r −1 (k ) = ∑ (z (k ) − z i =1 i r −1 ( k ))(z i ( k ) − z r −1 (k ))′, (3) 221 Session 5. Aviation then the zi(k) (i=1, …, n) are Np(a(k),Q(k)) if and only if wp+2(k), …, wn(k) are independently distributed according to the central F-distribution with p and 1, 2, . . . , n−(p+1) degrees of freedom, respectively. Proof. The proof is similar to that of [2] and so it is omitted here. ˛ Goodness-of-fit testing for the multivariate normality. The results of Theorem 1 can be used to obtain test for the hypothesis of the form H0: zi(k) follows Np(a(k),Q(k)) versus Ha: zi(k) does not follow Np(a(k),Q(k)), ∀i = 1(1)n. The general strategy is to apply the probability integral transforms of wk, ∀k = p+2(1)n, to obtain a set of i.i.d. U(0,1) random variables under H0 [2]. Under Ha this set of random variables will, in general, not be i.i.d. U(0,1). Any statistic, which measures a distance from uniformity in the transformed sample (for instance, Kolmogorov-Smirnov statistic) can be used as a test statistic. Testing for similarity of the two signals. In this paper, for testing that the two signals (target signal and reference signal) are similar, we propose a statistical approach that is based on the generalized maximum likelihood ratio. We have the following hypotheses: H0(k): Similarity is valid for the acceptable range of accuracy under a given experimental frame; H1(k): Similarity is invalid for the acceptable range of accuracy under a given experimental frame. Thus, for fixed n, the problem is to construct a test, which consists of testing the null hypothesis H0(k): zi(k) ∼ Np(0,Q(k)), ∀i = 1(1)n, (4) where Q(k) is a positive definite covariance matrix, versus the alternative H1(k): zi(k) ∼ Np(a(k),Q(k)), ∀i = 1(1)n, (5) where a(k)=(a1(k), ..., ap(k))′ ≠ (0, ..., 0)′ is a mean vector. The parameters Q(k) and a(k) are unknown. 3. GMLR STATISTIC In order to distinguish the two hypotheses (H0(k) and H1(k)), a generalized maximum likelihood ratio (GMLR) statistic is used. The GMLR principle is best described by a likelihood ratio defined on a sample space Z with a parameter set Θ, where the probability density function of the sample data is maximized over all unknown parameters, separately for each of the two hypotheses. The maximizing parameter values are, by definition, the maximum likelihood estimators of these parameters; hence the maximized probability functions are obtained by replacing the unknown parameters by their maximum likelihood estimators. Under H0(k), the ratio of these maxima is a Q(k)-free statistic. This is shown in the following. Let the complete parameter space for θ(k)=(a(k),Q(k)) be Θ={(a(k), Q(k)): a(k)∈Rp, Q(k)∈Qp}, where Qp is a set of positive definite covariance matrices, and let the restricted parameter space for θ, specified by the H0(k) hypothesis, be Θ0={(a(k),Q(k)): a(k)=0, Q(k)∈Qp}. Then one possible statistic for testing H0(k): θ(k)∈Θ0 versus H1(k): θ(k)∈Θ1, where Θ1=Θ−Θ0, is given by the generalized maximum likelihood ratio LR = max LH1 ( k ) (Z(k ); θ(k ) ) max LH 0 ( k ) (Z(k ); θ(k ) ). (6) θ ( k )∈Θ1 θ ( k )∈Θ 0 Under H0(k), the joint likelihood for Z(k) is given by ⎛ n ⎞ ∑ z′ (k )[Q(k )] − n/2 LH 0 ( k ) (Z(k ); θ(k )) = (2π ) − np/2 Q( k ) exp⎜ − ⎜ i −1 z i (k ) / 2 ⎟. ⎟ (7) ⎝ i =1 ⎠ 222 The 7th International Conference “RELIABILITY and STATISTICS in TRANSPORTATION and COMMUNICATION - 2007” Under H1(k), the joint likelihood for Z(k) is given by ⎛ n ⎞ ∑ (z (k ) − a(k ))′[Q(k )] − n/2 LH1 ( k ) (Z(k ); θ(k )) = (2π ) − np/2 Q(k ) exp⎜ − ⎜ i −1 ( z i (k ) − a(k ))/2 ⎟. ⎟ (8) ⎝ i =1 ⎠ It can be shown that − n/2 ˆ max LH j ( k ) (Z(k ); θ(k ) ) = (2π ) − np/2 Q (k ) exp(−np / 2), j = 0, 1, (9) θ ( k )∈Θ j j ˆ ˆ where Q 0 ( k ) =Z(k)Z′(k)/n, Q1 (k ) =(Z(k) − a (k)u′)(Z(k) − a (k)u′)′/n, and a (k)=Z(k)u/u′u are the well- ˆ ˆ ˆ known maximum likelihood estimators of the unknown parameters Q(k) and a(k) under the hypotheses H0(k) and H1(k), respectively, u=(1, ..., 1)′ is the n-dimensional column vector of units. A substitution of (9) into (6) yields n/2 −n / 2 ˆ LR = Q 0 (k ) ˆ Q1 (k ) . (10) Taking the (n/2)th root, this likelihood ratio is evidently equivalent to −1 ˆ ˆ LR • = Q 0 (k ) Q1 (k ) = Z(k )Z′(k ) / Z( k )Z′(k ) − (Z(k )u)(Z(k )u)′ / u′u . (11) Now the likelihood ratio in (11) can be considerably simplified by factoring out the determinant of the p × p matrix Z(k)Z′(k) in the denominator to obtain this ratio in the form ( LR • = 1 1 − (Z(k )u)′[Z(k )Z′(k )]−1 (Z( k )u) / n . ) (12) This equation follows from a well-known determinant identity. Clearly (12) is equivalent finally to the statistic ⎛n− p⎞ ⎛n− p⎞ −1 ⎜ p ⎟(LR • − 1) = ⎜ p ⎟na′(k )[T(k )] a(k ), Vn (k ) = ⎜ ⎟ ⎜ ⎟ (13) ⎝ ⎠ ⎝ ⎠ where T(k ) = nQ1 (k ) . It is known that (a(k ), T( k )) is a complete sufficient statistic for the parameter θ(k)=(a(k),Q(k)). Thus, the problem has been reduced to consideration of the sufficient statistic (a(k ), T( k )) . It can be shown that under H0, Vn is a Q(k)-free statistic which has the property that its distribution does not depend on the actual covariance matrix Q(k). This is given by the following theorem. Theorem 2 (PDF of the statistic Vn(k)). Under H1(k), the statistic Vn(k) is subject to a non- central F-distribution with p and n−p degrees of freedom, the probability density function of which is p −1 p ⎡ ⎛ p n − p ⎞⎤ ⎡ p ⎤ 2 −1 f H1 ( k ) (vn (k ); n, q ) = ⎢Β⎜ , ⎟⎥ ⎢ ⎥ vn (k ) 2 ⎣ ⎝2 2 ⎠⎦ ⎣ n − p ⎦ −n ⎛ p nq (k ) ⎡ n− p ⎤ ⎞ ⎡ ⎤ −1 p 2 × 1+ ⎢ vn (k ) ⎥ e − nq / 2 1 F1 ⎜ n ; ; 1+ ⎟, 0<vn(k)<∞. (14) 2 ⎢ pvn (k ) ⎥ ⎟ ⎢ ⎥ ⎢ n− p ⎥ ⎜2 2 ⎣ ⎦ ⎠ ⎣ ⎦ ⎝ 223 Session 5. Aviation where 1F1(b;c;x) is the confluent hyper-geometric function, q(k)=a′(k)[Q(k)]−1a(k) is a non-centrality parameter. Under H0(k), when q(k) = 0, (14) reduces to a standard F-distribution with p and n−p degrees of freedom, p −1 p −n ⎡ ⎛ p n − p ⎞⎤ ⎡ p ⎤ 2 −1 ⎡ p ⎤ 2 f H 0 ( k ) (vn (k ); n) = ⎢Β⎜ , ⎟⎥ ⎢ ⎥ vn (k ) 2 ⎢1 + vn (k )⎥ , 0<vn(k)<∞. (15) ⎣ ⎝2 2 ⎠⎦ ⎣ n − p ⎦ ⎣ n− p ⎦ Proof. The proof follows by applying Theorem 1 [3] and being straightforward is omitted here. ˛ 4. CFAR TEST The CFAR test of H0(k) versus H1(k), based on Vn(k), is given by ⎧ ≥ h(k ), then H1 (k ) Vn (k )⎨ (16) ⎩< h(k ), then H 0 (k ), where h(k)>0 is a threshold of the test which is uniquely determined for a prescribed level of significance α(k). It follows from (15) that this test achieves a fixed probability of a false alarm. If Vn(k)>h(k) then the kth reference vibration signal is eliminated from further consideration. If (m −1) reference vibration signals are so eliminated, then the remaining reference vibration signal (say, kth) is the one with which the target vibration signal may be identified. If all reference vibration signals are eliminated from further consideration, we decide that the target vibration signal cannot be identified with one of the m specified reference vibration signals. If the set of reference vibration signals not yet eliminated has more than one element, then we declare that the target vibration signal may be identified with the k*th reference vibration signal, where k * = arg max (h(k ) − Vn (k )), (17) k ∈D where D is the set of simulation models not yet eliminated by the above test. CONCLUSION The main idea of this paper is to find a test statistic whose distribution, under the null hypothesis, does not depend on unknown (nuisance) parameters. This allows one to eliminate the unknown parameters from the problem. References 1. Dimarogonas, A.D. Vibration of Cracked Structures: a State of the Art Review, Engineering and Fracture Mechanics, Vol. 55, 1996, pp. 831-857. 2. Nechval, N.A., Nechval, K.N. Characterization Theorems for Selecting the Type of Underlying Distribution. In: Abstract Book of Communications of the 7th Vilnius Conference on Probability Theory and Mathematical Statistics & the 22nd European Meeting of Statisticians. Vilnius, Lithuania: TEV, 1998, pp. 352-353. 3. Nechval, N.A. Radar CFAR Thresholding in Clutter under Detection of Airborne Birds. In: Proceedings of the 21st Meeting of Bird Strike Committee Europe. Jerusalem, Israel: BSCE, 1992, pp. 127-140. 224 The 7th International Conference “RELIABILITY and STATISTICS in TRANSPORTATION and COMMUNICATION - 2007” STOCHASTIC FATIGUE MODELS FOR DECISION-MAKING IN SERVICE OF AIRCRAFT STRUCTURE COMPONENTS Konstantin N. Nechval Mathematical Methods and Modelling Department, Transport and Telecommunication Institute Lomonosova 1, Riga, LV-1019, Latvia e-mail: konstan@tsi.lv For important fatigue-sensitive structures of aircraft whose breakdowns cause serious accidents, it is required to keep their reliability extremely high. In this paper, we discuss inspection strategies for such important structures against fatigue failure. The focus is on the case when there are fatigue-cracks unexpectedly detected in a fleet of aircraft within a warranty period (prior to the first inspection). The paper examines this case and proposes stochastic models for prediction of fatigue-crack growth to determine appropriate intervals of the inspections. We also do not assume known parameters of the underlying distributions, and the estimation of that is incorporated into the analysis and decision-making. Numerical example is provided to illustrate the procedure. Keywords: aircraft, fatigue crack, inspection interval 1. INTRODUCTION Fatigue is one of the most important problems of aircraft arising from their nature as multiple- component structures, subjected to random dynamic loads. The analysis of fatigue crack growth is one of the most important tasks in the design and life prediction of aircraft fatigue-sensitive structures (for instance, wing, fuselage) and their components (for instance, aileron or balancing flap as part of the wing panel, stringer, etc.). An example of in-service cracking from B727 aircraft (year of manufacture 1981; flight hours not available; flight cycles 39,523) [1] is given on Figure 1. Figure 1. Example of in-service cracking from B727 aircraft Several probabilistic or stochastic models have been employed to fit the data from various fatigue crack growth experiments. Among them, the Markov’s chain model [2], the second-order approximation model [3], and the modified second-order polynomial model [4]. Each of the models may be the most appropriate one to depict a particular set of fatigue growth data but not necessarily the others. All models can be improved to depict very accurately the growth data but, of course, it has 225 Session 5. Aviation to be at the cost of increasing computational complexity. Yang’s model [3] and the polynomial model [4] are considered more appropriate than the Markov’s chain model [2] by some researchers through the introduction of a differential equation which indicates that fatigue crack growth rate is a function of crack size and other parameters. The parameters, however, can only be determined through the observation and measurement of many crack growth samples. If fatigue crack growth samples are observed and measured, descriptive statistics can then be applied directly to the data to find the distributions of the desired random quantities. Thus, these models still lack prediction algorithms. Moreover, they are mathematically too complicated for fatigue researchers as well as design engineers. A large gap still needs to be bridged between the fatigue experimentalists and researchers who use probabilistic methods to study the fatigue crack growth problems. 2. PROBLEM DESCRIPTION Let us assume that a fatigue-sensitive component has been found cracked on n aircraft within a warranty period. The cracking has not yet caused an accident, but the safety experts have told the manager that if this item fails, an accident will be possible. It is clear that the part will have to be redesigned and replaced. The manager’s dilemma is that redesigning the part, manufacturing the new design, and installing it in the fleet will take, say, at least two years. The manager must decide how to manage risk for the next two years. The alternatives include doing nothing and accepting the risk of continued cracking and the possibility of an accident. An inspection program is usually instigated, which should reduce the risk of failure, but due to uncertainties in aircraft loading histories, provides no direct measurement of the criticality of the detected cracks. Generally, such a program will lead to some aircraft being grounded, eliminating risk for those aircraft and reducing overall risk, but reducing operational capability. This would leave precious few aircraft to spare before the service’s ability to accomplish its mission became impaired. In such a scenario, the decision process involves a complex probability problem concerning the likelihood of additional failures and acceptable risk. To compound the difficulty little guidance is provided in aircraft design specifications for this situation. The situation presented is not uncommon. The purpose of this paper is to present a more accurate stochastic crack growth analysis method, while maintaining the simplicity of the proposed stochastic fatigue models, for the above problem. We discuss the optimal relationship between the inspection time and the pre-specified minimum level of reliability. To illustrate the proposed technique, a numerical example is given. 3. PARIS-ERDOGAN’S LAW AS A STARTING POINT The basis of most of the fatigue models is Paris-Erdogan’s law [5] relating the rate of growth of crack size a to N cycles: da ( N ) = q[a ( N )]b (1) dN in which q and b are parameters depending on loading spectra, structural/material properties, etc. We fit da/dN vs a(N) with a function that we can integrate between limits (initial crack size, a0, and any given crack size, a) to get a life prediction. In the linear region (see Fig. 2) we use Paris-Erdogan’s Equation (1) as follows. Integrating N a( N ) da ∫ N0 dN = ∫ a0 qa b , (2) we have 226 The 7th International Conference “RELIABILITY and STATISTICS in TRANSPORTATION and COMMUNICATION - 2007” N − N0 = 1 q( −b + 1) [ − ] a( N ) − b +1 − a0 b +1 . (3) Thus, the crack growth equation representing the solution of the differential equation for Paris- Erdogan’s law is given by 1 ⎛ 1 1 ⎞ N − N0 = ⎜ b −1 − ⎟. (4) q(b − 1) ⎝ 0 ⎜a b −1 ⎟ a( N ) ⎠ Figure 2. Crack growth rate versus crack size curve (I = near-threshold region; II = linear region; III = instability region) 3.1. Sensitivity Analysis Consider the solution of the differential equation for Paris-Erdogan’s law written in the form of (4) as: 1 ⎛ 1 1 ⎞ N (a) = ⎜ b −1 − b −1 ⎟, (5) ⎜a q(b − 1) ⎝ 0 a ⎟ ⎠ where a0 is the initial crack size at N0=0. The derivatives of the number of load cycles with respect to the parameters q and b read: dN ( a ) N (a) =− (6) dq q and dN ( a ) 1 ⎡ 1 ⎛ ln a ln a0 ⎞ ⎤ = ⎢ ⎜ b −1 − b −1 ⎟ − N ( a)⎥. ⎜a ⎟ (7) db b −1 ⎢ q ⎝ ⎣ a0 ⎠ ⎥ ⎦ 227 Session 5. Aviation From this one can see that the number of cycles to reach a certain crack size is very sensitive to changes of the parameter q. 4. STATISTICAL VARIABILITY OF FATIGUE-CRACK GROWTH The traditional analytical method of engineering fracture mechanics (EFM) usually assumes that crack size, stress level, material property and crack growth rate, etc. are all deterministic values which will lead to conservative or very conservative outcomes. However, according to many experimental results and field data, even in well-controlled laboratory conditions, crack growth results usually show a considerable statistical variability (as it is shown on Fig. 3). Figure 3. Constant amplitude loading fatigue test data curves Yet more considerable statistical variability is the case under variable amplitude loading (as shown on Fig. 4). The basis of most data analyses seems to be to take logarithms in (1) and estimate b and q by least squares in the equation ln(da ( N ) / dN ) = ln q + b ln a ( N ). (8) Unfortunately to use this equation estimates of da(N)/dN are required. Estimates of derivatives are notoriously unreliable. If several repetitions of an experiment under the same conditions are made it is not always clear how to combine the results. Moreover, as regressions model the properties of the estimates of the coefficients in (8) are not the same as those of estimates of the coefficients in (4). Thus it is sensible to ask why the estimation does not proceed directly from the data on crack size and cycles through equation (4). It is interesting to note that if b were known q could be estimated from a straight line 1 1 − = (b − 1)q ( N − N 0 ) (9) a0 −1 b a b −1 and indeed such a plot for a few values of q is indicative of the nature of Paris-Erdogan’s equation in a particular case. 228 The 7th International Conference “RELIABILITY and STATISTICS in TRANSPORTATION and COMMUNICATION - 2007” Figure 4. Variable amplitude loading fatigue test data curves During the service of the components being assessed, there may be uncertainties in the applied loading conditions, extrapolation of the material data to service conditions, component dimensions, and nature, size and location of detected (postulated) defects, etc. These uncertainties/variations are critical inputs to the crack growth assessment and can be taken into account using probabilistic methodologies. There is now an extensive literature on the subject of the statistical nature of crack growth. Most of the literature is concerned with model building and the agreement between the general features of the model and the observed behaviour of the crack. However, little use has been made of the statistical nature of the models to analyse experimental results. While most industrial failures involve fatigue, the assessment of the fatigue reliability of industrial components subjected to various dynamic loading situations is one of the most difficult engineering problems that remain. Material degradation processes due to fatigue depend upon material characteristics, component geometry, loading history and environmental conditions. As a result, stochastic models for crack growth have been suggested by many investigators in the last 15 years. These include evolutionary probabilistic models, cumulative jump models and differential equation (DE) models. DE models are the most widely used models for predicting stochastic crack growth accumulation in the reliability and durability analyses of fatigue critical components. In practical applications of the stochastic crack growth analysis, either one of the following two distribution functions is needed: the distribution of the crack size at any service time or the distribution of the service time to reach any given crack size. Unfortunately, when the crack growth rate is modelled as a random process, these two distribution functions are not amenable to analytical solutions. As a result, numerical simulation procedures have been used to obtain accurate results. The simulation approach is a very powerful tool; in particular, with modern high-speed computers. However, it is a very time consuming procedure and therefore simple approximate analytical solutions are very useful in engineering. The purpose of this paper is to present a more useful stochastic fatigue crack growth models by using the solution of Paris-Erdogan’s law equation, which result in a simple analytical solution for either the distribution of the service time to reach any given crack size or the distribution of the crack size at any service time. The probability that crack size a(N) will exceed any given crack size a• in the service interval (N0,N·), Pr{a(N·)>a•}, is frequently referred to as crack exceedance probability and can be found based on the stochastic fatigue crack growth model. In addition to this probability distribution of crack size, the probability distribution of cycles (or time) for a crack to grow from size a0 to a•, Pr{N(a•)≤ N •}, can also be found based on the above model. In fact, the probability that service time N(a•) will be 229 Session 5. Aviation within the interval (N0, N •) for crack size to reach a• is identical to Pr{a(N·)>a•}.That is Pr{N(a•)≤ N •} = Pr{a(N·) > a•}. To summarize the concept of the above derivation, the readers can refer to Fig. 5. Figure 5. Schematic diagram of crack size distribution and random time distribution 5. STOCHASTIC FATIGUE-CRACK GROWTH PARAMETER VARIABILITY MODELS These models allow one to describe the uncertainties in one or two parameters of the solution (4) of the differential equation (1) for Paris-Erdogan’s law via parameters modelled as random variables in order to characterize the random properties, which seem to vary from specimen to specimen (see Fig. 3). In other words, the stochastic fatigue-crack growth parameter variability models (with respect to the parameters b and q modelled as random variables) are given by 1 ⎛ 1 1 ⎞ N − N0 = ⎜ B −1 − B −1 ⎟ , (10) Q(B − 1) ⎜ a0 ⎝ a ⎟ ⎠ where (N−N0) is a joint random variable of B and Q. In fact, these models are suited to account for this type of variability. The ones however cannot explain the variability of the crack growth rate during the crack growth process. In particular, crack growth data (crack size versus service time and vice versa) may be analysed using Eq. (10) by considering, for instance, two different approaches: (i) B is identical for each specimen and Q varies from specimen to specimen, referred to as Case 1; (ii) both B and Q vary from specimen to specimen, referred as Case 2. For Case 1, with B=1, the crack growth data for each specimen are best fitted by equation ⎡ a( N ) ⎤ ln ⎢ ⎥ a( N 0 ) ⎦ N − N0 = ⎣ (11) Q to obtain a sample value of Q, where a(N) ≡ a, a(N0) ≡ a0. For Case 2 equation (10) is used to best fit the crack growth data for each specimen to obtain a set of sample values of B and Q. From the statistical standpoint, B is considered to be a deterministic value and Q to be a statistical (random) variable in Case 1, while both B and Q are considered to be statistical variables in Case 2. It is found that the lognormal or Weibull’s distribution provides a reasonable fit for B and Q in both cases. 230 The 7th International Conference “RELIABILITY and STATISTICS in TRANSPORTATION and COMMUNICATION - 2007” 5.1. Weibull’s Crack Growth Parameter Variability Model Consider Case 1. The Weibull’s probability distribution function, F(q;σ,δ) , of Q is expressed as ⎧1 − exp[−(q / σ )δ ], q ≥ 0, F (q;σ , δ ) = ⎨ (12) ⎩0, otherwise, in which F(q;σ,δ) is the probability that Q is smaller than or equal to an arbitrary value q; σ and δ are distribution parameters representing the scale parameter and the shape parameter, respectively. 5.2. Crack Exceedance Probability For B=1, the probability that crack size a(N) will exceed any given (say, maximum allowable) crack size a• can be derived and expressed as ⎧ ln[a • / a ( N 0 )] ⎫ ⎡ ⎛ ln[a • / a ( N )] ⎞δ ⎤ ⎬ = exp ⎢− ⎜ • 0 ⎟ ⎥ Pr{a ( N ) > a } = Pr ⎨Q > . (13) ⎩ N − N0 ⎭ ⎢ ⎜ σ ( N − N0 ) ⎟ ⎥ ⎝ ⎠ ⎦ ⎣ For B=b≠1 the maximum allowable crack size exceedance probability for a single crack is given by ⎧ [a( N 0 )]− (b −1) − [a • ]− (b −1) ⎫ ⎡ ⎛ [a( N )]− (b −1) − [a • ]− (b −1) ⎞δ ⎤ ⎬ = exp ⎢− ⎜ ⎟ ⎥ • Pr{a( N ) > a } = Pr ⎨Q > 0 ⎜ σ (b − 1)( N − N ) ⎟ ⎥ . (14) ⎩ (b − 1)( N − N 0 ) ⎭ ⎢ ⎝ ⎠ ⎦ ⎣ 0 It will be noted that the crack exceedance probability can be used for assigning sequential in-service inspections [6]. 6. STOCHASTIC FATIGUE-CRACK GROWTH LIFETIME VARIABILITY MODELS These models allow one to characterize the random properties, which seem to vary during crack growth (see Fig. 4), via crack growth equation with a stochastic noise V dependent, in general, on the crack size a: 1 ⎛ 1 1 ⎞ ⎜ b −1 − b −1 ⎟ = N − N 0 + V (15) (b − 1)q ⎝ 0 a ⎟ ⎜a ⎠ ⎛ 1 ⎛ 1 1 ⎞⎞ ln⎜ ⎜ b −1 − b −1 ⎟ ⎟ = ln( N − N 0 ) + V (16) ⎜ (b − 1)q ⎜a a ⎟⎟ ⎝ ⎝ 0 ⎠⎠ 1 1 − = (b − 1)q( N − N 0 ) + V , (17) a0 −1 a b −1 b ⎛ 1 1 ⎞ ln⎜ b −1 − b −1 ⎟ = ln[(b − 1)q( N − N 0 )] + V , ⎜a (18) ⎝ 0 a ⎟ ⎠ and so on, where V~N(0,σ2(b,q,N)), a0 ≡ a(N0), a ≡ a(N). They are suited to account for this type of variability. The ones however cannot explain the variability of the crack growth rate from specimen to specimen. 231 Session 5. Aviation 6.1. Crack Exceedance Probability If V~N(0,σ2) in (15), then the probability that crack size a(N) will exceed any given (say, maximum allowable) crack size a• can be derived and expressed as ⎛⎡ [a ]− (b −1) − [a • ]− (b −1) ⎤ ⎞ Pr{a ( N ) > a • } = Φ⎜ ⎢ N − N 0 − 0 ⎥ σ⎟, (19) ⎜ (b − 1)q ⎟ ⎝⎣ ⎦ ⎠ where Φ(.) is the standard normal distribution function, η 1 Φ(η ) = ∫ exp(− x 2 / 2) dx. (20) 2π −∞ If V~N(0,[(b-1)σ(N-N0)1/2]2) in (17), then the probability that crack size a(N) will exceed any given (say, maximum allowable) crack size a• can be derived and expressed as ⎛ ⎡ (b − 1)q( N − N 0 ) − ([a0 ]− (b −1) − [a • ]− (b −1) ) ⎤ ⎞ Pr{a ( N ) > a • } = Φ⎜ ⎢ ⎥⎟ . (21) ⎜ (b − 1)σ ( N − N 0 )1 / 2 ⎟ ⎝ ⎣ ⎦⎠ In this case, the conditional probability density function of a is given by a −b ⎛ 1 ⎡ (a − (b −1) − a − (b −1) ) − (b − 1)q ( N − N ) ⎤ 2 ⎞ f ( a; a0 , N 0 , N ) = exp⎜ − ⎢ 0 0 ⎥ ⎟. ⎟ (22) σ [2π ( N − N 0 )]1 / 2 ⎜ 2 (b − 1)σ ( N − N 0 )1 / 2 ⎝ ⎣ ⎦ ⎠ 6.2. Data Analysis for a Single Crack Consider the regression model corresponding to (17). Because the variance is non-constant (17) is a non-standard model; however, on dividing by (N−N0)1/2 the model becomes a1− b − a1− b 0 1/ 2 = (b − 1)q( N − N 0 )1 / 2 + W , (23) ( N − N0 ) where W is normally distributed with mean zero and standard deviation (b−1)σ independent of N. Thus if b is known the estimator of (b−1)q is just the least-squares estimator of the coefficient in Equation (23) and the estimate of (b−1)σ is just the estimate of the variance of the regression. It remains to determine what to do about b. Given the data describing a single crack, say, a sequence {(ai , N i )}in=1 , it is easy to construct a log-likelihood using the density given by (22) and estimate the parameters b, q and σ by maximum likelihood. The log-likelihood is 2 n 1 n ⎛ a1− b − ai1− b − (b − 1)q( N i − N 0 ) ⎞ i =1 ∑ L(b, q,σ ;{( ai , N i )} = −b ln ai − n ln σ − 2 ∑ ⎜ 0 ⎜ i =1 ⎝ (b − 1)σ ( N i − N 0 )1 / 2 ⎟ . ⎟ ⎠ (24) Inspection shows that this differs from the standard least-squares equation only in the term –b∑lna, where the subscript i has been dropped. The likelihood estimators are obtained by solving the equations 232 The 7th International Conference “RELIABILITY and STATISTICS in TRANSPORTATION and COMMUNICATION - 2007” dL/db =0; dL/dq =0; dL/dσ =0. (25) In this case the equations have no closed solution. However, it is easy to see that the estimators for q and σ given b are the usual least-squares estimators for the coefficients in (23) conditioned on b, −1 1 ⎛ 1− b ⎞⎛ ⎞ n n q (b) = ⎜ na0 − b −1 ⎜ ⎝ ∑ i =1 ai1− b ⎟⎜ ⎟⎜ ∑ ⎠⎝ i =1 ( Ni − N0 ) ⎟ , ⎟ ⎠ (26) 1 n [a1− b − ai1− b − q (b)(b − 1)( N i − N 0 )]2 σ 2 (b) = n(b − 1) 2 ∑ i =1 0 Ni − N0 , (27) and on substituting these back in the log-likelihood gives a function of b alone, n L(b) = −b ∑ ln a − n ln[σ (b)] − n / 2. i =1 i (28) Thus the technique is to search for the value of b that maximizes L(b) by estimating q and σ as functions of b and substituting in L(b). In this study a simple golden-section search worked very effectively. 6.3. Pooling Data When several experiments have been performed it is possible to combine the log-likelihood from each experiment to give estimators of the parameters of interest. Suppose that several experiments have been performed. Each experiment is labelled with j, j runs from 1 to m, and yields nj observations. The data are then a set of sequences {(ajk,Njk)}, with j=1, …, m, k=1, …, nj. The log- likelihood for the whole set of experiments is simply the sum of the log-likelihood for the individual cracks; writing Lj(bj,qj,σj) for the log-likelihood for the jth crack gives 2 1 ⎛ a0 − (b j − 1)q j ( Nk − N0 ) ⎞ nj 1− b j 1− b j nj − ak ∑ L j (b j , q j , σ j ;{(a jk , N jk )}) = −b j ln a jk − n j ln σ j − ⎜ 2 k =1 ⎜ ∑ (b j − 1)σ j ( N − N0 )1 / 2 ⎟ (29) ⎟ k =1 ⎝ ⎠ and m L= ∑ L (b , q ,σ j =1 j j j j ;{( a jk , N jk )}). (30) The global log-likelihood can be used to investigate explicit parametric models for the parameters, or simply as a way to pool data. Estimation by maximum likelihood proceeds exactly as above; the qj and σj are obtained as ordinary least-squares estimators from equations like (26) and (27), one for each crack, and substituted back into the log-likelihood to yield m nj m m 1 L(b1 , b2 , ..., bm ) = − ∑ ∑j =1 bj k =1 ln a jk − ∑ j =1 n j ln[σ j (b j )] − 2 ∑n . j =1 j (31) When the cracks are all assumed to be independent with distinct parameters the estimators from the joint log-likelihood are precisely those obtained by estimating from each separately as outlined above. 233 Session 5. Aviation If a common value of b is used and the qj and σj are assumed to absorb most of the experimental variability, the joint log- likelihood reduces to m nj m m 1 L(b) = −b ∑∑ j =1 k =1 ln a jk − ∑ j =1 n j ln[σ j (b)] − 2 ∑n . j =1 j (32) 7. STOCHASTIC FATIGUE-CRACK GROWTH PARAMETER AND LIFETIME VARIABILITY MODELS These models allow one to describe the uncertainties in the fatigue-crack growth of Paris- Erdogan’s law via crack growth equation with a stochastic noise dependent, in general, on the crack size, and parameters modelled as random variables in order to characterize the random properties, which seem to vary both from specimen to specimen and during crack growth (see Fig. 4). In other words, the stochastic fatigue-crack growth parameter and propagation lifetime variability model (with respect to the parameters B and Q, modelled as random variables, and the stochastic noise V dependent, in general, on the crack size a) may be given, for example, as 1 ⎛ 1 1 ⎞ N − N0 = ⎜ B −1 − B −1 ⎟ + V . (33) (B − 1)Q ⎝ 0 ⎜a a ⎟ ⎠ 7.1. Crack Exceedance Probability In this case, the probability that crack size a(N) will exceed any given (say, maximum allowable) crack size a• can be derived and expressed as ⎧ ⎡ ⎛ δ ⎫ • ⎪ ⎢ ⎜ [a( N 0 )]− (b −1) − [a • ]− (b −1) ⎞ ⎤ ⎪ ⎟ ⎥⎬ . Pr{a ( N ) > a } = E ⎨exp − ⎜ ⎟ (34) ⎪ ⎢ ⎝ σ (b − 1)( N − N 0 + V ) ⎠ ⎥ ⎪ ⎩ ⎣ ⎦⎭ 8. EXAMPLE Let us assume that a fatigue-sensitive component (outboard longeron, Fig. 6)) has been found cracked on n=10 aircraft within a warranty period. Here a fleet of ten aircraft have all been inspected. Table 1. Inspection results Aircraft Flight hours Crack size (mm) (Ni) (ai) 1 3000 1 2 2300 0.5 3 2200 1 4 2000 2 5 1500 0.8 6 1500 1.5 7 1300 1 8 1100 1 9 1000 1 10 800 1 234 The 7th International Conference “RELIABILITY and STATISTICS in TRANSPORTATION and COMMUNICATION - 2007” Figure 6. Location of cracking on the F/A-18 Y497 upper outboard longeron It is assumed that cracks start growing from the time the aircraft entered service. For typical aircraft metallic materials, an initial discontinuity size (a0) found through quantitative fractography is approximately between 0.02 and 0.05 mm. Choosing a typical value for initial discontinuity state (e.g., 0.02 mm) is more conservative than choosing an extreme value (e.g., 0.05 mm). This implies that if the lead cracks can be attributed to unusually large initiating discontinuities then the available life increases. We test a goodness of fit of the data of Table 1 with the Weibull’s fatigue-crack growth parameter variability model (see Case 1), where a1−b − ai1−b Qi (b) = 0 , i = 1(1)n, (35) (b − 1)( N i − N 0 ) with a common value of b, a0=0.02, and N0=0. 8.1. Goodness-of-Fit Testing We assess the statistical significance of departures from the Weibull’s model by performing empirical distribution function goodness-of-fit test. We use the S statistic [7]. For complete datasets, the S statistic is given by n −1 ⎛ ln(Qi +1 (b) / Qi (b)) ⎞ ∑ ⎜ ⎜ i =[ n / 2 ]+1 ⎝ Mi ⎟ ⎟ ⎠ = 0.43, S (b) = n−1 (36) ⎛ ln(Qi +1 (b) / Qi (b)) ⎞ ∑i =1 ⎝ ⎜ ⎜ Mi ⎟ ⎟ ⎠ 235 Session 5. Aviation where [n/2] is a largest integer ≤ n/2, Qi is the ith order statistic, the values of Mi are given in Table 13 [7]. The rejection region for the α level of significance is {S>Sn;1-α}. The percentage points for Sn;1-α were given in [7]. The value of b is one that minimizes S(b). For this example, b = 0.87 and S=0.43 < Sn=10; 1-α=0.95=0.69. (37) Thus, there is not evidence to rule out the Weibull’s model. Using the relationship (4), the inspection results can be extrapolated from the expected initial crack size, a0, to the time of the next inspection when the maximum allowable crack size is equal to a•=10 mm as presented in Table 2. Table 2. Predicted next inspection results Aircraft Maximum allowable Next inspection time crack size a• (mm) (flight hours) 1 10 5626 2 10 5503 3 10 4126 4 10 3033 5 10 3030 6 10 2477 7 10 2438 8 10 2063 9 10 1875 10 10 1500 CONCLUSION The authors hope that this work will stimulate further investigation using the approaches on specific applications to see whether obtained results with it are feasible for realistic applications. ACKNOWLEDGMENTS This research was supported in part by Grant No. 06.1936 and Grant No. 07.2036 from the Latvian Council of Science and the National Institute of Mathematics and Informatics of Latvia. References 1. Jones, R., Molent, L. and Pitt, S. Studies in Multi-Site Damage of Fuselage Lap Joints, J. Theor. Appl. Fract. Mech.,Vol. 32, 1999, pp. 18-100. 2. Bogdanoff, J.L. and Kozin, F. Probabilistic Models of Cumulative Damage. New York: Wiley, 1985. 3. Yang, J.N. and Manning, S.D. A Simple Second Order Approximation for Stochastic Crack Growth Analysis, Engineering Fract. Mech., Vol. 53, 1996, pp. 677-686. 4. Wu, W.F., Ni, C.C. and Liou, H.Y. Random Outcome and Stochastic Analysis of Some Fatigue Crack Growth Data, Chin. J. Mech., Vol. 17, 2001, pp. 61-68. 5. Paris, R. and Erdogan, F. A Critical Analysis of Crack Propagation Laws, Journal of Basic Engineering, Vol. 85, 1963, pp. 528-534. 6. Nechval, N.A., Nechval, K.N. and Vasermanis, E.K. Statistical Models for Prediction of the Fatigue Crack Growth in Aircraft Service. In: Fatigue Damage of Materials 2003 (ed. by A. Varvani-Farahani & C. A. Brebbia). Southampton, Boston: WIT Press, 2003, pp. 435-445. 7. Kapur, K.C. and Lamberson, L.R. Reliability in Engineering Design. New York: Wiley, 1977. 236