PLANNING OF INSPECTION PROGRAM OF FATIGUE-PRONE AIRFRAME

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1 Yu. aramonov, A. Kuznetsov ANING OF INSECTION ROGRAM OF FATIGUE RONE AIRFRAME (Vol. 2008, Deember ANNING OF INSECTION ROGRAM OF FATIGUE-RONE AIRFRAME Yu. aramonov, A. Kuznetsov Aviation Institute, Riga Tehnial University, Riga, atvia omonosova Str., Riga, atvia Abstrat. To keep the atigue ageing ailure probability o an airrat leet on or below the ertain level an inspetion program is appointe to isover atigue raks beore they erease the resiual strength o the airrame lower the level allowe by regulations. In this artile the Minimax approah with the use one- an two-parametri Monte Carlo moelling or alulating ailure probability in the interval between inspetions is oere. Keywors: ailure probability, Minimax approah, inspetion program, approval test INTRODUCTION Inspetion program evelopment shoul be mae on the base o proessing o approval lietime test result, when we shoul make some reesign o the teste system i any requirement is not met. Here we onsier some example o p-set untion appliation to the problem o evelopment an ontrol o inspetion program. We make assumption that some Strutural Signiiant Item (SSI, the ailure o whih is the ailure o the whole system, is haraterize by a ranom vetor (r.v. ( T, T, where T is ritial lietime (up to ailure, T is servie time, when some amage (atigue rak an be etete. So we have some time interval, suh that i in this interval some inspetion will be ulille, then we an eliminate the ailure o the SSI. We suppose also that a require operational lie o the system is limite by so-alle Speiie ie (S, t S, when system is isare rom servie. -set untion or ranom vetor is a speial statistial eision untion, whih is eine in ollowing way. et Z an X are ranom vetors o m an n imensions an we suppose that it is known the lass {, Ω} to whih the probability istribution o the ranom vetor W=(Z,X is assume to belong. The only thing we assume to be known about the parameter is that it lies in a ertain set Ω, the parameter spae. I S ( x = S, ( x is suh set o isjoint sets o z values as i Z untion o x that sup ( Z S Z, i ( x p then statistial eision untion S z (x is p-set untion or r.v. Z on the base o a sample x=(x,...,x n. ater on the value x, observation o the vetor X, woul be interprete as estimate ˆ o parameter, Z woul be interprete as some ranom vetor-harateristi o some SSI in servie: Z = T, T. U i Z i ( FATIGUE CRACK GROWTH MODE The atigue rak growth proess lows in aorane with quite ompliate rules, whih epen on a big number o ators. An analytial approah esribing that proess oul be onsiere as almost impossible. Nevertheless, it an be shown, that in general ase rak growth proess oul be well enough approximate with the ormula: Qt a( t = α e,. where a(t is a atigue rak size at time t (the number o light bloks; α is so alle equivalent initial rak size (as i the airrame has been initially proue with the rak o suh small size; α orrespons to the best it o test ata; an parameter Q eines the spee o growth o atigue rak an epens on the loaing moe (on the stress range in ase o yling loaing. For urther nees, let us take a logarithm o both let an right sies o equation :

2 Yu. aramonov, A. Kuznetsov ANING OF INSECTION ROGRAM OF FATIGUE RONE AIRFRAME (Vol. 2008, Deember ln a ( t = lnα + Qt. 2. Thus, ln a( t lnα t =, 3. Q so the time when rak beomes etetable an the time when rak reahes its ritial size an be alulate as ln a C T = et lnα ln a =, rit lnα C T Q Q = =, 4. Q Q where a et is a rak size, at whih hanes to isover it tens to unit, a rit is a rak size, whih orrespons to the minimum resiual strength o an airrat omponent allowe by regulations, T is a time or rak to grow to its etetable size an T is a time or rak to grow to its ritial size, C an C are appropriate onstants. et us eine ailure as the situation, when we were unable to isover raks with the size a et a < a rit, or, in other wors, i there are no inspetions perorme in [T ; T ] time interval. It is lear, that varying the number o inspetions 0 we will n in the servie interval [ ] inspetions isover a ierent number o raks; thereore, the estimate o ailure probability will vary as well. Unortunately, we on t know the real values o parameters, so we are using theirs estimates rom a small number (one, selom two o available observations (atigue raks uring atigue test instea.,t S USING MONTE CARO MODEING TO ESTIMATE FAIURE ROBABIITY We use the Monte Carlo metho to generate a set o raks to be proesse in aorane with proeure, esribe in Setion 0. The parameters or moelling an be erive rom the ull-sale atigue tests or rom other real rak observations. We an never know how the ertain atigue rak urve will look like. Thus, perorming approximation o that atigue rak urve with a ertain moel, the atigue rak growth moel parameters (FCGM we have two FCGM parameters X = ln Q an Y = ln CC will vary as well, so they are ranom values, an these ranom values have theirs own parameters o istribution. To perorm Monte Carlo moelling o the atigue rak growth proess we have to know FCGMs istribution types an parameters, i.e.... o eah FCGM. From the analysis o the atigue test ata it an be assume, that the logarithm o time require the rak to grow to its ritial size is istribute normally: 2 lnt ~ N( μ T, σ. 5. T From ormulas 2 an 4 ollows: lnt = ln C ln Q. 6. From aitive property o normal istribution omes that ln T oul be normally istribute either i both ln C an ln Q are normally istribute: 2 2 X = ln Q ~ N( μ X, σ X, Y = ln C ~ N( μy, σ Y, 7. or i one o them is normally istribute while another one is a onstant. Thus, the value o logarithm o our FCGM parameters is istribute normally or, on other wors, FCGM parameters have a log-normal istribution. To get estimates o FCGM istribution parameters ( μˆ an σˆ we onsier statistis o several rak observations. For eah o those raks we alulate estimates o istribution parameters lnq an lnc, an then gather all ata together into the table with two olumns: lnq an lnc. From that table we then erive estimates o mean value an stanar eviation or eah olumn, as well as estimate o orrelation between lnq an lnc. The Monte Carlo moelling in at means the proess o getting a big number o pairs [T ; T ] with upper mentione speii istribution parameters. Having the array o [T ; T ] pairs we apply an inspetion

3 Yu. aramonov, A. Kuznetsov ANING OF INSECTION ROGRAM OF FATIGUE RONE AIRFRAME (Vol. 2008, Deember program looking or ailures situations, when both T an T are loate between two onsequent inspetions. For eah interval between inspetions [t i- ;t i ] ailure probability will be i = ( ti < T T < ti, 8. an or the entire inspetion program = i. 9. i MINIMAX DECISION MAKING AROACH As it was state above, the goal is to evelop an inspetion program, eine by the vetor o inspetion time moments t r = ( t, t2, t n, 0. i.e. to in a vetor untion t r (ˆ (ˆ is the estimate o FCGM istribution parameters, n is the total number o inspetions per inspetion program, so tn = t S that limits airrat ailure probability at the require level require with the minimum inspetions n unertaken in servie interval [ 0,t S ] ( tn = t S. In mathematial terms that an be presente as: r sup, t,. ( ( require where n, r =. 2. ( t ( Ti T T < Ti i= In expression 2 T, T2, K, T n are time moments o inspetions: ranom value r T = ( T,, ( ˆ T2, Tn = t, 3. where T 0 = 0, Tn = t S, an n = 0,, 2, K. The expression n = symbolially means that the airrat must be returne or reesign to the esign oie. The inspetion program einition vetor t r ( ˆ is a untion, where both number o inspetions uring servie interval n an isposition o inspetion time moments T, T2, K, T n uring [ 0,t S ] are to be hosen as a untion o ˆ an some limitations. It is lear, that there might be many ways how to position inspetion time moments on [ 0,t S ] or a partiular n. et us apply the ollowing inspetion time moment isposition rule R D : the time o the irst inspetion T ( T is a ranom value beause it is a untion o ˆ will be eine by proeure similar to the sae lie approah (probability o ailure without inspetions is less than some small value, while all remaining inspetions are istribute evenly in the interval [ T,t S ]. O ourse, this rule R D in general ase oes not minimise the total require number o inspetions n ; there are other rules that are more optimal, but our hoie o rule R D is ause by its simpliity or urther appliations; inspetion programs reate by this rule are urrently use in pratie or ommerial jet airrats. To apply a partiular rule R D we have to in the total require number o inspetions n, whih epens on the limiting value o the ailure probability = R, where require reliability require require R require is manate, or example, by JAR regulations. As it was shown above, the ailure probability is a untion o the number o inspetions n an, n, n monotonially ereases when the parameter ; let us enote it as (. We also suppose that (

4 Yu. aramonov, A. Kuznetsov ANING OF INSECTION ROGRAM OF FATIGUE RONE AIRFRAME (Vol. 2008, Deember number o inspetions n inreases (at least when n is large enough an lim (, n = 0 n or all. et n is a solution o the equation (, n =. 4. require Then let us enote (, = n( n =, 5. require require, n as the minimal inspetion number at whih ailure probability ( require n ˆ = n ˆ, an ˆ (, nˆ the in unknown, so (. But the true value o require = are ranom values. We suppose that we begin the ommerial proution an operation o airrats only i some speii requirements to reliability are met. For the simplest ase there is a limitation or the maximum allowe number o inspetions n max : we will return airrame projet or reesign as unproitable in ase, when the require number o inspetions in the inspetion program n exees n max (we nee to inspet airrat too oten to ensure require reliability. It an be assume, that the probability o ailure or the returne projets is equal to zero, i.e. ( ˆ ˆ ˆ, n, n nmax orrete =. 6. 0, nˆ > nmax In the more omplex ase there is a set o limitations. For example, in aition to limitation on the expete number o inspetions nalulate = nˆ we will return airrame projet or reesign i estimate o expetation value o T ( T alulate is too small in omparison with t S (breaking minimum threshol T min ; i estimate o time between two onsequent inspetions Δ talulate is smaller than a threshol Δ tmin ; i estimate o initial equivalent rak size α alulate exees rak etetable size a et an so on. et us enote r r the vetor o alulate values o limiting values = ( ˆ as nalulate r Δtalulate =, 7. T alulate α alulate an the set o its allowe values D as ( 0, nmax ] [ Δtmin, D =. 8. [ T, min [ 0, aet ] Atually, the number o elements in r an, thereore, the number o imensions in the set o the allowe values D may vary epening on moelling situation an speii requirements. For example, or inspetion programs with the equal time between inspetions in the whole servie interval [ 0,t S ] the time between two onsequent inspetions Δ t = ts / n, so it an be exlue rom the set o limitations, but it is important in programs when the time between inspetions may vary. I vetor o limiting values r oes not math the set o its allowe values D, then the projet is onsiere as unproitably an is returne bak or reesign in the esign oie. As we state above, the probability o ailure or returne projets is equal to zero, thus r r ( D ˆ,, orrete = r. 9. 0, D

5 Yu. aramonov, A. Kuznetsov ANING OF INSECTION ROGRAM OF FATIGUE RONE AIRFRAME (Vol. 2008, Deember The parameter, whih eines the... o vetor ( T, T, is a vetor parameter. For onsiere ase in this work, i both rak moel parameters are ranom an have normal istribution, it onsists o ive omponents: = [, 0,,, ], ln ln 0 ln r 20. C C Q ln Q where 0 stans or a loation an stans or a sale parameter o the appropriate rak growth moel parameter ln C or ln Q ; r is a oeiient o orrelation between ln C an ln Q, an Θ = (, ; [ 0, ; (, ; [ 0, ; [ 0, ]. As it was shown beore, we on t know the real value o, thus we use its estimate ˆ. A part o elements o ˆ may be assume as known. For example, ln C, ln Q an orrelation oeiient r an be onsiere as onstants, so proessing atigue rak growth ata we shoul estimate only two remaining parameters 0 ln C an 0 ln Q. It an be shown that or onsiere eision making proeure ranom variable ˆ orrete has expetation value, whih is a untion o, an this untion has a maximum value or Θ. To prove that let us ix one o two rak moel parameters an look how the probability o ailure epens on another one. et us onsier that the equivalent initial rak size is a onstant: α = onst, i.e. 0 α = μ α = onst, α = σ α = 0. In aorane with upper eine rules the probability o ailure tens to zero when the rak growth spee representing parameter 0 = Ε{ lnq} tens to zero: this is a ase when the item is extremely reliable ln Q an raks are growing so slowly, that have no hane to grow up to a rit in interval [ 0,t S ], thus there are no inspetions require. The ailure probability without inspetions is eine by ormula: lnt S μlnt = = Φ ( T wi ts, 2. σ lnt or, in terms o reliability, lnts μln T = > = = Φ R wi ( T ts ( T ts, 22. σ ln T 2 where ln T is istribute normally as lnt ~ N( μ lnt ; σ lnt. From other sie, i the 0 ln Q is high, then the probability o ailure tens to zero as well: with high r probability we return or reesign all items ue to the break o limiting rules, i.e. D (see ormulas 7, 8 an 9. Between these zero values o Ε{ orrete } there an be non-zero values somewhere in between, when the atigue raks maybe an reah theirs ritial size uring the time between inspetions, maybe not, but there are no suiient reasons to return projet or reesign so ar. et us all a value o ailure probability use or alulations (at the hoie o the number o inspetions require, or hoosing vetoruntion t r as al. The ollowing onlusion an be mae rom the upper mentione: the epenene o the probability o ailure as a untion o is a untion whih has a maximum, the value o that maximal value is unknown, but somehow epens on the value o ailure probability use or alulations. et us all the value o expetation o ailure probability or all allowable as Ε { } name it as orrete to istinguish it rom al goal is to in suh a maximum value o ailure probability or alulations al ˆ orrete. We have, beause we take into onsieration some limitations. The * al that the orrete value o

6 Yu. aramonov, A. Kuznetsov ANING OF INSECTION ROGRAM OF FATIGUE RONE AIRFRAME (Vol. 2008, Deember ailure probability orrete oes not exee the require limiting value o ailure probability require = R require : * ~ : ( al al, 23. require where ~ ( = max Ε ˆ al. 24. orrete Graphially this approah or a two-imensional ase (when either α = onst or Q = onst is presente in Figure 3: Figure 3. Minimax approah example ( α = onst or ln Q = onst For the more omplex ase we get a three-imensional piture like in Figure 4: Figure 4. Minimax approah example (general ase Depening on parameters the shapes o these two- or three-imensional ailure probability urves may vary, but this oes not aet our onlusions

lak o the initial atigue test ata. There are examples o numerial moelling or one- an two-parametri moels shown in Figure 5 an Figure 6 below (please note: in pitures Q=N(Q= 0 ln Q, C=N(C= 0 C.

7 Yu. aramonov, A. Kuznetsov ANING OF INSECTION ROGRAM OF FATIGUE RONE AIRFRAME (Vol. 2008, Deember NUMERICA EXAME AND CONCUSION The upper mentione approah lets us to ensure reliability o the airrame on or above the require level by eveloping appropriate inspetion program or the ase o lak o the initial atigue test ata. There are examples o numerial moelling or one- an two-parametri moels shown in Figure 5 an Figure 6 below (please note: in pitures Q=N(Q= 0 ln Q, C=N(C= 0 C..6 x 0-3 ailure as a untion o n(q or T m =.0 an N inspmax =999 00%.4.2 ailure Max( orr = req = al = a et =20.00mm a rit =237.84mm T sl =40000h N MC 7:02:33 orrete % reesigne % reesigne, Reliability WI Reliability WI % -7.8 Q Figure 5. One-parametri numerial example ( α = onst Figure 6. Two-parametri numerial example (3D an projetion

8 Yu. aramonov, A. Kuznetsov ANING OF INSECTION ROGRAM OF FATIGUE RONE AIRFRAME (Vol. 2008, Deember REFERENCES. MSG-3. Operator/Manuaturer Sheule Maintenane Development. Revision // Air Transport Assoiation o Ameria, In. Washington, D.C., Y.aramonov, A.Kuznetsov. Fatigue rak growth parameter estimation by proessing inspetion results // In: Thir International onerene on Mathematial Methos in Reliability MMR2002 Tronheim, Norway, Y.aramonov, A.Kuznetsov. Inspetion ata use or airrame inspetion interval orretion // In: Aviation Issue #6 Vilnius, Tehnika, Y.aramonov, A.Kuznetsov. lanning o inspetions o atigue-prone airrame // In: roeeings o International onerene Diagnosis o tehnial systems, numerial an physial non-estrutive quality testing 2004 Vilnius, April 23, Y.aramonov, A.Kuznetsov. Swithing to ouble airrat inspetion requeny strategy analysis or exponential atigue rak growth moel // In: roeeings o International onerene on ongevity, Ageing an Degraation moels in Reliability, Meiine an Biology (AD2004, Volume, pp St. etersburg, Russia,

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