RELIABILITY, RISK AND AVAILABILITY ANLYSIS OF A CONTAINER GANTRY CRANE ABSTRACT


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1 Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March RELIABILITY RIK AND AVAILABILITY ANLYI OF A CONTAINER GANTRY CRANE Joanna oszynskabudny. Gdyna Martme Unversty Gdyna Poland ABTRACT The jont model of the system operaton process and the system multstate relablty s appled to the relablty rsk and avalablty evaluaton of the contaner gantry crane. The contaner gantry crane s descrbed and the mean values of the contaner gantry crane operaton process uncondtonal sojourn tmes n partcular operaton states are found and appled to determnng ths process transent probabltes n these states. The contaner gantry crane dfferent relablty structures n varous ts operaton states are fxed and ther condtonal relablty functons on the bass of data comng from experts are approxmately determned. Fnally after applyng earler estmated transent probabltes and system condtonal relablty functons n partcular operaton states the uncondtonal relablty functon the mean values and standard devatons of the contaner gantry crane lfetmes n partcular relablty states rsk functon and the moment when the rsk exceeds a crtcal value are found. Next the renewal and avalablty characterstcs for the consdered gantry crane are determned. INTRODUCTION Most real techncal systems are very complex and t s dffcult to analyze ther relablty avalablty and safety. Large numbers of components and subsystems and ther operatng complexty cause that the dentfcaton evaluaton predcton and optmzaton of ther relablty avalablty and safety are complcated. The complexty of the systems operaton processes and ther nfluence on changng n tme the systems structures and ther components relablty characterstcs are very often met n real practce. Takng nto account the mportance of the safety and operatng process effectveness of such systems t seems reasonable to expand the twostate approach to multstate approach (Aven 98 Kolowrock Kołowrock oszyńskabudny ) n ther relablty analyss. The assumpton that the systems are composed of multstate components wth relablty states degradng n tme gves the possblty for more precse analyss of ther relablty and operaton processes effectveness. Ths assumpton allows us to dstngush a system relablty crtcal state (Kolowrock Kołowrock oszyńska Kołowrock oszyńskabudny oszyńska ) to exceed whch s ether dangerous for the envronment or does not assure the necessary level of ts operaton process effectveness. Then an mportant system relablty characterstc s the tme to the moment of exceedng the system relablty crtcal state and ts dstrbuton whch s called the system rsk functon. Ths dstrbuton s strctly related to the system multstate relablty functon that s basc characterstcs of the multstate system. The complexty of the systems operaton processes and these processes nfluence on changng n tme the systems structures and ther components relablty characterstcs (Kołowrock oszyńska Kołowrock oszyńskabudny Kołowrock oszyńskabudny oszynska ) s often very dffcult to fx and to analyse. A convenent tool for solvng ths problem s semmarkov (Grabsk Lmnos ) modellng of the systems operaton processes whch s proposed n the paper. Usng the jont model of the system multstate relablty and the system semmarkov operaton process (Kołowrock oszyńska Kołowrock oszyńskabudny oszyńska ) t s possble to pont out the varablty of system
2 Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March components relablty characterstcs by ntroducng the components condtonal multstate relablty functons determned by the system operaton states. Consequently t s possble to fnd the system condtonal relablty functon dependent on the operaton states and next t s possble to fnd ts uncondtonal relablty functon and renewal and avalablty characterstcs. Ths way the obtaned results concerned wth multstate systems n ts varyng n tme operaton states can be appled to the relablty rsk and avalablty evaluaton of the contaner gantry crane. CONTAINER GANTRY CRANE YTEM ANALYI We analyse the relablty of the contaner gantry crane that s operatng at the contaner termnal placed at the seashore (Kołowrock oszyńskabudny Kołowrock oszyńskabudny ). The consdered contaner termnal s engaged n transshpment of contaners. The loadng of contaners s carred out by usng the gantry cranes called hptohore (T). We consder the T contaner gantry crane that s composed of basc subsystems and havng an essental nfluence on ts relablty. Those subsystems are as follows:  the power supply subsystem  the control and montorng subsystem  the arm gettng up and gettng down subsystem  the transferrng subsystem  the loadng and unloadng subsystem. The gantry crane power supply subsystem conssts of:  a hgh voltage cable delverng the energy from the substaton to the gantry crane  a drum allowng the cable unreelng durng the crane transferrng  an nner crane power supply cable E E  a devce transmttng the energy from the hgh voltage cable to the nner crane cable  man and supportng voltage transformers E  a low voltage power supply cable E  relayng and protectve electrcal components E 7. E E The gantry crane control and montorng subsystem conssts of:  a crane software controller precsely analyzng the stuaton and takes sutable actons n order to assure correct work of the crane E  a measurng and dagnostc devce sendng sgnals about the crane state to the software controller E  a transmtter of sgnals from the controller to elements executng the set of commands E  devces carryng out the controller s orders (a permsson to work a blockade of work etc.) E  control panels (an engne room an operator s cabn a crane arm cabn) E  control and steerng cables connectons E. The gantry crane arm gettng up and gettng down subsystem conssts of: ()  a propulson unt (an engne a rope drum a transmsson gear a clutch breaks a rope) E ()  a set of rollers and multwheels E ()  a crane arm (jonts hooks fastenng the arm) E. The gantry crane transferrng subsystem conssts of: 7
3 Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March  a drvng unt (an engne a clutch breaks a transmsson gear gantry crane wheels) The gantry crane loadng and unloadng subsystem conssts of the wnch unt  a propulson unt (an engne a clutch breaks a transmsson gear ropes)  a wnch head (whch a contaner grab s connected to)  a contaner s grab  a contaner s grab stablzng unt () and the cart unt E composed of:  a propulson unt (an engne a clutch breaks a transmsson gear cart wheels ropes)  rals whch cart s movng on durng the operaton  a crane cart. E. E composed of: The subsystems are formng a general seres gantry crane relablty structure presented n Fgure. ()... Fgure. General scheme of gantry crane relablty structure CONTAINER GANTRY CRANE OPERATION PROCE CHARACTERITIC PREDICTION The contaner gantry crane relablty structure and the subsystems and components relablty depend on ts changng n tme operaton states. Takng nto account expert opnons on the varyng n tme operaton process of the consdered contaner gantry crane we fx the number of the system operaton process states v and we dstngush the followng as ts sx operaton states: an operaton state z the crane standby wth the power supply on and the control system off an operaton state z the crane prepared ether to startng or fnshng the work wth the crane arm angle poston of 9 o an operaton state z the crane prepared ether to startng or fnshng the work wth the crane arm angle poston of o an operaton state z the crane transferrng ether to or from the loadng and unloadng area wth the crane arm angle poston of 9 o an operaton state z the crane transferrng ether to or from the loadng and unloadng area wth the crane arm angle poston of o an operaton state z the contaners loadng and unloadng wth the crane arm angle poston of o. Moreover we fx that there are possble the transtons between all system operaton states. To dentfy all parameters of the contaner gantry crane operaton process the statstcal data about ths process was needed. All statstcal data are collected n (Kołowrock oszyńskabudny ). On the bass of statstcal data from (Kołowrock oszyńskabudny ) the followng matrx 8
4 Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March [ p bl ] of the probabltes p b l... of the contaner gantry crane operaton process transtons bl from the operaton state z b to the operaton state z l has been fxed. The mean values M bl E[ bl ] b l... b l of the contaner gantry crane operaton process condtonal sojourn tmes at the partcular operaton states accordng to (.) (Kołowrock oszyńskabudny ) are as follows: M.98 M.8 M M M 7.89 M 9. M. M M. M. M.8 M 7.8 M M. M M.9 M.8 M.. M. M.. After consderng the results and applyng the formula (.) from (Kołowrock oszyńska Budny ) the uncondtonal mean sojourn tmes of the contaner gantry crane operaton process at the partcular operaton states are gven by: M [ E ] M E ] [ M E[ ] M [ E ] M E ] [ M E ] [ nce accordng to (.) (Kołowrock oszyńskabudny ) from the system of equatons we get [ ] [ ] p bl x
5 Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March Then the lmt values of the transent probabltes p b (t) of the gantry crane operaton process at the operaton states z b accordng to (.) n (Kołowrock oszyńskabudny ) are gven by p.87 p.87 p. p. p.77 p.7. CONTAINER GANTRY CRANE IN VARIABLE OPERATION CONDITION RELIABILITY AND RIK EVALUATION After dscusson wth experts takng nto account the effectveness of the operaton of the contaner gantry crane we fx that the system and ts components have four relablty states.e. z. And consequently at all operaton states z b b... we dstngush the followng relablty states of the system and ts components: a relablty state the gantry operaton s fully effectve a relablty state the gantry operaton s less effectve because of ageng a relablty state the gantry operaton s less effectve because of ageng and more dangerous a relablty state the gantry s destroyed. We assume that there are possble the transtons between the components relablty states only from better to worse ones and we fx that the system and components crtcal relablty state s r. Consequently we assume that the gantry crane subsystems... are composed of fourstate components.e. z = wth the multstate relablty functons [ R ( ) ( t )] ( b) ( ) ( b) = [ [ R ( ) ( b) ( t)] [ R ( ) ( b) ( t)] [ R ( t)] ] b... wth exponental coordnates [ ( ) ( b) R ( t)] [ ( ) ( b) R ( t)] [ ( ) R ( t)] ( b) dfferent n varous operaton states z b b.... In (Kołowrock oszyńskabudny ) on the bass of expert opnons the relablty functons of the gantry crane components n dfferent operaton states are approxmately determned. We wll use them n our further system relablty analyss and evaluaton. At the system operaton state z the contaner gantry crane s composed of the subsystem whch s a seres system composed of n 7 components E.. 7 (subsystems) wth the structure showed n Fgure. Fgure. The scheme of the contaner gantry crane at operaton state z Thus at the system operaton state z the contaner gantry crane s dentcal wth subsystem that s a fourstate seres system wth ts structure shape parameter n 7 and accordng to (.) (.) (Kołowrock oszyńskabudny ) ts fourstate relablty functon s gven by the vector
6 Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March [ R ( t )] [ [ t)] [ R ( t)] [ R ( t)] ] t wth the coordnates [ R ( t)] = exp[.t]exp[.t]exp[.t]exp[.t]exp[.t] exp[.8t]exp[.t] = exp[.9t] () [ R ( t)] = exp[.t]exp[.t]exp[.t]exp[.t]exp[.t] exp[.8t]exp[.t] = exp[.t] [ R ( t)] = exp[.t]exp[.t]exp[.t] exp[.t]exp[.t] exp[.t] exp[.t] = exp[.t]. () The expected values of the contaner gantry crane condtonal lfetmes n the relablty state subsets { } { } { } at the operaton state z calculated from the results gven by ()() accordng to (.8) (Kołowrock oszyńskabudny ) respectvely are: ()..79 ().7 years. At the system operaton states z and z the contaner gantry crane s composed of the subsystems and formng a seres structure shown n Fgure. The subsystem s a seres system composed of n 7 components E.. 7 the subsystem s a seres system composed of n components E.. and the subsystem s a seres system () composed of n components E. Fgure. The scheme of the contaner gantry crane at operaton states z and z Thus at the system operaton state z the contaner gantry crane s composed of the subsystems and formng a seres structure. At ths operaton state the subsystem s a fourstate seres system wth ts structure shape parameter n 7 and accordng to (.)(.) (Kołowrock oszyńskabudny ) ts fourstate relablty functon s gven by the vector [ R ( t )] [ [ R ( t)] [ R ( t)] [ R ( t)] ] t
7 Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March wth the coordnates [ R ( t )] =exp[.t] exp[.t] exp[.t] exp[.t] exp[.t] exp[.8t] exp[.t] = exp[.9t] (7) [ R ( t )] = exp[.t] exp[.t] exp[.t] exp[.t] exp[.7t] exp[.8t] exp[.t] = exp[.t] (8) [ R ( t )] = exp[.t] exp[.t] exp[.t] exp[.t] exp[.8t] exp[.t] exp[.t] = exp[.7t]. (9) The subsystem at the operaton state z s a fourstate seres system wth ts structure shape parameter n and accordng to (.)(.) (Kołowrock oszyńskabudny ) ts fourstate relablty functon s gven by the vector [ R ( t )] [ [ R ( t)] [ R ( t)] [ R ( t)] ] t wth the coordnates [ R ( t )] = exp[.t]exp[.8t]exp[.8t] exp[.8t] exp[.t]exp[.8t] = exp[.t] () [ R ( t )] = exp[.9t] exp[.t] exp[.t] exp[.t] exp[.t] exp[.9t] = exp[.7t] () [ R ( t )] = exp[.t] exp[.9t] exp[.9t] exp[.9t] exp[.t]exp[.t] = exp[.t]. () The subsystem at the operaton state z s a fourstate seres system wth ts structure shape parameter n and accordng to (.)(.) (Kołowrock oszyńskabudny ) ts fourstate relablty functon s gven by the vector [ () () R ( t )] [ [ R ( t)] () () [ R ( t)] [ R ( t)] ] t wth the coordnates () R ( )] = exp[.t] exp[.t] exp[.t] = exp[.9t] () () R ( )] = exp[.t] exp[.t] exp[.t] = exp[.t]
8 Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March () R ( )] = exp[.t] exp[.t] exp[.t] = exp[.98t]. () Consderng that the contaner gantry crane at the operaton state z s a fourstate seres system composed of subsystems and after applyng (.)(.) (Kołowrock oszyńska Budny ) ts condtonal fourstate relablty functon s gven by the vector [ R ( t )] [ [ t)] [ R ( t)] [ R ( t)] ] t wth the coordnates R ( )] = exp[.9t] exp[.t] exp[.9t] = exp[.9t] () R ( )] = exp[.t] exp[.7t] exp[.t] = exp[.t] (7) R ( )] = exp[.7t] exp[.t] exp[.98t] = exp[.88t]. (8) The expected values of the contaner gantry crane condtonal lfetmes n the relablty state subsets { } { } { } at the operaton state z calculated from the results gven by ()(8) accordng to (.8) (Kołowrock oszyńskabudny ) respectvely are:.9. (). year. (9) After proceedng n the analogous way n the system relablty analyss and evaluaton at the remanng operaton states z z z and z we may determne the system condtonal relablty functon that are presented below. Fgure. The scheme of the contaner gantry crane at operaton states z and z Fgure. The scheme of the contaner gantry crane at operaton state z At the operaton state z the contaner gantry crane condtonal relablty functon of the system s gven by the vector [ () () R ( t )] [ [ t)] () () [ R ( t)] [ R ( t)] ] t
9 Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March wth the coordnates () R ( )] = exp[.9t]exp[.t] exp[.9t] = exp[.t] () () R ( )] = exp[.t] exp[.7t]exp[.t] = exp[.t] () () R ( )] = exp[.7t] exp[.t]exp[.98t] = exp[.889t]. () The expected values of the contaner gantry crane condtonal lfetmes n the relablty state subsets { } { } {} at the operaton state z calculated from the results gven by ()() accordng to (.8) (Kołowrock oszyńskabudny ) respectvely are:.9. ( ). year. () At the system operaton states z and z the contaner gantry crane s composed of the subsystems and formng a seres structure shown n Fgure. The subsystem s a seres system composed of n 7 components E.. 7 the subsystem s a seres system composed of n components E.. the subsystem s a seres system composed of () n components E and the subsystem conssts of a component E. Thus at the operaton state z the contaner gantry crane condtonal relablty functon of the system s gven by the vector [ R ( t )] [ [ t)] [ R ( t)] [ R ( t)] ] t wth the coordnates R ( )] = exp[.t] exp[.t] exp[.t] exp[.9t] = exp[.7t] () R ( )] = exp[.89t] exp[.78t] exp[.9t] exp[.t] = exp[.98t] () R ( )] = exp[.8t] exp[.8t] exp[.t] exp[.t]= exp[.9t]. () The expected values of the contaner gantry crane condtonal lfetmes n the relablty state subsets { } { } {} at the operaton state z calculated from the results gven by ()() accordng to (.8) (Kołowrock oszyńskabudny ) respectvely are:.8. (). year. (7) At the operaton state z the contaner gantry crane condtonal relablty functon of the system s gven by the vector [ () () R ( t )] [ [ t)] () () [ R ( t)] [ R ( t)] ] t wth the coordnates
10 Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March () R ( )] =exp[.t]exp[.t]exp[.t]exp[.t] = exp[.t] (8) () R ( )] =exp[.89t]exp[.78t] exp[.9t]exp[.9t] = exp[.87t] (9) () R ( )] =exp[.8t]exp[.8t]exp[.t]exp[.t] = exp[.99t]. () The expected values of the contaner gantry crane condtonal lfetmes n the relablty state subsets { } { } {} at the operaton state z calculated from the results gven by (8)() accordng to (.8) (Kołowrock oszyńskabudny ) respectvely are: ().8. (). year. At the system operaton state z the contaner gantry crane s composed of the subsystems and formng a seres structure shown n Fgure. The subsystem s a seres system composed of n 7 components E.. 7 the subsystem s a seres system composed of n components E.. the subsystem s a seres system composed of n () components E and the subsystem s a seres system composed of n components E. () Thus at the operaton state z the contaner gantry crane condtonal relablty functon of the system s gven by the vector [ () () R ( t )] [ [ t)] () () [ R ( t)] [ R ( t)] ] t wth the coordnates () R ( )] =exp[.t]exp[.t]exp[.87t]exp[.8t] =exp[.8t] () () R ( )] = exp[.7t] exp[.9t] exp[.t] exp[.t]= exp[.78t] () () R ( )] = exp[.9t] exp[.7t]exp[.t] exp[.t]= exp[.t]. The expected values of the contaner gantry crane condtonal lfetmes n the relablty state subsets { } { } {} at the operaton state z calculated from the results () accordng to (.8) (Kołowrock oszyńskabudny ) respectvely are: ()..8 ().98 year. In the case when the operaton tme s large enough the uncondtonal fourstate relablty functon of the contaner gantry crane s gven by the vector R ( t ) = [ R (t) R (t) R (t) ] t ()
11 Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March where accordng to (.)(.) (Kołowrock oszyńskabudny ) and consderng the vector coordnates are gven respectvely by t).87 [ t)].87 [ t)]. [ t)] (). [ t)] (7).77 [ t)] () ().7 [ t)] for t t ).87 [ t)].87 [ t)]. [ t)] (). [ t)].77 [ t)] () ().7 [ t)] for t (8) t ).87 [ t)].87 [ t)]. [ t)] (). [ t)].77 [ t)] () ().7 [ t)] for t (9) (b) and the coordnates [ R ( t u)] b... u are gven by ()() ()(8) ()() ()() (8)() (). The graphs of the coordnates of the contaner gantry crane relablty functon are presented n Fgure. Fgure. The graph of the contaner gantry crane relablty functon R ( t ) coordnates The expected values and standard devatons of the contaner gantry crane uncondtonal lfetmes n the relablty state subsets { } { } {) calculated from the results gven by (7) (9) accordng to (.7)(.9) (Kołowrock oszyńskabudny ) and consderng () (9) () (7) () () respectvely are: years ( ).7years
12 Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March ().7.8. years ( ). years () years ( ). years Further consderng (.) from (Kołowrock oszyńskabudny ) and the mean values of the uncondtonal lfetmes n the partcular relablty states respectvely are: ( ). ( ) ().8 ( ) (). years. () nce the crtcal relablty state s r = then the system rsk functon accordng to (.) (Kołowrock oszyńskabudny ) s gven by r(t) = t ) where t ) s gven by (8). Hence the moment when the system rsk functon exceeds a permtted level for nstance =. from (.) (Kołowrock oszyńskabudny ) s () = r (). year. The graph of the rsk functon r(t) of the contaner gantry crane operatng at the varable condtons s gven n Fgure 7. Fgure 7. The graph of the contaner gantry crane rsk functon r (t) CONTAINER GANTRY CRANE AVAILABILITY PREDICTION Usng the results of the contaner gantry crane relablty predcton gven by () and the results of the classcal renew theory presented n (Kołowrock oszyńskabudny ) we may predct the renewal and avalablty characterstcs of ths system n the case when t s reparable and ts tme of renovaton s ether gnored or nongnored. 7
13 Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March Frst assumng that the contaner gantry crane s repared after the exceedng ts relablty crtcal state r = and that the tme of the system renovaton s gnored we obtan the followng results: a) the tme untl the Nth exceedng by the system the relablty crtcal state r = for N suffcently large N has approxmately normal dstrbuton N (.N. N ).e. ( ) t.n F N ( t) P( N t) F N () ( ) t ( );. N b) the expected value and the varance of the tme N untl the Nth exceedng by the system the relablty crtcal state r = are respectvely gven by E[ N ].N D[ N ]. 7N ; c) the number N (t) of exceedng by the system the relablty crtcal state r = up to the moment t t for suffcently large t approxmately has the dstrbuton of the form.( N ) t P( N( t) N) F N () ( ) ).97 t (.N t F N () N... ;.97 t d) the expected value and the varance of the number N (t) of exceedng by the system the relablty crtcal state r = up to the moment t t for suffcently large t approxmately are respectvely gven by H( t).89 t D( t).9t. Further assumng that the contaner gantry crane s repared after the exceedng ts relablty crtcal state r = and that the tme of the system renovaton s not gnored and t has the mean value. 7 and the standard devaton. we obtan the followng results: a) the tme N untl the Nth exceedng by the system the relablty crtcal state r = for suffcently large N has approxmately normal dstrbuton N (.N.7( N ).7N.9( N )).e. ( ) t.7n.7 F N ( t) P( N t) F N () ( ) t ( );.7N.9 b) the expected value and the varance of the tme N untl the Nth exceedng by the system the relablty crtcal state r = for suffcently large N are respectvely gven by E[ N ].N.7( N ) D[ N ].7N.9( N ) ; c) the number N (t) of exceedng by the system the relablty crtcal state r = up to the moment t t for suffcently large t has approxmately dstrbuton of the form 8
14 Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March.7( N ) t.7 P( N ( t) N) F N () ( ).9 t.7.7n t.7 F N () ( ) N... ;.9 t. d) the expected value and the varance of the number N (t) of exceedng by the system the relablty crtcal state r = up to the moment t t for suffcently large t are respectvely gven by H ( t).9( t.7) D ( t).7( t.7); e) the tme untl the Nth system s renovaton for suffcently large N has approxmately N normal dstrbuton N (.7N.9 N ).e. ( ) t.7n) F N ( t) P( N t) F N () ( ) t ( );.9 N f) the expected value and the varance of the tme N untl the Nth system s renovaton for suffcently large N are respectvely gven by E[ N ]. 7N D[ N ]. 7N ; g) the number N (t) of the system s renovatons up to the moment t t for suffcently large t has approxmately dstrbuton of the form.7( N ) t.7n t P( N ( t) N) F N () ( ) F N () ( ) N... ;.9 t.9 t h) the expected value and the varance of the number N (t) of system s renovatons up to the moment t t for suffcently large t are respectvely gven by H ( t).8t D( t).7t; ) the steady avalablty coeffcent of the system at the moment t t for suffcently large t s gven by A ( t).9989 t ; j) the steady avalablty coeffcent of the system n the tme nterval t t ) for suffcently large t s gven by 9
15 Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March where R (t ) s gven by (7.97). CONCLUION A( t ).9 R ( t) dt t In the paper the multstate approach to the analyss and evaluaton of systems relablty and rsk has been practcally appled. The contaner gantry crane has been consdered at varyng n tme operaton condtons. The system relablty structure and ts components relablty functons were changng at varable operaton condtons. The paper proposed an approach to the soluton of practcally very mportant problem of lnkng the systems relablty and ther operaton processes. To nvolve the nteractons between the systems operaton processes and ther varyng n tme relablty structures a semmarkov model of the systems operaton processes and the multstate system relablty functons were appled. Ths approach gves practcally mportant n everyday usage tool for relablty evaluaton of the systems wth changng relablty structures and components relablty characterstcs durng ther operaton processes what exemplary was llustrated n ts applcaton to the gantry crane. The characterstcs of the gantry crane operaton process are of hgh qualty because of the very good statstcal data necessary for ther estmaton. Unfortunately the relablty characterstcs of the gantry crane components are evaluated on non suffcently exact data comng from experts and concerned wth the mean values of the components lfetmes only that because of the complete lack of statstcal data about ther falures are strongly lowered. Also the system and ts components relablty states are defned on a hgh level of generalty and should be descrbed more precsely. All these naccuraces causes that the evaluaton of the gantry crane relablty rsk and avalablty characterstcs should be consder as an llustraton of the proposed approach applcaton. Acknowledgements The paper descrbes part of the work n the Polandngapore Jont Research Project ttled afety and Relablty of Complex Industral ystems and Processes supported by grants from the Poland s Mnstry of cence and Hgher Educaton (MHE grant No. /Nngapore/7/) and the Agency for cence Technology and Research of ngapore (A*TAR ERC grant No. 7 ). 7 REFERENCE Aven T. 98. Relablty evaluaton of multstate systems wth multstate components. IEEE Transactons on relablty pp Aven T. Jensen U tochastc Models n Relablty. prngerverlag New York. Grabsk F.. emmarkov Models of ystems Relablty and Operatons. Warsaw: ystems Research Insttute Polsh Academy of cence. Kolowrock K.. Relablty of Large ystems. Elsever Amsterdam  Boston  Hedelberg  London  New York  Oxford  Pars  an Dego  an Francsco  ngapore  ydney  Tokyo. Kołowrock K. oszyńska J.. Integrated afety and Relablty Decson upport ystem I&RD. Tasks.. n WP: afety and Relablty Decson upport ystems for Varous Martme and Coastal Transport ectors. Polandngapore Jont Research Project. MHE Decson No. /Nngapore/7/. Gdyna Martme Unversty.
16 Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March Kołowrock K. oszyńskabudny J.. The contaner gantry crane operaton relablty rsk avalablty dentfcaton predcton and optmzaton  Testng I&RD. Task 9. n WP9: Applcatons and Testng of Packages of Tools n Complex Martme Transportaton ystems and Processes afety and Relablty Optmzaton. Polandngapore Jont Research Project. MHE Decson No. /Nngapore/7/.Gdyna Martme Unversty. Kołowrock K. oszyńskabudny J.. Relablty and safety of complex techncal systems and processes. ModelngIdentfcatonPredctonOptmzaton. prnger. Lmnos N. Oprsan G.. emmarkov Processes and Relablty. Brkhauser Boston. oszyńska J.. Relablty and rsk evaluaton of a port ol ppelne transportaton system n varable operaton condtons. Internatonal Journal of Pressure Vessels and Ppng Vol. 87 No
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