ScienceDirect. Load Description and Damage Evaluation using Vehicle Independent Driving Events

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1 Avalable onlne at ScenceDrect Proceda Engneerng (25 ) rd Internatonal Conference on ateral and Component Performance under Varable Ampltude Loadng, VAL25 Load Descrpton and Damage Evaluaton usng Vehcle Independent Drvng Events Roza aghsood a, FF*, Igor Rychlk a, Pär Johannesson b a Chalmers Unversty of Technology, SE Gothenburg, Sweden b SP Techncal Research Insttute of Sweden, SE-4 22 Gothenburg, Sweden Abstract We consder the loads that are related to steerng events, and focus on the events that cause hgh forces on steerng components. The load s smplfed by keepng the extreme force value for each drvng event. We defne a smplfed stochastc model for the load by modelng the extreme value for each drvng event by a random varable. We gve formulas to compute the theoretcal load spectrum and the expected fatgue damage caused by the drvng events. Further, n a senstvty study we nvestgate how much the expected damage depends on the varablty of parameters of the proposed model. 25 The The Authors. Publshed Publshed by Elsever by Elsever Ltd. Ths Ltd. s an open access artcle under the CC BY-NC-ND lcense ( Peer-revew under responsblty of the Czech Socety for echancs. Peer-revew under responsblty of the Czech Socety for echancs Keywords: Fatgue damage ndex; hdden arkov models; arkov chan; ranflow cycles; vehcle ndependent load models; steerng events.. Introducton In vehcle engneerng, durablty s an mportant aspect of desgnng a vehcle wth hgh qualty n ts components. Therefore, consderng the servce loadng condtons s necessary. In addton, n fatgue desgn the loads need to be assessed. By descrbng the load envronment, the customer usage and the vehcle dynamcs one can defne the load condtons []. * Correspondng author. Tel.: E-mal address: rozam@chalmers.se The Authors. Publshed by Elsever Ltd. Ths s an open access artcle under the CC BY-NC-ND lcense ( Peer-revew under responsblty of the Czech Socety for echancs do:.6/j.proeng

2 Roza aghsood et al. / Proceda Engneerng ( 25 ) Nomenclature hdden arkov model parameters d damage ntensty D (x) damage ndex ED ( X ) expected damage ndex, materal constants for Palmgren ner s rule N rfc ranflow countng dstrbuton u, v lower and upper ranges for cycle range ntensty of nterval crossngs For vehcle companes, t s mportant to characterze the way that the trucks have been used. They want to descrbe the usage of the trucks n a way that t s ndependent of the vehcle. The loads wll be dfferent for dfferent usage of trucks and for dfferent drver s behavor. A drver can affect the load by changng the speed, brakng or adaptng to the curves. These behavors can be characterzed as drvng events and can be assessed usng measurements obtaned from specally equpped vehcles on a test track. easurng the load on each truck s expensve. However, they want to measure and dentfy actvtes of the drver and specfy the relevant events occurrng on the road. To dentfy the events we need to use the nformaton avalable for all vehcles by means of CAN (Controller Area Network) bus data. If we defne the events such as statc steerng by usng the nformaton from CAN data, we can detect the amount of events that are occurrng n customer vehcles. Then, t s possble to calculate the forces generated from the same knd of events by repeatng the loads under well-defned condtons on a provng ground. By usng the force sgnal we can clarfy whch occasons wll generate hgh forces. We have proposed a stochastc model of loads related to the steerng events such as curves and maneuvers, whch cause large forces actng on steerng components. An explct formula for calculatng the expected fatgue damage based on dentfed drvng events s gven, see also aghsood and Rychlk [2]. The expected damage depends on the frequences of drvng events and the expected value of the extreme force durng an event. The model conssts of two parts; descrpton of the sequence of steerng events and the model for the extreme loads occurrng durng the events. The sequence of steerng events s modeled by means of a arkov chan. Ths s a vehcle ndependent part of the load. For smplcty, the extreme forces durng the events are assumed to be statstcally ndependent. Ther dstrbutons may depend on the type of steerng event, e.g. (left, rght) cornerng, slow maneuver to the rght or to the left etc. The parameters of the dstrbutons are vehcle dependent and need to be estmated usng dedcated measurement campagns or test track measurements. In the examples n Sectons 5 and 6, the Raylegh dstrbuton wll be used to descrbe the varablty of extreme forces. Further, the uncertanty n fatgue damage due to model parameters wll be dscussed. The paper s organzed as follows. In Secton 2 hdden arkov models (Hs) based algorthm to detect the steerng events s revewed. The proposed model for loads and means to calculate the expected damage are descrbed n Sectons and 4. Examples and ther results for measured data are shown n Secton 5. In Secton 6 the senstvty analyses are nvestgated. Conclusons are presented n Secton Detecton of the steerng events Hdden arkov models (Hs) have been proposed for detecton of steerng events such as curves and maneuverng usng on-board loggng sgnals avalable on trucks, such as lateral acceleraton, vehcle speed and steerng wheel angle. The dea s to consder the current drvng event as the hdden state and set up the model based on them, see aghsood and Johannesson [, 4]. We have used a dscrete H, ( A, B, ) where represents model parameters whch contan the transton matrx, the emsson matrx and the ntal state dstrbuton. The parameters must be estmated to characterze the model, see Rabner [5] for more detals. In an H, a tranng set s used to estmate the parameters of the model, whle a test set s used to valdate the model. A tranng set conssts of all necessary nformaton for estmatng the model parameters. In the examples, the tranng set contans all hstory about the curves such as the start and stop ponts of them. Fg. shows a lateral acceleraton sgnal and the correspondng dentfed hdden states process.

3 27 Roza aghsood et al. / Proceda Engneerng ( 25 ) Load Detected events LT SF RT Tme(s) Fg.. Lateral acceleraton sgnal and the correspondng detected events.. Random model of lateral loads based on steerng events odelng of the external loads s an mportant aspect n durablty studes of vehcle components. The approach taken here s to approxmate the load by a vehcle ndependent sequence of steerng events, here representng Left and Rght steerng (SL, SR) or Left and Rght turns (LT, RT). In both cases the two events are separated by a secton when wheels have approxmately zero turnng angle, whch s called Straght forward (SF). Thus, a reduced load can be defned by keepng the extreme value for each left and rght event and set zero for each straght forward event. The most extreme value of the load wll be modeled by a random varabley. Frst the varablty of the sequence of steerng events s modeled by a arkov chan Z havng two states "" and "2", then the values of extreme forces durng events wll be modeled. The arkov chan s defned by a transton matrx P ( p j ),, j, 2, where pj denotes the transton probabltes between the states. Let,,,2,... be a sequence of ndependent and dentcally dstrbuted (d) postve random varables whle,,,2,... m denotes the negatve random varables. Assume that the three sequences Z, m are ndependent. The process Z s vehcle ndependent whle The sequence of extreme loads Y,,,2,..., s defned by: and and m depend on the vehcle, drver etc., f Z, Y () m, f Z 2. Fnally, we can defne the random load X,,,2,... by addng zeros between Y and Y for each straght forward event:, f s odd, X (2) Y / 2, otherwse 4. Fatgue damage ndex The am s to compute the expected damage based on the detected drvng events. To evaluate the model, we wll compare the estmated damage ndex from the measured forces usng ranflow method wth the expected damage from the proposed load model. To calculate the damage, we have used forces whch are measured from specally equpped vehcles on a test track. Frst we wll revew some models and methods on fatgue damage. Assume that the measured load x s gven n form of tme seres x,,,2,..., n. The rsk for fatgue falure n the materal s often measured by means of a damage ndex whch can be computed by Palmgren-ner rule [6, 7], vz.

4 Roza aghsood et al. / Proceda Engneerng ( 25 ) N N D ( x) h () N where N s the number of cycles havng ranges h to falure estmated n constant ampltude tests and presented n form of S-N curve.. The parameter s the fatgue strength of the materal and s the damage exponent. The varablty of the load s modelled by means of random processes. Therefore, the measured load x s one of many possble realzatons of the process. For the random loads, the ranflow ranges become random varables and the damage ndex s a random quantty too. The varablty of the ranflow cycles can be descrbed usng a cumulatve hstogram N rfc whch s called the ranflow countng dstrbuton. The ranflow countng dstrbuton N rfc s equal to the number of tmes that the load x,,2,..., n u, v n upward,, crosses an nterval drecton, denoted by N n. The equalty between the ranflow countng dstrbuton and the nterval crossng was shown ndependently n [8] and [9]. The damage ntensty can be used to measure the severty of the random load and t can be computed usng the ntensty of nterval crossngs: EN n lm, (4) n n then the damage ntensty s d v 2 ( ) ( v u) dudv. (5) The man result s an explct formula for based on the random load X : 2 m u), u v m u) p, u v (6) 2 2 2, u v where p 2 s the soluton to the equaton system p( u, p p p( u, m u) p2 p2, (7) p2 p 2 p 2 p( u, m u) p22 p2. see [6] for more detals and prof of formula (6). 5. Example The results are presented for maneuverng events. The curves were also studed n [2] but the results wll not be presented here. The maneuverng events,.e. drvng n or out of a parkng lot, standng stll but turnng steerng wheel, Lateral Acceleraton(m/s 2 ).5.5 x x x.5 x 4 x 6 x Tme(s) x 5 Fg. 2. Reduced load represented by dots compared wth erved load, lateral acceleraton, represented by the rregular sold lne.

5 272 Roza aghsood et al. / Proceda Engneerng ( 25 ) are consdered as the events whch wll happen n speed less than about km/h. Here, three maneuverng events are consdered; Steerng Left (SL), Steerng Rght (SR) and Straght Forward (SF). The measured loads are denoted by x. Frst, the steerng events were detected usng H algorthm, then the extreme loads durng events were found. We assume that each event follows by a drvng straght secton. The sgnal consstng of the extreme loads durng steerng events and zeros for secton when vehcle s drvng straght wll be denoted by x x, x,..., x ) and called ( n the reduced load. In Fg. 2 part of measured load x (lateral acceleraton) s shown as the sold lne whle the reduced load x by dots. The lnk rod force s used as the load and t s shown n the top plots of Fg. a. The extreme forces are negatve, postve and zero n the three states SR, SL and SF, respectvely. In the fgure stars are the extreme rod forces, occurrng durng maneuvers, consttutng the reduced load. In the lower plot of Fg. a, the detected tme perods wth 2 detected maneuverng events are shown. (a) (b) Load 2 Lnk rod 5 Ranflow cycles States Tme(s) SL SF Detected maneuverng events ax SR Tme(s) n Fg.. (a) Top: sold rregular lne s the measured lnk rod force whle stars represent the reduced load. Bottom: Detected maneuvers. (b) Dots - the ranflow cycles found n the measured lnk rod force. Crcles - the ranflow cycles counted n the reduced load. The ranflow cycles have been found both n the load and n the reduced load and compared n Fg. b. The ranflow cycles found n the measured lnk rod force are marked as dots. One can see that there are few large cycles and many very small ones. The ranflow cycles found n the reduced load are presented as crcles. As can be seen n Fg. b, all large cycles found n the lnk rod force are also found n the reduced load and hence one can expect that the damage ndex computed for the measured load and the reduced load should be very close. The estmated transton matrx accordng to the detected maneuvers s P..9. The Raylegh dstrbutons have been ftted to postve and negatve values, respectvely. The estmates of the parameters of Raylegh dstrbutons were very close. The dfference between the parameter values were not sgnfcant hence the average value (6.) of the parameters have been used. Table shows a comparson of the damage ndexes D ( x ) computed for measured load, ( x) for the reduced load and the expected damage ndex D (X ) E for the random model of the reduced load. Damage ndces ( x ) and D ( x) are gven n columns 2 and. As expected these are almost dentcal. We conclude that the reduced load models well the varablty of the measured load. Further, the expected damage of the model s qute close to the measured one. D D

6 Roza aghsood et al. / Proceda Engneerng ( 25 ) Table. Comparson of damage ndces D ( x ) computed for the measured load, D ( x) for the reduced load and the expected damage E D ( X ). ndex Damage D ( x ) (x) D ED ( X ) In Fg. 4a, the load spectra estmated from the measured lnk rod force and the reduced load are compared wth the theoretcal load spectrum. As can be seen n Fg. 4b, where the load spectra for smulated loads are compared wth the theoretcal load spectrum and the load spectrum of the reduced load, the dfferences between the measured spectrum and the expected one does not seem to be sgnfcant. 5 (a) Load spectra 4 (b) Load spectra 5 25 Range 2 5 Range Cumulatve frequency of cycles 2 Cumulatve frequency of cycles Fg. 4. (a) The regular sold lne s the theoretcal load spectrum. The stars lke functons are the load spectra found n measured lnk rod force and the reduced load. (b) Load spectra for smulated loads compared wth the theoretcal load spectrum and the load spectrum of the reduced load (the thck stars lke lne). 6. Senstvty analyss of the damage ndex As t was mentoned before, the sequence of steerng events s modeled by a arkov chan wth transton matrx P. Ths sequence s a vehcle ndependent part of the load. The extreme forces durng the events are assumed to be statstcally ndependent, but ther dstrbutons depend on the type of steerng event. The parameters of the dstrbutons are vehcle dependent and need to be estmated usng dedcated measurement campagns or test track measurements. In the examples, Raylegh dstrbutons have been ftted to postve and negatve values. Now suppose that both dstrbutons have the same parameter, vz. v 2 ( ) u 2 ( ) 2 2 e, v m u) e, u (8) then the load can be wrtten as a scaled standard load, X Xˆ, where Xˆ s a reduced load wth standard Raylegh r / 2 random varables, P ( R r ) e 2. Therefore, the llaton ntensty can be wrtten as a scaled one, namely u v ˆ (, ). (9) Further, the damage ndex can be calculated as d d ˆ, where dˆ s the expected damage computed by the standard Raylegh dstrbuton. Ths means that the expected damage ndex s a factor of the parameter to the power.

7 274 Roza aghsood et al. / Proceda Engneerng ( 25 ) Here, we wll consder two types of uncertantes. The varablty of the load envronment wll manfest n the transton matrx P and the vehcle dependent varablty n the parameter of the Raylegh dstrbuton. In the followng subsectons we wll study how much the expected damage wll vary because of varablty of matrx P and parameter. We wll also nvestgate the statstcal uncertanty of the estmaton of. In fatgue relablty evaluaton usng the load-strength concept often the log-normal dstrbuton s used, see [, Chapter 7]. Therefore, the uncertanty n damage wll be measured n terms of the standard devaton of the logarthmc damage, whch corresponds to the relatve uncertanty n damage (or fatgue lfe). 6.. Varablty of the transton matrx P To examne how much the expected damage wll depend on the transton matrx P, three dfferent arkov chans have been used to model the sequence of drvng events. Frst, assume that we always go from left to left. Ths would be the case wth the smallest possble damage, snce all mnma are equal to zero. The expected damage s ED ne R, where n denotes the number of turns and R represents the standard Raylegh random varable for the maxmum force. Second, consder that the events change each tme. In ths case, p p, and we wll get the 2 2 maxmum damage for ths type of arkov chan. The llaton ntensty ( u, v ) gven n Eq. (6) can be smplfed to P ( m u ), u v m u ) (), u v 4 m u ), u v Fnally, assume that left and rght turns occur ndependently of the past wth probabltes p. 5, and Eq. (6) smplfes to (see also []) j 4 P ( m u ), P ( m u ), P ( P ( m u ), u v u v u v () The expected damage values for the three dfferent cases have been summarzed n Table 2. Here, we have consdered standard Raylegh dstrbutons for negatve and postve values and the number of events s n. Table 2. Expected damage calculated from the three dfferent arkov chans. Damage nmum Independent axmum The two extreme cases wll be used to calculate the uncertanty n damage by assumng a unform dstrbuton between the mnmum and maxmum values, vz. for ln d max ln d mn ln.7 ln.5.5 (2) P 2 2 whch can be nterpreted as correspondng to 5% relatve uncertanty n damage, as the natural logarthm s used.

8 Roza aghsood et al. / Proceda Engneerng ( 25 ) Statstcal uncertanty of parameter Suppose that we estmate the parameter based on n ervatons, then we may ask how much the estmaton uncertanty of parameter mpacts the damage. The estmate of parameter for a Raylegh dstrbuton s ˆ 2 / X and ts dstrbuton s approxmately normal 4 2 N (, ). Thus, the uncertanty n damage can n be approxmated usng Gauss approxmaton formula, vz. 4, stat Var lnˆ Varlnˆ.29. n () wth an example for a short sgnal wth n maneuverng events and, correspondng to a typcal length of the measurements. 6.. Varablty of parameter For dfferent measurements of maneuverng events, we have found dfferent estmates of parameter, say, 2,..., l for l dfferent measurements. The uncertanty n damage due to the varablty n the estmated s computed as the sample standard devaton, vz. std ln std ln (4) for an example wth and estmated -values 6.5, 6.5, 9.4, 8.96, 7.76, 6.7, However, the uncertanty ncludes both the varablty and the statstcal uncertanty of. Thus, the pure varablty can be estmated as , var, stat 7. Concluson A reduced load,.e. a sequence of the most extreme forces durng steerng events, was ntroduced. A random load modelng the varablty of the reduced load was proposed. The sequence of steerng events, whch s vehcle ndependent nformaton, was modeled usng a two states arkov chan. The extreme forces occurrng durng the steerng events were modeled by means of ndependent Raylegh dstrbuted varables. For the model, an explct formula for the expected fatgue damage was presented. The proposed random model depends only on four parameters whch could be used to classfy and compare the severty of drvng envronments. The results were valdated usng measured data. The slow speed maneuverng events were detected. All large ranflow cycles found n measured load were also counted n the reduced load. Hence the reduced load can be used to predct fatgue damage of steerng components. The erved load spectrum dd not sgnfcantly dffer from load spectra found n the smulated loads. We conclude that the proposed random load accurately descrbe the varablty of the ranflow ranges for the consdered measured loads. A senstvty study was conducted to see how much the expected damage depends on the parameters of the proposed model. Acknowledgements We would lke to thank Volvo Trucks for supplyng the data n ths study and to the members n our research group at Volvo for ther valuable advce. We gratefully acknowledge the fnancal support from VINNOVA. References [] P. Johannesson and. Speckert, edtors. Gude to Load Analyss for Durablty n Vehcle Engneerng, Wley:Chchester, 2.

9 276 Roza aghsood et al. / Proceda Engneerng ( 25 ) [2] R. aghsood and I. Rychlk, Estmaton of fatgue damage of steerng components usng vehcle ndependent load model, Submtted to Probablstc Engneerng echancs, August 24. [] R. aghsood and P. Johannesson, Detecton of the curves based on lateral acceleraton usng hdden arkov models, Proceda Engneerng, 66:425-44, 2. [4] R. aghsood and P. Johannesson. Detecton of the steerng events based on vehcle loggng data usng hdden arkov models. Submtted to Internatonal Journal of Vehcle Desgn, Aprl 24. [5] L. R. Rabner, A tutoral on hdden arkov models and selected applcatons n speech recognton, Proceedngs of the IEEE, 77(2): , 989. [6] A. Palmgren, De Lebensdauer von Kugellagern. Zetschrft des Verens Deutscher Ingeneure, 68:9-4, 924. In German. [7]. A. ner. Cumulatve damage n fatgue. Journal of Appled echancs, 2:A59-A64, 945. [8] A. Beste, K. Dressler, H. Kötzle, W. Krüger, B. aer, and J. Petersen. ultaxal ranflow - a consequent contnuaton of Professor Tatsuo Endo's work. In Y. urakam, edtor, The Ranflow ethod n Fatgue, pages - 4. Butterworth-Henemann, 992. [9] I. Rychlk. Note on cycle counts n rregular loads. Fatgue & Fracture of Engneerng aterals & Structures, 6:77-9, 99. []. Karlsson. Load odellng for Fatgue Assessment of Vehcles - a Statstcal Approach. PhD thess, Chalmers Unversty of Technology, Sweden, 27.