Parameter Sensitivity Analysis of a Tractor and Single Axle Grain Cart Dynamic System Model

Transcription

1 Agiulual and Biosysems Engineeing Confeene Poeedings and Pesenaions Agiulual and Biosysems Engineeing 6-29 Paamee Sensiiviy Analysis of a Tao and Single Axle Gain Ca Dynami Sysem Model Manoj Kakee Iowa Sae Univesiy Bian L. Sewad Iowa Sae Univesiy, bsewad@iasae.edu Follow his and addiional woks a: hp://lib.d.iasae.edu/abe_eng_onf Pa of he Bioesoue and Agiulual Engineeing Commons The omplee bibliogaphi infomaion fo his iem an be found a hp://lib.d.iasae.edu/ abe_eng_onf/38. Fo infomaion on how o ie his iem, please visi hp://lib.d.iasae.edu/ howoie.hml. This Confeene Poeeding is bough o you fo fee and open aess by he Agiulual and Biosysems Engineeing a Iowa Sae Univesiy Digial Reposioy. I has been aeped fo inlusion in Agiulual and Biosysems Engineeing Confeene Poeedings and Pesenaions by an auhoized adminisao of Iowa Sae Univesiy Digial Reposioy. Fo moe infomaion, please ona digiep@iasae.edu.

2 Paamee Sensiiviy Analysis of a Tao and Single Axle Gain Ca Dynami Sysem Model Absa Tao and owed implemen sysem models have beome ineasingly impoan fo model-based guidane onolle design, viual pooyping, and opeao-and-hadwae-in-loop simulaion. Vaious ao and owed implemen models have been poposed in he lieaue whih onain uneain o ime-vaying paamees. Sensiiviy analysis was used o idenify he effe of sysem paamee vaiaion on sysem esponses and o idenify he mos iial sysem paamees. Sensiiviy analysis was pefomed wih espe o hee ie oneing siffness paamees, hee ie elaxaion lengh paamees, and wo implemen ineial paamees. Oveall, he sysem was mos sensiive o he ie oneing siffness paamees and leas sensiive o he implemen ineial paamees. In geneal, he vaiaion in he inpu paamees and he sysem sae vaiables wee elaed in a non-linea fashion. Wih he nominal paamee values fo a MFWD ao, a single axle gain a, and on subble sufae ondiions, a % vaiaion in oneing siffness paamees aused a 5% aveage vaiaion in he sysem esponses wheeas an 8% hange in oneing siffness paamees aused an 8% aveage vaiaion a 4.5 m/s fowad veloiy. If a % aveage vaiaion in sysem esponses is aepable, he oneing siffness paamees and implemen ineial paamees mus be esimaed wihin 2% and 3% of aual/nominal values espeively. The elaxaion lengh paamees have o be wihin 75% of he nominal values. Keywods ao and implemen model, sensiiviy analysis, sensiiviy measues, paamee uneainy Disiplines Bioesoue and Agiulual Engineeing Commens ASABE Pape No This onfeene poeeding is available a Iowa Sae Univesiy Digial Reposioy: hp://lib.d.iasae.edu/abe_eng_onf/38

3 An ASABE Meeing Pesenaion Pape Numbe: Paamee Sensiiviy Analysis of a Tao and Single Axle Gain Ca Dynami Sysem Model Manoj Kakee Agiulual and Biosysems Engineeing Dep., Iowa Sae Univesiy, kakee@iasae.edu. Bian L. Sewad Agiulual and Biosysems Engineeing Dep., Iowa Sae Univesiy, bsewad@iasae.edu. Wien fo pesenaion a he 29 ASABE Annual Inenaional Meeing Sponsoed by ASABE Gand Siea Reso and Casino Reno, Nevada June 2 June 24, 29 Absa. Tao and owed implemen sysem models have beome ineasingly impoan fo model-based guidane onolle design, viual pooyping, and opeao-and-hadwae-in-loop simulaion. Vaious ao and owed implemen models have been poposed in he lieaue whih onain uneain o ime-vaying paamees. Sensiiviy analysis was used o idenify he effe of sysem paamee vaiaion on sysem esponses and o idenify he mos iial sysem paamees. Sensiiviy analysis was pefomed wih espe o hee ie oneing siffness paamees, hee ie elaxaion lengh paamees, and wo implemen ineial paamees. Oveall, he sysem was mos sensiive o he ie oneing siffness paamees and leas sensiive o he implemen ineial paamees. In geneal, he vaiaion in he inpu paamees and he sysem sae vaiables wee elaed in a non-linea fashion. Wih he nominal paamee values fo a MFWD ao, a single axle gain a, and on subble sufae ondiions, a % vaiaion in oneing siffness paamees aused a 5% aveage vaiaion in he sysem esponses wheeas an 8% hange in oneing siffness paamees aused an 8% aveage vaiaion a 4.5 m/s fowad veloiy. If a % aveage vaiaion in sysem esponses is aepable, he oneing siffness paamees and implemen ineial paamees mus be esimaed wihin 2% and 3% of aual/nominal values espeively. The elaxaion lengh paamees have o be wihin 75% of he nominal values. Keywods. ao and implemen model, sensiiviy analysis, sensiiviy measues, paamee uneainy 2

4 . Inoduion Off-oad vehile sysem models ae beoming ineasingly impoan as mehaoni enginees ineasingly ely on model-based onolle design, viual pooyping, and eal-ime hadwaeand-opeao-in-loop simulaion (Kakee and Sewad 28). As a emendous amoun of ompuaional powe has beome available fo numeial simulaion, vehile models have beome ineasingly omplex ofen possessing dozens of model paamees. Auae esimaion of hose paamees is ofen diffiul beause of he high vaiabiliy in field ondiions. Beause uneain paamee esimaes will have an effe on he model esponses, he offoad vehile simulaions may ofen be unealisi, whih limis paione s onfidene in he models and in model-based sudies (Kioussioukis, e al. 24). I is impoan o undesand and quanify he effe of hese paamee uneainies o vaiaions on he sysem esponse (Fales, 24). Sensiiviy analysis is one appoah o idenify and quanify hese elaionships (Xu and Gene, 27), whih an impove ou onfidene in off-oad vehile models. Sensiiviy analysis evaluaes he hanges in dynami model saes and oupu wih espe o (w...) hanges in model paamees (Coseo and Taanola, 2; Deif, 986). Thus, sensiiviy analysis an be used o pefom uneainy analysis, esimae model paamees, analyze expeimenal daa, guide fuue daa olleion effos, and sugges he auay o whih he paamees mus be esimaed (Rodiguez-Fenandez and Banga, 29). Tadiionally, sensiiviy analysis has been used o opimize he vehile sysem design (Jang and Han, 995; Pak e al., 23). Jang and Han (997) used a die diffeeniaion mehod fo sensiiviy analysis of vehile laeal dynamis. They pefomed a sae sensiiviy analysis w... ie oneing siffness, loaion of vehile ene of gaviy (CG), vehile mass, and vehile momen of ineia (MI) using a biyle model of a fon wheel seeing vehile. The sudy was pefomed a ypial on-oad vehile veloiies anging fom 9 m/s (2 mph) o 53 m/s (2 mph). Pak e al. (23) pefomed a dynami sensiiviy analysis fo a panogaph of a ail vehile. Dominan design vaiables wee idenified using deivaive-based sae sensiiviy measues and wee modified fo opimal design. Rua and Wojiki (23) also applied sensiiviy analysis o a dynami aiload model. They foused on developing an analyial soluion fo deivaive-based sensiiviy analysis of a sysem epesened by a se of diffeenial equaions. Boh fis and seond ode deivaives wee used o isolae a se of paamees o whih he model oupus wee he mos sensiive. Ebehad (27) pefomed a vehile model sensiiviy analysis w... vaious design vaiables. The vehile dynamis was highly sensiive o MI and he CG loaion. The sudy was ondued a m/s (~22mph) fowad veloiy. In off-oad vehile sysems, he appliaion of sensiiviy analysis o undesand he effe of uneain paamees on a ao and owed implemen laeal dynamis is impoan o suppo he emeging fam auomaion ehniques suh as implemen guidane and oodinaed guidane. Vaious ao and owed implemen seeing models have been poposed in he lieaue fo boh on-oad (Chen and Tomizuki, 995; Dag and Kang, 23; Kim e al., 27) and off-oad (Feng e al., 25; Kakee and Sewad, 28) opeaions. As suggesed by hese models, he laeal dynamis of an off-oad ao and owed implemen sysem depend on he laeal foes geneaed by soil-ie ineaions. The esponses may be diffeen wih diffeen soil ypes and/o diffeen soil moisue onen. Ohe soil paamees inluding inenal fiion angle, ohesion and poosiy may also affe he vehile esponses. Similaly, he esponses may vay wih he vaiaion in ie onsuion, size, inflaion pessue and nomal load. I may be had o esimae hese vaiables auaely, and i is also had o find a widely aeped model o elae hese vaiables o he ie laeal foe. The oneing siffness oeffiien, whih is he iniial sensiiviy of ie foe o side slip angle, is he lumped paamee used o epesen he ombined effe of hese vaiables (Mez, 993). Consequenly, his paamee ends o be 3

5 highly uneain. Vehile ie elaxaion lengh, whih is defined as he disane a ie olls befoe he seady sae side slip angle is eahed, is anohe paamee ha ould be highly uneain (Bevly e al., 22). In he ase of implemens suh as a gain a o a spaye, implemen mass and MI may also be uneain o may vay subsanially ove ime. In his wok, dynami model sensiiviy analysis was pefomed o quanify he degee of dependene of ao and single axle gain a sysem esponses on he uneain model paamees. This analysis will help us undesand he elaive signifiane of hese paamees so ha moe esoues an be alloaed o auaely esimae hose paamees o whih he sysem is moe sensiive. Speifi objeive will be: o evaluae he effe of paamee uneainies on he sysem esponses and o idenify he paamees o whih he model is mos sensiive 2. Mehods: In his eseah, sensiiviy of a ao and single axle gain a seeing sysem w... vaiaion in vaious model paamees was analyzed using deivaive-based loal sensiiviy analysis mehod. A dynami biyle model of he sysem (Kakee and Sewad, 28) was adaped o pefom he analysis. Gain a mass and MI, ie oneing siffness, and ie elaxaion lengh paamees wee assumed o have some level of uneainy and/o vaiaion ove he field opeaion. A ange of hese uneain paamee values wee used o define a paamee spae fo he sudy. A numeial simulaion-based appoah was used o alulae loal sensiiviies w... hose paamees. 2. Vehile Model A ao and single axle owed implemen model was equied o sudy he sensiiviy of he sysem saes and/o oupus w... vaious sysem paamees. Kakee and Sewad (28) sudied vaious models of a ao and single axle gain a sysem. Among he hee models used, a dynami biyle model wih ie elaxaion lengh dynamis epesened he sysem mos auaely. In his wok, hei dynami model was adjused as follows o sudy he sensiiviy of he sysem w... eigh sysem paamees, namely implemen mass, implemen MI, ao fon ie oneing siffness, ao ea ie oneing siffness, implemen ie oneing siffness, ao fon ie elaxaion lengh, ao ea ie elaxaion lengh, and implemen ie elaxaion lengh. Common vehile dynamis symbols wee used o desibe he dynamis of he sysem (fo noaion, see he lis of he vaiables). This model is desibed by Eqs. o 8: i i i i i i i ( m + m ) v& m & γ m d & γ = ( m + m ) u γ C α C α C α α, f f α, α, i 2 i i i i i i ( I + m )& γ m v& + m d & γ = m u γ ac α + bc α + C α z i i 2 i i i i i i ( I + m d )& m dv& + m d & γ = m du γ + ( d + e) C α z v & α f = σ f γ α, aγ u u + δ α f σ σ σ f f v bγ u & α = α σ σ σ f α, f f α, α, () (2) (3) (4) (5) 4

6 i v γ & α = i i σ σ & ψ = γ & i i ψ = γ i ( d + e) γ u u u i i σ i + ϕ ϕ α i i i σ σ σ Eqs. 8 an be epesened in maix fom as, M X& = NX + PU & =, i i i X [ ] T whee, v & & γ & γ & α & α & α & ϕ & ϕ f i i i X = [ γ γ α α α ϕ ϕ ] T v f and U = [ δ ] The paamee values suggesed by Kakee and Sewad (28) wee used as he nominal values in his sudy (Table ). The paamees wee based on a MFWD ao (model 793, Deee and Co., Moline, IL) and a single axle 5 bu gain a (model 5, Alliane Podu Goup, Kalida, OH). An uneainy egion of ±% was defined abou he nominal value fo eah paamee used in he sensiiviy analysis. Table : Dynami biyle model paamees and uneainies fo he JD 793 ao and Pake 5 gain a sysem. Tao. Implemen Paamees Values ± Uneainy Paamees Values ± Uneainy a.7 m d 3.62 m b.2 m e. m 2. m m 939 kg m i 227 ± 227 kg** I z C αf C α σ f σ 3579 kg-m 2 I z i 22* ± 22 KN/ad** 486* ± 486 KN/ad** C α i. # ±. m** σ i.5 # ±.5 m** 642 ± 642 kg-m 2 ** 67* ± 67 KN/ad**. # ±. m** *Values based on he wok of Mez (993) and Shwangha and Ro (984). #Values based on he wok of Bevly e al. (22). **Minimum values of zeo fo hese paamees wee no possible. Small posiive numbes wee used in he sudy insead of zeos. 2.2 Sensiiviy Analysis Sensiiviy is he measue of he hange in sysem esponses w... o small hanges in a sysem paamee. When using a dynami model, he moe a sae vaiable hanges wih a small hange in a model paamee, he moe sensiive i is o his paamee (Rodiguez- Fenandez and Banga, 29). The sensiiviy measue alulaed a a paiula loaion in he (6) (7) (8) (9) 5

7 paamee spae is alled he loal sensiiviy. This measue will be valuable in sudying he paamee uneainy abou a nominal poin o a poin of inees. The umulaive sensiiviy measue alulaed ove he enie paamee spae is alled he global sensiiviy. This ype of sensiiviy is impoan o idenify he oveall anking of paamees wihin he paamee spae as he mehod aouns fo he ineaion beween oelaed paamees (Kuheenko e al., 29). Vaious sensiiviy analysis ehniques have been poposed in he lieaue o alulae boh loal and global sensiiviy measues o indies. Pimaily, hese ehniques an be lassified ino hee goups, namely vaiane-based mehods, seening mehods, and deivaive-based mehods. In he vaiane-based mehods, he onibuion of a model paamee o he oupu vaiane is esimaed (Kiousioukis e al., 24). Some of he widely used vaiane based mehods ae Fouie Ampliude Sensiiviy Tes (FAST) (Cukie e al., 973; MRae e al., 982), he exended FAST (Salelli e al., 999) and he mehod of Sobol (Sobol, 993). The dawbak of he vaiane based mehods is hei ompuaional ineffiieny equiing a lage numbe of funion evaluaions o alulae he sensiiviy wih a easonable auay (Kuheenko e al., 29). Seening mehods ae used o qualiaively ank he model paamees fom he mos influenial o he leas influenial ode. These mehods ae ompuaionally aaive, bu ae less auae (Kuheenko e al., 29). The mehod of Mois (Mois, 99) is one of suh mehods. Deivaive-based mehods ake he fis ode diffeeniaion of he model oupus wih espe o he model paamees. Loal and global sensiiviy measues ae defined based on hose deivaives. Jang and Han (995) and Kuheenko e al. (29), among ohes, have poposed and used he deivaive-based mehods. Kuheenko e al. (29) used he mehod o analyze sensiiviy of seven algebai equaions and ompaed he esuls wih hose fom he mehod of Sobol and Mois. This mehod was moe auae han he mehod of Mois and was ompuaionally effiien han he Sobol s mehod. Moeove, deivaive-based sensiiviy analysis mehod was pefeed ove ohe mehods in analyzing dynami models (Jang and Han, 997; Pak e al., 23; Rua and Wojiki, 23; Ebehad e al., 27). In his wok, some modifiaions wee applied o he deivaive-based fomulaion developed by Kuheenko e al. (29) so ha he sensiiviy analysis of a omplex dynami model would be possible. Only he loal sensiiviy measues wee used Sensiiviy Measues A muli-inpu muli-oupu (MIMO) dynami sysem suh as one pesened in seion 2. an be epesened as, ( X, p ) X & = f, () whee, p is he paamee veo and is ime. A loal sensiiviy of he sysem o a paamee p i is defined as he paial deivaive of he funion f w... he paamee. The loal sensiiviy funion abou a nominal poin p* in he paamee spae is hen epesened as, X Si ( p*) = & () p i p= p* whee, i =,, n and n = size of paamee veo. As he poin p* moves in he paamee spae, he loal sensiiviy measues will also hange. 6

8 Beause he analyial soluion o he loal sensiiviy measue (Eq. ) was diffiul o obain fo he ao and implemen sysem used in his sudy, a numeial simulaion-based soluion was used. Tao laeal veloiy and yaw-ae, implemen yaw-ae, and hee ie side slip angle saes wee used as oupu vaiables, whih allowed us o use oupu sensiiviies and sae sensiiviies inehangeably. The ao and gain a dynami sysem (Eq. 9) was hen epesened in he sae spae fom as, X & = AX + BU (2) Y = CX + DU (3) whee, A = M N, B = M P, C = ideniy maix of size 8, and D=[]. M and N wee defined in Eq. 9. Fis, he model was simulaed wih a sep seeing inpu a he nominal poin p* in he paamee spae. The simulaion ime was wie he seling imes of he slowes non-zeo eigenvalue of he sysem model, whih wee 3 s, 6 s and 3 s espeively a. m/s, 4.5 m/s and 7.5 m/s fowad veloiies. Wih his simulaion ime, he sensiiviy analysis aouned fo boh he ansien and seady sae sensiiviies. The oupu esponses fom his simulaion wee used as he efeene. Paamees of inees wee hen vaied one a a ime by a small amoun o ge he peubed oupu esponses. Finally, he loal sensiiviy of an oupu vaiable w... he paamees a he nominal poin was alulaed using he finie diffeene equaion, S ij ( p Y * * * * * * ( p,..., p, p + p, p,..., p ) Y * ( ) * j i i i i+ n j ) = * pi whee, j =,..,m; m = numbe of oupu o sae vaiables; p i *= small hange in he paamee of inees. The vaiable Y j epesened a ime seies oupu daa of he dynami sysem. A mean of he absolue hange in oupu vaiable ove he simulaion ime was used o alulae he loal sensiiviy measue. This loal sensiiviy measue was alulaed fo all paamees of inees a vaious poins in he paamee spae. The loal sensiiviies a he nominal poin wee used o evaluae he effe of paamee vaiaions o he sysem esponses and model he elaionships beween he vaiaion in he inpu paamees and he oupu vaiables (Gau, 22). The hanges in inpu paamees wee divided by he nominal values of he paamees o nomalize he esuls as i was no appopiae o diely ompae o aveage he oupu sensiiviies w... vaious inpu paamees whose numeial anges diffeed by seveal ode of magniude. Oupu vaiables wee also of diffeen odes of magniudes. A good way o define he sensiiviy in his ase would also be o nomalize he hanges in he oupu vaiables by he nominal oupu values. Howeve, beause we wee dealing wih a dynami sysem wih zeo ossings, he oupu vaiables wee nomalized only afe he mean of he absolue hanges in oupu vaiables wee alulaed. Finally, he oveall loal sensiiviy o a paamee was alulaed using a mean ove he oupu vaiables, S = m i S ij m j = p (4) (5) 7

9 3. Resuls and Disussions: The loal sensiiviies a he nominal poin (Table ) in he paamee spae showed he effe of vaiaions in he sysem paamees on he sysem saes o oupus (Fig. - Fig. 3). A low fowad veloiies, ie side slip angles wee highly sensiive o all inpu paamees. As expeed, he implemen ie side slip angle was mos sensiive o he implemen mass, implemen MI, and implemen ie oneing siffness paamees. The ao laeal veloiy and yaw-ae saes wee mos sensiive o he fon and ea ie oneing siffness paamees and so wee he ao and implemen yaw ae paamees (Fig. 2). In geneal, loal sae/oupu sensiiviies w... implemen ineial paamees deeased as he fowad veloiy was ineased (Fig. ). Implemen ie side slip angle was he mos sensiive oupu o boh he implemen mass and implemen MI. This esul was expeed as he implemen ineial paamees will have a highe influene on he implemen dynamis. The implemen ie side slip sensiiviies w... implemen mass wee 2.5,. and. espeively a. m/s, 4.5 m/s and 7.5 m/s fowad veloiies. The same w... implemen MI was.7 a. m/s fowad veloiy. The sensiiviies of all ohe oupu vaiables w... boh implemen mass and MI wee less han.4 a all veloiies. Sensiiviy (%Oupu/%Inpu) Tao Laeal Veloiy Tao Yawae Implemen Yaw -ae Vel =. m/s Tao Fon Tao Rea Implemen Tie a) Loal sensiiviies of sae vaiables w... implemen mass Sensiiviy (%Oupu/%Inpu) Tao Laeal Veloiy Tao Yawae Implemen Yawae Tao Fon Vel =. m/s Tao Rea Implemen Tie b) Loal sensiiviies of sae vaiables w... implemen MI Fig. : Loal sensiiviies a he nominal poin in he paamee spae w... implemen ineial paamees; a) implemen mass, and b) implemen MI. The sensiiviies wee alulaed a. m/s, 4.5 m/s and 7.5 m/s fowad veloiies. Sensiiviies w... ie oneing siffness paamees showed a mixed vaiaion wih ineasing fowad veloiy (Fig. 2). In geneal, he ie side slip saes wee less sensiive o hese 8

10 paamees as he fowad veloiy was ineased. The opposie end was obseved wih he laeal and yaw veloiy saes. A all veloiies, he ie side slip angles wee he mos sensiive sae vaiable w.. he ie oneing siffness paamees. The sae sensiiviies wee as high as 2.6 fo he fon ie oneing siffness, 2.4 fo he ea ie oneing siffness and 3. fo he implemen ie oneing siffness. The sysem was moe sensiive o he oneing siffness paamees han o he implemen ineial paamees and elaxaion lengh paamees. Tie elaxaion lengh paamees mosly influened he oesponding ie side slip angles (Fig. 3). Sensiiviy (%Oupu/%Inpu) Tao Laeal Veloiy Vel =. m/s Tao Yawae Implemen Yawae Tao Fon Tao Rea Implemen Tie a) Loal sensiiviies of sae vaiables w... fon ie oneing siffness Sensiiviy (%Oupu/%Inpu) Tao Laeal Veloiy Tao Yawae Vel =. m/s Implemen Yawae Tao Fon Tao Rea Implemen Tie Sensiiviy (%Oupu/%Inpu) b) Loal sensiiviies of sae vaiables w... ea ie oneing siffness Tao Laeal Veloiy Tao Yawae Implemen Yawae Vel =. m/s Tao Fon Tao Rea Implemen Tie ) Loal sensiiviies of sae vaiables w... implemen ie oneing siffness Fig. 2: Loal sensiiviies a he nominal poin in he paamee spae w... oneing siffness paamees; a) fon ie oneing siffness, b) ea ie oneing siffness, and ) implemen ie oneing siffness. The sensiiviies wee alulaed a. m/s, 4.5 m/s and 7.5 m/s fowad veloiies. 9

11 Sensiiviy (%Oupu/%Inpu) Tao Laeal Veloiy Tao Yawae Vel =. m/s Implemen Yawae Tao Fon Tao Rea Implemen Tie a) Loal sensiiviies of sae vaiables w... fon ie elaxaion lengh Sensiiviy (%Oupu/%Inpu) Tao Laeal Veloiy Tao Yawae Vel =. m/s Implemen Yawae Tao Fon Tao Rea Implemen Tie b) Loal sensiiviies of sae vaiables w... ea ie elaxaion lengh Sensiiviy (%Oupu/%Inpu) Tao Laeal Veloiy Tao Yawae Implemen Yawae Vel =. m/s Tao Fon Tao Rea Implemen Tie ) Loal sensiiviies of sae vaiables w.. implemen ie elaxaion lengh Fig. 3: Loal sensiiviies a he nominal poin in he paamee spae w... o elaxaion lengh paamees; a) fon ie elaxaion lengh, b) ea ie elaxaion lengh, and ) implemen ie elaxaion lengh. The sensiiviies wee alulaed a. m/s, 4.5 m/s and 7.5 m/s fowad veloiies. The sysem was highly sensiive o all inpu paamees a. m/s fowad veloiy. Beause, he sysem had moe osillaoy ansien behavio a. m/s han a 4.5 m/s and 7.5 m/s fowad veloiies (Fig. 4), he ineaion beween ie elaxaion lengh and ohe paamees

12 aused a lage phase shif beween he efeene and peubed oupu esponses a. m/s. The ineased phase shif esuled o he highe sysem sensiiviy a he low fowad veloiy. Rea Tie (Deg) Refeene Peubed Time (s) a) Time hisoy of ea ie elaxaion lengh a. m/s Rea Tie (Deg) Refeene Peubed Time (s) a) Time hisoy of ea ie elaxaion lengh a 4.5 m/s Fig. 4: Time hisoy of ea ie side slip angle fo he nominal paamees (efeene) and % hange in he oneing siffness paamees (peubed); a) a. m/s fowad veloiy, b) a 4.5 m/s fowad veloiy. A he nominal poin, he oveall sysem was mos sensiive o he fon and ea ie oneing siffness paamees a all hee fowad veloiies (Fig. 5). The fon and ea ie elaxaion lenghs wee he nex wo influenial paamees. The sysem was leas sensiive o he implemen mass and MI. A. m/s, he sysem was almos equally sensiive o oneing siffness and oesponding elaxaion lengh paamees. A 4.5 m/s and 7.5 m/s, howeve, he ao-and-implemen sysem was wie as sensiive o oneing siffness paamees as i was o elaxaion lengh paamees. The sysem beame less sensiive o elaxaion lengh paamees as he fowad veloiy was ineased. This esul was expeed beause he ies avelled he elaxaion lengh equivalen disane soone as he fowad veloiy ineased.

13 .2 Vel =. m/s Sensiiviy (%Oupu/%Inpu) Fon Tie Coneing Siffness Rea Tie Coneing Siffness Fon Tie Relaxaion Lengh Rea Tie Relaxaion Lengh Implemen Tie Coneing Siffness Implemen Tie Relaxaion Lengh Implemen Mass Implemen MI Fig. 5: Loal sensiiviies a nominal poin in he paamee spae wih espe o vaious sysem paamees a. m/s, 4.5 m/s and 7.5 m/s fowad veloiies. The sysem beame less sensiive o he oneing siffness paamees a 4.5 m/s as ompaed o he sensiiviies a. m/s fowad veloiy. The aveage sae sensiiviy a. m/s was.8 w... he fon and ea ie oneing siffness paamees. Howeve, i dopped o.35 a 4.5 m/s fowad veloiy. When he fowad veloiy was ineased o 7.5 m/s, he sysem beame moe sensiive o all hee oneing siffness paamees. A his veloiy, he aveage sae sensiiviy ineased o.4 w... he fon ie oneing siffness and o.5 w... he ea ie oneing siffness. This esul an be explained beause he oneing siffness paamees ae linked o he highe ode sysem dynamis whih beome signifian a highe veloiies (Kakee and Sewad, 28). Howeve, beause he ineaion beween oneing siffness and elaxaion lengh paamees was subsanial a lowe veloiies (Fig. 2), he sysem was highly sensiive o oneing siffness paamees a. m/s fowad veloiy. Simila ends wee obseved wih he implemen ineial paamees. Wih espe o implemen mass and MI, he sysem was mos sensiive a. m/s fowad veloiy. The aveage sae sensiiviy was abou.6 w... implemen mass a. m/s. The sensiiviy dopped o.2 a 4.5 m/s and ineased slighly a 7.5 m/s. The sensiiviy w... he implemen MI was.25 a. m/s and less han. a 4.5 m/s and 7.5 m/s. In geneal, he sysem was highly sensiive o all sysem paamees a. m/s fowad veloiy. As disussed befoe (Fig. 4), highe sysem sensiiviy a he low fowad veloiy was aused by he lage phase shif beween he sysem ansien esponses. The ineaions beween inpu paamees wee obseved by ompuing loal sensiiviies a vaious poins in he paamee spae. A a sample poin wih small elaxaion lengh paamees (% of he nominal values), lage implemen ineial paamees (2% of he nominal values), and nominal oneing siffness paamees, he ineaion beween elaxaion lengh and ohe inpu paamees beame insignifian even a. m/s fowad veloiy (Fig. 6). As expeed, he sensiiviies a. m/s w... oneing siffness and implemen ineial paamees wee smalle a his poin. The sysem sensiiviies w... oneing siffness paamees ineased onsisenly as he fowad veloiy was ineased. In geneal, he sysem was mos sensiive o oneing siffness paamees, followed by he implemen ineial paamees and hen he elaxaion lengh paamees. 2

14 Sensiiviy (%Oupu/%Inpu) Rea Coneing Siffness Fon Coneing Siffness Implemen Mass Implemen Coneing Siffness Vel =. m/s Implemen MI Fon Relaxaion Lengh Rea Relaxaion Lengh Implemen Relaxaion Lengh Fig. 6: Loal sensiiviies a a sample poin in he paamee spae wih espe o vaious sysem paamees a. m/s, 4.5 m/s and 7.5 m/s fowad veloiies. The loal sensiiviy analysis mehod was used o sudy he elaionship beween sysem paamee vaiaions and sysem sae/oupu vaiaions (Fig. 7 and 8). The vaiaions in oupu vaiables wee no symmei abou he nominal poin. The oupus vaied by a diffeen amoun abou he oupu wih nominal paamee values when a 5% hanges wee applied o he oneing siffness paamees in he posiive and negaive dieion (Fig. 7). In geneal, he hange in oupu vaiables ineased as he fowad veloiy was ineased. The hanges in oupu vaiables a. m/s wee vey small and wee no appaen in he diagam. Bu he peenage hanges abou he efeene values wee sill subsanial as we an see in he loal sensiiviy plos (Fig. 5)..5 Oupu Value Vel =. m/s. Tao Laeal Veloiy (m/s) Tao Yaw-ae (ad/s) Im plem en Yaw -ae (ad/s) Fon Tie Side Slip Angle (ad) Rea Tie Side Slip Angle (ad) Implemen Tie Side (ad) Fig. 7: Changes in sae/oupu vaiables fo a 5% hange in oneing siffness paamees a. m/s, 4.5 m/s and 7.5 m/s. The solid bas epesened he values of he oupu vaiables fo he nominal paamees and he veial lines epesened he values of he oupu vaiables when he oneing siffness paamees wee ineased and deeased by 5% of he nominal values. The six sae/oupu vaiables (Fig. ) wee used o alulae he aveage oupu vaiaion of he sysem wih espe o he vaiaion in sysem inpu paamees (Fig. 8). A all veloiies, he aveage oupu vaiaion ineased exponenially wih he vaiaion in oneing siffness paamees (Fig. 8a). The esul showed ha if a % aveage vaiaion in sysem esponses is 3

15 aepable, he oneing siffness paamees mus be esimaed wihin 7%, 8% and 8% of aual/nominal values espeively a. m/s, 4.5 m/s and 7.5 m/s fowad veloiies. The elaionship beween aveage oupu vaiaion and vaiaion in elaxaion lengh paamees was faily linea a all veloiies (Fig. 8b). The oupu vaiaion was always moe han % a. m/s fowad veloiy. A ohe veloiies, a 75% vaiaion in he elaxaion lengh paamees aused abou % aveage oupu vaiaion. Similaly, he elaionship beween aveage oupu vaiaion w... he vaiaion in he implemen ineial paamees was also linea (Fig 8) a 4.5 m/s and 7.5m/s fowad veloiies. Up o 3% vaiaion in he implemen mass and MI was aepable onsideing a % oupu vaiaion o be aepable a hese veloiies. A. m/s, howeve, he aveage oupu vaiaion ineased apidly in he beginning. Only abou 2% vaiaion in he implemen ineial paamees was aepable fo a % aveage oupu vaiaion. Aveage oupu Vaiaion.2 Vel =. m/s Vaiaion in Coneing Siffness (a) Aveage Oupu Vaiaion.24.6 Vel =. m/s Vaiaion in Relaxaion Lengh (b).2 Aveage oupu Vaiaion Vel =. m/s Vaiaion in Implemen Mass and MI () Fig. 8: The elaionship beween aveage vaiaion ove he six oupu vaiables of he sysem and he vaiaion in a) oneing siffness paamees, b) ie elaxaion lengh paamees, and ) implemen ineial paamees (mass and MI). The vaiaions wee alulaed a. m/s, 4.5 m/s and 7.5 m/s fowad veloiies. 4. Conlusions: Loal sensiiviies of a ao and single axle gain a sysem wee alulaed using a deivaive-based mehod. Due o he omplexiy in deiving analyial sensiiviy indies, a dynami sysem numeial simulaion ehnique was used. The wok was impoan in undesanding he effe of paamee vaiaions on he ao and owed implemen sysem 4

16 esponses. The sysem paamees wee anked based on he elaive impoane of he paamees o he sysem esponses. Speifially, he following ae he onlusions fom his wok. In geneal, vaiaion in sysem paamees and sysem oupus wee elaed in a nonlinea fashion. A he nominal values and 4.5 m/s fowad veloiy, a % vaiaion in he oneing siffness paamees esuled o a 5% aveage oupu vaiaion whee as an 8% vaiaion in he oneing siffness esuled o a 8% vaiaion in he oupu. Oveall, oneing siffness was he mos influenial paamee of he ao and a single axle owed implemen sysem dynamis. A he nominal poin, he sysem was equally sensiive o boh of he ao fon and ea ie oneing siffness paamees. The sysem was leas sensiive o implemen ineial paamees. Tie elaxaion lengh paamees beame less influening o he sysem dynamis as he fowad veloiy was ineased whee as he oneing siffness paamees, in geneal, beame moe influenial wih he ineasing fowad veloiy. Aknowledgemens This eseah was suppoed by Hah A and Sae of Iowa funds. Jounal pape of he Iowa Agiulue and Home Eonomis Expeimen Saion, Ames, Iowa, Poje No

17 Noaion and Lis of Vaiables: Vaiable: Supesip: Subsip : v y, The vaiable iself. Big o bold lee veo o maix, small lee sala Denoes whehe he vaiable is elaed o ao o implemen. ao, i implemen Speifies he o-odinae axis he vaiable oesponds o. x x axis, y y axis, z z axis Subsip 2: Speifies he loaion he vaiable oesponds o. f fon ie axle, ea ie axle, ene of gaviy, p oe pin (hih poin) Lis of vaiables α side slip angle o he angle beween he dieion he ie is going and he dieion i is faing. The veloiy veo o he igh of he ie is posiive and evese is negaive. α seady sae side slip angle γ yaw ae δ λ σ φ a b C α d e F I z m p seeing angle Vaiable Subsip angle beween ao heading and implemen heading elaxaion lengh Supesip heading angle disane beween fon axle and CG of ao disane beween ea axle and CG of ao disane beween hih poin and CG of ao oneing siffness disane beween hih poin and CG of implemen disane beween ea axle and CG of implemen foe yaw momen of ineia mass paamee veo Subsip 2 6

18 S u v X-Y x -y y loal sensiiviy measue longiudinal veloiy laeal veloiy wold oodinaes vehile oodinaes posiion of CG in y- axis of he wold o-odinae sysem 7

19 Refeenes: Bevly, D. M., J. C. Gedes, and B. W. Pakinson. 22. A new yaw dynami model fo impoved high speed onol of a fam ao. Jounal of Dynami Sysems, Measuemen, and Conol 24: Chen, C., and M. Tomizuka Dynami modeling of ao-semiaile vehiles in auomaed highway sysems. Po. of Ameian Conol Confeene, Seale, WA, June 2-23, 995. Coseo, M. and S. Taanola. 2. Uneainy and sensiiviy analysis: ools fo GIS-based model implemenaion. Inenaional Jounal of Geogaphi Infomaion Siene 5(4): Cukie, R. I., C. M. Fouin, K. E. Shule, A. G., Peshek, and J. H. Shaibly Sudy of he sensiiviy of oupled eaion sysem o uneainies in ae oeffiiens - I: Theoy. Jounal of Chemial Physis 59: Dag, W., and X. Kang. 23. Paamei sudy on vehile-aile dynamis fo sabiliy onol. SAE Tansaions, Jounol of Passenge Cas, 23: Deif, A. S Sensiiviy analysis in linea sysems. Spinge-Velag, Belin, Heidelbeg. Ebehad, P, W. Shiehlen, and J. Sies. 27. Sensiiviy analysis of ineial paamees in mulibody dynamis simulaions. Twelfh IFToMM Wold Congess, Besanon, Fane, June 8-2, 27. Fales, R. 24. Robus onol design fo an eleohydauli wheel loade sysem wih a human-in-he-loop assessmen in viual envionmen. PhD disseaion, Iowa Sae Univesiy. Feng, L., Y. He, Y. Bao, and H. Fang. 25. Developmen of ajeoy model fo a aoimplemen sysem fo auomaed navigaion appliaions. Insumenaion and Measuemen Tehnology Confeene Oawa, Canada, May 7-9, 25. Gau, C. A. 22. A paamei sudy of he laeal dynamis of a nonlinea fou-wheel oad vehile. PhD disseaion, Univesiy of Cininnai. Jang, J. H., C. S. Han The sae sensiiviy analysis of he fon wheel seeing vehile. KSME Inenaional Jounal (6): Kakee, M. and B. L. Sewad. 28. Open and Closed Loop Sysem Chaaeisis of a Tao and an Implemen Dynami Model. ASABE Pape No S. Joseph, Mih.: ASABE. Kim, Y. C., K. H. Yun, K. D. Min, Y. S. Byun, and J. K. Mok. 27. A laeal dynami model of an all wheel seeing bimodal vehile. Inenaional Confeene on Conol, Auomaion and Sysems, Seoul, Koea, O. 7-2, 27. Kiousioukis, I., S. Taanola, A. Salelli, and D. Gaelli. 24. Uneainy and global sensiiviy analysis of oad anspo emission esimaes. Amosphei envionmen 38: Kuheenko, S., M. Rodiguez-Fenandez, C. Panelides, and N. Shah. 29. Mone Calo evaluaion of deivaive-based global sensiiviy measues. Reliabiliy Engineeing and Sysem Safey. Aile in pesss. MRae, G. J., J. W. Tilden, and J. H. Seinfield Global sensiiviy analysis-a ompuaional implemenaion of he Fouie ampliude sensiiviy es (FAST). Compuaional Chemial Engineeing 6: Mez, L. D Dynamis of fou-wheel-see off-highway vehiles. SAE Pape No Waendale, PA: SAE. 8

20 Mois, M. D. 99. Faoial sampling plan fo peliminay ompuaional expeimens. Tehnomeis 33: Pak, T. J., C. S. Han, J. H. Jang. 23. Dynami sensiiviy analysis fo he panogaph of a high-speed ail vehile. Jounal of Sound and Vibaion 266: Rodiguez-Fenandez, M., and R. Banga. 29. Global sensiiviy analysis of a biohemial pahway model. Advanes in Sof Compuing. Spinge-Velag, Belin, Heidelbeg. Rua, P. and Z. Wojiki. 23. Sensiiviy analysis applied o a dynami aiload model. Jounal of Sound and Vibaion 266: -3. Shwangha, H., and K. Ro The influene of he ie ead on he olling esisane and seeing foes on undiven wheels. In Po. 8h Inenaional Confeene of he Soiey of Teain Vehile Sysems, Cambidge, UK: ISTVS. Sobol, I.M. 993: Sensiiviy esimaes fo nonlinea mahemaial models. Mahemaial Modeling and Compuaional Expeimen : Sobol, I.M. 2: Global sensiiviy indies fo nonlinea mahemaial models and hei mone alo esimaes. Mahemais and Compue Simulaions 55: Xu, C. and G. Gene. 27. Exending a global sensiiviy analysis ehnique o models wih oelaed paamees. Compuaional Saisis and Daa Analysis 5: