Application of Gauge Sensitivity for Calculating Vehicle Body Natural Frequencies

Transcription

1 Inernaonal Journal of Mechancs and Applcaons 013, 3(6): DOI: /j.mechancs Applcaon of Sensvy for Calculang Vehcle Body Naural Frequences Shengyong Zhang College of Engneerng and Technology, Purdue Unversy Norh Cenral, Wesvlle Indana, 46391, USA Absrac Thckness of samped-shee seel panels ha comprse he majory of a modern vehcle s body-n-whe (BIW) has a drec effec on vehcle dynamc performance. Despe he ongong ncrease n worksaon processor speed, compuaonal me sll creaes lmaons on performng modal calculaons. These lmaons are even more pronounced han for sac and quas-sac srucural calculaons. sensvy mehodologes allow a desgner o make quck, accurae esmaes of srucure naural frequences based upon only a few baselne analyses. Ths paper performs gauge sensvy analyss o assess changes n he vehcle body naural frequences due o changes n maeral hckness. Correspondng algorhm for compuaonal mplemenaon s presened. sensvy resuls of rods n longudnal vbraon, beams n ransverse vbraon, and shafs n orsonal vbraon are provded o valdae he proposed gauge sensvy mehod. Applcaon sudes are conduced o calculae changes n he naural frequences of a commercal lgh duy ruck cab and s componens when subjeced o hckness modfcaons. sensvy resuls are compared o fne elemen based resuls and he agreemen s good. Keywords sensvy, Naural frequency, Vehcle body-n-whe, Wegh reducon 1. Inroducon Lghwegh vehcle desgn has been movaed by a need o reduce fuel consumpon whou compromsng vehcle s nose, vbraon and harshness (NVH) performance characerscs. Improvng fuel effcency n all vehcle classes has been made even more acue by recen ncrease n energy prces and requremen for low greenhouse gas emsson. The lgher a vehcle could be, he less energy would be desred for propulson, whch n urn perms he use of smaller, more effcen powerran componens. The vehcle s BIW ypcally accouns for one-hrd of he oal wegh and s herefore a prme arge for wegh reducon. A 10% reducon n a vehcle s BIW can lead o secondary wegh (e.g. chasss, powerran) reducons of 5-7% of vehcle prmary wegh, and a 10% reducon n mass yelds an ncreasng n he range of -5% n fuel economy[1], whch s que sgnfcan. The avalably of hgh-srengh seel (HSS) wh much hgher yeld srenghs[, 3], as well as an array of new manufacurng echnologes (e.g. laser weldng echnology [4], hydroformng[5], and alor-welded blanks[6]) are beng adoped by auomove ndusres, gvng desgners he opon of ncorporang more hn-walled componens n * Correspondng auhor: syzhang@pnc.edu (Shengyong Zhang) Publshed onlne a hp://journal.sapub.org/mechancs Copyrgh 013 Scenfc & Academc Publshng. All Rghs Reserved her vehcles. The fundamenal advanage of hn-walled srucures as found n vehcle bodes s ha he maeral s effcenly loaded, resulng n unformly dsrbued normal and shear sresses. These unform sress dsrbuons n urn yeld hgh sffness-o-wegh raos. However, such hckness decreases may adversely affec assocaed NVH characerscs, resulng n hgher levels of vehcle body vbraon and neror cabn nose. The vbraon envronmen s one of he mporan crera by whch people judge he desgn and consrucon qualy of a vehcle. To mprove he overall rde performance, s necessary o mgae he dynamc response of vehcle body n response o excaon from eher road roughness or on-board sources (re/wheel assembly, powerran and engne, for example). Mos mporanly, s desrable o avod excessve vbraon due o resonance. Ths can be acheved by desgnng he vehcle BIW so ha s naural frequences are ou of he range of dynamc excaons. Boh srucural dynamc modfcaon (SDM) mehod and sensvy mehod have been employed o sudy he effecs of desgn parameer modfcaons on vehcle BIW dynamc performances. The former deermnes he changes n srucural modal properes due o modfcaons of srucural properes (e.g. mass, sffness, and dampng coeffcen) usng eher es or fne elemen daa as he bass. The laer characerzes he sensvy of dynamc quanes o neresed desgn varables, focusng on he dervaves of sysem egenvalues and egenvecors wh respec o srucural parameers.

2 140 Shengyong Zhang: Applcaon of Sensvy for Calculang Vehcle Body Naural Frequences Despe he ongong ncrease n worksaon processor speeds, compuaonal cycle mes sll creae lmaons on usng fne elemen models for performng modal calculaons. These lmaons are more pronounced han for sac or quas-sac srucural calculaons. sensvy ndces are developed for assessng he effec of maeral hckness changes on vehcle BIW performance [1,7]. They allow quanave consderaon of he radeoffs beween wegh reducon and NVH performance resulng from desgn modfcaon nvolvng hckness changes. sensvy ndces are orgnally derved by approxmang a number of sffness-relaed srucural parameers (area momen of nera, bendng sffness, orsonal sffness, for example) as a power funcon of he shee meal hckness. The dervaon also suggess he algorhm for compuaonal mplemenaon, n whch he sffness-based parameers are used as he bass for deermnng gauge sensvy. Ths paper presens a mehod for calculang srucural gauge sensvy ndces based upon naural frequences and esablshes correspondng analyss algorhm. Analycal gauge sensvy analyses are performed for longudnally vbrang rods, ransversely vbrang beams, and orsonally vbrang shafs o valdae he proposed gauge sensvy mehod. The work concludes wh case sudes llusrang applcaon of gauge sensvy echnque for assessng vehcle body naural frequences.. Sensvy for Naural Frequency Analyss.1. Naural Frequency-based Sensvy Indces Whle mahemacal dervaons for gauge sensvy ndces are beyond he scope of hs arcle, he applcaon equaons are summarzed here. Mos modern auomobles and rucks have body srucures comprsed of samped or bonded panels conneced wh combnaons of faseners, ressance welds and bondng. For a sffness-proporonal parameer, f(), ha s a funcon of panel hckness, we can express he gauge sensvy λ as: or log f f ( ) ( ) / λ (1) ( f ( )) log( f ( 1 )) log( ) λ () λ n he above equaons governs he sensvy of he parameer f() o panel hckness and s referred o as he gauges sensvy. We can make he followng conclusons regardng λ: Eq. 1, he angen mehod, s appled o calculae λ when he sffness-relaed parameer f() can be analycally or numercally expressed as a funcon of shee meal hckness. ( ) f here represens he frs dervave of he parameer wh respec o hckness. λ can be graphcally nerpreed o 1 be he rao beween he slope of he angen lne o f() a he operang pon o he slope of he secan lne from he orgn o he operang pon. Eq., he secan mehod, s an alernave approach (n dscree form) for calculang λ usng wo known values of he hckness (one s slghly less han he nomnal hckness and he oher s slghly more han he nomnal hckness) and he correspondng parameer values. The nonlnear naure of he sffness-relaed parameer wll resul n varable λ a dfferen hckness. However, for realsc BIW gauge modfcaon range, λ can be assumed consan and be appled o nvesgae he nfluence of hckness modfcaons on he response parameer f(). Wh known and consan λ, he followng Eq. 3 can be used o esmae changes n he parameer f due o a unformly proporonal gauge modfcaon from 1 o. The erm α n he equaon s defned as gauge modfcaon facor, such ha α 1. λ λ (3) f f1 f1α 1 Excep of sffness-relaed parameers, gauge sensvy ndces can also be deermned based upon srucure s naural frequences. The h order naural frequency,, of a vehcle body srucure can be calculaed from correspondng modal sffness (K ) and modal mass (M ): K (4) M Snce K s a lnear funcon of λ and M s a lnear funcon of, Eq. 4 may be rewren as: C λ C λ 1 Here C s a consan. Takng dervave of Eq. 5 wh respec o hckness and elmnang he consan C yelds: d d λ 1 + (6) Eq. 6 can be used o calculae λ when he frequency and s dervave d /d can be expressed as funcons of hckness. λ can be alernavely calculaed n dscree form by fng a sragh lne beween a known par of naural frequency /hckness values evaluaed a wo nearby pons o he orgnal shee hckness. Correspondng o hckness change from 1 and ( α 1 ), he h order naural frequency wll have s value alered accordng o Eq. 5: ( ) ( ) 1 ( ) λ 1 ( λ 1) Takng he logarhm of Eq. 7 and solvng for λ, he gauge sensvy based on he h order naural frequency of he srucure may hus be calculaed as: 1 (5) (7)

3 Inernaonal Journal of Mechancs and Applcaons 013, 3(6): log λ 1+ ( ( ) ( )) 1 logα For many srucures and desgn cases, 10 % hckness ncrease and 10% hckness reducon ( and where 0 s he orgnal shee hckness of he srucure) work well for calculang λ. Wh s value beng deermned by eher Eq. 6 or Eq. 8, λ can hen be used o calculae changes n he h order naural frequency due o a unformly proporonal gauge modfcaon of he srucure from 1 o by: ( λ 1) ( ) ( ) α (8) 1 (9).. Sample Applcaons of λ To llusrae applcaon of he proposed echnque, consder he vbraon of a rod n he longudnal drecon. The one-dmensonal vbraon s conrolled by s wave equaon nvolvng boh spaal and emporal dervaves. The naural frequences of he rod can be solved analycally wh gven boundary and nal condons. The fxed-free confgured rod of lengh ll, for example, has he naural frequences evaluaed by[8]: ( 1) π E 1,,... (10) l ρ where E s he maeral s modulus of elascy and ρ s he maeral densy. E ρ defnes he velocy of propagaon of he dsplacemen (or sress wave) n he longudnal drecon n he rod. Eq. 10 denoes ha he h order naural frequency of a hollow rod s exclusvely deermned by s lengh and maeral properes (.e., E and ρ). Independence of s naural frequences wh respec o hckness resuls n λ value of uny from Eq. 6. sensvy value of uny mples he naural frequency s nsensve o hckness. In oher words, he modal sffness and modal mass relaed o hs naural frequency change a he same rae when hckness s beng changed. Modfcaons o hckness of such srucures may no be a sraegy for mprovng modal properes. Anoher example s based on he bendng vbraon of a canlever beam wh hollow crcular cross secon. Is naural frequency assocaed wh he h order vbraon mode s deermned by: EI C 1,,... (11) ρa where C s a consan, I s he area momen of nera abou he beam s neural axs, A s he cross seconal area, and E and ρ are he maeral s modulus of elascy and densy, respecvely. Consder a beam wh nner radus of r, wall hckness of, and lengh of ll, I and A can be evaluaed by 4 4 [( r + ) r ] I π (1) 4 and ( ) Aπ r+ r (13) Subsung Eqs. 1 and 13 n Eq. 11 yelds as a funcon of. λ s hen calculaed by Eq. 6: + r r λ 1+ (14) r r If plong λ, we see ha he values of λ range from 1.0 for very small hckness values, ncrease monoonously wh ncreasng hckness o 3.0 for large hckness, see Fgure 1. Consder a canlever beam wh nsde radus of 0.1m, hckness of 0.01m, and lengh of 1m. Whou loss of generaly, we ake s frs naural frequency 1 () as an example n he followng analyss. Expressng 1 () as a funcon of hckness and subsung n Eq. 6 yelds λ value of 1.1. λ may be consdered as consan f hckness modfcaon of he hn-walled beam s made nearby he orgnal hckness. Wh known and consan λ, we can evaluae he changes n he frs order naural frequency wh respec o hckness. The resuls correspondng o hckness rangng from one-half of he orgnal hckness (α 0.5) o double hckness (α ) are lsed n Table 1. The gauges sensvy resuls are n good agreemen wh correspondng analycal resuls. Fgure 1. Varaon of canlever beam gauge sensvy wh hckness

4 14 Shengyong Zhang: Applcaon of Sensvy for Calculang Vehcle Body Naural Frequences Table 1. Changes n he frs order naural frequency of a canlever beam wh respec o hckness modfcaons for he followng gven daa: ll 1m, r 0.1m, E 00GPa, and ρ 7850kg/m 3 (he naural frequency a /r 0.1 s used as reference value) /r Naural frequency (Hz) sensvy Analycal analyss Percen dfference Fnally gauge sensvy s appled o shafs n orsonal vbraon. In hs case he vbraon occurs n an angular drecon. The roaon of a shaf abou s cener axs s a funcon of boh he poson along he lengh of he shaf and he me. Eq. 4 can sll be used o esmae naural frequences by replacng K and M wh orsonal sffness and polar momen of nera of he shaf, respecvely. For a hollow shaf wh crcular cross secon, he naural frequences s calculaed analycally by he followng Eq. 15 f he shaf s fxed a one end and free a he oher end[8]: ( 1) π G 1,,... (15) l ρ where ll s he lengh of he shaf, G and ρ are maeral s shear modulus and densy, respecvely. Smlar o ha of longudnally vbrang rods, he orsonal naural frequences of hollow crcular shafs are ndependen of cross secon hckness, resulng n λ value of uny. I ndcaes ha modfcaons o hckness of such shafs do no affec orsonal naural frequences. For a hollow shaf of oher han crcular cross secon, we can replace G ρ n Eq. 15 wh GγρJ for calculang orsonal naural frequences, where γ s he cross secon- assocaed orsonal consan and J s polar mass momen of nera of he shaf per un lengh. A hn-walled recangular cross secon wh nsde wdh of w and hegh of h, for example, wll have s γ and J calculaed by ( w + ) ( h + ) γ (16) w + h + and ρ J ( w + )( h + ) ( w + ) + ( h + ) 1 ρ w ( w + h ) w h h w (17) ρ 3 3 w h wh + + ( w + h ) 16 Boh γ and J depend on hckness, and so does (). Subsung () n Eq. 6 wll yeld λ, a measure of he sensvy of he h order orsonal naural frequency of he recangular hollow shaf relave o wall hckness. Deermnaon and applcaon of λ for orsonal vbraons are n he same manner as ha for rods and beams. 3. Applcaon Sudes 3.1. B-pllar o Rocker Jon A vehcle body jon s radonally defned as he nersecon of wo or more beam-lke load-carryng members, such as he fron hnge pllar o rocker jon, B-pllar o rocker jon, C-pllar o roof ral jon, ec. Body jons are complan and her flexbly affecs sgnfcanly vehcle body dynamc behavour. Reference[7] presens applcaon of he gauge sensvy echnque o he B-pllar o rocker jon of a lgh-duy ruck and demonsraed he effec of modfcaons o he panel hckness on he jon sffness. The same jon model s used n hs paper o demonsrae applcaon of he proposed naural frequencybased gauge sensvy for evaluang changes n he jon naural frequences due o changes n panel hckness. Fgure a shows he B-pllar o rocker jon whch s comprsed of exernal and nernal meal shees fasen ogeher by spo welds along he boundary edges. Fgure b shows he jon fne elemen (FE) model n whch all he perpheral nodes a boh rocker cus are fxed. The gauge sensvy analyss s performed as follows: perform FE modal analyss a he orgnal jon hckness confguraon o denfy he neresed vbraon mode (fundamenal bendng n he In/Ou drecon, for example) and correspondng naural frequency value. Perform oher wo baselne FE modal analyses usng wo known values of he jon hckness (one s 10% less han he nomnal hckness and he oher s 10% more han he nomnal hckness). Calculae λ usng he secan mehod (Eq.8). Wh λ deermned, he changes n he naural frequency of hs vbraon mode due o changes n panel hckness can be esmaed by usng Eq. 9. For he B-pllar o rocker jon, we are neresed n he wo fundamenal bendng modes (one n he In/Ou drecon and he oher n he Fore/Af drecon) and he fundamenal orsonal mode abou he pllar drecon. We make use of he hree λs assocaed wh each of hese fundamenal vbraon

5 Inernaonal Journal of Mechancs and Applcaons 013, 3(6): modes and esmae changes n he naural frequency when he hckness s ncreased by 30 and 50%, separaely. The resuls are lsed n Tables o 4 and compared wh FE-based resuls. They are n good agreemen. 3.. Lgh-duy Truck Cab The gauge sensvy echnque as descrbed s ndependen of vehcle ype. A lgh-duy ruck cab s used for applcaon sudy, see Fgure 3. Reference[7] presened gauges sensvy resuls of he same cab model from bendng and orsonal sffness pon of vew. Excep of sffness, he naural frequences assocaed wh crcal vbraon modes are also an mporan consderaon n he vehcle BIW desgn. In hs sudy, we focus on he fundamenal naural frequences, snce lower order modes have a pronounced effec on he rde and handlng performance and NVH characerscs. (a) Fgure 3. FEA model of a lgh-duy ruck cab (b) Fgure. (a) B-pllar o rocker jon and (b) jon fne elemen model Table. Fundamenal bendng naural frequency n he In/Ou drecon for dfferen hckness modfcaons of he jon Naural frequency (Hz) Modfcaon Dfference facor α FEA (Hz) sensvy Table 3. Fundamenal bendng naural frequency n he Fore/Af drecon for dfferen hckness modfcaons of he jon Naural frequency (Hz) Modfcaon Dfference facor α FEA (Hz) sensvy Table 4. Fundamenal orsonal naural frequency abou B pllar pah for dfferen hckness modfcaons of he jon Naural frequency (Hz) Modfcaon Dfference facor α FEA (Hz) sensvy sensvy analyss for he cab s n a manner dencal o ha for he B-pllar o rocker jon. Whou loss of generaly, we analyze he fundamenal bendng vbraon abou y-axs. The value of λ assocaed wh hs bendng mode s calculaed o be 1.86 hrough wo baselne FE modal analyses. We hen make use of he deermned λ o esmae changes n he naural frequences when he cab hckness s modfed by a facor of 0.5, 1.5,.0,.5, and 3.0, separaely. To check he accuracy of he foregong gauge sensvy resuls, FE modal analyss s performed a each modfed hckness confguraon. The resuls are conssen wh each oher, see Table 5. Table 5. Changes n he naural frequency of he fundamenal bendng mode abou y-axs due o changes n he cab hckness, calculaed by gauge sensvy analyss and fne elemen analyss Naural frequency (Hz) Modfcaon Percen facor α FEA dfference sensvy Conclusons The fundamenal concep of gauge sensvy orgnaes from approxmang sffness-relaed srucural parameers as a polynomal funcon of he shee meal hckness. Ths paper pus forward gauge sensvy ndces based upon

6 144 Shengyong Zhang: Applcaon of Sensvy for Calculang Vehcle Body Naural Frequences srucure s naural frequences. The compuaonal procedure s provded for deermnng λ values and calculang naural frequency changes resulng from hckness modfcaons. A seres of analycal examples, ncludng longudnal vbraon of rods, bendng vbraon of beams, and orsonal vbraon of shafs are presened o valdae gauges sensvy resuls. A lgh-duy ruck cab and s B-pllar o rocker jon are ulzed n he applcaon case sudes. All gauge sensvy resuls are compared o eher analycal resuls or FE-based resuls and he agreemen s good. Fuure research ncludes applcaons of gage sensvy o non-body vehcle srucures. For example, ruck frames and cross members are ofen fabrcaed from samped or rolled members and fall no hs archecure caegory. I should be possble o employ gauge sensvy o hese componens wh lle or no modfcaons o he mehodologes, however, hese componens may yeld gauge sensvy values que dfferen from hose seen for body srucures. REFERENCES [1] Paon, R. and Edwards, M. (00) Esmang wegh reducon effecs of maeral subsuon on jons wh consan sffness, SAE paper number [] Marn, D.C. (1997) ULSAB: a progress repor, Advanced Maerals & Processes, Vol.15, No. 3, pp [3] Praer, G., Azzouz, M., Furman, V., Shahhossen, A. and Sae, M. (00) Use of FEA concep models o develop lgh-ruck cab archecures wh reduced wegh and enhanced NVH characerscs, SAE paper number 0M-145. [4] Schuber E, Klassen M, Zerner I, Walz C, Sepold G. (001) Lgh-wegh srucures produced by laser beam jonng for fuure applcaons n auomoble and aerospace ndusry, Journal of Maerals Processng Technology, Vol. 115, pp. 8. [5] Teng BG, Yuan SJ, Wang ZR. (001) Expermenal and numercal smulaon of hydroformng orodal shells wh dfferen nal srucure, Inernaonal Journal of Pressure Vessels and Ppng, Vol. 78, pp [6] Kusuda, H., Takasago, T.and Nasumm F. (1997) Formably of alored blanks. Journal of Maerals Processng Technology, Vol. 71, pp [7] Praer, G., Zhang, S.Y., Shahhossen, A., Rchards, C. and Osborne, G. (009) sensvy ndces for vehcle body srucure assessmen and opmzaon, Inernaonal Journal of Vehcle Sysems Modellng and Tesng, Vol. 4, Issue 1/, pp [8] Inman D. J. (013) Engneerng vbraon, New Jersey: Prence Hall.