Time-Frequency Analysis for Nonstationary Random Response of Vehicle

Transcription

1 IW - Intitutmitteilung N. 8 (3) 3 Time-Fequency Analyi o Nontationay Random Reone o Vehicle Lijun Zhang; Tiania Zhang; Hui He In thi ae the nontationay andom eone o baing vehicle in time domain a it obtained by uing nontationay oad oughne model and vehicle model ith 5DOF. To obtain the eult o nontationay eone in equency domain, the maimum entoy method hich had oved to have moe advantage ove taditional FFT method a ued o oceing nontationay eone o baing vehicle in equency domain. Finally, the thee-dimenional tanient maimum entoy ectum (ES) o eone a given. Intoduction It i non that mot eeach o on Random vibation o vehicle caued by the ecitation o ough oad in equency domain a baed on to aoache. One aoach i diect method in hich equency chaacteitic can be obtained diectly by uing tane unction accoding to the theoy o andom vibation (Doainih,.A. and Elmadany,., 98). The othe i an indiect method in hich the eone o vehicle in time domain i it obtained, and then the eone o vehicle in equency domain can be obtained by u- ing FFT method hich i idely ued o oceing tationay ignal (Bigham, E.O.988; Welch, P.D, 967). While a ca i taveling at vaiable eed uch a tating, acceleating a ell a baing, the unction o oad oughne i a nontationay andom oce in time domain deite it i tationay andom oce in atial domain. Theeoe the eone o vehicle to the ecitation o ough oad hould be nontationay andom oce. It i not oe to ue FFT to oce inut and eone o baing vehicle, becaue FFT method i jut uitable o oceing tationay ignal. In thi ae, maimum entoy method in hich contain moe inomation ith le data i ued to obtain tanient equency chaacteitic. It i neceay to oint out that the nontationay eone o vehicle in time domain hould be given a biely tatement it. Nontationay Reone o Vehicle In geneal cae, the unction o oad oughne i egaded a a tationay andom oce in ace domain. A a ca i moving ith contant velocity, it i alo a tationay andom oce in time domain. Hoeve oad oughne i a non-tationay andom oce in time domain hile a vehicle i taveling at vaiable eed. Theeoe the vibation caued by ough oad uace hould be alo conideed a a non-tationay andom oce. A tate-ace aoach a eented to analyze the eone o a vehicle taveling on homogenou ough oad (Hammond and Haion, 98). In that o, the dynamic ee modeled by line odinay dieential equation in time domain hile the ecitation oce a modeled by a dieential e- quation in atial domain. The vaiance o eone a obtained by uing o-called covaiance equivalent modeling. Baed on thi o, an imoved method ith moe comutational eiciency a ooed by uing comle modal analyi (Hang, J. H. and Kim, J. S, ). Thee a a- nothe method o olving thi oblem (Nigam and Yaday, 974; Fang, T., 997), in hich dieential equation ith vaiable coeicient ee etablihed it in ace domain, and then, the time changing covaiance a comuted. Hoeve, the amount o comutation o thi method i lage. Thi ection mainly invetigate a ne time method by uing the nontationay ecitation model o a ough oad.. odeling In thi ection, a ne method o olving nontationay vibation o a vehicle i mainly invetigated, hich can be alo ued in vaiou ind o vehicle model. Theeoe, a an eamle, a vehicle model ith 5 degee o eedom (DOF) i etablihed. The vehicle model i hon in Figue.

2 3 IW - Intitutmitteilung N. 8 (3) Fig. Vehicle model ith 5 degee o eedom Whee: Z -vetical dilacement o eat, Z b -vetical dilacement o vehicle body at cente o gavity, Z -itch angula dilacement o vehicle body, Z -bouncing dilacement o unung ma o ont uenion, -ma o human body and eat, b -ma o vehicle body, -moment o inetia o vehicle body aound y ai, K -tine coeicient o eat,, -unung mae o ont and ea uenion, eectively, K, K -tine coeicient o ont and ea uenion, eectively, K t, K t -tine coeicient o ont and ea tie, eectively, C -daming coeicient o eat, C, C -daming coeicient o ont and ea uenion, eectively, q (, q (-dilacement o oad inut at ont and ea tie, eectively, l -ditance om the cente o gavity to eat, l, l 3 -ditance om the cente o gavity to ont and ea tie, eectively, L-heelbae. In thi model, daming o tie i neglected, ince it i o mall. The nonlinea daming and tine o uenion can be alo conideed i neceay. The dieential equation o the vehicle model can be decibed a ollo: [ Z&& + [ C Z& + [ K Z [ FQ hee: Z-outut vecto Z z z z [ z z T [ -mati o ma b () [ C C Cl C + C + C l l + C l [ K -tine mati [ K K lk K + K + K l l + K l K l l + C l C l C l + C l C l l + K l l + C l l l + K l K l [ F -ecitation oce mati [ F Kt Kt [ Q -ecitation vecto [ Q [ q T q l K K l K l C C l K l K + Kl K + K t l3 C l3 K + Kt When a ca i unning at vaiable eed, unction o q ( and q ( ae non-tationay in time domain, but it i noted that they ae tationay in atial domain.. State Equation To obtain the eult o equation () in time domain, tate vecto ae utilized a ollo: T [ [ z z z z z b (-a) T [ [ b (-b) Hence [ & & & & & T [ T [ & & & & & [&& z && z && z && z & z (-c) b (-d) Subtituting thee tate vecto into Eq. (), then the olloing tate equation can be obtained: X & AX + BU hee: X -tate vaiable [ T X () [ b [ C -daming mati.3 Simulation o Road Roughne In geneal cae, the oe ectal denity (PSD) o ough oad in equency domain can be eeed a: ω Sq ( ω ) Sq ( Ω ) v /( ω + ) (3)

3 IW - Intitutmitteilung N. 8 (3) 33 hee, S q( Ω ) i the coeicient o oad oughne, Accoding to the claiication o oad, the value o S q ( Ω ) can be obtained om ome eeence. Ω i the eeence value o atial angula equency, Ω ( ad / m), v i taveling eed o vehicle, and ω i the loet cut-o angula equency. Eq.(3) can be conideed a a eone o a it ode linea ytem to hite noie ecitation. Baed on the theoy o andom vibation, olloing elationhi i obtained q ( ) H ( ω) S S ω (4) hee H (ω ) i the tane unction, and S i the PSD o hite noie hee nomally S. Fom Eq (3) and (4), H (ω ) i itten by Sq ( Ω) v H ( ω) (5) ω + jω Fom Eq.(5), the dieential equation about oad oughne i eeed a q& ( +ω q( Sq ( Ω ) v( (6) While a ca i unning ith vaiable eed, deite oad oughne i tationay andom oce in ace domain, it i non-tationay andom oce in time domain. Note that ω πnv( (7) hee n i ace equency. Theeoe, Eq.(6) become a ollo q& ( + π n v( q( π S ( n ) n v( ( ) (8) q t hee v ( v + at, a i the deceleation o ca baing, t i baing time. Accoding to Eq.(8), the numeical imulation can be caied out. The Figue and Figue 3 eeent the imulation conequence o nontationay ough oad and it PSD eectively. Fom Figue, it i concluded that ith ineaing o time, the amlitude o oad oughne in time domain decee in both lo equency and high equency..4 Simulation o Nontationay Reone Inutting the value o oad oughne into Eq.(), the non-tationay eone o the vehicle ytem to ecitation o oad oughne can be calculated. The model aamete eeenting 5 degee o eedom ae hon in Table. Figue 4 eeent the eone o vehicle ytem to non-tationay ecitation, i.e., the eone o vehicle baing ith a-3m/ V 6m/h. It ha been een that at the beginning o baing, the vibation acceleation o eat and vehicle body, elative dilacement and itch acceleation ae all andom ocee. When the baing time i at about t6, i.e., the velocity o vehicle i equal to zeo, All the eone o vehicle become ee vibation, hich can clealy elect the natual equencie o eone o vehicle. Ro ug hn e OF oa d( m) PSD o ough oad(m /Hz) Gade C a-3.m/ V6m/ Fig. Non-tationay oad oughne Gade C a-3m/ V6m/ - Fig. 3 Fequecy(Hz) PSD o nontationay oad (g) b (g) (g.m) (g) (g) K K K Kt Kt C C C 55 8 Table Paamete o vehicle model ith 5 DOF Pitch acceleation(ad/ ) Relative dilacement(m) Acceleation o eat(m/ ) Dynamic load o tie(n) Acceleation o body(m/ ) acceleation o ale(m/ ) Fig. 4 Nontationay eone o vehicle (V 6m/h, a-3.m/ )

4 34 IW - Intitutmitteilung N. 8 (3) 3 aimum Entoy Sectal Analyi aimum entoy ectal analyi a it ooed by Bug in 967, hich ha develoed in the at tenty yea. It i alo called moden ectum comaed ith claical FFT method. Although thee ee much moe ucceul alication o FFT to oceing tationay ignal, the equency eolution i conlict ith amling time inteval t in uing FFT method. Futhemoe the aumtion that thee i no data beyond the amling time length T eult in eduction o equency eolution. Theeoe it i not uitable to ue FFT method to oce hot time data. aimum entoy ectal analyi i aamete etimation method in hich a data model i it etablihed and then the etimation o PSD i caied out ith model aamete. Theeoe the equency eolution i not limited by length o data, and highe equency ditinguihing ability can be obtained. 3. odeling and Paamete Etimation Given amled dicete eie (, hee n i amling numbe, n,,3, N. Auming that ( i the eone o a linea model ubjected to ecitation o hite noie. Theeoe it can be eeed a: ( a ( n ) + ( (9) hee ( i hite noie ith zeo mean and σ vaiance, i the numbe o model ode, and a i model aamete,,, 3. Accoding to the deinition o autocoelation R ( m) E[ ( ( n + m) () and conideing Eq.(9), the olloing equation i obtained R ( m) E( a ( n + m) + ( n + m) Simliying Eq. (), and it can be eeed a [ ( (( n m) R ( m) a R ( m ) + E + hen m ( i ielevant to ( n + m) Hence E ( m > ) ( σ ( m ) [ ( n + m) () () (3) Subtituting Eq.(3) into Eq.(), the olloing equation ae obtained. R ( m) + R () + a R ( m ) ( m > ) a R ( ) σ ( m ) Eq.(4) can alo be itten in tem o mati R () R () R ( ) R ( ) R () R ( ) L L K R ( ) σ R ( + ) a R () a While given amled dicete eie (, autocoelation R and covaiance σ (4) (5) can be calculated. Accoding to Eq.(5), model aamete can be obtained. So a thee ae many algoithm o comuting model aamete among hich Bug algoithm i a at algoithm imila to FFT. While moe data need to be calculated, thi method i much eective. 3. Bug Fat Algoithm Bug at algoithm can obtain the model aamete diectly ith amled data ithout calculating the autocoelation. It i noted that the value obtained om Eq.(9) ae the evaluated value o ~ ) amled data. Deine (n a the evaluated value. Theeoe, the oad eo o data model ith ode i eeed a: e ( ) n ~ ( ) ( ( ( + a ( n ) (6) Similaly, the bacad eo o data model ith ode i itten by e ( ) bn ~ ( ) ( ( n ) ( + a ( n + ) (7) The elationhi beteen high ode eo and lo ode eo can be eeed a e e + ρ e (8) ( ) ( ) ( ) n n bn e e + ρ e (9) ( ) ( ) ( ) bn bn bn hee ρ i the electing coeicient. Deine ES a the um o eo quae. Hence

5 IW - Intitutmitteilung N. 8 (3) 35 N n ( ) ( ) {[ e n + [ ebn } ES () E Let ( ) ρ hence ( ) ( n ) E[ e n ebn ( ) ( ) [ + E[ e bn () ρ () E e n The elationhi beteen model aamete and electing coeicient can be eeed a: a ( ) ρ ( ) ( ) ( ) (3) a a + ρ a,,3... (4) Accoding to Eq.() and Eq.(4), the ecuive calculation can be accomlihed. thee dimenional tanient maimum entoy ectum o vehicle body acceleation, ale acceleation and itch angula acceleation in baing, eectively. Fom Figue 9, it i concluded that ith the inceaing o time i.e. the deceaing o velocity, the amlitude o ES o vehicle body acceleation ha the tendency o eduction, and ha ome luctuation hich elect the ave hae o ough oad. Figue illutate the amlitude o tanient ES o ale acceleation deceae ith inceaing o time. Figue. ho that tanient ES o itch acceleation ha to main ea in both lo and high equency. The ea in in lo equency educe ith the inceaing o time, but doe not alay educe. The ea in high equency deceae ith the inceaing o time aimum Entoy Sectal Etimation Taing Z tanom to both ide o Eq.(9), the dicete tane unction i eeed a H ( z) X ( z) W ( z) A( z) + a z hence the PSD o the ignal can be obtained P ( ω) H ( e (5) jω σ (6) ) σ j + ae In Figue 5 i the oceing eult o vehicle body acceleation by uing maimum entoy method ith data o 3 oint. It i een that maimum entoy ectal etimation ell elect to main ea ith le amled data. Fit ea eeent eonant equency o vehicle body, and econd ea i cloe to the eonant equency o unung ma. oeove the equency eolution i continuou. Figue 6 i the PSD o vehicle body acceleation by uing FFT method ith ame amled data. It illutate that FFT method cannot elect to main ea, and the equency eolution i bigge. 6. Hz T N t Theeoe maimum entoy method ha moe advantage than FFT method in oceing hot time data. Figue 7 and Figue 8 eeent ES and PSD o ale acceleation eectively. Data o oint ae ued in both method. It conclude that maimum entoy method can give the eonant equency o ale eactly and FFT method cannot do. Figue 9, Figue and Figue ae the ES o acceleation o body(m/ ) PSD o acceleation o vehicle body(m/ ) ES o acceleation o ale(m/ ) Fequency (Hz) Fig ES o vehicle body acceleation Fequency (Hz) Fig PSD o vehicle body acceleation Fequency (Hz) Fig.7 ES o ale acceleation

6 36 IW - Intitutmitteilung N. 8 (3) 7 PSD o acceleation o ale(m/ ) Fequency (Hz) Fig. 8 PSD o ale acceleation ES o itch accleation (m / 4 /Hz) Fequency(Hz) Fig. Tanient ES o itch acceleation 5 4 CONCLUSIONS Thi ae it eom numeical imulation o nontationay eone o vehicle in time domain by uing nontationay ecitation model o oad and vehicle model ith 5DOF. To obtain equency chaacteitic, maimum entoy method i ued to oce nontationay eone in equency domain. The eult ho that maimum entoy method ha moe advantage than taditional FFT method in oceing le data. The equency eolution i ucceive. The ES method i uitable o oceing hot time data, theeoe the intantaneou ectal chaacteitic can be obtain, hich clealy elect nontationay chaacteitic o vehicle in baing. ES o accleation o body(m / 4 /Hz) Fig.9 ES o accleation o ale (m / 4 /Hz) Fequency(Hz) Tanient ES o vehicle body acceleation 5 5 Fequency(Hz) Fig. Tanient ES o ale acceleation REFERENCES // Bigham, E.O. (988). The Fat Fouie Tanom and It Alication, Pentice- Hall, Engleood Cli,Ne Jeey // Bug, J.P.(967). aimum Entoy Sectal Analyi, 37th Ann. Inten. eet., Soc. Etlo. Geohy., Oct. /3/ Doainih,.A. and Elmadany,..(98). Random Reone o Tacto-emitaile Sytem. Veh. Sy. Dyn. 9:87-. /4/ Fang, T., Sun,.N. and Song, C.Q.(997). Non-Stationay Reone to the Evolutionay Random Ecitation Fom A Same Souce. Chinee Jounal o Alied echanic. Vol.4.No.: 7-3. /5/ Hammond, J.K.and Haion, R.F.(98). Nontationay Reone o Vehicle on Rough Gound-A State Sace Aoch, Taniction o The ASE Vol. 3: /6/ Hang, J.H. and Kim, J.S.(). On The Aoimate Solution Aicat Landing Gea Unde Nontationay Random Ecitation KSE Intenational Jounal. Vol 4, No 9, /7/ Nigam, N.C. and Naayanan,S.(994). lication o andom vibation. Rajamal Electic Pe. /8/ Nigam, N.C. and Yadav, D.(974).Dynamic Reone o Acceleating vehicle to Gound Roughne. Poc. Noie, Shoc and Vibation Coneence, onah Univeity:8-85. /9/ Welch, (967). The ue o FFT o The Etimation o Poe Sectal-A method Baed on Time Aveaging Ove Shot odiied eiodogam, IEEE Tan, AV-5.