NUMERICAL SIMULATION OF VEHICLE MOTION ALONG THE ROAD STRUCTURE

Transcription

1 37 ROCZNIKI INŻYNIERII BUDOWLANEJ ZESZYT 2/22 Komsja Inżyner Budowlanej Oddzał Polskej Akadem Nauk w Katowcach NUMERICAL SIMULATION OF VEHICLE MOTION ALONG THE ROAD STRUCTURE Jozef MELCER and Gabrela LAJČÁKOVÁ Unversty of Zlna, Faculty of Cvl Engneerng, Department of Structural mechancs. Introducton The roads are the transport structures subjected to drect dynamc effect of movng vehcles. The knowledge of the real load actng on the roads, ts varablty n tme and frequency composton, s needed for the soluton of many engneerng tasks as desgn, fatgue, lfetme, relablty, mantenance, structure development, mcro-tremor, etc. The task can be solved by expermental or by numercal way. But the most effectve way s the combnaton of the both mentoned advances. The submtted paper s dedcated to the descrpton of facltes how to obtaned the requred data by numercal way. Ths process requres creaton the computng models of vehcles, the computng models of the roads and to pay attenton to the numercal soluton of equatons of moton and the analyss of obtaned results n tme and n frequency doman. 2. Vehcle computng model The computng models of vehcles can be created on varous levels as -dmensonal, 2-dmensonal or 3-dmensonal []. For the purpose of ths contrbuton the 3-dmensonal space computng model of a lorry was adopted, Fg.. It s dscrete computng model wth 5 degrees of freedom. The 9 mass degrees of freedom correspond to the vertcal dsplacements r (t) of the mass objects m. The mass-less degrees of freedom correspond to the vertcal movements of the contact pont of the model wth the road surface. The vbraton of the mass objects of the model s descrbed by the 9 functons of tme r (t), ( = 9). The mass-less degrees of freedom are assocated wth the tre forces F j (t), (j = 5) actng at the contact ponts. The equatons of moton have the form of ordnary dfferental equatons and wth respect on the used method of numercal soluton they can be wrtten n the form [ m ] {& r } + [ b] r& } + [ k] r} = { F}, () m ] {& r } = { F} [ b] r& } [ k] r} = { F} { F } { F } = { F }. (2) [ d re v Soluton of equatons of moton s realzed numercally n the envronment of programmng system MATLAB.

2 38 3(I x ) (m ) T 2(I y ) 3 7(I y4 ) 8 4 6(m 4 ) 7 c c 9(I y5 ) 8(m 5 ) 9 e e b 5 a 4(m 2 ) 2 5(m 3 ) 6 d d Fg. Space model of the vehcle TATRA Road surface unevenness The rgd pavement wth random road profle s assumed for the purpose of numercal soluton. The random road profle h(x) s assumed as statonary ergodc functon wth zero mean value and normal dstrbuton. The propertes of the road profle are descrbed by Power Spectral Densty functon (PSD) n the form k Ω S h ( Ω) = S h ( Ω), (3) Ω where Ω n [rad/m] denotes the wave number, Ω = rad/m s the reference wave number and the wavness k = 2. Accordng to the nternatonal drectve ISO 868 [2], typcal road profles can be grouped nto classes from A to E. Each class s smply defned by ts reference value S Ω ), Fg. 2, Table. h( Table : Classfcaton of pavements accordng to road unevenness [2] Class S h (Ω ) [m 2 /(rad/m)] at Ω = rad/m lower bound geometrc average upper bound A B C D E

3 39 S h (Ω) - S h (n) Wave length λ (m), Power spectral densty (m 2 /(rad/m)) Class of road H G F E D C B A ,, Spatal frequency n (cykly/m), Angular spatal frequency Ω (rad/m) Fg. 2 Power spectral denstes of road profles A random profle of a sngle track can be approxmated as N h ( x) = 2 S( Ω ) Ω cos( Ω x + φ ), (4) where φ s the unformly dstrbuted phase angle. 4. Results of numercal smulaton of the tre forces Numercal soluton was carred out n the envronment of the program system MATLAB. The 4 st order Runge-Kutta step-by-step ntegraton method was used for the soluton of equatons of moton [3]. As the result of numercal smulaton the movement of vehcle along the road profle wth the speed V = 36 km/h was smulated. In the frst step the random road profle h(x) on the bass of known power spectral densty for the value S h (Ω ) = m 2 /(rad/m), category B, was generated. The generated road profle s shown n Fg. 3.

4 4 Road profle h(t) 2 h(t) [mm] Tme t [s] Fg. 3 Random road profle, S h (Ω ) = m 2 /(rad/m) The correspondng tre forces under rear wheel of rear axle of the vehcle computng model are shown n Fg. 4. Tre force F8(t) -4 F8(t) [kn] Tme t [s] Fg. 4 Tre forces under rear wheel of rear axle Smlar results can be obtaned for varous road profles, for varous speeds of vehcle moton and for varous vehcle computng models. At the certan category of road the tre forces are theoretcally postve. In realty the contact between the wheel and the road s lost. The wheel wll be bounce from the road. The contact s always replaced by the mpact. It s very bad stuaton not only for the pavement but also for the vehcle. The soluton can by carred out n the tme or n the frequency doman. In the frequency doman the frequency composton of vbraton s nterestng. It can be assessed for example through Power Spectral Densty functons (PSD). The PSD functon of the road profle nforms us about the frequency composton of the road profle and the PSD of the tre force nforms us about the frequency composton of the pavement load. The PSD of the road profle from Fg. 3 s plotted n Fg. 5 and the PSD of the dynamc component of tre force from Fg. 4 s plotted n Fg. 6. As we can see n the frequency composton of the road profle the low frequences domnate. In the frequency composton of the tre forces the frequences n the nterval from 6 to 2 Hz domnate. It relates wth natural frequences of the vehcle.

5 4 4 x -4 PSD of the road profle PSD h [m 2 /Hz] Frequncy f [Hz] Fg. 5 PSD of the road profle 3 PSD of the tre force dynamc component F8d PSD Fd [kn 2 /Hz] Frequncy f [Hz] Fg. 6 PSD of tre force F 8d (t) Road profle h(t) represents the nput functon and tre force represents the output functon. The Frequency Response Functon (FRF) between the two quanttes can be derved. The second power of the modulus of FRF s denoted as Power Response Factor (PRF). It can be obtaned by dvdng the PSD of output and nput functons. 5 x 6 PRF Power Response Factor Fd(.f) 2 PRF Fd/h [kn 2 /m 2 ] Frequncy f [Hz] Fg. 7 Power response factor of the tre force F 8d (.f) 2 Smoothng of PRF obtaned by dvdng the PSD of output and nput functons (red lne) we receve the same results as at drect soluton n frequency doman for example by Fourer or Laplace ntegral transform.

6 42 4. Conclusons Numercal modelng of the problems of vehcle - road nteracton s an effectve tool for the soluton of real tasks of engneerng practce. Qualty of obtaned results s dependent on the qualty of nput data. The present state of computng technque enables the numercal processng of solved problems n real tme. From the practcal pont of vew the nfluence of road profle on vehcle vbraton and tre forces s nterested. The results obtaned from numercal analyses are used n the process of desgn and optmzaton of road structures wth respect to lfetme, relablty and envronmental aspects. References [] Melcer, J. Martncká, I.: Roll effect on the force and knematcal values of vehcle. Cvl and Envronmental Engneerng, Vol. 6, No. 2, 2, ISSN , p [2] ISO 868, Mechancal vbraton road surface profles reportng of measured data. Internatonal standard, 995. [3] Melcer, J. Lajčáková, G.: Dynamc effect of movng load on asphalt pavement. Proceedngs of the 2 nd Internatonal Conference on Road and Ral Infrastructure, CETRA 22, Road and Ral Infrastructure II, Croata, Dubrovnk, May 7-9, 22, Unversty of Zagreb, 22, ISBN , p Acknowledgement The Authors are grateful for support from the Grant Agency VEGA of the Slovak Republc, project No. /295/2. NUMERICKÁ SIMULÁCIA POHYBU VOZIDLA PO CESTNÝCH KOMUNIKÁCIÁCH Resumé Predkladaný príspevok je venovaný metódam numerckého modelovana pohybu vozdel po cestných komunkácách. Uvažuje prestorový výpočtový model vozdla TATRA 85 a výpočtový model tuhej vozovky s náhodným proflom povrchu jazdnej dráhy. Prezentované je numercké rešene pohybu vozdla po vozovke v časovej vo frekvenčnej oblast. Rešene pohybových rovníc je robené v prostredí programovaceho jazyka vyššej úrovne MATLAB. Získané výsledky sú zobrazené v grafckej forme.