STEADY STATE MOTION OF RAIL VEHICLE WITH CONTROLLED CREEP FORCES ON CURVED TRACK

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1 OL. 6, NO., FEBUAY ISSN Asian eseah Publishing Netwok (APN). All ights eseved. STEADY STATE MOTION OF AIL EHICLE WITH CONTOLLED CEEP FOCES ON CUED TACK M. Messoui Univesity M Hamed Bougaa, dépatement de méanique, FHC, Boumedès, Algeia messoui_ma15@live.f ABSTACT A linea analysis of steady state uve tavesing is developed fo ail vehile with wheels oupled with a tosional visous dampe. The govening equations ae given and ae solved numeially. We onside the equilibium of the bogie with two wheel sets ated upon by the foes of gavity and the tansvese foe. The appoah is based on poviding guidane by eep foes in onjuntion with wheel oniity, so that flange ontat is nomally avoided. It desibes the behavio of the vehile in the uves of diffeent adii. It is, theefoe, impotant to define the haateistis of the tosional dampe. It gives the modulus (magnitude) of the moment of fition in the dampes podued by the elative motion of the wheels as well as the bogie yaw, the elative lateal and yaw displaement of the leading and tailing wheel set at eah uve tak of onstant uvatue. Keywods: ail vehile, steady state motion, uved tak, fition, tosional visous dampe, linea analysis. 1. FOEWOD In evey ounty, a onsideable popotion of the total length of tak on a ail system is omposed of uved taks. In the uved taks, exessive longitudinal stiffness in the pimay suspension o high entifugal foes has a detimental effet on the insiption of the vehile. They upset the nomal guidane assued by the eep foes. This happen beause the foes balane is distubed and demand a geate esultant eep foes than the limiting fition foe. As a esult the wheel sets slip. The ail vehile annot peseve the dietion of motion. It skids. The flange ontat ous allowing the flanges foes to estoe the equilibium. The wheels and the ail annot be safeguaded against aeleated wea. This fat geneates a need fo a new vehile design. The vehile with ontolled eep foes is noted fo its stability even with a soft longitudinal pimay suspension. The tehnial innovation is to inopoate a tosional visous dampe between wheels of the split up wheel sets. Wothy of notie in the design of ail vehile is that the smalle the adius of uve, the less should be the longitudinal stiffness o the yaw stiffness in the pimay suspension. The vehile haateistis aepted fo alulation in this study ae given in the nomenlatue. We give and solve equations with aount of the design paametes of the dampe and the basi haateistis of the uved tak (the ant angle and the adius) whih ae kept onstant. We give an oveall pitue of how the example vehile behaves though uves and the ouple developed in the dampe.. INTODUCTION It must be eognized that in piniple the pevailing goal is to design a vehile so that the eep foes ating on the wheel tead insue guidane in the uve, that is, the flange ontat is avoided. Cuve tavesing unde the ontol of eep foes of an adjusted magnitude has the advantages of minimizing wea of wheel and ail unde nomal onditions of iulation. It was unde these onditions that linea analysis is valid; Boook developed it [1]. The equations of motion of the vehile tavelling in uves ae idential to the equations obtained by alignment unless it appeas the funtions of foes in the ight hand of the equations. The effets of the uvatue and the ant of the tak ae to povide soliitations fom outside the vehile and its stati and dynami stability is theefoe the same [, 3]. In the ase of stable taffi at speeds slightly highe than those owing to the foes balane (low ant defiieny), we an neglet in the fist appoximation, the foes of inetia and damping in espet with suspension foes. Moeove, we believe that the entifugal foes ae small. In these iumstanes, we an assume that the oll angles ae vey small. The vehile moves at a onstant speed along the tak, whih has a onstant ente-line adius and unifom angle of ant. This study shows the behaviou of the ail vehile in the uved taks, its advantages and its limitations. It undelies the detemination (the hoie and alulation) of the tosional dampe as well. 3. EHICLE ACHITECTUE The vehile used onsists of a body, two idential bogies and fou-wheel sets. Figue-1 depits a sheme of one bogie suounded by the povision of suspensions. 31

2 OL. 6, NO., FEBUAY ISSN Asian eseah Publishing Netwok (APN). All ights eseved. The pimay suspension between the bogies and the wheel sets is made by weightless spings in seies with ubbe pads to damp the noise. Fou dampes in paallel with spings omplement the suspension. The same goes fo the seonday suspension. The body/bogie oupling link allows to the body to pefom an osillatoy motion aound a ylindial and longitudinal hinge plaed undeneath. The lowest end of this oupling link slides in a ball-and-soket joint judiiously loated in the bogie fame. The wheel set onsists of two wheels igidly linked to two semi hollow shafts tuning aound an inne ommon axle. A tosional visous dampe onnets the two half-hollow shafts. Thus, the otation of one wheel affets the otation of the othe. The elative otation between wheels will be moe o less onstained depending on the oeffiient of esistane value o damping oeffiient C φ of the tosional dampe. This onsolidation povides an oppotunity to the mehanism to etun to its equilibium position: the wheel that limbs up the flange taking the advane on the othe wheel auses a otation of the axle, whih tends to bing it to the ental position in the tak. This onstutive featue gives us, in fat, the ability to ontol the eep foes magnitude geneated at the wheels. 4. OLLING CONTACT THEOY The Kalke theoy of fitional olling ontat allows the evaluation of the ontat foes on the wheel [4,5]. He defined a digital poess fo evaluation these foes. The explanation of eep foes lies in the elasti popeties of the wheel and the ail: owing to this elastiity, these two bodies touh along a ontat aea. This poblem was analysed by Hetz. Tangential foes appea in the tiny ontat aea when the elasti wheel olls on the elasti ail owing the small deviation fom pue olling of two elasti bodies. Convesely, if we apply extenal tangential effots, deviations fom ontinuous olling an ou. Fo low values of these movements, thee is a pue adhesion ove the entie sufae of ontat: the sliding at the point of ontat of the wheel with the ail is negligible and the wheel hanges the dietion of motion mainly due to elasti defomation. That is, thee is a dependene of the elasti defomations on the tangential foes magnitudes in the ontat sufae between the two bodies: etain onstitutive elations deived fom the theoy of elastiity onnet them. If the deviations fom pue olling gow, the bodies touh along a ontat aea, whih pemit onditions intemediate between pue adhesion and pue slip. If the deviations ae no longe small, the full slipping of the wheel ommenes. In the ontat aea, Coulomb s law onnets total slip and the tangential foes. The tangential foes have a limiting value, whih is estited by the fition oeffiient of the wheel with the ail sufae. The wheel meets the ail at infinitely lage numbe points of an ellipse. At eah point, the Hetz theoy allows the detemination of the nomal pessue in the ontat. As the wheel olls, the defomations inease in the font pat of the wheel and diminish at the ea potion. Hene, the elementay eations in the font aea of ontat ae lage than that in the ea and the line of ation of the esultant is displaed with espet to the ente of the ellipse. This esultant builds up moment ating on the wheel. We have to add the eation moment of the onstaint applied to the wheel, whih is elative to the spin effet. The onnetion between the omponents of defomation and the tangential foes is alled the eepfoe law, that is, it has the fom: the longitudinal eep foe: T x j = - C v j. x (1) the lateal eep foe : y j j ( ) T = - C v. u - C φ () the moment: j 3 j 3

3 OL. 6, NO., FEBUAY ISSN Asian eseah Publishing Netwok (APN). All ights eseved. ( ) j M = C v. u - C φ (3) j 3 j j 33 In whih: C, C and C 3, C 33 ae the eep and spin Kalke oeffiients. They ae funtions of the elasti popeties (Young modulus E=.1 x1 N.m - ; Poisson s atio σ =.5 fo both the wheels and the ail) and of the atio of the axes of the ontat ellipse. These oeffiients vay as (wheel load) /3 ν epesents the eepage o eep atio at the point of j ontat point I j, φ j means the spin at the of ontat point I j, x and ( j ) u vetos defining the ontat plane, x the x-axis pointing along the dietion of the steady motion, u( j ) is lying on the ontat plane it is pependiula to the dietion of motion. We will use equations (1), () and (3) and we will onside the ail and the wheels absolutely igid. Theefoe, the numeial value of the wheel adius is assumed onstant. Ceep foes an be lage. Thei magnitude is satuated at the limiting fition foe. They ae able to exet a stong guiding influene duing uve tavesing. Ceepage o eep atio. It is the atio between the sliding veloity g of the ail to the wheel at the point of ontat I j and the module of the olling veloity. is oughly the tanslation speed of the wheel as a whole. j ( ail wheel I j ) O ( I j ) g /, v = (4) Spin The spin is onneted with the oniity of the wheel. Aoding the definition given by Kalke, the spin is equal to the omponent ω N of ate of otation on the nomal axis to the ontat plane divided by the module of an aveage speed of the two bodies in ontat. The ail being fixed, the latte is equivalent to the advane speed of the vehile (Figue-). ωn j φ j = (5) 5. EQUATIONS OF MOEMENT It will be noted that the wheel set displaements ae speified with espet to the pue olling line. We show in [1, 3] that the unestained wheel sets adopt adial positions in the ak along the pue line olling in absene of ant defiieny. They ae displaed lateally outwads fo a pue olling and fo a powe dissipated in the tosional dampe equal to zeo. The lateal displaement of the pue olling line fom the tak ente-line may be witten as: y e = (6) Whee e e +' e = -' e - y The value of is positive meaning that the y- axis is dieted outwads the uve. The elative oodinates fo the leading bogie may desibe the onfiguation of the vehile as follows: α 1 Yaw displaement of bogie fom lie joining wheel sets entes, y y 1 α α 1 Lateal displaement of the leading wheel set fom pue olling line, Lateal displaement of the tailing wheel set fom pue olling line, Yaw displaement of leading wheel set fom adial lines, (7) Yaw displaement of tailing wheel set fom adial lines. A simila set of o-odinates applies fo the tailing bogie. We obtain the system given in the appendix in whih: ϑ d = g ϑ (8) - 33

4 OL. 6, NO., FEBUAY ISSN Asian eseah Publishing Netwok (APN). All ights eseved. M M W = + + m+ mˆ g Wheel sets load 4 Kinematis elations bind the oll angle y ki ψ ki to the swing. The motion is studied fom geometi standpoint in [6]: ψ = with ki Γyki Γ= (9a, b) e- In addition, the vetial position of the wheel set inetia ente is given by the geomety study: z ki= f( y ki,αki) o with e yki αki ki -ε χ = ; ε ( ) e - ξ z = ξ (1a, b) = e (a, b) 1 e + = ; Γ ' e e = (1a, b) It is shown in [1, ] that the equations fo the leading and tailing bogies and vehile body may be deoupled. The vehile body lies appoximately tangential to the tak at its mid-length. The motion an be desibed in tems of only nine equations, whih ae appliable fo both leading and tailing bogies. The elative lateal displaements of the vehile body and the bogie fame ae given in [1]. The emaining displaements ae yielded by nine equations given in the appendix. 5.1 Foes ating in the vehile The majo lateal foes and ouples ating on the vaious mass elements of the vehile given in [1] ae: - the entifugal foes, - the gavitational foes due to tak ant, - the suspensions foes: -the seonday yaw suspension ouple C in equation (A.1) is: * ( ) A * * C=-Kxd -α 1+ y-y 1 / a (13) -the pimay suspension exets the following yaw ouple C on the leading wheel set; given in equation (A.4): * ( ) a * * * C=-Kxd +α1 -α + y-y 1 / a (14) In addition to these foes, the patiula longitudinal and lateal eep foes of the leading wheel set an be alulated fom the following elations (15, and 18). With appopiate indies, they ae appliable to the tailing wheel set. They ae the same than fo a staight line exept the tems elating the uvatue effet [3]. ϕ 1 χy * ν1=- x+ - α ( ) u 1 ; (15) ϕ χy * ν = - x+ - α u( ) ; (16) ϕ 1 e e ee 1 1= y y y φ + ; (17) ϕ = e e e e 1 y y y φ (18) Whee ( ) is the usual time deivative y ki lateal displaement fom tak ente line In uvilinea motion, the inne wheels with espet to the ente of tun ae unloaded unde the gyosopi and entifugal foes upon the wheel set. The oute wheels bea geate load. The hange T Z an be alulated by the fomula (19). 1 m ρ T y Z= Wϑd + (19) e Due to a small hange as ompaed with the weight load and fo simpliity, the numeial values of the Kalke oeffiients ae assumed equal fo the ight and left hand wheels of the wheel set. This diffeene in wheel loads auses an inease in longitudinal eep oeffiient at the lighte-loaded wheel and a deease at the heavieloaded wheel until the assoiated eep foes ae equal in magnitude but opposite in sign. Let us wite the equations of foe moments ating upon eah wheel of the leading wheel set. Fo simpliity, let us disegad the spin effet. Theefoe, as given in equations (A.6) and (A.7) we have: C ϕ = C e 1 ϕ ϕ ϕ + y 1 C ϕ = C ϕ1 ϕ -- e y ϕ () (1) The longitudinal foes C ϕ 1 of the oute wheel and C ϕ of the inne wheel ae numeially equal but ae opposite in sign. They ae able to exet a stong guiding influene duing uve tavesing. Fom these equations, we may dedue that ϕ an appea only with yki. The ability to tun is a esult of lateal slip fom pue olling line. kij 34

5 OL. 6, NO., FEBUAY ISSN Asian eseah Publishing Netwok (APN). All ights eseved. We an show fom the equations and 1 that ϕ is positive fo y 1 >. This ase always happens with the leading wheel sets. Similaly, we an show fom equations A.8 and A.9 that ϕ is negative fo y 1< with ϕ < y 1. This ase may happen with the taining wheel set fo e. and a small adius Tables (4, 5). That is, the ouple developed by the longitudinal eep foes is then opposite to the leading wheel set one. e popotion of the esultant eation foe is then available to equilibate unbalaned entifugal foes. 6. ESULTS AND DISCUSSIONS The tabulated data (Table-1) ae taken fom the gaph of dependene of the itial speed on the oeffiient Cϕ fo thee values of e (Figue-3) [3,7]. Table-1. Admissible value of Cϕ with the speed. Cuved taks with a high adius Fo these uves, we an keep the vehile unning at high speed. Thee is no need fo suh uves fo high longitudinal eep foes to stain the yaw soft suspension spings though small angles of ±a / C. See the equations (A.4) and (A.5) in the appendix. Cuved taks with a small adius Thee is a need fo geate longitudinal foes so to stain the yaw soft suspensions spings though athe highe angles ±a / C. See equations (A.4) and (A.5). Centifugal foes The ation of entifugal foe may ause the wheels to slip in a tansvese dietion. The unbalaned entifugal foes an be eated by lateal eep foes emaks The linea dependene of the equations (1, and 3) is valid only if the wheel slip and flange ontats ae avoided. The bonds of the linea theoy will be taken to slip ondition [; 3]. Fo a wheel affeted by the longitudinal foe T x and the tansvese foe T y, the esultant eation appeas in the ontat aea. = (T X +T Y ) 1/ () T x and T y an appea only if the wheel is ated upon by the thid omponent: the nomal eation of the ail to the load. The following ondition should be obseved fo a wheel to move without hoizontal and tansvese slip: µt Z (3) Whee µ is the oeffiient of fition between wheel and ail ;T Z is the wheel load. If this inequality is violated, the vehile skids and looses its stee ability. In onfomity with the above fomula, the following ondition is to be obseved fo a wheel to oll without slipping: T Y (µ T Z -T X ) 1/ (4) Thus, the geate the foe µ T Z and the smalle the foe T X, the highe will be the tansvese foe that an be applied to wheel without ausing it to slip. Highe Figue-3. Dependene of the itial speed on Cφ fo 3 values of e and a patiula value of Kx, Ky, Cx and Cy. Aoding to this Figue, we must enslave the damping effet between wheels at to desied speed of the vehile so to exeed the speed patied to day. The key is to edue the damping oeffiient as speed ineases. This edution must be highest so to offset the negative effet in pefomane of a stong e. As egad, dynami behaviou the wheels povided with adjustable tosional oeffiient allows geatest itial speeds with a soft pimay suspension. This is an advantage ompaed to the wheels made integal with a ommon axle o to the wheels made independent on a ommon axle. The esults given in Tables, 3, 4 and 5 with thee values of e and thee adius pompt the onlusions that the longitudinal eep foes aising fom the displaements yij align the wheel sets oetly almost in a adial position by staining the soft yaw suspension and so enable the neessay lateal eep foe to be geneated. All these esults hold with no wheel set sipping fo the ase µ =.7. The ant angle tak is set equal to 6. It is lesse than the itial ant angle fo toppling onditions. 35

6 OL. 6, NO., FEBUAY ISSN Asian eseah Publishing Netwok (APN). All ights eseved. Table-. esults fo = 1 km and = 1 m.s -1. e =.1 e =. e =.3 α ad α ad y 1 m α ad α ad g 1 ad.s g ad.s C 1 N.m C N.m C ϕ = 45 (N.m.s.ad -1 ) =. ϑ d Table-3. esults fo = km and = 7 m.s -1. e =.1 e =. e =.3 α ad y m y 1 m α ad α ad g 1 ad.s g ad.s C 1 N.m C N.m C ϕ =3 1 4 (N.m.s.ad -1 ) ϑ =.14 Table-4. esults fo =1km and = 5 m.s -1. e =.1 e =. e =.3 α ad y m y 1 m α ad α ad g 1 ad.s g ad.s C 1 N.m C N.m C ϕ = (N.m.s.ad -1 ) d ϑ d =.15 Table-5. esults fo = 1km and = 35 m.s -1. e =.1 e =. e =.3 α ad y m y 1 m α ad α ad g 1 ad.s g ad.s C 1 N.m C N.m C ϕ = (N.m.s.ad -1 ) ϑ d =. Whee C = g 1 C ϕ 1 (5) g 1 = 1 e ϕ ϕ + y (6) and C = g C ϕ (7) g = 1 e ϕ ϕ + y 1 (8) 6.1 Cant defiieny The alulations show that it is possible to tavel faste. The estitions will appea fist fom the equiements of omfot befoe those of safety taffi. An idea of the hange in ϑ d is given in Tables 4 and 5. A highe ant defiieny auses the bogie to yaw moe and inwads and to podue ountebalaning lateal eep foes. 6. Wheel sets displaements The leading wheel set is always displaed lateally outwads but the tailing wheel set may be displaed inwads o outwads (Tables 4 and 5). The tailing wheel set outwad displaement appeas only with the smalle oniity in uved tat with a small adius and with all the oniity values fo the highe tak adius (Table-). The value of these deviations is too small even with a highe speed. Theefoe, thee is no need fo a lage leaane between the wheel set and the tak. 6.3 Effetive oniity The lateal displaement and the yaw of both the bogie fame and the wheel set ae sensitive to vaiation of oniity. Thee numeial values ae less with a highe 36

7 OL. 6, NO., FEBUAY ISSN Asian eseah Publishing Netwok (APN). All ights eseved. oniity. The eason is that the absolute value of the eep oeffiients is highe. It is known that the geat oniity pomotes good uving pefomane [1]. 6.4 Yaw displaement The Tables to 5 indiate that the wheel sets depat moe and moe fom adial alignment as both and the speed ae ineased. The lage yaw angles oespond to the smalle effetive oniity. Fom Tables 4 and 5 the wheel sets yaw in phase at the speed of 5m.s -1 and yaw in anti-phase at 35m.s -1. It follows fom the ompaison of Tables (4 and 5) with the edution of the speed the leading wheel set yaws outwads. That is, the leading wheel set stee slightly leftwads to tun the ail vehile ightwads. In addition, its lateal foe (not given) is highe. Theefoe, the leading wheel set is the fist to slip duing uve tavesing with adius smalle than 1km. 6.5 Dampe The best design is the one that keeps the oveall size and mass of the tosional dampe to a minimum. It has to be designed so to ensue good opeating ability, bette eliability and a longe sevie life. The lok is then of pime impotane. The ouple developed by the dampe is equal to 8471 N.m in a adius tat of 1km; it oesponds to g 1 = ad.s -1 and e =.1 (Table-4). It is equal to 64 N.m fo the adius of 1 km with g 1 =.587 ad.s -1 and e =.1 (Table-). Compaing the ouple developed at 1 km adius uved tak to the ouple developed at 1 km adius uved tak, it an be lealy seen that the ouple at 1km is oughly seven times lage. Besides, the itial speed is elatively onstant fo the high value of Cϕ (Figue-3) and eah wheel set behaves as a single piee in staight line. An eletomagneti powe luth may be used. 6.6 Damping oeffiient With the inease of Cϕ the lateal displaement of the leading wheel set is moe and moe edued. The value of Cϕ has to lie within a magin so the loss of stability an be peseved and safe motion ensued. 6.7 emaks The minimum uve adius, whih an be negotiated without slip at any ant defiieny, is 1 km fo the assumed oeffiient of fition µ =.7 Thee is a neessity to lok the liaison between the wheels so to ut down on the tosional dampe failues. The vehile an be made to tavel as faste as possible in the tak with a small uvatue and to slow down in the geatest with loked wheel sets. The vehile speed aveage will be still at a high level. The speed inease imposes high demands on the baking systems. Fo the so-alled "TG", the slip of the wheels annot be avoided fo a uved tak whose adius is below 8 km. When the ail vehile moves, the situation in the line inessantly hanges whih alls fo ineased attention on the pat of the dive and a onstant eadiness to hange Cϕ value. 7. CONCLUSIONS We gave an oveview of the theoetial uve behaviou of the vehile with spings stiffness Kx = 5. MN.m -1 and Ky = 7.85 MN.m -1 at the pimay suspension. This alulation puts the poblem of seletion and ode of magnitudes haateistis of the tosional dampe. Moeove, it shows that the tajetoy of a small adius is possible. Beause the impoved egistation of the wheel sets in the tak, the isk of deailment and the aggessiveness in view of the tak ae smalle. It an theefoe tavel faste though a biggest distane than the "TG" fo whih the aveage ouse is km. It may supplant all the othe ail vehiles. EFEENCES [1] D. Boook Steady-state motion of ailway vehiles on uved tat. Jounal Mehanial Engineeing Siene. ol. 6. []. Joly Stabilité tansvesale et onfot vibatoie en dynamique feoviaie. Thèse de dotoat d Etat, Univesité de Pais I. [3] M. Messoui Pefomanes d un véhiule feoviaie muni de oues à pseudo glissement ontôlé, stabilité tansvesale en alignement et insiption en oube, Thèse, ENSAM, Pais. [4] J.J. Kalke On the olling ontat of two elasti bodies in the pesene of dy fition. Dotoate Thesis. Delft Univesity. [5] J.J. Kalke Suvey of Wheel-ail olling Contat Theoy. ehile System Dynamis. ol. 5, pp [6] A.D. Pate Péis de la théoie de l inteation ente la voie et le véhiule du hemin de fe. Doument OE UTCHT. [7] M. Messoui. 9. Lateal stability of ail vehiles-a Compaative study. APN Jounal of Engineeing and Applied Sienes. 4(3):

8 OL. 6, NO., FEBUAY ISSN Asian eseah Publishing Netwok (APN). All ights eseved. Appendix α 1 Equation ( ) α d d d d ( a a ) ( a a ) x + x + y 1+ x + x x + x 1 x A x α1 Kd x Kd Kd Ka K K y K K y Kdα Kd = y Equation (A.1) (A.) K a + W C ( ) y C C ( ) = W W y C e χ + 3 y+ C 3 χ χ χ y α1 ξ 3e χ α 3 ϕ1 ϕ ϑd ξ y 1 Equation (A.3) K a W C ( ) y C C ( ) W W y χ χ χ y α1 + ξ 3e 1 χ α1 3 ϕ ϕ1 = ϑd ξ + C e y + C χ 3 3 α Equation (A.4) d 1 d x α 1 x 33 a e x 13 x ε a ) α e e 1 1 a 33 ϕ1 33 ϕ 33e 3 x Kd + K C ( ) y + K y + (C + Kd W ( ) ( ) C C + C C = C y + C + K d α 1 Equation (A.5) d d 1 x α 1 x a x 33 a e 13 x ε) α 1 e e 1 1 a 33 ϕ 33 ϕ1 33e 3 x Kd K y + K C ( ) y + (C + Kd W ( ) ( ) C C + C C = C y + C K d ϕ 1 Equation (A.6) ( ) C + y C + e 1 e C ϕ C 33e y C3α C ϕ C C33 ϕ 1 C ϕϕ 1 e 33 e 33 = ϕ Equation (A.7) C y C e 1 e C ϕ C 33e ( ) y C3α C 1 33 ϕϕ Cϕ C C ϕ 1 e 33 e + 33 = ϕ Equation (A.8) ( ) C + y C + e 1 e C ϕ C 33e y 1 C3α1 C 33 1 ϕ C C ϕ C ϕϕ 1 e 33 e 33 = 38

9 OL. 6, NO., FEBUAY ISSN Asian eseah Publishing Netwok (APN). All ights eseved. 1 ϕ Equation (A.9) C y C e 1 e C ϕ C 33e ( ) y 1 C3α1 C 33 1 ϕϕ Cϕ C C ϕ 1 e 33 e + 33 = Notations and paametes fo example vehile A distane between bogies entes m a longitudinal distane of the spings and dampes m a wheelbase of bogie 3m b distane between the devie leves of anti-yaw body/bogie 1.3m C 1 modulus (magnitude) of the moment of fition in the leading N.m wheel set dampe C modulus (magnitude) of the moment of fition in the leading wheel set dampe N.m 4Cx longitudinal damping oeffiient of the pimay suspension vaiable 4Cy tansvesal damping oeffiient of the pimay suspension vaiable 4Cx longitudinal damping oeffiient of the seonday suspension N.s.m -1 4Cy lateal damping oeffiient of the seonday suspension 4x35 N.s.m -1 Cz vetial damping oeffiient of the seonday suspension 15 N.s.m -1 Cϕ oeffiient of ésistane o damping between wheels of a wheel set vaiable d tansvese distane of the spings and dampes fo the pimay suspension m d tansvese distane of the spings and dampes fo the seonday m suspension d tansvese distane between inetia entes of wheels beaings box (~ d) e semi-tak of wheel set 1.5 m g gavitational aeleation 9.81m.s - g 1 wheels elative veloity of the leading wheel set ad.s -1 g wheels elative veloity of the tailing wheel set ad.s -1 h position of seonday plane of suspension elative to G.88 m h 1 half height of the seonday suspension.1 m tansvese dampe position in the seonday suspension elative to h seonday plan of suspension m h position of pimay plane of suspension elative to G k.1 m h 1 half height of the pimay suspension m H vetial distane between spings points attahments of seonday suspension elative to the pimay plane of suspension,467 m H 1 vetial distane between tansvese dampes of seonday suspension elative to the pimay plane of suspension.67 m H p distane between the lowe opeative point in bogie fame of the taining devie with the pimay tansvese plane of suspension 4Kx longitudinal stiffness of the pimay suspension 5. MN.m -1 4Ky tansvesal stiffness of the pimay suspension 7.85 MN.m -1 4Kz vetial stiffness of the pimay suspension 4 x. 975 MN.m -1 39

10 OL. 6, NO., FEBUAY ISSN Asian eseah Publishing Netwok (APN). All ights eseved. 4 Kx longitudinal stiffness of the seonday suspension 4 x.175 MN.m -1 4Ky tansvesal stiffness of the seonday suspension 4 x.173 MN.m -1 4Kz vetial stiffness of the seonday suspension 4 x.53 MN.m -1 K o leves stiffness of anti -yaw devie between body and bogie 3.6 MN.m -1 l wheel-axle set diamete.165m m mass of a wheel set 15 kg m ) mass of a wheel set box o jounal beaing 5 kg M mass of a body 43 kg M mass of a bogie 3 kg uvatue adius of a ail pofile,3 m uvatue adius of wheel pofiles funtion of e o mean adius of wheel.45 m s o longitudinal eentiity of G.1m T Z the wheel load vehile speed m.s -1 itial speed m.s -1 e angle of the tangential plane to the ommon points of ontat wheel/ail with the hoizontal plane when the wheel set is ented in the midline of the ail effetive oniity.5 ϑ ant angle of tak 6 angle of ant defiieny µ oeffiient of fition.7 ϑ d ρ y adius of inetia of the wheel set aound Gki yki axis ϕ inement in the angle of otation aound the axis of the wheel S kij elatively to an imaginay wheel oiniding with the latte and olling without slipping Subsipts k (k = 1 fo the leading tuk and k = fo the tailing tuk). i (i = 1 fo the leading wheel set of a tuk and i = fo the tailing wheel set of a tuk). j (j = 1 fo the left hand side and j = fo the ight hand side). 4