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1、SAE TECHNICAL PAPER SERIES The Role of Steer and Sideslip in the Mechanism of Slip Angle Walter Bergman (Retired) Ford Motor Co. Dirk Pelargus Ford-Werke Aktiengesellschafi The Engineering Society 61rFor Advancing MOW, and Sea Air and Space, International Congress however, in the vehicle it is gener
2、ally produced by steer andlor the sideslip. Therefore, in most cases lhe slip angle in the vehicle is composed of two components: the steer angle and the sideslip angle. Each of these components of the slip angle has a different polarity and a different function. The steer component of the slip angl
3、e is a control input producing the lateral force, which changes the direction of vehicles motion. This force always acts in the direction of steer: (he positive steer component of the slip angle produces a positive lateral force. However. the sideslip component of the slip angle is the vehicles resp
4、onse to control or disturbance inputs. The sideslip angle tends to oppose the change in the direction of the vehicles motion necessary lo maintain the vehicles dircctiolal stability. This force acts in the opposite direction of lateral velocity, which is used as a paranleter indicating the change in
5、 the direction of the vehicles motion. Therefore, lhe positive sideslip angle produces a negative lateral force. In the existing Lire terminologies the slip angle is defmed as an angle from the wheel plane ( the line of intersection of the wheel plane with the road plane ) to the direction of the wh
6、eel travel. According to this definition the positive slip angle produces a negative lateral force. This implies that the lateral force generated by lhe slip angle tends to oppose the change in the direction of the vehicles motion. This shou7s that the polarity of the slip angle has been improperly
7、adopted in this definition of the slip angle. However. the slip angle defined in such a manner has been used as the control input in vehicle dynamics studies. Applying such sign convention for the slip angle in vehicle dynamics will reverse the direction of steer, i e steer to the right will turn th
8、e vehicle to the left and vice versa To the best of the authorss knowledge. the users of tire data employing such sign convention have provisions in their vehicle dynamics computer programs to correct for the slip angle sign. However, the rationale for such a correction has never been formally discl
9、osed in the SAE Technicals Reports. It should be noled that the problem with the sign convention used in the existing tire terminologies is not limited to the slip angle alone, but also extends lo lhe inclination angle. The problem nith the inclination angle resulted from an improper choice of a coo
10、rdinate system, in which lhe inclination angle was defined. In the SAE J670 tire axis system Ule positive direction of the vertical direction ( Z axis ) is downward. Therefore, the inclination angle was improperly defined as an angle from the negative branch of the Z axis to the wlleel Author:Gillig
11、an-SID:1178-GUID:19091828-141.213.232.87 plane. As a result of this error the positive inclination angle produces a positive lateral force, which is incorrect. SLIP ANGLE AS A hIECHANISh.1 PRODUCING LATERAL FORCE The slip angle is an angle determining the orientation of the rolling wheel with respec
12、t to its direction of uavel. This angle produces the lateral deformation ( perpendicular to the wheel plane ) of the tire with respect lo the wheel rinx which results in lateral force. ?his process is illustrated in Figure 1, which shorn the lateral deformation of the tires equatorial line uith resp
13、ect to the contact line. The equatorial line and k e contact line are used as h e reference lines in the description of the mechanism of the formation of slip angle and lateral force. resulting from the slip angle. The equatorial line is a line on Ihe tires uead surface around the circumference of t
14、he tire. On the unloaded tire, lhis line is determined as the intersection of the wheel plane wilh the uead surface. The contact line is the intersection of the wheel plane with the road plane. In the suaight moving wheel at zero inclination angle and zero lateral force the equatorial line coincides
15、 with the contact line and also with the direction of wheel uavel. Steering of the wheel results in a lateral displacement of the equatorial line from the contact line. It is generally assumed that the slip angle is produced by rotation of the wheel about the vertical axis through the contact center
16、 ( center of the tire contact as= ). The contact center is defined as the point of intersection of the contact line with the normal projection of the spin axis ( the a i s of wheel rotation ) onto the road plane. It is assumed, that the slip angle. as shoun in Figure 1, resulu exclucively from steer
17、. Because of this assumption the slip angle is equal to the steer angle and is defined as an angle from the direction of wheel travel to the contact line. The slip angle is generally used as an input producing lateral force. The slip angle results Gom a lateral displacement of the equatorial line fr
18、om the contact line in the direction of steer. i.e. steer to the right diplaces the equatorial line to the right and vice versa. The lateral deformation of the tire tread and carcass results in a lateral force acting in the direction of this deformation. Therefore, lateral force acts in the directio
19、n of steer. i.e. steer to the right produces lateral force lo the right and vice versa. Because of fiction between the tire mead and lhe road surface each point on the equatorial line tends to follow the diection of wheel uavel. To do this, it has to deviate from the contact line. The magnitude of t
20、his deviation linearly inmeass ac lhis point moves rearward from point of entry into the contact uith the road surface. until the lateral force at this point reaches Ule value of friction linlits. Within these limits. the lateral force at each point on the equatorial line linsasly increases with the
21、 increase of lateral deflection. Xner the lateral force at the tire element reaches the friction limifi, the tire element swts to slide back toward the wheel rim. and the force at this element will non-linearly decrezse a s this element continues to move reanvard. This resulu in an unequal disuibuti
22、on of the lateral force and the lateral deformtion of the equatorial line along the contact line. Because of such unequal disuibution. the point of application of the resultant lateral force is usually located behind the contact center. This resuIU in a moment about the vertical 2xis lhrough the COP
23、CI center. This moment tends to align the wheel rim nith the direction of wa.el. Therefore. this moment is usually called the aligning moment ( aligning torque ). Figure 1: Deformed Equatorial Line of a Tie i n the Rightflrfl Tums Individual uead elements constitute an integral past of the uead and
24、the carcass. u W h can be consitlered as an elastic band around the wheel rim. Therefore, deflection of these elements uihin the conlact area results in the deflection of this elastic band around the circumference of the wheel rim This deflection extends far beyond the liniits of the contact area. T
25、o produce this deflection, the tire has to wave1 a distance which is approximately equal to the tire circumference. This results in a distance lag between the slip angle input and the lateral lire force andlor aligning moment response. The relationship between the slip angle and the lateral force an
26、d aligning moment are greally influenced by wheel load, tire inflation pressure. read depth and the coemcient of frictioli between the tire and lhe road surface. Measurenents are usually performed at different wheel loads on high Giction road surfaces. The difference between the values of coefficien
27、t of friction on different road surfaces are usually not accounted (as far as hard dr) road surfaces are concerned). primarily because of difficulties associated with Giction measuremen its posi.tive direction is f o m d . The Y axis is positiwe to the righL Because the X anis is fixed in the wheel
28、plane, the lvheel plane cannot rotate with respect to this axis. Therefore. the slip angle is determined in this system as an angle resulting from a rotation of ffie road plane with respect to the nheel plane instead of a rotalion of the wheel plane nith respect to the road plane used in vehicle sle
29、er and inclination angle nwsuremcnls. Such inversion of rotation results in an inversion of the slip angle sign. ConsequenUy, according to this s i p convention. the positive slip angle produces nagative lateral force. Therefore. lateral force acts in the opposile direction of steer. i.e. steer to t
30、he right tuns the vehicle to the lea and ice versa. Therefore, tire data specified in this format must be corrected before they can be used in vehicle dynamic simulation. This corrections are not limited lo the slip angle alone, but also should he applied to the inclination angle. In the SAE J670 ti
31、re axis system a positive inclination angle results from clocLuise rotation of he wheel plane about the X wis. The sign of inclination angle does not change, whether it is nimured from the positive or negative branch of the Z axis; honever. the sign of the lateral force does change (see Fillre 9 ) .
32、 The hleral force always acls in the direction of the lateral displacement of the top of Ule wheel uith respect to the Z axis. Fgure 9: Lated Force produced by positive Inclination Angle dctcrmincd in the XYZ Tire Axis System from the a) negative and b ! positive Branch of the Z Axis In this axis sy
33、stem. the inclination angle is determined from the negative branch of the Z axis. which is mathematically incorrect. Consequenuy, the positive inclination angle ill displ2ce the top of the wheel in the positive direction of the Y axis. This sill result in a positive lateral force (see Fi y r e 9a).
34、To correct this error, the inclination angle should be measured lrom the positive branch of the Z axis as it is shown in Figure 9b. In order to do this, the wheel should be located underneath the road surface as it is illusualed in Figure 9bThat shows that because of the peculiar orientation of Uie
35、SAE J670 tire axis systenS it is practically impossible to correcUy deternine the larrral force produced by the inclination angle. The problem asociated with the inclination angle have been resol%.ed in the IS0 8855 wheel axis system (XW YW Zw ) (see Fi-me 10). This system was evolved from Ule SAE 1
36、670 tire axis s)xtern but the direction of Ule Zn. axis has been changed from positive downward to positive upuwd, and the positive direction of the YW axis has been changed from 6 posilive to the right to positive to thc left in order to comply Author:Gilligan-SID:1178-GUID:19091828-141.213.232.87
37、with generally accaptable conventions used in the textbooks. With these modifications. the error associated ulth the inclination angle has been corrected. but the error associated with the slip angle still remained. N0lE - D l i s d n a n io thir Figurcrchrrrmiolly idintrr Cawiguralior of thcTwezd i
38、rr m o n w i i h r c r p m t o i h ; r Spa*mand rhovld not k inluprclcd u ihc Lpatorial Lim Figure 10: Wheel Axis Syslem (IS0 8855) By using the sign convention established in the IS0 8855 wheel axis system, a positive slip angle produces a negative lateral force, which will result in the right turn
39、. A positive inclination angle also produces a negative lateral force, which acts toward the inside of the turn. Therefore, lateral forces produced by a positive slip angle and a positive inclination angle act in the same direction (toward inside the turn). Such behaviour is contrary to the behaviou
40、r of a vehicle in cornering maneuvers. In the cornering maneuver, the wheels are usually inclined toward the outside of the turn (because of vehicle roll). Therefore. the lateral force produced by the inclination angle is directed toward the outside of the turn, thus opposing the lateral force due t
41、o the slip angle, which acts toward the inside of the turn. This incompatibility between the IS0 8855 wheel axis system and vehicle dynamics resulted from an improper definition of the slip angle, which was adopted from the SAE J670 tire axis system. In the SAE J670 tire axis system the positive sli
42、p angle produces a negative lateral force. Such force will turn the vehicle in the opposite direction of steer, i.e. steer to the right turns the vehicle to the lefl and vice versa. It is quite obvious that such behaviour is contrary to the actual behaviour of the vehicle in a cornering maneuver. In
43、 the SAE J670 tire a s system the lateral force due to the inclination angle acts in the opposite direction lo Ule lateral force due to slip angle. However, such behaviow resulted from an error caused by an improper measurement of the inclination angle from the negative branch of the Z axis. By corr
44、ecting this error by measuring the inclination angle from the positive branch of the Z axis the positive inclination angle will produce a negative lateral force. Therefore, the lateral force due to inclination angle and slip angle will act in the same direction. Such behaviour is contrary to the beh
45、aviour of a vehicle in a cornering maneuver, where the lateral forces due to slip angle and inclination angle act in the opposite direction to each other. The lateral force due to the slip angle acts toward the inside of the turn; however, the lateral force due to the inclination angle acts toward t
46、he outside of the turn. Such incompatibility between the lateral force produced by the slip angle and Lbe lateral force produced by the inclination angle resulted from an improper determination of the slip angle by rotation of the road plane with respect to the eel plane instead of rotation of the w
47、heel plane with respect to the road plane. Therfore, by using such an axis system it is not possible to deiie the slip angle in the manner compatible with vehicle dynamics. Since the X (XW) axis is fixed in the wheel plane, a slip, angle cannot be produced in a correct manner by rotation of the u,he
48、el plane with respect to the road plane. To resolve the problem it has been proposed to introduce Ihe trajectory axis system XT YT ZT (see Figure 11) as a supplement to the IS0 8855 wheel axis system This system is the same as the IS0 8855 wheel axis system except the XT axis, which coincides with t
49、he tangent to the trajectory of the center of tire contact and the YT axis is perpendicular to the direction of the wheel travel. This axis system is used exclusively as a reference system for angular orientation of the wheel plane, produced by its rotation about the Zw (G) axis. VOTE . Ultpc dram in rhlt Cigwcschcmurcally #dint= (:odtgmtiun Ithc1:rs and irr h u o n wlrh = p a torhc Aitr Sjnrrnard 8.1ld n a bc apracd as ihc consequenlly, the dFy I d a becomes a negative quantity and therewith the cornering stiffness C, a positive quantity.
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