A SIMPLE STATIC ANALYSIS OF MOVING ROAD VEHICLE UNDER CROSSWIND

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1 A SIMPLE STATIC ANALYSIS OF MOVING OAD VEHICLE UNDE COSSWIND Milan Batist Marko Perkoič Uniersity of Ljubljan Faculty of Maritime studies and Transport Pot pomorščako 4, 630 Portorož, Sloeni EU Dimitrij Najdoski X3DATA, Ljubljana ABSTACT In the article a static model of ehicle for the determination of critical ind speed for oerturn, sideslip and rotation is considered. The basic equations are setup on the assumptions of uniform straight line motion of the ehicle. Explicit formulas for critical ind speed for all three possible ind induced accidents are deried. Numerical examples compares present static model ith simple dynamic model. Keyords: road ehicle, crossind 1 INTODUCTION According to Baker ([1]) crossind accidents may be classified by three types (Figure 1): rolloer accidents sideslip accidents rotating accidents Figure 1: Vehicle accidents in crossind according to Baker In the first type of accident, a ehicle is blon oer; in the second type a ehicle is blon a significance distance sideays; and in the third kind a ehicle is rotating through a significant 1

2 angle around its ertical axis. The proposed criteria for detecting the risk of possible type of accident hen a ehicle enters a sudden crossind are: the contact force falls to zero ithin 0.5 s the lateral displacement of the ehicle exceeds 0.5 m ithin 0.5 s the absolute alue of ya angle exceeds (0. rad) ithin 0.5 s In this paper the Baker dynamic criteria for particular accident detection hich are a bit artificial ill be replaced ith static criteria hich are the folloing: the contact force falls to zero all the heels reach the friction limit a ehicle heel reach the friction limit For determine the critical ind speed that may cause a particular type of crossind accident a simplest possible ehicle model ill be used; namely, a ehicle ill be considered as a single rigid body ith a gien mass and dimensions executing straight line motion. BASIC EQUATIONS Equilibrium equations. Consider the ehicle subject to the uniform crossind that executes a steady straight line motion. To remain on a straight path under the aerodynamic loads produced by ind, the friction forces should be generated in the contact of ehicle heels and road (see Figure ). Figure : Dimensions, reaction forces and ind induced resultant forces and moments on the ehicle. Note that the drag force, the rolling moment and the pitching moment are assumed to act in negatie directions ith respect to ehicle HS ehicle coordinate system. If a ehicle is treated as a rigid body then the equilibrium conditions of forces and moment ith respect to ehicle center of graity yields six equilibrium equations. These are the equilibrium of forces F F F F F i T T i T T (1) D x1 x x3 x FS Fy 1 Fy Fy 3 Fy 4 0 ()

3 mg FL Fz 1 Fz Fz 3 Fz 4 0 (3) and the equilibrium of moments c M Fz Fz 1 Fz 4 Fz 3 hfy 1 Fy Fy 3 Fy 4 0 (4) M a F F b F F h F F F F (5) P z1 z z3 z4 x1 x x3 x4 0 MY afy1 Fy bfy 3 Fy 4 c Fx Fx 1 Fx 4 Fx 3 (6) c c i1 T1 T i T3 T4 0 here F D is drag force, F S is side force, F L is lift force, M is rolling moment, M P is pitching moment, M Y is yaing moment, T 1,T,T 3,T 4 are heels traction forces and i 1 and i takes alue 1 or 0, depends if axle is drien or not. For gien aerodynamics load the aboe equations constitute a set of six equations for sixteen unknons: tele reaction forces and four traction forces. The nine additional equations hae to be supplied by constraint and constitutie assumptions. x, y, z are coordinates of centre of ith heels here z i is Constraint equation. Let ( ) i i i displacement of heel center from its equilibrium position in ertical direction. Since ehicle is rigid all the coordinates of heels centre must all the time lay on the same plane hich is gien by sayax + By + Cz + D = 0. Substituting coordinates of heels centre x1 x a, x3 x4 b, y1 y3 c and y y4 c into the plane equation one obtain a homogeneous system of four linear equations for parameters of the equation. For nontriial solution the determinant of the system must be equal to zero. This yield condition z1 z z4 z3 0 (7) If e future assume that each ertical force is proportional to the displacement and all the heels suspension has same stiffness then constraint equation (7) leads to ertical forces constraint equation Fz 1 Fz Fz 4 Fz 3 0 (8) The unilateral contact beteen heels and road demands that F 0 j 1,,3,4 (9) zj If Fzj 0 then jth heel loose contact. For static consideration the reaction side force on a ehicle heel is restricted by ell knon Coulomb friction la Tj Fxj Fyj Fzj j 1,,3,4 (10) here is static friction coefficient. When inequality holds then a heel is stick ith the road, hen equality holds then heel heels sliding just begins. Note also that unilateral contact beteen heel and road implicate that if Fzj 0 then also Fyj 0. Constitutie equations. The rolling resistances are gien by ([]) F f F j 1,,3,4 (11) xj zj here rolling resistance coefficient f is assumed to be a constant. Also it is assumed that the traction forces hae a form purposed by Baker ([1]) 3

4 T qf j 1,,3,4 (1) j zj here q is unknon traction parameter. The purposed form is a bit artificial but takes into account that if heel lose contact then there is no traction force. Figure 3: Absolute (true) ind and relatie (apparent) ind Aerodynamic forces and moments. When a ehicle is moing ith elocity 0 and the true (ambient, absolute) ind is aligned to the direction of a ehicle moing at angle then the apparent (relatie) ind that acts on the ehicle has elocity a and is aligned to the direction of the ehicle moing by angle. The angles and are taken to be positie in the compass sense (Figure 3). From the figure one may see that ector equation 0 a holds. From this it follos that hen true ind speed and angle are gien then the apparent ind speed and its angle are a 0 cos sin (13) sin arctan (14) 0 cos It is seen from the aboe equations that the apparent ind speed ill be greatest hen true ind direction is perpendicular to the direction in hich the ehicle is traeling. No, the result of the interaction of a ehicle and air are normal pressure and shear stresses on the ehicle surface ([]), and these produce the aerodynamic forces and moments (Figure ). The total aerodynamic force and moment acting on a ehicle may be obtained by integrating the stresses oer the ehicle surface. This leads to a complex air flo around the ehicle so theoretical formulas are replaced by semi-empirical formulas of the form a a a FD CD A FS CS A FL CL A (15) a a a M CAh M P CPAh MY CYAh here C D, C S, C L, C, C P and C Y are aerodynamic load coefficients, A is the characteristic area of the ehicle, hich is usually taken as the projection of front ehicle area and h is characteristic length hich is usually taken as distance beteen road and ehicle center of mass. 3 SOLUTION The fifteen equations, namely six equilibrium equations (1)-(6), constraint equation (8) and eight constitutie equations (11) and (1), include seenteen unknons, namely tele reaction 4

5 forces, four traction forces and traction parameter q. The system is clearly indeterminate and if Coulomb conditions (10) for each heel are included then it become oerdeterminate. Consequently from the system one cannot determine all the unknons. Hoeer, by the inspection of the system, one may see that it is complete if only resultant side force for each ehicle axis is included as unknons. In this case the solution of the system is the folloing expressions for ertical reaction forces 1 bmg FL 1 hfs M 1 hfd M P Fz 1 a b c a b (16) 1 bmg FL 1 hfs M 1 hfd M P Fz a b c a b (17) 1 amg FL 1 hfs M 1 hfd M P Fz 3 a b c a b (18) 1 amg FL 1 hfs M 1 hfd M P Fz 4 a b c a b (19) the folloing expressions for the resultant lateral force on each ehicle axis bfs MY hfs M Fy 1 Fy q a b a b (0) afs MY hfs M Fy3 Fy4 q a b a b (1) here i1 i q q f () and the traction parameter is q a b F D f mg FL i b i a mg F i i hf M (3) 1 L 1 D P By using expressions for aerodynamic forces and moments (15) the traction parameter may be also ritten as A a a b fgm CD fcl q (4) Aa i1b ia mg i1 i hcp CD i1b iacl Note that ertical reactions are different from those gien by Baker ([1]). Namely present static ertical reactions (16) contain side force F S hile in dynamical case the heels side force are proportional to sideslip angle hich is assumed to be zero at oerturning condition. Obsere also that 1 Fz1 Fz 4 Fz Fz 3 mg FL (5) This means that antisymetric ehicles heels support half of the ehicle eight reduced by lift force. 5

6 The conditions for oerturning, rotation and sideslip ill no be treated separately. It hat follos e ill assume that ind blos from ehicle right so indard heels are 1 and 3, and leeard heels are and 4. This assumption implies that F, M ł 0. Oerturning. First indication of possible ehicle oerturn is that one of its heels loses contact ith the ground. If all alues of aerodynamics forces and moments are positie then, as it is seen from (16)-(19), the minimal ertical force is on heel 1. If drag force and pitch moments turns its sign then the minimal ertical force is on heel 3. By substituting expressions for aerodynamic loads (15) into (16) and (18) e find the critical apparent ind speed for one heel to lose contact S oerturn1 oerturn3 mg bc A h a bcs C hccd CP bccl mg ac A h a bcs C hccd CP accl (6) (7) This formula become Baker condition for oerturn if one set CS 0. The (6)-(7) gie the limit apparent ind speed for one heel lost contact. In literature ([3]) hoeer the condition for oerturn usually demands that resultant ertical force on heels on indard lose contact ith the road. From (16) and (18) this condition is 1 hfs M F F mg F 0 (8) z1 z3 L c By substituting expressions for aerodynamic loads (15) into (8) yield the ell knon critical apparent ind speed for oerturn oerturn mg c A hcs C ccl (9) Obsere that the limit speed for oerturn (9) does not depend on drag force and pitch and ya moment neither on dimension of ehicle heelbase. Also in any case a heel ill lose contact before the ehicle reach oerturn condition that is as expected. oerturn 1 oerturn otation. Since heels on each ehicle axis are assumed to be rigid connected the sliding of heels on axis ill be reached hen both heels satisfy equality (10). From (0)-(1) and (16)-(19) the condition for heels on axle to slip are therefore here bf M q hf M b mg F hf M S Y S 1 L D P af M q hf M a mg F hf M S Y S L D P i q f i q f 1 1 (30) (31) By substituting expressions for aerodynamic loads (15) yield critical apparent ind speed 6

7 slip1 slip mg 1b A bcs h CY qcs C 1 bcl hcd CP mg a A acs h CY qcs C acl hcd CP (3) (33) Note that these equations are alid only if all the ertical reaction forces are positie and q f. Sideslip. When all the ehicle s heels reaction side forces simultaneously reaches its maximal alues permitted by friction then the ehicle is just to beginning to slide. In this case F F F F F F F F (34) y1 y y3 y4 1 z1 z z3 z4 and this by (15) leads to critical apparent ind speed sideslip mg a 1b A a b C a C b hc C S 1 L 1 D P (35) Note than hen q f is small then 1 and the aboe equation reduce to sideslip mg A C C It is seen from this expression that in simplified ersion only side force and lift force contribute to critical apparent ind speed. Discussion. If e take most critical situation hen ind blos perpendicular to ehicle path then by using (13) all the deried formulas for critical apparent ind speed hae the form mg 0 f CD, CS, CL, C, CP, CY, b, c, h,, f (37) A When aerodynamic coefficients are knon one may from the formula calculate the limit ehicle speed for gien ind speed for particular type of accident. If for example one takes all the aerodynamic coefficients as constant then the safe driing is restricted to circle. This is hoeer unrealistic since the formula restrict ehicle speed een in the case hen there is no ind. In general, the aerodynamic coefficients depend on S L (36) and this angle depends, as it seen from (14), on ehicle speed, true ind speed and true ind direction. For gien true ind speed direction (37) represent a highly transcendent equation for calculation of critical ehicle speed. Hoeer, an relatiely simple explicit formulas for critical ind speed for sideslip and oerturn may be obtained if coefficient of lift force is taken to be constant and the side and roll coefficient are distributed sinusoidally C C sin C C sin C C (38) S S0 0 L L0 Note that the condition for oerturn and simplified sideslip has identical structure and may be ritten as here * a mg A C C sin S 0 L0 (39) 7

8 sideslip CS 0 sideslip CS 0 (40) c h oerturn CS0 C0 oerturn When ehicle is at rest then this formula gie the ultimate ind speed at hich ill oerturn or sideslip hich is mg A C C S0 L0 By using trigonometry identities sin tan 1 tan and expression (14) one may rerite (39)to * mg C * L0a CS 0sin a 0 A (4) This is quadratic equation for unknon critical apperent ind speed. No to possibilities hae to be considered. If the lift force is zero then the critical apparent ind speed is mg * a A CS 0sin (43) and hen it is not zero then S0 CS0sin C sin 8 CL0 * a (44) CL0 The aboe expressions may be soled for critical ehicle speed, hoeer the result are relatiely complex expressions. These are simplified in special most critical case hen ind blos perpendicular to ehicle traeling path. In this case the critical ehicle speed hen there is no lift force is mg * 0 ACS0 and hen one has also lift force the critical ehicle speed is 4 EXAMPLE 1 mg mg * 0 CS 0 CL0 CS 0 CS 0 8 CL0 4 CL0 (46) C A A L0 As the numerical example e ill calculate dependence of critical ehicle speed on true ind speed for to trucks. The data for them are gien by Baker ([1],[4]) and are shon in Table 1. Note that the coefficient of friction is calculated from the Baker s coefficient m.5 by assuming that sideslip saturation angle is The results of calculations are present as graphs on Figure 4. mg A (41) (45) 8

9 Table 1: Vehicle data Parameter Unit Value Case 1 Case m kg a m b m c m.0. h m A m f μ sin sin3 C D C S sin C L sin.sin C C P C Y sin sin 6.0sin Figure 4: Dependence of ehicle speed on ind speed. Baker s oerturn and sideslip correspond dynamical model 5 CONCLUSION The present static model clearly simplifies real situation of driing in a strong crossind since many influence factors are ignored. Hoeer, as as pointed by Lemay ([3]), luck of releant data needed for more sophisticated modelling make the static analysis a useful start point for predication of critical ehicle speed in strong crossind. 9

10 EFEENCES 1. C.J.Baker, A Simplified Analysis of Various Types of Wind-induced oad Vehicle Accidents. Journal of Wind Engineering and Industrial Aerodynamics, : p T.D.Gillespie, Fundamentals of Vehicle Dynamics. 199: SAE Inc. 3. J.Lemay, eue de littérature portant sur la problématique de perte de contrôle des éhicules lourds causée par la présence de ents latéraux. 010, Ministère des transports du Québec. 4. Baker, C.J., High Sided Articulated oad Vehicles in Strong Cross Winds. Journal of Wind Engineering and Industrial Aerodynamics, (1): p