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Linköpings Institute of Technology Campus Norrköping Accident and Injury Prevention Course instructor: Lennart Strandberg Mini-Thesis Jan Capek (Czech Republic) janca380@student.liu.se Jiri Sloup (Czech Republic) jirsl795@student.liu.se December 11th 2003 1 CONTENTS CONTENTS ....................................................................................................................... 2 1 DEFINITION ............................................................................................................... 3 2 DECELERATION PERFORMANCE ......................................................................... 5 3 YAW AND ROLL STABILITY.................................................................................. 6 3.1 Tires ..................................................................................................................... 6 3.2 Analysis of the yaw stability, critical speed ........................................................ 9 3.3 Oversteer and understeer vehicle ........................................................................ 13 3.4 Discussion how to improve/decrease yaw stability of the vehicle ...................... 16 3.5 Analyses of Roll Stability ................................................................................... 17 4 VARIOUS TYPES OF ANTI-LOCK BRAKING SYSTEMS (ABS) ........................ 19 4.1 Definition ............................................................................................................ 19 4.2 Types of ABS ...................................................................................................... 20 4.2.1 Passenger vehicles .................................................................................... 20 4.2.2 Trucks ....................................................................................................... 21 4.2.3 Trailers ..................................................................................................... 22 5 BRAKING .................................................................................................................... 22 5.1 Braking time and braking deceleration ............................................................... 22 5.2 Speed as a function of position during braking with constant deceleration ........ 23 6 BRAKING PERFORMANCE - DYNAMIC LOAD TRANSFER ............................. 24 7 ADVANTAGE OF CONVENTIONAL BRAKES ..................................................... 26 8 EVALUATION ............................................................................................................ 27 8.1 What is the most important for the safety of cars: Stability, Steerability or Braking performance? ......................................................................................... 27 8.2 Which type of Anti-lock braking system would be the most effective? ............. 27 REFERENCES ................................................................................................................... 28 2 1 DEFINITIONS Camber Camber is the angle formed by the inward or outward tilt of the wheel referenced to a vertical line. It is measured in degrees. Camber is positive when the wheel is tilted outward at the top and is negative when the wheel it tilted inward at the top. Caster Caster is the forward or rearward tilt of the steering Figure 1.1 ([10]): Camber axis in reference to a vertical line. It is measured in degrees. Caster is positive when the top of the steering axis is tilted rearward and is negative when the tilt is forward. Proper caster is important for directional stability and returnability. Figure 1.2 ([10]): Caster Steerability Steerability is ability to react to steering input. G-Force G-Force is a unit of force equal to the force exerted by gravity; used to indicate the force to which a body is subjected when it is accelerated. Centrifugal Force Centrifugal force causes passengers to slide across the seat when cornering at high speeds and happens when a moving object, such as a vehicle, changes direction. The weight of a vehicle means that when travelling in a straight line it will endeavour to continue in that direction, even when the driver turns the steering wheel. Changing direction causes the vehicle‟s weight to move to the outside of the turn, which, unless the driver controls its speed, can lead to rollover or sliding out. Centripetal Force For an object to move along a curved path a force must be applied to it. This force only changes the direction, not the speed of the object and is called a centripetal force. Note that the direction of the force (and hence the acceleration) is towards the centre of the curve. Centripetal means "centre seeking". v2 Fcentripetal m r Braking Force Coefficient (Braking Coefficient) 3 Braking Force Coefficient is the ratio of the braking force to the vertical load. Braking Distance The braking distance is the distance that a vehicle travels while slowing to a complete stop. Figure 1.3: Vehicle axis system Yaw Velocity Yaw velocity is the angular velocity about the z-axis. Roll Roll is the angular component of ride vibrations of the sprung mass about the vehicle x-axis. Pitch Pitch is the angular component of ride vibrations of the sprung mass about the vehicle y-axis. Sideslip Angle (Attitude Angle) Sideslip angle is the angle between the traces on the X-Y plane of the vehicle x-axis and the vehicle velocity vector at some specified point in the vehicle. Sideslip angle can be calculated from the lateral velocity vy and longitudinal velocity vx. vy tan 1 vx Steer Angle Steer angle is the angle between the projection of a longitudinal axis of the vehicle and the line of intersection of the wheel plane and the road surface. Longitudinal Acceleration Longitudinal acceleration is the component of the vector acceleration of a point in the vehicle in the x-direction. Lateral Acceleration Lateral acceleration is the component of the vector acceleration of a point in the vehicle perpendicular to the vehicle x-axis and parallel to the road plane. Centre of Mass (Centre of gravity) A point in the vehicle reference frame that coincides with the centre of mass of the entire vehicle when the suspensions are in equilibrium and the vehicle is resting on a flat level surface. 4 Braking Force Braking force is the negative longitudinal force resulting from braking torque application. Driving Force Driving force is the longitudinal force resulting from driving torque application. Lateral Force (Fy) Lateral force is the component of tire force vector in the Y direction. Load or Normal Force Load is the component of the force vector in the z direction. 2 DECELERATION PERFORMANCE Different combinations of vehicles In the Figure 2.1 we can see that deceleration of trucks is much more slowly than of other road users. Figure 2.1 ([8]): Stopping distance of various vehicles 5 Different combinations of road surface The acceleration of a braking vehicle depends on the frictional resistance and the grade of the road. Figures 2.1 and 2.2 show values of coefficient of friction for various kinds of roads. Table 2.1 ([8]): Road Surface Classification and Coefficient of Friction Table 2.2 ([8]): Values of Coefficient of Friction 3 YAW AND ROLL STABILITY 3.1 Tires Yaw stability of a vehicle is highly dependent on its tires performance. It is very important to discuss this topic first, only then we can understand the analyses of the yaw stability of a vehicle well. When the tire rolls at a slip angle ( ), side force (S) is generated as a function of the slip angle. The mechanism of the generation of side force is described in fig. 3.1. 6 Fig. 3.1 ([3]): Rolling tire deformation under a lateral force In the beginning of the contact patch of the tire tread elements are undeflected and therefore produce no lateral force. But, as the tire advances further, the tread elements are deflected and the lateral force builds up as the element moves rearward in the contact patch up to the point where the lateral force acting on the element overcomes the friction available and slip occurs ([3]). The relationship between the tire side force and the slip angle can be seen in the tire characteristic (fig. 3.2) that is assessed by road experiments. Fig. 3.2 ([6]): Tire characteristic The characteristic is not linear but the linearization is usually applied and then we can say: S C , (3.1) where C is called “cornering stiffness” (slope of the characteristic at 0 ). Another form of the equation for side forces: S K N , (3.2) where N is a normal force acting on the tire and K is “cornering stiffness coefficient”. The increase of side force as higher loads is degressive, that means the cornering stiffness coefficient decreases when the normal force (load of the tire) increases. 7 But in fact, the tire characteristic (fig. 3.2) is not linear and it may cause (especially for heavy trucks) yaw instability of the vehicle. When a heavy vehicle is cornering the vertical load acting on the tire on the inside of the curve is greater then the vertical load acting on the outer tire. This is caused by lateral acceleration while cornering. Then we can see in Fig 3.3 that the average cornering stiffness of these two tires is smaller compared to the situation without any load transfer. This loss in cornering capacity can be large especially for tires on the rear axle of heavy trucks because its rear suspension is significantly stiffer than the front suspension ([6]). Then a much greater load is transferred at the rear axle than at the front axle and it leads to the greater loss in side force capacity at the rear tires. Fig. 3.3 ([6]): Cornering stiffness vs. normal force If a tire operating under circumferential forces (braking or driving force) or slip is subjected to an additional slip angle, at all slip rates the usable circumferential forces will decrease with increasing slip angle ([1]). When we draw the diagram for side vs. circumferential force (at constant wheel load) we will get the elliptical curve called a friction ellipse. The curve represents the limits of longitudinal and lateral forces due to friction. The disadvantage of the friction ellipse diagram is that it is valid only for a certain load, tire pressure and temperature. A simplified version of the friction ellipse is friction circle (fig. 3.4). The limit, which is shown as a circle, is defined by the product of friction coefficient of the road and the vertical load of the vehicle on the tire. The friction circle does not take into account slip angle and slip ratio. 8 Fig. 3.4 ([18]): Friction circle 3.2 Analysis of the yaw stability, critical speed In the analysis the most simplified vehicle dynamic model will be used. It is a two-degree-of- freedom bicycle model that represents the lateral and yaw motions only. It means the longitudinal direction is in this case neglected. However in the chapter 3.3 the influence of the longitudinal direction (while braking) on the yaw stability will be discussed. Fig. 3.5 shows the two-degree-of-freedom model. 9 Fig. 3.5: A bicycle model It consists of a rigid mass with the center of gravity (CG) that is the origin of the vehicle fixed coordinate system. denotes the wheel steer angle, a the longitudinal distance from front axle to CG, m the mass of the vehicle, J z the yaw moment of inertia, v x the longitudinal velocity (that is assumed to be constant), v y the lateral velocity and z is the rotation velocity of the vehicle. 12 , 34 are the slip angles of the front and rear tires, S12 and S 34 are the front and rear side forces produced at the tires due to the slip angles. The aerodynamical and driving/braking forces are neglected. Then the force and moment equations will be: mv y mv x S 34 S12 cos 0 J z aS12 cos bS34 0 By using eq. (3.1) and assuming small angles we obtain: mv y mv x C 34 34 C12 12 0 (3.3) J z aC12 12 bC34 34 0 (3.4) The sideslip angles (with the assumption of small angles) will be: 10 v y a 12 (3.5) vx b v y 34 . (3.6) vx Then by insertion these equations into eq. (3.3), (3.4) we obtain: C12 C 34 C 34b C12 a mv x 2 m 0 v y vx vx v C 0 J bC aC y 12 a 2 C12 b 2 C 34 (3.7) z 34 12 aC12 vx vx that can be modified to its normal form: A B ([19]) (3.8) where: v y represents the state vector, C12 C 34 C 34 b C12 a mv x 2 mv x mv x A , bC34 aC12 a 2 C12 b 2 C 34 J z vx J z vx C12 m B aC . 12 Jz These equations represent a linear control system where necessary and sufficient condition to be stable is that none of the eigenvalues of the matrix A has a nonnegative real part ([19]). The eigenvalues s 1, 2 can be obtained from the characteristic equation: det(sI A) 0 (3.9) where I is the identity matrix with ones on the main diagonal and zeroes on all other elements. After solving eq. (3.9) we obtain the second order equation: s 2 s 0 (3.10) 11 where C12 C34 a 2 C12 b 2 C34 (3.11) mv x J z vx C12 C34 a 2C12 b 2C34 aC12 bC34 C12 a C34b mvx2 2 (3.12) mJ z v x Now, we can calculate the eigenvalues: 2 1, 2 (3.13) 2 4 From eq (3.11) it is obvious that 0 . Now we can see that only 1 (with + sign before square root) has got a real part that can be also positive. The real part of the eigenvalue is negative if: 0 (3.14). After solving it the unequality is valid if: K 34 K 12 (3.15) or C12 C 34 a b K 12 K 34 a b g 2 vx (3.16) maC12 bC34 K 12 K 34 where eq (3.2) were used while normal forces are equal to: b N12 mg ab a N 34 mg . ab The expression (3.16) means that the vehicle is stable at every speed if the cornering stiffness coefficient K of the rear wheals is larger than that of the front wheels. Otherwise, when the condition (3.15) is not valid, the vehicle becomes unstable after reaching critical speed v c : K12 K 34 a b g vc . (3.17) K12 K 34 12 3.3 Oversteer and understeer vehicle When we use again eq (3.16) for the critical speed and modify a little its form: C12 C34 a b a b g 2 vc (3.18) maC12 bC34 where is understeer gradient that is equal to: b 1 a 1 N12 N 34 mg (3.19). a b C12 a b C34 C12 C34 And because for a cornering vehicle this equation is applied: S12 S v2 34 ([6]) (3.20) N12 N 34 gR by using this equation in eq (3.19) we obtain: gR S12 N12 S 34 N 34 gR 1 2 (3.21) v 2 N12 C12 N 34 C34 v 2 A vehicle is said to be understeer when, as lateral acceleration increases, the slip angle at the front axle increases more than at the rear axle. Since in this case the understeer gradient is still positive, the vehicle is stable at every speed. When opposite situation occurs then the vehicle is oversteer, the understeer gradient is negative and yaw instability occur when the vehicle reaches critical speed. vx By using eq (3.5) and (3.6) and also by using equation (R is radius of the curve) we R obtain: L v2 (3.22), R gR where L = (a + b) . . . wheelbase. (3.23) Then, using eq (3.22) we can plot a diagram called handling characteristic where the slope determines the steering character. Diagram in fig. 3.6 represents the handling characteristic for three simple linear vehicles. 13 Fig. 3.6 ([6]): Handling characteristics The vehicle‟s steering behaviour is not always the same at all rates of lateral acceleration. It can be caused due to a loss in cornering capacity (as discussed before). If the loss of cornering capacity is relatively greater at the rear axle than at the front axle, the difference 2 1 is larger and the understeer (slope of curve in handling characteristic) changes as the level of lateral acceleration increases ([6]). Fig 3.7 is an example of a handling curve typical for heavy trucks. Fig. 3.7 ([6]): Handling characteristic of a heavy vehicle The change of vehicle‟s steering behaviour can be also caused by braking. The inertia forces associated by retardation unloads the rear axle, cornering stiffness, that is dependent on the 14 normal load, decreases at the rear and increases at the front. This can be shown at a 4x2 tractor: Fig. 3.8: A 4x2 tractor Force and moment equations are: N12 N 34 mg (3.24) N12 a b mga mh 0 x (3.25) By solving these equations we obtain for normal forces: mga mh mg 1 x N12 mg (3.26) ab mga mh mg x N 34 (3.27) ab where: a L x g h . L Now, by using eq (3.26), (3.27) understeer gradient may be rewritten: 15 1 g (3.28) 1 c12 c34 where cornering stiffnesses are assumed to be linearly dependent on the load (normal force) and c i used in eq (3.28) is cornering stiffness divided with the static normal load over the gravity constant ([2]). The eq (3.28) shows that an understeering vehicle changes to oversteering during braking and reaches critical velocity under extended braking. There is an example of plotting in fig 3.9. Fig. 3.9 ([2]): Understeer gradient versus non-dimensional retardation 3.4 Discussion how to improve/decrease yaw stability of the vehicle When two tires of better condition are mounted only at the front axle, cornering stiffness coefficient is then at the front axle greater than at the rear axle. According to the condition (3.15) it leads to yaw instability. On the other hand steerability increases. If braking or driving forces are applied to a wheel, the cornering stiffness will decrease. It means, when the car starts to turn by itself (losing its yaw stability) with front-wheel-drive and the driver press the clutch pedal, the side force acting at the front wheels increases and it also may lead to the yaw instability. If the same situation occurs with rear-wheel-drive then after pressing the pedal yaw stability of the vehicle is even better. From fig 3.2 we know that cornering stiffness coefficient decreases as the normal load of a tire increases. That means if we put an extra load in the rear end of the car, K at the rear axle is smaller and it may also lead to yaw instability. And when we consider the loss in cornering capacity due to load transfer during cornering, the influence of the extra load on yaw stability is more severe. 16 3.5 Analyses of Roll Stability Roll-over of a vehicle traveling on a road is caused by the centrifugal force acting through the vehicle‟s center of gravity when the centrifugal force is sufficiently large. The force increases with speed and with the curvature of the road. Mostly heavy trucks with a high center of gravity have a tendency to roll-over. On the other hand, automobiles with a low center of gravity have a tendency to slide out of the curve rather than roll-over. In this chapter the mechanics of roll-over will be investigated using simplified vehicle models, that means vehicle suspended on compliant passive suspensions and tires, as shown in fig. 3.10. Fig. 3.10 ([16]): Simplified suspended vehicle model It is also assumed that the total mass of the vehicle is in the sprung mass, the compliance of the suspensions and tires is lumped into a single equivalent compliance, and that the roll of the sprung mass takes place about the point on the ground plane at the mid-track position. Moment equation in this point is then: ma y hcm Fz T mgh cm , where the left side of the equation is the primary overturning moment due to the lateral acceleration, the right side is the net restoring moment which is the sum of the restoring moment ( Fz T , arises from the lateral load transfer from the inside tires to the outside tires) and the lateral displacement moment ( mgh cm , due to the roll motion that displaces the centre of mass laterally from the center line of the vehicle). The roll-over of the vehicle occurs if the primary overturning moment exceeds the net stabilizing moment that can be provided 1 by the vehicle. That means when the restoring moment is equal to mgT . The roll-over 2 threshold is then: 17 Tg ay * g , 2hcm where * is the critical roll angle at wheel lift-off. In reality, the sprung mass rolls about a suspension roll centre that is not a ground level as shown in Fig. 3.10. This problem is discussed in Individual Task V7. In fig. 3.11 an example of a roll response graph of a vehicle with multiple axles is shown. It represents the balance between the primary overturning moment and the net restoring moment. The primary overturning moment is plotted against lateral acceleration (on the left side of the graph) while the net restoring moment (sum of the restoring moment and the lateral displacement moment) is plotted against roll angle (on the right side of the graph). Fig. 3.11 ([16]): Roll response graph In this example, the trailer axle has got the highest stiffness-to-load ratio, the tractor steer axle has got the lowest stiffness-to-load ratio. The trailer axles are also the most heavily laden, followed by the tractor drive axle and the tractor steer axle. The maximum restoring moment that can the axle provide depends on the load condition of the axle. The more heavily laden the axle, the greater the maximum restoring moment it can provide. That is why the maximum suspension moment is higher for the trailer axle group (A) than for the tractor drive axle (B) or the tractor steer axle (C). The roll angle at wheel lift-off for a given axle depends on the ratio of effective roll stiffness to vertical load, axles with a higher stiffness-to-load ratio lift off at smaller roll angles. This is 18 also obvious from the roll response graph where the trailer axle group lifts off at the lowest roll angle (A). At point E the inside tires of the trailer axle group and of the tractor drive axle lift off and because the tractor steer axle is not sufficiently stiff to provide a restoring moment, the roll over threshold is then defined by this point E and not by the lift off of the inside wheel of the tractor steer axle (F). 4 VARIOUS TYPES OF ANTI-LOCK BRAKING SYSTEMS (ABS) 4.1 Definition Anti-lock braking system is device that monitors, control and regulates breaking system during braking. During braking wheels can decelerate too rapidly and they may lock. This is monitored by ABS and when the wheel is on the lockup limit ABS regulates pressure in braking system. ABS consists of: - an electronic unit (ECU) - a solenoid for releasing and replying pressure to a brake - a wheel speed sensor ECU monitors vehicle speed through the wheel speed sensor and after brake application it begins to compute an estimate of the diminishing speed of the vehicle. Actual wheel speed can be compared against the computed speed to determine whether a wheel is skipping excessively, or the deceleration rate of a wheel can be monitored to determine when the wheel is advancing toward lockup. Figure 4.1 ([13]): Wheel speed cycling during ABS operation In the figure 4.1 is shown braking performance with ABS. If the brakes are applied to a high level, or the road is slippery, the speed of one or more wheels begins to drop rapidly (point 2), indicating that the tire has gone through the peak of the slip curve and is leading toward lockup. At this time ABS starts regulate and release the brakes on those wheels before lockup occurs (point 3). Aim of the ABS is to keep each tire on the vehicle operating near the peak of the slip curve for that tire. 19 Lockup of front wheels Lockup of front wheels causes loss of the ability to steer the vehicle and the vehicle will continue straight ahead despite any steering inputs. Lockup of rear wheels If it lockup, any yaw disturbance will initiate a rotation of the vehicle. The front wheels, which yaw with the vehicle, develop a cornering force and the yaw angle continues grow. It is mainly problem of smaller vehicles (passenger vehicles), because by long vehicles the growth is slow and the driver can apply correcting steer and prevent the full rotation. 4.2 Types of ABS Types of ABS differ in monitored wheels (number and which are controlled) and how to regulate brake pressure. The term channel refers to either an input or output path to the ECU. 4.2.1 Passenger vehicles One channel – Rear axle control This system was used in the end of 1960‟s. It monitors speed of the two rear wheels. The brake pressure to both of them is monitored by whichever wheel begins to lock first. It is known as „Select Low. This system ensures that neither rear wheel can lock. Straight line stability should be better with this system, but front wheels are not monitored and regulated and they can lock and driver can not steer the car. This is uncontrollable but stable situation. Figure 4.2 ([7]): One channel system Figure 4.3 ([7]): Two channel system Front wheel control Two channel – Front wheel control This system is used on some front wheel drive car with diagonally split breaking system. It monitors speed of the front wheels independently and regulates brake pressure to their diagonally opposite rear wheel via an apportioning valve. The advantage of this system is that when is a car braking on a split-coefficient surface, one wheel may lock but never both. Two channel – Front and rear axle control This system monitors all four wheels and each axle. Rear axle is controlled by „Select Low‟ and front axle by „Select High‟. This allows one wheel lock and starts regulate the brake pressure to the front wheels when the second front wheel is on the lockup limit. 20 Figure 4.4 ([7]): Two channel system Figure 4.5 ([7]): Three channel system Front and rear axle control Three channel – Front wheel and rear axle control This is the most common Anti-lock braking system in passenger cars. Usually all four wheels are monitored, the front wheels are controlled independently and rear axle is controlled by „Select Low‟. 4.2.2 Trucks There are a number of types of ABS used in trucks. The type of ABS that will operate on truck the best depends on number of axles, axle configuration, axle load, brake circuit distribution and brake force distribution. Because the more severe braking, the greater role plays front axle brake and so the steering wheels should be included in ABS. Six channel system This system was used at the beginning of ABS technology and now it is not so common. It is often called as a 6S/6M system, where S means sensor and M modulator. In this system each wheel is monitored by speed sensor and air pressure to the service chamber can be regulated based on exact condition at each wheel. Six/four channel system (6S/4M) Figure 4.6 ([17]): Six-channel ABS configuration This system uses sensors on each wheel, but it has only 4 outputs, because 4 wheels are regulated in pairs. If either of the wheels grouped in a pair is on limit of lockup, then the system regulates brake pressure in both wheels. Four channel system (4S/4M) Wheel speed sensor is located on each wheel of the steering axle and on 2 of 4 rear wheels. Wheels sensors on rear axles have to be located on either side. 21 Figure 4.7 ([17]): Four-channel ABS configuration Two channel system It can be used on trailer or straight truck. 4.2.3 Trailers Anti-lock braking system controls the wheel rotation under braking. Trailer has its own ECU and anti-lock braking is independent to the tractor. In trailers are used these configurations: 2S/1M – 2 wheel speed sensors and single modulator valve 2S/2M – 2 wheel speed sensors and 2 modulator valves 4S/2M – 4 wheel speed sensors and 2 modulator valves 4S/3M – 4 wheel speed sensors and 3 modulator valves 6S/3M – 6 wheel speed sensors and 3 modulator valves 5 BRAKING 5.1 Braking time and braking deceleration Braking time consists of these times - reaction time (reaction of the driver) ...................... t1 - reaction of braking mechanism ............................... t2 - time when pressure in braking system increases .... t3 - constant braking pressure ........................................ t4 Then braking time is: t t1 t 2 t 3 t 4 When braking, we need to counteract kinetic energy of the moving vehicle. E k Wt E k ..... kinetic energy Figure 5.1: Distance-time, W t ..... work of friction speed-time graph 1 m.v0 m.g. .sb 2 2 m ...... mass of vehicle v 0 ....... initial speed ....... coefficient of friction 22 s b ....... braking distance v0 a.t v0 t a 2 1 1 v0 sb a.t 2 2 2 a a g. 5.2 Speed as a function of position during braking with constant deceleration a ...... deceleration ...... change of velocity per time dv dv ds dv ds dv vdv dv 2 a . . .v dt dt ds ds dt ds ds 2ds v ...... velocity ............. change of position per time ds v dt 2 dv a 2ds separation of variables 2ads dv 2 integration s1 v1 2ads dv 2 0 v0 s1 ........ position during braking v0........ velocity before braking v1........ velocity after braking on braking distance s1 a const. s1 v1 2a. ds dv 2 0 v0 2a.s v 2 s1 0 v1 v0 2a.s1 v v0 2 2 1 v12 2a.s1 v0 2 v1 2a.s1 v0 2 23 35 v1 [m/s] 30 25 20 15 10 5 0 0 50 100 150 200 250 a=2m/s^2 a=3m/s^2 s1 [m] Figure 5.2: Course of velocity during braking commensurate with braking distance 6 BRAKING PERFORMANCE - DYNAMIC LOAD TRANSFER Influence on braking performance from dynamic load transfer If the load is not balanced correctly along vehicle‟s length, wheel lockup during braking becomes more likely. Figure 6.1: Load distribution of vehicle during braking braking performance Total braking force can be calculated: B m.a x m.D x BF BR Da R f G. sin where: D x ...... linear deceleration BF ...... front axle braking force 24 BR ...... rear axle braking force Da ...... aerodynamic drag force R f ...... rolling resistance force ........ uphill grade If we neglect aerodynamic drag force, rolling resistance force and uphill grade, we have in x direction equation of equilibrium: BF BR m.Dx (6.1) Load distribution can be determined from the moment equations around centre of gravity and equations of equilibrium in directions x and z. Z1 Z 2 m.g (6.2) BR BF .h Z 2 .r Z1 . f J . (6.3) If we neglect pitch motion, then the equation is: BR BF .h Z 2 .r Z1 . f 0 (6.4) By inserting of equations (6.1) and (6.2) into (6.4) and modifying we can obtain equations for load distribution on front and rear wheels: r h Z1 G. m.Dx . l l f h Z 2 G. m.Dx . l l where: Z 1 ....... front axle load Z 2 ....... rear axle load G ........ the weight of the vehicle acting at its centre of gravity G m.g r ......... longitudinal distance from centre of gravity to front axle f ........ longitudinal distance from centre of gravity to rear axle l .......... wheelbase D x ...... longitudinal deceleration J y ....... moment of inertia…. ........ pitch angular acceleration According to mass distribution to front and rear axles it can be three events after releasing of some deceleration: - both front and rear wheels lockup simultaneously - front wheels lockup first - rear wheels lockup first If we have vehicle that is optimized for the situation with all wheels lockup simultaneously, then we have maximal braking forces on both axles before lockup: BF , BR . From this we can calculate brake force distribution in the brake system: BF pF BF BR where maximal braking forces on front and rear axles are: r h BF .Z 1 . G. m.D x . l l f h BR .Z 2 . G. m.D x . l l 25 After changing of centre of gravity, p F is staying the same, but then front or rear wheels may lockup first. Next we consider coefficient of friction constant, because its change has also influence for lockup of wheels, but together with influence of change of load, it is difficult to define which wheels lockup first. It can be defined only by calculation. Change of load 1) Front axle – more load Rear axle – less load This leads to change of position of centre of gravity nearer to the front axle. Because the higher load is on the front wheels, they can transfer greater braking force without lockup. On the rear wheels is lower normal force and they can transfer lower braking force before lockup. From this results that first lockup rear wheels. Then braking force on rear wheels when lockup is: BR .Z 2 and braking force on front wheels can be calculated from BF pF BF BR B .p BF R F 1 pF 2) Front axle – less load Rear axle – more load This leads to change of position of centre of gravity nearer to the rear axle. First lockup front wheels because we decrease normal force on the front wheels and they can transfer lower braking force while on the rear wheels is greater normal force and they can transfer greater braking force. Then braking force on front wheels when lockup is: BF .Z 1 and braking force on front wheels can be calculated from BF pF BF BR 1 pF BR BF . pF 7 ADVANTAGE OF CONVENTIONAL BRAKES In what circumstances might conventional brakes have an advantage over ABS? There are some conditions where stopping distance may be shorter without ABS. For example, in cases where the road is covered with loose gravel or freshly fallen snow, the locked wheels of a non-ABS car build up a wedge of gravel or snow, which can contribute to a shortening of the braking distance. 26 8 EVALUATION 8.1 What is the most important for the safety of cars: Stability, Steerability or Braking performance? Everything from this is certainly very important for safety of cars. But braking performance has influence for both stability and steerability. Because when the front wheels lockup, the car losses ability to steer and when the rear wheels lockup, the car losses stability and becomes yaw. So it is very useful to provide device that prevent wheel‟s lockup – Anti-lock braking system. With this comes next task. 8.2 Which type of Anti-lock braking system would be the most effective? The best system would be probably with sensors of all wheels and modulating braking pressure to all wheels individually. But this would be very expensive and this system is seldom used. In A study of various car anti-lock braking systems is shown comparison of several various systems and their performance during braking. Some of these systems can provide longer braking distance with ABS operating (Ford Escort with mechanical two channel system and Fiat Uno with electronic variant of two channel system), but their braking performance on a curve and on split surface is better than without ABS operating. So using these systems gives more controllability when panic braking on these conditions. Next systems that were compared: Honda (two channel system with front and rear axle control), BMW, Mazda and Ford Granada (all have three channel electronic system): These systems have better braking performance on a curve and on split surface than without ABS operating and also their braking distance (deceleration) is usually shorter (larger) on various road surfaces and various weather conditions. Finally it has to be said that when Anti-lock Braking system is disabled, then braking system works like conventional braking system. 27 REFERENCES [1] BOSCH (1996). Automotive Handbook, Rober Bosch GmbH, Germany [2] Dahlberg, E. (2001). Commercial Vehicle Stability, KTH, Sweden, [page 1-36] [3] Gillespie, T. D. (1992). Fundamentals of Vehicle Dynamics, SAE, U.S.A., [page 348- 351] [4] Kullberg, G. (1977). Anti-Lock Braking System for Passenger Cars Development of a Brake System giving Yaw Stability and Steerability during Emergency Braking, VTI, Sweden, [page 1-18] [5] Limpert, R. (1999). Brake Design and Safety, SAE, U.S.A. [6] Pacejka, H. B. (1983). Handling and Stability of Commercial Vehicles, Delft, [page [7] Robinson, B.J., Riley, B.S. (1991). A study of various car anti-lock braking systems. Transport and road research laboratory Crowthorne, Berkshire, [page 1-5] [8] Tokunaga, R.A., Hagiwara, T., Onodera, Y. (2000). 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