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  1. Vehicle Dynamics (MV6131) Lateral Dynamics Part Part- -two Steady state Handling Characteristics two Steady state Handling Characteristics Department of Motor Vehicle Engineering Defence University, College of Engineering 2019-20 A.Y Semester I

  2. Out-line Steady State cornering Steady State Cornering Characteristics Steady state Vehicle respond to steering input Yaw rate, Lateral acc. Curvature, side slip angle response Neutral Steer point & Static Margin Testing of Handling Characteristics: Constant Radius Test Constant Speed test Constant steering Angle Test Chassis System Effects on Handling Characteristics Lateral force transfer, camber change, Roll steer, Lat. force compliance Steer, Alignment Torque and Tractive Force Effects Reference: Karnopp, Jazar, Wong Gillespie Vehicle Stability, Chapter 6 section VI , Vehicle Dynamics, Chapter 10, page 620- Theory of Ground vehicle, Chapter 5, Page 339- Fund. Of vehicle Dynamics, Chapter 6, Page 195-

  3. Introduction • In a previous section • stability was analyzed only for motion in a straight line • In this Section • By allowing to have nonzero steering input values (δ), the model can also be used to study cornering behavior • Assumptions in steady turns, i.e., • turns taken at a constant speed and • having a constant turn radius • Advantages • The terms understeer and overseer, are most easily related to steady turn behavior. • Relation for the steer angle • in terms of the slip angles and the yaw rate =0 −αr= − βr− ?? = −βr U = rR → R =U r→1 R=r U Naturally, such turns do not strictly occur in normal driving but often there is a period of time in actual turns when the car nearly is in a steady state. βf=ar u= δf− αf L R −αr= − β −br Rear Wheel βr=br u= αr u −αf= − βf− δf = − β +ar −αf= − −αr+br Front Wheel but β = −αr+br + δf Without considering sign of slip angle u u δf=L δf=L u+ar + δf= αr−L R+ αr− αf R+ αf− αr R+ δf u

  4. Steady State Handling Characteristics Steady State Handling Characteristics • Dynamic Equation of Motion in Body centered Coordinate System • Translational Equation of Motion Fy= m ሶV + rU But ሶV = 0 for Steady State Fy= may= mrU=mU2 Where r =U R= Fyf+ Fyr R • Rotational Equation of Motion Mz= Izzሶ r = aFyf− bFyr But ሶ r = 0 for Steady State Mz= 0 = aFyf− bFyr • Solve for the tire forces mU2 mU2 R mU2 R R= Fyf+ Fyr b L a L Fyf= Fyr= ൢ 0 = aFyf− bFyr • These lateral forces at the front and rear axles are distributed in a similar fashion to the normal or weight forces supported by the axles • Normal Force at the Front and Rear wheel, take moment about point 1 and 2 • Then The Tire Force become b L L mg mg b a W=mg 2 1 →b L=Wf a →a L=Wr L Wf= mg Wr= mg Wr Wf U2 R U2 R Wf g Wr g Fyf= Fyr=

  5. Steady State Handling Characteristics • Slip Angle in terms of Tire Force (assume small α and Fy)→ linearized relations between lateral tire forces and slip angles → αr= −Fyr → αf= −Fyf Fyr= −Cαrαr Fyf= −Cαfαf Cαr Cαf • Now Substitute the above slip angle relation in the following Steer angle relation mU2 R R+ −Fyr − −Fyf δf=L ? ? δf=L δf=L R+ − + R+ αr− αf LCαr LCαf Cαr Cαf δf=L U2 R δf=L U2 R U2 R δf=L b a δf=L bCαr− aCαf LCαfCαr R+ K1?? R+ K1 R+ m − R+ m LCαf LCαr • Another version using the weight forces ay g δf=L U2 Rg δf=L U2 Rg R+ K2 δf=L Wf Cαf −Wr Cαr R+ K2 R+ −Fyr − −Fyf δf=L R+ Cαr Cαf • Where, K1and K2are under steer gradient/coefficient bCαr− aCαf LCαfCαr Wf Cαf −Wr Cαr K3=m bCαr− aCαf L???Cαr K1= K3= K2 and K2= K1= m g rad/(m/s2) rad/‘‘g’’ rad/(m/s2)

  6. Steady State Handling Characteristics Steady State Handling Characteristics • Dependent on the values of the understeer coefficient K the steady-state handling characteristics may be classified into three categories: • Neutral steer, Understeer, and Oversteer • Neutral Steer • When the understeer coefficient K = 0, • which is equivalent to the slip angles of the front and rear tires being equal, αr= αfor Wf Cαf Cαr Relationships between steer angle and speed =Wr or bCαr= aCαf • The steer angle δfrequired to negotiate a given curve is independent of forward speed and is given by δf=L = 0 δf=L R+ αr− αf R δf K > 0 Understeer • The forces at the front and rear can be adjusted to give a larger lateral acceleration by simply changing the car attitude without any change in steer angle Neutral steer K = 0 Over steer Ucrti2/R U2 R K <0 Relationships between steer angle and Lat.Acc.

  7. Steady State Handling Characteristics Curvature responses • Neutral Steer • when it is accelerated in a constant radius turn the driver should maintain the same steering wheel position • In other words • when it is accelerated with the steering wheel fixed, the turning radius remains the same, as illustrated in Fig ⚫ When a neutral steer vehicle originally moving along a straight line is subjected to a side force acting at the center of gravity, equal slip angles will be developed at the front and rear tires αf= αr ⚫ As a result, the vehicle follows a straight path at an angle to the original Directional responses

  8. Steady State Handling Characteristics Steady State Handling Characteristics • Under Steer • When the understeer coefficient K > 0, • which is equivalent to the slip angle of the front tire is grater than a rear tires αf> αr or Wf Cαf Cαr Relationships between steer angle and speed >Wr or bCαr> aCαf • The steer angle δfrequired to negotiate a given curve increases with the square of the speed (lateral acceleration) • Characteristic Speed • A speed at which the steer angle required to negotiate a turn is equal to 2L/R U2 Rg δf=2L R=L Lg R+ K2 ?????= ൗ K2 δf K > 0 Understeer • Car requires a larger change in slip angle at the front than the rear to maintain equilibrium for a higher lateral acceleration. This can be accomplished by increasing the steer angle. Neutral steer K = 0 Over steer Ucrti2/R U2 R K <0 Relationships between steer angle and Lat.Acc.

  9. Steady State Handling Characteristics • Under Steer Curvature responses • when it is accelerated in a constant radius turn the driver must increasethe same steering angle. • In other words when it is accelerated with the steering wheel fixed, the turning radius increases • At the same steering wheel position and vehicle forward speed, the turning radius of an understeer vehicle is larger than that of a neutral steer vehicle. ⚫ When a side force acts at the C.G of an under steer vehicle originally moving along a straight line the front tires will develop a slip angle greater than that of the rear tire αf> αr ⚫ As a result, a yaw motion is initiated and the vehicle turns away from the side force Directional responses

  10. Steady State Handling Characteristics Steady State Handling Characteristics • Over Steer • When the understeer coefficient K < 0, • which is equivalent to the slip angle of the front tire is less than a rear tires αf< αr or Wf Cαf Cαr Relationships between steer angle and speed <Wr or bCαr< aCαf • The steer angle δfrequired to negotiate a given curve decreases with an increase vehicle speed (lateral acceleration) • Critical Speed (Ucrit) • is the speed at which an over steering car can negotiate a turn with no steer angle at all δf K > 0 Understeer U2 Rg δf= 0 =L Lg R+ K2 ?????= ൗ −K2 Neutral steer K = 0 • Car requires a smaller change in slip angle at the front than the rear to maintain equilibrium for a higher lateral acceleration. This can be accomplished by decreasing the steer angle. Over steer Ucrti2/R U2 R K <0 Relationships between steer angle and Lat.Acc.

  11. Steady State Handling Characteristics • Over Steer • when it is accelerated in a constant radius turn the driver must decrease the same steering angle. In other words • when it is accelerated with the steering wheel fixed, the turning radius decreases • At the same steering wheel position and vehicle forward speed, the turning radius of an over steer vehicle is smaller than that of a neutral steer vehicle. Curvature responses ⚫ When a side force acts at the C.G of an under steer vehicle originally moving along a straight line the front tires will develop a slip angle less than that of the rear tire αf< αr ⚫ As a result, a yaw motion is initiated and the vehicle turns into the side force ⚫ Ucritrepresents the speed above Which an oversteer vehicle exhibits directional instability L2CαfCαr m bCαr− aCαf −L This implies that the driver must steer left to turn right Ucrit= = − K3 Directional responses

  12. Steady State Handling Characteristics Demo-Video-1 Demo-Video-2 • Over Steer • For speeds above the critical speed, • there is a control reversal, which means that the driver must counter steer. • This implies that the driver must steer left to turn right. • The reason for this is that, • for an over steer car, the rear slip angle magnitude increases faster than the front slip angle magnitude as the lateral acceleration increases. • Thus the driver must reduce the front slip angle by reducing the steer angle as the car rotates with respect to the turn radius to decrease the rear slip angle when increasing speed above the critical speed. • Above the critical speed, the actual steer angle is in the reverse direction to the turn direction. • Terminal over steer or loose • Automobile racers refer the limit speed in a turn occurs when the rear axle reaches its limiting lateral force before the front axle • The subjective feeling is that the rear of the car is only loosely connected to the road. Counter steering necessary when an over steering car exceeds the critical speed car also becomes unstable

  13. Steady State Handling Characteristics • The primary factors controlling the steady state handling characteristics of a vehicle are • How the power is delivered • Weight distribution of the vehicle • Cornering stiffness of the tire • Front Engine Front Wheel Drive Vehicle • with a large proportion of the vehicle weight on the front tires may tend to exhibit understeer behavior. • A rear- engine, rear-wheel-drive car • with a large proportion of the vehicle weight on the rear tires, on the other hand, may tend to have oversteer characteristics • Effect of Load shifting when a vehicle accelerate during turning • Due to load transfer from the front to the rear, the slip angles of the front tires increase (decrease cornering stiffness) while those of the rear tires decrease, Consequently, the vehicle tends to exhibit understeer characteristics • Effect of Load Shifting during when a vehicle decelerate during turning • Due to load transfer from the rear to the front, the slip angles of the front tires decrease (increase cornering stiffness) while those of the rear tires increase. As a result, the vehicle tends to exhibit oversteer oversteer oversteer) ) bCαr− aCαf LCαfCαr Wf Cαf −Wr Cαr K1= m K2= Wf Cαf >Wr Cαr Wf Cαf <Wr Cαr αf> αr understeer characteristics. αf< αr oversteerbehavior. (lift behavior. (lift- -off off

  14. Steady State Handling Characteristics • Turning Behavior of 4WD Car • Depends on the ratio of torque applied at the front axle to that applied at the rear axle • with an equal distribution of driving torque between the front and rear axles, • at a fixed steering wheel position • During acceleration (a=+0.2g) the car exhibits increased understeer behavior. • • During deceleration (a=-0.2g) • when the lateral acceleration is up to approximately 0.7g, the vehicle demonstrates oversteer characteristics Variation of turning radius with lateral acceleration of a four-wheel- drive car at various longitudinal accelerations

  15. Parameters affect Cornering Stiffness Effect of Inf. Pressure on Cα Effect of vertical Load on Fy Effect of Side Slip Angle on Fy Effect of tire size on Cα Effect of vertical Load on Cα 2 Cα= aFz+ bFz cornering stiffness, divided by the corresponding tire load, is called the cornering stiffness coefficient 2nddegree Polynomial

  16. Steady State Handling Characteristics • Operating and Deign parameters affect the cornering stiffness of the tire • Effect of Radial & Bias-ply tires • Installing laterally stiff radial-ply tires on the front and relatively flexible bias-ply tires on the rear may change an understeer vehicle to an oversteer • Effect of Inflation Pressure • The cornering stiffness of a tire usually decreases with a decrease of inflation pressure. • Lowering the inflation pressure in the rear tires compared with front tire makes the vehicle to show can Oversteer behavior • often the tire pressures are nearly identical for New vehicle, When this is the case, • A forward C.G tends to promote understeer (bCαr> aCαf) and a rearward C.G tends to result in oversteer (bCαr< aCαf). • Racing Cars: C.G is placed at the center of Wheel base, makes Neutral steer for better maneuverability • Passenger car: C.G is placed towards front, makes understeer for safety reason • Overloaded pickup with low tire pr at the rear causes over steer & almost un drivable at moderate speed to avoid such problem designers usually gives more under steer for empty pickups • Application of a driving or braking torque to the tire during a turn • For a rear-wheel-drive vehicle • the application of tractive effort during a turn reduces the effective cornering stiffness of the rear tires, producing an oversteering effect. • For a front-wheel-drive car, • the application of tractive effort during a turn reduces the effective cornering stiffness of the front tires, thus introducing an understeering effect.

  17. Steady State Handling Characteristics • Variation of understeer coefficient with Operating Conditions • Number of design and operational factors that would affect the understeer coefficient • Effect of lateral acceleration on K for various vehicles • Curve 1 represents • the characteristics of a conventional Front-engine RWD car. • the understeer coefficient increases sharply with an increase of lateral acceleration. • Curve 2 represents • The behavior of a European front-engine FWD Car • It exhibits similar characteristics. • Curve 3 represents • The behavior of a European rear-engine RWD Car • exhibits understeer behavior up to lateral acc. of approximately 0.5 g, above which it tends to become oversteer. • Curve 4 represents • The behavior of an American rear-engine RWD Car exhibits oversteer characteristics in the operating range shown.

  18. Steady State Handling Characteristics • Handling Diagram • To illustrate the changes in the handling behavior of road vehicles with operating conditions • The vehicle lateral acceleration in g-units, • During a turning maneuver, the turning radius R may be difficult to measure directly. However, it can be readily determined from the yaw velocity r (measured using a rate-gyro) and the forward speed U of the vehicle (R = • The relationship between Τ ayg and (L( Τ r U) - δf), is expressed U2gR), is plotted as a function of the parameter ( Τ Τ Τ ayg ( L R − δf) ) Τ U r). ay g L R− δf = − δf=L ay g R+ K2 d = −1 The Slope of the curve is given by drL ay g= − rL U− δf K2 U− δf K2 • This indicate that the handling behavior of a road vehicle can be identified by the slope of the curve • If the Slope is –ve then K2will be +ve • vehicle exhibits under steer behavior • If the slope is infinite, then K2will be Zero • The Vehicle is Neutral Steer • If the slope is +ve, then K2will be -ve • The vehicle is Over steer it is possible to change a single car from understeer to neutral to oversteer by changing the loading, the tires, or even the tire pressure.

  19. Steady State Response • Yaw Velocity Response Gyaw/Gr • Defined as the ratio of the steady-state yaw velocity to steering angle • Is important during a sharp( 90o) cornering, gives ability to regain its orientation r =U R r U r δf U U U2 gR δf=L Gyaw= = Gyaw= = = U2 g R+ K2 L R+ K2U2 δf Rδf L + K2 R gR U2 R r U U δf=L Gyaw= r U = = R+ K1 U2 R Gyaw= L R+ K1 = δf Rδf R L + K1U2 δf • Using Cramer’s Rule • When the Laplace variable s is set equal to zero in transfer functions, the steady state relation of output to input is given • r δfS=0 ∆ U ma???S +????Cαr ma???S +????Cαr Cαf+ Cαr U U U = = ቤ mIzz??+ ma2Cαf+ b2Cαr CαfCαr?2 U2 + Izz ? + + m bCαr− aCαf ????Cαr U r δfS=0 U U ቤ = = = CαfCαr?2 U2 ? +mU2bCαr− aCαf ????Cαr m bCαr− aCαf ????Cαr U2 ? + + m bCαr− aCαf r δfS=0 U ቤ = L + K3U2

  20. Steady State Response • Yaw Velocity Response Gyaw/Gr • If the yaw Velocity Gain w.r.t steer wheel angle is desired where G﷮s is the steering gear ratio r Grf= δfGs • For Neutral Steer vehicle, K=0 • Yaw Velocity Gain Gyawincrease linearly with increase of U r δfk=0 U =? Gyaw= = ቤ L + K2U2 0= U L + K2U2 ? g r δfk>0 • For Understeer vehicle, K>0 Gyaw= = ቤ Gyawincrease with increase of U & reaches max at Characteristic Speed UChar= • g 1/2 Τ Lg K2 Gyaw max=r U ? ?? Oversteer = = ቤ Lg K2g δfUChar (rad/s/rad) Neutral steer L + K2 • For Oversteer vehicle K<0 K=0 K<0 r δfk<0 U 1/2 Τ UCrit= Lg −K2 Gyaw= = ቤ L − K2U2 ????/?? g Gyawincrease with increase of U & reaches infinity at Critical Speed r δfUCrit L − K2 K2g K>0 U =? Gyaw max= = ?= ∞ ቤ Under steer ? Lg UChar UCrit If the measured yaw velocity r > U/L the vehicle is oversteer

  21. Steady State Response • Yaw Velocity Response Gyaw/Gr • yaw velocity r can be measured using a rate-gyro • For a neutral steer car • if a specific yaw rate is desired, smaller steering inputs are required at high speed than at low speed. • For the understeering car, • is stable at high speed, but requires large steer inputs to achieve a given yaw rate at very high speeds (become less responsive at high speed) • This is the common property of many vehicles that, a high degree of stability at the expense of maneuverability • The oversteering car • yaw rate is infinitely sensitive to steering inputs at the critical speed. • Above the critical speed, control reversal occurs and the driver must countersteer as well as to attempt to stabilize an unstable vehicle.

  22. Steady State Response • Lateral Acceleration Response Gacc/Ga • Defined as the ratio of the steady-state lateral acceleration ayto steering angle • Is important during change lanes on a straight freeway when traveling at high speed U2 ay= R U2 r U2 Τ ayg δf U2 gR δf=L Gacc= = Gacc= = Lg + K2U2 R+ K2 L R+ K2U2 δf gR gR U2 Gacc=ay U2 Gacc=r U2 R δf=L = = R+ K1 L R+ K1U2 δf L + K1U2 δf R R • Using Cramer’s Rule • When the Laplace variable s is set equal to zero in transfer functions, the steady state relation of output to input is given • ay δfS=0 ∆ U ?b???Cαr U ?b???Cαr U Cαf+ Cαr U Izz???S?+ Izz???S?+ S + ????Cαr S + ????Cαr ቤ = = mIzz??+ ma2Cαf+ b2Cαr CαfCαr?2 U2 + Izz ? + + m bCαr− aCαf U2 U2 ay δfS=0 ????Cαr ቤ = = = CαfCαr?2 U2 L +mU2bCαr− aCαf L???Cαr m bCαr− aCαf ????Cαr U2 L + + m bCαr− aCαf U2 Gaf=ay = ቤ L + K3U2 δfS=0

  23. Steady State Response • Lateral Acceleration Response Gacc/Ga • If the lateral acceleration gain w.r.t steer wheel angle is desired where G﷮s is the steering gear ratio ay δfGs Gaf= • For Neutral Steer vehicle, K=0 • Gaccincrease with the square of increasing U L + K1U2=?? 0= U2 L + K1U2 U2 Gacc=ay = ቤ δfk=0 ? • For Understeer vehicle, K>0 Gacc=ay = ቤ δfk>0 • at very high Speed the 1stterm in the denominator is much smaller than 2ndterm, then Gaccapproaches a value of 1/k1 asymptotically Gacc @ high U= L + K1U2= K1 U2 ? Oversteer Neutral steer U2 Gacc=ay • For Oversteervehicle K<0 = ቤ ????/?? L − K1U2 δfk<0 K=0 • Gaccincrease with increase of U at critical speed denominator become zero, then Gaccapproaches infinity UChar K<0 K>0 ? K1 U2 =? 2 Uchar 2L Ga @ UCrit= ?= ∞ 1/2 Τ L UCrit= L −K2 L − K1 Understeer UCrit K1 ?

  24. Steady State Response • Lateral Acceleration Response Gacc/Ga • Lateral acceleration can be measured using accelerometer • The three cases shown have no particular relation to each other. They are for three separate automobiles. There is no obvious connection between the characteristic speed for an understeering car and the critical speed for an oversteering car • it is possible to change a single car from understeer to neutral to oversteer • by changing the loading, the tires, or even the tire pressure. • Most cars are designed to be as insensitive to these changes as possible • For a Neutral car • lateral acceleration becomes very sensitive to steer angle at high speed. • For an oversteer car, • lateral acceleration sensitivity even becomes infinite as the finite critical speed is approached • For an understeer car • the acceleration gain approaches a finite limit at speeds that are high compared to the characteristic speed.

  25. Steady State Response K<0 K=0 • Curvature Response Gcur • Is the ratio of steady-state curvature Τ • Is another parameter commonly used for evaluating the response characteristics of a vehicle Τ 1 R δf R R+ K2 gR K>0 1 R to the steer angle response 1 1 R δf Τ δf 1 R 1 U2 = = δf=L = L + K1U2 OR = L R+ K2U2 L + K2U2 L + K1U2 gR g If the curvature response w.r.t steer wheel angle is desired the above equation should be divided by the steering gear ratio Τ δf 1 R =1 the curvature become independent of U (Ackerman steering geometry) • For Neutral Steer vehicle, K=0 L • For Understeer vehicle, K>0 Τ δf the curvature decrease as U increase 1 R 1 = L + K1U2 Τ δf 1 R 1 the curvature increase as U increase • For Oversteer vehicle K<0 = L − K1U2 @ Ucritthe curvature approaches infinity or turning radius become zero & the vehicle spins out of control

  26. Steady State Response • Side Slip Angle Response Gβ • Using Cramer’s Rule Izz???S − maCαfU +?b???Cαr ∆ Izz???S − maCαfU +?b???Cαr Cαf+ Cαr U V δfS=0 U U ቤ = = mIzz??+ ma2Cαf+ b2Cαr CαfCαr?2 U2 + Izz ? + + m bCαr− aCαf U b −ma b −ma ?CαrU2 ?CαrU2 ?b???Cαr− maCαfU2 CαfCαr?2 U V δfS=0 ቤ = = = L U+ K3U m bCαr− aCαf L???Cαr L U+ U + m bCαr− aCαfU b −ma LCαrU2 L + K3U2 b −ma b −ma ?CαrU2 LCαrU2 L + K3U2 β δfS=0 β δfS=0 =1 V δfS=0 =1 ቤ = ቤ ቤ = L U+ K3U U U U2−ma L + K3U2 U2 b ay δfS=0 b ma LCαr U2 β δfS=0 ay δfS=0 LCαr But ቚ β S=0= ቤ U2− = ቤ ቤ = L + K3U2

  27. Steady State Response • Side Slip Angle Response Gβ • Is the ratio of steady-state curvature β to the steer angle response b −ma Τ δf V U V V LCαrU2 L + K3U2 U2 gR = = δf=L OR β δf L R+ K2U2 Lr + K2rU2 R+ K2 = • For Neutral Steer vehicle, K=0 U gR g β δf =b ma L2CαrU2 b L @ U=0 L− • For Understeer vehicle, K>0 βo= δf b L βo= δf b −ma LCαrU2 L + K3U2 β δf b 2L− ma β δf @ U=Uchra 2 • At very high U = UChra = 2L2Cαr b −ma LCαrU2 L + K3U2 β δf −ma LCαrK3 −aCαf bCαr− ?Cαf β δf = = • For Oversteer vehicle K<0 = aCαf aCαf− bCαr b −ma b −ma LCαrU2 L − K3U2 LCαrU2 0 β δf @ U=Ucrit = = ∞ @ Ucritthe side slip angle approaches infinity or the vehicle spins out of control

  28. Low-speed or kinematic steering • Low speed or kinematic steering • is defined as the motion of a wheeled vehicle determined by pure rolling of the wheels. • The velocities of the centers of all the wheels lie in their mid plane, i.e, the sideslip angles α are vanishingly small. • In these conditions the wheels can exert no cornering force to balance the centrifugal force due to the curvature of the trajectory • Kinematic steering is possible only if the velocity is vanishingly small • No actual steering mechanism allows to follow exactly Ackerman steering law ? ? tanδ1= tanδ2= R1−t R1+t R1+t ? 2 2 R1−t ? 2 2 Cotδ1= Cotδ2= R1+t 2− R1+t ? 2 Cotδ2− Cot?1= Cotδ2− Cot?1=? ? Ackerman steering or Ackerman geometry.

  29. Low-speed or kinematic steering • Curvature Response(curvature gain) • Using Steady State Approach Τ δf 1 R 1 Τ δf 1 R =1 Neutral Steer Vehicle = L + K1U2 L • Using Kinematic Analysis in case radius of the trajectory is large tanδ=L L Cotδ =R1 ?=cot δ2+ Cot?1 R ≈ 2= b2+ R1 b2+ L2cot2δ R1= Lcotδ R = δ 2 Rδ=1 1 R =L δ =L Τ δ 1 R =1 L δ R L • Side Slip Angle Response • Steady state approach for Neutral Steering& U=0 = 0 β δf =b ma L2CαrU2 β δf =b L− L • Using Kinematic Approach β β =bδ β δ=b b L tanδ=btanδ b tanβ = = Τ L L R1 L

  30. Steady State Response R = L + K1U2δf 1 R= 1 • Curvature Response Τ δf 1 R 1 1 R= 1 = Τ δf δf 1 R 1 δf L + K2U2 L + K2U2 = L + K1U2 L + K1U2 g g U r δf U U r δf U = r = δf • Yaw Velocity Response r = L + K1U2δf = L + K2U2 L + K2U2 L + K1U2 g g U2 U2 U2 U2 ay δf ay δf r = δf = ay= L + K1U2δf = • Lateral Acceleration Response L + K2U2 L + K2U2 L + K1U2 g g b −ma b −ma LCαrU2 L + K2U2 b −ma b −ma LCαrU2 L + K2U2 LCαrU2 L + K1U2 LCαrU2 L + K1U2 β δf β δf β = δf = = β = δf • Side Slip Angle Response g g b ma LCαr β = ay U2− δf

  31. Steady State Response =Lg K2 L L Lg −K2 2 2 2 UChar 2 UCrit = UChar = UCrit = −K1 K1 Under Steer Over Steer Response Neutral Steer Generally @ Char. Speed Generally @ Crit. Speed L − K1U2 R δf 0 L + K1U2 L 2L Turning Radius 1 1 2L 1 1 L 1 0= ∞ Τ δf r δf ay δf 1 R Curvature L + K1U2 L − K1U2 U L U2 L U U 2L U2 2L U U 0= ∞ U2 0= ∞ Yaw L + K1U2 L − K1U2 U2 U2 Lateral acceleration L + K1U2 b −ma LCαrU2 L + K1U2 L − K1U2 b −ma LCαrU2 L − K1U2 b L− ma L2CαrU2 β δf b 2L− ma b −ma 2L2CαrU2 LCαrU2 0 Side Slip Angle = ∞

  32. Steady State Response • Summary • An extremely stable car • may not respond to steering inputs extremely well in the sense of generating lateral acceleration or yaw rate particularly at high speeds. • may not be able to swerve to avoid an obstacle & hard for driver to control at high speed • a car that is very responsive to steering inputs • may become unstable at high speed. • Neither extreme situation is desirable. • a neutral or near-neutral steer automobile represents a sound compromise in steering response • No fear from an oversteering vehicle • if the critical speed is significantly higher than the vehicle’s top speed. • understeer car will remain responsive to steering inputs • as long as the char speed is not too low relative to the car’s normal operation speed range. • Automobile designers have a number of means to adjust the handling of a car • The location of the center of gravity, • the spring rates at front and rear, • antisway bar rates at the front and rear, • the tire sizes and pressures at front and rear, and even • the shock absorber characteristics.

  33. Neutral Steer Point (NSP) • The vehicle steer characteristics, determined by the sign of have a fundamental influence on vehicle steady-state cornering • Angular Equation of motion ( ሶ r) a2Cαf+b2Cαr U r + aCαfδf • For steady state ( ሶ r = 0) and zero steering input(δf=0) but perturbed vehicle motion bCαr− ?Cαf • Izzሶ r = − aCαf− bCαrβ − a2Cαf+ b2Cαr U a2Cαf+ b2Cαr U 0 = − aCαf− bCαrβ − r bCαr− aCαfβ = r • If β is positive • If bCαr− aCαf< 0, then a moment that produces -ve r acts around the C.G. • If bCαr− aCαf>0, then a moment that produces +ve r acts around the C.G • If bCαr− aCαf= 0, then there is no moment • In other words, • If bCαr− aCαf< 0, then Fy= Fyf+Fyracts in front of C.G. • If bCαr− aCαf> 0, then Fy= Fyf+Fyracts behind of C.G • If bCαr− aCαf= 0, then Fy= Fyf+Fyracts at C.G • Neutral Steer Point (NSP) • Is the point where the resultant force (Fy) is acting r

  34. Static Margin(SM) • If the center of gravity has a side-slip angle of β and the lateral force acting on the front and rear wheels will be • Fy= Fyf+Fyr= −Cαfαf− Cαrαr=−Cαfβ − δf − Cαrβ − δr = −Cαfβ − Cαrβ • Fy= −Cαfβ − Cαrβ • The moment by Fyfand Fyraround NSP • Taking the distance of NSP from the vehicle C.G as e as shown in Fig, =0 =0 -Cαfβ a + e = −Cαrβ b − e Fyfa + e = Fyrb − e Cαf a + e = Cαr b − e e Cαf+ Cαr = bCαr− aCαf NSP is behind of the C.G. when (bCαr−aCαf)is -ve, NSP is in front of the C.G. when (bCαr−aCαf) is +ve NSP coincides with the C.G. when (bCαr− aCαf) =0 e =bCαr− aCαf Cαf+ Cαr e • Static Margin (SM) • Is the dimensionless quantity of the ratio of e to wheelbase, L SM =e L=bCαr− aCαf L Cαf+ Cαr

  35. Static Margin(SM) SM =e L=bCαr− aCαf L Cαf+ Cαr • Substitute b=L-a SM =e L=LCαr− a Cαf+ Cαr L Cαf+ Cαr Cαr −a SM = −a Cαr = L+ Cαf+ Cαr L Cαf+ Cαr • The vehicle steer characteristic is defined using the SM • the quantity of bCαr− aCαfwhich determines the vehicle steer characteristics, can be rewritten in the form of static margin, SM • when SM > 0 → the C.G is behind the NSP→ αf> αr, then the vehicle is Under Steer, • when SM = 0 → The C.G coincides with NSP → αf= αrthen the vehicle Neutral Steer • when SM < 0→ the C.G is in front of the NSP→ αf< αr, then the vehicle is Oversteer • The stability limit velocity, Ucritis expressed using the term SM, L L 1 LCαfCαr m Cαf+ Cαr 1 LCαfCαr m Cαf+ Cαr 2 UCrit = = − = − = − m bCαr− aCαf LCαfCαr bCαr− aCαf L Cαf+ Cαr −K1 SM 1 LCαfCαr m Cαf+ Cαr Ucrit= − SM

  36. Vehicle Dynamics (MV6131) End of Lecture-17 Department of Motor Vehicle Engineering Defence University, College of Engineering 2019-20 A.Y Semester I

  37. Testing of Handling Characteristics • To measure the handling behavior of a road vehicle under steady-state conditions, various types of test can be conducted on a skid pad • Three types of test can be distinguished: • Constant radius test, • Constant forward speed test, • Constant steer angle test. • During the tests, • the steer angle, forward speed, and yaw velocity (or lateral acceleration) of the vehicle are usually measured • Yaw velocity (r)can • be measured by a rate-gyro or • determined by the lateral acceleration divided by vehicle forward speed. • Lateral acceleration • can be measured by an accelerometer or • determined by the yaw velocity multiplied by vehicle forward speed r =U ay= rU R • Based on the relationship between the steer angle and the lateral acceleration or yaw velocity obtained from tests, the handling characteristics of the vehicle can be evaluated.

  38. Testing of Handling Characteristics • Constant Radius Test Constant Radius Test • the vehicle is driven along a curve with a constant radius at various speeds. • The steer angle δfor the angle of the steering wheel • required to maintain the vehicle on course at various forward speeds is measured • The lateral acceleration( ay) • corresponding to the steering angle is measured • deduced from the vehicle forward speed and the known turning radius. • The handling behavior of the vehicle can then be determined from the slope of the steer angle-lateral acceleration curve. ay= rU=U2 R For a constant turning radius, the slope of the curve is given by d δf K2= U2 gR ay g δf−L δf=L ay g R= K2 R+ K2 d the slope of the curve represents the value of the understeer coefficient. • • • (K2=0) - (K2>0) - (K2<0) - Zero slope +ve slope -ve slope for Neutral Steer Vehicle for under steer vehicle for Over steer vehicle

  39. Testing of Handling Characteristics • Constant Speed Test Constant Speed Test • The vehicle is driven at a constant forward speed at various turning radii • The steer angle and the lateral acceleration or yaw velocity are measured. • The handling behavior of the vehicle can then be determined from the slope of the steer angle-lateral acceleration curve • for a constant speed turn, the slope of the curve is given by dδf ay =gL U2+ K2 U2 gR δf=L R+ K2 d • If the vehicle is neutral steer, • K2=0 and the slope will be a constant of gL g U2 • If the vehicle is understeer • the slope will be greater than that for the neutral steer • the value of understeer coefficient K2is positive • If the vehicle is oversteer • the slope of the curve is less than that for the neutral steer • the value of understeer coefficient K2is negative, • When the slope is zero gL U2+ K2→ U = the oversteer vehicle is operating at the critical speed, and that the vehicle is at the onset of directional instability −gL • 0 = K2= Ucrit This indicate that •

  40. Testing of Handling Characteristics • Constant Steer Angle Test Constant Steer Angle Test • the vehicle is driven with a fixed steering wheel angle at various forward speeds. • The lateral accelerations at various speeds are measured. • From the test results, the curvature 1/R, which can be calculated from the measured lateral acceleration and forward speed by ay=U2 R→ ay dδf ay U2 gR 1 R=U2 ay g δf=L δf=L K2= R+ K2 R+ K2 d g • The handling behavior can then be determined by the slope of the curvature-lateral acceleration curve • If the vehicle is Neutral steer, • the understeer coefficient K2will be zero, • the slope of the curve is zero. represented by a horizontal line, • If the Vehicle is understeer • the slope of the curve is negative, • the understeer coefficient K2is positive, • If the Vehicle is oversteer • the slope of the curve is positive, • the understeer coefficient K2is negative.

  41. Testing of Handling Characteristics • Summary • The constant radius test is the simplest and requires little instrumentation. • The steer angle of the front tire (or the steering wheel angle) and • Forward speed are the only essential parameters to be measured during the test, • the steady-state lateral acceleration can be deduced from vehicle forward speed and the given turning radius. • The constant speed test is more representative of the actual road behavior of a vehicle than the constant radius test • as the driver usually maintains a more or less constant speed in a turn and turns the steering wheel by the required amount to negotiate the curve. • The constant steer angle test, on the other hand, is easy to execute. • Both the constant speed and constant steer angle tests would require, the measurement of the lateral acceleration or yaw velocity

  42. Vehicle Stability Control (VSC) • In a turning maneuver where tire forces are approaching or at the limit of road adhesion, the vehicle may deviate significantly from its intended direction of motion and path • ABS and TCS are incapable of actively controlling the directional behavior of the vehicle • Antilock Brake System (ABS) • can prevent the wheel from locking during braking and the • Traction Control System (TCS) • can prevent the driven wheel from spinning during acceleration, • With the increasing emphasis on road vehicle safety, a variety of systems designed for vehicle directional control have emerged. • Vehicle Dynamics Control (VDC (ASTC ASTC), Direct Yaw Moment Control (DYC VDC), Electronic Stability Program (ESP DYC), ESP), Vehicle Stability Assist (VSA VSA), Advanced Stability Control • They are intended to enhance vehicle stability and tracking performance to enhance vehicle stability and tracking performance in all operating conditions, including severe turning maneuvers. • The vehicle stability control system is normally integrated with • the brake system and powertrain and usually shares components with the antilock brake system and traction control system

  43. Vehicle Stability Control (VSC) • Basic Operating Principles • The intended (nominal or desired) motion of the vehicle is first established from • Driver's input, such as; steering wheel angle, accelerator pedal position, and brake pressure • Vehicle motion variables, such as angular speed of the wheels. • The actual directional behavior of the vehicle is deduced from • measured motion variables, such as yaw rate, lateral and longitudinal accelerations of the vehicle • The intended motion is then compared with the actual behavior of the vehicle. • If a noticeable difference between them is found, the vehicle stability control system will regulate • The brake pressure on select tires and/or • Reduce engine torque transmitted to the driven wheels, • So as to generate a restoring yaw moment to minimize the deviation from the intended motion

  44. Vehicle Stability Control (VSC) • Basic Operating Principles • basic parameters used for identifying the directional behavior of the vehicle are Sensing Driver inputs & vehicle Motion • Vehicle yaw rate r and • vehicle sideslip angle β Estimating intended Determining Actual (nominal) Vehicle behavior Vehicle behavior • Continuously monitored parameters, • the steering wheel angle, • accelerator pedal position, Comparing the intended & actual vehicle behavior • brake pressure, • wheel angular speeds, Estimating the restoring yaw moment • lateral acceleration /Yaw velo. Required to minimize the difference between • The nominal values of yaw rate and sideslip angle, which represent the intended motion of the vehicle, are derived from The intended and actual vehicle behavior • the steering wheel angle Regulating individual wheel brake pressure and /or engine output torque • accelerator pedal position • brake pressure, and • estimated vehicle longitudinal speed

  45. Vehicle Stability Control (VSC) • The estimated longitudinal speed • may be derived from the measured wheel angular speeds and steering wheel angle. • The nominal yaw rate and Sideslip angle • Yaw rate may be deduced from the estimated vehicle forward speed and the turning radius, which is related to steering wheel position. • Side slip angle may be estimated from the longitudinal and lateral speed of the vehicle. • The actual value of the yaw rate and Side slip angle • The actual yaw rate is directly measured, while • The actual sideslip angle is derived from the measured yaw rate, lateral acceleration, and steering wheel position, as well as estimated values of longitudinal speed of the vehicle and braking efforts on the tires • In some systems, to determine the actual value of vehicle sideslip angle, the lateral acceleration is first integrated with respect to time to obtain vehicle lateral speed. • The sideslip angle is then determined from the lateral speed and the estimated longitudinal speed of the vehicle • After determining the nominal and actual yaw rates and vehicle sideslip angles, • they are then compared • If the differences between them are found to be higher than prescribed values, • the control unit of the vehicle stability control system will send command signals to the actuators to modulate the brake pressures on select tires and/or reduce engine torque so that a restoring yaw moment is generated to keep the vehicle on the intended direction and path.

  46. Vehicle Stability Control (VSC) • If a rear-wheel-drive vehicle on a left-hand turn • is detected at the verge of losing directional stability due to the driven rear tires sliding outward, the vehicle stability control system will then • apply appropriate brake pressure on the outside front tire outside front tire. • At the same time, the driving torque transmitted to the driven rear tires may be reduced by means of • adjusting the throttle valve or spark retardation of the engine or shutting off fuel supply to a certain number of cylinders. • This will generate a yaw moment to restore the vehicle to its intended direction and path,

  47. Vehicle Stability Control (VSC) • if a front-wheel-drive vehicle on a left-hand turn • is found to be at the verge of losing directional control due to the driven front tires sliding outward, the vehicle stability control system will then • apply appropriate brake pressure on the inside rear tire and inside rear tire and • at the same time may reduce the driving torque on the front tires. • This will generate a yaw moment to bring the vehicle back to its intended direction and path, • It should be noted that the response of vehicle stability control to brake pressure modulation on the tire is much faster than that to engine intervention.

  48. Vehicle Dynamics (MV6131) End of Lecture-18 Department of Motor Vehicle Engineering Defence University, College of Engineering 2019-20 A.Y Semester I

  49. Gillespie- Chap 6, page:209-226 Chassis System Effect on Handling • The previous Bicycle Model and equation of motions employed many assumptions and simplification to the real vehicle system. • understeer gradient derived in bicycle model was determined for vehicle response to disturbance in a straight- ahead driving or steady state cornering. Wf Cαf −Wr Cαr K3=m bCαr− aCαf L???Cαr bCαr− aCαf LCαfCαr K1= K3= K2 K2= K1= m g • Cornering Compliance • Is the ratio of axle load (Wfor Wr) and cornering stiffness (Cαf or Cαr) for front and rear wheel Front Cornering Compliance=Wf Cαf Rear Cornering Compliance=Wr Τ deg g Cαr • Any factor influencing the cornering force developed at the wheel will have a direct effect on the directional response (understeer gradient) • Suspension and steering are among the primary sources of these influences. • Numerous secondary effects to vehicle handling are caused by kinematics and compliances in the chassis system, the effects include • Roll steer, Roll camber, Suspension compliance steer, Suspension compliance camber, Steering kinematics and Steering compliance steer • Now we the effect of these parameters will be included to modify the above understeer gradient coefficient

  50. Chassis System Effect on Handling • Effect of Lateral Force Transfer • The cornering forces of pneumatic tires depend on load • In hard cornering • On the outside wheels • The wheel load increases to a higher values • On the inside wheels • The wheel load reduce to a lower values • Large slip angle will be develop either on front or rear tire to maintain the lateral force necessary for turn. • If the front tire slip more the vehicle will understeer • If the rear tire slip more the vehicle will oversteer • This mechanism is at work on both axles of all vehicles but whether it contributes to understeer or oversteer depends on the balance of the roll moments distributed on the front and rear axle. • More roll moment on the front axle contributes to understeer • More roll moment on the rear axle contributes to oversteer • Stabilizer bar (roll stiffeners) works through this mechanism • The Lateral Load transfer has 2 different mechanisms • Due to cornering force • Due to vehicle roll 2 Cα= aFz+ bFz 2nddegree Polynomial

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