Dynamics of vehicles with controlled limited-slip differential

Chapter 6

Conclusions and final

remarks

In this thesis the directional behaviour of vehicles fitted with electronically controlled limited-slip differential is deeply analyzed.

A critical review of the concept of understeer-oversteer in steady-state conditions is performed for vehicles with locked differential. A theoretical analysis of the steady-state cornering behaviour of rear-wheel drive vehi-cles fitted with locked differential is first presented. Results obtained for some classical manoeuvres, with either constant forward speed, steer angle or turning radius, clearly show that, in case of locked differential, the cor-nering behaviour is strongly affected by the manoeuvre. As an important consequence, the handling diagram is not unique and the understeer gradi-ent is no longer dependgradi-ent only upon the lateral acceleration, like in vehicles fitted with an open differential.

The new concept of handling surface is then introduced, as a tool for the complete description of the understeer-oversteer behaviour in steady-state conditions of a general vehicle, like, e.g., one with locked differential. It is shown that such surface is a generalization of the classical handling diagram. However, unlike the handling diagram, which in general depends on the particular manoeuvre performed, the handling surface only depends on the vehicle parameters and hence is unique. Accordingly, it is suggested here to replace the handling diagram by the handling surface for the analysis of vehicle dynamics. A new definition of understeer gradient is also proposed, which is the gradient of the handling surface, and hence a vector.

136 Conclusions and final remarks The bases for a systematic reformulation of the classical concept of understeer-oversteer have been laid, even if for a quite particular condition: the steady-state motion. However, it should be noted that a precise, theoret-ical and, therefore, quantitative definition of the understeer-oversteer char-acteristics of vehicles under general dynamic conditions, such as transient manoeuvres, has not been given yet, neither in the scientific literature. In fact, the understeer gradient, which is a quantitative index of the understeer-oversteer behaviour of vehicles, is defined only in steady-state conditions. At present, during transient manoeuvres, like those analyzed in chapter 5, for the understeer-oversteer characteristics only a qualitative description seems to be possible, which is mainly based on professional drivers opinions. Fur-ther investigations on this topic may represent a future development of the analysis presented in this thesis, in order to have a clear and unambiguous picture of the understeer-oversteer phenomenon in every driving condition.

The thesis also presents a qualitative description of an advanced control strategy, developed at the University of Pisa in cooperation with Ferrari S.p.a. for the rear electronically controlled limited-slip differential of F1 race cars, in order to optimize both handling and stability under severe driving conditions. This control logic was implemented on a F1 race car and some road tests were conducted, showing satisfactory results.

List of main symbols

a1 longitudinal distance between the centre of mass of the vehicle

and the front axle;

a2 longitudinal distance between the centre of mass of the vehicle

and the rear axle;

a_G acceleration of the centre of mass of the vehicle;

ax longitudinal acceleration;

ay lateral acceleration;

˜a_y steady-state lateral acceleration;

d height of the roll axis measured at the centre of mass;

d1 height of the roll axis measured at the front axle; d₂ height of the roll axis measured at the rear axle; e gradient of the handling surface A(u, δ);

f gradient of the handling surface Y (u, δ);

g acceleration of gravity;

h height of the centre of mass of the vehicle; i, j, k versors of axes x, y and z;

k_φ vehicle roll stiffness;

kφ1 front axle roll stiffness;

kφ2 rear axle roll stiffness;

l wheelbase of the vehicle;

m mass of the vehicle;

p normal force applied to the clutch surfaces within the electronically controlled limited-slip differential;

138 List of main symbols

s Ackermann steer angle: s = l/R;

sx longitudinal practical slip of the generic tyre;

t vector tangent to the curve representing the manoeuvre;

t1 front track; t2 rear track;

u longitudinal speed;

v lateral speed;

w gradient of the handling surface H(˜ay, s);

(x, y, z) reference frame attached to the vehicle;

(x0, y0, z0) reference frame attached to the case of the differential;

A handling surface: A(u, δ) = α1− α2;

C instantaneous centre of rotation of the vehicle;

C0 _{stiffness of the generic tyre at (σ = 0, ∆F}

z= 0);

Cx aerodynamic drag coefficient;

F1, F2 forces which the differential pinion transmits to the side gears; Fa longitudinal component of the aerodynamic drag force;

F_t total tangential force acting on the generic tyre;

Ft magnitude of the total tangential force Ft;

Fx longitudinal force acting on the generic tyre;

F_x1 longitudinal force acting on the front axle;

Fx2 longitudinal force acting on the rear axle;

Fy lateral force acting on the generic tyre;

F_y1 lateral force acting on the front axle;

Fy2 lateral force acting on the rear axle;

Fz vertical load acting on the generic tyre;

F0

z static vertical load acting on the generic tyre;

G centre of mass of the vehicle;

H handling surface: H(˜ay, s) = α1− α2; J yaw moment of inertia of the vehicle;

Ja moment of inertia of a half-axle, including the contribution

of the wheel and planetary gear;

J_c moment of inertia of the differential case;

Je moment of inertia of the flywheel;

Jw moment of inertia of a wheel;

List of main symbols 139

K0 understeer gradient of the single track model with linear tyres; Kδ understeer gradient in manoeuvres with constant steer angle;

Km understeer gradient along the generic manoeuvre;

KR understeer gradient in manoeuvres with constant turning

radius;

Ku understeer gradient in manoeuvres with constant forward

speed;

Mb brake torque applied to the generic wheel;

Mz1 yaw moment of the front axle;

Mz2 yaw moment of the rear axle;

O centre of the differential;

R distance between the instantaneous centre of rotation of the vehicle and the longitudinal vehicle axis;

R1 rolling radius of the front tyres; R2 rolling radius of the rear tyres; S frontal area of the vehicle;

Te engine torque;

T_f friction torque between the discs of the clutch within the electronically controlled limited-slip differential;

Tfmax maximum allowed value for the friction torque Tf;

VG speed of the centre of mass of the vehicle;

W₁ static vertical load acting on the front axle;

W2 static vertical load acting on the rear axle; Y handling surface: Y (u, δ) = ˜ay;

α slip angle of the generic tyre;

α₁ slip angle of the front axle;

α2 slip angle of the rear axle; β slip angle of the vehicle;

δ front steer angle;

ε angle between vectors f and e;

ζ angle between vectors e and t;

θ angle between vectors f and t;

µ friction coefficient of the generic tyre;

ρ air density;

140 List of main symbols

σ magnitude of the theoretical slip σ;

σx longitudinal theoretical slip of the generic tyre;

σy lateral theoretical slip of the generic tyre;

τ gear ratio between the flywheel and the differential case;

φ roll angle of the sprung mass;

χ non-dimensional quantity related to the slips of rear tyres;

ψ gas pedal position;

∆Fz vertical load variation of the generic tyre with respect to the

static load F0

z;

∆Fz1 lateral load transfer of the front axle;

∆Fz2 lateral load transfer of the rear axle;

∆T difference between the torques transmitted to the output shafts of the differential;

∆Testim estimate of the differential torque ∆T ;

∆Tmax maximum allowed value for the differential torque ∆T ;

Φ1 cornering stiffness of the front axle;

Φ₂ cornering stiffness of the rear axle; Ω vector angular velocity of the vehicle;

Ω angular velocity of the rim of the generic tyre; Ω_c angular velocity of the differential case; Ωe angular velocity of the engine.

Bibliography

[1] Abe, M., 1986, A theoretical analysis on vehicle cornering behaviors in acceleration and in braking. Dynamics of Vehicles on Roads and on

Tracks, Paper presented at the Proceedings of 9th IAVSD Symposium,

Linkoeping, Sweden, 24–28 June, 1985, 1–14.

[2] Abe, M., Ohkubo, N., Kano, Y., 1996, A direct yaw moment control for improving limit performance of vehicle handling - comparison and cooperation with 4WS. Vehicle System Dynamics, 25, 3–23.

[3] Amato, T., Frendo, F., Guiggiani, M., 2002, Handling behaviour of racing karts. SAE Transactions, sec. 6, 111, 2096–2103.

[4] Amato, T., Frendo, F., Guiggiani, M., 2004, Handling behaviour of vehicles with locked differential system. Paper presented at Fisita 2004, Barcelona, Spain, 23–27 May 2004.

[5] Balluchi, A., Bicchi, A., Sou`eres, P., 2001, Optimal feedback control for line tracking with a bounded-curvature vehicle. International Journal

of Control, 74(10), 1009–1019.

[6] Balluchi, A., Bicchi, A., Sou`eres, P., 2005, Path-following with a bounded-curvature vehicle: a hybrid control approach. International

Journal of Control, 78(15), 1228–1247.

[7] Cossalter, V., 1997, Cinematica e dinamica della motocicletta (Padova: Edizioni Progetto).

[8] Dixon, J.C., 1991, Tyres, Suspension and Handling (Cambridge: Cam-bridge University Press).

142 BIBLIOGRAPHY [9] Ellis, J.R., 1994, Vehicle Handling Dynamics (London: Mechanical

Engineering Publications).

[10] Frendo, F., Ghelardi, E., Leoncini, A., Vitale E., 2001, A lumped parameter model for the analysis of kart dynamics. Paper presented at Florence ATA 2001: the Role of Experimentation in the Automotive Product Development Process, 23–25 May 2001.

[11] Frendo, F., Greco, G., Guiggiani, M., 2006, Critical review of handling diagram and understeer gradient for vehicles with locked differential.

Vehicle System Dynamics, 44, 431–447.

[12] Frendo, F., Greco, G., Guiggiani, M., Sponziello, A., 2007, The han-dling surface: a new perspective in vehicle dynamics. Paper accepted for publication in the journal Vehicle System Dynamics.

[13] Furukawa, Y., Abe, M., 1997, Advanced chassis control systems for vehicle handling and active safety. Vehicle System Dynamics, 28, 59– 86.

[14] Genta, G., 1993, Meccanica dell’autoveicolo (Torino: Levrotto & Bella).

[15] Gillespie, T.D., 1992, Fundamentals of Vehicle Dynamics (Warrendale: Society of Automotive Engineers).

[16] Guiggiani, M., 2007, Dinamica del veicolo (Torino: Citt`a Studi Edi-zioni).

[17] Hall, L.C., 1986, The influence of limited slip differentials on torque distribution and steady state handling of four-wheel drive military ve-hicles. IMechE Conference Publications, 59–66.

[18] Huchtkoetter, H., Klein, H., 1996, The effect of various limited-slip differentials in front-wheel drive vehicles on handling and traction.

SAE Special Publications, 1154, Transmission and Driveline Systems Symposium: Efficiency, Components, and Materials, 131–141.

[19] International Standard ISO/TR 3888, 1975, Road vehicles - Test

pro-cedure for a severe lane-change manoeuvre (Geneva: International

BIBLIOGRAPHY 143 [20] International Standard ISO 4138, 1982, Road vehicles - Steady-state

circular test procedure (Geneva: International Organization for

Stan-dardization).

[21] International Standard ISO/CD 7401, 1995, Road vehicles - Lateral

transient response test method (Geneva: International Organization

for Standardization).

[22] International Standard ISO 7975, 1985, Road vehicles - Braking in a

turn - Open-loop test procedure (Geneva: International Organization

for Standardization).

[23] International Standard ISO 8855, 1991, Road vehicles - Vehicle

dy-namics and road holding ability - Vocabulary (Geneva: International

Organization for Standardization).

[24] International Standard ISO 9816, 1993, Passenger cars - Power-off

reactions of a vehicle in a turn - Open-loop test method (Geneva:

In-ternational Organization for Standardization).

[25] Iwata, T., Murakami, T., Tamura, M., 1991, Development and analysis of new traction control system with rear viscous LSD. SAE

Transac-tions, 100, 935–945.

[26] Koepf, P., Escher, M., Gazyakan, Ue., Oberhauser, M., 1990, Op-timization of traction and driving stability in 2-wheel drive cars by means of electro–hydraulic limited-slip differentials. Proceedings - So-ciety of Automotive Engineers, 813–818.

[27] Lorenz, K., Dietrich, C., Donges, E., 1986, Einfluss des Sperrdiffer-entials auf Traktion und Fahrverhalten von Fahrzeugen in Standard-bauweise - Teil 1. (Influence of a limited slip differential on the traction and handling of rear axle driven cars). Automobiltechnische Zeitschrift, 88(2), 95–100.

[28] Lorenz, K., Dietrich, C., Donges, E., 1986, Einfluss des Sperrdiffer-entials auf Traktion und Fahrverhalten von Fahrzeugen in Standard-bauweise - Teil 2. (Influence of a limited slip differential on the traction and handling of rear axle driven cars - Part 2). Automobiltechnische

144 BIBLIOGRAPHY [29] Mastinu, G., Battistini, E., 1993, The influence of limited-slip dif-ferentials on the stability of rear-wheel-drive automobiles running on even road with dry surface. International Journal of Vehicle Design, 14(2/3), 166–183.

[30] Milliken, W.F., Milliken D.L., 1995, Race Car Vehicle Dynamics (War-rendale: Society of Automotive Engineers).

[31] Mokhiamar, O., Abe, M., 2002, Combined lateral force and yaw mo-ment control to maximize stability as well as vehicle responsiveness during evasive maneuvering for active vehicle handling safety. Vehicle

System Dynamics, 37, 246–256.

[32] Mokhiamar, O., Abe, M., 2004, Simultaneous optimal distribution of lateral and longitudinal tyre forces for the model following control.

Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, 126(4), 753–763.

[33] Motoyama, S., Uki, H., Isoda, K., Yuasa, H., 1993, Effect of traction force distribution control on vehicle dynamics. Vehicle System

Dynam-ics, 22, 455–464.

[34] Pacejka, H.B., 1973, Simplified analysis of steady-state behaviour of motor vehicles. Vehicle System Dynamics, 2, 161–172, 173–183, 185– 204.

[35] Pacejka, H.B., Sharp, R.S., 1991, Shear force development by pneu-matic tyres in steady state conditions: a review of modelling aspects.

Vehicle System Dynamics, 20, 121–176.

[36] Pacejka, H.B., 2002, Tyre and Vehicle Dynamics (Oxford:

Butterworth–Heinemann).

[37] Shibahata, Y., Shimada, K., Tomari, T., 1993, Improvement of ve-hicle maneuverability by direct yaw moment control. Veve-hicle System

Dynamics, 22, 465–481.

[38] Turco, P., Borodani, P., Klaarenbeek F.W.G., 1997, Algorithms for a vehicle dynamics monitoring system, based on model reference struc-ture. Meccanica, 32, 449–457.

BIBLIOGRAPHY 145 [39] Winkler, C.B., 1998, Simplified analysis of the steady-state turning of

complex vehicles. Vehicle System Dynamics, 29, 141–180.

[40] Yamamoto, M., 1991, Active control strategy for improved handling and stability. SAE Transactions, 100, 1638–1648.