Performance of vehicles

Performance of vehicles

Pierre Duysinx

Research Center in Sustainable Automotive Technologies of University of Liege

Academic Year 2010-2011

Lesson 3

Performance criteria

Outline

„ STEADY STATE PERFORMANCES

„ Maximum speed

„ Gradeability and maximum slope

„ ACCELARATION AND ELASTICITY

„ Effective mass

„ Acceleration time and distance

References

„ T. Gillespie. « Fundamentals of vehicle Dynamics », 1992, Society of Automotive Engineers (SAE)

„ R. Bosch. « Automotive Handbook ». 5th edition. 2002. Society of Automotive Engineers (SAE)

„ J.Y. Wong. « Theory of Ground Vehicles ». John Wiley & sons.

1993 (2nd edition) 2001 (3rd edition).

„ W.H. Hucho. « Aerodynamics of Road Vehicles ». 4th edition.

SAE International. 1998.

„ M. Eshani, Y. Gao & A. Emadi. Modern Electric, Hybrid Electric and Fuel Cell Vehicles. Fundamentals, Theory and Design. 2^nd Edition. CRC Press.

Max speed and gradeability

Vehicle performances

„

Vehicle performance are dominated by two major factors:

„ The maximum power availableto overcome the power dissipated by the road resistance forces

„ The capability to transmit the tractive forceto the ground (limitation of tire-road friction)

„

Performance indices are generally sorted into three categories:

„ Steady state criteria: max speed, gradebility

„ Acceleration and braking

„ Fuel consumption and emissions

Study of performances with tractive force diagrams

„ The steady state performances can be studied using the tractive forces / road resistance forces diagrams with respect to the vehicle speed

„ Stationary condition a_x=0

„ Equilibrium writes

Study of performances with tractive force diagrams

„ One generally defines the net force

„ On also can use the net force diagram to calculate

„ The maximum speed

„ The maximum slope

„ The reserve acceleration available

Study of performances with tractive force diagrams

v I

II

III

vmax

IV Fa

F_rlt Max speed of the vehicle

Study of performances with tractive force diagrams

v

I

II

III

IV

Vmax

mg sinθ3 max

mg sinθ4 max

Fa

F_rlt Max slope that can be overcome with a give gear ratio

Study of performances with tractive force diagrams

v

I

II mg sinθ1

max

mg sinθ₁^maxmax Sliding clutch

Fa

F_rlt

Maximum slope in 1st gear

Maximum speed

„ For a given vehicle, tires, and engine, calculate the transmission ratio that gives rise to the greatest maximum speed

„ Solve equality of tractive power and dissipative power of raod resistance

„ with

„ As the power of resistance forces is steadily increasing, the maximum speed is obtained when using the maximum power of the power plant

Maximum speed

„ Iterative scheme to solve the third order equation (fixed point algorithm of Picard)

„ Once the maximum speed is determined the optimal

transmission ratiocan be easily calculated by since it occurs for the nom rotation speed:

Max speed for given reduction ratio

Proues(v)

v ηPmax

vmax(long)

Présistance(v)

Rapport plus long Optimal Rapport plus court

v_max^max vmax(court)

Rapport plus court Optimal

Rapport plus long

Max speed is always reduced compared to v_max^max

Max speed for given reduction ratio

„ Solve equation of equality of tractive and resistance power, but this time, the plant rotation speed is also unknown.

„ Numerical solution using a fixed point algorithms (Picard iteration scheme)

Maximum slope

„ For the maximum slope the vehicle can climb, two criteria must be checked:

„ The maximum tractive force availableat wheel to balance the grading force

„ The maximum force that can be transmitted to the road because of tire friction and weight transfer

Longitudinal force generation in tires

„ When braking the tire rotation speed is lower than the equivalent velocity in the contact patch

„ Tread and side walls radial fibers are sheared because of the local friction between road and tire

„ Slip occurs only at the end of the contact patch

„ The resultant of the shear forces in contact patch is the longitudinal force developed by the tire

Gillespie Fig 10.6:

Deformation due brakingin contact patch

“Brush” model

Tractive force

Blundel. Fig. 5.21

Braking force

Blundel. Fig. 5.18

Longitudinal force generation in tires

„ Longitudinal slip (Slip ratio)

SR = (Ω-Ω₀)/Ω₀= Ω/Ω₀– 1 If Re is the effective rolling radius of the tire

SR = Ω R_e/V – 1

„ Free rolling SR = 0

„ Wheel blocked in braking SR = -1

„ Spinning : SR = +1

„ Free slipping: SR →∞

Longitudinal force generation in tires

Milliken. Fig. 2.17 Milliken. Fig. 2.16

Longitudinal force generation in tires

Wong Fig 1.18 Wong Fig 1.16

Longitudinal force generation in tires

Influence of road condition on the longitudinal forces

Maximum slope

Maximum slope

„ Limitation due to the friction coefficient

„ Normal forces under the front and rear wheel sets

„ At low speed (F_aero~0) and constant speed (a_x=0)

Maximum slope

FOUR-WHEEL DRIVE with electronic power split

Maximum slope

Maximum slope

FRONT WHEEL DRIVE

Maximum slope

Maximum slope

REAR WHEEL DRIVE

Maximum slope

Selection of first gear ration

„ Maximum slope to be overcome, for instance θ_max= 33%

„ Tractive force at wheels

„ Sizing of first gear ration

Gradability

Characteristic curves a 3-gear vehicle

Wong, Fig. 3.29

Accelerations and elasticity

Acceleration performance

„ Estimation of acceleration and elasticity is based the second Newton law

„ Warning:when accelerating, one is also increasing the rotation speed of all driveline and transmission components: wheel sets, transmission shafts, gear boxes and differential, engine…

ÎEffective massto account for the kinetic energy of all

Effective mass

„ Total kinetic energy of the vehicle and its driveline :

ω_{bo ite0}

ωmot ωtrans

ωroues

i0

i /i1 0 i2

Effective mass

„ The rotation speed of the driveline components is linked to the longitudinal speed of the vehicle

„ The kinetic energy writes

Effective mass

„ One defines an effective mass

„ The calculation of the effective mass requires the knowledge of the geometry of all the driveline components

„ Empirical formulafor preliminary design of cars by Wong

Velocity as a function of time

„ We now proceed to time integration of Newton equation.

„ Time to accelerate form V₁to V₂.

Velocity as a function of time

„ Time to accelerate from V1 to V2:

„ Alternatively

Genta Fig 4.20 : 1/F as function of time

¢tV1!V2 = mef f

Z V2

V1

dv Fnet(v)

¢tV1!V2 = mef f

Z V₂ V1

V dV P^net(V)

Velocity as a function of time

„ Criteria for gear ratio up shift in order to minimize the acceleration time

„ If two curves intersects each other: change the ratio at curve intersection

„ If there is no intersection, then it is necessary to push the ratio up to maximum rotation speed

„ Lower limit is given by an infinite number of gear ratio, that is a continuous

Variables Transmission Genta Fig 4.20 : 1/F as function of time

Velocity as a function of time

„ The solution of differential

equation yields the t as a function of the velocity

„ The reciprocal function V=g(t) requires to invert the relation

„ Gear ratio changes must be taken into account

Velocity as a function of time

Distance as a function of the speed

„ The distance from start qui be evaluated as well by a second integration of the Newton equation

„ Velocity and distance are linked by the kinematic relation

„ It comes

Distance as a function of the time

„ One can eliminate the velocity V between the two curves t=f(V) and d=h(V)

„ On gets the distance as a function of the time:

Change of gear ratio

„ Criteria for changing the gear ratio.

„ Gear ratio changing is a delicate operation that needs being studied in details:

„ Gear box changing takes some time

„ Tractive force is interrupted

„ The vehicle is coating and slows down

„ For an expert driver

„ Small time to change the gear

„ Reduction of velocity given by the first order estimation

Change of gear ratio

„ When several gear change are necessary, the integration needs to be carried out by parts

„ For instance

„ with