NPTEL – Electrical Engineering – Introduction to Hybrid and Electric Vehicles
Total driving resistance
The traction force (Ft) required at the drive wheels is made up of the driving resistance forces
and is defined as
Fresis tan ce Fr Fw Fg Fa (14)
Substituting the values of all the forces in equation 14, gives
1 dV
Fresis tan ce Mgf r cos(a ) Af CDV 2 Mg sin(a ) M (15)
2 dt
The equation 15 may be used to calculate the power required (Preq):
Preq Fresis tan ceV (16)
Dynamic equation
In the longitudinal direction, the major external forces acting on a two axle vehicle (Figure 1)
include:
the rolling resistance of the front and rear tires (Frf and Frr), which are represented by
rolling resistance moment, Trf and Trr
the aerodynamic drag (Fw)
grade climbing resistance (Fg)
acceleration resistance (Fa)
The dynamic equation of vehicle motion along the longitudinal direction is given by
Ftf Ftr Frf Frr Fw Fg Fa
dV
M (17)
dt
The first term on the right side is the total tractive effort and the second term is the total
tractive resistance. To determine the maximum tractive effort, that the tire ground contact can
support, the normal loads on the front and rear axles have to be determined. By summing the
moments of all the forces about point R (centre of the tire-ground area), the normal load on
the front axle Wf can be determined as
dV
MgLb cos(a ) Trf Trr Fw hw Mghg sin(a ) Mhg
Wf dt (18)
L
Similarly, the normal load acting on the rear axle can be expressed as
dV
MgLa cos(a ) Trf Trr Fw hw Mghg sin(a ) Mhg
Wr dt (19)
L
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In case of passenger cars, the height of the centre of application of aerodynamic resistance
(hw) is assumed to be near the height of centre of gravity of the vehicle (hg). The equation18
and 19 can be simplified as
Lb hg rdyn dV
Wf Mg cos(a ) Fw Fg Mgf r cos(a ) M (20)
L L hg dt
and
La hg rdyn dV
Wr Mg cos(a ) Fw Fg Mgf r cos(a ) M (21)
L L hg dt
Using equation 5, 17, 20 and 21 can be rewritten as
La hg rdyn
Wr Mg cos(a ) Ft Fr 1 (22)
L L hg
La hg rdyn
Wr Mg cos(a ) Ft Fr 1 (23)
L L h
g
The first term on the right hand side of equation 22 and equation 23 is the static load on the
front and the rear axles when the vehicle is at rest on level ground. The second term is the
dynamic component of the normal load.
The maximum tractive effort (Ftmax) that the tire-ground contact can support is described by
the product of the normal load and the coefficient of road adhesion (m). In Table 3, the values
of coefficient of adhesion are given for different speeds of the vehicle and different road
conditions. For the front wheel drive vehicle, Ftmax is given by
L hg rdyn
Ft max W f b Mg cos(a ) Ft max Fr 1 (24)
L
L hg
Mg cos(a ) Lb f r hg rdyn / L
Ft max (25)
1 hg / L
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For the rear wheel drive vehicle, Ftmax is given by
L hg rdyn
Ft max Wr b Mg cos(a ) Ft max Fr 1 (26)
L
L hg
Mg cos(a ) La f r hg rdyn / L
Ft max (27)
1 hg / L
Table 3: Coefficient of road adhesion
Road speed Coefficient of road Coefficient of road
[km/h] adhesion for dry adhesion for wet
roads roads
50 0.85 0.65
90 0.8 0.6
130 0.75 0.55
Adhesion, Dynamic wheel radius and slip
When the tractive effort of a vehicle exceeds the maximum tractive effort limit imposed by
the adhesive capability between the tyre and ground, the driven wheels will spin on the
ground. The adhesive capability between the tyre and the ground is the main limitation of the
vehicle performance especially when the vehicle is driven on wet, icy, snow covered or soft
soil roads.
The maximum tractive effort on the driven wheels, transferred from the power plant
through the transmission should not exceed the maximum values given by equation 25 and
equation 27. Otherwise, the driven wheels will spin on the ground, leading to vehicle
instability. The slip between the tyres and the surface can be described as:
R rdyn V
drive slip ST
R rdyn
where (28)
R angular speed of the tyre [rad / s]
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The dynamic wheel radius (rdyn) is calculated from the distance travelled per revolution of the
wheel, rolling without slip. The dynamic wheel radius is calculated from a distance travelled
at 60km/h. The increasing tyre slip at higher speeds roughly offsets the increase in rdyn. The
values of rdyn for different tyre sizes are given in table 4.
Table 4: Dynamic wheel radius of common tyre sizes
Rolling Rolling
Circumference Rdyn Tyre Circumference
Tyre Size [m] [m] Size [m] Rdyn [m]
Passenger cars Passenger cars
205/65
135 R 13 1.67 0.266 R15 1.975 0.314
195/60
145 R 13 1.725 0.275 R15 1.875 0.298
205/60 R
155 R 13 1.765 0.281 15 1.91 0.304
145/70 R
13 1.64 0.261 Light commercial vehicles
155/70
R13 1.68 0.267 185 R 14 1.985 0.316
165/70 R
13 1.73 0.275 215 R 14 2.1 0.334
175/70 R
13 1.77 0.282 205 R 14 2.037 0.324
195/75 R
175 R 14 1.935 0.308 16 2.152 0.343
205/75 R
185 R 14 1.985 0.316 16 2.2 0.35
195/70 R
14 1.94 0.309 Trucks and buses
185/65 R 12 R
14 1.82 0.29 22.5 3.302 0.526
185/60 R 315/80 R
14 1.765 0.281 22.5 3.295 0.524
195/60 R 1.8 0.286 295/80 R 3.215 0.512
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14 22.5
195/70 R 215/75 R
15 2 0.318 17.5 2.376 0.378
185/65 275/70 R
R15 1.895 0.302 22.5 2.95 0.47
195/65 305/70 R
R15 1.935 0.308 19.5 2.805 0.446
References:
[1] M. Ehsani, Modern Electric, Hybrid Electric and Fuel Cell Vehicles: Fundamentals,
Theory and Design, CRC Press, 2005
Suggested Reading:
[1] I. Husain, Electric and Hybrid Electric Vehicles, CRC Press, 2003
[2] C. C. Chan and K. T. Chau, Modern Electric Vehicle Technology, Oxford Science
Publication, 2001
[3] G. Lechner and H. Naunheimer, Automotive Transmissions: Fundamentals, Selection,
Design and Application, Springer, 1999
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