__Car Performance Formulas__

__Air Resistance (Air Drag) __(*AR)*:

_{}

where:

*AR* = air resistance [N]

*ρ* = air density [kg/m^{3}] ≈ 1.202
kg/m^{3}, at sea level and at 15^{o }C

*A _{f}* = car frontal area [m

*C _{d}* = coefficient of
aerodynamics resistance (drag coefficient) ≈ 0.2 : 0.5 for cars

*v* = car relative velocity [km/h]

*v _{car}*

*v _{wind}* = wind velocity [km/h]

* ( __+__ ), (+) with **head wind**, the wind velocity
**opposite** the car velocity direction (**against**) .

(-) with **tail wind**, the wind velocity **in
the same **direction** **as** **the car velocity (**with**).

__Rolling Resistance __(*RR*):

_{}

where:

*RR* = rolling resistance [N]

*f _{r}* = coefficient of rolling
resistance ≈ 0.015 : 0.02 (hard surface) 0.2 : 0.3 (sand)

*w* = car weight [N] ≈ 10000 : 24000 N, for
small and mid size cars

*m* = car mass [kg] ≈ 1000 : 2400 kg, for
small and mid size cars

*g* = acceleration due to gravity = 9.81
[m/s^{2}]

__Gradient Resistance__ (*GR*):

_{}

where:

*GR* = gradient resistance [N]

*w* = car weight [N] = mg [kg m/s^{2},^{
}(N)]

*θ* = gradient angle < 2.3^{o}
(highways), 5.7 : 6.9^{o}_{ }(local roads), ≈ 11.5^{o}
(max. grad)

*S* = gradient percentage [%]

*S* < 4% (highways), <10% : 12%
(local roads), ≈ 20% (maximum gradient)

*G* = gradient ratio [1: n = 1/n]

<1:25 (1/25) highways, < 1:8 (1/8) local roads, 1:5 (1/5) maximum grad.

* ( __+__ ), (+) with the car going **up hill** (**ascending**).
{resistance effort}

(-) with the car going **down hill **(**descending**).
{tractive effort}

__Inertia Resistance__ (*IR*):

_{}

where:

*IR* = inertia resistance [N]

*m* = car mass + equivalent
mass of rotating parts [kg]

*a* = car acceleration [m/s^{2}],
(from 0 to 100 km/h in: 6 s (4.63 m/s^{2}), 18 s (1.543 m/s^{2}))

*m _{car}* = car mass
[kg]

*m _{eq}* = equivalent
mass of rotating parts [kg]

= [ *I _{w} (1/r_{w})^{2}
+ I_{p} *

where:

*I _{w}* =
polar moment of inertia of wheels and axles ≈ 2.7 [kg m

*I _{p}*

*I _{e}* = polar moment
of inertia of engine ≈ 0.2 [kg/m

*h _{f}*
= mechanical efficiency of final drive

*h _{t}*
= mechanical efficiency of transmission system (h

*i _{g}* = gearbox
reduction ratio [i

*i _{f}* = final drive
reduction ratio

*r _{w}* = tire radius
[m]

* ( __+__ ), (+) with the car in **acceleration**. {tractive
resistance}

(-) with the car in **deceleration. **{tractive
effort}

__Total Resistance__ (*TR*):

_{}

where:

*TR* = total resistance [N]

*RR* = rolling resistance [N]

*AR* = air resistance [N]

*GR* = gradient resistance [N]

*IR* = inertia resistance [N]

__Car Velocity__ (*v*):

_{}

where:

*v* = car velocity [km/h]

*r _{w}* = rolling radius of the
tire [m]

*N _{w}* = tire rotational speed
[rpm]

*N _{e}*

*i _{g}*

*i _{f}* = final drive reduction
ratio

__Wheel torque__ (*T _{w}*):

_{}

where

*T _{w}* = wheel torque [Nm]

*T _{e}* = engine torque [Nm]

*i _{g} *= gearbox reduction ratio

*i _{f}* = final drive reduction
ratio

*h _{t}*
= total transmission efficiency

*P _{w}* = wheel power [kW]

*P _{e}* = engine power [kW]

__Wheel Power__ (*P _{w}*):

_{}

where:

P_{w} = wheel power [kW]

T_{w} = wheel torque [Nm]

N_{w} = wheel rotational speed [rev/s,
(rpm)]

N_{e} = engine rotational speed [rev/s,
(rpm)]

h_{t}
= total transmission efficiency

__Car Tractive Effort__ (*TE*):

_{}

where:

*TE* = tractive effort [N]

*T _{w}* = wheel torque [Nm]

*T _{e}* = engine torque [Nm]

*P _{w}* = wheel power [kW]

*P _{e}* = engine power [kW]

*v* = car velocity [km/h]

*i _{g}* = gearbox reduction ratio

*i _{f}* = final drive reduction
ratio

*v* = car velocity [km/h]

__Surplus Effort__ (*SE*):

_{}

where:

*SE* = surplus effort [N]

*TE* = tractive effort [N]

*TR* = total resistance [N]

* At maximum car speed, SE = 0