Car Performance Formulas

 

 

Air Resistance (Air Drag) (AR):

 

where:

AR = air resistance [N]

ρ = air density [kg/m3] ≈ 1.202 kg/m3, at sea level and at 15o C

Af = car frontal area [m2] ≈ 1.2 : 3.2 m2, for small and mid size cars

Cd = coefficient of aerodynamics resistance (drag coefficient) ≈ 0.2 : 0.5 for cars

v = car relative velocity [km/h]

vcar = car velocity [km/h] (vcar = v, at stand still wind, vwind = 0)

vwind = 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]

fr = 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/s2]

 

 

 

Gradient Resistance (GR):

 

where:

GR = gradient resistance [N]

w = car weight [N] = mg [kg m/s2, (N)]

θ = gradient angle < 2.3o (highways), 5.7 : 6.9o (local roads), ≈ 11.5o (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/s2], (from 0 to 100 km/h in: 6 s (4.63 m/s2), 18 s (1.543 m/s2))

mcar = car mass [kg]

meq = equivalent mass of rotating parts [kg]

= [ Iw (1/rw)2 + Ip hf (if /rw)2 + Ie ht (if ig / rw)2]

where:

Iw = polar moment of inertia of wheels and axles ≈ 2.7 [kg m2]

Ip = polar moment of inertia of propeller shaft ≈ 0.05 [kg m2]

Ie = polar moment of inertia of engine ≈ 0.2 [kg/m2] + polar moment   of inertia of flywheel and clutch ≈ 0.5 [kg m2]

hf = mechanical efficiency of final drive

ht = mechanical efficiency of transmission system (hg x hf)

ig = gearbox reduction ratio [ig1 or ig2 or .]

if = final drive reduction ratio

rw = 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]

rw = rolling radius of the tire [m]

Nw = tire rotational speed [rpm]

Ne = engine rotation speed [rev/s, (rpm)]

ig = gearbox reduction ratio

if = final drive reduction ratio

 

 

Wheel torque (Tw):

 

where

Tw = wheel torque [Nm]

Te = engine torque [Nm]

ig = gearbox reduction ratio

if = final drive reduction ratio

ht = total transmission efficiency

Pw = wheel power [kW]

Pe = engine power [kW]

 

 

Wheel Power (Pw):

 

where:

Pw = wheel power [kW]

Tw = wheel torque [Nm]

Nw = wheel rotational speed [rev/s, (rpm)]

Ne = engine rotational speed [rev/s, (rpm)]

ht = total transmission efficiency

 

Car Tractive Effort (TE):

 

where:

TE = tractive effort [N]

Tw = wheel torque [Nm]

Te = engine torque [Nm]

Pw = wheel power [kW]

Pe = engine power [kW]

v = car velocity [km/h]

ig = gearbox reduction ratio

if = 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