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