Road loads
Tractive Resistance (TR):
When a vehicle is traveling at constant speed, its resistance to motion, termed the tractive resistance, it consists of:
Rolling resistance (RR): This depends mainly upon the nature of the ground, the tires used, the weight of the vehicle, and to a lesser extent, the speed (the last variation is usually ignored).
Air resistance (AR): Air resistance (wind resistance) depends upon the size and shape of the vehicle its degree of streamlining and increases approximately as the square of the speed through the air.
Gradient resistance (GR): This is determined by the steepness of the hill and the weight of the vehicle, which must, in effect, be lifted from the bottom to top.
Air Resistance (AR)
Aerodynamics effects on vehicle functions:
Air forces and Moments:
Directional Control (Driving Safety) [pitching, yaw, and rolling moments]left and cross wind force.
Driving Performance and Fuel consumption [air resistance] tangential forces.
Air flow behavior, and pressure distribution:
Comfort [wind noises, passenger compartment ventilation, dirty interior].
Clear Visibility [Dirty windows and lamps, Prevention of windshield misting].
Auxiliary equipment functions [engine cooling, engine compartment ventilation, brake cooling, air conditioning].

Aerodynamic forces and moments:
Aerodynamic resistance, aerodynamic lift, and aerodynamic pitching moment have significant effects on vehicle performance at moderate and higher speeds. The increase emphasis on fuel economy and on energy conservation has stimulated new interest in improving the aerodynamic performance of road vehicles.
As a result of the air stream interacting with the vehicle, forces and moments are imposed. These may be defined systemically as the three forces and three moments, acting about the principal axes of the car. The reactions are as follows:
Direction 
Force 
Moment 
Longitudinal (xaxis, positive rearward) 
Drag 
Rolling moment 
Lateral (yaxis, positive to the right) 
Side force 
Pitching moment 
Vertical (zaxis, positive upward) 
Lift 
Yawing moment 

Aerodynamics drag:
Drag is the largest and most important aerodynamic force encountered by passenger cars at normal highway speeds. The overall drag on a vehicle derives from contributions of many sources.
Approximately 65% (.275/.42) of the drag arises from the body (fore body, after body, underbody and skin friction). The major contributor is the after body because the drag produced by the separation zone at the rear.
Effect of different factors on the car drag:
Fore body and after body (depend upon the car shape) 55%  60%
Skin friction (depend upon the car finish and the long of the body) 8%l0%
Flow resistance in the front grille and radiator 10%  15%
Air hits the outer components and car openings (luggage rack, side mirror and windows)
8%  12%
The air force equation is usually expressed in the following form (semiempirical formula):
AR=(1/2) ρ C_{d }A_{f} v^{2}
where:
AR = air resistance [N]
ρ = air density [kg/m^{3}]
A = the car frontal area [m^{2}]
v = car speed [m/s]
C_{d} = the coefficient of aerodynamic resistance [N/[(kg/m^{3}) x m^{2} x (m/s^{2})], [N/(kg m/s^{2}), [dimensionless]
OR
AR = [1 / (2 x 3.6 x 3.6)] ρ C_{d} A_{f} v^{2} = 0.0386 ρ C_{d} A_{f} v^{2}
where:
v = car speed [km/h]
the term [(1/2 ρ v^{2} ] in the above equation is the dynamic pressure of the air [(kg/m^{3}) (m^{3}/s^{2})], [N/m^{2}]. The drag properties of the car sometimes characterized by the value of (C_{d} A_{f}).
Air density (ρ):
The air density is usually equal to 1.202 kg/m^{3} and it depends on the air temperature and pressure:
= 1.225 (p_{r} / 101.325) (288.16/(273.16+Tr)) [kg/m^{3}]
= 3.48 (p_{r} / (273. 16+Tr))
where:
Pr = atmospheric pressure [kPa]
T_{r} = air temperature [degree Celsius °C]
Coefficient of aerodynamic resistance (C_{d}):
C_{d} is the coefficient of aerodynamic resistance that represents the combined effects of form drag, skin friction, and resistance due to air flow through the radiator and interior of the vehicle.
Typical values of C_{d}

Factors affecting the value of C_{d}:

DRAG COEFFICIENT COMPONENT 
TYPCAL VALUE 
Fore body 
0.05 
After body 
0.14 
Under body 
0.05 
Skin Friction 
0.025 
Total Body Drag 
0.275 
Wheel and wheel wells 
0.09 
Drip rails 
0.01 
Window recesses 
0.01 
External mirrors 
0.01 
Total Protuberance Drag 
0.12 
Cooling system 
0.025 
Total Internal Drag 
0.025 
Overall Total Drag 
0.42 
VEHICLE OF THE 1980s Cars Vans Pickup trucks 
0.30 0.35 0.330.35 0.42 0.46 
Main source of drag on a passenger car. 




Modification 
DC_{d }% 

Lower the car height by 30 mm 

5 
Wheel 
1 
3 
Wide tires 
+2 
+4 
Under body panels 
1 
7 
Moving head lamps (open) 
+3 
+10 
Side mirror 
+2 
+5 
Flow through radiator 
+4 
+14 
Opened windows 

+5 
Opened sunroof 

+2 
Luggage rake 

+40 
The relative velocity between car and air (v):
Still air: v = V_{car}
Head wind: v = v_{car} + v_{wind}, and v^{2 }= (v_{car} + v_{wind})^{2}
Tail wind: v = v_{car}  v_{wind}, and v^{2 }= (v_{car}  v_{wind})^{2}
* (a + b)^{2} ≠ a^{2} + b^{2}
The. car frontal area (A_{f}): A_{f} = the cross section area of the car A_{f} ≈ 0.8 (B x H)


* The power needed to overcome the air resistance = AR x v = constant x v^{3} That means double the car speeds will double the power needed to overcome the air resistance 8 times.
Rolling Resistance (RR)
The other major vehicle resistance force on level ground is the rolling resistance of the tires. At low speeds on hard pavement, the rolling resistance is the primary motion resistance force. In fact, aerodynamic resistance becomes equal to the rolling resistance only at speeds of 5060 mph (80100 km/h). For offhighway, level ground operation; the rolling resistance is the only significant retardation force.
While other resistances act only under certain conditions of motion, rolling resistance is present from the instant the wheels begin to turn. Rolling resistance, in addition, has another undesirable property, a large part of the power expended in a rolling wheel is converted into heat within the tire. The consequent temperature rise reduces both the abrasion resistance and the flexure fatigue strength of the tire material, and may become the limiting factor in tire performance.
There are at least seven mechanisms responsible for rolling resistance:
1) Energy loss due to deflection of the tire sidewall near the contact area
2) Energy loss due to deflection of the tread elements
3) Scrubbing in the contact patch
4) Tire slip in the longitudinal and lateral directions
5) Deflection of the road surface
6) Air drag on the inside and outside of the tire
7) Energy loss on bumps
Considering the vehicle as a whole, the total rolling resistance is the sum of the resistance from all the wheels:
RR = f_{r} w = f_{r} mg
where:
f_{r} = rolling resistance coefficient (dimensionless)
w= weight of the vehicle (N)
Factors affecting rolling resistance
The value of the coefficient of rolling resistance affected by the structure of the ground material, composition of the rubber, design elements of the tire, temperature, tire inflation pressure etc.
Tire temperature
In a typical situation where a tire begins rolling from a cold condition, the temperature will rise and the rolling resistance will diminish over a first period of travel.


Tire inflation pressure
To large extent, the tire inflation pressure determines the tire elasticity and, in combination with the load, determines the deflection in the sidewalls and contact region. The overall effect on rolling resistance also depends on the elasticity of the ground.


On soft surfaces like sand, high inflation pressures result in increased ground penetration work and therefore higher coefficients.
On medium plastic surfaces such as dirt, the effects of inflation pressure on tire and ground approximately balance.
On hard (paved) surfaces, the coefficient decreases with higher inflation pressure since the flexure work of the tire body will greatly reduced.
Velocity
The coefficient is directly proportional to speed because of increased flexing work and vibration in the tire body, although the effect is small at moderate and low speeds and is often assumed to be constant for calculation. From the figure, it can be seen that the coefficient is constant for different tires types till approximately speed of 60 mph (120 km/h).
Tire diameter
The effect of tire diameter is negligible on hard surfaces but it increase with smaller diameter on soft surfaces.
Tire slip
The driving tires have a greater coefficient value than the driven ones because the tire slips.
Typical Coefficients values:
Vehicle Type 
Surface Type 

Concrete 
Medium Hard 
Sand 

Passenger cars 
0.015 
0.08 
0.30 
Heavy trucks 
0.012 
0.06 
0.25 
Tractors 
0.02 
0.04 
0.20 
Gradient Resistance (GR)
The gradient resistance (climbing resistance, inclined road force) depends on the angle of the road inclination and the weight of the car.
GR = w sin θ = mg sin θ
where:
w = the car weight (N) = mg
θ = the angle of road inclination
Road inclination (gradient):
The road gradient can be described as 1 in N, this description can be either:

The description (b) is not suitable incase of level road the gradient will be 1 in ∞, especially when using the computer. So, the description of the gradient will be as in (a), and the gradient will be (G = sin θ). The gradient can be written in as a percentage (G = sin θ = S%).
* For small angle (sin θ = tan θ)
* The road gradient in the highway usually does not exceed 4% and on the local roads it could reach 10%12%.
* The steepest gradient the vehicle is expected to climb (this may normally be taken as 20%, that is 1 in 5).
Total Resistance Effort (TR)
TR = AR + RR + GR

Surplus Effort (SE)
The surplus effort is the difference between the tractive effort and the total resistance.
SE = TETR = TE(AR+RR+GR)
Where:
SE = surplus effort (N)
TE = tractive effort (N)
TR = total resistance (N)
AR = air resistance (N)
RR = rolling resistance (N)
GR = gradient resistance (N)
* Maximum speed occurs when SE = 0
Inertia Force (F_{i})
Change the forward speed of the car:
To accelerate the car speed it needs a force, this force is represented by the car resistance to change its speed (inertia force). This force depends on the mass of the car and the value of the car acceleration. The surplus effort is the source of this force.
F_{i} = ma
Where:
F = inertia force (N)
m = car mass (kg)
a = car acceleration (m/s^{2})
Change the angular speed of the car rotatingparts
v = ω r
where: v = linear velocity (m/s) ω = the angular velocity (1/s) r = radius of rotation of the moving point (m) 
The relationship between the linear and angular velocity is:
a = α r
where:
a = linear acceleration (m/s)
r = radius of rotation of the moving point (m)
a = the angular acceleration (1/s^{2})
To increase the angular speed of a rotating body a torque should be exerted on this body.
T = I α
where:
T = the torque (N m)
I = the polar moment of inertia (kg m^{2})
α = the angular acceleration (1/s^{2})
The equivalent mass of rotating parts:
T = F r = (ma) r = m (α r) r = m r^{2} α
and
T = I α
then
T = m r^{2} α = I α
m r^{2} = I
the equivalent mass of rotating body (part) will be equal to:
m_{rot }= I / r^{2}
To accelerate the car we need a force F to accelerate the (car and the rotating parts). The rotating parts of the car are the engine, the flywheel, the clutch, the gearbox, the propeller shaft, the final drive, the axles, and the wheels.
m_{eq} = m_{car} + [(I_{w})(1/r_{w})^{2} + (I_{p}) η_{f} (i_{f} / r_{w})^{2} + (I_{e})_{ } η_{t }(i_{f} i_{g} / r_{w})^{2}]
where:
m_{eq} = the equivalent mass (kg)
m_{car }= the car mass (kg)
I_{w} = polar moment of inertia of the wheels and axles (≈ 2.7 kg m^{2})
I_{p} = polar moment of inertia of propeller shaft (≈ 0.05 kg m^{2})
I_{e} = polar moment of inertia of engine (≈ 0.2 kg m^{2}), flywheel and clutch (≈ 0.5 kg m^{2})
R_{w} = radius of tire (m)
η_{f} = mechanical efficiency of the final drive
η_{t} = mechanical efficiency of the transmission (η_{f} x η_{g}), where η_{g} is the gearbox mechanical efficiency.
i_{f} = final reduction ratio
i_{g }= gear box reduction ratio
* The equivalent mass at the first gear is equal approximately to double the car mass (m_{eq} = 2 m_{car}). The value of equivalent mass is decrease with higher gears.
Energy distribution for road loads:

Worked example:
(1) A vehicle mass is 1500 kg has a coefficient of rolling resistance of 0.015. The transmission has a final drive ratio 4.07: 1 and overall mechanical efficiency of 85%. If the engine develops a maximum torque of 100 Nm and the effective road wheel radius is 0.27 m. Assume the steepest gradient to be encountered is a 1 in 4. Find the gearbox ratio.
RR=f_{r} w= f_{r} mg = 0.015 x 1500 x 9.81 = 220.7 N
GR = w (1/4) = mg /4 = 1500 x 9.81/4 = 3678.8 N
TR = AR + RB. + GR = negligible + RR + GR = 220.7 + 3678.8 = 3899.5 N
TE = TR
T_{e} η_{t} i_{g} i_{f} / R_{w} = 3899.5 N
i_{g} = (3899.5 R_{w}) / (T_{e} η_{t} i_{f}) = 3899.5 x 0.27 /(100 x 0.85 x 4.07 x 0.27)
i_{g }= 3.04: 1
(2) A car is moving on top gear (i_{g} = 1) with a constant speed of (v_{car} = 108 km/h) against wind (v_{wind} = 9 km/h). The car has a frontal area (A_{f}= 1.88 m) and the coefficient of air resistance (C_{d} = 0.6). Calculate the air resistance (AR).
AR = 1/2 ρ C_{d} A_{f }[v_{car} + v_{wind}) /3.6]^{2}
AR = 0.5 x 1.2 x 0.6 x 1.88 x [(108 + 9) /3.6]^{2} = 714.88 N
Aerodynamic effects on vehicle functions:

Actual Values of air resistance parameters (Toyota):