Vehicle Motion Resistances

The following resistance act on a vehicle when it starts motion (Equation of motion of a vehicle):-

1. Rolling resistance
2. Aerodynamic drag resistance
3. Grading resistance
4. Acceleration resistance

Three Types of Rotational Motions

1. Rotation About X-axis (Roll angle)
2. Rotation About Y-axis (Pitch angle)
3. Rotation About Z-axis (Yaw angle)

• Longitudinal acceleration causes pitch coupling.
• At the time of cornering, yaw angle and roll angle are is coupled simultaneously.
• When One wheel among four wheels goes over bump or sump, we will have roll angle, pitch angle coupling simultaneously.

Three Types of coordinate systems

1. Inertial coordinate system -Coordinate system of frame in which the car is observed. Inertial frame means that the frame is in rest or in uniform velocity (E.g Earth)
2. Body coordinate system -It's origin is at C.G of the car.
3. Wheel coordinate system -It's origin is at the contact point of the wheel.

Rolling resistance

The force that resists the rolling of the circular object (Tyre) on the surface

The above figure is a free body diagram of the rolling tyre.

Fr- Rolling resistance.

Rst-Static radius of tyre.  

Fz-Resultant of the pressure force distribution shown by red colour arrows.

Ew-Eccentricity of the resultant force.

Asymmetric pressure distribution

When the car is stationary, the pressure distribution on tyres is caused due to weight of car only, thus will be symmetric but when the tyres are rolling, the front motion and weight will form pressure distribution which will be asymmetric thus, shift in the position of resultant pressure by Ew

Calculation of Rolling resistance

As the resultant force Fz is not on axis so it will form a couple.The force required to pull the wheel Fr is the rolling resistance of the wheel.

Balancing the couples,

Fr * Rst=Fz * Ew

Fr=Ew/Rst * Fz

Ew/Rst=Mur  is the Rolling resistance coefficient

Mur ~ 0.01

Reading of tyre data

185/60R14

185- Width of tyre in millimeter 

14-Diameter of rim in inches

R-Radial construction of tyre
60-percentage ratio of Side wall of tyre/Width of tyre
=> 60/100=side wall/185
=>side wall=111 mm
Radius of tyre=side wall+radius of rim

R=111+177.8 (7 inches)
R=288.8

Try this yourself-175/65R14

Rolling resistance on wet road

Frw=b*(V)^n

b-width of tyre

V=velocity of tyre

n~1.6 for water of height 0.5mm

Bearing friction resistance on Tyre

Mb=Mub*R*r 

Mb-bearing resistance.

Mub-coefficient of friction in bearing.

R-resultant of y (weight on bearing) and x (longitudinal force on bearing)-sqrt(x^2+y^2)

r-radius.

Aerodynamic Resistance

Aerodynamic drag resistance

The atmosphere on earth consists of air which is a fluid. When the vehicle passes through this air the aerodynamic drag resistance acts on it due to the friction of air with vehicle surface and friction between layers of air.

Its composed of:

1.Turbulent air flow around vehicle body (85%)

2.Friction of air over vehicle body (12%)

3.Vehicle component resistance, from radiators and air vents (3%)

Fd=Cd/2 * ρ *A*(Vr)^2

Fd-drag force on the vehicle

Cd-coefficient of aerodynamic drag=0.2~0.3 for car

Vr-relative velocity between vehicle and air

A-surface area of car

ρ-mass density of air

Grade resistance

It's the resistance to the motion of the vehicle when climbing a slope or hill

As seen from figure the grade resistance

Fg=m*g*sinθ

Fg=grade resistance
m=mass of vehicle
g=gravitational acceleration-9.81m/s2
θ=angle of slope

Acceleration resistance

It's the resistance of vehicle to change in the state of rest or uniform motion (also known as inertial resistance)

By definition it's

Fi=m*a

Fi=inertial or acceleration resistance

m=mass of vehicle

a=acceleration of vehicle

but this is not true for vehicles

As vehicles have rotating parts too e.g. Engine, Drive shaft etc. So the mass correction factor λ is introduced.

Let moment of inertia and angular velocity for engine be Ie and ωe respectively

For transmission parts and clutch-Ic and ωc

Acceleration resistance due to rotating parts-F*

For rear wheel driven vehicle angular velocity of rear wheel-ω2

We know Driven wheel slip

Driven wheel slip- (Vc-Vd)/Vc

1-S=Vd/Vc

As, Vc=R*ω2

So, ω2=Vd/(1-S)R

Cardan shaft and rear axle angular velocity is not same as it passes through differential

Let 

differential ratio be id.

gear box ratio be ig.

ωc-cardan shaft angular velocity.

ωe-Engine angular velocity.

so,

ωe=ig*id*ω2

ωe=ig*id*Vd/(1-S)R

Using, kinetic energy in case of rotation(K.E)=1/2*I*(ω)^2 for engine and cardan shaft rotation

total kinetic energy=K.E(vehicle)+K.E(engine rotation)+K.E(cardan shaft rotation)

For inertial resistance F*= d/dt(∂/∂V(K.E(total))

F*= M[1+Ic(id/(1-S)R)^2/M+Ie(id*ig/(1-S)R)^2/M]*a

F*=M *λ*a