FORCES ACTING ON A VEHICLE
AIM:
TO CALCULATE MAXIMUM SPEED OF THE MOTOR AND ITS PROPULSION POWER AND TRACTION POWER FROM THE GIVEN DATA
OBJECTIVE:
1) Calculate the maximum speed of the motor from the given Tire data.
2) Create an excel calculator to find out the propulsion power of the vehicle based on given inputs and outputs.
3) Calculate traction power difference when the tire pressure is reduced by 15% which leads to an increase in the rolling resistance.
4) Range rover sport dragon specification details
GOVERNING EQUATIONS:
1) Diameter of Tire
Tire Diameter = 2*(Tire Width/2.54)*(Aspect Ratio/100) + Rim Diameter
`Dt = (2 . Hs) + Dr`
Where,
Dt - Dia. of the tire (m)
Ws - Section width (m)
AS – Aspect Ration
Hs - Section Height (m) (Hs = Ws * AS/100)
Dr - Dia. of the rim (m)
2) Angular Velocity
`omega = (v/r)*G`
Where,
`omega` - Angular Velocity (rad/sec)
`v`- Velocity (m/s)
r - Radius (m)
G - Gear Ratio
3) Rolling Resistance
`Frr = murr *m*g`
Where,
Frr - Rolling Resistance Force (N)
`murr` - Rolling Resistance Coefficient
m - Mass (Kg)
g - Gravity (9.81)
4) Aerodynamics Drag Force
`Fad = 0.5*rho*A*Cd*v^2`
Where,
Fad - Aerodynamic Drag Force (N)
`rho` - Air Density (Kg/m^3)
Cd - Coefficient of Drag
A - Area (m^2)
v - Velocity (m/s)
5) Tractive Force
`Ft = Frr + Fad`
Where,
Ft - Traction Force (N)
Frr - Rolling Resistance Force (N)
Fad - Aerodynamic Drag Force (N)
GIVEN DATA:
a. Velocity - 90 Kmph = 25 m/s
b. Code1 - 90/100 R10 53J
c. Code2 - 90/90 R18 51P
d. Gear Ratio (G) - 7
1) CALCULATIONS:
READING THE DATA INSIDE GIVEN CODE1:
CODE1: 90/100 R10 53J
Section Width (Ws) - 90mm = 0.09 m
Aspect Ratio - 100%
Section Height (Hs) - 0.09 m
Rim Diameter - 10 inch = 0.254 m
Load Index - 53 N
Speed Symbol - J
Calculating Tire Diameter:
Dt = 2*Hs + Dr
Dt = 2*(0.09) + 0.254
Dt = 0.434 m
Calculating Tire Radius:
Rt = Dt/2
Rt = 0.217 m
Calculating angular velocity:
`omega = (v/r)*G`
`omega` = (25/0.217)*7
`omega` = 806.45 rad/sec
Calculating max. speed of the motor:
`vt1`max. = `omega * r` (r = Rt)
`vt1`max. = 806.45*0.217
`vt1`max. = 174.99 m/s
`vt1`max. = 629.964 Kmph
READING THE DATA INSIDE GIVEN CODE2:
CODE2: 90/90 R18 51P
Section Width (Ws) - 90 mm = 0.09 m
Aspect Ratio - 90%
Section Height (Hs) - 0.081 m
Rim Diameter - 18 inch = 0.4572
Load Index - 51 N
Speed Symbol - P
Calculating Tire Diameter:
Dt = 2*Hs + Dr
Dt = 2*(0.081) + 0.4572
Dt = 0.6192 m
Calculating Tire Radius:
Rt = Dt/2
Rt = 0.3096 m
Calculating angular velocity:
`omega = (v/r)*G`
`omega` = (25/0.3096)*7
`omega`= 565.25 rad/sec
Calculating max. speed of the motor:
`vt2`max. = `omega * r` (r = Rt)
`vt2`max. = 565.25*0.3096
`vt2`max. = 175 m/s
`vt2`max. = 630 Kmph
Result:
`vt1`max. = 629.964 Kmph
`vt2`max. = 630 Kmph
The 10% reduction in aspect ratio and increase in the size of rim diameter does not lead to a significant change max. speed of the motor.
2) PROPULSION POWER:
Now, by considering all the forces acting on the vehicle, we will calculate propulsion power with the help excel calculator.
Assuming there are no hill climbing actions, (Fhc = 0)
Thus, the Tractive Force:
`Ft = Frr + Fad`
`Ft` = `Frr = murr *m*g` + `Fad = 0.5*rho*A*Cd*v^2`
Calculation in the excel sheet,
Fig.1. Excel sheet for Vehicle1
Result:
Output - Total Propulsion Power = 5.55 KW
Limitations:
Our assumption is based only on the movement of the car on the plain surfaces, we have neglected hill-climbing force and all the data considered are ideal parameters for a vehicle.
3) TIRE PRESSURE REDUCTION:
ASSUMING THE 15% REDUCTION IN TIRE PRESSURE LEADING TO TWICE THE INCREASE IN ROLLING RESISTANCE.
BY KEEPING THE OTHER PARAMETERS UNCHANGED.
DIFFERENCE OF TOTAL TRACTION POWER (PROPULSION POWER) = P_V1 - P_V2,
`murr` = 0.02
`murr` = 2*0.02
`murr` = 0.04
Fig.2. Power difference of Vehicle1 and Vehicle2
Result:
Output - Power Difference = 1.95 KW
4) RANGE ROVER SPORT DRAGON CHALLENGE
Fig.3. Range Rover Sport
Specifications:
a. Vehicle Type - Plug-in hybrid electric vehicle
b. Maximum Speed - 209 Kmph
c. Acceleration (0-100 kmph) - 8.5 sec
d. Capacity - 2997 (cc)
e. Maximum Power - 221/4000 (kW/rpm)
f. Gross Vehicle Weight - 3100 Kg
The Range Rover Sport Dragon Challenge demonstrates the powerful performance of the car at extreme conditions, climbing at Tianmen mountain with gradability angle of 45 Degree, with an average speed of 50-55 Kmph covering the 99 turns and 999 steps, completing the task in 22 mins 41 secs.
Forces acting on the Vehicle while climbing on the hill:
a. Rolling Resistance Force (Frr)
b. Aerodynamic Drag Force (Fad)
c. Hill Climbing Force (Fhc)
d. Linear Acceleration Force (Fla)
e. Angular Acceleration (F`omega`)
EXCEL SHEET: https://drive.google.com/file/d/13oGqOzgsXJbjvI5hY3nCR9wpB4qboySi/view?usp=sharing
REFERENCES:
1. https://www.landroverusa.com/experiences/stories/dragon-challenge.html