Forces Acting on a Vehicle
FORCES ACTING ON VEHICLE
AIM :-
To calculate the maximum speed of a vehicle and propulsion power.
PLATFORM USED :-
- Software – Microsoft Office Excel
SYSTEM SPECIFICATIONS :-
- Manufacturer - DELL
- Model – Vostro 3478
- Operating System - Windows 10
- Processor - Intel(R) Core(TM) i5-8250U CPU @ 1.60GHz 1.80GHz
- RAM – 8 GB
OBJECTIVE :-
- To calculate the maximum speed of the vehicle from the given tire specification.
- Prepare a excel calculator to evaluate vehicle propulsion power.
GOVERNING EQUATIONS :-
1.Diameter of the tire,
`D_t = (2* H_s) + D_r`
Where,
`D_t` - Diameter of Tire (m)
`W_s` - Section Width (m)
AR - Aspect Ratio
`H_s` - Section Height (AR% of W_s)
`D_r` - Diameter of Rim (m)
2.Angular Velocity,
`omega = frac{v}{r}*G`
Where,
`omega` - Angular Velocity (rad/s)
v - Velocity (m/s)
r - Radius (m)
G - Gear Ratio
3.Rolling Resistance Force,
`F_(rr) = mu_{rr}*m*g`
Where,
`F_(rr)` - Rolling Resistance Force (N)
`mu_(rr)` - Rolling Resistance Co-efficient
m - Mass (Kg)
g - Gravity (m/s^2)
4.Aerodynamic Drag Force,
`F_(ad) = 0.5*rho*A*C_d*v^2`
Where,
`F_(ad)` - Aerodynamic Drag Force (N)
`rho` - Air Density (Kg/m^3)
A - Area (m^2)
`C_d` - Co-efficient of Drag
v - Velocity (m/s)
5. Tractive Effort Force,
`F_(te) = F_(rr) + F_(ad)`
Where,
`F_(te)` - Traction Effort Force (N)
`F_(rr)` - Rolling Resistance Force (N)
`F_(ad)` - Aerodynamic Drag Force (N)
SOLUTIONS :-
- Maximum Speed of Electrical Scooter :-
Given Data :-
- Tire Code 1 – 90/100 R10 53J
- Tire Code 2 – 90/90 R18 51P
- Velocity – 90 Kmph = 25 m/s
- Gear Ratio – 7
To Find :-
- `v_max`of tire 1 = ?
- `v_max`of tire 2 = ?
Calculations :-
TIRE 1 :-
Extracting the data from tire specification,
90/100 R10 53J
Section Width = 90 mm = 0.09 m
Aspect Ratio = 100%
Section Height = 90*100% = 0.09 m
Rim Diameter = 10 inch = 0.254 m
Calculating the Tire Diameter,
D_t = (2* H_s) + D_r
D_t = (2 * 0.09) + 0.254
D_t = 0.18 + 0.254
D_t = 0.434 m
Calculating the Tire Radius,
R_t = D_t/2
R_t = 0.434/2
R_t = 0.217 m
Calculating the Angular Velocity,
`omega = frac{v}{r}*G`
`omega = (25/0.217)*7`
`omega = 806.45` rad/s
Calculating the Maximum Speed,
`v_{max of T1} = omega*r`
`v_{max of T1} = 806.45*0.217`
`v_{max of T1} = 174.99` m/s
`v_{max of T1} = 48.61 Kmph`
TIRE 2 :-
Extracting the data from tire specification,
90/90 R18 51P
Section Width = 90 mm = 0.09 m
Aspect Ratio = 90%
Section Height = 90*90% = 0.081 m
Rim Diameter = 18 inch = 0.4572 m
Calculating the Tire Diameter,
D_t = (2* H_s) + D_r
D_t = (2 * 0.081) + 0.4572
D_t = 0.162 + 0.4572
D_t = 0.6192 m
Calculating the Tire Radius,
R_t = D_t/2
R_t = 0.6192/2
R_t = 0.310 m
Calculating the Angular Velocity,
`omega = frac{v}{r}*G`
`omega = (25/0.310)*7`
`omega = 564.51` rad/s
Calculating the Maximum Speed,
`v_{max of T2} = omega*r`
`v_{max of T2} = 564.51*0.310`
`v_{max of T2} = 174.99` m/s
`v_{max of T2} = 48.61 Kmph`
Results :-
a) `v_{max of T1} = 48.61 Kmph`
b) `v_{max of T2} = 48.61 Kmph`
2) Vehicle Propulsion Power :-
The vehicle propulsion power is calculated by considering the forces acting on the vehicle. Here the hill climbing action is not considered so no hill climbing force (`F_(hc)`). So the tractive effort force becomes,
`F_(te) = F_(rr) + F_(ad)`
`F_te = (mu_{rr}*m*g) + (0.5*rho*A*C_d*v^2)`
Excel Calculator Screen Shot,
Results :-
Vehicle Propulsion Power = 5.55 KW
Limitations :-
This power is only required during the vehicle is moving in a plain surfaces in a constant speed and not in hill regions. And also this power is to move the vehicle and the power required during acceleration is different.
3) Assuming that tire pressure reduction by 15% results in increase of rolling resistance effect by two times and evaluating the difference in total traction power keeping all other parameters same :-
Then the co-efficient of rolling resistance becomes,
`mu_{rr} = 0.02`
`mu_{rr} = 0.02*2`
`mu_{rr} = 0.04`
Excel Calculator Screen Shot,
Results :-
Difference in Power = 1.94 KW
4) Gradeability, Average Speed and Forces acting on the Vehicle :-
The vehicle considered here is Range Rover Sport.
Vehicle Specifications,
- i) Vehicle Type – PHEV (Plug-in Hybrid Electric Vehicle)
- ii) Maximum Speed – 220 Kmph
Gradeability,
Gradeability of a car is that the ability to climb slope. It is measured in degrees or percentage.
Gradeability = `45°`
Average Speed,
Average Speed = 40 Kmph
Forces Acting on car,
- Rolling Resistance Force (F_rr)
- Aerodynamic Drag Force (F_ad)
- Hill Climbing Force (F_hc)
- Linear Acceleration Force (F_la)
- Angular Acceleration Force (`F_omega`)
LINK TO THE EXCEL CALCULATOR :-
REFERNCES :-