To Calculate Forces Acting on a Vehicle & Power calculations required to run a vehicle.

Solution 1 : -

Numerical calculation

• What should be the maximum speed of the motor used in an electric scooter capable to run at 90 kmph, if the fixed gear ratio is 7 and Tyre size is 90/100 R10 53J ? How to read a Tyre ?
• Recalculate the same value if Tyre code is 90/90 R18 51P ?

Speed Rating of Tyre Table :-

Load Carrying Capacity chart for Tyres:-

Given data for electric scooter

For Calculation 1 (90/100 R10 53J) (To find Max Speed of Motor)

Velocity = 90 kmph

Gear Ratio = 7

Tyre size = 90/100 R10 53J

Step 1: - Converting km/hr to m/sec, so for that we will proceed with following calculation :-

Then for 90 km/hr
= 90 * 5/18 (m/sec)
= 25 m/s

Now, we are calculating Tyre Diameter “90/100 R10”
Here, we have
Sectional Width of Tyre = 90 mm
Aspect Ratio = 100
Rim diameter (inches) = 10

So, converting

D(mm) = ('d') * 25.4
= 10 * 25.4
= 254 mm

Now for calculating Tyre Diameter we will use given formula

D = 2*W*(A.R./100) + d

Where,

W = width of Tyre
A.R. = Aspect ratio
d = Rim diameter

So,

Tyre Diameter (D) = (2*w*(AR/100)) + d
= 2 * 90 (100/100) + 254
= 180(1) + 254
= 434 mm

Now for Tyre Radius,

Radius = Diameter (D) / 2
= 434 /2
= 217 mm

= 217/1000
= 0.217 m

Another step is to calculate speed of motor angular velocity,

ω = G * V / r
= 7 * 25 / 0.217
= 806.45 rad/s

Now, converting rad/s to rpm,

we know that angular velocity (ω =(2*pi*speed/60))

Hence, speed in rpm = (60*ω/(2*pi))

speed in rpm = 7700.05 rpm

Therefore, the maximum speed of the motor is 7700.05 rpm or 806.451 rad/s for tyre size 90/100 R10 53J used by electric scooter.

For Calculation 2 (90/90 R18 51P) (To find Maximum Speed of Motor )

Velocity = 90 kmph

Gear Ratio = 7

Tyre size = 90/90 R18 51P

Step 1: - Converting km/hr to m/sec, so for that we will proceed with following calculation :-

Then for 90 km/hr
= 90 * 5/18 (m/s)
= 25 m/s

Now, we have to calculate Tyre Diameter for tyre size specification “90/90 R18”
Here,
1.Width of Tyre = 90 mm

2. Aspect Ratio = 90

3. Rim diameter (inches) = 18 mm

4. Construction = Radial

So, Here we have to convert Rim diameter (inches) into mm

D(mm) = ('d') * 25.4
= 18 * 25.4
= 457.2 mm

Now for calculating Tyre Diameter we will use given formula

D = 2*w* (A.R./100) + d

Where,

w = width of Tyre
A.R. = Aspect ratio
d = Rim diameter

So,

Tyre Diameter (D) = 2*w*(AR/100) + d
= 2 * 90*(90/100) + 457.2
= 180(0.9) + 457.2
= 162 + 457.2
= 619.2 mm

Now for Tyre Radius (R)

Radius = Diameter (D) / 2
= 619.2 /2
= 309.6 mm

= 309.6 * 10^(-3)m
= 0.3096 m

Another step is to calculate speed of motor angular velocity

ω = G * V / r
= 7 * 25 / 0.3096
= 565.25 rad/s

Now, converting rad/s to rpm,

we know that angular velocity (ω =(2*pi*speed/60))

Hence, speed in rpm = (60*ω/(2*pi))

speed in rpm = 5397 rpm

Therefore, the maximum speed of motor is 5397 rpm or 565.25 rad/s for 90/90 R18 51P used by electric scooter.

Solution 2 : -

Prepare a simple excel calculator to identify vehicle propulsion power based on given inputs and outputs. Implement formulas in cells.

Inputs: Kerb Weight (kg), Payload (kg), Coefficient of rolling resistance, Air density (kg/m3), Width (m), Height (m), Drag coefficient, Acceleration (0 to top speed in specified seconds), Hill climbing angle, Speed

Output: Total power in kW

What are the limitations of this calculation?

Formula:-

• Rolling Resistance (Frr) = μrr * Total Weight
• Total weight = kerb weight + payload weight
  μrr = Coefficient of rolling resistance
• Aerodynamic Drag Force (Fad) = ½ * rho * A * (V^2) *Cd
• Total Tractive Force (Fte) = Frr + Fad + Fhc + Facclrn
• Total Power (kW) = [ Fte *Velocity in (m/s) /1000 ]

Inputs :-

• Rolling resistance coefficient (Urr) for concrete road = 0.011
• Kerb weight = 120 kg for vehicle 1 & 140 kg for vehicle 2
• Payload weight = 90 kg for vehicle 1 & 80 kg for vehicle 2
• We have considered electric scooter as upright commuter, so Drag Coefficient (Cd) = 1.1
• Width = 0.5 m & Height = 1 m
• We assume that velocity is similar throughout the process. Hence, dv/dt=0. So, we have not considered acceleration force.
• Hill climbing angle = 10 degree considered.
• Speed (kmph) = 72 kmph

Hill Climbing Force Calculations :-

Vehicle Power Calculations :-

1. Rolling Resistance Force (N) (Frr) = Vehicle 1 is 22.66 N

= Vehicle 2 is 23.74 N

2. Aerodynamic Drag Force(N)(Fad) = Vehicle 1 is 137.5 N

= Vehicle 2 is 137.5 N

3. Hill Climbing Force (Fhc) (N) = Vehicle 1 is 357.553 N

= Vehicle 2 is 374.579 N

4. Total Tractive Power required = Vehicle 1 is 10.354 kW

= Vehicle 2 is 10.716 kW

Limitations of this calculation :-

• The coefficient of rolling resistance is constant but it depends on tire pressure & other road conditions.
• Air density is also taken as constant but it varies with respect to temperature, pressure and other factors.
• Drag coefficient also varies on the aerodynamics of the car which is also considered as constant.

Solution 3 : -

• Assuming that Tyre pressure reduction by 15% results in an increase of rolling resistance effect by two times, how much will be the difference in total traction power keeping all other parameters same ?

Given condition states that for decrease in tyre pressure by 15%, there is an increase of rolling resistance coefficent by 2 times. Hence, new Urr = 2 * old Urr. Due to this, rolling resistance forces (Frr) also doubles.

Accordingly, we have calculated tractive power as under;

The following results calculated;

1. Rolling resistance Force (Frr) (N):-

• Vehicle 1 = 45.322 N
• Vehicle 2 = 47.480 N

2. Aerodynamic Drag Force (Fad) (N) :-

• Vehicle 1 = 137.5 N
• Vehicle 2 = 137.5 N

3. Hill Climbing Force (Fhc) (N) :-

• Vehicle 1 = 357.553.5 N
• Vehicle 2 = 374.579 N

4. Total Tractive Power (Pte) :-

Vehicle 1 = 10.808 kW
Vehicle 2 = 11.191 kW

Tractive Power Difference :-

While going through it, we observe that for given hill climb condition, total tractive power required for vehicle 1 is 10.354 kW and if the tyre pressure drops by 15%, then the same vehicle will require the tractive power of 10.808 kW. That means tractive power difference for vehicle 1 is 0.453 kW which states that vehicle 1 will require more 0.453 kW of tractive power if tyre pressure drops by 15%.

Also, total tractive power required for vehicle 2 is 10.716 kW and if the tyre pressure drops by 15%, then the same vehicle will require the tractive power of 11.191 kW. That means tractive power difference for vehicle 2 is 0.475 kW which states that vehicle 2 will require more 0.475 kW of tractive power if tyre pressure drops by 15%.

Solution 4 : -

• See the video of Range rover sport dragon challenge. How much is the gradeability? What was the average speed? Which
  are the forces acting on car ?

Video Link of Range Rover Sports Dragon Challenge :

The challenge consisted of climbing 999 steps at a 45 degrees angle. It began at the bottom of the Tianmen Mountain Road, which has a stairway and 99 dizzying turns. It took Range Rover Sports 22 minutes 41 second to cover 11 km (7 mi) road with 99 turns and 999 Steps to complete and reaches the top of the mountain Tianmen cave (Heaven's gate), a natural arch in the mountain of a height of 131.5 m (431.4 ft).

Gradeability :-

1.Gradeability is well-defined as the highest grade a vehicle can go up maintaining a particular speed. In other words, it is the ability of vehicle to climb on slope at given constant speed.

Gradeability is dependent on:

• Tractive force
• Rolling resistance
• Adhesion (friction)
• Engine torque
• Transmission, transfer case, final drive and tyre ratio
• Overall combined mass properties

Greadeability can be defined in two ways;

1. Degree :- theta = tan inverse(rise/run)
2. Percentage :-

To convert, use this angle degrees to the percentage of slope formula:

Percentage = [ Tan ( Degrees ) ] x 100

For Range Rover Hill Riding an angle of 45 degrees would be:

Percentage of Slope = [ Tan ( 45 ) ] x 100

After calculating, it gives a slope percentage of 100.00 %

This show that gradeability of range rover sport is 100% or 45 degree angle or having 1:1 gradient ratio.

GRADEABILITY CHART :-

AVERAGE SPEED OF RANGE ROVER SPORTS :-

During 99 turns, average speed of = 85-90 kmph

During 999 steps, average speed of car = 35-40 kmph

Total time taken to complete the challenge = 22 min 41 sec

Forces Acting on Range Rover Sports During Challenges:-

1. Traction force (Fte)
2. Aerodynamic Drag Force (Fad)
3. Rolling resistance Forece (Frr)
4. Hill Climbing Force (Fhc)
5. Vertical & Longitudinal Force

Forces Acting on Range Rover :-

Excel Calculator Google Drive Link :-

https://docs.google.com/spreadsheets/d/1mprmYcYoLdo7U7muHxFNpUWvQDOU2-l7VgVrhrC3_nk/edit?usp=sharing