Resistive Forces, Acceleration, Gear Ratio, Braking Forces and FEA analysis of Brake Rotor
AIM: To find the Resistive forces, Acceleration, Gear ratio, Braking forces and FEA brake analysis for an urban hybrid vehicle.
Formulas and concepts
Resistive Forces:
- Rolling Resistance
- It is defined as the force resisting a body rolling on a surface. It opposes the motion of the body.
Fr =rolling resistance (N)
Crr=coefficient of rolling resistance
m=mass of the vehicle
g=gravitational acceleration
- Aerodynamic Resistance:
- It is the resistance created by the fluid surrounding the body which is in motion relative to the fluid.
Fa= drag force (N)
ρ = destiny of air
v = velocity
Cd = drag coefficient
A=cross sectional area
- Gradient Forces:
- When the vehicle travels uphill, a component of weight works in the direction opposite to the motion.
Fg= Gradient resistive forces (N)
m= mass of the vehicle
g= gravitational acceleration
`theta`= slope of the hill.
RPM at wheel
- According to the velocity the wheel RPM can be calculated by using the following formula.
RPM= (v∗60)/(π∗d)
v=velocity of the vehicle (m/s)
d=rolling diameter of the tire (m)
Load Distribution and Transfer:
- Load Ditribution:
- Rear Axle Load is given by W_R=W∗ l_m/L
W= Total weight of car
l_m= Horizontal Distance of CG
L= Wheel base
- Front Axle Load is given by W_F=W-W_r
Traction force
- It is given by F=(W_R∗μ)/(1-(h_m∗μ)/L)
W_R=static load on rear axle
μ = Coefficient of friction
h_m= vertical distance of CG
Load Transfer:
- It is given by ∆W=±(Fh_m)/L
Braking:
- Deceleration is given by a=(V^2-U^2)/(2∗s)
v = Final Velocity(m/s)
u = Initial Velocity (m/s)
s = stopping distance(m)
Braking Torque
- It is given by τ_B= W_F∗μ∗d/2
μ= coefficient of friction
d= rolling diameter of tire (m)
Caliper force: F_B= τ_B/(d_B∗μ_B )
dB=Rotor Diameter(m)
μ_B=Coefficient of friction of brakes
Procedure
- All the resistive forces are calculated using the given parameter.
- Add all resistive forces to get the total resistive forces.
- RPM of the wheels is calculated.
- Load Distribution and Load Transfer is calculated.
- Accelerative force is calculated.
- Gear ratios are calculated based on available torque, required torque and RPM is also considered.
- Braking forces are calculated like Braking Torque, caliper force
Input
|
Location of centre of gravity |
||||
|
Horizontal Distance (lm) |
1140 |
mm |
||
|
Vertical Distance (hm) |
400 |
mm |
||
|
Wheel Base (L) |
1900 |
mm |
||
|
Track Width (T) |
1160 |
mm |
||
|
Coefficient of friction |
1.5 |
|||
|
Mass of the vehicle |
420 |
KG |
||
|
Performance Requirements |
||||
|
Top Speed |
100 |
kmph |
||
|
acceleration |
0-40 |
8 |
sec |
|
| Motor Parameters | ||||
| Rated Torque | 33.9 | Nm | ||
| Rated RPM | 2431 | RPM | ||
- here are two motors are being used.
Calculations
|
Resistive forces |
||||
|
Rolling Resistance |
||||
|
Crr |
Mass(kg) |
g(m/s^2) |
Velocity (kmph) |
Frr(N) |
|
0.010422798 |
420 |
9.81 |
10 |
42.94401121 |
|
0.010575893 |
420 |
9.81 |
20 |
43.57479459 |
|
0.010831052 |
420 |
9.81 |
30 |
44.62610022 |
|
0.011188274 |
420 |
9.81 |
40 |
46.0979281 |
|
0.01164756 |
420 |
9.81 |
50 |
47.99027823 |
|
0.01220891 |
420 |
9.81 |
60 |
50.30315062 |
|
0.012872323 |
420 |
9.81 |
70 |
53.03654525 |
|
0.0136378 |
420 |
9.81 |
80 |
56.19046214 |
|
0.01450534 |
420 |
9.81 |
90 |
59.76490128 |
|
0.015474944 |
420 |
9.81 |
100 |
63.75986267 |
|
Aerodynamic Resistance |
|||||
|
Cd |
density(kg/m^3) |
Area(m^2) |
velocity(kmph) |
m/sec |
Fa(N) |
|
0.6 |
1.12 |
0.19127 |
10 |
2.777777778 |
0.495885185 |
|
0.6 |
1.12 |
0.19127 |
20 |
5.555555556 |
1.983540741 |
|
0.6 |
1.12 |
0.19127 |
30 |
8.333333333 |
4.462966667 |
|
0.6 |
1.12 |
0.19127 |
40 |
11.11111111 |
7.934162963 |
|
0.6 |
1.12 |
0.19127 |
50 |
13.88888889 |
12.39712963 |
|
0.6 |
1.12 |
0.19127 |
60 |
16.66666667 |
17.85186667 |
|
0.6 |
1.12 |
0.19127 |
70 |
19.44444444 |
24.29837407 |
|
0.6 |
1.12 |
0.19127 |
80 |
22.22222222 |
31.73665185 |
|
0.6 |
1.12 |
0.19127 |
90 |
25 |
40.1667 |
|
0.6 |
1.12 |
0.19127 |
100 |
27.77777778 |
49.58851852 |
- Gradient forces are zero as the vehicle is considered to be traveling on a flat road.
Total Resistive Forces:
|
velocity(kmph) |
Total Resistive forces (N) |
|
10 |
43.4398964 |
|
20 |
45.5583353 |
|
30 |
49.0890669 |
|
40 |
54.0320911 |
|
50 |
60.3874079 |
|
60 |
68.1550173 |
|
70 |
77.3349193 |
|
80 |
87.927114 |
|
90 |
99.9316013 |
|
100 |
113.348381 |
|
RPM at Wheel |
|||
|
velocity(kmph) |
m/sec |
Diameter of wheel(m) |
RPM at wheel |
|
10 |
2.777777778 |
0.544 |
98 |
|
20 |
5.555555556 |
0.544 |
195 |
|
30 |
8.333333333 |
0.544 |
293 |
|
40 |
11.11111111 |
0.544 |
390 |
|
50 |
13.88888889 |
0.544 |
488 |
|
60 |
16.66666667 |
0.544 |
585 |
|
70 |
19.44444444 |
0.544 |
683 |
|
80 |
22.22222222 |
0.544 |
780 |
|
90 |
25 |
0.544 |
878 |
|
100 |
27.77777778 |
0.544 |
975 |
|
Weight Distribution and Load Transfer |
|||||||
|
Rear axle Load |
252 |
kg |
2472.12 |
N |
Weight Distribution Percentage |
||
|
Front Axle Load |
168 |
kg |
1648.08 |
N |
Front |
Rear |
|
|
Traction Force |
5419.64769 |
N |
40 |
60 |
|||
|
Load Transfer |
1140.97846 |
N |
|||||
|
Max Torque can be given to tires (with out slip) |
1474.14417 |
Nm |
|||||
|
Accelerative Resistance |
|||||
|
Acceleration time |
6 |
sec |
|||
|
Intial velocity |
final velocity (KMPH) |
change in velocity (m/s) |
time (sec) |
Acceleration |
Accelerative Resistance (N) |
|
0 |
10 |
2.7777778 |
6 |
0.462962963 |
194.4444444 |
|
0 |
20 |
5.5555556 |
6 |
0.925925926 |
388.8888889 |
|
0 |
30 |
8.3333333 |
6 |
1.388888889 |
583.3333333 |
|
0 |
40 |
11.111111 |
6 |
1.851851852 |
777.7777778 |
|
0 |
50 |
13.888889 |
6 |
2.314814815 |
972.2222222 |
|
0 |
60 |
16.666667 |
6 |
2.777777778 |
1166.666667 |
|
0 |
70 |
19.444444 |
6 |
3.240740741 |
1361.111111 |
|
0 |
80 |
22.222222 |
6 |
3.703703704 |
1555.555556 |
|
0 |
90 |
25 |
6 |
4.166666667 |
1750 |
|
0 |
100 |
27.777778 |
6 |
4.62962963 |
1944.444444 |
Total tractive Effort:
|
Total Tractive Effort |
|
|
Velocity (kmph) |
Total Tractive Effort (N) |
|
10 |
237.8843408 |
|
20 |
434.4472242 |
|
30 |
632.4224002 |
|
40 |
831.8098688 |
|
50 |
1032.60963 |
|
60 |
1234.821684 |
|
70 |
1438.44603 |
|
80 |
1643.48267 |
|
90 |
1849.931601 |
|
100 |
2057.792826 |
Gear Ratio:
|
Velocity to be Accelerated |
60 |
kmph |
Required Top Speed |
100 |
kmph |
||
|
Resistive Force with Acceleration(N) |
1234.8217 |
N |
Required RPM at wheel |
975 |
RPM |
||
|
Required torque |
335.8715 |
Nm |
Rated Motor RPM |
2431 |
RPM |
||
|
Torque Available at motor |
69.44943 |
Nm |
|||||
|
Recommended Gear Ratio For Acceleration |
Recommended Gear Ratio For Top Speed |
||||||
|
4.8 |
2.5 |
||||||
|
Braking Torque |
||||||
|
Initial velocity |
27.8 |
m/s |
||||
|
Final Velocity |
0 |
m/s |
||||
|
Braking Distance |
25 |
m |
||||
|
Deceleration |
15.43209877 |
m/s^2 |
||||
|
Load Transfer |
1364.522417 |
N |
||||
|
Front Wheel |
Rear Wheel |
|||||
|
Load after transfer |
3012.602417 |
N |
Load after Transfer |
1107.597583 |
N |
|
|
Front Brake Torque |
1229.141786 |
Nm |
Rear Brake Torque |
451.8998138 |
Nm |
|
|
Front caliper Force |
10781.94549 |
N |
Rear caliper Force |
3964.033454 |
N |
|
|
Brake Bias |
||||||
|
Front |
Rear |
|||||
|
73 |
27 |
|||||
Graphs
FEA Analysis on Brake Rotor:
For the FEA analysis of a Brake Rotor
- The rotor is 285 mm in diameter.
- The material used is cast iron which has a friction coefficient of 0.4.
- For the analysis the condition consider is the vehicle is moving at top speed and the brakes are applied with max braking force and the Rotor is applied with high caliper force and the wheels are locked.
Design of the Rotor:
Load case:
Factor of Safety:
Stress:
Displacement:
Summary:
