This document discusses braking analysis of a four-wheeled vehicle moving up an inclined plane. It describes three different types of braking systems - rear wheel braking, front wheel braking, and all-wheel braking. For each system, it provides the equations of motion and equilibrium to calculate the braking force and retardation. All-wheel braking provides the shortest braking distance since braking forces act on both the front and rear wheels. The document references sources for further information on braking systems and theory of machines.

1. Gandhinagar Institute of Technology
Subject :- Theory of Machine (2151902)
MECHANICAL ENGINEERING
5th - B : 2
Braking analysis of four wheels
– Malaviya Bhautik [130120119094]
Pavan Narkhede [130120119111]
Darshit Panchal [130120119114]
:
Prof. Prananjay Patel

2. Brake is a device used for slowing ,stopping & controlling the vehicle.
Braking operation based on kinetic energy of vehicle is to converting
into heat, which dissipated into atmosphere.
While driving the vehicle, torque of the engine produces the tractive
effort due to periphery of driving vehicle.
When the brakes are applied it produces negative tractive effort on
wheel.

3. HYDRAULLIC
BRAKING
SYSTEM
PNEUMATIC
BRAKING
SYTEM
MECHANICAL
BRAKING
SYSTEM
DISC BRAKE
SYSTEM
ANTI BRAKING
SYSTEM
(ABS)

4.  In a four wheel moving vehicle, the brakes may be applied to
• 1. the rear wheels only,
• 2. the front wheels only, and
• 3. all the four wheels.

5. Now, consider a vehicle moving up an inclined plane,
• α = Angle of inclination of the plane to the horizontal,
• m = Mass of the vehicle in kg
• x = Perpendicular distance of C.G. from the rear axle in metres,
• L = Distance between the centres of the rear and front wheels (also called
wheelbase) of the vehicle in metres,
• RA = Total normal reaction between the ground and the front wheels in newtons,
• RB = Total normal reaction between the ground and the rear wheels in newtons,
• µ = Coefficient of friction between the tyres and road surface, and
• a = Retardation of the vehicle in m/s

6.  It is a common way of braking the vehicle in which the braking force acts at the
rear wheels only
 Let FB= Total braking force (in newtons) acting at the rear wheels due to the
application of the brakes. Its maximum value is µ.RB.
 For the equilibrium of the vehicle, the forces acting on the vehicle must be in
equilibrium.
 Resolving the forces parallel to the plane,
FB + mg sinα = ma
 Resolving the forces perpendicular to the plane,
RA + RB = mg cosα

7. • Taking moments about G, the centre of gravity of the vehicle,
FB × h + R × x = RA(L – x)
• Substituting the value of F in the above expression, we have FB= µ.RB, and RA =
m.g cosα – RB

9. • It is a very rare way of braking the vehicle, in which the braking force acts at the
front wheels only.
• Let FA= Total braking force (in newton's) acting at the front wheels due to the
application of brakes. Its maximum value is µ.RA.

12. • This is the most common way of braking the vehicle, in which the braking force
acts on both the rear and front wheels.
• Let FA = Braking force provided by the front wheels = µ.RA
• Let FB = Braking force provided by the front wheels = µ.RB
• A little consideration will show that when the brakes are applied to all the four
wheels, the braking distance (i.e. the distance in which the vehicle is brought to
rest after applying the brakes) will be the least. It is due to this reason that the
brakes are applied to all the four wheels.

13. • Resolving the forces parallel to the plane,
FA  FB  m.g sin   m.a
• Resolving the forces perpendicular to the plane,
RA  RB  m.g cos 
• Taking moments about G, the centre of gravity of the vehicle,
(FA  FB )h  RB x  RA (L  x)
• Substituting the value of FA  .RA , FB  .RB and RB  m.g cos   RA in the above
expression,
 (RA  RB )h  (m.g cos   RA ) x  RA (L  x)
 (RA  m.g cos   RA )h  (m.g cos   RA )x  RA (L  x)
 .m.g cos   h  m.g cos   x  RA L

14. Now from equation (i),
.RA   RB  m.g sin   m.a
 (RA  RB )  m.g sin   m.a
.m.g. cos   m.g sin   m.a

15. • Wikipedia
• A Text book of theory of machine by R.S.KHURMY and J.K.GUPTA
• www.sae.org.in