After studying this lecture, students will be able to explain the stability of a four-wheel vehicle when taking a turn. The vertical reaction of each wheel must be positive to prevent lifting. Forces like gyroscopic couples and centrifugal forces cause reactions on the inner and outer wheels. The limited speed of a vehicle around a curve can be calculated by ensuring the vertical reaction of the inner wheels remains upward.

1. Lecture Outcomes
After studying this lecture, a student will be able to-
explain about stability of four-wheel vehicle when taking a
turn.
1

2. In case of a four-wheel vehicle, it is an essential that no wheel is
lifted off the ground while the vehicle takes a turn.
The condition fulfilled as long as the vertical reaction of the ground
on any of the wheels is positive (or upwards).
A four-wheeled vehicle having a mass m. Assuming that the weight
is equally divided among the four wheels,
Weight on each wheel =
𝑊
4
=
𝑚𝑔
4
(downwards).
Reaction of ground on each wheel, 𝑅𝑤 =
𝑊
4
=
𝑚𝑔
4
(upwards).
2

3. 3

4. Effect of gyroscopic couple
Gyroscopic couple due to four wheels,
𝐶𝑤 = 4𝐼𝑤ω𝑤ω𝑝 = 4𝐼𝑤
𝑣2
𝑟𝑅
where
𝐼𝑤 = mass MOI of each wheel
ω𝑤 = angular velocity of wheels =
𝑣
𝑟
ω𝑝 = angular velocity of precession =
𝑣
𝑅
v = linear velocity of vehicle
R = radius of curvature
Gyroscopic couple due to engine rotating parts,
𝐶𝑒 = 𝐼𝑒ω𝑒ω𝑝 = 𝐼𝑒𝐺ω𝑤ω𝑝
G = gear ratio =
ω𝑒
ω𝑤
4

5. Total gyroscopic couple,
𝐶𝐺 = 𝐶𝑤 ± 𝐶𝑒 = (4𝐼𝑤 ± 𝐼𝑒𝐺)ω𝑤ω𝑝
Positive sign is used when the engine parts rotate in the same direction as the
wheels and the negative sign when they rotate in opposite.
Assuming that 𝐶𝐺 is positive and the vehicle takes left turn, the reaction
gyroscopic couple on it is clockwise when viewed from the rear of the vehicle.
The reaction couple is provided by equal and opposite force on the outer and the
inner wheels of the vehicle.
Forces on the two outer wheels =
𝐶𝐺
𝑤
(downwards)
Forces on the two inner wheels =
𝐶𝐺
𝑤
(upwards)
Forces on the each outer wheel =
𝐶𝐺
2𝑤
(downwards)
Forces on the each inner wheel =
𝐶𝐺
2𝑤
(upwards)
5

6. Thus the force on each of outer wheel is similar to the weight. On the inner
wheels it is in the opposite direction. Thus,
Reaction of ground on the each outer wheel, 𝑅𝐺 =
𝐶𝐺
2𝑤
(upwards)
Reaction of ground on the each inner wheel, 𝑅𝐺 =
𝐶𝐺
2𝑤
(downwards)
Effect of centrifugal couple
As the vehicle moves on a curved path, a centrifugal force also acts on the
vehicle in the outward direction at the centre of mass of the vehicle.
Centrifugal force, 𝑚𝑅𝜔𝑝
2
= mR(
v
R
)
2
= m
v2
R
This force would tend to overturn the vehicle outwards and the overturning
couple will be
𝐶𝑐 = mRωp
2*h = m
v2
R
h
This is equivalent to a couple due to equal and opposite forces on outer and inner
wheels.
6

7. Force on each outer wheel =
𝐶𝑐
2𝑤
(downwards)
Force on each inner wheel =
𝐶𝑐
2𝑤
(upwards)
Again, the force on each of the outer wheels is similar to the weight and on each
of the inner wheels, it is opposite.
Reaction of ground on the each outer wheel, 𝑅𝑐 =
𝐶𝑐
2𝑤
(upwards)
Reaction of ground on the each inner wheel, 𝑅𝑐 =
𝐶𝑐
2𝑤
(downwards)
7

8. Vertical reaction on each outer wheel =
𝑊
4
+
𝐶𝐺
2𝑤
+
𝐶𝑐
2𝑤
(upwards)
Vertical reaction on each inner wheel =
𝑊
4
+
𝐶𝐺
2𝑤
+
𝐶𝑐
2𝑤
(upwards)
It can be observed that there are chances that the reaction of the ground on the
inner wheels may not be upwards and thus the wheels are lifted off from the
ground.
For positive reaction, the conditions will be
𝑊
4
−
𝐶𝐺
2𝑤
−
𝐶𝑐
2𝑤
≥ 0
𝑊
4
≥
𝐶𝐺
2𝑤
+
𝐶𝑐
2𝑤
𝑅𝑊 ≥ 𝑅𝐺 + 𝑅𝐶
8

9. 9
𝑅𝑊 ≥ 𝑅𝐺 + 𝑅𝐶

10. Q : 1. Each wheel of a four-wheeled rear engine automobile has a MOI of 2.4
kg.𝒎𝟐 and an effective radius of 0.33m. The rotating part of the engine have a
MOI of 1.2 kg.𝒎𝟐
. The gear ratio of engine to the back wheels is 3 to 1. The
engine axis is parallel to the rear axle and the crankshaft rotates in the same sense
as the road wheels. The mass of the vehicle is 2200kg and the centre of mass is
0.55m above the road level. The track width of the vehicle is 1.5m. Determine
the limited speed of the vehicle around a curve with 80m radius so that all the
four wheels maintain contact with the road surface.
10

11. Q : 1. Each wheel of a four-wheeled rear engine automobile has a MOI of 2.4
kg.𝒎𝟐 and an effective radius of 0.33m. The rotating part of the engine have a
MOI of 1.2 kg.𝒎𝟐
. The gear ratio of engine to the back wheels is 3 to 1. The
engine axis is parallel to the rear axle and the crankshaft rotates in the same sense
as the road wheels. The mass of the vehicle is 2200kg and the centre of mass is
0.55m above the road level. The track width of the vehicle is 1.5m. Determine
the limited speed of the vehicle around a curve with 80m radius so that all the
four wheels maintain contact with the road surface.
11

12. Q : 1. Each wheel of a four-wheeled rear engine automobile has a MOI of 2.4
kg.𝒎𝟐 and an effective radius of 0.33m. The rotating part of the engine have a
MOI of 1.2 kg.𝒎𝟐
. The gear ratio of engine to the back wheels is 3 to 1. The
engine axis is parallel to the rear axle and the crankshaft rotates in the same sense
as the road wheels. The mass of the vehicle is 2200kg and the centre of mass is
0.55m above the road level. The track width of the vehicle is 1.5m. Determine
the limited speed of the vehicle around a curve with 80m radius so that all the
four wheels maintain contact with the road surface.
12

13. Summary of lecture
Stability of a four-wheel vehicle
Solved Numerical Problem.
13

14. References
S.No. Title of Book Author(S) Publication
1 Theory of Machines S S Rattan
Tata McGraw
Hill
2 Theory of Machines Bevan, T.
Pearson
Education,New
Delhi(India)
3.
Theory of Machines
and Mechanisms
Uicker, J.J.,
Pennocle, G.R, and
Shigley, J.E
Oxford
University Press
4
Mechanism And
Machine Theory
Ambekar , A. G.
Prentice-hall Of
India
14

15. Be Happy
and
Continue Learning
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