Engineering Mechanics

Chapter 32 : Motion of Vehicles „„„„„ 657

= 2

2

• hI P Py
  xxr I
  M
  r

×

⎛⎞

⎜⎟+

⎝⎠^2

P

a

I

M

r

⎡⎤

⎢⎥=

⎢⎥⎛⎞

⎜⎟+

⎢⎥⎣⎦⎝⎠

Q

= 2 –

P hI

y

x Mr I

⎡⎤

⎢⎥+

⎣⎦

...(ix)

Adding equations (viii) and (ix),

RF= 2

(^1) –
2
P hI
Mg y
x Mr I
⎡⎤⎛⎞
⎢⎥+ ⎜⎟
⎣⎦⎝⎠+
and RR= 2
(^1) –

• 2

P hI

Mg y

x Mr I

⎡⎤⎛⎞
⎢⎥⎜⎟
⎣⎦⎝⎠+
Note: If the tractive force passes through a point above the centre of gravity of the vehicle,
then the value of y is taken as positive.

Example 32.2. A four-wheeled vehicle of total mass 5000 kg has two pairs of wheels of
500 mm radius. Each pair of wheels with axles has mass of 750 kg, the radius of gyration of each
wheel being 400 mm. The axles are 2 metres apart and the c.g. of vehicle is mid-way at a height of
1·2 m above the road surface. Find the reactions of the front and rear wheels, if tractive force of
6 kN is acting 50 mm below the centre of gravity of vehicle.

Solution. Given: Mass of the vehicle (M) = 5000 kg = 5t; Radius of each wheel (r) = 500 mm
= 0·5 m; Mass of the each pair of wheel with axle (m) = 750 kg = 0·75 t; Radius of gyration of each
wheel (k) = 400 mm = 0·4 m; Horizontal distance between the centre of axle (2x) = 2 m or x = 1 m;
Height of the centre of gravity of the vehicle above road surface (h) = 1·2 m; Tractive force (P) = 6 kN
and distance between the centre of gravity of the vehicle and the point through which the tractive
force acts (y) = 50 mm = 0·05 m

We know that mass moment of inertia of a pair of wheels,

I=mk^2 = 0·75 (0·4)^2 = 0·12 t-m^2

∴ Reaction at the front wheels,

RF= 2

1

(^21)
PhI
Mg y
x Mr
⎡⎤⎛⎞
⎢⎥+ ⎜⎟
⎣⎦⎝⎠+
RF= 2
161·20·12
59·8 –0·05 kN
(^21) 5 (0·5) 0·12
⎡ ⎛⎞× ⎤
⎢⎥×+⎜⎟
⎣⎦⎝⎠+
= 24·67 kN Ans.
and reaction at the rear wheels,
RR= 2
1
––
(^21)
PhI
Mg y
x Mr
⎡⎤⎛⎞
⎢⎥⎜⎟+
⎣⎦⎝⎠
= 2
161·20·12
59·8– –0·05 kN
(^21) 5 (0·5) 0·12
⎡ ⎛⎞× ⎤
⎢⎥× ⎜⎟
⎣⎦⎝⎠+
= 24·33 kN Ans.
Example 32.3. A vehicle of mass 1200 kg is moving on a level surface by a tractive force of
1800 N acting horizontally at a height of 0·75 m above the road surface. The centre of gravity of the
vehicle is 1 m above the road surface and midway betweeen front and rear wheels which are 2·4 m
apart. The resistance to motion, at each pair of wheels, is 1/12 of the corresponding reaction.