Mechanical Engineering Principles

196 MECHANICAL ENGINEERING PRINCIPLES

Figure 17.8, where the frequency of

oscillation was found to be 1.26 Hz. If the

mass of the ring is 10.5 kg, determine the

mass moment of inertia about the centre of

gravity,IG.Takeg= 9 .81 m/s^2.

150 mm

xx

y

y

Figure 17.8

By inspection of Figure 17.8,

h=75 mm= 0 .075 m.

Now frequency, f=

1

2 π

√

gh

(k^2 G+h^2 )

i.e. 1. 26 =

1

2 π

√

9. 81 × 0. 075

(k^2 G+ 0. 0752 )

i.e. ( 1. 26 )^2 =

1

( 2 π)^2

×

9. 81 × 0. 075

(k^2 G+ 0. 0752 )

from which,

(k^2 G+ 0. 005625 )=

0. 73575

1. 5876 ×( 2 π)^2

= 0. 011739

k^2 G= 0. 011739 − 0. 005625

= 0. 006114

from which, kG=

√

0. 006114 = 0. 0782

The mass moment of inertia about the centre of
gravity,

IG=mk^2 G= 10 .5kg× 0 .006114 m^2

i.e. IG= 0 .0642 kg m^2

17.6 Torsional vibrations

From equation (17.7), it can be seen that for SHM

in a linear direction,

a+ω^2 y= 0

For SHM in a rotational direction,

αr+ω^2 y= 0

or α+ω^2

(y

r

)

= 0

or α+ω^2 θ= 0

i.e θ ̈+ω^2 θ= 0 ( 17. 18 )

whereθ=yr=angular displacement, and

θ ̈=α=angular acceleration

Now try the following exercises

Exercise 86 Further problems on pendu-

lums

1. Determine the period of oscillation of a
   pendulum of length 2 m ifg= 9 .81 m/s^2.
   [0.3525 Hz]


2. What will be the period of oscillation
   if g = 9 .78 m/s^2 for the pendulum of
   Problem 1? [0.3519 Hz]


3. What will be the period of oscillation
   ifg = 9 .832 m/s^2 for the pendulum of
   Problem 1? [0.3529 Hz]


4. What will be the value of the mass
   moment of inertia through the centre of
   gravity,IG, for the compound pendulum
   of worked problem 4, if the inner
   diameter of the disc of Figure 17.8 were
   100 mm? [0.0559 kg m^2 ]

Exercise 87 Short answer questions on

simple harmonic motion

1. State the relationship between the dis-
   placement (y) of a mass and its accelera-
   tion (a) for SHM to take place.