72 Theory of Machines
Simple
Harmonic
Motion
4
Features
1. Introduction.
2. Velocity and Acceleration of
a Particle Moving with
Simple Harmonic Motion.
3. Differential Equation of
Simple Harmonic Motion.
4. Terms Used in Simple
Harmonic Motion.
5. Simple Pendulum.
6. Laws of Simple Pendulum.
7. Closely-coiled Helical
Spring.
8. Compound Pendulum.
9. Centre of Percussion.
10. Bifilar Suspension.
11. Trifilar Suspension
(Torsional Pendulum).
4.1.
Introduction
Consider a particle
moving round the circumfer-
ence of a circle in an
anticlockwise direction, with
a constant angular velocity,
as shown in Fig. 4.1. Let P
be the position of the particle
at any instant and N be the
projection of P on the diam-
eter X X ′ of the circle.
It will be noticed that when
the point P moves round the
circumference of the circle from X to
Y, N moves from X to O, when P
moves from Y to X ′, N moves from O
to X ′. Similarly when P moves from
X ′ to Y ′, N moves from X ′ to O and
finally when P moves from Y ′ to X, N
moves from O to X . Hence, as P
completes one revolution, the point N
completes one vibration about the
Fig. 4.1. Simple harmonic
motion.
A clock pendulum
executes Simple
Harmonic Motion.
72
Chapter 4 : Simple Harmonic Motion 73
point O. This to and fro motion of N is known as simple
harmonic motion (briefly written as S.H.M.).
4.2. Velocity and Acceleration of a
Particle Moving with Simple
Harmonic Motion
Consider a particle, moving round the circumfer-
ence of a circle of radius r, with a uniform angular velocity
ω rad/s, as shown in Fig. 4.2. Let P be any position of the
particle after t seconds and θ be the angle turned by the
particle in t seconds. We know that
θ = ω.t
If N is the projection of P on the diameter X X ′,
then displacement of N from its mean position O is
x = r.cos θ = r.cos ω.t ... (i)
The velocity of N is the component of the velocity
of P parallel to XX ′ , i.e.
2
2
N
sin
. sin
v
v
r
r
x
=
θ = ω
θ = ω
−
... (ii)
2
2
...
. , and
sin
v
r
r
NP
r
x