Mechanical Engineering Principles

17

Simple harmonic motion

At the end of this chapter you should be

able to:

• understand simple harmonic motion


• determine natural frequencies for simple
  spring-mass systems


• calculate periodic times


• understand the motion of a simple
  pendulum


• understand the motion of a compound
  pendulum

17.1 Introduction

Simple harmonic motion is of importance in a num-
ber of branches of engineering and physics, includ-
ing structural and machine vibrations, alternating
electrical currents, sound waves, light waves, tidal
motion, and so on.

17.2 Simple harmonic motion (SHM)

A particle is said to be under SHM if its accel-
eration along a line is directly proportional to its
displacement along that line, from a fixed point on
that line.
Consider the motion of a particleA, rotating in a
circle with a constant angular velocityω, as shown
in Figure 17.1.
Consider now the vertical displacement ofAfrom
xx, as shown by the distanceyc.IfPis rotating at
a constant angular velocityωthen the periodic time
Tto travel an angular distance of 2π, is given by:

T=

2 π

ω

( 17. 1 )

Letf=frequency of motionC(in Hertz), where

f=

1

T

=

ω

2 π

( 17. 2 )

0

yc

c

y

−y

A

B

vA

vAsinq

vAcosq

w

−xxr

r

q

Displacement (yc)

Accelration (−ac)

Velocity (vc)

time't'

(a) (b)

Figure 17.1

To determine whether or not SHM is taking place,

we will consider motion ofAin the directionyy.

NowyC=OAsinωt,

i.e., yc=rsinωt ( 17. 3 )

wheret=time in seconds.

Plotting of equation (17.3) againsttresults in the

sinusoidal variation for displacement, as shown in

Figure 17.1(b).

From Chapter 11,vA = ωr, which is the tan-

gential velocity of the particleA. From the veloc-

ity vector diagram, at the pointAon the circle of

Figure 17.1(a),

vC=vAcosθ=vAcosωt ( 17. 4 )

Plotting of equation (17.4) againsttresults in the

sinusoidal variation for the velocityvC, as shown in

Figure 17.1(b).

The centripetal acceleration ofA

=aA=ω^2 r

Now aC=−aAsinθ

Therefore, aC=−ω^2 rsinωt ( 17. 5 )

Plotting of equation (17.5) againsttresults in the

sinusoidal variation for the acceleration atC,aC,as

shown in Figure 17.1(b).

Substituting equation (17.3) into equation (17.5)

gives:

aC=−ω^2 yC ( 17. 6 )