Module -1 - Oscillations
Syllabus
Simple Harmonic Motion – Preliminary concepts, superposition of SHMs in two mutually perpendicular
directions – Lissajous figures.
Damped vibrations – Differential equation and its solution, Logarithmic decrement, quality factor
Forced vibrations – Differential equation and its solution, Amplitude and velocity resonance, sharpness
of resonance, application to LCR circuit.
Introduction
In our day-to-day life we come across many type of motions that are mainly of two types (i) where a
body moves about a mean position known as vibrational or oscillatory motion i.e. an oscillating
pendulum, vibrations of stretched string, vibrations of electrons etc. and (ii) where a body moves from
one place to another with respect to time known as translational motion i.e. a moving train, flying
aeroplane, moving ball etc.
It is to be noted that any oscillatory motion should always be periodic, but a periodic motion may not
always be oscillatory in motion.
Simple Harmonic Motion
If any particle moves in such a manner that its acceleration is always directly proportional to its
displacement from its equilibrium position and is always directed towards equilibrium position, then the
motion of the particle is said to be Simple Harmonic Motion (SHM).
It is classified into various categories
(a) Linear SHM – where displacement is linear e.g. motion of simple pendulum, motion of a
small mass ties with a spring
(b) Angular SHM – where the displacement is angular e.g. torsional oscillations, oscillations of
a compound pendulum.
Essential conditions of SHM
If a is the linear acceleration and x is the displacement from the equilibrium position, then essential
condition for linear SHM is a   x and if a is angular acceleration with θ as angular displacement for a
angular SHM then a   .
The time elapsed to complete one oscillation of SHM is called Time Period (T) and the number of
oscillations per unit time is called Frequency (f) of the SHM. Hence T=1/f.
1
The maximum displacement of particle from its mean position is called as Amplitude of SHM and the
quantity which tells us about the instantaneous position and direction of motion of a SHM is known as
Phase of the motion. The general equation of motion for any SHM can be given as y  A sin(t   )
where A is amplitude, ω as angular frequency and φ as initial phase of the motion.
Hence if a particle of mass m takes SHM and x be the displacement from mean position then its
acceleration a is given by
a  x
a  kx
And force acting on the particle F = ma (also known as restoring force)
So
F  mkx   x (here µ=mk is called as force constant i.e. restoring force per uit displacement
of the particle.
Characteristics of SHM
(i) motion is linear
(ii) motion is periodic and oscillatory
(iii) and the restoring force is proportional to and acting in the direction opposite to the
displacement from mean position.
Let us consider a particle moving on the circumference of a circle of radius r with a uniform velocity v
Equation of SHM and its solution
Let us consider a particle of mass m moving under the influence of any SHM along a straight line with x
as displacement from mean position at any time t. Then using the basic condition of SHM
F  x
or
F  kx
where k is force constant.
…………..(1)
If a is the acceleration at any time t then
ma  kx
Or
m
d2x
 kx
dt 2
Replacing k/m with ω2
or
d2x k
 x0
dt 2 m
…………..(2)
d2x
 2 x  0
dt 2
…………..(3)
Equation (ii) and (iii) are known as equations of SHM. To find its solution let us consider
x  Cet
(where α and C are constants)
…………..(4)
2
So
And
dx
 C e t
dt
d2x
 C 2 e t
dt 2
…………..(5)
Using eq (4) and (5) in (3)
C 2et   2Cet  0
Ce t  2   2   0
But Cet  0
hence  2   2   0
So possible solutions can be x  Ceit
or   i
or
x  Ceit
And the combination of the above will give general solution of (3) i.e.
x  C1eit  C2 e it
(where C1 and C2 are constants)
Or
x  C1 (cos t  i sin t )  C2 (cos t  i sin t )
Or
x  C1  C2  cos t  iC1  iC2  sin t
Let us consider C1  C2  A sin  and
…………..(6)
…………..(7)
iC1  iC2  A cos 
Hence x  A sin  cos t  A cos  sin t
x  A sin t   
…………..(8)
Eq. (8) gives the solution of equation of SHM.
We can use  2 
k
m
or  
And time period T 
And frequency f 
2

m
k
 2
1
1

T 2
k
m
k
m
…………..(9)
…………..(10)
Here the quantity t    is known as the phase of the vibrating particle. If t =0 then φ is the initial
phase of the particle.
The particle taking SHM is called harmonic oscillator.
3
Velocity and Acceleration
As we know that the displacement of the particle taking SHM is given by
x  A sin(t   )
…………..(1)
Differentiating it w.r.t. time then we get velocity as
v
dx
 A cos(t   )
dt
…………..(2)
 A 1  sin 2 t   
  A2  A2 sin 2 t   
v   A2  x2
…………..(3)
The above equation represents the velocity of the particle at any displacement x from the mean position
when it is taking SHM. The maximum velocity is obtained at x = 0 i.e. vmax   A and the minimum
velocity is at x = A i.e. at the maximum displacement from mean position.
Gin differentiating equation (2) w.r.t. time, we get the acceleration of the particle at any instant of time
and at any displacement from mean position when it is taking SHM as
a
Or
dv d 2 x

  A 2 sin t   
dt dt 2
a   2 x
…………..(4)
The above equation represents the acceleration of the oscillating particle at any displacement from
mean position. Equation (4) is the standard equation of SHM.
Acceleration is maximum when x = A and it is minimum when x = 0.
Energy of a Harmonic Oscillator
4
5