Waves: Oscillations
Oscillations
Introduction: Mechanical vibration
Simple Harmonic Motion
Some oscillating systems
Damped Oscillations
Driven oscillations and resonance
Traveling waves
Wave motion. The wave equation
Periodic Waves: on a string, sound and electromagnetic waves
Waves in Three dimensions. Intensity
Waves encountering barriers. Reflection, refraction, diffraction
The Doppler effect
Superposition and standing waves
Superposition and interference
Standing waves
Oscillations
•
Simple Harmonic Motion. Energy
•
Some oscillating systems
Vertical String
The simple pendulum
The physical pendulum
•
Damped Oscillations
•
Driven (Forced) oscillations and resonance
INTRODUCTION. MECHANICAL VIBRATIONS
A mechanical vibration is the motion of a particle or a body
which oscilates about a position of equilibrium.
A mechanical vibration generally results when a system is displaced
from a position of stable equilibrium. The system tends to return to this
position under the action of restoring forces (either elastic forces as the
case of springs or gravitational forces, as the case of pendulum)
Period of vibration. The time interval required by the system
to complete a full cycle of motion.
Frequency: The number of cycles per unit of time
Amplitude: The maximum displacement of the system from its
position of equilibrium
Most vibrations are undesirable, wasting energy and creating
unwanted sound – noise. For example, the vibrational motions of
engines, electric motors, or any mechanical device in operation are
typically unwanted. Such vibrations can be caused by imbalances in
the rotating parts, uneven friction, the meshing of gear teeth, etc.
Careful designs usually minimize unwanted vibrations.
The study of sound and vibration are closely related. Sound,
or "pressure waves", are generated by vibrating structures
(e.g. vocal cords); these pressure waves can also induce the
vibration of structures (e.g. ear drum). Hence, when trying to
reduce noise it is often a problem in trying to reduce vibration.
Drum vibration
VIBRATION
Free
(Driven) Forced
Undamped
Damped
SIMPLE HARMONIC MOTION
Visualizing the simple harmonic motion through the motion of a
block on a spring
Consider the forces exerted on the block that is placed
above a table without friction.
F k x
Constant of the spring
The net (resultant) force on the block is that exerted by the
spring. This force is proportional to the displacement x,
measured from the equilibrium position.
Applying the Newton´s Second Law, we have
This equation is a second-order
linear constant coefficient ordinary
differential equation describing the
harmonic oscillator
d 2x
F m 2 k x
dt
d 2x
k
 x
2
dt
m
Verify that each of the functions
x1  C1 cos k t 
 m 
x2  C2 sin  k t 
 m 
satisfies the differential equation
A differential equation is a mathematical equation for an unknown function of one or several
variables that relates the values of the function itself and its derivatives of various orders. Differential
equations play a prominent role in engineering, physics, economics, and other disciplines
Simple Harmonic Motion
x,position; A, amplitude,
(ωt+δ) phase of the motion
v, velocity
acceleration
f , frequency, T period,
ω, angular frequency (natural
circular frecuency),
δ, phase angle or constant phase
Case study: harmonic motion of an object on a spring
 km
Simple Harmonic Motion and Circular Motion
Simple harmonic motion can be visualized as the motion of the projection onto the x axis
from a point which moves in a circular motion at constant speed
Position, [m]; Amplitude [m];
phase (ωt+δ) [rad]
Velocity, [m/s]

k
m
f , frequency, [cycles/s], T period,[s]
ω, [rad/s]angular frequency (natural
circular frecuency),
δ, phase angle [rad]
1.-A 0.8-kg object is attached to a spring of force constant k = 400 N/m. The block is held a distance 5 cm from
the equilibrium position and is released at t =0. Find the angular frequenccy and the period T. (b) Write the
position x and velocity of the object as a function of time.(c) Calculate the maximum speed the block reaches. (d)
The energy of the oscillating system
2.- An object oscillates with angular frequency 8.0 rad/s. At t = 0, the object is at x = 4 cm with an initial velocity
v = -25 cm/s. (a) Find the amplitude and the phase constant for the motion; (b) Write the position x and velocity of
the object as a function of time.(c) Calculate the maximum speed the object reaches (e) The energy of the
oscillating system
Simple Harmonic Motion. Energy
Potential
Energy
x
U    ( k x )dx 
x 0
Kinetic
energy
1
k x2
2
1
1
2
2
K  m v  m  A sin( t   ) 
2
2
Total mechanical energy in
Simple Harmonic Motion
1
1
2
Etotal  U  K  k A  m A2 2
2
2
The total mechanical energy in simple harmonic
motion is proportional to the square of the
amplitude
Some oscillating systems
Spring
The simple pendulum
The physical pendulum
Free-body diagram
  k m;
T  2 m
F
T
Free-body diagram
k
 m aT
 gL
T  2 L
g
mg sin   m L
d 2
mg sin   mL 2
dt
d 2
g
g


sin




2
dt
L
L
The motion of
a pendulum
approximates
simple
harmonic
motion for
small angular
displacements
  I
d 2
MgD sin   I 2
dt
d 2
MgD


sin 
2
dt
I
MgD


I
Show that for the situations depicted
the object oscillates, in the case (a)
as if it were a spring with a force
constant of k1+k2, and, in the case
(b) 1/k = 1/k1 +1/k2
Find the resonance frequency for each of the three
systems
The figure shows the pendulum
of a clock. The rod of length
L=2.0 m has a mass m = 0.8
kg. The attached disk kas a
mass M= 1.2 kg, and radius
0.15 m. The period of the clock
is 3.50 s. What should be the
distance d so that the period of
this pendulum is 2.5 s
Damped Oscillations
 F  k x  b v  m a
Equilibrium
position
Spring
force
x
d 2x
dx
m 2 b
k x 0
dt
dt
x  Ao e
Viscous
force

2m

t
2m
 b
A  Ao e
 b
t
cos(´t   )
and ´ o
E  mA   mA e
1
2
2
2
 b 

1  
 2mo 
1
2
2
o
 m t
 b
  Eo e
2
 m t
 b
2
Driven (Forced) Oscillations and resonance
In addition to restoring forces
and dumping forces are acting
external (periodic) forces
External driving force
(harmonic)
F ext  Fo cos t