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Waves: Oscillations
OscillationsIntroduction: Mechanical vibrationSimple Harmonic Motion Some oscillating systems Damped Oscillations Driven oscillations and resonance
Traveling wavesWave motion. The wave equationPeriodic Waves: on a string, sound and electromagnetic wavesWaves in Three dimensions. IntensityWaves encountering barriers. Reflection, refraction, diffractionThe Doppler effect
Superposition and standing waves Superposition and interferenceStanding waves
Oscillations • Simple Harmonic Motion. Energy
• Some oscillating systemsVertical StringThe simple pendulumThe physical pendulum
• Damped Oscillations
• Driven (Forced) oscillations and resonance
INTRODUCTION. MECHANICAL VIBRATIONSA 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
Drum vibration
energy and creating unwanted sound noise– . For example, the vibrational motions of engines, electric motors mechanical device, or any in operation are typically unwanted. Such vibrations can be caused by imbalances in the rotating parts, uneven friction gear, the meshing of teeth, etc. Careful designs usually minimize unwanted vibrations.
The study of sound and vibration are closely related. Sound, or "pressure
VIBRATION
Free
(Driven) Forced
Undamped
Damped
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.
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.
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
xkF −=Constant of the spring
xmk
dtxd
xkdt
xdmF
−=
−==
2
2
2
2
This equation is a second-order linear constant coefficient ordinary differential equation describing the harmonic oscillator
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
Verify that each of the functions
satisfies the differential equation
⎟⎠⎞⎜
⎝⎛=
⎟⎠⎞⎜
⎝⎛=
tmkCx
tmkCx
sin
cos
22
11
Simple Harmonic Motion
x,position; A, amplitude, (ωt+δ) phase of the motion
v, velocity
acceleration
Case study: harmonic motion of an object on a spring
f , frequency, T period, ω, angular frequency (natural circular frecuency), δ, phase angle or constant phase
mk=ω
Simple Harmonic Motion and Circular Motion
Position, [m]; Amplitude [m]; phase (ωt+δ) [rad]
Velocity, [m/s]
f , frequency, [cycles/s], T period,[s] ω, [rad/s]angular frequency (natural circular frecuency), δ, phase angle [rad]
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
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
mk=ω
Simple Harmonic Motion. Energy
2
0 21)( xkdxxkU
x
x
=−−= ∫=
Potential Energy
Kinetic energy ( )22 )sin(
21
21 δωω +== tAmvmK
Total mechanical energy in Simple Harmonic Motion 222
21
21 ωAmAkKUEtotal ==+=
The total mechanical energy in simple harmonic motion is proportional to the square of the amplitude
Some oscillating systemsSpring The simple pendulum The physical pendulum
kmT
mk
π
ω
2
;
=
=Free-body diagram
The motion of a pendulum approximates simple harmonic motion for small angular displacements
Free-body diagram
φφφ
φφ
αφ
Lg
Lg
dtd
dtdmLmg
Lmmg
amF TT
−≈−=
=
=
=∑
sin
sin
sin
2
2
2
2
gLT
Lg
π
ω
2=
=
φ
φφ
φφ
ατ
IMgD
IMgD
dtd
dtdIMgD
I
−≈
−=
=
=
sin
sin
2
2
2
2
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
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
Find the resonance frequency for each of the three systems
Damped Oscillations
Spring force
Viscous force
( )
( )
( ) ( )tmb
otm
b
o
oo
tmb
o
tmb
o
eEemAmAE
mbandeAA
teAx
xkdtdxb
dtxdm
amvbxkF
−−
−
−
===
⎟⎟⎠
⎞⎜⎜⎝
⎛−==
+=
=+−
=+−=∑
222122
21
2
2
2
2
2
21´
)´cos(
0
ωω
ωωω
δωx
Equilibrium position
Driven (Forced) Oscillations and resonance
tFF oext ωcos=
External driving force (harmonic)
In addition to restoring forces and dumping forces are acting external (periodic) forces
