Waves: Oscillations

Oscillations Introduction: Mechanical vibration

Simple Harmonic Motion Some oscillating systems Damped Oscillations Driven oscillations and resonance

Traveling waves Wave 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 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

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

xm

k

dt

xd

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

Constant of the spring

Visualizing the simple harmonic motion through the motion of a block on a spring

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

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

Simple Harmonic Motion. Energy

2

0 2

1)( xkdxxkU

x

x

Potential Energy

Kinetic energy 22 )sin(

2

1

2

1 tAmvmK

Total mechanical energy in Simple Harmonic Motion

222

2

1

2

1 AmAkKUEtotal

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

kmT

mk

2

;

Free-body diagram

The motion of a pendulum approximates simple harmonic motion for small angular displacements

Free-body diagram

L

g

L

g

dt

d

dt

dmLmg

Lmmg

amF TT

sin

sin

sin

2

2

2

2

gLT

Lg

2

I

MgDI

MgD

dt

d

dt

dIMgD

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

tmbo

tmb

o

oo

tmb

o

tmb

o

eEemAmAE

m

bandeAA

teAx

xkdt

dxb

dt

xdm

amvbxkF

222122

21

2

2

2

2

2

21´

)´cos(

0

x

Equilibrium position

Driven (Forced) Oscillations and resonance

tFF oext cos

External driving force (harmonic)

In addition to restoring forces and dumping forces are acting external (periodic) forces