PS217 - Vibrations and Waves

Section I Introduction to Oscillatory Motion

Section II The Simple Harmonic Oscillator (SHO)

Section III The Damped Harmonic Oscillator (DHO)

Section IV Some Complex Algebra

Section V Forced Oscillations

Section VI Coupled Oscillators

Section VII Waves and the Wave Equation

Section VIII Periodic and Non–Periodic Waves and the

Fourier Representation of Waves

Indicative Syllabus

2. The Simple Harmonic Oscillator or ‘SHO’

2.1 One example is that of a mass on a horizontal spring.

(Transparency 1)

Image dource: http://www.cs.wright.edu/~jslater/SDTCOutreachWebsite/images/gif/spring_mass_dia.gif

We assume a weightless spring and frictionless surface, e.g.the object (mass ‘m’) moves on an air track.

NB: In physics we usually start with an idealised system,and add the complications later, i.e. friction, gravity andweight of real springs. What would friction do ?

The force produced by the spring acts to push or pull themass back to its equilibrium position at x = 0 !!

The restoring force, F, on the mass ‘m’ is given by the equation:

F = - kx, which is commonly known as Hooke’s Law

where x is the displacement from equilibrium and k is the spring constant,

[ i.e. the restoring force /unit displacement].

The negative sign shows that restoring force acts in the a direction

opposite to the displacement ‘x’ !!

NB: In many complex physical situations, e.g., diatomic molecules like

H2, N2, O2…., Hookes law is obeyed (for small displacements) which is

why the present discussion is so valuable and far reaching.

The restoring force, F, on the mass ‘m’ is given by the equation:

F = - kx, which is commonly known as Hooke’s Law

where x is the displacement from equilibrium and k is the spring constant,

[ i.e. the restoring force /unit displacement].

The negative sign shows that restoring

force acts in the a direction opposite to

the displacement ‘x’ !!

NB: In many complex physical situations, e.g., diatomic

molecules like H2, N2, O2…., Hookes law is obeyed (for small

displacements) which is why the present discussion is so truly valuable

and far reaching.

Newton's Second Law states that: Force = Mass (m) x Acceleration (a)

This is the basic equation of simple

harmonic motion (SHM) and is the

basis for its description and

understanding.

(Transparency 2.)

The motion is oscillatory and best

described by sinusoidal functions

like ‘Sine’ and ‘Cosine’…..

!

F = ma = md2x

dt2

= "kx(Hooke'sLaw)

!

" a =d2x

dt2

=F

m=#kx

m

2.2 Find solutions for x(t), v(t) and a(t)……..

We want expressions for displacement - x(t), velocity - v(t) and

acceleration - a(t), all as functions of time.

Observing periodic motion of mass on spring, we look for a solution for

x(t) that is also periodic. Any suggestions ?

Lets try a sine orcosine function, c.f: ‘Cos(!)‘

So try the specific function, x = A.Cos (2"t/T), where A is amplitude.

NB: ! = 2"t/T is an angle (with units of radians),

As t goes from 0 # T, the angle ! goes from 0 # 2"......

Letting $ = (2"/T) where $ is the angular frequency of the oscillator (with

units of radians per second), we obtain, x(t) = A.Cos($t)

So SHM occurs when there is a restoring force (i.e. acting towards the

equilibrium position) that is directly proportional to the displacement

from equilibrium. The resulting motion is periodic !!

A: amplitude of oscillation

T: period of oscillation (i.e. the period of one complete cycle in units of

seconds)

f: frequency of oscillation = 1/T [(units of cycles per second or Hertz (Hz)]

% : is the phase (in the previous image - % = 0)

NB: Boundary condition: x = A at t = 0

In the next few slides we will develop this further !!

The ‘solution’ to the equation of simple harmonic motion (SHM) is

given by the expression:

!

x t( ) = ASin 2"ft + #( ) = ASin2"t

T+ #

$

% &

'

( )

Does this fit our observation ?

Yes, in particular x = A at t = 0 !!

Does this fit our differential equation ? Yes it does…..

and

or

i.e., x(t) is a solution of the second order differential equation of simpleharmonic motion, where !2 = k/m. So a mass ‘m’ of (m kg) attached to a

spring of spring constant ‘k’ will oscillate

with angular frequency: or frequency:

!

x t( ) = ACos2"t

T

#

$ %

&

' (

!

dx

dt= x'= v t( ) = "#ASin #t( )

!

d2x

dt2

= x"= a t( ) = "# 2ACos #t( ) = "# 2

x

!

d2x

dt2

= x"= "# 2x

!

" =k

m

!

f ="

2#=1

2#

k

m

Some physics observations:

T = 2"/$ and f = 1/T = $/2"

& f = 1/2".(k/m)1/2

Can you spot an important difference between A and f ?

Note that the frequency f is determined by the properties of the

oscillator, I.e., k and m, and does not depend at all on the amplitude, ‘A’

of the oscillation.

Why would that be useful in practice ?

+ guitar demo of decaying A but constant f.

f ' 1/(m)1/2, heavier things vibrate at lower freq.

f ' (k)1/2, the stronger the spring the higher the freq.

Clock pendulum: 1 Hz

Crab Nebula Radio Pulsar: 33 Hz

Middle C on piano: 256Hz

Crystal oscillator in Wristwatch: MHz (Megahertz)

Crystal oscillator in Laptop: GHz (Gigahertz)

Molecular vibration: 1012 - 1014 Hz

Earthquake: Minutes to Hours

Oscillating universe: Tens of billion years ?

Examples of frequencies and periods for

diverse periodic oscillating systems……

2.3 General solutions for SHMIn general, x(t) ( 0 at t = 0 and the motion will look like, e.g.

Cosine curve has been displaced horizontallyby angle, % called the phase angle.

x(t) is described by: x(t) = A cos ($t - %)and doesn’t reach its maximum value until $t = %

This is the general solution to our DE, x’’ = - $2x

[NB. Solns to 2nd order DE always contain 2 arbitrary constants]

NB $t # ($t + %) would shift curve to left.

Since Cos(a-b) = Cos(a)Cos(b) + Sin(a)Sin(b)

Acos ($t - %) = Acos $t cos % + Asin $t.sin % =

a cos $t + b sin $t, where a = Acos %, and b = Asin %

So the general solution of our DE call also be written as:

x(t) = a cos $t + b sin $t

where a and b are again determined by the initial (boundary) conditions

Demonstration of pendulum with x = A at t = 0. What isappropriate solution for x = 0 at t = 0 ?

2.4 Energy considerations in SHM

Demonstration of pendulum and exchange of PE and KE

Consideration of the energy in a system (kinetic plus potential) is apowerful tool in Vibrations and Waves. For one thing we are dealingwith scalar rather than vector quantities.

For a mass ‘m’ on a spring:

Kinetic energy: K = ! mv2

The potential energy ‘U’ stored in the spring is equal to the workdone in extending or compressing the spring, i.e., force ) distance.

When a spring is extended the force exerted by the spring is:F = – kx, but the force exerted on the spring is kx

* The potential energy ‘U’ = o+x k.xdx & U = ! kx2

Total energy, (K + U) must be constant, let’s call it ‘E’.

When mass is at max displacement, i.e. x = AK = 0 since mass is at rest and U = !kA2

* Total energy, E = K + U = !kA2

and so !mv2 + !kx2 = !kA2

which is true for any time t.

So for a periodic system ocillating at angular frequency ‘$’:x = A cos $t and v = -A$sin$t

* U(t) = !kx2 = !kA2cos2$t

and K(t) = !mv2 = !m$2A2sin2$t

* E = K + U = !kA2cos2$t + !kA2sin2$t = !kA2

The energy flows between kinetic and potential energies.

We can write total energy as

E = !m(x’)2 + !k(x)2 where x’ = dx/dt !!where the mass m holds KE and the spring constant k stores PE.

NB. In general for a SHO we have:

E = ,(x’)2 + -(x)2

where , and - are constants- another signature of SHM to look outfor.

[Aside: Just for fun we can also plot U and K with respect todisplacement - What will these plots (traces) look like ??]

Hint: At what value of ‘x’ is ‘U’ a maximum ? What value has ‘K’ there ?

2.5 Other examples of SHM

(a) Mass on a vertical spring - add gravity

When mass is attached to spring, its length isextended by .l, by Hooke’s law (F = -kx), thechange in restoring force is ‘-k .l ‘

*At the equilibrium position,the force on mass ‘m’ = mg - k "l = 0

{taking displacements {gravitational {force exerteddownwards as positive} force} upwards byspring}

*k .l = mg

When mass is displaced downwards by distance, x,

F = m x’’ = mg – k(.l + x) = mg – k.l - kx,

i.e. m x’’ = - kx

or

Which is of course just the equation of motion for SHM !!

NB. Force (m x’’) is in opp. direction to displacement, x.

and so we get SHM with $ = (k/m)1/2, as before……!

md2x

dt2

= "kx

(b) The simple pendulum

This time let’s find solution by considering energy of system. (Usefulbecause scalar quantities)

We have l2 = (l /y)2 + x2

l2 = l2 + y2 / 2ly + x2

and 2ly = y2+ x2

For small !, x >> y, so that y2 is negligible

So that y 0 x2/2l

Total energy, E = ! mv2 + ! mg(x2/l) KE PE

y

x

l

l

Note similarity with expression for total energy that we had alreadya few slides ago…..!!!!

Inspecting each term we can write that:

E = ! m(x’)2 + ! mg/l.(x)2 for a pendulum

Which we can compare with:

E = ! m(x’)2 + ! k (x)2 for a mass on spring

- a clear signature of SHM. Note that at x = A (amplitude), v = x’= 0

& E = ! mg(A2/l)

! mg(A2/l) = ! mv2 + ! mg(x2/l) - true for all times ‘t’

Which allows us to obtain an expression for the amplitude:

A2 = (l/g)v2 + x2 and so A = [(l/g)v2 + x2 ]1/2

Next we want to find ‘x’ - an expression for the displacementof the pendulum bob at any time ‘t’:

Substituting v = x’ = dx/dt in A2 = (l/g)v2 + x2

We get that A2 = l/g (dx/dt)2 + x2

& (dx/dt) = (g/l)1/2.[A2- x2]1/2

Rearranging & dx/[A2- x2]1/2 = (g/l)1/2 + dt

& + dx/[A2- x2]1/2 = (g/l)1/2 + dt

+ dx/[A2- x2]1/2 is a standard integral and = sin-1 x/A

* sin-1 x/A = (g/l)1/2.t + %

% is a constant of integration which we recognise as the phase angle

Rearranging we get that x = A sin ($t + %) , where $ = (g/l)1/2

i.e. We have SHM with ! = (g/l)1/2 and T = 2#/! = 2# (l /g)1/2

Note how the period [T = 2#/! = 2# (l /g)1/2] depends only on thelength of the pendulum ‘l’ and the acceleration due to gravity ‘g’…….

Measuring the period of a pendulum is a good way of determining, gc.f: 1st year experiment.Note that for l = 1 meter,T = 2"1l/9.81 ~ 2 seconds,In fact the original definitionof the second was one halfthe period of a 1m simplependulum

(c) The physical pendulum

In a physical pendulum the mass is not concentrated at one point as insimple pendulum, e.g a uniform rod of length l.

Considering forces this time,and the physical pendulumas a rotating system:

Then our usual equation of SHM: m x’’ = Fbecomes: I !’’ = 2, I = moment of inertia and! = angular displacement………..…

The moment of inertia for a rod is givem by I=1/3 ml2

The torque is given by the expression: 2 - mg sin!.l/2

So we can write I !’’ = 2 = 1/3 ml2. !’’ = - mg sin!.l/2

And so we obtain: !’’=-(3g/2l) ! (called the small angle approximation)

So writing !’’=-(3g/2l) ! explicitly as:

compare with

i.e. for a pivoted rod we still have SHM with angular frequency‘$’ = (3g/2l)1/2 and period ‘T’ = 2"[2l/3g]1/2

!

d2"

dt2

= #3g

2l

$

% &

'

( ) "

!

d2x

dt2

= "k.x

!ll

(d) The LC (resonant) circuit

The capacitor is initially charged to some voltage and then the switchis closed. No resistance included in this circuit - simple & undamped….

Kirchoff’s law: “the sum of the voltages around any loop of the circuitis zero.”

Vc + VL = 0 with Vc = q/C and VL = L(dI/dt)

* q/C + LdI/dt = 0

& q/C + L.q’’ = 0,since I = q’ (i.e., dq/dt)

q’’ = -q/LC , i.e.,

Once again we have an equation of SHM with $ = (1/LC)1/2 = (LC)-1/2

!

d2q

dt2

="1

LC

#

$ %

&

' ( .q

From energy considerations:

E = ! CVc2 + ! LI2 = constant

Total energy = energy stored in C + energy stored in L

i.e., a continuous exchange between electrostatic and magnetic energy.

Note also the important similarity between the DE for the LCResonant Circuit and the Mass on a Spring of spring constant ‘k’..

i.e., L q’’ + q/C = 0 and m x’’ + kx = 0

where: q 3 x, L 3 m and 1/C 3 k ….. !!!!!!!

Note also that ! CVc2 + ! LI2 = E

becomes [1/(2C)] q2 + ! L(q’)2 = E

c.f: ! k(x)2 + ! m (x’)2 = E

NB. Similarities in physics: The SHM equations for allexamples have exactly the same form, namely:

Y’’ = -,Y

from which the angular frequency ‘$’ of the ‘periodic’ or‘simple harmonic’ motion (oscillation) is given by:

$ = ,1/2

If you understand one system then you can understandand use/ find applications for lots of others with sameDifferential Equation !!

Name: John T. Costello

Function: Associate Dean (Research) - Faculty of Science

and Health

Email: john.costello@dcu.ie

URL*:http://www.physics.dcu.ie/~jtc

Phone: 7005304

Fax: 7005383

* Lecture notes will be uploaded onto this webpage after

each section is completed.

Acknowledgment: OHPs based on the textbook and lecture notes of Prof. G. King to whom the author expresses his gratitude.

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