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VIBRATIONSVIBRATIONS

FreeFree

damped anddamped and forcedforced

AP3204 Topic 1 (1/2)AP3204 Topic 1 (1/2)

ExamplesExamples

4.Torsional Pendulum

2. Compound pendulum

1. Simple pendulum

3. Mixed modes

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ExamplesExamples

6. LCR circuit

5. Between two atoms

8. A mass attached to a spring on a

frictionless track moves in simple

harmonic motion.

7. piston moving in a cylinder

Q Did you notice

these examples have much in common? What are they?

Simple Harmonic MotionSimple Harmonic Motion ((s.h.m.)s.h.m.)

Definition:

Simple harmonic motion of a

mechanical system corresponds

to the oscillation of an object

between two points for an indefiniteperiod of time, withno loss in

mechanical energy.

An object exhibits simple harmonic

motion if the net external force acting

on it is a linear restoring force:

F = -kx

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Simple Harmonic MotionSimple Harmonic Motion ((s.h.m.)s.h.m.)-- Sign and feature of the forceSign and feature of the force

Q Why minus sign?Q Why minus sign?

Without the minus sign the

acceleration will continually

increase as x, so the particle

will keep on moving faster

and faster away from its

original position.

With the minus sign an

oscillation will occur.

Q What kind of force is needed?Q What kind of force is needed?

x-F k=From

Simple Harmonic MotionSimple Harmonic Motion ((s.h.m.)s.h.m.)--Mathematical expressionMathematical expression

== xdt

xd

2

2

aand Newton's 2

nd

law: ma=F

: Angular frequency, the constant in the s.h.m.

equation. It defines the time taken for an oscillation

we have

0xx 2 =+

mk

=2

T

2=

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0xx2

=+

Solution of

)sin( += tAx

Q Can you give other solutions of x?

Simple Harmonic Motion (Simple Harmonic Motion (s.h.m.)s.h.m.)--Mathematical expressionMathematical expression

)t(A +cos

tCtB sincos +

ti*ti eDDe +

QQ Check that these satisfy the equation.Check that these satisfy the equation.

QQ How are B and C related to A andHow are B and C related to A and ??

]Re[ tiAe

0xx 2 =+

Simple Harmonic Motion (Simple Harmonic Motion (s.h.m.)s.h.m.)--Alternative solutions for s.h.m.Alternative solutions for s.h.m.

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)sin( += tAx

)(

)cos(v

22xA

tAx

=

+==

)sin(2 +== tAxa &&

Simple Harmonic Motion (Simple Harmonic Motion (s.h.m.)s.h.m.)-- Velocity and accelerationVelocity and acceleration

From

Simple Harmonic MotionSimple Harmonic Motion ((s.h.m.)s.h.m.)

-- TerminologyTerminology

A few terms relative to harmonic motions:

1. The amplitude A:maximum distance that an object

moves away from its equilibrium positions.

2. The period T: the time for one complete cycle of themotion.

3. The frequency : the number of cycles or vibrationsper unit of time.

4. Phase angle : the fraction of a cycle (2); theoscillation is out of phase with some reference time.

)sin( += tAx

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Simple Harmonic MotionSimple Harmonic Motion ((s.h.m.)s.h.m.)--Graphical representationAn experimental apparatus

for demonstrating simple

harmonic motion.

- A pen attached to the

oscillating mass traces out a

sine wave on the moving

chart paper.

Q Can you find if its

trace is consistent with

its mathematical

representation?

)sin( += tAyT

2

=

y0=

The value of the

phase constant

depends on the

initial displacement

and initial velocity

of the body.

Simple Harmonic MotionSimple Harmonic Motion ((s.h.m.)s.h.m.)--Graphical representation

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)cos(v +==

tAx

The velocity graph will be a cosine curve with amplitude A.

)sin(2 +==

tAxa

The acceleration graph will be a minus sine curve with amplitude

2A.

Simple Harmonic MotionSimple Harmonic Motion ((s.h.m.)s.h.m.)--What will graphs for velocity and acceleration look like?What will graphs for velocity and acceleration look like?

Yes.

Angular velocity is the rate of rotation around a fixed point.

For instance a swinging pendulum has a variable angular

velocity but a constant angular frequency.

A mass vibrating on a spring has no angular velocity.

Only in the case of a particle moving steadily around a

circle are the two quantities correlated constants.

Q Is angular frequency different fromangular velocity?

~ v

v

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QQ Why is s.h.m. important?Why is s.h.m. important?

Any oscillation can be modelled as

being made up of a number of s.h.m. components

which can be analysed separately and

then recombined using the Principle of Superposition.

etc(t)x(t)xx(t) 21 ++=

Fourier theoremFourier theorem

Fourier theorem:Fourier theorem: Any periodic function can beAny periodic function can be

expressed as a sum of the sine and cosine functionsexpressed as a sum of the sine and cosine functions

whose frequencies increase in the ratio of naturalwhose frequencies increase in the ratio of natural

numbers. i.e.numbers. i.e.

f(t+f(t+nTnT)=f(t);)=f(t); n=0,n=0,1,1,2,2,3,3,

can be expanded in the form:can be expanded in the form:

( )

=

=

++=11

0 )sin()cos(2

1

n

n

n

n tnbtnaatf

T

2=

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( )

=

=

++=11

0 )sin()cos(2

1

n

n

n

n tnbtnaatf

where

+

=

Tt

t

n tdtntfT

a0

0

cos)(2

n=0, 1, 2,

+

=

Tt

t

n tdtntfT

b0

0

sin)(2

n=1, 2, 3,

(t0 is arbitrary, may be 0)

The important predictions of

Fourier theorem can be illustrated

by discussing the analytically simple

example of a square vibration.

ExampleExample

Modeling a Square WaveModeling a Square Wave

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Sample problem:Sample problem:

Fourier expand the function defined by the following

equation:

f(t) = -A for T/2 < t < 0

= +A for 0 < t < T/2

f(t+T) = f(t)Ans: As the function is an odd function, an = 0

[ ]nT

nn

AtdtnA

Tb )1(1

2sin2

22/

0

==

- 1.5 - 0.5- 1 1.510.50

T

t

f(t)

S3S2

S1

[ ]=

=,...3,2,1

sin)1(112

)(n

ntn

n

Atf

[ ]...5sin3sinsin4

51

31 +++= ttt

A

tA

s

sin4

1=

[ ]ttA

s

3sinsin4

31

2 +=

...

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Other cases of SHM2. Compound pendulum1. Simple pendulum

A uniform rod of

mass m and length

L is pivoted about

one end and

oscillates in a

vertical plane.

Assume its moment

of inertia is J.

3.Torsional Pendulum

A rigid body suspended by a wire attached at the top of afixed support. When the body is twisted through some small

angle , the twisted wire exerts a restoring torgue on the

body proportional to the angular displacement.

(A) Write down the equation of motion for(A) Write down the equation of motion for

each of the following cases.each of the following cases.

(B) What are their natural frequencies?(B) What are their natural frequencies?

Sample problem:Sample problem: Simple pendulum

2

2

sindt

sdmmgFs ==

L

g=)sin( += tm

Ls =

0=+

mgmL

For small : sin ~

0=+

gL

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Sample problem:Sample problem: Compound pendulum

0

2

=+

mgL

J

J

mgL

2=

)sin( += tmA uniform rod of mass m and length L

is pivoted about one end and oscillates

in a vertical plane.

Assume its moment of inertia is J.

Moment produced by gravity:

mg sinL/2

Rotation: Jd2/dt2

Sample problem:Sample problem: Torsional Pendulum

0=+

cJ

J

c=

)sin( += tm

A rigid body suspended by a

wire attached at the top of a

fixed support. When the body is

twisted through some small

angle , the twisted wire exerts

a restoring torgue on the body

proportional to the angular

displacement.

The angular displacement: -c

c is torsion constant.

According to Newtons law for

rotational motion

-c= Jd2/dt2

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Energy in anEnergy in an S.H.M.S.H.M.

The maximum KEThe maximum KE

S.H.M. represents an ideal vibration where there

is no energy loss, so the sum of KE to PE stays

constant..

Continual cycle from KE to PE and back again.

At any one time the total energy is the sum of the

kinetic energy plus the potential energy.

22

02

1Am=max(KE)

QQ What is the value ofWhat is the value of PEPEmaxmax??


QQ Where is KE a maximum?Where is KE a maximum?

QQ Where is PE a maximum?Where is PE a maximum?

Variation of potential energy, kinetic energy and total energy

with displacements for a pendulum

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How do these energies vary with time?

The variation will be assin2t andcos2t

i.e. Epot

= E0

sin2t

Ekin

= E0

cos2t

Etotal

= E0

sin2t + E0

cos2t = E0

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Sample problem:Sample problem: Oscillations on a SurfaceOscillations on a SurfaceA 0.5 kg cube connected to a light spring for which theforce constant is 20.0 N/m oscillates on a horizontal,frictionless track. (a) Calculate the maximum speed of thecube and the total energy of the system if the amplitudeof the motion is 3.0 cm. (b) What is the velocity of thecube when the displacement is 2.0 cm?

Solution:(a)

Vmax= A=0.03(20.0/0.5)1/2

=0.190m/s

E=K+U=Kmax=UmaxKmax=m(Vmax)

2=0.50.50.1902=9.010-3J

sm

k

/32.65.0

20

===

(b) x=2.cm

x=Acostsint=(1-x2/A2)1/2

V=Asint=0.036.320.745=0.141 m/s

SUMARRYSUMARRY Free vibrationsvibrations

Simple Harmonic Motion (s.h.m.)Simple Harmonic Motion (s.h.m.)

Graphical representation of system behaviourGraphical representation of system behaviour

Examples of s.h.m. systemsExamples of s.h.m. systems

Vibration energy - Energy in anEnergy in an S.H.M.S.H.M.

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