8. Reporter: Filip Landek - PowerPoint Presentation

Slide1

8.

Reporter: Filip Landek

Sci-Fi Sound

Slide2

Table of

contents

2

Sci

–

Fi

Sound

– Table of

contents

Slide3

Problem

description:

3

„

Tapping a

helical spring

can make a

sound

like a “

laser shot

” in a science-fiction movie.

” „Investigate and explain this phenomenon.”

Sci

–

Fi

Sound

– Problem

description

Slide4

Table of

contents

4

Sci

–

Fi

Sound

– Table of

contents

Slide5

Bending

free

vibrations of

Slinky wire

5

Long

thin

beam

free bending vibrations Euler-Bernoulli theoryE – Young’s modulusρ – density (g/cm3)I – moment of inertiaS – cross-section areaSci – Fi Sound

– Theoretical model

bending

in space

bending

in time

= 0

 

Slide6

Dispersion

relation: bending wave

angular velocity 

n  wave number kn

 

n

= 0, 1,

2

, …,



6

Equation of acoustic

dispersion

ω

–

angular

velocity

(rad/s)

k –

wavenumber

(rad/m)

E –

Young

’s

modulus

ρ – material density

(g/cm3)I – moment of inertiaS –

beam cross-section area

ω

(rad/s)

k (rad/m)Sci – Fi Sound – Theoretical model

Slide7

7

Propagation

of

waves in different

mediums

 

 

 

v –

phase

velocity

f –

frequency

λ –

wavelenght

ω –

angular

frequency

k –

wavenumber

ω

(rad/s)

k

(rad/m)

Sci

–

Fi

Sound

– Theoretical model

Slide8

8

Propagation

of

waves

in

different

mediums

Dispersive

medium

Non-dispersive medium

Sci

–

Fi

Sound

– Theoretical model

Lenght

(m)

A

(mm)

Lenght

(m)

A

(mm)

Slide9

Initial

disturbance

of Slinky

wire9

w

(

x

,

t

)

+x

-xx=0propagating of initial displacement w(x,t

)

 =

D´ -

bending

stiffness

m´ -

mass

per

unit

lenght

t – time

a –

geometrical

parametar

E –

Young

’s modulusI – moment of inertia ρ – densityx - displacement Sci – Fi Sound – Theoretical model

Slide10

10

Dispersive

medium

Higher

frequencies

travel

faster

Observable time delay td 2. Time delay td will be bigger for: Longer Slinky Subsequent

echoes

3

.

Shape

of the

Slinky

is

irrelevant

Hypotheses

Sci

–

Fi

Sound

– Theoretical model

Slide11

Table of

contents

11

Sci

–

Fi

Sound

– Table of

contents

Slide12

Experiment

setup

12

Slinky

spring

Microphone

Metal

stand

Polyurethane

foam(sound isolation)

Metal base

Pendulum

Sci

–

Fi

Sound

–

Experiment

setup

Slide13

Experiment

setup

13

Sci

–

Fi

Sound

–

Experiment

setupUnstreched Slinky springSlinky stand

Slide14

Experimental

measurements

14

Qualitative

analysis

:

Case

1:

Slinky

(48 coils)a) Clamped end – Clamped endb) Clamped end – Free end Case 2: Round steel wire (19 m)a) Straight wire b) Hand-made helical spring Quantitative analysis:a) Dependency frequency  time delayb) Dependency time delay  echoc) Dependency time delay  number of coilsSci

– Fi Sound

–

Experiment

setup

Slide15

Table of

contents

15

Sci

–

Fi

Sound

– Table of

contents

Slide16

Quantitative

experimental

proof

of

acoustic

dispersion

Frequency

(Hz)

Time (s)

Sound intensity (dB)Time (s)16Sci – Fi Sound – Analysis of experimental

results

Slide17

Phase

III

Only

low frequencies remain

II

III

IV

1 a) The

anatomy

of

typical

sound recorded on Slinky clamped at both ends Analysis of

experimental results

1

17

Sci

–

Fi

Sound

–

Analysis

of

experimental

results

Phase

1

Intial

disturbance

Phase II Bending waves propagationEchoes modulated waveDampningAcoustic

disperision

Phase

IV

Silence

phase

Slide18

Phase

1

Intial

disturbance

Phase

II

Bending

waves

propagationEchoes modulated waveDampningAcoustic disperisionPhases IV & VIISilence phasePhases III & V & VI Secondary (internal) disturbances

II

1

III

IV

1 b) The

anatomy

of

typical

sound

recorded

after

hits

free

hanging

Slinky

Analysis of experimental results

VII

18

Sci

–

Fi

Sound

–

Analysis

of

experimental

results

V

VI

Slide19

Sound

comparison

19

2 b) Round wire

made

into

a

helical

spring

Frequency (Hz)Time (s)Frequency (Hz)Time (s)Time (s)Sound intensity (dB)

Time (s)

Sound

intensity

(dB)

2 a)

Straight

round

wire

Sci

–

Fi

Sound

–

Analysis

of

experimental

results

Slide20

Dependence

of time

delay

on frequency20

(

Clamped

-

Clamped

Slinky

with 80

coils)

 

t

d

– time

delay

L

Slinky

–

lenght

of the

Slinky

wire

v

f

–

velocity

of a

frequency

Sci

– Fi Sound – Analysis of experimental results

Slide21

Time

delay

between

higher and lower frequencies

in

echoes

21

 

Sci

–

Fi

Sound

–

Analysis

of

experimental

results

t

d

– time

delay

s – distance a

wave

has

travelled

v

f

– velocity of a frequencyne – number of echo

Slide22

Time

delay

between

higher and lower frequencies

in

echoes

Frequency

(Hz)

Time (s)

22

Sci

– Fi Sound – Analysis of experimental results

Slide23

Dependency

of frequency

delay

on the number of

coils

23

Sci

–

Fi

Sound

– Analysis of experimental results

Slide24

Table of

contents

24

Sci

–

Fi

Sound

– Table of

contents

Slide25

Theoretical model:

Free flexural vibrations

of a long thin beam

(

Euler-

Bernoulli

theory

)

Propagation of initial disturbance

Acoustic di

spersion Experimental results:Qualitative confirmation of the theory Delay time between higher and lower frequenciesQuantitative analysis close congruence to the computer simulationDependency time delay  no. of coils linearConclusions

25

Sci

–

Fi

Sound

–

Conclusions

and

references

Slide26

References

26

[1]

P. Gash:

Fundamental Slinky Oscillation Frequency using a

Center

-of-Mass Model

[2]

V. Hen

č

-

Bartolić, P.Kulušić: Waves and optics, School book, Zagreb, 3rd edition (in Croatian), 2004[3] A. Nilsson, B. Liu: Vibro-Acoustics, Vol.1, Springer-Verlag GmbH, Berlin Heidelberg, 2015[4] F. S. Crafword: Slinky whistlers, Am. J. Phys. 55(2), February 1987, p.130-134[5] F. S. Crafword: Waves, Berkeley Physics Course, Vol.3, Berkely, 1968

[6] W. C. Elmore, M.A. Heald:

Physics

of

waves

,

McGraw-Hill Book Company

, New York,

[7] J

. G

. Guyader:

Vibration in continuous media

,

ISTE

Ltd, London, 2002[8] G. C. King: Vibrations

and waves, John Wiley & Sons Ltd, London,

2009[9] Th. D. Rossing,

N. H., Fletcher: Principles of vibration and

sounds, Springer-Verlag New York, lnc., 2004[10]

L.E. Kinsle et.all: Fundamentals of Acoustics, 4th ed., John Willey & Sons, Inc, New York, 2000 [11] M. Géradin, D.J. Rixen: Mechanical Vibrations: Theory and Application to Structural Dynamics, 3rd ed., John Wiley & Sons, Ltd, Chichester, 2015[12] C.Y. Wang , C.M. Wang: Structural Vibration - Exact Solutions for

Strings

, Membranes

,

Beams

, and

Plates

,

CRC

PressTaylor

&

Francis

Group, Boca

R

aton

, 2014

[13] A.

Brandt

:

Noise and vibration analysis : signal analysis and experimental

procedures

,

John Wiley & Sons Ltd

, Chichester

, 2011

[14]F. S.

Crawford

, Jr. :

Waves

– Berkeley

Physics

Course

Volume

3,

Education

Development

Centre, 1965

[15]

L. E.

Kinsler

, A. R. Frey, A. B. Coppens, and A. V. Sanders,

Fundamentals of Acoustics

,

John Wiley

N

ew

York,

2000

Sci

–

Fi

Sound

–

Conclusions

and

references

Slide27

Thank you for your

attention!

27

Slide28

28

Slide29

Free hanging

Slinky

parametars

=

 

 

Coil

displacement

Centre of

mass

Slide30

Wave equation

derivation

= 0

 

1)

 

II)

Slide31

Ad.1. For

low frequency

vibration, when

the thickness (h) of beam’s cross

-

section

is

smaller

than

the vibrations wavelength n (e.g. h = 0.0025 m  kn < 420) or the dispersion relation and the phase

velocity relation have the

folowing

forms

[3,4,7]:

 

 

n

= 0, 1,

2

, …,



r

S

…

the

radi

us

of

gyration

of

beam

cross

-

section

(m)

c

S

…

the

phase

velocity

of

a

particular

point

in

a

beam

material

(m/s)

k

n

…

the

wave

number

of

n

th

bending

wave

I

…

the

rotational

inertia

moment

of

a

beam

’s

cross

-

section

surface

S …

area

of

a

beam

cross

-

section

surface

(m

2

)



n

…

the

n

th

eigencircular

frequency

of

a

bending

beam

(rad/s)

 

Low

frequency

bending

movement

of

a

beam

cross

-

sections

[

7

]

31

Slide32

Bending

at high and

low

frequencies

 

 

Low

frequency

vibrations

High

frequency

vibrations

 

ω

–

angular

frequency

E –

Young

’s

modulus

ρ

–

density

I – moment of

inertia

S –

cross

-

section

area

k -

wavenumber

Slide33

Ad. 2. For

high frequency waves

, the beam

deflection is completely determined by

transversal

and

longitudinal

waves and the dispersion and phase velocity relations showed non dispersive behaviour of the beam cross-section [11].n = 0, 1, 2, …, 

 

 

 

or

 

High

frequency

transverse

and

quasi

-

longitudinal

movement

of

a

beam

cross

-

sections

[

3

,7]

Due

to

dispersion

effect

the

lower

frequencies

had

been

recorded

,

and

heard

,

with

delayed

time

after

high

frequencies

.

33

Slide34

Dependency

of frequency

delay

on the number of

coils

34

Sci

–

Fi

Sound

– Analysis of experimental results

Slide35

35

For

both

modeled cases

the

time

function

g

n

(t) is builded from a harmonic and vanishing wave subfunctions [3]:exp

 

Mathematically

modelling

of

wave

damping

and

emitted

sound



..

t

he

wave

loss

factor

In the acoustic consideration the Slinky

wire

is

modeled as

continuous

line

sound

source

under

transversal

oscillations

.

Each

segment

of

line

(

x)

is

an

unbaffled

simple

source

which

generate

the increment of sound

pressure

pressure

level

(SPL)

in

the

air

[10].

The far field acoustic field at point p(r,

,t)

produced by line source of length L and radius

a

[10]

 

 

p(r,

,t

) …

sound

pressure

(Pa); j =

U

0,n

…

the

amplitude

of

the

wave

velocity



0

…

the

density

of

air

(



1.2 kg/m

3

)

c

a

…

the

velocity

of

sound

in

air

(



343 m/s)