Problem 8: „ Sci-Fi Sound - PowerPoint Presentation

Slide1

Problem 8: „Sci-Fi Sound”

IYPT

2019

Team Croatia

Slide2

Table of contents

Slide3

Problem description:

„

Tapping a helical spring can make a sound like a “laser shot” in a

science-fiction

movie.

”

„

Investigate and explain this phenomenon.

”

Slide4

Table of contents

Slide5

The sound of a laser shot from SF

movies

can be produced by causing high-frequency vibrations in a long

rod

or a wire

,

made of

dispersive medium with property that the velocity of sound waves v is increased for higher frequencies and decreased for lower frequencies with an measurable delaying time TD between higher and lower part of the frequency spectrum. The delay time TD depends from …

Hypothesis

Slide6

Propagation

of

vibration

in

a

dispersive mediumTheoretical modelNon dispresive materialPropagation of disturbances

Dispresive

material

Example

of

a

dispersion

diagram

 

 

 

 

 

Slide7

The

sound

dispersion

can

be

observed from the following facts:The all vibrations in impacted material were caused

at

the

same

initial

time, but

in

a

dispersive

material

,

the

vibrations

, and their

sounds, on

higher frequencies (H) traveled

faster along

rod and

they, and

they have been heared

before vibrations

at smaller frequencies (L) [4].The phase velocity of movement

(v, m

/s)

of

a particular point on the vibrating rod depends on circular frequency (), or wave number (k) of a wave which is traveled along rod. 3. The movement of a group of waves with different phase velocities form an envelope which contains all waves and traveled along rod with the group velocity (vg, m/s) which is:

The propagation of the modulated wave (x,t) in a dispersive medium [8]

 

Slide8

Slinky

spring

as a

long

and

stiff vibrating rod Unstrenched SlinkyFree

hanging

Slinky

Spring

The

Slinky

spring

is

made

from a stiff

prestressed wire, so it is modeled as a long stiff beam with cros

-sectional dimensions

b and h and length L

[4, 5].

b

h

F

v

FhDirections of forces during impact Slinky wireImpulsive excitation of the vertically

hanging Slinky spring

was by made by

a simple pendulum impact

Slide9

Bending free

vibrations

of

Slinky

wire

Deformation due to bending of a beam element

with constant cross-section [7,11]Initially tapped or impacted Slinky without additional external force, can be described as a long thin beam with propagating an initial impuls and free bending vibrations in accordance to the Euler-Bernoulli theory while taking into account rotational inertia of the

cross

-

section

beam

[3

]:

w

(

x

,

t

)

+x

-x

x=0Propagating of

initial

displacement w(x,t

) in a

dispersive meadia dependent from

time and

position on the rod [3]

= 0

 

 

=EI  bh= S 

Slide10

Slowdown

(2x)

show

of

the

Slinky’s bottom end vibrations after hits with pendulum.The solutions of the wave differentially equation for free bending beam consists from the product of the time harmonic time functions g(t) and the space functions (x) whose coefficients can be determined from boundary and initial conditions.

[3,7]:

const

.

2

 

n

= 0, 1,

2

, …,



 

+

B

 

+

D

n

+

E

n



cosh

+ Fnsinh n = 0, 1, 2, …, The beam free bending vibrations can have only some special values of angular frequencies n known as natural or eigenfrequencies n (n = 0, 1,…) which are constants of the observed vibratory system and connect time and space functions by relation [7] :

Slide11

Also, dependence between

bending

wave

eigenfrequency

n and the wave number kn is given with the dispersion relation [4 - 7]:

 

n

= 0, 1,

2

, …,



The

phase

velocity

v

b

of harmonic movement of an

ith point, at the beam length

0< x<L [3,7]:

 

In

dependence

from

the

ratio of the terms in the nominator the two border cases can be formulated: one for lower frequencies, and the other for higher. n = 0, 1, 2, …, 

Slide12

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 radius of gyration of beam cross-section (m)cS … the phase velocity of

a particular point in a beam

material (m/s)kn …

the

wave number of nth bending waveI … the rotational inertia moment of a beam’s cross-section surfaceS … area of a beam cross-section surface (m2)n … the nth eigencircular frequency of a bending beam (rad/s) Low frequency bending movement of a beam cross-sections [7]

Slide13

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.

Slide14

 

 

Slide15

Mathematically

modeled

bending

cases

CASE

1: Bending waves on a vertically free hanging

Slinky

modeled

as a

beam

with

upper

clamped

and

bottom free end (a uniform

cantilever beam) with the wave number

kn

 

n

= 0, 1, 2, …, 

Example

of the 1st to 5th bending modes  (x/L) for

a free vibrating clamped-free

end beam

CASE

2: Bending waves on a vertically hanging and full elongated Slinky modeled as a beam with both end clamped (a uniform clamped-clamped beam) with the wave number knExample of the 1st to 5th bending modes  (x/L) for a free vibrating clamped-free end beam n = 0, 1, 2, …, 

Slide16

For

both

modeled

cases

the

time function gn(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 = U0,n … the amplitude of the wave velocity0 … the density of air (1.2 kg/m3)c

a … the velocity of sound in air (343 m/s)

 

Slide17

Table of contents

Slide18

Case

Number

of Slinky

coils

included

in

bending

vibrations, NfreeNo. 1: Slinky helical spring with Clamped end - Free end48No.2: Slinky helical spring with Clamped end – Clamped end48Experimental measurementsCaseLength or

number

of

coils

No.3

Straight

steel

wire

clamped

on both endsd = 1.20 mm L = 1.55 m No. 4 Home-made

hellical spring with Clamped end - Free endSteel wire d = 1.20 mm with the Inner diameter of helix D = 72 mm Number of coils:

N = 86Lfree hanging = 1.880 mLtotal

= 18.80 m

Slide19

CASE 2. Number of coils free for bending N

free

at the vertically hanging full elongated Slinky at distance

L

12

= 1.885 m

:

2.1) 80

separated coils, 1 coil clamped at the bottom and 5 coil clamped at the upper end2.2) 70 separated coils, 1 coil clamped at the bottom and 15 coils clamped at the upper end2.3) 60 separated coils, 1 coil clamped at the bottom and 25 coils clamped at the upper endCASE 1. Number of coils free for bending Nfree at the vertically free hanging Slinky:1.1) 80 separated coils, 5 unseparated coils at the bottom

end

, 1

coil

clamped

at

the

upper

end

1.2) 70

separated

coils, 5 unseparated coils at the bottom end, 11 coils clamped at the

upper end1.3) 60 separated coils, 5 unseparated coils at the bottom end, 21 coils clamped at

the upper endAnalysed

cases (by mathematical modeling and experimentally testing)

Slide20

Properties of the used

Slinky

helixoidal

spring

Total

number of coils: N = 86The out diameter of unstrenched slinky Dout = 68.95 mmThe measured total mass m = 0.2156 kgDimension of cross-section: b x h = 2.50 x 0.50 mmThe total length of Slinky wire L = 18.8293 mThe single coil Slinky’s spring constant (calculated) kc = 75.125 N/m The Slinky spring constant (calculated) Kq =

2.046 N/m

The

Young

modulus

of

steel

E = 2

10

11

PaSteel density  = 7800 kg/m3Poisson’s

coefficient  = 0.3

Case 1. Free hanging Slinky

Case 2. Slinky with clamped ends

Slide21

The measured

mass

of

steel

pendulum

ball mp = 0.03267 kgThe measured diameter of steel pendulum ball dp = 20 mm.The measured length of pendulum string Lp = 0.845 mThe measured distance between pendulum at rest and Slinky Lps = 0.100 mThe measured pendulum oscilation amplitude La = 0.200 mThe calculated velocity of pendulum ball at the impact point v = 0.3395

m/s

The

calculated

forces

of

the

pendulum

impact

Fh = 0.0648 N, Fv =

0.0077 NThe duration of impact between pendulum ball and Slinky, ti = 0.170 sThe kinetic energy of pendulum ball transmitted to the

Slinky Ek= 0.0056 J

Properties of the pendulum

Slide22

Table of contents

Slide23

Proof

of

acoustic

dispersion

Frequency

(Hz)

Time (s)Sound intensity (dB)Time (s)

Slide24

Phase

1.

Intial

disturbance

of the

slinky

wire and decomposition of elastic deformations in a wire segment Phase 3. The waves continue to propagate throughout the Slinky but are almost inaudible due to dampningPhase 2. Resulting bending waves propagate through the Slinky wire, rebounding from it’s ends and forming a modulated

wave

which

is

constantly

dampned

Acoustic

dispersion

takes

place,

resulting in a audible laser shot sound.Phase

4.Silence phase with low frequency oscilations

2

34

The anatomy of typical

sound recorded on Slinky clamped at

both ends Analysis

of experimental results

1

Slide25

Phase

1.

Intial

disturbance

of the

slinky

wire and decomposition of elastic deformations in a wire segment Phase 2. Resulting bending waves propagate through the Slinky wire, rebounding from it’s ends and forming a modulated wave which is constantly dampned

Acoustic

dispersion

takes

place,

resulting

in a

audible

laser

shot

sound.

Phase 4.Silence phase with

low frequency oscilations , without impacts between coils and with

Phase 3. Formation of the secondary

(internal) disturbances by impacts last sepparated coils with group of unsepparated coils

or wire holders.

2

1

3

4

The anatomy of typical sound recorded after hits free hanging Slinky Analysis of experimental results

3

4

3

Slide26

Time delay

between

higher

and

lower

frequencies

Frequency (Hz)Time (s)

Slide27

Dependency of frequency

delay

on

the

number

of

coils

Slide28

Table of contents

Slide29

The

Slinky

spring

as a

high

fequency

sound source was mathematically modelled using: wave equation for free flexural (F-wave) vibrations of a thin long beam with upper end clamped and bottom end freeRelation

for

acoustic

dipersion

Equation for acoustic pressure at the free space.

The

experiments

qualitatively

confirmed

the theoretical model,

showing

the phenomenom of acoustic dispersion which is visible in

delay time between higher and lower frequencies.Conclusions

Frequency

(Hz)Time (s)

Slide30

The

experimental

results

show

a

linear

-like relation between echoes and depedency time as well as frequency delay time and number of coils.Conclusions

Slide31

REFERENCES

[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-VerlagNew 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 forStrings, Membranes,Beams, and Plates, CRC PressTaylor & Francis Group, Boca Raton, 2014[13] A.

Brandt: Noise and vibration analysis : signal analysis and experimental procedures, John Wiley & Sons Ltd, Chichester, 2011

Slide32

Slide33

A sound record in

deaf

room

.

Free

hanging

Slinky with Nfree=75 hits by steel pendulum (hanging on the string 0.6 m long, max. amplitude of pendulum before hit 0.48 m, distance between vertically hanging pendulum and Slinky before impact 0.34 m). Sound analysis /wave form and frequency spectrum conducted in the software Sonic Visualiser.

Sound

waveform

in

the

Phase

1

-

Formation

of an initial (external)

disturbances

Slide34

Slide35

Sound waveform in

the

Phase

2

-

Decomposition

of elastic deformations

Slide36

Slide37

Sound

waveform

in

the

Phase 3 – Generation, reflexing (and attenuation ) of bending vibrations

Slide38

Slide39

Sound waveform in

the

Phase

4

-

Silence

phase

Slide40

Sound

waveform

in

the

Phase 5 - Formation of the secondary (internal) disturbances

Slide41

Slide42

Phase

1.

Formation

of

a

n

initial

(external) disturbances, by impacts or tapping a Slinky wire .Wire material was elastic deformed around impact point. Phase 3. Generation (and attenuation ) of free

bending

vibrations

superposed

with

reflections

of

disturbances

and waves from beam ends, with frequency twinning

and separation

Phase 2. Decomposition of elastic

deformations in a wire segment, with high speed propagation of disturbance over whole

wire length Phase

4.Silence phase with low frequency

oscilations , without impacts between coils

and with Phase 5. Formation of the secondary (internal) disturbances

by impacts last sepparated coils

with group of unsepparated coils or wire holders.

2

1

3

453The anatomy of typical sound recorded after hits free hanging Slinky Analysis of experimental results