Investigating Sci-Fi Laser Shot Sounds in Helical Springs

1. Problem 8: „Sci-Fi Sound” IYPT 2019 Team Croatia

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3. Problem description: • „Tapping a helical spring can make a sound like a “laser shot” in a science-fiction movie.” • „Investigate and explain this phenomenon.”

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5. The sound of a laser shot from SF moviescan be produced by causing high-frequency vibrations in a long rod or a wire, made of dispersivemediumwith propertythatthe velocity of soundwaves v is increased for higherfrequenciesanddecreasedforlowerfrequencieswithanmeasurabledelaying time TDbetweenhigherandlowerpartofthefrequencyspectrum. Thedelay time TDdependsfrom … Hypothesis

6. Propagationofvibrationin a dispersivemedium Theoretical model Nondispresivematerial Dispresivematerial Propagationofdisturbances Exampleof a dispersiondiagram

7. The sounddispersioncanbeobservedfromthefollowingfacts: • The all vibrationsinimpactedmaterialwerecaused at the same initial time, but in a dispersivematerial, thevibrations, andtheirsounds, on higherfrequencies (H) traveledfasteralong rod andthey, andtheyhavebeenhearedbeforevibrations at smallerfrequencies (L) [4]. • The phasevelocity ofmovement (v, m/s) ofa particularpoint on thevibrating rod depends on circularfrequency (), or wavenumber (k)of a wavewhich is traveledalongrod. • 3. Themovementof a groupofwaveswithdifferentphasevelocitiesformanenvelopewhichcontains all wavesandtraveledalong rod withthegroupvelocity (vg, m/s) which is: The propagation of the modulated wave(x,t) in a dispersive medium [8]

8. Slinkyspring as a longandstiffvibrating rod TheSlinkyspring is madefrom a stiffprestressedwire, soit is modeled as a longstiffbeamwithcros-sectionaldimensionsbandhandlengthL [4, 5]. ImpulsiveexcitationoftheverticallyhangingSlinkyspringwasbymadebya simplependulumimpact b Fh h Fv DirectionsofforcesduringimpactSlinkywire UnstrenchedSlinky FreehangingSlinkySpring

9. Bendingfreevibrations of Slinkywire Deformation due to bending of a beam elementwithconstantcross-section [7,11] Initiallytapped or impacted Slinky without additional external force, can be described as a long thin beam with propagating an initial impuls and freebendingvibrationsin accordance to the Euler-Bernoullitheorywhiletakingintoaccountrotationalinertia of the cross-sectionbeam [3]: w(x,t) = 0 =EI bh= S -x x=0 +x Propagatingofinitialdisplacementw(x,t) in a dispersivemeadiadependentfrom timeandposition on therod [3]

10. The solutions of the wave differentially equation for free bending beam consists from the productofthetime harmonic time functions g(t) and the space functions (x)whosecoefficientscanbedeterminedfromboundaryandinitialconditions.[3,7]: n = 0, 1, 2, …,  + B + Dn + Encosh + Fnsinh n = 0, 1, 2, …,  Thebeamfreebendingvibrationscan haveonly some specialvaluesofangularfrequenciesnknown as naturalor eigenfrequenciesn (n = 0, 1,…) which are constantsoftheobservedvibratorysystemandconnect time andspacefunctionsbyrelation [7] : const.2 Slowdown (2x) showoftheSlinky’s bottomendvibrationsafterhitswithpendulum.

11. Also, dependencebetweenbendingwaveeigenfrequencynandthewavenumber knis givenwiththedispersionrelation [4 - 7]: n = 0, 1, 2, …,  Thephasevelocity vbofharmonicmovementofanithpoint, at thebeamlength 0< x<L [3,7]: n = 0, 1, 2, …,  Independencefromtheratioofthetermsinthenominatorthetwobordercasescanbeformulated: one for lowerfrequencies, andtheother for higher.

12. Ad.1. For lowfrequencyvibration, whenthethickness(h) ofbeam’s cross-section is smallerthanthevibrationswavelengthn (e.g. h = 0.0025 m  kn < 420) or thedispersionrelationandthephasevelocityrelationhavethefolowingforms [3,4,7]: n = 0, 1, 2, …,  rS… the radius of gyrationofbeamcross-section (m) cS… thephasevelocityof a particularpointin a beammaterial (m/s) kn … thewavenumberofnthbendingwave I … therotationalinertia moment of a beam’s cross-sectionsurface S … areaof a beamcross-sectionsurface (m2) n… thentheigencircularfrequencyof a bendingbeam(rad/s) Lowfrequencybendingmovementof a beamcross-sections [7]

13. Ad. 2. For highfrequencywaves, thebeamdeflection is completelydeterminedbytransversalandlongitudinalwavesandthedispersionandphasevelocityrelationsshowednondispersivebehaviourofthebeamcross-section [11]. or n = 0, 1, 2, …,  Due to dispersioneffectthelowerfrequencies had beenrecorded, andheard, withdelayed time afterhighfrequencies. Highfrequencytransverseandquasi-longitudinalmovementof a beamcross-sections [3,7]

15. Mathematicallymodeledbendingcases CASE 1: Bendingwaves on a verticallyfreehangingSlinkymodeled as a beamwithupperclampedandbottomfreeend (auniformcantilever beam) withthewavenumberkn n = 0, 1, 2, …,  Exampleofthe 1st to 5thbendingmodes (x/L)for a freevibratingclamped-freeendbeam CASE 2: Bendingwaves on a verticallyhangingandfullelongatedSlinkymodeled as a beamwithbothendclamped (auniformclamped-clampedbeam) with the wave number kn n = 0, 1, 2, …,  Exampleofthe 1st to 5thbendingmodes (x/L)for a freevibratingclamped-freeendbeam

16. For bothmodeledcasesthe time functiongn(t) is buildedfrom a harmonicandvanishingwavesubfunctions [3]: Mathematicallymodellingofwave damping andemittedsound  .. thewavelossfactor In the acoustic consideration the Slinky wire is modeled as continuouslinesoundsourceundertransversaloscillations. Each segment ofline (x) is anunbaffledsimplesourcewhichgeneratethe increment of sound pressurepressurelevel (SPL) in theair [10]. exp p(r,,t) … soundpressure (Pa); j = U0,n … the amplitude ofthewavevelocity 0 … thedensityofair (1.2 kg/m3) ca … thevelocityofsoundinair (343 m/s) The far field acoustic field at point p(r,,t) produced by line source of length L and radius a[10]

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18. Experimentalmeasurements

19. Properties of theusedSlinkyhelixoidalspring • Total numberofcoils: N = 86 • The out diameter of unstrenched slinky Dout = 68.95 mm • Themeasured total mass m= 0.2156kg • Dimensionofcross-section: bxh = 2.50 x 0.50 mm • The total length of Slinky wire L = 18.8293 m • ThesinglecoilSlinky’s springconstant (calculated) • kc = 75.125 N/m • The Slinky springconstant (calculated) Kq= 2.046 N/m • TheYoungmodulusofsteel E = 21011 Pa • Steel density  = 7800 kg/m3 • Poisson’s coefficient  = 0.3 Case 1. Freehanging Slinky Case 2. Slinky with clampedends

20. Propertiesofthependulum • Themeasuredmassofsteelpendulumballmp = 0.03267 kg • Themeasureddiameterofsteelpendulumballdp = 20 mm. • Themeasuredlengthofpendulumstring Lp = 0.845 m • The measured distance betweenpendulum at rest and Slinky • Lps = 0.100 m • Themeasuredpendulumoscilation amplitude La = 0.200 m • Thecalculatedvelocity of pendulum ball at the impactpoint • v = 0.3395 m/s • Thecalculatedforcesofthependulumimpact • Fh= 0.0648 N, Fv= 0.0077 N • The duration of impact between pendulum ball and Slinky, • ti= 0.170 s • The kinetic energy of pendulum ball transmitted to the Slinky • Ek= 0.0056 J

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22. Proofof acousticdispersion Frequency (Hz) Time (s) Soundintensity (dB) Time (s)

23. Analysisofexperimentalresults The anatomy of typicalsoundrecordedonSlinkyclamped at bothends 4 3 1 2 Phase 2. Resultingbendingwavespropagatethrough the Slinkywire, rebounding from it’s ends and forming a modulatedwavewhich is constantlydampned Acousticdispersiontakes place, resulting in a audible laser shotsound. Phase 1. Intialdisturbance of the slinkywire and decompositionof elasticdeformations in a wire segment Phase 3. The wavescontinue to propagatethroughout the Slinky but are almostinaudibledue to dampning Phase 4. Silencephase with lowfrequencyoscilations

24. Analysisofexperimentalresults TheanatomyoftypicalsoundrecordedafterhitsfreehangingSlinky 1 3 3 4 4 3 2 Phase 2. Resultingbendingwavespropagatethrough the Slinkywire, rebounding from it’s ends and forming a modulatedwavewhich is constantlydampned Acousticdispersiontakes place, resulting in a audible laser shotsound. Phase 3. Formationofthesecondary (internal) disturbancesbyimpactslastsepparatedcoilswithgroupofunsepparatedcoilsor wireholders. Phase 1. Intialdisturbance of the slinkywire and decomposition of elasticdeformations in a wire segment Phase 4. Silencephasewithlowfrequencyoscilations , withoutimpactsbetweencoilsandwith

25. Time delaybetweenhigher and lowerfrequencies Frequency (Hz) Time (s)

26. Dependency of frequencydelayon the number of coils

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28. Conclusions The Slinkyspring as a highfequencysoundsource was mathematicallymodelledusing: • waveequation for freeflexural (F-wave) vibrations of a thinlongbeam with upperendclamped and bottomendfree • Relation for acousticdipersion • Equation for acoustic pressure at the free space. The experimentsqualitatively confirmedthe theoretical model, showing the phenomenom of acousticdispersionwhich is visibleindelay time betweenhigher and lowerfrequencies. Frequency (Hz) Time (s)

29. Conclusions The experimentalresultsshow a linear-likerelationbetweenechoes and depedency time as well as frequencydelay time and number of coils.

30. REFERENCES [1] P. Gash: Fundamental Slinky Oscillation Frequency using a Center-of-Mass Model [2] V. Henč-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: Physicsofwaves, McGraw-Hill Book Company, New York, [7] J. G. Guyader: Vibration in continuous media, ISTE Ltd, London, 2002 [8] G. C. King: Vibrationsandwaves, John Wiley & Sons Ltd, London, 2009 [9] Th. D. Rossing,N. H., Fletcher: Principlesofvibrationandsounds, 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: StructuralVibration- Exact Solutions forStrings, Membranes,Beams, and Plates, CRC PressTaylor & FrancisGroup, Boca Raton, 2014 [13] A. Brandt: Noise and vibration analysis : signal analysis and experimental procedures, John Wiley & Sons Ltd, Chichester, 2011

32. SoundwaveforminthePhase 1 - Formation of an initial (external) disturbances A soundrecord in deafroom. FreehangingSlinkywithNfree=75 hitsbysteelpendulum (hanging on thestring 0.6 m long, max. amplitude ofpendulumbefore hit 0.48 m, distance betweenverticallyhangingpendulumandSlinkybeforeimpact 0.34 m). Soundanalysis /waveformandfrequencyspectrumconductedinthesoftwareSonicVisualiser.

34. SoundwaveforminthePhase 2- Decompositionofelasticdeformations

36. SoundwaveforminthePhase 3 – Generation, reflexing(and attenuation ) of bending vibrations

38. SoundwaveforminthePhase 4- Silence phase

39. SoundwaveforminthePhase 5- Formation of the secondary(internal) disturbances

41. Analysisofexperimentalresults TheanatomyoftypicalsoundrecordedafterhitsfreehangingSlinky 5 2 4 3 1 3 Phase 4. Silencephasewithlowfrequencyoscilations , withoutimpactsbetweencoilsandwith Phase 3. Generation (andattenuation ) offreebendingvibrationssuperposedwithreflectionsofdisturbancesandwavesfrombeamends, withfrequencytwinningandseparation Phase 5. Formationofthesecondary (internal) disturbancesbyimpactslastsepparatedcoilswithgroupofunsepparatedcoilsor wireholders. Phase 1. Formationof an initial (external) disturbances, byimpacts or tapping a Slinkywire . Wirematerialwaselasticdeformedaroundimpactpoint. Phase 2. Decompositionofelasticdeformationsin a wire segment, withhighspeedpropagationofdisturbanceoverwholewirelength