Fermi Lab. Colloquium May 23, 2007 W. B. Atwood

Transcription

1 Fermi Lab. Colloquium May 23, 2007 W. B. Atwood

2 A Brief History of the Instrument Violin Given its Final Form Gasparo da Salo & Maggini (~1550) The Amati Family - Cremona, Italy Andrea ( ) - Start of School Heronymus & Antonio ( ) - 2nd Generation Nicola ( ) The Grand Amati s Best Known Antonio Stradivarius ( ) violins! Giuseppe Guarnerius ( ) - Preferrred?

3 Details of the Sound Box

4 Overview of the Structure Mersenne s Law (1636): f = 1 2L where f = Frequency; L = String Length; T = Tension; σ = Linear Mass Density T σ String σ(mg/cm) f(hz) T(dynes) Downward Force: 20 + lbs E x10 6 Tension A x10 6 D x10 6 G x10 6 TOTAL 27.3x10 6 = 61.4 lbs

5 Violin Sound: The Driver and the Response Helmholtz String Motion Bow Speed: v Bow f (fraction of string length) F(t) saw tooth Contact Point String Bridge Stop F(ω) Harmonics α 1/Ν Bow Guarneri Violin Frequency (Hz)

6 Drive + Response Different Notes Have Different Harmonic Content C (512 Hz) Since Harmonics Drop off As 1/N log( Amp ) log( ω ) Response of Box Distorts Spectrum E (660 Hz)

7 Acoustic Basics Mechanical Oscillators have the same mathematics as RLC circuits = Inductance L Mass M Resistance R Mechanical Resistance R m Capacitance 1/C Stiffness: K } { IMPEDANCE Z Z Zm jωl + R = iω L+ R+ = iωm + Rm i ωc C i K ω + j 1 Stiffness Dominated Resistance Dominated Mass Dominated Violin: Amplitude 0.1 COUPLED LINEAR OSCILLATORS Frequency (Hz)

8 Regions of the Response Spectrum Below 1000 Hz: Size of violin sets scale Only modes in which the VOLUME of the Violin changes contribute: 1 foot λ - QUASI-MONOPOLAR MODES Names: A 0, B 1-, B 1+, etc. Helmholtz Mode: A0 ( Hz) Some Whole Body Modes Below 1000 Hz Two Two good Poor sound Sound Radiators Not Radiating Excellent truely monopolar Modes Sound Radiator Large Frequency ΔV over Hz Plates move a cycle. together Small ΔV over a cycle Nodal Lines Air Movement Nodal Lines Nodal wrap Lines around Nodal wrap around Lines cut body body into cut body body into regions Freq Hz Freq. regions Hz Freq Hz Freq Hz Helmhotz Mode

9 The A 0 Mode in Motion Per Gren et al, Meas. Sci. Tech., 17 (2006) Simple Radiation Pattern Approx. uniform in all directions Simple mono-polar radiator The body is expanding and contracting (pumping)

10 Region Above 1000 Hz 1000 Hz 2500 Hz 2500 Hz Top & Back Tend to Become Decoupled Frequency Above First Bridge Resonance Modes Become Modes even more Blurred Overlapped blurred and Not Easily Modes have many Distinguishable nodal/anti-nodal areas Back Quiet Typical Mode at ~ Hz Bridge an active Radiated Sound Comes Radiator Primarily from the Top

11 Motion at 1415 Hz (Several Modes Contribute) Per Gren et al, Meas. Sci. Tech., 17 (2006) Complex Radiation Pattern Presents different sound depending on direction which is constantly changing " as if the sound was coming for everywhere at once, dancing and sparkling in the listeners ear " G. Wienwreich

12 The Violin Maker s Role Making the Minimal Structure Objective: Making the most efficient radiator or Making the lightest structure consistent with the strength and resonance requirements Wood Selection Grain Orientation - To Split or Not? - Book matching Shaping Arching Pattern Thickness

13 Circuits Violins CIRCUITS VIOLINS R, L, and C M, R m, and K INDEPENDENT COUPLED TOGETHER one piece of WOOD Example: Stiffness of a beam: K H 3 Mass of a beam: M H More Complications: Wood is NON-ISOTROPIC Strength properties described by Tensor } Natural Freq. f = K M H Diag. Elements = Young s Modulus Off Diag. Elements = Shear Mod.

14 Over-Arching Physics Principles Scaling laws: Material Properties Frequency and Response Properties (Originally due to John Schelleng, E.E.) Overall Dimensions: H = Thickness & L = Length Y = Young s Modulus (Elastic Modulus), ρ = Density, Mass M = ρhl 2 Y 3 2 c s = Speed of Sound= Stiffness (K) Freq. f = ρ H YH L cs 2 L Example: Impedance Z (keeping f & L fixed) Z = F v 1 x& && ρ ~ ( Mx) ~ ( ρhl ) f ~ ( ) f L cs Want Low Impedance More Motion More Sound

15 Wood Properties Selecting the Wood BEST BACKS BOSIAN MAPLE r = gr/cc BEST TOPS ITALIAN SPRUCE BALSA POPLAR NRW SPRUCE LINDEN USE WOOD WITH THE HIGHEST Cs/ρ VALUE PADOUK WHITE FIR PINE LARCH WALNUT CONSISTANT WITH MAHOGANY ASH STRUCTURAL REQUIREMENTS NRW MAPLE SYC MAPLE Sound Velocity Cs (cm/sec) x x10 5 C s /ρ 25 TOPS 12 BACKS r = gr/cc SCYMORE OAK 1 PEAR PERNAMBUCO DENSITY ρ (gr/cc)

16 Grain Orientation HEART WOOD SAP WOOD XYLEM (WOOD) CAMBIUM PHLOEM (BARK) TOPS Spruce: Uniform Growth ~ 1mm Annual Ring Spacing Aging: Best if > 5 Years! TWO HALVES OF A VIOLIN TOP

17 Grain Orientation Loss Factor Speed of Sound Martin Schleske, Speed of Sound and Damping..., CAS Vol 1, No 6 (Series II), 1990 Investigation of speed of sound (C s ) and damping (loss factor) as a function of cut angle: run-out Book Ending Notice the V Pattern Sets grain lines ~ parallel to local normals

18 Shaping the Plates: Part I Overall Shape (as seen for the outside): Arch or Arching Pattern Thickness Pattern (Inside Arch): Graduations Arching is sculpting in sound! Tom Croen, 1996 Cross Section of the Back of a Violin Channel Recurve Thickness C L Height Flat Arch: (Height mm)- not strong REQUIRES GOOD WOOD

19 Characterizing the Arching A s = (A + -A - ) / (A + + A - ) A s has a range from -1 to +1 A + When A s = 0 the curve is "balanced" A s > 0 the curve is "full" A s < 0 the curve is "swoopy" A - Cross Arching Locations Top/Back Averaged Area Asymmetries Results of a study on 25 Guarneri Violins. Data extracted from Guiseppe Guarneri del Gesu Peter Biddulph, London, Upper Bout Upper Block C Bout Lower Block Lower Bout S S Stradivarius: Top S Back Top S S Back S S S

20 Time Works Wonders? Top Lower Block Asymmetries Violins start life ~ symmetrically Over time the body distorts Due to the unbalanced forces generated by the bridge acting on the sound post and bass bar. This may contribute to asymmetries in modes below 1000 Hz SOUND POST Treble Bass Back Lower Bout Asymmetries Treble Bass

21 Shaping the Plates: Part II (Graduations Thicknessing) Goals: To promote Resonance (Enhance Q) Set Resonances in Assembled Instruments at Particular Frequencies. Tradition & Intuition: Use resonance properties of components as a guide - Plate tuning: Tap tones Chaldini Patterns

22 Where to remove wood: Traditional Mode 5: Receipt: 5 th Eigenmode of a Sensitivity ~.5 Violin Plate Hold Plate at Tap Plate at Desired Note ~ F-F# ( Hz) Mode 2: Tap Pate at Desired Note ~ an octave lower 2 nd Eigenmode of a Sensitivity ~ 1. Violin Plate Eigenmode Plate Tuning ( Hz) Sensitivity = f f m m Chladini Patterns

23 Same Modes Found in Top Plate 5 th Eigenmode of a Violin Plate Freq Hz 2 nd Eigenmode of a Violin Plate Freq Hz Bass Bar Profile Where to Remove Wood Plate ~ uniform thickness Bass Bar Profile and Angle Controls Modes

24 Free Plate to Assembled Instrument What matters are the Establish Eigenmodes Relationship in the Assembled by Instrument Correlation Studies Allows Relationship Maker to to Free Build Plate Eigenmodes in not obvious! Specific Frequencies F R 2 = Nodal Lines wrap around body Freq Hz R 2 = B1+ (Hz) F2 Back F2 Top Linear (F2 Back) Linear (F2 Top)

25 More Mode Correlations Small if any Correlation to B 1+ Recall that Mode Freq. go as plate thickness Creates false Correlation! F R 2 = R 2 = F5 Back F5 Top 330 Linear (F5 Back) 325 Linear (F5 Top) B1+ (Hz) R 2 = R 2 = F F2 Back 150 F2 Top Linear (F2 Back) 145 Linear (F2 Top) B1- (Hz) Both Mode 2 in Top & Back Appear to be Correlated Mode 5 again shows little Correlation

26 Mode Adjustment in Assembled Instruments Frequencies of Certain Modes can Example: B 0 & B -1 be moved in assembled instruments (poor sound radiators) Nodal Line Motion Both Modes are Bending Modes Finger Board acts as a Adding Weight to end of coupled Oscillator Finger Board Lowers Frequencies

27 Fixed Support Coupled Linear Oscillators A Simple Model Eff. FB Mass Effective Body Mass KBody KFB (Mass 1 Amp)2 (db) Masses move in Phase (Lower Frequency) Masses move with opposite Phase (Higher Frequency) 1313 Eff. FB Mass (gr) Two Modes Driving Freq. (Hz)

28 How Well does the Model Fit? Data Masses: 0 gr 8 gr added to end of FB B 0 Recorded B 0 & B -1 Adjusting Parameter M eff FB = 11 gr ν FB = 221 Hz Freq. (Hz) B Added Mass (gr) Test: What are ν FB &M eff FB Separate from the Violin? M eff FB = 10 gr Bench Added Mass ν FB = 230 Hz ω ω = m 2m

29 The Player s Domain: Controlling the Helmholtz String Motion Main Components Speed Dictates String Amplitude Pressure Too little no stick Too much no release In between - changes harmonic content Contact Point Close to Bridge increased harmonics Close to Finger Board decrease harmonics Pressure Bridge Contact Point Bow Speed: v Bow ALFs mf f (fraction of string length) Bridge f String Mid-Point Stop Stick Time: f round trip time mf Release Time: (1-f) round trip time (round trip time or period Finger = 1/freq. Board= 1/pitch) Bow ppp mp Stick Distance: v Bow Stick Time Speed Contact Point fff

30 G - Ref Contact Point Sound Comparisons G - Bridge G - ALF Contact Point Anomalous Low Frequencies ALF s In the late 1980 s a New Violin Sound was discovered by researchers and used in Concert by Mari Kimura in 1994 Defeat Helmhotz Motion by using Too Much Pressure... 2X Harmonics String Torsional Mode Increased Harmonics Bow Hair

31 Many of the Traditional Methods of Making are found to be well based on Physics Principles. The Violin is a complex, interlocking system of coupled oscillators. Disentangling Material Properties, Shaping, and Thicknessing is challenging. Beginning to understand how to quantifiably control results. Both Maker and Player have significant roles in the resulting sound.

32 Non-traditional materials Balsa wood, Composites (carbon fiber, etc.) Electric Violins More complete understanding of mode structure But the renaissance in traditional making is over. No longer economically viable.