Double-reed physics Almeida Motivations Objective
Double-reed physics and sound synthesis
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
A. Almeida Directeurs: X. Rodet, R. Caussé (IRCAM) Encadrant: C. Vergez (LMA)
Flow model Synthesis Implementation Results
Summary Perspectives
IRCAM – Centre Georges Pompidou Instrument Acoustics and Analysis / Synthesis Teams Université Pierre et Marie Curie – Paris 6
June 26, 2006 – PhD dissertation defence
Motivations Double-reed physics Almeida Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Growing interest in physical model based sound synthesis Publications on double reeds are scarce (compared to single reeds and work on resonators) Increase knowledge
Interest from musicians using and making double-reeds
Objectives of the PhD Double-reed physics Almeida Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Do measurements to (in)validate existing reed models Propose modifications to generic model Verify the importance of these modifications in simulated sounds
Outline Double-reed physics Almeida
1
Introduction What are double reeds? Reed instrument models
2
Experiments Global reed characteristics Reed mechanics Flow details
3
Flow model
4
Sound synthesis Implementation Results
Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Outline Double-reed physics Almeida
1
Introduction What are double reeds? Reed instrument models
2
Experiments Global reed characteristics Reed mechanics Flow details
3
Flow model
4
Sound synthesis Implementation Results
Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Double-reed instruments Double-reed physics Almeida
Instruments in the orchestra using double-reeds:
Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
oboe
bassoon
Double-reed Role in the instrument Double-reed physics Almeida Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
How are they played? Double-reed physics Almeida Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Reed is soaked to adjust elastic properties Teeth gently press lips against the reed blades A tip of 1 to 2 mm is left free inside the mouth
How are they played? Double-reed physics Almeida Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Reed is soaked to adjust elastic properties Teeth gently press lips against the reed blades A tip of 1 to 2 mm is left free inside the mouth
Teeth
Reed blade
Lip Staple
How are they played? Double-reed physics Almeida Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Reed is soaked to adjust elastic properties Teeth gently press lips against the reed blades A tip of 1 to 2 mm is left free inside the mouth
1 ~ 2 mm
Outline Double-reed physics Almeida
1
Introduction What are double reeds? Reed instrument models
2
Experiments Global reed characteristics Reed mechanics Flow details
3
Flow model
4
Sound synthesis Implementation Results
Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Elementary model Mechanics Double-reed physics
Linear spring model for the reed opening
Almeida
Motivations
pm
Objective
Introduction Double reeds Models
Experiments
S
p r
Global reed characteristics Mechanics Flow details
Flow model
pm − pr = kS (S0 − S)
Synthesis Implementation Results
Summary Perspectives
Questions Is the linear model accurate? Can the reed opening adjust immediately to a change in pressure?
Flow Double-reed physics
Bernoulli model at the reed entrance
Almeida Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics
pm
p r
?
p
(∆p)r
Mechanics Flow details
Flow model Synthesis
1 2 1 = pr + ρur2 pm + ρum 2 2
Implementation Results
Summary Perspectives
Questions Is energy conserved? Is the pressure maintained along the reed and staple? What happens for time variations in pressure?
Non-linear characteristics Double-reed physics
PT
Almeida
0.25
Introduction Double reeds
(l/s)
Motivations Objective
Global reed characteristics Mechanics
Flow
Models
Experiments
0.2 0.15 0.1 0.05
Flow details
0
Flow model Synthesis
0
5
10 Pressure
Implementation Results
Summary Perspectives
PM
Remarks Dynamic effects neglected PT = PM /3
15 (kPa)
20
Outline Double-reed physics Almeida
1
Introduction What are double reeds? Reed instrument models
2
Experiments Global reed characteristics Reed mechanics Flow details
3
Flow model
4
Sound synthesis Implementation Results
Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Reed characteristics Overview Double-reed physics Almeida
Objective
Motivations Objective
Introduction Double reeds
Measure the non-linear reed characteristics in quasi-static regime, and compare it to the elementary model
Models
Experiments Global reed characteristics Mechanics Flow details
Flow model
Requirements Reed must not oscillate Volume flow measurements
Synthesis Implementation Results
Summary Perspectives
Implementation Diaphragm method, used by S. Ollivier and J.-P Dalmont (LAUM) [Dalmont et al., 2003]
Flow measurement Pressure drop in a diaphragm Double-reed physics Almeida
(∆p)s
(∆p)r
Objective
(∆p)d
Introduction Double reeds Models
pm
pr
Experiments Global reed characteristics
1
Bernoulli model
Flow model Synthesis Implementation Results
Summary Perspectives
(∆p)d
10
patm
Mechanics Flow details
In practice
= pr − patm 2 q 1 ρ = 2 Sdiaph
Volume Flow (q) in l/s
Motivations
1.0 mm (exp) 1.3 mm (exp) 1.8 mm (exp) 1.0 mm (theor) 1.3 mm (theor) 1.8 mm (theor) Rec limit
0
10
−1
10
−2
10
0
10
1
10 Pressure drop (∆ p)d in kPa
2
10
Preventing oscillations Double-reed physics Almeida Motivations Objective
Introduction
Added reed masses
Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Increased reed mass More inertia Reduced tendency for auto-oscillations [C. Frappé, LAUM]
Experiment pm
Double-reed physics Almeida Motivations
Camera
Lens
pr Reed
Controllable leak
Objective
Introduction
Diaphragm
Double reeds Models
Experiments Global reed characteristics Mechanics
Manometer
Compressed air source
Flow details
Flow model Synthesis
Artificial mouth
Humidifier
Implementation Results
Summary Perspectives
Reed opening measurements: Image analysis [Almeida et al., 2006]
Reed non-linear characteristics Results Double-reed physics Almeida Motivations
50
Objective
Introduction
Mechanics Flow details
Flow model
0.25
30 20 10
Results
Summary Perspectives
−10 0
0.2 0.15 0.1
0
0.05
Synthesis Implementation
0.3
flow (l/s)
Global reed characteristics
pressure (kPa)
Experiments
pm
40
Double reeds Models
0.35
pr
20
40
60 80 time (s)
100
120
140
0 −10
0
10 20 30 pressure difference (kPa)
40
50
Reed non-linear characteristics Results Double-reed physics Almeida Motivations
50
Objective
Introduction
Mechanics Flow details
Flow model
0.25
30 20 10
Results
Summary Perspectives
−10 0
0.2 0.15 0.1
0
0.05
Synthesis Implementation
0.3
flow (l/s)
Global reed characteristics
pressure (kPa)
Experiments
pm
40
Double reeds Models
0.35
pr
20
40
60 80 time (s)
100
120
140
0 −10
0
10 20 30 pressure difference (kPa)
40
50
Reed non-linear characteristics Results Double-reed physics Almeida Motivations
50
Objective
Introduction
Mechanics Flow details
Flow model
0.25
30 20 10
Results
Summary Perspectives
−10 0
0.2 0.15 0.1
0
0.05
Synthesis Implementation
0.3
flow (l/s)
Global reed characteristics
pressure (kPa)
Experiments
pm
40
Double reeds Models
0.35
pr
20
40
60 80 time (s)
100
120
140
0 −10
0
10 20 30 pressure difference (kPa)
40
50
Reed non-linear characteristics Results Double-reed physics Almeida Motivations
50
Objective
Introduction
Mechanics Flow details
Flow model
0.25
30 20 10
Results
Summary Perspectives
−10 0
0.2 0.15 0.1
0
0.05
Synthesis Implementation
0.3
flow (l/s)
Global reed characteristics
pressure (kPa)
Experiments
pm
40
Double reeds Models
0.35
pr
20
40
60 80 time (s)
100
120
140
0 −10
0
10 20 30 pressure difference (kPa)
40
50
Reed non-linear characteristics Results Double-reed physics Almeida Motivations
50
Objective
Introduction
Mechanics Flow details
Flow model
0.25
30 20 10
Results
Summary Perspectives
−10 0
0.2 0.15 0.1
0
0.05
Synthesis Implementation
0.3
flow (l/s)
Global reed characteristics
pressure (kPa)
Experiments
pm
40
Double reeds Models
0.35
pr
20
40
60 80 time (s)
100
120
140
0 −10
0
10 20 30 pressure difference (kPa)
40
50
Reed non-linear characteristics Results Double-reed physics Almeida Motivations
50
Objective
Introduction
Mechanics Flow details
Flow model
0.25
30 20 10
Results
Summary Perspectives
−10 0
0.2 0.15 0.1
0
0.05
Synthesis Implementation
0.3
flow (l/s)
Global reed characteristics
pressure (kPa)
Experiments
pm
40
Double reeds Models
0.35
pr
20
40
60 80 time (s)
100
120
140
0 −10
0
10 20 30 pressure difference (kPa)
40
50
Reed non-linear characteristics Results Double-reed physics Almeida Motivations
50
Objective
Introduction
Mechanics Flow details
Flow model
0.25
30 20 10
Results
Summary Perspectives
−10 0
0.2 0.15 0.1
0
0.05
Synthesis Implementation
0.3
flow (l/s)
Global reed characteristics
pressure (kPa)
Experiments
pm
40
Double reeds Models
0.35
pr
20
40
60 80 time (s)
100
120
140
0 −10
0
10 20 30 pressure difference (kPa)
40
50
Reed non-linear characteristics Results Double-reed physics Almeida Motivations
50
Objective
Introduction
Mechanics Flow details
Flow model
0.25
30 20 10
Results
Summary Perspectives
−10 0
0.2 0.15 0.1
0
0.05
Synthesis Implementation
0.3
flow (l/s)
Global reed characteristics
pressure (kPa)
Experiments
pm
40
Double reeds Models
0.35
pr
20
40
60 80 time (s)
100
120
140
0 −10
0
10 20 30 pressure difference (kPa)
40
50
Reed non-linear characteristics Results Double-reed physics Almeida Motivations
50
Objective
Introduction
Mechanics Flow details
Flow model
0.25
30 20 10
Results
Summary Perspectives
−10 0
0.2 0.15 0.1
0
0.05
Synthesis Implementation
0.3
flow (l/s)
Global reed characteristics
pressure (kPa)
Experiments
pm
40
Double reeds Models
0.35
pr
20
40
60 80 time (s)
100
120
140
0 −10
0
10 20 30 pressure difference (kPa)
40
50
Reed non-linear characteristics Results Double-reed physics Almeida Motivations
50
Objective
Introduction
Mechanics Flow details
Flow model
0.25
30 20 10
Results
Summary Perspectives
−10 0
0.2 0.15 0.1
0
0.05
Synthesis Implementation
0.3
flow (l/s)
Global reed characteristics
pressure (kPa)
Experiments
pm
40
Double reeds Models
0.35
pr
20
40
60 80 time (s)
100
120
140
0 −10
0
10 20 30 pressure difference (kPa)
40
50
Reed non-linear characteristics Results Double-reed physics Almeida Motivations
50
Objective
Introduction
Mechanics Flow details
Flow model
0.25
30 20 10
Results
Summary Perspectives
−10 0
0.2 0.15 0.1
0
0.05
Synthesis Implementation
0.3
flow (l/s)
Global reed characteristics
pressure (kPa)
Experiments
pm
40
Double reeds Models
0.35
pr
20
40
60 80 time (s)
100
120
140
0 −10
0
10 20 30 pressure difference (kPa)
40
50
Reed non-linear characteristics Results Double-reed physics 50
pm
40
Motivations
0.35
pr
Almeida
0.3
Double reeds Models
Experiments
0.25
30 flow (l/s)
Introduction
pressure (kPa)
Objective
20 10
0.15 0.1
Global reed characteristics Mechanics
0.2
0
0.05
Flow details
Flow model
−10 0
20
40
60 80 time (s)
100
120
140
0 −10
0
10 20 30 pressure difference (kPa)
40
Synthesis Implementation Results
Summary Perspectives
Remarks Flow not completely stopped Different paths for increasing / decreasing pressures
50
Comparison to model Double-reed physics
0.35
Almeida
0.3
Motivations
0.25
Double reeds Models
flow (l/s)
Objective
Introduction
experimental increasing decreasing
Parameters Increasing:
0.2 0.15
PM = 35 kPa ks = 10.4 × 109 kg m−3 s−2
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis
0.1
Decreasing PM = 27 kPa ks = 8.9 × 109 kg m−3 s−2
0.05 0 −10
0
10 20 30 pressure difference (kPa)
40
50
Implementation Results
Summary Perspectives
Remark Theory: PT /PM = 1/3 Measurements: PT /PM ' 1/5
Questions Double-reed physics Almeida Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Displacement of the maximum of the characteristics: why? Hysteresis: why? Role of the subsystems in the overall curve: Reed mechanics Flow
Outline Double-reed physics Almeida
1
Introduction What are double reeds? Reed instrument models
2
Experiments Global reed characteristics Reed mechanics Flow details
3
Flow model
4
Sound synthesis Implementation Results
Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Overview Double-reed physics Almeida Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics
Problem Model for the reed elasticity is simplistic and considers only the displacement at the reed tip: In a solid, the displacement at one point may not depend linearly on the force.
Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Objectives Check the reed elastic model (slow variations) Apply pressure difference between inside and outside Measure reed opening
Elasticity measurements Double-reed physics Almeida Motivations Objective
Introduction
pr
Double reeds Models
Experiments
pm
Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Implementation Block flow – homogeneous pressure distribution inside the reed
Summary Perspectives
Plastic film covering the reed
Elasticity Dry reed Double-reed physics
3.5 3
Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Reed opening area (mm²)
Almeida Motivations
Low pressures: ks = 4.5 × 109
2.5 2
High pressures: ks = 22 × 109
1.5 1
(SI = kg m−3 s−2 )
0.5
Flow model Synthesis Implementation
0 0
10
20 30 Pressure difference (kPa)
40
Results
Summary Perspectives
Remarks Stiffness increases for high pressures Noticeable hysteresis
Elasticity Soaked reed
3.5
Motivations
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation
Reed opening area (mm2)
Almeida
Objective
anche4mouillée
4
Double-reed physics
3 2.5
ks = 7 × 109
2 1.5
(SI = kg m−3 s−2 )
1 0.5 0 0
10
20 30 Pressure difference (kPa)
40
Results
Summary Perspectives
Remarks Stiffness is almost constant along the pressure range Hysteresis increases (characteristic relaxation τ ∼ 30 s)
Conclusions (Elastic model) Double-reed physics Almeida Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Conclusions Viscoelasticity can explain hysteresis on the characteristic curve
Linear spring suitable for soaked reeds Cannot explain shifting in maxima of the characteristic curves
Outline Double-reed physics Almeida
1
Introduction What are double reeds? Reed instrument models
2
Experiments Global reed characteristics Reed mechanics Flow details
3
Flow model
4
Sound synthesis Implementation Results
Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Static flow Preventing oscillations Double-reed physics Almeida Motivations Objective
Artificial mouth walls
Introduction Double reeds Models
Experiments Global reed characteristics
Positioning Scew
Mechanics Flow details
Flow model
Pinhead Epoxy Joint
Synthesis Implementation Results
Summary Perspectives
Reed blade
Screws turn in the mouthpiece walls Screw tip attached to reed blade through joint Reed opening adjustable from outside
Schlieren visualisations Double-reed physics Almeida Motivations Objective
Introduction
pm = 15 Pa Re = 700
Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model
pm = 300 Pa Re = 3400
Synthesis Implementation Results
Summary Perspectives
pm = 15 kPa Re = 19700 Collaboration with B. Fabre, LAM
Normal playing pressure (oboe): 4 ∼ 12 kPa
Hot-wire setup Double-reed physics
Artificial mouth
Almeida Motivations Objective
Introduction Double reeds Models
Manometer
Compressed air CO2
Experiments Global reed characteristics Mechanics
Camera
Reed
Flow details
Hot−wire
Flow model Synthesis Implementation
Lens
Results
Summary Perspectives
Collaboration with B. Fabre, LAM
Results Symmetry: variable profile direction, constant pressure and opening Double-reed physics
1
Normalised flow velocity
Almeida Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics
parallel perpendicular
0.8 0.6 0.4 0.2
Mechanics Flow details
0 0
Flow model
5 10 Probe position (mm)
15
Synthesis Implementation Results
Summary
Remarks Normalised profiles (correct slight variations in pressure)
Perspectives
Similar profiles (∼ 95%) along asymmetrical duct directions Axisymmetric flow despite of initial duct asymmetry
Results Variation with pressure (constant opening and measurement direction) Double-reed physics Almeida 50
Objective
Introduction
40
Experiments Global reed characteristics Mechanics Flow details
Flow velocity (m/s)
Double reeds Models
15 mBar 130 mBar 65 mBar
30 20 10
1
Normalised flow velocity
Motivations
15 mBar 130 mBar 65 mBar
0.8 0.6 0.4 0.2
Flow model Synthesis
0 4
6
8 10 12 Probe position (mm)
14
16
0 4
6
8 10 12 Probe position (mm)
Implementation Results
Summary Perspectives
Remarks Pressure scales velocity magnitudes (u 2 ∝ pm ) Profile shape remains constant
14
16
Turbulent flow profile Comparison with fully developed turbulent flow in cylindrical tube Double-reed physics Almeida
Models
Experiments Global reed characteristics Mechanics Flow details
uUavg
Double reeds
Normalised flow velocity
1.2
Objective
Introduction
1
1.4
Motivations
1 0.8 0.6 0.4 0.2 -1
-0.5
0 rR
0.5
1
0.6 0.4 0.2 0 4
Flow model Synthesis Implementation Results
Summary Perspectives
15 mBar 130 mBar 65 mBar
0.8
6
8 10 12 Probe position (mm)
Remarks Similar narrow boundary layers Velocities grow faster towards the axis
14
16
Flow profiles Conclusions Double-reed physics Almeida Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model
Observations Flow is almost axisymmetric at the reed output Both pressure and opening scale the profiles proportionally Profile related to developed turbulent flow in cylinder but modified due to tapering
Synthesis Implementation Results
Summary Perspectives
Hypothesis Turbulent flow in conical duct can induce pressure recovery along the reed
Pressure recovery measurements Overview Double-reed physics Almeida Motivations Objective
Introduction Double reeds
Objective Measure the pressure recovery inside the reed
Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Method Pressure measurements inside the reed and at the reed output Oscillations not artificially prevented Pressure range underneath the oscillation threshold
Pressure recovery Measurements Double-reed physics
Method Pressure measurements:
Almeida Motivations Objective
Introduction
in the mouth: pm inside the reed, near the tip: p2
Double reeds
(∆p)rec
Models
Experiments Global reed characteristics
Mechanics
Flow details
Flow model
pm
p2
patm
Synthesis
Implementation
Results
Summary Perspectives
Pressure at the reed output considered constant = patm Mouth pressures below oscillation threshold Recovered pressure (∆p)rec = patm − p2
Reed pressure vs Mouth pressure Double-reed physics Almeida
0.2
Motivations
0.1
Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Reed (2) pressure (kPa)
Objective
Introduction
0 −0.1 −0.2 −0.3 −0.4 −0.5 −1
0
1 2 Mouth pressure (kPa)
3
4
Summary Perspectives
Time fluctuations of pressure indicated by error bars
Pressure recovery Explanation Double-reed physics Almeida
u1
u2
Motivations
p2 = p 1
p2 = p1 + 12 ρu21
0
1
Objective
Introduction
Cr
Double reeds Models
Experiments Global reed characteristics
p2 = p1 + Cr 21 ρu21
Mechanics Flow details
Flow model
p1
p2
Synthesis Implementation Results
Summary Perspectives
Limit cases Cr = 0: Kinetic energy is completely lost Cr = 1: Kinetic energy is completely converted in potential energy
Pressure recovery coefficient Determined from measurements Double-reed physics
2
Almeida
Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation
Recovery coefficient (CP)
Motivations
1.5 1 0.5
−0.5 −1 0
Results
Summary Perspectives
Cr =
0
1000
2000 3000 Reynolds number
Turbulent regime Cr ' 0.7
4000
5000
p2 − pr 1/2ρu12
Flow regions Double-reed physics Almeida Motivations Objective
Introduction
Reed entrance
p
m
1
Double reeds Models
Experiments Global reed characteristics
p1 = pm −
p
m
ps
Mechanics Flow details
p
Flow model Synthesis Implementation Results
Summary Perspectives
p
r
Reed (cane) duct ps = p1
p
m
p
rec
1 q 2 ρ 2 S
Staple p
1
=
p
s
1 pr = ps + Cr ρ 2
q Ss
2
Pressure drop corrected for pressure recovery Double-reed physics
0.35
experimental increasing decreasing
Almeida 0.3
Motivations Objective
0.25
Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
flow (l/s)
Introduction
0.2 0.15 0.1 0.05
Flow model Synthesis
0 −10
0
Implementation
10 20 30 pressure difference (kPa)
Results
Summary Perspectives
(∆p)c = kS (S0 − S), pm − pr = (∆p)c − Cr 12 ρ
q Sc
2
40
50
Double reed model Double-reed physics Almeida Motivations Objective
Introduction
Conclusion Measurements show deviations from the characteristic curves predicted by the elementary model Pressure recovery can explain these differences
Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation
Questions Oscillating regimes — which description for the conical diffuser: quasi-static model for the pressure recovery? acoustic propagation in a conical resonator?
Results
Summary Perspectives
Answers Numerical implementation of the model Effects of pressure recovery on simulated reed behaviour and sound
Outline Double-reed physics Almeida
1
Introduction What are double reeds? Reed instrument models
2
Experiments Global reed characteristics Reed mechanics Flow details
3
Flow model
4
Sound synthesis Implementation Results
Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Resonator Propagation Double-reed physics Almeida
p−
Motivations
S(x2 )
Objective
Introduction
S(x1 )
x
Double reeds Models
Experiments
p+
x1
Global reed characteristics Mechanics
x2
Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Conical resonator ⇒ Spherical traveling waves Propagation = delay: P ± (ω, x) = e∓ık (x−x0 ) P ± (ω, x0 ) Implemented using fractional delay lines [G. Peeters, IRCAM]
Resonator Viscothermal losses Double-reed physics Almeida
Friction of the air against the walls Described using a complex k (ω):
Motivations Objective 0
Double reeds Models
Experiments
Magnitude (dB)
Introduction
Global reed characteristics
theory fit
−10 −20 −30 −40 0
0.5
Mechanics
Implementation Results
Summary Perspectives
Phase (degrees)
Synthesis
1.5
1 Frequency (Hz)
1.5
2
4
x 10
400
Flow details
Flow model
1 Frequency (Hz)
300 200 100 0 0
0.5
2
4
x 10
ω i 3/2 − ηcω 1/2 c 2 (Kirchhoff’s theory) Implemented using IIR filter Coefficients optimised using Least Square fitting
Resonator Losses through resonator termination Double-reed physics Almeida
Radiation losses Reflection at the resonator termination (localised) Theory:
Motivations Objective 0
Double reeds Models
Experiments
Magnitude (dB)
Introduction
Global reed characteristics
theory fit
−10 −20 −30 −40
0.5
Mechanics
Synthesis Implementation Results
Summary Perspectives
Phase (degrees)
Flow model
1 Frequency (Hz)
1.5
1 Frequency (Hz)
1.5
2
4
x 10
200
Flow details
150 100
Implementation:
50 0
Radiation of a flanged piston
0.5
2
4
x 10
IIR filter Coefficients optimised using Least Square fitting
Reed Double-reed physics Almeida Motivations Objective
Introduction
Reed dynamics Reed as a damped harmonic oscillator:
Double reeds Models
Experiments Global reed characteristics
pr (t) − pm = kS (S(t) − SO ) + rs
∂S ∂2S + ms 2 ∂t ∂t
Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Implementation Impulse invariance method Allows explicit resolution for S(t) as a function of past values of pr (t)
Reed Double-reed physics Almeida Motivations Objective
Flow Combined flow equation with pressure recovery:
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
(∆p)r (t) = pm − p(t) 1 q(t) 2 = ρ 2 S(t)
S(t)2 1 − Cr S32
!
Flow model Synthesis Implementation Results
Summary Perspectives
S(t) varies from sample to sample both q(t) and (∆p)r (t) are unknown Equation needs to be solved together with the resonator (coupled resolution)
Coupling the reed to the resonator General case Double-reed physics Almeida Motivations Objective
Problem Resonator equation depends on past values of q and p which depend on time
Introduction Double reeds Models
General solution
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis
pn
M M X X = 2pn+ − b0 qn − ( bj qn−j − ai yn−i ) j=1
i=1
= 2pn+ − b0 qn + Λ
Implementation Results
Summary Perspectives
p+ is the incoming spherical wave ai and bj are coefficients given by the transformation between flow / pressure variables and the traveling waves (for instance, spherical)
Outline Double-reed physics Almeida
1
Introduction What are double reeds? Reed instrument models
2
Experiments Global reed characteristics Reed mechanics Flow details
3
Flow model
4
Sound synthesis Implementation Results
Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Sound example Effect of pressure recovery Double-reed physics Almeida
Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
0.5
0
−0.5 0
Adimensioned reed position
Objective
Introduction
Adimensioned pressure
Motivations
500
1000 time (samples)
1500
2000 CP=0
1
CP=0.8 0.5
0 0
500
1000 time (samples)
1500
2000
Without recovery: Cr = 0 With maximum recovery: Cr = 0.8
Summary Double-reed physics Almeida Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis
Review Unprecedented measurements of the non-linear characteristics curve of double reeds show deviations from the elementary model Well explained by the pressure recovery of a turbulent flow in the conical staple
Sound synthesis: pressure recovery produces noticeable changes in sound less perceptible than changes in the resonator
Implementation Results
Summary Perspectives
Main conclusion Elementary reed model characterises well the double reed in static regime, provided that the staple is not considered as part of the exciter. Opposed to friction loss model [Hirschberg, 1995]
Summary Double-reed physics Almeida Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis
Other results Reed opening area (S) is mostly proportional to inter-blade distance (h), opposed to classical models [Barjau and Agulló, 1989] Statically, the reed behaves mostly like a linear spring Viscoelasticity is important, especially for soaked reeds, but should be compensated by musician adjustments during performance
Implementation Results
Summary Perspectives
Turbulent flow in the reed erases initial asymmetry Synthesis model: developments used in industrial “digital saxophone”
Questions Double-reed physics Almeida Motivations Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Questions Should the staple be part of the exciter in a full instrument model? Simulations suggest that this is not very important
How should the model be adapted for dynamic (oscillating) regimes? Inclusion of the reed dynamics (done in synthesis model) Adimensional analysis (Strouhal number) suggests that unstationnary term in NS equation should be taken into account Flow induced by reed motion should have same magnitude as main flow but was not detected in dynamic flow measurements
Perspectives Double-reed physics Almeida Motivations Objective
Introduction Double reeds Models
Dynamic flow measurements: Extend to normal playing regimes, with resonator
Experiments Global reed characteristics Mechanics Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Propose dynamic fluid-structure model Explore diversity of reeds and variability in experiments Synthesis model Parameter and mapping tuning
Bibliography Double-reed physics Almeida Motivations
Almeida, A., Vergez, C., and Caussé, R. (2006). Experimental investigation of reed instrument functionning through image analysis of reed opening. Submitted to Acustica.
Objective
Introduction Double reeds Models
Experiments Global reed characteristics Mechanics
Dalmont, J. P., Gilbert, J., and Ollivier, S. (2003). Nonlinear characteristics of single-reed instruments: quasi-static volume flow and reed opening measurements. J. Acoust. Soc. Am., 114(4):2253–2262. Okwuobi, P. A. C. and Azad, R. S. (1973). Turbulence in a conical diffuser with fully developed flow at entry. J. Fluid Mech., 57(3):603–622.
Flow details
Flow model Synthesis Implementation Results
Summary Perspectives
Hirschberg, A. (1995). Mechanics of Musical Instruments, chapter 7: Aero-acoustics of Wind Instruments, pages 229–290. Springer-Verlag. Nederveen, C. J. (1998). Acoustical Aspects of Musical Instruments. Northern Illinois Univ. Press. Barjau, A. and Agulló, J. (1989). Calculation of the starting transients of a double-reed conical woodwind. Acustica, 69:204–210.
Reed cross section profile Double-reed physics Almeida Opening geometry Opening geometry
Flow vs pressure
8
dimension (mm)
Flow vs pressure
Flow vs pressure
4 2 0
0
10
20
0
10
20
30 40 axial distance (mm)
50
60
70
50
60
70
15
area (mm2)
Determination of the input velocity to the staple
height width
6
10
5
0
30
40
Overview Double-reed physics Almeida Opening geometry Opening geometry
Problem Many authors propose that the reed opening area is not proportional to the distances between reeds
Flow vs pressure Flow vs pressure Flow vs pressure Determination of the input velocity to the staple
Proportional / power-law models h
h
1 – proportional
2 – diamond-shape
Reed opening – results Double-reed physics Almeida
4
Opening geometry
Flow vs pressure Flow vs pressure Flow vs pressure Determination of the input velocity to the staple
3 2.5 Area (m2)
Opening geometry
−6
x 10
3.5
2 1.5 1 0.5 0 0
Area=5.183492e−03*width+−6.253252e−08
1
2
3 4 Width (m)
5
6
7
−4
x 10
Conclusion Reed opening area mostly proportional to distance between reeds
Relation between static flow and pressure Overview Double-reed physics Almeida Opening geometry Opening geometry Flow vs pressure
Concept Velocity profiles at reed output seen to grow proportionally with pressure and reed opening Volume flow q scales in the same proportion as velocities
Flow vs pressure Flow vs pressure Determination of the input velocity to the staple
Objective Trace the velocity and volume flow scaling as a function of pressure reed opening
Relation between static flow and pressure Scaling with pressure Double-reed physics
6
Almeida
5
Opening geometry Flow vs pressure Flow vs pressure Flow vs pressure Determination of the input velocity to the staple
volume flow (m3/s)
Opening geometry
−4
x 10
4 3
0.3 0.9 1.4 1.9 2.6
2 1 0 −1 −2 0
5000 10000 pressure (Pa)
15000
Remark Flow (velocities and volume) scale as p1/2
Relation between static flow and pressure Scaling with reed opening Double-reed physics
2
Almeida Opening geometry
Flow vs pressure Flow vs pressure Flow vs pressure Determination of the input velocity to the staple
1.5 (q/s)2 (SI)
Opening geometry
5
x 10
0.3 0.9 1.4 1.9 2.6
1
0.5
0 0
5000 10000 pressure (Pa)
15000
Remark Opening areas not suitable to describe the velocity scaling according to a Bernoulli model
Hot-wire setup (close-up) Double-reed physics Almeida Opening geometry Opening geometry
1
reed output
Flow vs pressure
2
microphone
Flow vs pressure
3
hot-wire probe
4
millimetric screw
Flow vs pressure Determination of the input velocity to the staple
Viscoelasticity
Opening geometry Opening geometry Flow vs pressure Flow vs pressure Flow vs pressure Determination of the input velocity to the staple
Pressure difference (kPa)
Almeida
4
50
3.5
40
0.4 4
30
3
20
2.5
2
0 0
100
1.5
0 200
0.2
0.15
1
Relaxation: τ ∼ 30 s
0.1
0.5 0 110
0.3 0.25
10
2
0.35
opening (mm2)
Double-reed physics
120
130
140
150 160 Time (s)
170
Pressure 0.05 Opening 0 180 190 200
Temporary deformation can explain the shifting of frequency / amplitude when playing the instrument in an artificial mouth compensated by an adaptation of the musician control in normal conditions
Pressure recovery coefficient Experimental determination of u1 Double-reed physics Almeida Opening geometry
u1
ur
Opening geometry Flow vs pressure Flow vs pressure
pm
pr
= p1
1 2 ρu = pm − pr 2 r patm
ur Sr = u1 S1
Flow vs pressure Determination of the input velocity to the staple
Remarks Sr measured from reed opening photos (variable) S1 – staple entrance cross-section (constant)
Results Variation with reed opening (constant pressure and measurement direction) Double-reed physics Almeida 35
open half small
Opening geometry Flow vs pressure Flow vs pressure Flow vs pressure Determination of the input velocity to the staple
Flow velocity (m/s)
30 25 20 15 10 5 0 0
2
4 6 8 Probe position (mm)
10
12
1
Normalised flow velocity
Opening geometry
open half small
0.8 0.6 0.4 0.2 0 0
2
4 6 8 Probe position (mm)
Remarks Reed opening scales velocity magnitudes Profile shape remains constant
10
12
