How well can linear stability analysis predict the behavior of an outward valve brass instrument model ?

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How well can linear stability analysis predict the

behavior of an outward valve brass instrument model ?

Lionel Velut, Christophe Vergez, Joël Gilbert, Mithra Djahanbani

To cite this version:

Lionel Velut, Christophe Vergez, Joël Gilbert, Mithra Djahanbani. How well can linear stability

analysis predict the behavior of an outward valve brass instrument model ?. Acta Acustica united

with Acustica, Hirzel Verlag, 2017, 103 (1), pp.132-148. �10.3813/AAA.919039�. �hal-01245846v4�

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behaviour of an outward-striking valve brass instrument 2 model? 3 Lionel Velut 1 , Christophe Vergez 1 , Joël Gilbert 2

, and Mithra Djahanbani

1 4

1

LMA, CNRS, UPR 7051, Aix-Marseille Univ., Centrale Marseille, F-13453

5

Marseille edex 13, Fran e.

6

2

Laboratoire d'A oustique de l'Université du Maine, UMR CNRS-6613, Avenue

7

Olivier Messiaen, 72085 Le Mans edex 9, Fran e

8

De ember 13, 2016

9

Abstra t 10

Aphysi almodelofbrassinstrumentis onsideredinthispaper: aonedegree-of-freedom

11

outward-striking valve for thelips, non-linearly oupled to amodalrepresentation of theair

12

olumn. It is studied through Linear Stability Analysis (LSA) of the equilibrium solution.

13

Thisapproa hprovidesthethresholdblowingpressurevalue,at whi hinstabilityo urs,and

14

the instability frequen y value. The relevan e of the results of this method is theoreti ally

15

limitedto the neighbourhoodof theequilibriumsolution. Thispaper he ks thee ien y of

16

LSA to understand thebehaviour of the model omputed through time-domain simulations.

17

As expe ted, a good agreement is observed between LSA and numeri al simulations of the

18

ompletenonlinearmodelaroundtheos illationthreshold. Forblowingpressuresfarabovethe

19

os illation threshold, the pi ture is more ontrasted. In most of the ases tested, a periodi

20

regime oherent with the LSA results is observed, but over-blowing, quasi-periodi ity and

21

period-doubling also o ur. Interestingly, LSA predi ts theprodu tion of the pedal note by

22

a trombone, for whi h only nonlinear hypotheses have been previously proposed. LSA also

23

predi ts the produ tion of a saxhorn note whi h, although known to musi ians, has barely

24

been do umented.

25

1 Introdu tion

26

LinearStabilityAnalysis(LSA) anbeusedtoanalysethebehaviourofdynami alsystemsaround

27

equilibriumpoints(i.e. non-os illatingsolutions). LSA onsistsinwritingalinearisedversion ofa

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to harmoni perturbations. 30

LSA has already been applied to physi al models of musi al instruments, su h as woodwind

31

instruments [Wilson and Beavers, 1974, Chang,1994, Silva etal., 2008, Karkaret al.,2012℄,

32

ute-like instruments [Terrien et al.,2014℄ and brass instruments [Cullenet al.,2000,

33

Lopez et al.,2006, Silvaetal., 2007℄. By denition, the domain of relevan e of the LSA

re-34

sults is theoreti ally limited to the neighbourhood of the equilibrium solution. However, re ent

35

results on utes have highlighted that LSA an predi t important features of periodi regimes,

36

su h as their frequen ies [Terrien et al.,2014℄. This paper examines to what extent LSA an be

37

used to understandsome aspe ts of the behaviourof a physi almodelof brass instrument.

38

Physi al models of brass instuments have been proposed in multiple

stud-39

ies [Eliottand Bowsher, 1982, Flet her, 1993, Ada hi and Sato, 1996, Cullenet al.,2000,

40

Campbell,2004, Silvaetal., 2007℄. Sin e our fo us in this study is a simple model, a one

41

degree-of-freedom system is retained to model the player's lips: the outward-striking valve,

42

also referred to as "

(+, −)

" in some publi ations. The same goal of simpli ity makes us

43

ignore nonlinear propagation in the bore of the instrument, whi h is responsible for "brassy

44

sounds" at high sound levels [Myers et al.,2012℄. The oupling by the airow blown

be-45

tween the lips and the air olumn inside the bore is modelled through a usual nonlinear

46

algebrai equation [Hirs hberg et al.,1995℄. This model is detailed in Se tion 2.1. Even

47

su h a simple brasswind model has more parameters needing to be tuned than the

sim-48

plest models of woodwind instruments, whi h is based on two dimensionless parameters

49

only [Hirs hberg et al.,1995, Dalmontet al.,1995, Taillard etal.,2010, Bergeot et al.,2013℄.

50

However, brasswind players make their instrument os illate on several modes, whi h implies a

51

signi ant modi ation of the me hani al hara teristi s of their lips. In musi al terms, this

52

orresponds to playing multiple notes without pulling a slide nor depressing a valve, whi h is

53

part of the playing te hnique of all brass instruments. Therefore, the lip dynami s annot be

54

ignored,whi h impliesanin rease inthe numberof parameters totune. A bibliographi alreview

55

is given in Se tion 2.2 to give grounds to the values hosen for ea h parameter of the model. In

56

Se tion 2.3, details are given on how LSA is applied to the model. There are several possible

57

approa hes to highlighting nonlinear model behaviours to ompare them with LSA results. For

58

instan e, theHarmoni Balan e MethodgivesaFourierseriesapproximationofthesteady stateof

59

periodi regimes, in ludingunstable ones [Gilbert etal., 1989, Co helin and Vergez, 2009℄. Sin e

60

the pioneering work des ribed in [S huma her, 1981, M Intyre et al.,1983℄, it is also possible to

61

arry out time-domainsimulationsatmoderate omputational ost,providinga esstotransients

62

and possibly non-periodi solutions. The se ond approa h is retained here (see Se tion 2.4).

63

Se tion 3 ompares LSA results and numeri al simulations for dierent sets of parameter values.

64

Periodi regimes, orresponding to the usual sound of the instrument, are explored, along with

65

less ommonregimes su h asquasi-periodi ityand period-doubling. InSe tion 4,we fo us on the

66

lowest a ousti resonan e of brass instruments, alled the pedal note, a parti ularly interesting

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2 Tools 69

2.1 Brass instrument model

70

In most windinstruments[Flet her, 1993,Chaigne and Kergomard, 2016℄, in ludingbrass

instru-71

ments[Eliott and Bowsher, 1982,Yoshikawa, 1995,Cullen etal.,2000℄,theos illationresultsfrom

72

the oupling between an ex iter and a resonator. More generally, the losed-loop system

repre-73

sentation shown in Figure 1 has been widely used by the musi al a ousti s ommunity sin e the

74

seminal work of Helmholtz[Helmholtz,1877, M Intyre et al.,1983℄.

75

76

Figure 1: (Color online) Closed-loop model in free os illation, suitable for the des ription of

most self-sustained musi alinstruments. Self-sustained os illationsare generated by the lo alised

nonlinear oupling between a linear ex iter and a linear resonator. For brass instruments, the

ex iteristhe lipreedwhilethe resonatoristheair olumninsidethebore,and the ouplingisdue

to the airow between the lips.

77

For brass instruments, the ex iter is the lips of the musi ian. It is represented by a linear,

78

os illator-likevalvelinkingthe height of the hannelbetween the lips

h(t)

andthe pressure

dier-79

en e a ross the lips

δp(t) = p

b

− p(t)

, where

p

b

is the blowing pressure, and

p(t)

is the os illating

80

pressure signalinside the mouthpie e (the input of the bore).

81

A one degree of freedom valve (referred to hereafter as "1-DOF valve") [Flet her, 1993℄ is

82

enough to model the lips for ommon playing situations [Yoshikawa, 1995℄ with a manageable

83

number of parameters. Two kinds of 1-DOF valves an be onsidered : the "outward-striking"

84

valvetends to open when

δp

grows, while the "inward-striking" valve tends to lose.

85

While it is now admitted that woodwind reeds an be satisfa torily modelled by inward

86

striking valves [Wilson and Beavers, 1974, Dalmont etal., 1995℄, there is no onsensus about

87

the modelling of the lip reed, as neither the outward-striking nor the inward-striking valve

88

model reprodu es all the behaviours observed with real musi ians. Parti ularly, brass players

89

are able to rea h a playing frequen y

f

osc

above and below the

n

th

bore resonan e frequen y

90

f

ac,n

[Campbell, 2004℄, while a 1-DOF inward-striking oroutward-striking valve model is limited

91

to playing frequen ies respe tively below or above

f

ac,n

to meet the regeneration ondition

ex-92

plained in [Eliott and Bowsher,1982℄. Moreover, measurements of the me hani al response of

93

arti ial[Cullen etal., 2000, Nealet al.,2001℄ and natural lips[Newton etal.,2008℄ revealed the

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oexisten eofbothinward-strikingandoutward-strikingresonan es-this oexisten eallowing

f

osc

95

to bebelowor above

f

ac,n

.

96

However, situationswhere

f

osc

isbelow

f

ac,n

(inward-strikingbehaviour) are mostly spe i to

97

some musi al ee ts. For normal playing situations, the playing frequen y is above

f

ac,n

, and an

98

outward-strikingvalvemodelispreferred. Moreover,thegeometryofhumanlipsmakesthemopen

99

when the pressure in the mouth in reases, whi h is onsistent with the behaviour of the

outward-100

striking valve model. The relevan e of this hoi e will be reinfor ed throughout this arti le, by

101

omparing the results of the modelanalysis with experimentalbehaviours of brasswinds.

102

The outward-striking valve model gives the relation below, linking the height of the hannel

103

between the lipsand the pressure dieren e a ross the lips:

104

d

2 _h

dt

2 +

ω

l

Q

l

dh

dt

+ ω

2 l

(h − h

0 ) =

1 µ

(p

b

− p(t)),

(1) where

ω

l

= 2πf

l

(rad · s

−1

₎

is the lip resonan e angular frequen y;

Q

l

the (dimensionless) quality

105

fa tor of the lips;

h

0

the value of

h(t)

at rest;

µ

a lip surfa e mass equivalent

(kg · m

−2

₎

. The

106

variables are reported onthe sket h of the lipregion inFigure 2:

107 108

p

b

h(t)

u(t)

p(t)

lip

mouth

mouthpiece

109

Figure 2: (Colour online) Sket h of the mouth and lips of the musi ian and the instrument

mouthpie e. The mouth (left) is onsidered as a avity under a stati pressure

p

b

. The lips

(ellipses)separatethemouthfromthemouthpie e. The heightbetween thelipsis

h(t)

,the airow

between the lipsis

u(t)

and the pressure inthe mouthpie e is

p(t)

.

110

This modelassumes the mouth pressure tobe onstant, even thoughthe existen e of an

os il-111

lating omponentinthemouthhasbeendemonstratedexperimentally[Fréour and S avone, 2013℄.

112

A morepre isemodelwould onsider this os illating omponent,whi hisdue tothe tunablepipe

113

formedby thevo altra t[Eliott and Bowsher,1982℄. Asigni antrole ofthe vo altra thasbeen

114

shown for saxophone and larinet playing [Clin het al.,1982, Fritz, 2005, S avone et al.,2008,

115

Guillemainetal., 2010, Chenet al.,2011℄. However, for brass instrument playing, the role of the

116

vo al tra t does not seem to be signi ant when playing periodi regimes in the usual musi al

117

rangeofthe instrument-althoughitsintera tionwiththe lipshas beenhighlightedby

experimen-118

talstudies[Kaburagiet al.,2011,Chen etal., 2012,Fréourand S avone, 2013,Fréouret al.,2015,

119

Boutinet al.,2015℄.

120

The resonator is the air olumn inside the bore of a trombone or a saxhorn (see Se tion 4.2).

121

It ismodelledby itsinputimpedan e, whi histhe ratio between pressure

P (ω)

and a ousti ow

(6)

U(ω)

inthe mouthpie e. Its expression inthe frequen y domainis: 123

Z(ω) =

P (ω)

U(ω)

.

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Nonlinear ee ts in the resonator should be taken into a ount to a urately des ribe

124

the behaviour of brass instruments at medium/high playing levels [Hirs hberg et al.,1996,

125

Myers et al.,2012℄ parti ularlythe "brassy sound"relatedto the formationofsho k waves.

How-126

ever, themainobje tiveofthisworkisthestudy ofos illationatlowlevels. Thereforethea ousti

127

propagation along the bore is assumed tobe linear and thus the input impedan e fully des ribes

128

the resonator in our model. Here, input impedan es of a Courtois "T149" tenor trombone (and

129

when mentioned, aCouesnon "Ex elsior"baritone-saxhorn in

B♭

) are used. Impedan es are

mea-130

suredwiththe impedan esensor des ribedin[Ma aluso and Dalmont,2011℄. They are ttedbya

131

sum of omplex modes (pole-residuefun tions) using aLeast Mean Squares method, asdes ribed

132

in[Silva,2009,p.2840℄. The hara teristi impedan e of the resonator is

Z

c

= ρc/S

,

S

being the

133

input ross se tion of the bore at the mouthpie e rim. The modal-tted impedan e is written:

134

Z(ω) = Z

c

N

X

n=1

C

n

jω − s

n

+

C

n

∗

jω − s

∗

n

,

(3)

s

n

and

C

n

beingthe omplexpolesandthe omplexresiduesofthe

n

th

omplexmode,respe tively.

135

Translation ofeq.(3)inthe timedomainandde ompositionof

p(t)

intoitsmodal omponents

p

n

,

136

su h as

p(t) = 2.

P

N

n=1

Re(p

n

)

results in anordinary dierential equationfor ea h

p

n

:

137

dp

n

dt

= Z

c

.C

n

.u(t) + s

n

.p

n

∀n ∈ [1, N].

(4)

0

200

400

600

800 −180

−90

0

90

180 frequency (Hz)

arg(Z) (deg)

10

3

10

4

10

5

10

6

10

7 |Z| (kg.m

−4

.s

−1

)

138

Figure3: ( olouronline)Magnitude(top)andphase(bottom)oftheinputimpedan eofaCourtois

tenor trombone with the slide in itsrst position. The dashed (blue) urve depi tsthe measured

impedan e,thesolid(red) urveisthetted urvewith18 omplexmodes. Thedieren ebetween

tand measurement isalsoplotted (magenta).

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maximum relative dieren e between the measured and the tted urves, for frequen ies above 141

30Hz

, is lower than

2.6 %

for the magnitude, and

4.7 %

for the phase. Measurement in low

142

frequen y islimited by the impedan e sensor pre ision.

143

Those two linear elements (ex iter and resonator) are non-linearly oupled by the airow

144

through the lip hannel. The air jet is assumed to be laminar in the lip hannel, but

tur-145

bulent in the mouthpie e, all its kineti energy being dissipated without any pressure

re ov-146

ery. Applying the Bernoulli law and the mass onservation law gives the following expression

147

of the ow between lips, depending on the pressure dieren e and the height of the lip

han-148

nel[Wilson and Beavers, 1974, Eliott and Bowsher,1982, Hirs hberg et al.,1995℄:

149

u(t) =

r 2

ρ

W h(t)pp

b

− p(t),

(5)

where

u(t)

is the airow(

m

3 _{· s}

−1

),

h(t)

the height of the hannel between the lips(

m

),

ρ = 1.19

150

kg · m

−3

the density of the airat

20 ◦

_C

and

W

the width of the lip hannel (

m

).

151

The dynami s of the system des ribed by (5), (1)and (4) an be put intoa state-spa e

repre-152

sentation

X = F (X)

˙

, where F is a nonlinear ve tor fun tion, and X the state ve tor, ontaining

153

the observables of the system. Sin e

p(t) =

P

N

n=1

2Re(p

n

(t))

, this results inthe following system: 154







d

2 _h(t)

dt

2 = −ω

l

2 h(t) −

ωl

Ql

dh(t)

dt

−

p(t)

µ

+ ω

2 l

h

0 +

pb

µ

dpn

dt

= s

n

p

n

(t) + Z

c

C

n

q

2 ρ

W h(t)pp

b

− p(t)

for

n ∈ [1, N].

(6)

This leads to the followingstate ve tor, similar tothe one proposed in[Silva etal.,2014℄:

155

X =

h(t);

dh

dt

; {p

n

(t), n ∈ [1, N]}

′

,

(7)

and the fun tion

F

an bewritten as:

156

dX

dt

=







dh

dt

d

2 _h

dt

2 dp

1 dt

. . .

dp

n

dt







= F (X) =







X(2)

−ω

2 l

X(1) −

ω

l

Q

l

X(2) −

1 µ

P

N+2

k=3

2Re[X(k)] + ω

l

2 h

0 +

p

b

µ

s

1 X(3) + C

1 .Z

c

.

r 2

ρ

W X(1)

q

p

b

−

P

N

+2

k=3

2Re[X(k)]

. . .

s

N

X(N + 2) + C

N

.Z

c

.

r 2

ρ

W X(1)

q

p

b

−

P

N

_k=3

+2

2Re[X(k)]







.

(8)

2.2 Choi e of lip parameters

157

Setting the values for the parameters of the lip model is not obvious, be ause measuring the

158

me hani aladmittan e(velo ity overfor eratio) under playing onditions(os illatinglips) seems

159

out of rea h, even if some experiments tend to it [Newton etal.,2008℄. Adjusting parameters to

(8)

a one-DOF model depends on a small number of parameters, dierent sets of parameter values 162

mayleadtosimilarresults[Hélieetal., 1999℄. Moreover, lipvalveparametersareexpe tedtovary

163

far more than reed valveparameters, parti ularlythe lipresonan e frequen ies.

164

A bibliographi al review on lip parameter values has been done. Results from the literature

165

are gathered in Table 1 along with abrief summaryof the methodused inthe reviewed arti les.

166

Referen e

h

0 (m)

W (m)

f

l

(Hz)

µ

−1

_(m

2 _{· kg}

−1

₎

_Q

l

Summary

[Eliott and Bowsher, 1982℄ N/A N/A 200 0.2 0.5

Q

l

measured on heek

[Cullen etal.,2000℄

1 st

(Outward) mode Embou hure: Soft

6.3 · 10

−4

_{18 · 10}

−3

189 0.07 10.5 Arti ial lips Medium

5.3 · 10

−4

_{12 · 10}

−3

203.5 0.11 6 3 embou hures Tight

4.4. · 10

−4

_{11 · 10}

−3

222 0.09 9 [Lopez etal., 2006℄

2 · 10

−4

_{30 · 10}

−3

162 0.03 5 Arti ial lips

[Gazengel etal.,2007℄ Human lips;

Embou hure: Soft N/A N/A 115.7 N/A 0.79 saxophone-like

Medium N/A N/A 479.9 N/A 0.46 position;

Tight N/A N/A 1073 N/A 0.46 3 embou hures

[Newtonet al.,2008℄ N/A N/A 32 N/A 1.21.8 Human lips

High-speed amera

[Ri hards, 2003℄

5 · 10

−4

_{7 · 10}

−3

162 0.19 3.7 Arti ial lips

tfor goodresults

[Rodet and Vergez, 1996℄ N/A N/A 428.4 0.67 2.88 Trumpet; adjusted

forsimulation

[Ada hi and Sato, 1996℄

1 · 10

−3

_{7 · 10}

−3

60700 variable 0.53 Trumpet; adjusted

forsimulation

167

Table1: Reviewofdierentvaluesoflipparametersfromliterature,alongwithabriefexplanation

of the method. In some arti les, ertain values are not available (N/A). For papers presenting

2-DOFlip models, onlythe rst, outward-strikingDOFis reported. Allbut the lasttwo referen es

deal with trombone parameter values.

168

This work omplements the review published in [Newton, 2009, p.119℄. Many authors do not

169

provide the parameter values they use, nor do they give explanations about their method to get

170

these values, ex ept the fa t that these parameters allow periodi self-sustained os illationof the

171

model. Measurements on human or arti ial lips were made in onditions as similar as possible

172

to the playing onditions. The list of publi ations is not exhaustive: we left aside most of the

173

publi ations sin e they do not justify their values or donot t their measurements with a modal

174

lip-reed model.

175

Geometri parameters (lip hannelwidth, and lip hannel heightwhen the player is not blowing)

176

given inallstudies are onsistent,around

W = 12.10

−3

_m

and

h

0 = 5.10

−4

_m

. Parametri studies

(9)

the model: numeri al values hange but the overall behaviour is the same. Similar observations 179

havebeen madeabout

µ

,even thoughtherange ofthe valuesgathered isalittlewider(

µ ∈ [5, 33]

180

for the trombone).

181

Measurements from [Gazengel et al.,2007, Newton etal., 2008℄ tend to give low quality-fa tor

182

values between

0.5

and

2

for human lips. However, preliminary analysis arried out with

Q

l

≈ 1

183

showed very unrealisti pressure thresholds (order of magnitude :

10

4

to

10

5 _Pa

). Thus, a value

184

for

Q

l

= 7

was hosen, losertothe values measuredon arti iallips(

Q

l

∈ [3.7, 10.5]

). The set of

185

parameters used for simulationand LSA throughout this paperis given inTable 2:

186

h

0 (m)

W (m)

1/µ (m

2 kg

−1

) Q

l

5.10 −4

_12.10

−3

0.11 7

187

Table 2: Lip parametersretained inthis study.

188

Thevalueof

f

l

is onstantlyadaptedbythemusi ianwhileplaying. Forthisreason,weperformed

189

LSAwith

f

l

valuesrangingfrom

20 Hz

to

500 Hz

. Thisallows os illationontherst eightregimes

190

of the instrument, whi h orrespond to the usual notes of the trombone, from

B♭1

to

B♭4

with

191

the slide in rst position.

192

2.3 Stability of the equilibrium solution

193

Linearisinga losed-loopsystemtoassesspotentialinstabilitiesisawidelyusedmethod,inthe

dy-194

nami alsystems ommunity[Bergé etal., 1995℄aswellasinmusi ala ousti sforbrasswind,

wood-195

wind and ute-like instruments [Wilson and Beavers, 1974, Cullenet al.,2000, Silvaet al.,2008,

196

Auvray et al.,2012, Terrienet al.,2014℄. Basi ally, the equations modelling the system are

lin-197

earised arounda known equilibriumsolution. Then,the stability of this solution isdetermined.

198

When the system des ribed in Se tion 2.1 is in stati equilibrium, the lip opening position has

199

a stati value

h(t) = h

e

. This equilibrium position is slightly larger than the lip opening at

200

rest

h

0

, due to the onstraint of the blowing pressure on the inner sides of the lips. Similarly,

201

there is a smallstati overpressure

p

e

at the input of the bore of the instrument, as

Z(ω = 0)

is

202

nonzero. This is related to the pressure loss in the instrument. Mathemati ally, this equilibrium

203

isobtained by an elling alltime derivativesinthe system,asdes ribed inappendixA. Thevalue

204 of

A =

√

_p

b

− p

e

is obtained by solving: 205

A

3 +

A

2 β

+ h

0 µω

2 l

A −

p

b

β

= 0,

(9) with

β =

W Z(ω=0)

µω

2 l

q

2 ρ

. The value of

Z(ω = 0)

is extrapolated from the tted version of the

206

impedan e. Equation (9) has 1 or 3 real roots. In the latter ase, the smallest real positive root

207

should be onsidered to ompute

p

e

= p

b

− A

2

[Silva,2009℄, as

Z(ω = 0)

issmall. The lip hannel

208

height at equilibrium

h

e

is then given by (1)with

¨h = ˙h = 0

.

(10)

In the vi inity of the equilibriumsolution

X

e

, the linearisedfun tion

F

˜

an be writtenas: 210

˜

F (X) = F (X

e

) + J

F

(X

e

)(X − X

e

),

(10)

where

J

F

(X)

is the Ja obianmatrix of the fun tion

F

and

X

e

the state ve tor at the equilibrium

211

solution. The solutions of

X = ˜

˙

F (X)

are under the form:

212

X(t) − X

e

=

N

X

i=1

U

i

e

λi

·t

,

(11)

where

λ

i

are the eigenvalues of

J

F

(X

e

)

and

U

i

the orresponding eigenve tors.

213

Thus,theeigenvaluesoftheJa obianmatrixgiveinformationaboutthestabilityoftheequilibrium

214

solution for a given set of parameters. If at least one of these eigenvalues

λ

has a positive real

215

part, the amplitude of the linearisedsolution tends to innity while time in reases, whi h means

216

the equilibriumis unstable and the solution starts os illating. Referring to (11), this means that

217

one ofthe termsof the sumdominates thesolution,allotherterms being de reasingexponentials.

218

As a rst approximation, the solutionof the linearisedsystem an be written:

219

X(t) − X

e

=

X

Re(λ

i

)>0

U

i

e

λi

·t

(12)

Thedevelopedtoolndsthelowestvalueof

p

b

atwhi htheequilibriumsolutionbe omesunstable,

220

i.e. the value atwhi hone eigenvalue

λ

with positivereal part appears. This value of

p

b

isfurther

221

referredtoas

p

thresh

theos illationthreshold(orthresholdpressure). Duringthetransientphaseof

222

the os illation,the exponentialgrowth of the amplitude isdetermined by the positive real part of

223

λ

,andtheangularfrequen yisgivenbyitsimaginarypart

ω = Im(λ)

. However, thenonlinearities

224

ofthe systemlimitthenalamplitude andalsoae tthe os illationfrequen y ofthesteady state.

225

This method only dete ts instabilitiesemerging from the equilibrium solution. If a stable

os il-226

lating regime oexists along with the stable equilibrium solution, it will not be dete ted. This

227

situation o urs for example in ertain woodwind instruments, where the Hopf bifur ation

( on-228

ne tingtheequilibriumsolutiontotheos illatingone)isinverse insome ases [Grandet al.,1997,

229

Dalmontet al.,2000, Farner etal., 2006,Ri aud et al.,2009℄.

230

2.4 Time-domain simulation

231

Another approa h forstudying musi al instrumentsrelies ontime-domainab initio simulationsof

232

the hosen model, for a given set of parameters.

233

Multiple numeri al methods have been developed to simulate wind instruments with models

234

similar to the one presented in Se tion 2.1. Various approa hes have been proposed to

imple-235

ment the resonator a ousti behaviour. The ree tion fun tion of the bore has been widely

236

used [S huma her, 1981, M Intyre et al.,1983, Ada hi and Sato, 1995, Vergezand Rodet, 1997,

237

Gilbert and Aumond, 2008℄. The modal de omposition of the bore has been hosen for this

(11)

able [MoReeSC, 2013℄. Its prin iples and results have been des ribed in [Silva etal.,2014℄. This 240

simulation tool uses the state-spa e paradigm, similar to the one presented in Se tion 2.1. It

al-241

lowed us to simulate the behaviour of the model with a high number of a ousti modes for the

242

resonator (18 in this study), and oers the ne essary exibility to modify the modelparameters,

243

in ludingthe resonator parameters,as itis done inSe tion 4.

244

3 Results

245

3.1 Linear Stability Analysis

246

The LSA method detailed in Se tion 2.3 is applied to the modeldened in Se tion 2.1, with the

247

set of lip parameters dened in Table 2. The resonator is modelled with a modal t (N=18 in

248

Equation (3)) of a measured impedan e (

B♭

trombone, rst position).

249

Forea hvalue of

f

l

onsidered,the eigenvaluesof the Ja obianmatrix

J

F

(X

e

)

presented in

Equa-250

tion (10)are omputedforin reasing values of

p

b

,untilarst instability, hara terizedby atleast

251

oneeigenvaluewithpositiverealpart,o urs. Forea hvalueof

f

l

,Figure4arepresents

p

thresh

,the

252

lowest value of

p

b

givingrise to an unstable equilibriumsolution (see se tion 2.3). Figure4b

rep-253

resents the imaginarypart of the orresponding eigenvalue divided by

2π

, whi h is the os illation

254

frequen y at threshold, further alled

f

thresh

. Ea h horizontal dotted linein Figure 4brepresents

255

the

n

th

a ousti resonan e frequen y of the instrument

f

ac,n

, given by the lo almaximum of the

256

input impedan e amplitude.

257

It should be noted that, for

p

b

values higher than

p

thresh

, other pairs of onjugate eigenvalues

258

may have a positive real part, whi h implies a system with multiple instabilities. If dierent

259

os illating solutions are stable with these parameters, the system is able to start os illating on

260

dierent a ousti resonan es. In Figure 4, and later g. 11 and g. 14, the rst instability (the

261

one orresponding to

p

b

= p

thresh

) is shown for ea h

f

l

value (solid urve). The se ond instability

262

is reportedonly for anarrowrange of

f

l

(dashed urve).

263

Between

20

and

500

Hz, the two urves of Figure 4 an be divided into 8 parts. Ea h part

264

orrespondstoarangeof

f

l

asso iatedtooneregimeofos illation,relatedtoonea ousti resonan e

265

of the instrument:

[30, 63

Hz℄ (rst regime),

[72, 123

Hz℄ (se ond regime),

[124, 179 Hz]

,

[180, 234

266

Hz]

,

[235, 288 Hz]

,

[289, 352 Hz]

,

[353, 404 Hz]

,

[405, 460 Hz]

. InFigure4b, anos illatingfrequen y 267

plateau is maintained just above ea h value of

f

ac,n

. This is the usual behaviour of an

outward-268

striking valve oupled to anair olumn: when playing on the

n

th

a ousti mode of the bore, the

269

os illationfrequen y atthreshold

f

thresh

isjustabove

f

ac,n

,whi histhe resonan efrequen yof the

270

n

th

a ousti mode [Campbell,2004℄. For ea h regime,

f

thresh

monotonously follows the variation

271

of

f

l

. This mat hes the experien e of the brass player, who an slightly "bend" the pit h up and

272

down, i.e. in rease or de rease the pit h, by adjusting

f

l

through the mus ular tensionof the lips,

273

and by adaptingthe blowing pressure to the hange in

p

thresh

. The range of ea hplateau,i.e. the

(12)

quality fa tor

Q

l

, as detailed in [Silva etal.,2007℄. These frequen y limits are plotted as plain 276

(blue) lines on Fig. 4b). Between

64

Hz and

71

Hz, the equilibrium solution is un onditionally

277

stable whatever the value of

p

b

: this frequen y range overs the impedan e minimum between

1 st

278

and

2 nd

peaks, whi harefartherapartfromoneanotherthantheotherpeaksdue tothe rstpeak

279

inharmoni ity.

280

It an be observed in Figure 4a that the os illation threshold globally in reases with the rank of

281

the a ousti resonan e. A larger

p

b

value isrequired to rea h the higher notes of the instrument,

282

in a ordan e with the musi al experien e. For ea h regime, the

p

thresh

urve is U-shaped, as

283

alreadyobserved in[Silvaet al.,2007℄. Itsminimumvalue

p

opt,n

,marked witha ir le inFigure4,

284

is known to depend signi antly onthe quality fa tor of the lips

Q

l

. In the following, we assume

285

asin[Lopez etal.,2006℄ that

p

opt,n

and the asso iatedlipresonan efrequen y

f

opt

l,n

and os illation

286

frequen y atthreshold

f

opt

thresh,n

representtheoptimalplaying ongurationforahumanperformer. 287

This hypothesis isinlinewith whatmusi iansreport, i.e. they develop astrategytominimizethe

288

eorttoprodu easoundonagivenregime. Thevaluesof

p

opt,n

,between

500 Pa

and

15.3

kPahave

289

thesame orderofmagnitudeasblowingpressure measured by[Bouhuys, 1968℄and [Fréour, 2013℄.

290

The pressure threshold in reases faster when

f

l

grows above

f

opt

l,n

rather than when it de reases

291

below

f

opt

l,n

,asillustratedbytheinsetinFigure4a. Theseresultsare ompatiblewiththeexperien e

292

of brass players, who reportthat "bending down" a note requires less eortthan bendingit up.

293

The rest of this se tion fo uses on some examples of

[p

b

, f

l

]

points to illustrate the dierent

be-294

haviours observed with the model. For ea h ase, the agreement between LSA results and the

295

sound produ edby the time-domainsimulation des ribed in Se tion2.4 isdis ussed.

(13)

50

100

150

200

250

300

350

400

450 1000

5000

10000

20000

f

l

(Hz)

p

thresh

(Pa)

50

100

150

200

250

300

350

400

450

0

100

200

300

400

500 f

l

(Hz)

f

thresh

(Hz)

180 195

215

235 2000

5000

10000

180 195

215

235

230

240

250 b)

a)

297

Figure4: ( olour online)Resultsof LSAappliedto the modeldetailedin Se tion2.1with

param-eters from Table 2. For a range of lip resonan e frequen ies

f

l

, (a) shows the threshold pressure

p

thresh

, while (b) shows the orresponding os illationfrequen y

f

thresh

. Dotted lines are the

val-ues of

f

ac,n

. Cir les indi ate the "optimal" values

p

opt,n

and

f

opt

thresh,n

as dened in the text. The

magnied subplot (zoomon

4 th

regime) highlights the asymmetri al

p

thresh

behaviour above and

below

p

opt,n

. For illustration, the se ond destabilisation threshold (a) and the orresponding

fre-quen y (b) arealsoplottedbetween

f

l

= 109

Hzand

123

Hz. Diagonalsolid(blue)lines in(b) are

analyti allimitsto

f

thresh

for a lossless model.

298

3.2 Exa t mat h between simulation and LSA

299

The simulated pressure atthe input of the instrumentis ompared with the LSA results. In

par-300

ti ular, the os illation threshold is assessed by performing simulations with

p

b

in the vi inity of

301

p

thresh

. The orrespondingfrequen ies, alled

f

osc

,arealso omparedto

f

thresh

givenby LSA.This 302

latterquantity ismeasuredbyapplyingazero- rossingalgorithm[Wall,2003℄,witha sliding

Han-303

ning window(width

0.3s

, overlapping

99%

). This method results in small omputationartefa ts,

304

whi hshould not be taken intoa ount.

305

A simulation with the exa t value of

p

thresh

would theoreti ally lead to aninnite transient time,

306

dened asthe time ittakestorea hsteady state. Therefore, values of

p

b

slightlybelowand above

307

p

thresh

are tested. To illustrate a periodi os illationof the model, the lip resonan e frequen y is 308

(14)

set to

f

l

= 90 Hz

, everything else being given inTable 2. The orresponding mouthpie epressure 309

waveforms are represented in the rst two plots in Figure 5. The third plot shows a situation

310

where

p

b

ismu h higher than

p

thresh

.

311

When the mouth pressure is below the threshold (

p

b

= 1210 Pa

whereas

p

thresh

= 1222 Pa

)

312

(Fig.5 a),the os illationde reases exponentially towards the stati , non-os illatingsolution. The

313

mouthpie epressure onverges towards

115.5 Pa

,whi histhevalueof

p

e

omputedwithLSA.The

314

thi k line represents the exponential de rease in the amplitude

X

a

.e

Re(λ)t

(amplitude of solutions

315

taken from Eq. (12)), where

X

a

is an arbitrary onstant. In this ase, all eigenvalues of

J

F

316

have negative real parts:

λ

is the eigenvalue of

J

F

whi h real part is the losest to zero. The

317

al ulated os illation frequen y (dash-dotted line) is almost onstant and equal to

f

thresh

= 116

318

Hz = Im(λ)/2π

. 319

When the mouth pressure is slightly above the threshold (

p

b

= 1234 Pa

)(Fig. 5b), the simulated

320

pressure waveformenvelopein reases exponentiallyduringthe transientphase, inagreement with

321

Equation(12). However,whentheamplitudein reases,thesignalenvelopeisnolongerexponential

322

andnallystabilizesinasteady-stateregime. The al ulatedos illationfrequen y

f

osc

(dash-dots)

323

begins at

f

thresh

= 116 Hz

; it be omes quite higher inthe permanent regime(

126 Hz

, that is,

8.6

324

%

or 143 musi al ents above

f

thresh

).

325 326 a) b) )

0

5

10

15

50

100

150 time (s)

Pressure (Pa)

p

b

=1210Pa

0

5

10

15

110

120

130

140 f

osc

(Hz)

0

5

10

15 −2000

−1000

0 1000

2000

time (s)

Pressure (Pa)

p

b

=1234Pa

0

5

10

15

110

120

130

140 f

osc

(Hz)

327

Figure5: ( olouronline)Time-domainsimulationswithparametersfromTable 2and

f

l

= 90 Hz

,

with mouth pressure

p

b

(horizontal solid line) lower (a) and higher (b) than the linearised model

threshold (

p

thresh

= 1222 Pa

). Mouth pressure (steady) and mouthpie e pressure (os illating) are

plotted (left verti al axis) along with the exponential growth/diminutionof amplitude al ulated

using LSA (thi k urves: envelope of Equation (12)). The dash-dotted urve depi ts the

instan-taneous playing frequen y (right verti al axis). The expe ted os illation frequen y at threshold

is

f

thresh

= 116 Hz

. The third plot ( ) orresponds to a blowing pressure mu h higher than the

threshold (

p

b

= 3 kPa

;zoomonrst se ond of signal).

328

As expe ted, the behaviourof time-domain simulationsis a urately predi ted by LSA aslong as

329

p

b

remains in the vi inity of

p

thresh

(Figure5a and 5b). The value of

p

thresh

given by LSA is in

330

agreementwith simulations. The eigenvalue with the largest real part predi ts the frequen y and

331

the amplitude of the os illationat the beginning of the simulation. However, above the pressure

332

threshold in Fig.5b, after

t = 8

s,the simulatedamplitude gets ae ted by nonlinear phenomena

(15)

signal, but is obviouslyunable tofully predi t the amplitude of the sustained regime waveform. 335

The thirdplot shows the results with

p

b

= 3 kPa

mu hhigherthan

p

thresh

. LSA andtime-domain

336

simulation give roughly oherent information. As in Figure 5b, the os illating frequen y of the

337

established regime

f

osc

= 130.5 Hz

is

8 %

higher than

Im(λ)/(2π) = 120.8 Hz

. The dieren e

338

is 134 musi al ents, larger than a semitone. This dieren e is lower when

p

b

is loser to

p

thresh

.

339

Despite this dieren e,

f

thresh

predi ts whi ha ousti al resonan e supports the os illation. An in

340

vivo experiment has also shown that the pit h rises when the player in reases the blowing

pres-341

sure [Campbell and Greated, 1994℄. However, this remark shouldbe onsidered arefullybe ause

342

duringpra ti eabrassplayeralwaysapply orrelated ontrolovermouthpressureandlipmus ular

343

a tivity. 344

p

b

(Pa)

Re(λ)

Im(λ)/2π

f

osc

(Hz) measured transient duration(s)

1234

0.2864

116.74

126.5

9.71 1500

5.5591

117.66

127.6

0.74 2000

12.0262

118.99

128.9

0.31 2500

16.0891

120.01

129.7

0.215 3000

18.8507

120.82

130.5 0.1675

345

Table3: Valuesoftherealpartofthe destabilisingeigenvalue

λ

,itsimaginarypartdividedby

2π

,

theos illationfrequen yoftheestablishedregime,andthedurationofthetransient(bothmeasured

onsimulations)fordierent valuesof theblowing pressure (allother parametersun hanged). The

real part of

λ

in reases with

p

b

, whi h implies a faster-growing envelope as

p

b

in reases. This

is onsistent with the transient duration measured with MIRonsets

1

fun tion estimating the time

needed torea h

95%

of the maximum amplitudeof

p(t)

.

346

Transient time, i.e. the time needed for the amplitude to rea h

95%

of its nal value, have been

347

measured with dierentvalues of

p

b

. The values are reported inTable 3.

348

The transienttime de reases while

Re(λ)

in reases, whi h anbemodelled: a ording toEq. (12)

349

the amplitude grows exponentially with

Re(λ)

. Thus, under the assumption that

p

e

is negligible

350

ompared to

95%

of the nal amplitude (hereinafter noted

p

95%

), one an write:

351

p

95%

= B.e

Re(λ).transient

,

(13)

where

B

is areal onstant and

transient

the transient time (s).

352

Furthermore,a ordingto[Bergé etal., 1995,p.40℄inthe vi inityofadire tHopfbifur ation,the

353

maximum amplitudeof the os illationisproportionaltothe square rootof the dieren ebetween

354

theparametervalueandthethreshold value,whi hmeans

√

_p

b

− p

thresh

here. Therefore, thevalue 355

of the pressure at

t = transient

is:

356

1

Part of MIRtoolbox: https://www.jyu.fi/hum/laitokset/musiikki/en/resear h/ oe/materials/

(16)

p

95%

= 0.95.C.

√

p

b

− p

thresh

,

(14)

where Cis areal onstant.

357

Introdu ing this expression of

p

95%

in the natural logarithm of Eq. (13) results in the following

358

analyti alexpression of the transient time where

A =

0.95.C

B

: 359

transient =

1 Re(λ)

· ln(A

√

p

b

− p

thresh

).

(15)

With

A = 4.75

tted on values measured on time-domain simulations, this model mat hes very 360

wellwith the evolutionof transient durations measured onsimulationswith dierent values of

p

b

,

361 as shown inFigure 6. 362

0

5

10

15

20

0

2

4

6

8

10 Re(

λ

)

transient duration (s)

363

Figure6: ( olouronline)Transientdurations measuredontime-domainsimulations,plottedalong

the

Re(λ)

value (

∗

symbols). The solid lineisthe transientduration modeldes ribed by Eq.(15). 364

The os illationfrequen y alsoin reases with

p

b

. An estimateofthe frequen y is alsogiven

(imag-365

inary partof

λ

divided by

2π

)whi h mat hes wellthe pseudo-frequen y of the transient phase of

366

ea h signal.

367

This example is representative of most ases tested: LSA orre tly predi ts whether the solution

368

is os illating, with an a eptable estimation of the os illation frequen y. The transient duration

369

an be a urately predi ted with the real part of

λ

, asdes ribed in Eq. (15) even for

p

b

far above

370

the threshold. However, the a ura y of the os illationfrequen y predi tion is limited, and LSA

371

an predi tneitherthe steady-state waveformnor the natureof the os illationregime. This latter

372

observation willbefurther highlighted inthe followingsub-se tion.

373

3.3 Unforeseen behaviours

374

LSAprovidesalotofrelevantinformationaboutthe os illationthresholdand thetransientphase.

375

This is parti ularly true when

p

b

is near

p

thresh

. However, some simulations (detailed below)

(17)

in ludequasi-periodi os illations. Bydenition, theseare deterministi os illationswhose energy 378

is lo ated at frequen ies whi h are integer ombinations of base frequen ies, whose ratio is an

379

irrationalnumber.

380

381

Quasi-periodi os illations

382

Firstly, the previous omparison between LSA and time-domain simulation is reprodu ed with a

383

dierent lip resonan e frequen y. Three simulations are performed with the parameters given in

384

Table 2 and

f

l

= 110 Hz

. Forthese parameters,

p

thresh

is equalto

711 Pa

. Again, three dierent 385

p

b

values aretested:

p

b

= 701 Pa

,

p

b

= 720 Pa

toillustratethe behaviourjustbelowand abovethe

386

threshold, and

p

b

= 2 kPa

foranexamplefarabovethe threshold. Resultsare plottedinFigure7.

387

When

p

b

is under the threshold, results are very similar to the previous ase with

f

l

= 90 Hz

388

(Fig. 7a and 7d). However, when

p

b

be omes large enough to indu e an os illatingsolution, the

389

os illation of the mouthpie e pressure be omes quasi-periodi instead of periodi (Figure 7b, 7e,

390

7 and 7f). The quasi-periodi natureof the signal is learly visibleonthe spe tra (Figure7e and

391

7f) with se ondary peaksaround the prin ipalfrequen y peaks.

392 393 a) b) )

0

5

10

15

0

20

40

60

80

100

120 time (s)

pressure (Pa)

p

b

=701Pa

0

5

10

15 −1000

−500

0

500 time (s)

pressure (Pa)

p

b

=720Pa

0

0.5

1

1.5

2 −4000

−3000

−2000

−1000

0 1000

2000

time(s)

Pressure (Pa)

p

b

=2000Pa

d) e) f)

0

200

400

600

800 1000

−40

−20

0

20

40

60

80

100 Frequency (Hz)

20.log[FFT(p(t)]

0

200

400

600

800 1000

−60

−40

−20

0

20

40

60

80

100 Frequency (Hz)

20.log[FFT(p(t)]

0

200

400

600

800 1000

−40

−20

0

20

40

60

80

100 Frequency (Hz)

20.log[FFT(p(t)]

394

Figure7: ( olouronline)Simulationresultsfor

f

l

= 110

Hz,the pressurethresholdbeing

p

thresh

=

711

Pa. Like in Figure 5 three simulations are shown with

p

b

= 701

Pa (a),

p

b

= 720

Pa (b)

and

p

b

= 2

kPa, mu h higher than

p

thresh

( ).

p

b

is plotted as an horizontal solid (red) line. The

envelopeofEq.(11)isplottedinplain(bla k)line. Otherparameters(lip hara teristi s)aregiven

inTable 2. Figures (d), (e) and (f)are the spe tra orresponding to(a), (b) and ( ), respe tively

((e) and (f) al ulated using steady regimes of (b) and ( ).

(18)

is attested in the vi inity of the bifur ation, and the pressure threshold

p

thresh

is a urately 397

predi ted, but the o urren e of aquasi-periodi regime annotbe predi ted.

398

399

Period doubling

400

When

f

l

is equal to

55

Hz,

p

b

to

400

Pa (

p

thresh

being

161 Pa

), and the other parameters are the

401

values given inTable 2,the simulationresultos illatesat

f

osc

= 32.5 Hz

, far below

f

thresh

= 59.78

402

Hz. This is a pe uliar behaviour, as this os illationfrequen y is signi antly under the trombone

403

rst a ousti resonan e (

f

ac,1

= 38 Hz

). Indeed, the hosen model indu es playing frequen ies

404

above the a ousti resonan e frequen y (

f

osc

> f

ac,n

), at least near the pressure threshold, to

405

omply with the regeneration ondition [Eliott and Bowsher, 1982℄.

406

Figure 8 ompares the spe trum of the mouthpie e pressure simulated with the aforementioned

407

parameters and

f

l

= 55

Hz (dotted line) and then with

f

l

= 50

Hz (solid line). For

f

l

= 50 Hz

,

408

f

osc

= 65 Hz

is higher than

f

thresh

= 56.3

Hz, like in previous simulations in Se tion 3.2. For 409

f

l

= 55

Hz,areasonable expe tationwould beanos illationfrequen y slightlyhigher than

65

Hz,

410

as

f

osc

tends to in rease with

f

l

. However, the simulation os illationfrequen y at

f

l

= 55

Hz is

411

f

osc

= 32.47

Hz, lose to halfof its value at

f

l

= 50

Hz. 412

0

50

100

150

200

40

60

80

100

120

140 frequency (Hz)

Magnitude (dB)

413

Figure8: ( olouronline)Spe tra ofthe simulatedtrombone mouthpie epressures, with

p

b

= 400

Paforbothlipresonan efrequen ies,

f

l

= 50

Hz(solid)and

f

l

= 55

Hz(dotted)(otherparameters

fromTable 2). Crossmarkers givethe values of

f

thresh

= 56.3

Hzfor

f

l

= 50

Hzand

f

thresh

= 59.8

Hz for

f

l

= 55

Hz. The solid verti al line indi ates the rst a ousti resonan e frequen y of the

trombone bore,

f

ac,1

= 38

Hz.

414

Furthersimulationswere arriedout, with

f

l

goingfrom

50

to

61

Hzinsteps of

1

Hz,

p

b

= 400

Pa

415

and the others parameters set as in Table 2. Table 4 reports the os illationfrequen y measured

416

on the simulated signals, along with the

f

thresh

value predi ted by LSA. Between

54

and

55

Hz,

417

the os illation frequen y is almost halved. Then, between

56

and

57

Hz, the frequen y is again

418

halved, be omingaquarterofitsvaluefor

f

l

< 55

Hz. For

f

l

= 59

Hzand above,thefundamental

(19)

inthe spe trum. 421

f

l

(Hz) 50 51 52 53 54 55 56 57 58 59 60 61

f

osc

(Hz) 65.45 65.48 65.49 65.49 65.46 32.53 32.54 16.32 16.32 65.1 65.1 65.1

f

thresh

(Hz) 56.3 56.97 57.71 58.36 59.08 59.78 60.51 61.27 62 62.77 63.58 64.44 422

Table4: Os illationfrequen iesmeasuredonthesimulatedmouthpie epressure,forlipfrequen ies

from 50 to 61 Hz,

p

b

= 400

Pa and other parameters from Table 2. Os illation frequen ies at

threshold given by LSA are also reported.

423

These results are lose to those reported in [Gibiat and Castellengo, 2000℄, with a trombone

424

player performing two su essive period doublings. When in reasing

f

l

in this range, the model

425

undergoes multiple period-doubling bifur ations. Similar s enarios have been observed on

nu-426

meri al models of woodwind instruments [Gibiat, 1988, Kergomard etal.,2004℄. This su ession

427

of period doublings is also known as subharmoni as ade or Feigenbaum s enario and leads to

428

haoti behaviour, whi hmay explainthe noisinessof signalsabove

f

l

> 58

Hz. Again,explaining

429

the o urren eof su h phenomena isout of rea h with LSA.

430

431

Overblowing 432

Besidesthesetwononlinearphenomena,otherdieren esbetweeneigenvalue-basedLSAand

time-433

domain simulation an be observed. Another example is given with

f

l

= 120 Hz

, the parameters

434

given inTable2andahighblowingpressure:

p

b

= 6.5 kPa

whilethe thresholdis

p

thresh

= 1056 Pa

.

435

While

f

thresh

= 128.4 Hz

isjustabovethe

2 nd

a ousti resonan efrequen y ofthebore (

f

ac,2

= 112

436

Hz

), the simulation os illation frequen y ex eeds the

3 rd

:

f

osc

= 187.5

Hz >

f

ac,3

= 170

Hz. 437

Figure9 shows the spe trumof asimulation os illatingonthe third a ousti resonan e, while the

438

predi ted os illationat threshold orresponds tothe se ondone.

439

0

100

200

300

400

500

600

700

60

80

100

120

140

160

180 fréquence (Hz)

pression (dB)

f

thresh

440

Figure9: ( olouronline)Spe trumofsimulatedmouthpie epressurefor

f

l

= 120 Hz

and

p

b

= 6.5

kPa

withotherparameterstakenfromTable2. Theself-sustainedos illationo ursat

f

osc

= 187.5

Hz

, orresponding to the third a ousti resonan e, while LSA predi ts an os illation at

f

thresh

=

128.4 Hz

(thi k verti al line) with

p

thresh

= 1056 Pa

. Ea h dash-dotted line represents the

n

th

a ousti resonan e frequen y

f

ac,n

of the trombone bore.

(20)

The method previously used, whi h onsists in retaining the lowest

p

b

value ausing a destabil-442

isation, does not predi t the behaviour of the system with su h a high blowing pressure. Yet,

443

this os illation on the third regime an be understood, sin e another pair of eigenvalues of the

444

Ja obian matrix with a positive real part appears for

p

b

> p

thresh

. The dashed lines in Figure4a

445

and 4bshows the pressure threshold orresponding to the se ond pair of su h eigenvalues ( alled

446

λ

2

and

λ

∗

2

), and the asso iated os illation frequen y. For

f

l

= 120 Hz

the se ond threshold is

447

6116 Pa

with an os illation frequen y equal to

Im(λ

2 )/2π = 172 Hz

, orresponding to the third 448

regimeofos illationofthesystem. Thisis onsistentwiththebehaviourobserved inthenumeri al

449

simulation. 450

3.4 Open-loop transfer fun tion

451

For a better understanding of the origin of the dierent instabilities, another LSA formalism is

452

used,whi hgivesvisualinformationaboutthestabilitymarginsofthedierentos illationregimes.

453

It onsistsinstudyingalinearisedversionof theopen-looptransferfun tion(OLTF)ofthe system

454

dened by Equation (5), (1) and (3) [Saneyoshi et al.,1987, Ferrandet al.,2010℄. This OLTF is

455

divided into two parts: the ex iter admittan e

Y

a

whi h des ribes the lip reed behaviour, from

456

Equation (5) and (1), and the resonator input impedan e, whi h is modelled with a modal t of

457

itsinput impedan e

Z

likeinthe otherformalism (see Equation (3)).

458

The linearisationof theex iter admittan e

Y

a

simpliestoa

1 st

degree Taylor expansionof

Equa-459

tion (5)near the equilibriumpoint;Equation (1) isthen put intothe result. Details an befound

460

inAppendix B about the al ulationwhi h leads tothe followingexpression of

Y

a

:

461

Y

a

= W h

e

s

2δp

e

ρ

D(ω)

Kh

e

−

1 2δp

e

,

(16)

where

D(ω)

represents the dynami sof the lipreed.

462

The stability of the OLTF, alled

H

OL

, is then evaluated with the Barkhausen

rite-463

rion [Wangenheim,2011℄, whi h points to possibly unstable solutions when

H

OL

= Y

a

.Z = 1

.

464

OnaBodediagram,pointswith

H

OL

havinga

0

dBmagnitudeand0

◦

phasearelimitsofstability.

465

This methodhas already been used for larinetmodels with inward-strikingvalves, and for brass

466

and ute-likeinstruments [Saneyoshi et al.,1987, Ferrand etal.,2010,Terrien et al.,2014℄.

467

Figure10shows the Bode diagramofthe OLTF of thesystem fed withthe same parametersasin

468

Figure9. The stabilitylimitsare indi ated with rosses.

(21)

Figure10: ( olouronline)Bodediagramoftheopen-looptransferfun tionofthe trombonemodel

with the parameters spe iedin Table 2,

f

l

= 120

Hz and

p

b

= 6.5

kPa. There are two instability

points ( rosses), with a

0

dBmagnitude and a zero phase.

471

Here, the Bode diagram shows two points of

0

dB magnitude and 0 degree phase at

132

Hz and

472

172

Hz. In terms of the eigenvalues-based LSA tool des ribed in Se tion 2.3, these frequen ies

473

orrespond to the imaginary part of the eigenvalues of

J

F

al ulated with

p

b

= 6500

Pa and

474

havingapositiverealpart. Thefrequen y obtained withOLTFdiers fromthe oneobtained with

475

eigenvalues of the Ja obian matrix, be ause

f

thresh

= 128 Hz

is obtained at

p

b

= p

thresh

= 1056

476

Pa

while the OLTF value is obtained with

p

b

= 6.5

kPa. The real part ofthe se onddestabilising

477

pair of eigenvalues be omes positive above

6116 Pa

, whi h is ompatible with an os illation on

478

this regime at

p

b

= 6.5

kPa. The related frequen y at threshold is

172.9

Hz orresponding to an

479

os illationonthe third a ousti resonan e.

480

Both LSA methods show multiple instabilities of the stati solution, that is, multiple possible

481

regimes of os illation. The predi tions of threshold pressures and possible os illation frequen ies

482

aresatisfa tory. Buttheygivenoinformationeitheraboutthestabilityoftheseos illationregimes,

483

or about whi h regime the instrument will a tually os illate on. This is determined by initial

484

onditions and by the stability of the dierent os illating solutions, whi h depends on nonlinear

485

elements out of rea hof the method.

486

4 Lowest regime of os illation

487

This hapter fo uses on the results of LSA and time-domain simulation on the lowest regime,

488

relatedtothe rst a ousti resonan e of theair olumninsidethe bore. This lowest playable note

489

is alled "pedal note" by musi ians. Forthe trombone in rst position, and the saxhorn with no

490

valvedepressed (neutralposition),the pedal note is a

B♭1

at

58 Hz

inthe musi al s ale.

(22)

To ompare the behaviourof thedierentregistersof thetrombone, the ratiobetween the thresh-493

old frequen y

f

thresh

and the resonan e frequen y of the orresponding a ousti al mode

f

ac,n

is

494

omputed. Figure 11a and 11b give

p

thresh

and

f

thresh

like in Figure 4 but on a smaller

f

l

range,

495

and Figure 11 gives the

f

thresh

/f

ac,n

ratio.

496

At the lip frequen ies orresponding to the pressure threshold minima, alled

f

opt

l,n

(see ir les in

497

Figure 11), this ratio appears to be signi antly higher for the rst a ousti resonan e than for

498

the other ones:

f

opt

thresh,1

/f

ac,1

= 55.6/38 = 1.46

while

f

opt

thresh,n

/f

ac,n

∈ [1.04, 1.1]

for

n ≥ 2

as shown 499

inTable 5.

500

It anbe notedthat,atleast forthe ve lowest resonan es,

f

opt

thresh,n

isingoodagreementwith the 501

note supposed tobe played onthe instrumentfor this resonan e, a ording tothe tempered s ale

502

(seeTable5). Therefore,LSAgivesareliableestimationofthenote orrespondingtothesea ousti

503

resonan es,in ludingthe pedalnote, witharelativeerror between

f

opt

thresh,n

and thetempereds ale 504

note smallerthan

5.5%

. However,

f

opt

thresh,n

underestimatesthe playingfrequen y of the pedal note 505

while it overestimates the othernotes.

506 507 Regime

f

opt

l,n

(Hz)

f

opt

thresh,n

(Hz) tempered s ale (Hz) relativeerror

f

ac,n

(Hz)

f

opt

thresh,n

/f

ac,n

1 49 55.6 58.27

−4.6%

38 1.46 2 110 122.9 116.54

5.4%

112 1.1 3 162 180.0 174.81

2.9%

170 1.06 4 215 238.9 233.08

2.5%

228 1.05 5 271 301.6 291.35

3.5%

290 1.04 508 Table 5:

f

opt

thresh,n

valuesfor the velowest regimesofthe trombone, omparedwith thefrequen y

of the expe ted note. The a ousti resonan e frequen y of the orresponding mode, the

f

opt

l,n

value

and the

f

opt

thresh,n

/f

ac,n

ratio are alsogiven.

f

opt

thresh,n

isasuitable predi tionofthe played note. The

f

_thresh,n

opt

/f

ac,n

ratio is parti ularlyhigh for the rst os illationregime. 509

For illustration, a simulation is arried out with the usual parameters from Table 2 with