simulations of modal active control applied to the self-sustained...

ACTA ACUSTICA UNITED WITH ACUSTICAVol 100 (2014) 1149 ndash 1161

DOI 103813AAA918794

Simulations of Modal Active Control Applied tothe Self-Sustained Oscillations of the Clarinet

Thibaut Meurisse1) Adrien Mamou-Mani1) Reneacute Causseacute1) Baptiste Chomette2)lowast David B Sharp3)

1) UMR STMS IRCAM - CNRS - UPMC - 1 place Igor Stravinsky 75004 Paris Francethibautmeurisseircamfr

2) CNRS - UMR 7190 - Institut Jean Le Rond drsquoAlembert - 4 place Jussieu 75005 Paris France3) Acoustics Research Group Faculty of Maths Computing and Technology DDEM MCT The Open University

Walton Hall Milton Keynes MK7 6AA Bucks Great Britain

SummaryThis paper reports a new approach to modifying the sound produced by a wind instrument The approach isbased on modal active control which enables adjustment of the damping and the frequencies of the differentresonances of a system A self-sustained oscillating wind instrument can be modeled as an excitation sourcecoupled to a resonator via a non-linear coupling The aim of this study is to present simulations of modal activecontrol applied to a modeled self-sustained oscillating wind instrument in order to modify its playing propertiesThe modeled instrument comprises a cylindrical tube coupled to a reed and incorporates a collocated loudspeakerand microphone it can thus be considered to approximate a simplified clarinet Modifications of the pitch thestrength of the harmonics of the sound produced by the instrument and of the oscillation threshold are obtainedwhile controlling the first two resonances of the modeled instrument

PACS no 4350Ki 4375Pq

1 Introduction

Active control is a means of altering the sound and vibra-tions of a mechanical system by automatic modificationof its structural response [1] The important componentsfor an active control set-up are a sensor to detect the vi-bration (usually a microphone for acoustical systems) acontroller to manipulate the signal from the sensor and anactuator to influence the response of the system (usually aloudspeaker for acoustical systems)

Two different approaches to active control are com-monly used the classical approach and the state-spaceapproach [2] The classical approach to active control isbased on transfer functions and involves applying gainsphase shifting filters and delays to the vibration signalmeasured by the sensor and then sending the resultant sig-nal to the actuator Classical active control can be appliedto a system either over specific frequency ranges or overthe whole bandwidth but it cannot modify the modal pa-rameters of the individual resonances of the system in atargeted manner Classical active control techniques wereoriginally developed as a means of reducing vibrationsin mechanical systems [3] For example using bandpassfilters and time delays self-sustained oscillating systems

Received 13 January 2014accepted 01 August 2014lowast Also at UPMC - Universiteacute Pierre et Marie Curie 06 - Institut Jean LeRond drsquoAlembert 4 place Jussieu 75005 Paris France

such as cavity flow oscillations can be controlled [4] Mu-sical wind instruments such as the clarinet which can bemodeled as an excitation (often referred to as a disturbancein the context of active control) coupled to a resonatorvia a non-linear coupling [5 6 7 8] (see Figure 1) areself-sustained oscillating systems In principle thereforeit is possible to apply classical active control techniques towind instruments to modify their playing properties Overthe past decade some preliminary work in this area hasbeen carried out (this work has followed previous appli-cations of classical active control to percussion and stringinstruments [9 10 11]) In particular classical active con-trol has been used to play a complete octave on a flute withno holes [12] To achieve this the incident wave was ab-sorbed by a speaker at the end of the tube and replaced bya chosen reflected wave The classical approach has alsobeen used to modify all the resonances of a simplified clar-inet with gain and phase shifting [13] but the control couldnot modify the resonances independently

The state-space approach to active control involves es-tablishing a state space representation of the system Thisis a mathematical model which when projected on themodal base represents the system in terms of state vari-ables (expressed as vectors and matrices) containing themodal parameters (frequency damping) of the systemSuch a control is called modal active control [1] Unlikethe classical approach modal active control enables themodal parameters of a system to be modified so that in-dividual resonances can be adjusted to reach target fre-

copy S Hirzel Verlag middot EAA 1149

ACTA ACUSTICA UNITED WITH ACUSTICA Meurisse et al Self-sustained oscillations of clarinetVol 100 (2014)

quency and damping values [14 15] In principle there-fore modal active control should enable the frequenciesand the damping factors of the resonances of a wind instru-ment to be altered in a specified manner However the be-havior of a self-sustained system with modal active controlapplied is unknown and needs to be investigated Therehave been relatively few applications of modal active con-trol to musical instruments [16 17] and no application towind instruments to the authorsrsquo knowledge

This paper reports a first step towards applying modalactive control to real musical wind instruments In this pa-per simulations of modal active control applied to a modelof a self-sustained simplified clarinet (ie a cylindricaltube coupled to a reed) are presented The control is im-plemented with the intent of adjusting the resonances ofthe virtual instrument thereby altering the timbre of thesound it produces as well as its playing properties Suchadjustments provide similar effects to those that would re-sult from modifying the instrumentrsquos bore profile

In section 2 a model of a simplified clarinet with an in-corporated active control system is presented The clarinetpart of the model is made up of a reed model non-linearlycoupled to a state-space model of the resonator Mean-while the active control part of the model adds the com-ponents (microphone and loudspeaker) that are requiredto enable active control to be applied to the virtual instru-ment In section 3 simulations are carried out to demon-strate the effects of the control Four sets of simulationsare presented The first set provides examples of the con-trol of the frequency and damping of the first resonance ofthe tube The second and third sets provide maps showingcontrol possibilities in terms of both frequency and damp-ing for the first resonance of this self-sustained oscillatingsystem The fourth set provides an example of the controlof the frequency and damping of the second resonance ofthe tube Finally section 4 provides some conclusions andoutlines the proposed next steps for the research

2 Modeling a self-sustained oscillatingwind instrument with incorporatedmodal active control system

Models of self-sustained wind instruments such as theclarinet have been reported for over 30 years [6 18 7 85] In such models the instrument is described in termsof both linear (reed resonator) and non-linear (coupling)elements (see Figure 1)

In this section a complete model of a simplified clarinetwith an incorporated modal active control system is devel-oped (see Figure 2) First a model of the clarinet reed andthe non-linear coupling is described Then a state-spacerepresentation of the resonator is presented This represen-tation takes into account possible excitation both from thereed and from the speaker of the active control system aswell as monitoring of the pressure by the microphone ofthe active control system Finally the designs of the ob-server and controller elements of the active control systemare described

Figure 1 Model of a self-sustained wind instrument [6 19]

Figure 2 Model of a self-sustained wind instrument with controlsystem w y and us are defined in equation (24) y and x are theobserver estimations of y and x

21 Reed and non-linear coupling

The model of reed and non-linear coupling used in thispaper is the one described by Karkar et al [8]

In a clarinet a single reed controls the flow of air fromthe playerrsquos mouth into the instrument Let h(t) be the po-sition of the reed at time t Then

1

ω2r

h(t) +

qr

ωr

h(t) + h(t) minus h0 = minus 1

KrPm minus P (t) (1)

where ωr is the resonance frequency of the reed qr itsdamping h0 its equilibrium position and Kr its stiffnessThe and symbols represent respectively the first andsecond order time derivatives Pm is the pressure in theplayerrsquos mouth and P (t) the pressure in the mouthpiece

Using the same assumptions as described in [19] andfollowing Hirschberg [20] namely thatbull the flow is quasi-staticbull the kinetic energy of the flow is entirely dissipated by

turbulencebull the pressure is uniformly distributed in the mouthpiece

and equal to that of the input of the resonatorbull the velocity of the air in the musicianrsquos mouth is negli-

gible compared to that of the air in the flowthe flow can be determined from

U (t) = H h(t) sign Pm minus P (t)

middot W h(t)2|Pm minus P (t)|

ρ0 (2)

1150

Meurisse et al Self-sustained oscillations of clarinet ACTA ACUSTICA UNITED WITH ACUSTICAVol 100 (2014)

where W is the width of the reed duct and ρ0 the den-sity of the air The Heaviside function H (h(t)) has beenadded to model the contact with the mouthpiece in orderto deal with possible negative h values which may happennumerically but are not physical (they would involve thereed passing through the solid walls of the mouthpiece)[21] In a previous paper [22] the Heaviside function wasnot used and resulted in non-physical results

To simplify the formulation of the equations the follow-ing dimensionless parameters are introduced

xh(t) = h(t)h0

yh(t) =h(t)v0

p(t) = P (t)PM

u(t) = U (t)ZcPM

(3)

where v0 = h0ωr is characteristic of the reedrsquos velocityduring free motion PM = Krh0 is the pressure requiredto completely close the reed channel in static regime andZc = ρ0cS is the characteristic impedance of the tubewith c the velocity of sound in the medium and S the sec-tion of the tube The motion of the reed and the non-linearcoupling can therefore be described by the following equa-tions

1ωr

xh(t) = yh(t) (4a)

1ωr

yh(t) = 1 minus xh(t) + p(t) minus γ minus qryh(t) (4b)

u(t) = H xh(t) ζ sign γ minus p(t)

middot xh(t) |γ minus p(t)| (4c)

where ζ = ZcW 2h0ρ0Kr is the dimensionless reedopening parameter and γ = PmPM represents the dimen-sionless pressure in the musicianrsquos mouth Without con-trol the oscillation threshold is at γ 13 + with 1[7]

22 State-space model of the resonator

Modal active control makes it possible to control the fre-quencies and damping factors of the modes of a system Tobe able to apply modal active control however the sys-tem must be expressed in terms of a state-space modelcomprising vectors and matrices which describe the sys-temrsquos modal parameters Here a state-space model ofthe resonator of the simplified clarinet is described Themodel also incorporates the control system with micro-phone (sensor) and speaker (actuator) included as well asthe disturbance (excitation) produced by the reed (see sec-tion 21)

The state-space model of the cylindrical tube used inthis paper is derived in [14] The diameter 2R of the tubeis sufficiently small compared with its length Lt that thetube can be considered to be a one-dimensional waveguidewith spatial coordinate z where 0 le z le Lt In the modelthe speaker element of the control system is positioned atz = zs the microphone at z = zm and the disturbance is

produced at z = zd They are all at the same position atthe input of the tube (zs = zm = zd = 0)

The pressure in the tube with the speaker incorporatedin the tube wall and the reed at the entrance of the tube isdescribed by the nonhomogeneous equation

1c2

p(z t) = p (z t) + ρ0

vs(t)δ(z minus zs)

+ρ0

SZc

u(t)δ(zminuszd) (5)

where p is the dimensionless acoustic pressure vs thespeaker baffle velocity δ the Kronecker delta and u thedimensionless flow at the input of the instrument Theand symbols represent respectively the first and secondorder spatial derivatives The second and third terms of theright-hand side of the equation represent respectively thecontribution of the vibration of the control speaker and thecontribution of the flow coming from the reed channel atthe end of the tube

Using separation of variables let

p(z t) = V (z) q(t) (6)

and consider the homogeneous equation

1c2

p(z t) = p (z t) (7)

Using equations (6) and (7) gives

V (z) +λ

c2V (z) = 0 (8)

Hence

V (z) = α sin(kz) + β cos(kz) (9)

with k2 = λc2 where λ is a positive constant As thetube is modeled as being closed at one end and open at theother the following boundary conditions apply

V (Lt) = 0 (10)

V (0) = 0 (11)

This implies α = 0 and cos(kLt) = 0 then ki = (2i minus1)π2Lt Therefore

Vi(z) = β cos(kiz) (12)

Vi(z) is scaled so that

Vi(z)1c2

Vj(z) =1c2

Lt

0Vi(z)Vj(z) dz

= δij (13)

where δij is the Kronecker delta This implies that

β = c2Lt

(14)

1151


and thus

Vi(z) = c2Lt

cos(kiz) (15)

where Vi(z) is the amplitude of the mode i at position zReturning to the nonhomogeneous equation (5) let

p(z t) =j

Vj(z) qj(t) (16)

so that

1c2

j

qj(t) + λjqj(t) Vj(z) =

ρ0

vs(t)δ(z minus zs) +ρ0

SZc

u(t) δ(zminuszd) (17)

where λj = k2j c

2 Taking the inner product of both sides of(17) with Vi(z) as in (10) and (11) yields

qi + λiqi(t) = bius(t) + giw(t) (18)

where

us = ρ0

vs (19)

is the command sent to the speaker while the control sys-tem is operating and

bi = Vi(zs) (20)

with Vi described in equation (15)

w(t) =ρ0

SZc

u (21)

is the disturbance signal sent by the reed through the non-linear coupling while the musician is blowing and

gi = Vi(zd) (22)

Equation (18) is the undamped normal mode equation forthe cylindrical tube To obtain a state-space description ofthe cylindrical tube with r modes without considering themode of the speaker and including a proportional modaldamping let

x(t) =qiqi

i = 1r (23)

where x(t) is the state vector used in the linear model sothat

x(t) = Ax(t) + Bus(t) +Gw(t)

y(t) = Cx(t) (24)

where us is defined in equation (19) w(t) is the distur-bance signal at zd = 0 y(t) is the output of the systemthe acoustic pressure measured at the position of the mi-crophone so that

y(t) = p(zm t) =i

Vi(zm)qi (25)

and

A =0rr Irr

minus diag (ω2i ) minus diag (2ξiωi)

(26)

is the dynamical matrix which contains the frequenciesand damping factors of the resonances of the cylindricaltube (with ωi and ξi respectively denoting the frequencyand the damping factor of the ith mode) For a real instru-ment over an infinite bandwidth the dynamical matrix Ais infinite with an infinite number of modes In this pa-per where a modeled instrument is studied over a limitedbandwidth the number of modeled modes is finite Themodal parameters of these modes are extracted from theinput impedance of the tube calculated using transmis-sion matrices with electroacoustical analogues [23] usinga Rational Fraction Polynomials (RFP) algorithm [24]

Meanwhile I is the identity matrix From equation (18)and (19) it follows that

B =

0r1

b1br

(27)

the actuator matrix with bi described in equation (20)From equation (24) and (25) it follows that

C = V1(zm) middot middot middotVr(zm) 01r (28)

the sensor matrix with Vi described in equation (15) Fi-nally from equation (18) and (21) it follows that

G =

0r1

g1gr

(29)

the disturbance matrix with gi decribed in equation (22)As with the A matrix the B C and G matrices are in-finitely large for a real instrument Here where a modeledinstrument is studied they are finite in size

23 Modal Control

Modal active control is used to modify the modal parame-ters of the resonances of a system In other words to mod-ify their frequencies and damping factors

To apply modal active control to the resonator of thesimplified clarinet a Luenberger [25] observer and a con-troller are implemented (see Figure 2) The observer di-rectly receives what is measured by the microphone itsrole is to rebuild the state vector x using a model of thesystem and the measurement y = p Let Am Bm and Cm

be the modeled identified matrices of the system used bythe observer In this paper the model used by the observerhas the same number of modes as the clarinet model de-veloped in section 22 (thus Am = A Bm = B and Cm =C) Meanwhile let x(t) be the built state vector estimatedby the Luenberger observer x(t) is used by the controller

1152


to generate a command us(t) that is transmitted throughthe speaker in order to apply the control to the resonatorThis command us(t) the same as in equation (24) can beexpressed as

us(t) = minusKx(t) (30)

where K is the control gain matrix used to move the polessi of the system K is determined together with the Am

and Bm matrices using a pole placement algorithm (inthis work the algorithm developed by Kautsky et al [26]is used)

The frequencies and damping factors of the resonatormay be defined through the poles si of the Am matrix thatis its eigenvalues so that [2]

Re(si) = minusξiωi

Im(si) = plusmnωi 1 minus ξ2i

(31)

By using control gains K these poles can be moved in or-der to reach new target values for the resonatorrsquos frequen-cies (angular frequencies ωit ) and damping factors (ξit )These new poles are the eigenvalues of Am minus BmK

Control of this type is referred to as proportional statecontrol The dynamics of this observer can be written

x(t) = Amx(t) + Bmus(t) + L y(t) minus y(t)

y(t) = Cmx(t)(32)

where L is the observer gain vector L is chosen suchthat the error between the measurement and its estimationey = yminusy converges to zero It is calculated using a LinearQuadratic Estimator (LQE) algorithm Theˆsymbol repre-sents the estimations of the observer us(t) is calculated inequation (30)

In the next section transfer functions are carried out forboth without control and with control (closed loop) condi-tions Without control the system of Figure 1 is describedby

x(t) = Ax(t) +Gw(t)

y(t) = Cx(t)(33)

The transfer function HW C of such a system is then [27]

HW C (s) =y

w= C sI minus A minus1G (34)

With control applied the system of Figure 2 is describedby equation (24) (32) and (30) The closed loop transferfunction HCL is then [27]

HCL(s) =y

w(35)

= C sIminus AminusBK sIminus(AmminusBmKminusLCm)minus1LC

minus1G

The modal parameters of the transfer functions calculatedwith equation (34) and (35) are extracted using a RFP al-gorithm as for equation (26)

The sounds and attack transients resulting from the sim-ulations are also presented and discussed over the remain-der of the paper

0 05 1 15 2 25 3Time (s)

Pressure

γ03s

Figure 3 Shape of the dimensionless pressure input used for thesimulations

3 Simulations

In order to validate the control and to observe its possibleeffects on the playing frequency timbre and playability ofnotes produced by the modeled clarinet a series of sim-ulations is presented The simulations are carried out us-ing the Bogacki-Shampine method (ode3 under Simulink)with a sample time of 144100 seconds and a simulationtime of 3 seconds

In the simulations the controlled resonator described byequation (24) (30) and (32) is excited by a flow u (equa-tion 21) obtained through the non-linear coupling (equa-tion 4c) between it and the reed model (equations 4a and4b) on which a mouth pressure is applied Figure 3 showsthe shape of the dimensionless input pressure used The re-sultant pressure generated within the resonator (cylindricaltube) is then fed back into the reed model and the non-linear coupling thereby ensuring the feedback between thereed and the resonator is accurately modeled

In addition the Luenberger observer measures the acou-stic pressure at the input of the resonator throughout thesimulation and reports to the controller which provides asecond excitation for the resonator

The characteristics of the modeled tube and reed are re-ported in table I[19] Only the first 8 resonances of the tubeare modeled

Four sets of simulations are described The first set pro-vides examples of the control of the frequency and damp-ing of the first resonance in order to establish the effi-ciency of the control The second and third sets providemaps which show the effects of the control of both the fre-quency and the damping of the first resonance on the noteproduced by the virtual instrument Finally the fourth setprovides an example of the control of the frequency anddamping of the second resonance of the modeled tubein order to verify the efficiency of the control on a reso-nance other than the first one The effects of the control onthe sound spectrum (calculated with the measured acous-tic pressure) and the attack transient are studied To em-

1153


0 500 1000 1500 2000-50

0

50

Amplitude(dB)

0 500 1000 1500 2000-3

-2

-1

0

Frequency (Hz)

Phase(rad)

0

1

2

3

Figure 4 Top Transfer functions of the system in cases 0 (un-controlled solid black line) 1 (f1 = 120 Hz solid grey line) 2(ξ1n reduced by a factor of 2 dash black line) and 3 (f1 = 120 Hzand ξ1n reduced by a factor of 2 dash grey line) The transferfunctions are calculated using equation (34) (case 0) and equa-tion (35) (cases 1 to 3) Bottom Phases of the transfer functionsin cases 0 (solid black line) 1 (solid grey line) 2 (dash blackline) and 3 (dash grey line)

phasise their musical meaning frequency differences aredescribed in terms of cents (a semi-tone is 100 cents)

31 Examples of control of the frequency anddamping of the first resonance

In this first set of simulations the efficiency of the con-trol of the first resonance in terms of both frequencyand damping is investigated Four different cases are pre-sentedbull Case 0 No controlbull Case 1 The control is applied with the intention of

changing the frequency of the first resonance from149 Hz to 120 Hz

bull Case 2 The control is applied with the intention of re-ducing the damping of the first resonance ξ1n

by a factorof 2

bull Case 3 The control is applied with the intention ofchanging the frequency of the first resonance from149 Hz to 120 Hz and reducing its damping by a fac-tor of 2

Figure 4 shows the transfer functions of the tube in thefour cases These transfer functions have been determinedusing equation (34) (case 0) and equation (35) (cases 12 and 3) with the modal parameters extracted from calcu-lated input impedance using an RFP algorithm Table IIshows the modal parameters of the first resonance of thetransfer functions for the four cases For cases 1 to 3 thetable also shows the differences between the intended (tar-get) frequencies and damping factors and the frequenciesand damping factors that were actually observed The con-troller is very accurate with no error on the frequency andonly 06 error for the damping In all the cases the am-plitude is also affected with an increase of between 41 dB(case 1) and 101 dB (case 3) When the control changes

Table I Parameters used to characterise the tube and the reed[19]

Lt 057 m R 0007 mfr 1500 Hz qr 1Kr 8 times 106 Nm h0 3 times 10minus4 mW 0015 m

Table II Modal parameters (frequency amplitude damping) ofthe first resonance of the transfer functions in Figure 4 in cases 0(uncontrolled) 1 (f1 = 120 Hz) 2 (ξ1n reduced by a factor of 2)and 3 (f1 = 120 Hz and ξ1n reduced by a factor of 2) and differ-ences between attempted and obtained frequency and dampingin cases 1 to 3 Frequencies and damping factors are extractedusing a RFP algorithm

Case 0 1 2 3

Frequency (Hz) 149 120 149 120Differences (Cents) NA 0 0 0

Damping 00179 00180 00089 0009Differences () NA 06 06 06

Amplitude (dB) 225 266 285 326

0 500 1000 1500 2000

-100

-80

-60

-40

-20

Frequency (Hz)

Amplitude(dB)

0

1

2

3

Figure 5 Enveloped sound spectra of the self-sustained oscillat-ing system in cases 0 (uncontrolled solid black line) 1 (f1 =120 Hz solid grey line) 2 (ξ1n reduced by a factor of 2 dashblack line) and 3 (f1 = 120 Hz and ξ1n reduced by a factor of 2dash grey line) with γ = 04

the frequency (cases 1 and 3) the amplitude of the thirdresonance is also affected with a decrease of 03 dB

Further explorations of the effects of the control haveshown that increasing the frequency of the first resonanceresults in a decrease in its amplitude while decreasing thefrequency of the first resonance results in an increase in itsamplitude

Figure 5 shows the sound spectra of the steady-states ofthe sound produced by the system in the four cases witha dimensionless mouth pressure γ = 04 With the controlapplied the simulation still produces a sound whose pitchis based on the first resonance of the system Envelopesare drawn on the four sound spectra to help illustrate thenumber of apparent harmonics (ie harmonics that are vis-

1154


ible above the background noise level) in each sound Itcan be seen that when the control is applied with the in-tention of changing the frequency of the first resonance(cases 1 and 3) the number of apparent harmonics in thesound spectrum is reduced The reason for this is that theshifting of the first resonance frequency has caused it tobe no longer harmonically related to the frequencies of thehigher resonances As a result the upper harmonics in thesound are weakened Conversely when the control is ap-plied with the intention of reducing the damping of the firstresonance the number of apparent harmonics in the soundis increased It can be seen that the sound spectrum in case2 has more apparent harmonics than the spectrum in case0 and the sound spectrum in case 3 has more apparent har-monics than the spectrum in case 1 The effect of decreas-ing the damping is similar to that of increasing the mouthpressure With the observed changes in the relative ampli-tudes of the harmonics in the sound spectra across the fourcases it is clear that the control has an effec t (in terms ofboth intonation and timbre) on the sound produced by thevirtual instrument

Figure 6 shows the attack transients of the self-sustainedsystem in the four cases with γ = 04 (remember that ineach case the mouth pressure applied to the reed model isas shown in Figure 3) It can be seen from Figure 6 thatthe attack transient is shorter in the cases where the con-trol is applied it is 105 seconds in duration when there isno control 012s shorter (11 decrease) in case 1 021sshorter (19 decrease) in case 2 and 034s shorter (31decrease) in case 3 The amplitude of the signal is alsoaffected with increases of 19 (case 1 and 2) and 32(case 3) but the waveform (not shown on the figure) isnot affected With the observed changes in the attack tran-sient of the sound produced across the four cases it is clearthat the control has an effect on the onset of the note andappears to be affecting the playing characteristics of thevirtual instrument

32 Control of the frequency of the first resonance

This second set of simulations further explores the effectsof the control on the frequency of the first resonance In thesimulations the control is applied such that the frequencyof the first resonance f1 is varied from 20 Hz to 300 Hzin 1 Hz steps For each frequency the value of the dimen-sionless mouth pressure γ is varied from 0 (null pressure inthe mouth of the musician) to 1 (the reed channel is closedin static regime) in 0002 steps In total 140781 separatesimulations are performed

Figure 7 shows the entire search space covered by thesimulations The shaded regions correspond to conditions(first resonance frequencies and mouth pressure) whichprovide self-sustained oscillation That is conditions un-der which a note is produced The depth of the shading(from light gray to black) indicates the playing frequen-cies of the note produced (In this paper the fundamen-tal frequency of any given note produced by the virtualinstrument is considered to be the playing frequency of

04 06 08 1-04

-02

0

02

04

0

04 06 08 1-04

-02

0

02

04

1

04 06 08 1-04

-02

0

02

04

Time (s)

2

04 06 08 1-04

-02

0

02

04

Time (s)

3

Figure 6 Attack transients of the self-sustained oscillating sys-tem with γ = 04 Top left Case 0 (uncontrolled) Top right Case 1 (f1 = 120 Hz) Bottom left Case 2 (ξ1n reduced by afactor of 2) Bottom right Case 3 (f1 = 120 Hz and ξ1n reducedby a factor of 2)

γ

f1(H

z)

p

0 02 04 06 08 1

50

100

150

200

250

300

50

100

150

200

250

300

350

400

450

Figure 7 Map of the playing frequencies fp (indicated by thedepth of shading) of the notes played by the simulations plottedwith respect to the frequency of the first resonance f1 (ordinate)and to the pressure in the mouth of the musician γ (abscissa)White regions correspond to conditions under which no note isproduced

that note) The white regions correspond to conditions un-der which no note is produced That is conditions whichdo not support self-sustained oscillation It can be seenthat provided the dimensionless mouth pressure is greaterthan approximately 04 stable solutions exist for all thefirst resonance frequencies tested in the simulations Thisobservation regarding the oscillation threshold (minimummouth pressure which provides self-sustained oscillation)corroborates section 21

Examination of Figure 7 reveals two main shaded re-gions a black region in the bottom right corner of the mapand a largely gray region in the top right corner of the map

The black region corresponds to conditions that resultin the playing frequency (and therefore the pitch) of thenote produced by the virtual clarinet being based on thefirst resonance of the system Broadly speaking these con-ditions occur when the dimensionless mouth pressure falls

1155


0 500 1000 1500 2000-150

-100

-50

0

Frequency (Hz)

Amplitude(dB)

Uncont rolledf 1 = 30Hzf 1 = 194Hz

Figure 8 Enveloped sound spectra of notes produced by the un-controlled (solid black line) self-sustained oscillating system andby the system with 2 cases of control applied f1 = 30 Hz (dashblack line) and f1 = 194 Hz (solid gray line) and with γ = 05

in the range 034 lt γ lt 1 and when the first resonance fre-quency falls in the range 20 Hz lt f1 lt 196 Hz Howeverclose inspection of the left hand side of the black regionreveals that the oscillation threshold (ie the mouth pres-sure at which the first non-white pixel is reached as onemoves horizontally across the figure) actually increasesfrom 034 to 04 as the frequency of the first resonanceis increased Within the black region the playing frequen-cies range from 10 Hz to 1943 Hz

The gray region corresponds to conditions under whichthe playing frequency (and thus the pitch) of the note pro-duced by the virtual clarinet becomes based on the sec-ond resonance of the system (The effect is similar tothat which results from opening a register key in a realwind instrument where the amplitude of the first reso-nance is reduced and its frequency is altered so that thereis no longer a harmonic relationship with the higher reso-nance frequencies [28]) In general these conditions oc-cur when the dimensionless mouth pressure falls in therange 0362 lt γ lt 1 and when the first resonance fre-quency falls in the range 197 Hz lt f1 lt 300 Hz Howeverclose inspection of the left hand side of the gray region re-veals that the oscillation threshold actually decreases from04 to 0362 as the frequency of the first resonance is in-creased Without any control applied the first two reso-nance frequencies of the cylindrical tube are 149 Hz and448 Hz respectively The control is applied with the inten-tion of shifting the first resonance frequency the secondresonance frequency of the resonator should remain un-changed at 448 Hz However within the gray region wherethe note produced is based on the second resonance ofthe system the playing frequencies do not match this sec-ond resonance frequency exactly instead they range from4349 Hz to 4446 Hz

Within the gray region of Figure 7 there is a smalldarker region corresponding to specific conditions wherethe playing frequency (pitch) of the note produced by thevirtual clarinet once again becomes based on the first res-onance frequency of the instrument These conditions oc-

cur when the dimensionless mouth pressure is in the range0474 lt γ lt 0516 and when the first resonance frequencyis in the range 208 Hz lt f1 lt 224 Hz The reason that theplaying frequency becomes based on the first resonancefrequency under these conditions can be understood if itis recalled that the resonator of the virtual instrument is aclosed-open cylindrical tube Without control applied thefrequencies of the first two resonances of the tube have aharmonic relationship with f2 = 3 times f1 When the con-trol is applied the frequency of the first resonance is al-tered so that in general it no longer has a harmonic rela-tionship with the higher resonance frequencies Howeverunder the conditions corresponding to the small darker re-gion the first resonance frequency is shifted upwards suchthat the frequencies of the first two resonances of the tubeare once more approximately harmonically related withnow f2 = 2 times f1

Figure 8 shows the sound spectra of three notes ob-tained from the simulations with a dimensionless mouthpressure of γ = 05 with no control applied and with con-trol applied to give first resonance frequencies of 30 Hzand 194 Hz

When f1 = 30 Hz the playing frequency of the note isbased on the first resonance of the system The low firstresonance frequency results in closely spaced harmonicsin the spectrum However these harmonics tend to be lowin amplitude because the majority are not supported by thehigher resonances of the tube (which are not harmonicallyrelated in frequency to the first tube resonance) It is onlywhen the frequencies of the harmonics happen to coincidewith those of the higher resonances of the tube (at around448 Hz for example which is the second resonance fre-quency) that their amplitudes become greater It should benoted that in the simulations the dynamics of the loud-speaker are not modeled This makes it possible to alterthe first resonance frequency by such a large amount andstill produce a sound In reality the loudspeaker dynamicswill prevent the control from achieving such a large degreeof variation of the resonance frequencies

When f1 = 194 Hz with a mouth pressure of γ = 05the playing frequency is based on the first resonance ofthe system and it is at the edge of the register changeThe high first resonance frequency results in an increase inthe frequency separation of the harmonics in the spectrumIt can be seen that at lower frequencies (under 1650 Hz)these harmonics have a lower amplitude compared to theuncontrolled system because they are not supported bythe higher resonances of the tube (which are not harmon-ically related in frequency to the first tube resonance)At higher frequencies especially for the harmonics withfrequency 1940 Hz and 2328 Hz this amplitude becomeshigher compared to the system with no control suggestingthat these harmonics may coincide in frequency with (andhence be supported by) higher resonances of the tube

Figure 9 shows the playing frequencies fp observed inthe black region of Figure 7 (20 Hz lt f1 lt 196 Hz) plot-ted with respect to the mouth pressure γ Each line corre-sponds to a single value of f1 (ie a single target first res-

1156


onance frequency) and the lines go up in 2 Hz steps Fig-ure 9 therefore shows the behaviour of the system when themouth pressure is increased There are three zones wherethe lines converge highlighted by arrows on the right sideof the figure

The region indicated by the first arrow corresponds to afrequency range where the controlled f1 is approximatelyharmonically related to the other resonances of the tube(that is f2 = 3 times f1 f3 = 5 times f1 and so on) There-fore over this range the reed will be strongly supported(by all the resonances) in opening and closing at a rate ap-proximately equal to f23 that is at a frequency close tothe value of f1 when no control is applied Therefore theplaying frequency fp in this region is approximately equalto f23

Similarly the region indicated by the second arrow cor-responds to a frequency range where the controlled f1

is approximately harmonically related to the second res-onance of the tube (f2 = 4 times f1) Over this range the reedwill be strongly supported (by both the first and secondresonances) in opening and closing at a rate approximatelyequal to f24 Therefore the playing frequency fp in thisregion is approximately f24

And the region indicated by the third arrow correspondsto a frequency range where the controlled f1 is approxi-mately harmonically related to the third resonance of thetube (f3 = 4times f1) For this frequency range the reed willbe strongly supported by the first and third resonances inopening at a rate approximately equal to f34 Thereforethe playing frequency fp in this region is approximatelyf34

33 Control of the Damping of the First Resonance

This third set of simulations further explores the effectsof the control on the damping of the first resonance Inthe simulations the control is applied such that the naturaldamping of the first resonance ξ1n

isbull Increased by factors from 0 to 115 (times the natural

damping (ξ1n)) in 005times steps

bull Increased by factors from 12 to 54 in 02times stepsbull Increased by factors from 55 to 105 in 05times stepsbull Increased by factors from 11 to 20 in 1times stepsFor each damping value the value of the dimensionlessmouth pressure γ is varied from 0 (null pressure in themouth of the musician) to 1 (the reed channel is closedin static regime) in 0002 steps In total 33567 separatesimulations are performed

Figure 10 shows the entire search space covered by thesimulations As before the shaded regions correspond toconditions (first resonance frequencies and mouth pres-sure) which provide self-sustained oscillation That isconditions under which a note is produced The depth ofthe shading (from light gray to black) indicates the playingfrequencies of the note produced The white regions cor-respond to conditions under which no note is producedThat is conditions which do not support self-sustained os-cillation It can be seen that provided the dimensionlessmouth pressure is greater than approximately 04 stable

04 05 06 07 08 09 10

50

100

150

200

1

2

3

γ

fp(H

z)

Figure 9 Playing frequencies fp observed in the black regionof Figure 7 (20 Hz lt f1 lt 196 Hz) plotted with respect to themouth pressure γ Each line stands for a single value of f1 (ie asingle target first resonance frequency) The first arrow indicatesa region such that f1 asymp f23 the second arrow indicates a regionaround f1 asymp f24 and the third arrow indicates a region aroundf1 asymp f34

0 02 04 06 08 10

5

10

15

20

γ

Target

ntimes

ξ 1n

p

150

200

250

300

350

400

450

Figure 10 Map of the playing frequencies (indicated by the depthof shading) of the notes played by the simulations plotted withrespect to the target modification of the damping of the first res-onance (ordinate ξ1 = value times ξ1n ) and to the pressure in themouth of the musician γ (abscissa) White regions correspond toconditions under which no note is produced

solutions exist for all the first resonance damping factorstested in the simulations This observation regarding theoscillation threshold also corroborates section 21

Examination of Figure 10 reveals two main shaded re-gions a black region in the bottom right corner of the mapand a large gray region in the top right corner of the map

The black region corresponds to conditions that result inthe playing frequency (and therefore the pitch) of the noteproduced by the virtual clarinet being based on the firstresonance of the system Broadly speaking these condi-tions occur when the dimensionless mouth pressure fallsin the range 034 lt γ lt 1 and when the first resonancedamping factor falls in the range 0 times ξ1n

lt ξ1 lt 42 times ξ1n

However close inspection of the left hand side of the blackregion reveals that the oscillation threshold (ie the mouthpressure at which the first non-white pixel is reached

1157


as one moves horizontally across the figure) actually in-creases from 034 to 041 as the damping of the first res-onance is increased Within the black region the playingfrequencies range from 1467 Hz to 1479 Hz which is alittle under the frequency of the first resonance 149 Hz

The gray region corresponds to conditions under whichthe playing frequency (and thus the pitch) of the note pro-duced by the virtual clarinet becomes based on the secondresonance of the system In general these conditions occurwhen the dimensionless mouth pressure falls in the range039 lt γ lt 1 and when the first resonance damping factorfalls in the range 2 times ξ1n

lt ξ1 lt 20 times ξ1n However close

inspection of the left hand side of the gray region revealsthat the oscillation threshold actually decreases from 041to 039 as the damping of the first resonance is increasedWithin the gray region the playing frequencies range from4403 Hz to 444 Hz which is close to the frequency of thesecond resonance 448 Hz

The black region extends up to a damping value of42 times ξ1n

at a mouth pressure γ = 085 This is the largestdamping value possible before a change in register occursIndeed at this point increasing or decreasing the mouthpressure results in a register change from the first regis-ter to the second register (ie the playing frequency (andthus the pitch) of the note produced by the virtual clarinetbecomes based on the second resonance)

Figure 11 shows the sound spectra of three notes ob-tained from the simulations with a dimensionless mouthpressure of γ = 041 with no control applied and withcontrol applied to give first resonance damping factors re-duced by a factor of 100 and increased by a factor of 10

When ξ1nis reduced by a factor of 100 the playing fre-

quency of the note is based on the first resonance of thesystem Reducing the damping in this way results in an in-crease of the amplitude of all the harmonics (from +25 dBto +125 dB) with one more apparent harmonic (ie har-monic that is visible above the background noise level)This effect is similar to increasing the mouth pressure

When ξ1nis increased by a factor of 10 the playing fre-

quency of the note is based on the second resonance of thesystem The last apparent harmonic in the sound spectrumis the same as for the sound produced with no control ap-plied but as the playing frequency is now based on thesecond resonance of the system the number of harmonicsin the spectrum is decreased by a factor of 3

34 Example of control of the frequency and damp-ing of the second resonance

Sections 31 32 and 33 explored the effects of the controlof the frequency and damping of the first resonance of themodeled tube It is also possible to apply the active controlto the other tube resonances

In this fourth set of simulations an example of the con-trol of the second resonance in terms of both frequencyand damping is presented The control is applied with theintention of modifying the frequency of the second reso-nance from f2 = 448 Hz to f2 = 470 Hz and reducing itsdamping value by a factor of 2

0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Frequency (Hz)

Amplitude(dB)

Uncont rolledξ1 10

ξ1times 100

Figure 11 Enveloped sound spectra of notes produced by theuncontrolled (solid black line) self-sustained oscillating systemand by the system with 2 cases of control applied ξ1n reduced bya factor of 100 (dash black line) and ξ1n increased by a factor of10 (solid gray line) and with γ = 041

0 500 1000 1500 2000-50

0

50

Amplitude(dB)

0 500 1000 1500 2000

-2

0

2

Frequency (Hz)

Phase(rad)

Uncont rolledf 2 = 470 Hz ξ2 2

Figure 12 Top Transfer functions of the system without control(solid black line) and with control applied with the aim of mod-ifying the frequency of the second resonance from f2 = 448 Hzto f2 = 470 Hz and reducing its damping by a factor of 2 (dashblack line) Bottom Phases of the transfer functions with nocontrol (solid black line) and with control applied (dash blackline)

Figure 12 shows the transfer functions of the uncon-trolled tube and the tube with control applied These trans-fer functions have been determined using equation (34)(uncontrolled tube) and equation (35) (control applied)with the modal parameters extracted from the calculatedinput impedance using an RFP algorithm Table III showsthe modal parameters of the first 8 resonances of the trans-fer functions without control and with control applied Thetable also shows the differences between the intended (tar-get) frequencies and damping factors and the frequenciesand damping factors that were actually observed Modi-fications of the amplitude due to the control are also ob-served As only the second resonance is controlled forall the other seven resonances the intended values are thesame as the values when uncontrolled The controller isvery accurate for the frequency and damping with no er-ror on the frequency of the controlled resonance and no

1158


Table III Modal parameters (frequency amplitude damping) of the resonances of the transfer functions (Figure 12) of the uncontrolled(U ) and controlled (C) tubes and differences between intended (T ) and obtained values For resonances 1 and 3 to 8 intended valuesand values when uncontrolled are the same Frequencies and damping are extracted using a RFP algorithm

Resonance 1 2 3 4 5 6 7 8

Frequency (Hz) U 149 448 746 1046 1344 1643 1942 2242C 149 470 746 1046 1344 1643 1942 2242

(T 470)Difference (cents) 0 0 0 0 0 0 0 0

Damping U 00178 00105 00082 00071 00065 0006 00057 00054C 00178 00055 00083 00072 00067 00059 00056 00053

(T 00053)Difference () 0 3 1 1 1 1 1 1

Amplitude (dB) U 225 81 13 -33 -68 -96 -12 -14C 218 131 19 -3 -66 -95 -119 -139

Difference (dB) -07 +5 +06 +03 +02 +01 +01 +01

0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Amplitude(dB)

Frequency (Hz)


Figure 13 Enveloped sound spectra of the notes produced bythe self-sustained oscillating system without control (solid blackline) and with control applied with the aim of modifying the fre-quency of the second resonance from f2 = 448 Hz to f2 =470 Hz and reducing its damping by a factor of 2 (dash blackline) with γ = 04

more than 3 error for the damping The amplitude of thecontrolled resonance is however affected with an increaseof 5 dB The amplitude of all the other resonances is alsoaffected with a decrease of 07 dB of the first resonanceand increases of between 01 dB (resonances 6 to 8) and06 dB (resonance 3)

Figure 13 shows the sound spectra of the steady-statesof the sound produced by the system when uncontrolledand with the control applied with a dimensionless mouthpressure γ = 04 With the control applied the simulationstill produces a sound whose pitch is based on the first res-onance of the system However it can be seen that with thecontrol the number of apparent harmonics (ie harmonicsthat are visible above the background noise level) in thesound spectrum is reduced The reason for this is that theshifting of the second resonance frequency has caused itto be no longer harmonically related to the frequency ofthe first resonance As a result the strength of the secondharmonic in the sound is weakened as well as the strengthof the upper harmonics With the observed changes in the

05 06 07 08 09 1 11 12

-02

0

02

04Uncontrolled

05 06 07 08 09 1 11 12

-02

0

02

04

Controlled

Time (s)

Figure 14 Attack transients of the self-sustained oscillating sys-tem with γ = 04 Top uncontrolled Bottom control appliedwith the aim of modifying the frequency of the second resonancefrom f2 = 448 Hz to f2 = 470 Hz and reducing its damping by afactor of 2

relative amplitudes of the harmonics in the sound spectrait is clear that the control of the second resonance also hasan effect (in terms of timbre) on the sound produced bythe virtual instrument

Figure 14 shows the attack transients of the self-sus-tained system when uncontrolled and with the control ap-plied with γ = 04 (the mouth pressure applied to the reedmodel is as shown in Figure 3) Remember that even withthe control applied the model still produces a sound whosepitch is based on the first resonance of the system It can beseen from Figure 14 that the attack transient is longer whenthe control is applied it is 105 seconds in duration whenthere is no control 012 s longer (11 increase) when thecontrol is applied The amplitude of the signal is also af-fected with an increase of 4 and the waveform is modi-fied from a square-like signal to a more sinusoidal-shapedsignal With the observed changes in the attack transient ofthe sound produced it is clear that the control of the sec-ond resonance also has an effect on the onset of the note

1159


and appears to be affecting the playing characteristics ofthe virtual instrument

4 Conclusion and perspectives

In this paper simulations of modal active control usinglinear quadratic estimator and pole placement algorithmsapplied to a model of a self-sustained simplified clarinetin order to modify its emitted sound and playability havebeen presented To the authorsrsquo knowledge it is the firsttime that such a control has been applied to a self-sustainedsystem Effects of this control have been studied with foursets of simulations Modifications were observed ofbull the pitch of the sound produced by the instrumentbull the strength of the harmonics of the soundbull the oscillation thresholdThe controller has been shown to be very accurate withrespect to the control of the damping and frequency ofthe first resonance Similar levels of accuracy have beendemonstrated when controlling a higher resonance (thesecond resonance) although in this case the control ledto unintended modifications of the amplitudes of the un-controlled resonances Meanwhile the control of the firstand second resonances has been shown to lead to modifi-cations in the sound spectra and attack transients of notesproduced by the modeled simplified clarinet The harmon-ics in the sound spectra were observed to be weakened asa result of either frequency changes or increases in damp-ing of the resonances and strengthened as a result of de-creases in damping Moreover large increases in the fre-quency andor damping of the first resonance resulted in areduction in its amplitude leading to a change in the reg-ister of the note produced This effect is the same as thatresulting from opening a register hole in a real wind in-strument

Although the simulations have demonstrated the greatpotential of modal active control for modifying the fre-quencies and damping values of the resonances of a windinstrument it should be noted that no consideration has sofar been given to the dynamics of the loudspeaker used toapply the control These loudspeaker dynamics will placerestrictions on the extent to which the frequencies anddamping values of a given resonance can be modified Itis planned to take into account the loudspeaker propertiesin the simulations in order to make them more realisticA study of the limits and possibilities of the control of allthe resonances (ie the limits of the pole placement withrespect to the mouth pressure) will then be carried out Inaddition the optimal positions of the sensor (microphone)and actuator (loudspeaker) will be investigated It is thenplanned to apply the control to a real simplified instrument

Acknowledgement

This work was carried out during the PhD of ThibautMeurisse funded by Agence Nationale de la Recherche(ANR IMAREV project) and Universiteacute Pierre et MarieCurie (UPMC Paris 6) We thank gratefully Sami Karkar

and Fabrice Silva for their help with their clarinet modelWe also thank Simon Benacchio for his support aboutthe LQE algorithm We thank the Newton InternationalFellowship scheme for funding the collaboration betweenFrance and UK

References

[1] C R Fuller S J Elliott P A Nelson Active control ofvibration Academic Press Harcourt Brace and CompanyLondon 1996 reprinted 1997 Chap3 pp59-90

[2] A Preumont Vibration control of active structures an in-troduction Third edition Springer Verlag Berlin Heidel-berg 2011 1-432 p197

[3] P A Nelson S J Elliot Active control of sound Aca-demic Press Harcourt Brace and Company London 1992reprinted 1995

[4] C W Rowley D R Williams T Colonius R M MurrayD G Macmynowski Linear models for control of cavityflow oscillation J Fluid Mech 547 (2006) 317ndash330

[5] R T Schumacher Self-sustained oscillations of the clar-inet An integral equation approach Acustica 40 (1978)298ndash309

[6] M E McIntyre R T Schumacher J Woodhouse On theoscillations of musical instruments J Acoust Soc Am 74(1983) 1325ndash1345

[7] F Silva J Kergomard C Vergez J Gilbert Interactionof reed and acoustic resonator in clarinetlike systems JAcoust Soc Am 124 (2008) 3284ndash3295

[8] S Karkar C Vergez B Cochelin Oscillation threshold ofa clarinet model A numerical continuation approach JAcoust Soc Am 131 (2012) 698ndash707

[9] C Besnainou Modal stimulation a sound synthesis newapproach Proc of the Int Symp on Music Acoustics 1995Dourdan France 1995 434ndash438

[10] H Boutin Meacutethodes de controcircle actif drsquoinstruments demusique Cas de la lame de xylophone et du violon (Musi-cal instruments active control methods Cases of the xylo-phone bar and of the violin) Phd Thesis Universiteacute Pierreet Marie Curie - Paris VI 2011

[11] E Berdahl J Smith Feedback control of acoustic musicalinstruments Collocated control using physical analogs JAcoust Soc Am 131 (2012) 963ndash973

[12] J Gueacuterard Modeacutelisation numeacuterique et simulation expeacuteri-mentale de systegravemes acoustiques Application aux instru-ments de musique (digital modeling and experimental sim-ulation of acoustical systems Application to musical in-struments) Phd Thesis Universiteacute Pierre et Marie Curie -Paris VI 2002

[13] T Meurisse A Mamou-Mani R Causseacute D Sharp Ac-tive control applied to simplified wind musical instrumentProc Int Cong on Acoustics 2013 19 2pEAb7 MontreacutealCanada 2ndash7 June 2013

[14] J Hong J C Akers R Venugopal M-N Lee A GSparks P D Washabaugh D S Bernstein Modelingidentification and feedback control of noise in an acousticduct IEEE Transactions on Control Systems Technology 4(1996) 283ndash291

[15] B Chomette D Remond S Chesneacute L Gaudiller Semi-adaptive modal control of on-board electronic boards us-ing identication method Smart Material and Structures 17(2008) 1ndash8

1160


[16] S Hanagud S Griffin Active structural control for a smartguitar Proc of the 4th European Conference On SmartStructures and Materials Harrogate United Kingdom July6-8 1998 1ndash7

[17] S Benacchio A Mamou-Mani B Chomette F OllivierSimulated effects of combined control applied to an exper-imentally identified soundboard Proc of the StockholmMusic Acoustics Conference 2013 (SMAC 2013) Stock-holm Sweden 2013 601ndash606

[18] F Silva Emergence des auto-oscillations dans un instru-ment de musique agrave anche simple (Appearance of self-sustained oscillation in a single reed musical instrument)Phd Thesis Universiteacute de Provence - Aix-Marseille I 2009

[19] S Karkar Meacutethodes numeacuteriques pour les systegravemes dy-namiques non lineacuteaires Application aux instruments demusique auto-oscillants (Digital methods for non linear dy-namical systems Application to self-sustained oscillatedmusical instruments) Phd Thesis Universiteacute de Provence- Aix-Marseille I 2012

[20] A Hirschberg Aero-acoustics of wind instruments ndash InMechanics of Musical Instruments volume 335 of CISMCourses and Lectures G Weinreich A Hirschberg J Ker-gomard (eds) Springer-Verlag Wien - New York 1995291ndash369

[21] S Karkar C Vergez B Cochelin Toward the systematicinvestigation of periodic solutions in single reed woodwindinstruments Proc of the 20th Int Symp on Music Acous-ticas Associated Meeting of the International Congress on

Acoustics International Commission fos Acoustics Syd-ney Australia 2010 1ndash7

[22] T Meurisse A Mamou-Mani R Causseacute D Sharp Simu-lations of modal active control applied to the self-sustainedoscillations of the clarinet Proc of the Stockholm Mu-sic Acoustics Conference 2013 (SMAC 2013) StockholmSweden 30 Julyndash3 August 2013 425ndash431

[23] M Rossi Traiteacute drsquoeacutelectriciteacute Volume XXI Eacutelectroacou-stique (Electricity tract Volume 21 Electroacoustic)Presses polytechniques romandes premiegravere eacutedition Lau-sanne 1986 Chap5 pp185-232

[24] T Richardson D Formenti Parameter estimation fromfrequency response measurements using rational fractionpolynomials First International Modal Analysis Confer-ence (1eIMAC) Orlando USA 1982 167ndash180

[25] D Luenberger An introduction to observers IEEE Trans-actions on Automatic Control 16 (1971) 596ndash602

[26] J Kautsky N K Nichols Robust pole assignment in linearstate feedback Int J Control 41 (1985) 1129ndash1155

[27] B Chomette S Chesneacute D Reacutemond L Gaudiller Damagereduction of on-board structures using piezoelectric com-ponents and active modal control Application to a printedcircuit board Mechanical Systems and Signal Processing24 (2010) 352ndash364

[28] A Mamou-Mani D Sharp T Meurisse W Ring Investi-gating the consistency of woodwind instrument manufac-turing by comparing five nominally identical oboes JAcoust Soc Am 131 (2012) 728ndash736

1161


quency and damping values [14 15] In principle there-fore modal active control should enable the frequenciesand the damping factors of the resonances of a wind instru-ment to be altered in a specified manner However the be-havior of a self-sustained system with modal active controlapplied is unknown and needs to be investigated Therehave been relatively few applications of modal active con-trol to musical instruments [16 17] and no application towind instruments to the authorsrsquo knowledge

This paper reports a first step towards applying modalactive control to real musical wind instruments In this pa-per simulations of modal active control applied to a modelof a self-sustained simplified clarinet (ie a cylindricaltube coupled to a reed) are presented The control is im-plemented with the intent of adjusting the resonances ofthe virtual instrument thereby altering the timbre of thesound it produces as well as its playing properties Suchadjustments provide similar effects to those that would re-sult from modifying the instrumentrsquos bore profile

In section 2 a model of a simplified clarinet with an in-corporated active control system is presented The clarinetpart of the model is made up of a reed model non-linearlycoupled to a state-space model of the resonator Mean-while the active control part of the model adds the com-ponents (microphone and loudspeaker) that are requiredto enable active control to be applied to the virtual instru-ment In section 3 simulations are carried out to demon-strate the effects of the control Four sets of simulationsare presented The first set provides examples of the con-trol of the frequency and damping of the first resonance ofthe tube The second and third sets provide maps showingcontrol possibilities in terms of both frequency and damp-ing for the first resonance of this self-sustained oscillatingsystem The fourth set provides an example of the controlof the frequency and damping of the second resonance ofthe tube Finally section 4 provides some conclusions andoutlines the proposed next steps for the research

2 Modeling a self-sustained oscillatingwind instrument with incorporatedmodal active control system

Models of self-sustained wind instruments such as theclarinet have been reported for over 30 years [6 18 7 85] In such models the instrument is described in termsof both linear (reed resonator) and non-linear (coupling)elements (see Figure 1)

In this section a complete model of a simplified clarinetwith an incorporated modal active control system is devel-oped (see Figure 2) First a model of the clarinet reed andthe non-linear coupling is described Then a state-spacerepresentation of the resonator is presented This represen-tation takes into account possible excitation both from thereed and from the speaker of the active control system aswell as monitoring of the pressure by the microphone ofthe active control system Finally the designs of the ob-server and controller elements of the active control systemare described

Figure 1 Model of a self-sustained wind instrument [6 19]

Figure 2 Model of a self-sustained wind instrument with controlsystem w y and us are defined in equation (24) y and x are theobserver estimations of y and x

21 Reed and non-linear coupling

The model of reed and non-linear coupling used in thispaper is the one described by Karkar et al [8]

In a clarinet a single reed controls the flow of air fromthe playerrsquos mouth into the instrument Let h(t) be the po-sition of the reed at time t Then

1

ω2r

h(t) +

qr

ωr

h(t) + h(t) minus h0 = minus 1

KrPm minus P (t) (1)

where ωr is the resonance frequency of the reed qr itsdamping h0 its equilibrium position and Kr its stiffnessThe and symbols represent respectively the first andsecond order time derivatives Pm is the pressure in theplayerrsquos mouth and P (t) the pressure in the mouthpiece

Using the same assumptions as described in [19] andfollowing Hirschberg [20] namely thatbull the flow is quasi-staticbull the kinetic energy of the flow is entirely dissipated by

turbulencebull the pressure is uniformly distributed in the mouthpiece

and equal to that of the input of the resonatorbull the velocity of the air in the musicianrsquos mouth is negli-

gible compared to that of the air in the flowthe flow can be determined from

U (t) = H h(t) sign Pm minus P (t)

middot W h(t)2|Pm minus P (t)|

ρ0 (2)

1150




xh(t) = h(t)h0

yh(t) =h(t)v0

p(t) = P (t)PM

u(t) = U (t)ZcPM

(3)


1ωr

xh(t) = yh(t) (4a)

1ωr










1c2

p(z t) = p (z t) + ρ0

vs(t)δ(z minus zs)

+ρ0

SZc




p(z t) = V (z) q(t) (6)


1c2

p(z t) = p (z t) (7)


V (z) +λ

c2V (z) = 0 (8)

Hence



V (Lt) = 0 (10)

V (0) = 0 (11)




Vi(z)1c2

Vj(z) =1c2

Lt

0Vi(z)Vj(z) dz

= δij (13)


β = c2Lt

(14)

1151


and thus

Vi(z) = c2Lt

cos(kiz) (15)


p(z t) =j

Vj(z) qj(t) (16)

so that

1c2

j


ρ0


SZc


where λj = k2j c



where

us = ρ0

vs (19)


bi = Vi(zs) (20)


w(t) =ρ0

SZc

u (21)


gi = Vi(zd) (22)


x(t) =qiqi

i = 1r (23)



y(t) = Cx(t) (24)


y(t) = p(zm t) =i

Vi(zm)qi (25)

and

A =0rr Irr


(26)



B =

0r1

b1br

(27)




G =

0r1

g1gr

(29)


23 Modal Control




1152









(31)




y(t) = Cmx(t)(32)



x(t) = Ax(t) +Gw(t)

y(t) = Cx(t)(33)


HW C (s) =y



HCL(s) =y

w(35)


minus1G



0 05 1 15 2 25 3Time (s)

Pressure

γ03s


3 Simulations






1153


0 500 1000 1500 2000-50

0

50

Amplitude(dB)

0 500 1000 1500 2000-3

-2

-1

0

Frequency (Hz)

Phase(rad)

0

1

2

3







by a factorof 2






Case 0 1 2 3



Amplitude (dB) 225 266 285 326

0 500 1000 1500 2000

-100

-80

-60

-40

-20

Frequency (Hz)

Amplitude(dB)

0

1

2

3





1154







04 06 08 1-04

-02

0

02

04

0

04 06 08 1-04

-02

0

02

04

1

04 06 08 1-04

-02

0

02

04

Time (s)

2

04 06 08 1-04

-02

0

02

04

Time (s)

3


γ

f1(H

z)

p

0 02 04 06 08 1

50

100

150

200

250

300

50

100

150

200

250

300

350

400

450





1155


0 500 1000 1500 2000-150

-100

-50

0

Frequency (Hz)

Amplitude(dB)











1156













04 05 06 07 08 09 10

50

100

150

200

1

2

3

γ

fp(H

z)


0 02 04 06 08 10

5

10

15

20

γ

Target

ntimes

ξ 1n

p

150

200

250

300

350

400

450







1157
















0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Frequency (Hz)

Amplitude(dB)

Uncont rolledξ1 10

ξ1times 100


0 500 1000 1500 2000-50

0

50

Amplitude(dB)

0 500 1000 1500 2000

-2

0

2

Frequency (Hz)

Phase(rad)




1158



Resonance 1 2 3 4 5 6 7 8

Frequency (Hz) U 149 448 746 1046 1344 1643 1942 2242C 149 470 746 1046 1344 1643 1942 2242


Damping U 00178 00105 00082 00071 00065 0006 00057 00054C 00178 00055 00083 00072 00067 00059 00056 00053

(T 00053)Difference () 0 3 1 1 1 1 1 1

Amplitude (dB) U 225 81 13 -33 -68 -96 -12 -14C 218 131 19 -3 -66 -95 -119 -139

Difference (dB) -07 +5 +06 +03 +02 +01 +01 +01

0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Amplitude(dB)

Frequency (Hz)





05 06 07 08 09 1 11 12

-02

0

02

04Uncontrolled

05 06 07 08 09 1 11 12

-02

0

02

04

Controlled

Time (s)




1159






Acknowledgement



References
















1160
















1161




xh(t) = h(t)h0

yh(t) =h(t)v0

p(t) = P (t)PM

u(t) = U (t)ZcPM

(3)


1ωr

xh(t) = yh(t) (4a)

1ωr










1c2

p(z t) = p (z t) + ρ0

vs(t)δ(z minus zs)

+ρ0

SZc




p(z t) = V (z) q(t) (6)


1c2

p(z t) = p (z t) (7)


V (z) +λ

c2V (z) = 0 (8)

Hence



V (Lt) = 0 (10)

V (0) = 0 (11)




Vi(z)1c2

Vj(z) =1c2

Lt

0Vi(z)Vj(z) dz

= δij (13)


β = c2Lt

(14)

1151


and thus

Vi(z) = c2Lt

cos(kiz) (15)


p(z t) =j

Vj(z) qj(t) (16)

so that

1c2

j


ρ0


SZc


where λj = k2j c



where

us = ρ0

vs (19)


bi = Vi(zs) (20)


w(t) =ρ0

SZc

u (21)


gi = Vi(zd) (22)


x(t) =qiqi

i = 1r (23)



y(t) = Cx(t) (24)


y(t) = p(zm t) =i

Vi(zm)qi (25)

and

A =0rr Irr


(26)



B =

0r1

b1br

(27)




G =

0r1

g1gr

(29)


23 Modal Control




1152









(31)




y(t) = Cmx(t)(32)



x(t) = Ax(t) +Gw(t)

y(t) = Cx(t)(33)


HW C (s) =y



HCL(s) =y

w(35)


minus1G



0 05 1 15 2 25 3Time (s)

Pressure

γ03s


3 Simulations






1153


0 500 1000 1500 2000-50

0

50

Amplitude(dB)

0 500 1000 1500 2000-3

-2

-1

0

Frequency (Hz)

Phase(rad)

0

1

2

3







by a factorof 2






Case 0 1 2 3



Amplitude (dB) 225 266 285 326

0 500 1000 1500 2000

-100

-80

-60

-40

-20

Frequency (Hz)

Amplitude(dB)

0

1

2

3





1154







04 06 08 1-04

-02

0

02

04

0

04 06 08 1-04

-02

0

02

04

1

04 06 08 1-04

-02

0

02

04

Time (s)

2

04 06 08 1-04

-02

0

02

04

Time (s)

3


γ

f1(H

z)

p

0 02 04 06 08 1

50

100

150

200

250

300

50

100

150

200

250

300

350

400

450





1155


0 500 1000 1500 2000-150

-100

-50

0

Frequency (Hz)

Amplitude(dB)











1156













04 05 06 07 08 09 10

50

100

150

200

1

2

3

γ

fp(H

z)


0 02 04 06 08 10

5

10

15

20

γ

Target

ntimes

ξ 1n

p

150

200

250

300

350

400

450







1157
















0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Frequency (Hz)

Amplitude(dB)

Uncont rolledξ1 10

ξ1times 100


0 500 1000 1500 2000-50

0

50

Amplitude(dB)

0 500 1000 1500 2000

-2

0

2

Frequency (Hz)

Phase(rad)




1158



Resonance 1 2 3 4 5 6 7 8

Frequency (Hz) U 149 448 746 1046 1344 1643 1942 2242C 149 470 746 1046 1344 1643 1942 2242


Damping U 00178 00105 00082 00071 00065 0006 00057 00054C 00178 00055 00083 00072 00067 00059 00056 00053

(T 00053)Difference () 0 3 1 1 1 1 1 1

Amplitude (dB) U 225 81 13 -33 -68 -96 -12 -14C 218 131 19 -3 -66 -95 -119 -139

Difference (dB) -07 +5 +06 +03 +02 +01 +01 +01

0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Amplitude(dB)

Frequency (Hz)





05 06 07 08 09 1 11 12

-02

0

02

04Uncontrolled

05 06 07 08 09 1 11 12

-02

0

02

04

Controlled

Time (s)




1159






Acknowledgement



References
















1160
















1161


and thus

Vi(z) = c2Lt

cos(kiz) (15)


p(z t) =j

Vj(z) qj(t) (16)

so that

1c2

j


ρ0


SZc


where λj = k2j c



where

us = ρ0

vs (19)


bi = Vi(zs) (20)


w(t) =ρ0

SZc

u (21)


gi = Vi(zd) (22)


x(t) =qiqi

i = 1r (23)



y(t) = Cx(t) (24)


y(t) = p(zm t) =i

Vi(zm)qi (25)

and

A =0rr Irr


(26)



B =

0r1

b1br

(27)




G =

0r1

g1gr

(29)


23 Modal Control




1152









(31)




y(t) = Cmx(t)(32)



x(t) = Ax(t) +Gw(t)

y(t) = Cx(t)(33)


HW C (s) =y



HCL(s) =y

w(35)


minus1G



0 05 1 15 2 25 3Time (s)

Pressure

γ03s


3 Simulations






1153


0 500 1000 1500 2000-50

0

50

Amplitude(dB)

0 500 1000 1500 2000-3

-2

-1

0

Frequency (Hz)

Phase(rad)

0

1

2

3







by a factorof 2






Case 0 1 2 3



Amplitude (dB) 225 266 285 326

0 500 1000 1500 2000

-100

-80

-60

-40

-20

Frequency (Hz)

Amplitude(dB)

0

1

2

3





1154







04 06 08 1-04

-02

0

02

04

0

04 06 08 1-04

-02

0

02

04

1

04 06 08 1-04

-02

0

02

04

Time (s)

2

04 06 08 1-04

-02

0

02

04

Time (s)

3


γ

f1(H

z)

p

0 02 04 06 08 1

50

100

150

200

250

300

50

100

150

200

250

300

350

400

450





1155


0 500 1000 1500 2000-150

-100

-50

0

Frequency (Hz)

Amplitude(dB)











1156













04 05 06 07 08 09 10

50

100

150

200

1

2

3

γ

fp(H

z)


0 02 04 06 08 10

5

10

15

20

γ

Target

ntimes

ξ 1n

p

150

200

250

300

350

400

450







1157
















0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Frequency (Hz)

Amplitude(dB)

Uncont rolledξ1 10

ξ1times 100


0 500 1000 1500 2000-50

0

50

Amplitude(dB)

0 500 1000 1500 2000

-2

0

2

Frequency (Hz)

Phase(rad)




1158



Resonance 1 2 3 4 5 6 7 8

Frequency (Hz) U 149 448 746 1046 1344 1643 1942 2242C 149 470 746 1046 1344 1643 1942 2242


Damping U 00178 00105 00082 00071 00065 0006 00057 00054C 00178 00055 00083 00072 00067 00059 00056 00053

(T 00053)Difference () 0 3 1 1 1 1 1 1

Amplitude (dB) U 225 81 13 -33 -68 -96 -12 -14C 218 131 19 -3 -66 -95 -119 -139

Difference (dB) -07 +5 +06 +03 +02 +01 +01 +01

0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Amplitude(dB)

Frequency (Hz)





05 06 07 08 09 1 11 12

-02

0

02

04Uncontrolled

05 06 07 08 09 1 11 12

-02

0

02

04

Controlled

Time (s)




1159






Acknowledgement



References
















1160
















1161









(31)




y(t) = Cmx(t)(32)



x(t) = Ax(t) +Gw(t)

y(t) = Cx(t)(33)


HW C (s) =y



HCL(s) =y

w(35)


minus1G



0 05 1 15 2 25 3Time (s)

Pressure

γ03s


3 Simulations






1153


0 500 1000 1500 2000-50

0

50

Amplitude(dB)

0 500 1000 1500 2000-3

-2

-1

0

Frequency (Hz)

Phase(rad)

0

1

2

3







by a factorof 2






Case 0 1 2 3



Amplitude (dB) 225 266 285 326

0 500 1000 1500 2000

-100

-80

-60

-40

-20

Frequency (Hz)

Amplitude(dB)

0

1

2

3





1154







04 06 08 1-04

-02

0

02

04

0

04 06 08 1-04

-02

0

02

04

1

04 06 08 1-04

-02

0

02

04

Time (s)

2

04 06 08 1-04

-02

0

02

04

Time (s)

3


γ

f1(H

z)

p

0 02 04 06 08 1

50

100

150

200

250

300

50

100

150

200

250

300

350

400

450





1155


0 500 1000 1500 2000-150

-100

-50

0

Frequency (Hz)

Amplitude(dB)











1156













04 05 06 07 08 09 10

50

100

150

200

1

2

3

γ

fp(H

z)


0 02 04 06 08 10

5

10

15

20

γ

Target

ntimes

ξ 1n

p

150

200

250

300

350

400

450







1157
















0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Frequency (Hz)

Amplitude(dB)

Uncont rolledξ1 10

ξ1times 100


0 500 1000 1500 2000-50

0

50

Amplitude(dB)

0 500 1000 1500 2000

-2

0

2

Frequency (Hz)

Phase(rad)




1158



Resonance 1 2 3 4 5 6 7 8

Frequency (Hz) U 149 448 746 1046 1344 1643 1942 2242C 149 470 746 1046 1344 1643 1942 2242


Damping U 00178 00105 00082 00071 00065 0006 00057 00054C 00178 00055 00083 00072 00067 00059 00056 00053

(T 00053)Difference () 0 3 1 1 1 1 1 1

Amplitude (dB) U 225 81 13 -33 -68 -96 -12 -14C 218 131 19 -3 -66 -95 -119 -139

Difference (dB) -07 +5 +06 +03 +02 +01 +01 +01

0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Amplitude(dB)

Frequency (Hz)





05 06 07 08 09 1 11 12

-02

0

02

04Uncontrolled

05 06 07 08 09 1 11 12

-02

0

02

04

Controlled

Time (s)




1159






Acknowledgement



References
















1160
















1161


0 500 1000 1500 2000-50

0

50

Amplitude(dB)

0 500 1000 1500 2000-3

-2

-1

0

Frequency (Hz)

Phase(rad)

0

1

2

3







by a factorof 2






Case 0 1 2 3



Amplitude (dB) 225 266 285 326

0 500 1000 1500 2000

-100

-80

-60

-40

-20

Frequency (Hz)

Amplitude(dB)

0

1

2

3





1154







04 06 08 1-04

-02

0

02

04

0

04 06 08 1-04

-02

0

02

04

1

04 06 08 1-04

-02

0

02

04

Time (s)

2

04 06 08 1-04

-02

0

02

04

Time (s)

3


γ

f1(H

z)

p

0 02 04 06 08 1

50

100

150

200

250

300

50

100

150

200

250

300

350

400

450





1155


0 500 1000 1500 2000-150

-100

-50

0

Frequency (Hz)

Amplitude(dB)











1156













04 05 06 07 08 09 10

50

100

150

200

1

2

3

γ

fp(H

z)


0 02 04 06 08 10

5

10

15

20

γ

Target

ntimes

ξ 1n

p

150

200

250

300

350

400

450







1157
















0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Frequency (Hz)

Amplitude(dB)

Uncont rolledξ1 10

ξ1times 100


0 500 1000 1500 2000-50

0

50

Amplitude(dB)

0 500 1000 1500 2000

-2

0

2

Frequency (Hz)

Phase(rad)




1158



Resonance 1 2 3 4 5 6 7 8

Frequency (Hz) U 149 448 746 1046 1344 1643 1942 2242C 149 470 746 1046 1344 1643 1942 2242


Damping U 00178 00105 00082 00071 00065 0006 00057 00054C 00178 00055 00083 00072 00067 00059 00056 00053

(T 00053)Difference () 0 3 1 1 1 1 1 1

Amplitude (dB) U 225 81 13 -33 -68 -96 -12 -14C 218 131 19 -3 -66 -95 -119 -139

Difference (dB) -07 +5 +06 +03 +02 +01 +01 +01

0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Amplitude(dB)

Frequency (Hz)





05 06 07 08 09 1 11 12

-02

0

02

04Uncontrolled

05 06 07 08 09 1 11 12

-02

0

02

04

Controlled

Time (s)




1159






Acknowledgement



References
















1160
















1161







04 06 08 1-04

-02

0

02

04

0

04 06 08 1-04

-02

0

02

04

1

04 06 08 1-04

-02

0

02

04

Time (s)

2

04 06 08 1-04

-02

0

02

04

Time (s)

3


γ

f1(H

z)

p

0 02 04 06 08 1

50

100

150

200

250

300

50

100

150

200

250

300

350

400

450





1155


0 500 1000 1500 2000-150

-100

-50

0

Frequency (Hz)

Amplitude(dB)











1156













04 05 06 07 08 09 10

50

100

150

200

1

2

3

γ

fp(H

z)


0 02 04 06 08 10

5

10

15

20

γ

Target

ntimes

ξ 1n

p

150

200

250

300

350

400

450







1157
















0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Frequency (Hz)

Amplitude(dB)

Uncont rolledξ1 10

ξ1times 100


0 500 1000 1500 2000-50

0

50

Amplitude(dB)

0 500 1000 1500 2000

-2

0

2

Frequency (Hz)

Phase(rad)




1158



Resonance 1 2 3 4 5 6 7 8

Frequency (Hz) U 149 448 746 1046 1344 1643 1942 2242C 149 470 746 1046 1344 1643 1942 2242


Damping U 00178 00105 00082 00071 00065 0006 00057 00054C 00178 00055 00083 00072 00067 00059 00056 00053

(T 00053)Difference () 0 3 1 1 1 1 1 1

Amplitude (dB) U 225 81 13 -33 -68 -96 -12 -14C 218 131 19 -3 -66 -95 -119 -139

Difference (dB) -07 +5 +06 +03 +02 +01 +01 +01

0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Amplitude(dB)

Frequency (Hz)





05 06 07 08 09 1 11 12

-02

0

02

04Uncontrolled

05 06 07 08 09 1 11 12

-02

0

02

04

Controlled

Time (s)




1159






Acknowledgement



References
















1160
















1161


0 500 1000 1500 2000-150

-100

-50

0

Frequency (Hz)

Amplitude(dB)











1156













04 05 06 07 08 09 10

50

100

150

200

1

2

3

γ

fp(H

z)


0 02 04 06 08 10

5

10

15

20

γ

Target

ntimes

ξ 1n

p

150

200

250

300

350

400

450







1157
















0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Frequency (Hz)

Amplitude(dB)

Uncont rolledξ1 10

ξ1times 100


0 500 1000 1500 2000-50

0

50

Amplitude(dB)

0 500 1000 1500 2000

-2

0

2

Frequency (Hz)

Phase(rad)




1158



Resonance 1 2 3 4 5 6 7 8

Frequency (Hz) U 149 448 746 1046 1344 1643 1942 2242C 149 470 746 1046 1344 1643 1942 2242


Damping U 00178 00105 00082 00071 00065 0006 00057 00054C 00178 00055 00083 00072 00067 00059 00056 00053

(T 00053)Difference () 0 3 1 1 1 1 1 1

Amplitude (dB) U 225 81 13 -33 -68 -96 -12 -14C 218 131 19 -3 -66 -95 -119 -139

Difference (dB) -07 +5 +06 +03 +02 +01 +01 +01

0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Amplitude(dB)

Frequency (Hz)





05 06 07 08 09 1 11 12

-02

0

02

04Uncontrolled

05 06 07 08 09 1 11 12

-02

0

02

04

Controlled

Time (s)




1159






Acknowledgement



References
















1160
















1161













04 05 06 07 08 09 10

50

100

150

200

1

2

3

γ

fp(H

z)


0 02 04 06 08 10

5

10

15

20

γ

Target

ntimes

ξ 1n

p

150

200

250

300

350

400

450







1157
















0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Frequency (Hz)

Amplitude(dB)

Uncont rolledξ1 10

ξ1times 100


0 500 1000 1500 2000-50

0

50

Amplitude(dB)

0 500 1000 1500 2000

-2

0

2

Frequency (Hz)

Phase(rad)




1158



Resonance 1 2 3 4 5 6 7 8

Frequency (Hz) U 149 448 746 1046 1344 1643 1942 2242C 149 470 746 1046 1344 1643 1942 2242


Damping U 00178 00105 00082 00071 00065 0006 00057 00054C 00178 00055 00083 00072 00067 00059 00056 00053

(T 00053)Difference () 0 3 1 1 1 1 1 1

Amplitude (dB) U 225 81 13 -33 -68 -96 -12 -14C 218 131 19 -3 -66 -95 -119 -139

Difference (dB) -07 +5 +06 +03 +02 +01 +01 +01

0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Amplitude(dB)

Frequency (Hz)





05 06 07 08 09 1 11 12

-02

0

02

04Uncontrolled

05 06 07 08 09 1 11 12

-02

0

02

04

Controlled

Time (s)




1159






Acknowledgement



References
















1160
















1161
















0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Frequency (Hz)

Amplitude(dB)

Uncont rolledξ1 10

ξ1times 100


0 500 1000 1500 2000-50

0

50

Amplitude(dB)

0 500 1000 1500 2000

-2

0

2

Frequency (Hz)

Phase(rad)




1158



Resonance 1 2 3 4 5 6 7 8

Frequency (Hz) U 149 448 746 1046 1344 1643 1942 2242C 149 470 746 1046 1344 1643 1942 2242


Damping U 00178 00105 00082 00071 00065 0006 00057 00054C 00178 00055 00083 00072 00067 00059 00056 00053

(T 00053)Difference () 0 3 1 1 1 1 1 1

Amplitude (dB) U 225 81 13 -33 -68 -96 -12 -14C 218 131 19 -3 -66 -95 -119 -139

Difference (dB) -07 +5 +06 +03 +02 +01 +01 +01

0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Amplitude(dB)

Frequency (Hz)





05 06 07 08 09 1 11 12

-02

0

02

04Uncontrolled

05 06 07 08 09 1 11 12

-02

0

02

04

Controlled

Time (s)




1159






Acknowledgement



References
















1160
















1161



Resonance 1 2 3 4 5 6 7 8

Frequency (Hz) U 149 448 746 1046 1344 1643 1942 2242C 149 470 746 1046 1344 1643 1942 2242


Damping U 00178 00105 00082 00071 00065 0006 00057 00054C 00178 00055 00083 00072 00067 00059 00056 00053

(T 00053)Difference () 0 3 1 1 1 1 1 1

Amplitude (dB) U 225 81 13 -33 -68 -96 -12 -14C 218 131 19 -3 -66 -95 -119 -139

Difference (dB) -07 +5 +06 +03 +02 +01 +01 +01

0 500 1000 1500 2000-120

-100

-80

-60

-40

-20

Amplitude(dB)

Frequency (Hz)





05 06 07 08 09 1 11 12

-02

0

02

04Uncontrolled

05 06 07 08 09 1 11 12

-02

0

02

04

Controlled

Time (s)




1159






Acknowledgement



References
















1160
















1161






Acknowledgement



References
















1160
















1161
















1161

simulations of modal active control applied to the self-sustained...

Documents