design of an active controller for the enhancement of the...

Design of an active controller for the enhancement of the sound quality of aclassical violin

Gregory Pinte, Peter Deckers, Jan De Caigny, David Moens, Jan Swevers, Paul SasKULeuven - Department of Mechanical Engineering - Division PMA, Celestijnenlaan 300B, B-3001 Heverlee, Belgium, e-mail:

[email protected]

The structural design of classical musical instruments has a century long tradition. The violin is a typicalexample of an instrument of which the mechanical construction is based on this tradition. However, eventhough nowadays all classical violins look almost identical, the sound quality among them varies strongly.This sound quality is the key-property in determining the value of a classical violin. The fact that the designof the classical violin has not evolved much over the past decades indicates that an optimum has beenobtained within the reach of the classical mechanical construction with purely passive acoustical soundgeneration. The aim of this paper is to study the applicability of an active controller for the enhancementof the sound quality generated by the classical violin. More in particular, this study envisages a modularcontroller which does not affect the classical mechanical construction of the violin. The first section of thepaper briefly discusses the typical design aspects influencing the acoustical properties of the violin, and anobjective measurement technique for violin sound quality. Next, the design of the controller is discussed.A preliminary structural modal analysis on a classical violin forms the basis for the choice of the placementof the sensors and actuators on the structure. An optimal controller is defined based on a control objectivefunction incorporating the objectively determined sound quality indicators. Finally, the actively controlledviolin is validated experimentally.

1 Introduction

The violin is a musical instrument with a long and richtradition. Throughout the history, the violinmakers cre-ated an instrument of high acoustical and dynamic qualitywithout the knowledge of any numerical modelling tech-niques or modal analysis methods. The current structuralshape is the result of a series of design modifications inorder to improve the radiated sound quality of the violin.Figure 1 shows the different parts of a violin. When theplayer strikes the strings by means of the bow, a stick-slipphenomenon sets the string into vibration. The vibratingstring produces a force on the bridge, which is transmittedvia the bridge to the top plate. The bass bar is used to setthe whole top plate into vibration. The function of the F-holes in the top plate is double. They reduce the stiffnessof the top plate, such that the amplitude of the plate vi-brations grows and, consequently, the radiated sound in-creases. Next to this, the F-holes also change the acousti-cal properties of the violin cavity: a Helmholtz resonancearises around 300 Hz. The sound post, which propagatesthe vibrations to the back plate, is very important for thetimbre of the violin. Small changes in the position of thesound post can create a significant decrease of the qualityof the violin [1].

The accuracy and the composition of the different partsdefine the quality of a violin. In the first part of this arti-cle, some tests are described to define the ideal structuralproperties of a violin. Because the dynamic and acous-tic properties of a violin are extremely sensitive to small

Bridge

Bass barSound

postF-holes

Tailpiece

Bridge

Figure 1: Outline of a violin.

structural deviations from the ideal design, the construc-tion of a good violin is very labour intensive and conse-quently also very expensive. Therefore, in order to in-crease the sound quality of a mediocre violin, the possi-bilities of active control are explored in the second partof the article. The traditional sound post is replaced byan active piezo-element, which is controlled such that thesound quality of the active violin approximates the qual-ity of an ideal violin.

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Forum Acusticum 2005 Budapest Pinte, Deckers, De Caigny, Moens, Swevers, Sas

2 Tests of the sound quality of a vi-olin

Although the definition of the sound quality of a musi-cal instument is very subjective, most people agree thata Stradivarius sounds much better than a standard violin.Therefore, a lot of research has been carried out to char-acterize the specific dynamic and acoustical properties ofa Stradivarius. In acoustic tests, the sound radiated by aviolin is measured by one or more microphones. In struc-tural tests, vibrations are measured at certain locations onthe violin to get more insight in the dynamic behaviourgenerating the sound. In this research project, some ex-isting tests were validated and some new tests were de-veloped.

Three violins are compared in this project. The first vi-olin (violin A), the best one and considered as the refer-ence violin in this project, is an authentic instrument ofthe Flemmish school, which was developed in Brusselsaround 1760. The price of this violin is valued at 8000euro. The quality of the two other violins (violins B andC) is much lower. The active sound post is placed in vio-lin C.

2.1 Acoustic tests

The only acoustic test, applied in this project, is the loud-ness test, which is developed by C.M. Hutchins [2]. Inthis relatively simple test, a microphone measures the ra-diated sound pressure when the violin player plays 1 toneas loud as possible. The peak level of the sound pressureis plotted on the loudness curve at the corresponding basefrequency. In the same way, all tones from 200 Hz up till2000 Hz are measured.

Table 1: Frequencies of the open strings.

Name of the string Frequency of base tone (Hz)G-string 196.00D-string 293.67A-string 440.00E-string 659.26

In literature, it is demonstrated that, in case of a Stradivar-ius, peaks appear in the loudness curve at the frequenciesof the open strings (table 1). The first, third and fourthpeak are caused by structural resonances of the violin,the second peak is the Helmholz resonance frequency ofthe F-holes. In the loudness curve of violins of lowerquality, the peaks in the loudness curve show up at dif-ferent frequencies. Due to this, the amplification of theviolin will be too small in certain frequency bands, re-sulting in frequency gaps. Similar results could be drawnbased on the experiments on the three investigated violins

in this project. Figure 2 shows the loudness curves of ref-erence violin A and of violin B of lower quality. Whilethe curve of violin A exhibits the same characteristics asthe Stradivarius, important differences appear in the loud-ness curve of violin B. Because the Helmholz resonanceappears at a lower frequency and the second structuralresonance shows up at a higher frequency, compared to aStradivarius, violin B has a clear frequency gap betweenboth resonances, where the violin doesn’t amplify suffi-ciently. This frequency gap is the reason for the lowersound quality of this violin.

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peak

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Violin AViolin B

G D A E

Figure 2: Comparison of the loudness curves of violin Aand B.

2.2 Structural tests

Two structural quality tests are carried out to compare thedynamics of violins of different quality. In the first test,the mode shapes of the violin panels are defined. The cor-respondance of the resonance frequencies and the modeshapes to those of a Stradivarius gives an indication of thesound quality of the violin. In the second test, which wasdeveloped in this project, frequency response functionsare measured between certain locations on the violin. Es-pecially transfer functions between points on the vibra-tion transmission path (from the strings via the bridge andthe sound post to the top and back plate) were studied,because the developed active control strategy (paragraph3.1) aims at the adaptation of the dynamic properties ofthis path.

2.2.1 Resonance frequencies and mode shapes

Different methods exist to define the mode shapes of aviolin. The simplest method is to put a layer of sandon the plates of the violin. When the plates are acous-tically excited by a loudspeaker, at resonance frequen-cies the sand will vibrate away from the antinodes andgather at the nodes. In this way, the corresponding modeshapes (’Chladni patterns’, [3]) can easily be identified.

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Although this test is very simple, it is quite time consum-ing because a lot of frequencies should be tested. Next tothis, the test can only be carried out on the seperate platesand not on the whole violin.

Figure 3: Photo of the laser scanning test set-up.

For these reasons, more sophisticated methods have beendeveloped: accelerometer measurements of a hammerimpact exciation, holografic interferometry with acousticexcitation,... In this project, an automatic laser scanningunit was used to measure the velocities at 100 points onthe top and back plate of the violin, while a loudspeakerexcites the structure at a resonance frequency (figure 3).In this way, the resonance frequencies and mode shapesof violin A and B could be defined and compared to thoseof a Stradivarius [3].

Table 2: Comparison of the resonance frequencies of vi-olin A and a Stradivarius.

Resonance Resonance Absolut Relativefrequency frequency difference difference

(ideal) (Hz) (violin A) (Hz) (Hz) (%)409 402 6.75 1.65448 489 40.7 8.33513 544 31.4 5.77817 835 17.55 2.10

Table 3: Comparison of the resonance frequencies of vi-olin B and a Stradivarius.

Resonance Resonance Absolut Relativefrequency frequency difference difference

(ideal) (Hz) (violin B) (Hz) (Hz) (%)409 430 21.2 4.93448 500 51.9 10.38513 748 234.6 31.38817 814 2.7 0.33

Tables 2 and 3 enumerate the first 4 identified structuralresonance frequencies of both violins and show the dif-ference from the frequencies measured on a Stradivarius.

It is clear that the resonance frequencies from referenceviolin A (maximum difference of 8 %) correspond muchbetter to a Stradivarius than those of violin B (maximumdifference of 31 %). Also the mode shapes of violin Aresemble much better the mode shapes of a Stradivarius,especially for the lowest modes: figure 4 compares thefirst structural mode, measured on a Stradivarius, on vi-olin A and on violin B. At higher frequencies, the modeshapes of violin A also differ from a Stradivarius. How-ever, in literature [4], it is mentioned that a correspon-dance of resonance frequencies, which is the case for vi-olin A, is much more important for the sound quality thansimilar mode shapes.

Violin A Violin B

Stradivarius

Figure 4: Comparison of the mode shapes of violin A,violin B and a Stradivarius.

2.2.2 Frequency responce functions

A second test is the measurement of frequency responsefunctions (frf’s) between an input force and an accelera-tion at certain locations on the vibration transmission pathof the violin. No results on similar tests were found in lit-erature. Therefore, violin A, which was the best violinaccording to the other quality tests, is used as the ref-erence in this test. Because the measured frf’s are usedfor the control design, these tests are carried on violin C(with the active sound post) instead of violin B.

To simulate the force of the strings, which is the sourceof vibrations in a violin (see introduction), the input ofthe frf is a transverse force on the bridge (3 on figure 5).A shaker excitation is used for the generation of this in-put force. Two response points are measured with an ac-celerometer:

• point 1 on the top plate next to the bridge, to inves-tigate the vibration transmission through the bridgeto the top plate (figure 5)

• point 2 on the back plate under the sound post, toinvestigate the vibration transmission through the

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1

2

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Figure 5: The measurement of frequency response func-tions with shaker excitation.

sound post to the back plate (figure 5)

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Violin AViolin C

Figure 6: The measured frequency response function tothe accelerometer on the top plate.

While in the frf to the top plate of violin A, the reso-nance frequencies, which were identified in the loudnesstest (2.1) and in the first structural test (2.2.1), cannot beobserved, some of these resonance frequencies are clearlypresent in the frf to the back plate. Especially at thefrequency of the E-string (659 Hz) and at the first andfourth identified resonance frequencies (402 Hz and 835Hz) peaks are visible in the frf to the back plate of refer-ence violin A. In the frf of violin C, the frequencies of thepeaks deviate significantly from those of violin A. There-fore the measurement of the frf from a force on the bridgeto an acceleration on the back plate can be considered asa good measure of the sound quality of a violin. This frf,measured on violin A, will serve as the goal function inthe control design (paragraph 4).

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Violin AViolin C

403 Hz 659 Hz 835 Hz

Figure 7: The measured frequency response function tothe accelerometer on the back plate.

3 Design of the active sound post

3.1 Control configuration

Two different strategies can be applied for the active con-trol of a violin: an active element can be introduced inthe vibration transmission path or the radiating plates canbe actively controlled. In the former case, discrete actu-ators can only influence the frf of the transmitted vibra-tions from the bridge to the unaltered radiating plates. Inthe latter case, the resonance frequencies as well as themode shapes of the radiating surfaces can be modified.However, because the resonance frequencies determinemuch more the sound quality of a violin than the modeshapes (paragraph 2.2.1), an active element in the vibra-tion transmission path, which is practically more feasibleto place, is sufficient. In the last structural test, it wasclear that the transmission of vibrations to the back plateis a better measure of the quality of a violin than the trans-mission to the top plate. Therefore the sound post seemedthe most convenient place for an actuator. Next to this,there is enough space in the sound post for an active ele-ment, that can deliver the desired stroke (±5µm), whileat other possible positions (between the left or right con-nection of the bridge to the top plate,...) the space for anactive element is too limited.

3.2 Active sound post

A normal sound post is a cylindrical wooden bar whichconnects the top and back plate of the violin. Figure 8shows a drawing of the active sound post. The activepiezo-element (table 4) is fixed between two woodenparts of the original sound post. The two aluminium partsguarantee a good connection between the piezo-elementand the wooden bars. This active sound post is placed inthe assembled violin by a professional violinmaker, suchthat a correct axial alignment is achieved.

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Wooden bar

Aluminium part

Piezo-element

Figure 8: A drawing of the built-in active sound post (left)and of the active sound post itself (right).

Table 4: Properties of the piezo-element.

manufacturer Piezomechanik GmbH Munichtype Pst 150/2x3/20

dimensions 18 mmmaximum stroke 30 µm

electrical capacity 340 nFstiffness 12 N/µm

resonance frequency 35 kHzmaximum force 300 N

4 Control algorithm

The conclusion of the last structural test in paragraph2.2.2 is that the frf between the force on the bridge andthe acceleration on the back plate is a good measure forthe violin sound quality. In this test, violin A was consid-ered as the reference violin of high quality. The goal ofthe active control algorithm is to modify the frf bridge-back plate of violin C such that the difference with themeasured frf on violin A decreases. Based on the differ-ence between the frf’s of violin A and C, a control signalis calculated and sent to the active sound post in order tominimize the difference.

Due to the high modal density, it is, in first instance, toocomplicated to design a control strategy which improvesthe sound quality over the whole frequency range. There-fore, in this paper, a controller is developed for a limitedfrequency range from 400 to 500 Hz based on the follow-ing considerations. In the frequency range below 400 Hz,it is difficult to improve the sound quality with a struc-tural actuator because the acoustical Helmholz-resonancedominates the radiated noise spectrum. In all the qual-ity tests (acoustic as well as structural), it was clear thatsome important structural frequencies were present be-tween 400 and 500 Hz. The goal of the control design isto replace the structural resonance frequencies of violinC between 400 and 500 Hz with these of reference violinA. First, a feedforward controller is developed to achievethis goal, which is afterwards extended with a feedbackcompensation.

4.1 Feedforward control without feedbackcompensation

Figure 9 describes the control configuration with the ex-citation (4, a shaker or the bow), the piezo-actuator (3)and the two accelerometers (1 and 2). Figure 10 showsthe control scheme of the feedforward controller with-out feedback compensation. Accelerometer 1, on the topplate next to the bridge, is the reference signal of the feed-forward algorithm, while accelerometer 2 is the error ac-celerometer, which is only used during the design and theevaluation of the controller and can be omitted in the finalconfiguration. This accelerometer could be used later inthe design of adaptive control algorithms. Based on thereference signal of accelerometer 1, the controller calcu-lates the input signal for the piezo-actuator to compen-sate the difference between the identified and the ideal frfbridge-accelerometer 2.

F

1

2

4

3

Figure 9: The control configuration for the feedforwardcontroller (without and with feedback compensation).

Bridge - acc1

++

+

Excitation Bridge - acc2

Ident

Ident

Bridge - acc2

Ideal

FF controller1 Piezo - acc2

IdentAcc1

++

-

Acc2 Error

Figure 10: The control scheme of the feedforward con-troller without feedback compensation.

The feedforward controller is designed such that the con-trolled plant bridge-accelerometer 2 becomes equal to theideal plant, measured in the reference violin A.

(bridge − acc2)ideal = (bridge − acc2)ident+

(bridge − acc1)ident · CFF · (piezo− acc2)ident

(1)

where

• (bridge − acc2)ideal is an identification of the frfbridge-accelerometer 2 in violin A,

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• (bridge − acc2)ident is an identification of the frfbridge-accelerometer 2 in violin C,

• (bridge − acc1)ident is an identification of the frfbridge-accelerometer 1 in violin C,

• (piezo − acc2)ident is an identification of the frfpiezo-accelerometer 2 in violin C,

• CFF is the frf of the feedforward controller.

The feedforward controller then becomes:

CFF =(bridge − acc2)ideal − (bridge − acc2)ident

(bridge − acc1)ident · (piezo− acc2)ident

(2)

In figure 11, the transfer function of the implementedcontroller is compared with the ideal frf’s for the con-troller. Although, in simulation, the performance of thiscontroller is very good in the frequency range between400 and 500 Hz, in the practical experiments, the con-troller is unstable. This unstability is caused by the un-modelled feedback path from the piezo-actuator to thereference accelerometer 1, which was neglected in thecontrol scheme in figure 10.

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Figure 11: The identified frf’s and the frf of feedforwardcontroller 1.

4.2 Feedforward control with feedbackcompensation

In the control scheme in figure 12, the feedback ef-fect from the piezo-actuator to accelerometer 1 is alsomodelled. The goal is again to minimize the differ-ence between the identified and the ideal frf bridge-accelerometer 2.

Bridge - acc1

++

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Excitation Bridge - acc2

Ident

Ident

Bridge - acc2

Ideal

FF controller1 Piezo - acc2

IdentAcc1

++

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Acc2 Error

++

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Piezo - acc1

Ident

Feedback compensation

Figure 12: The control scheme of the feedforward con-troller with feedback compensation.

(bridge − acc2)ideal = (bridge − acc2)ident+

(bridge − acc1)ident · CFF · (piezo− acc2)ident

1 − (piezo − acc1)ident · CFF

(3)

This results in the following controller:

CFF =(bridge − acc2)ideal − (bridge − acc2)ident

(bridge − acc1)ident · (piezo− acc2)ident

+(bridge− acc2)ideal · (piezo− acc1)ident

−(bridge − acc2)ident · (piezo− acc1)ident

(4)

Figure 13 compares the identified frf of the controllerwith the ideal frf for the controller: a clear correspon-dence can be observed. In contrast with the feedforwardcontroller without feedback compensation, no instabilityoccurs when this controller was tested in practice. Theperformance of this controller was evaluated by a mea-surement of the frf bridge-accelerometer 2 (shaker excita-tion) and by a test of a violin player. Figure 14 shows thechange of the optimised frf when the controller is turnedon: the correspondence with the frf, measured on the ref-erence violin A, is much better with control than withoutcontrol. In the test by the violin player, no audible im-provement can be detected because only a small part ofthe frequency domain (400-500 Hz) has been controlledand the overtones, which are also important for the soundquality, are not modified. To get an audible result, a morecomplicated controller should be developed in the sameway, which can tackle the whole frequency range from200 to 2000 Hz. Next to this, with the actual set-up, theplayer should play rather quiet to avoid saturation of thepiezo-actuator.

4.3 Conclusion and future work

This paper has shown the possiblities of active controltechniques to improve the sound quality of a violin oflower quality. The introduction of an active element (inthis case an active sound post) can significantly change

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Figure 13: The identified and the ideal frf of feedforwardcontroller 2.

the sound radiation of a violin. In this research, a con-troller has been developed to improve the sound qualityin the frequency range from 400 to 500 Hz. This con-troller will be extended in the future to a broader fre-quency range such that the sound radiation of the over-tones can also be modified and an audible improvementof the sound quality should be observable. In the newdesign, a piezo-actuator with a higher stroke should beplaced in the sound post, because in the operational tests,when the violin player plays loudly, the piezo actuatorwas sometimes saturated. In the future, active controlcan also be applied to improve the quality of other in-struments (e.g. a guitar).

References

[1] I. Johnston, ’Measured tones: The interplay ofphysics and music’, Adam Hilger, Bristol and NewYork (1989)

[2] C. M. Hutchins, ’The physics of Music: readingsfrom Scientific American’, W. H. Freeman and Com-pany, San Francisco (1978)

[3] M. Schleske, ’http://www.schleske.de/’

[4] E. Jansson, , ’Acoustics for violin and guitar mak-ers’,Kungl Tekniska Hogskolan (2002)

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ideal plant (violin A)without control (violin C)with control (violin C)

ideal plant (violin A)without control (violin C)with control (violin C)

Figure 14: Measurement of the frf bridge-accelerometer2 on violin A (ideal), violin C (without control) and violinC (with control).

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design of an active controller for the enhancement of the...

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