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Models for Vibration and Radiationof two stringed Instruments

Svein BergeApril 1, 1996

Abstract

This report summarizes the findings and explorations that were done duringa six month internship at the acoustics group of the Signal department at EcoleNationale Superieure des Telecommunications in Paris. The exploration consistedin literature studies, experiments and conversations on the vibration and radiationof stringed musical instruments. The measurement and control over the vibration ofstrings were investigated. For two particular instruments, numerous measurementswere done and analyzed: A somewhat special classical guitar and a small, Celticharp. Models for their vibration and radiation are investigated and compared withmeasurements.

Chapter 1

Preface

All work described in this diploma thesis has been done during a six monthinternship at the acoustics group of the Signal department at Ecole NationaleSuperieure des Telecommunications in Paris.

The acoustics group at ENST has for several years been working on the mod-elling of vibration and radiation of musical instruments. One of my contributionshas been to check the validity of certain models for sound radiation.

1.1 Motivation

During the six months this project has lasted, a great number of measurements havebeen done and these have been analyzed and treated to give as much informationas possible about the vibration and radiation of the two instruments that have beenstudied.

Some of the results are presented in the current report. Hopefully, they revealinteresting pieces of information, both regarding the measurement techniques andthe particular instruments that have been studied. Before spending six months ona project, it should also be set into a broader perspective. Can it have any practicaluse? Possible utilizations are found in three directions:

Sound field regeneration: Imagine placing a loudspeaker on the stage of a con-cert hall. A recorded piece of music, played on a single instrument, isreplayed over the loudspeaker. Regardless of the quality of the equipment,the reproduction will always be somewhat unrealistic. Choosing a particu-larly good listening position can give a good result, but one loudspeaker cannot generally reproduce good, natural sound everywhere in a listening room.To do so, it would have to mimic the directivity of the particular musicalinstrument which is played. This is feasible with an arrangement of severalloudspeakers and a knowledge about the instrument’s radiation. The latteris studied, for two particular instruments, in this project.

Instrument improvement: The manufacturing of musical instruments is a workgoverned by traditions that carry centuries of experience between generationsof instrument makers. Improvements are done by trial and error. Acous-ticians may contribute with measurements and suggestions, but are always

1

2 CHAPTER 1. PREFACE

limited by the simplicity and inaccuracy of their models. However, as tech-nology evolves in other domains — especially electronics and computers —better, more detailed measurements can be done, and more elaborate modelsmay be used. Finite element analysis is a result of automated numerical anal-ysis, and allows the effect of proposed changes to an instrument be evaluatedwith good precision at low cost. This does not eliminate the need for detailedmeasurements, though. In order to give realistic results, the parameters ofthe original model must be matched with reality. The best verification of afinite element model is to compare its modes with those found in a modalanalysis of the real instrument, which is one of the results in this project.

Sound synthesis: As the cost of computation has dropped dramatically and con-tinuously for decades, digital synthesizers have become toys that can literallybe picked up at the grocery store. Among the models in the opposite endof the price range, new approaches, based on physical models of musicalinstruments, have been introduced. These are computationally far more de-manding than the signal-oriented approaches used in all previous electronicsynthesizers, but are capable of generating more realistic sounds. Such anapproach demands good knowledge of the physical system which is mod-eled, and all parameters involved. Some of the measurements that havebeen done during this project gives the necessary information to synthesizethe sound of the studied instruments. For instance, an experimental systemcalled MODALYS, which has been developed at IRCAM, uses the modalanalysis of an instrument to synthesize its sound.

Since this project was the final part of my education, the motivation was, aboveall, to learn. The project has undoubtedly been successful in that sense, since thelargest fields that have been treated — the measurement techniques and modalanalysis — were totally unknown to me before I started. Within other subjects,such as signal processing and numerical methods in acoustics, the project has givenme valuable experience.

Some of the methods and models that have been used are only describedsuperficially, since they are commonly used and only small adaptations have beennecessary to utilize them in the current work. Detailed descriptions of these can befound by following the references.

1.2 Acknowledgements

There are a large number of people that have been of great help to me and thisproject. I wish to thank them all, and would like to mention a few of them inparticular, more or less in “order of appearance”:

Firstly, my supervisor at NTH, prof. Ulf R. Kristiansen, who was more thanhelpful in connecting me with suitable laboratories where I could have my in-ternship, and who later took on the responsibility of supervising this project.Ass. prof. Jan Tro at NTH was the one who suggested to take contact with theacoustics group at ENST, and I owe both of them deep gratitude for displaying theconfidence in my motivation necessary to let me do my internship here.

1.2. ACKNOWLEDGEMENTS 3

Furthermore, I wish to thank ass. prof. Andrew Perkis from NTH for effectivelyhandling the communication with ENST, where Jennifer Molet made sure that Igot all necessary documents, and who gave me a warm welcome. I also owe herand Claude Chouraqui a big thanks for giving me, free of charge, a room at ENST’sstudents’ home during two weeks when I was hopping around on crutches.

It was a pleasure to meet prof. Antoine Chaigne during the InternationalCongress on Acoustics in Trondheim the summer of 1995. I was convincedthat my stay here would be a pleasant one, and was not mistaken. Chaigne leadsthe acoustics group at ENST, and has made sure that I have always had the bestconditions to work under, and given me the advice and motivation necessary to usethe available time efficiently, from day one.

The daily work has been done in close collaboration with Alexis Le Pichon, whois in his 3rd year of studies for his Ph.D degree. We have had countless discussionsall along, concerning all the topics treated in this report. The calculations ofradiation which are presented here are done by him, and he did a good portion ofthe measurements, particularly when I was disabled by my injured foot.

The other members of the acoustics group, particularly Christophe Lambourgwho works on the vibration of plates, and also Dr. Vincent Doutaut, Dr. DenisMatignon, Dr. Jean Laroche, Vincent Rousarri and Dr. Ioannis Stylianou, have beenmost helpful in discussing my work, and have taught me a lot through discussions oftheir own projects. They have all contributed to a friendly and relaxed atmosphere,where it has been a pleasure to work and study. Prof. Thomas D. Rossing, whopaid us a visit towards the end of my stay, gave me valuable comments on the workI had done, and helped me with the language in this report.

Dr. Rene Causse and Philippe Derogis at IRCAM have also contributed withvaluable experience. In particular, Derogis’ work on modal analysis, which heused an afternoon to explain to me, has been highly useful. The Matlab code inappendix D.5 is based on a script written by him.

Contents

1 Preface 11.1 Motivation � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 11.2 Acknowledgements � � � � � � � � � � � � � � � � � � � � � � � � 2

I Theoretical introduction 7

2 Plucked strings 82.1 String models � � � � � � � � � � � � � � � � � � � � � � � � � � � 8

2.1.1 Linear, first-order � � � � � � � � � � � � � � � � � � � � � 82.1.2 Model refinement � � � � � � � � � � � � � � � � � � � � � 9

2.2 String — soundboard interaction � � � � � � � � � � � � � � � � � � 10

3 Vibrating instrument body 123.1 Modal Behavior � � � � � � � � � � � � � � � � � � � � � � � � � � 123.2 Modal testing and analysis � � � � � � � � � � � � � � � � � � � � � 13

4 Instrument body radiation 164.1 Plane surfaces � � � � � � � � � � � � � � � � � � � � � � � � � � � 164.2 Three-dimensional instruments � � � � � � � � � � � � � � � � � � 16

4.2.1 Baffle model � � � � � � � � � � � � � � � � � � � � � � � � 174.2.2 Baffleless model � � � � � � � � � � � � � � � � � � � � � � 174.2.3 Rigid-back model � � � � � � � � � � � � � � � � � � � � � 184.2.4 Several faces � � � � � � � � � � � � � � � � � � � � � � � � 18

II Preliminary experiments 19

5 String excitation 205.1 Electromagnetic excitation � � � � � � � � � � � � � � � � � � � � � 20

5.1.1 Chaotic oscillations � � � � � � � � � � � � � � � � � � � � 225.2 Modeling the string’s resonance � � � � � � � � � � � � � � � � � � 22

5.2.1 Experimental results � � � � � � � � � � � � � � � � � � � � 235.2.2 String variations � � � � � � � � � � � � � � � � � � � � � � 23

5.3 Photodetector � � � � � � � � � � � � � � � � � � � � � � � � � � � 265.4 Magnetic sensor � � � � � � � � � � � � � � � � � � � � � � � � � � 285.5 Wiener bridge � � � � � � � � � � � � � � � � � � � � � � � � � � � 30

4

CONTENTS 5

5.6 Self-oscillation � � � � � � � � � � � � � � � � � � � � � � � � � � � 325.6.1 Linear feedback � � � � � � � � � � � � � � � � � � � � � � 325.6.2 Nonlinear feedback � � � � � � � � � � � � � � � � � � � � 33

5.7 External reference signal � � � � � � � � � � � � � � � � � � � � � � 345.8 String — soundboard interaction � � � � � � � � � � � � � � � � � � 365.9 Conclusions � � � � � � � � � � � � � � � � � � � � � � � � � � � � 37

6 Velocity distribution 386.1 Devices � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 38

6.1.1 Accelerometer � � � � � � � � � � � � � � � � � � � � � � � 386.1.2 Laser vibrometer � � � � � � � � � � � � � � � � � � � � � � 386.1.3 Gramophone pickup � � � � � � � � � � � � � � � � � � � � 39

7 Measurement of pressure fields 407.1 Direct approach � � � � � � � � � � � � � � � � � � � � � � � � � � 407.2 Turntable setup � � � � � � � � � � � � � � � � � � � � � � � � � � � 40

7.2.1 Choice of excitation signal � � � � � � � � � � � � � � � � � 427.2.2 Extraction of data � � � � � � � � � � � � � � � � � � � � � 427.2.3 Linearity � � � � � � � � � � � � � � � � � � � � � � � � � � 427.2.4 Angular resolution � � � � � � � � � � � � � � � � � � � � � 427.2.5 Graphical presentation � � � � � � � � � � � � � � � � � � � 43

III The Laplane guitar 47

8 First setup 48

9 Second setup 509.1 Soundhole velocity � � � � � � � � � � � � � � � � � � � � � � � � � 529.2 Experimental setup and measurements � � � � � � � � � � � � � � � 54

9.2.1 Excitation signal � � � � � � � � � � � � � � � � � � � � � � 559.2.2 Sound-field measurement � � � � � � � � � � � � � � � � � 56

Signal processing � � � � � � � � � � � � � � � � � � � � � 569.2.3 Velocity measurement � � � � � � � � � � � � � � � � � � � 56

Hand-supported laser � � � � � � � � � � � � � � � � � � � 57Synchronization � � � � � � � � � � � � � � � � � � � � � � 57Signal processing � � � � � � � � � � � � � � � � � � � � � 58Interpolation � � � � � � � � � � � � � � � � � � � � � � � � 59Measured vibration � � � � � � � � � � � � � � � � � � � � 60

9.3 Radiation field calculation � � � � � � � � � � � � � � � � � � � � � 629.3.1 Some results � � � � � � � � � � � � � � � � � � � � � � � � 629.3.2 Sources of error � � � � � � � � � � � � � � � � � � � � � � 63

9.4 Near-field intensity � � � � � � � � � � � � � � � � � � � � � � � � � 659.5 Conclusion � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 66

6 CONTENTS

10 Modal analysis 6710.1 Extraction of eigenvalues � � � � � � � � � � � � � � � � � � � � � 6710.2 Reconstruction of transfer functions � � � � � � � � � � � � � � � � 6810.3 The phases of the modes � � � � � � � � � � � � � � � � � � � � � � 7010.4 Mode shapes � � � � � � � � � � � � � � � � � � � � � � � � � � � � 7110.5 Conclusion � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 72

IV The Celtic harp 77

11 Measurements 7811.1 Vibrational measurements � � � � � � � � � � � � � � � � � � � � � 7811.2 Intensity measurements � � � � � � � � � � � � � � � � � � � � � � 7911.3 Radiation measurements � � � � � � � � � � � � � � � � � � � � � � 79

12 Data analysis 8312.1 Modal analysis � � � � � � � � � � � � � � � � � � � � � � � � � � � 83

12.1.1 Eigenvalue extraction � � � � � � � � � � � � � � � � � � � 83Manual clustering � � � � � � � � � � � � � � � � � � � � � 83

12.1.2 Eigenvector extraction � � � � � � � � � � � � � � � � � � � 8412.1.3 Mode frequencies � � � � � � � � � � � � � � � � � � � � � 86

12.2 Input admittance � � � � � � � � � � � � � � � � � � � � � � � � � � 8612.3 Discussion � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 8812.4 Conclusions � � � � � � � � � � � � � � � � � � � � � � � � � � � � 89

V Appendices 95

A Mesh grossness 96A.1 Measuring k � � � � � � � � � � � � � � � � � � � � � � � � � � � � 96

A.1.1 Measurements � � � � � � � � � � � � � � � � � � � � � � � 98

B Instrument excitation 100B.1 Shaker excitation � � � � � � � � � � � � � � � � � � � � � � � � � � 100B.2 Impact hammer � � � � � � � � � � � � � � � � � � � � � � � � � � 101

C Manual plucking 102

D Matlab code 104D.1 Eigenvalues.m � � � � � � � � � � � � � � � � � � � � � � � � � � � 106D.2 fesprit.m � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 107D.3 ampl.m � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 108D.4 Mancluster.m � � � � � � � � � � � � � � � � � � � � � � � � � � � � 109D.5 Eigenvectors � � � � � � � � � � � � � � � � � � � � � � � � � � � � 111D.6 Reconstruct.m � � � � � � � � � � � � � � � � � � � � � � � � � � � 112D.7 Quality.m � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 113D.8 Compliance.m � � � � � � � � � � � � � � � � � � � � � � � � � � � 115

Part I

Theoretical introduction

This part summarizes a few of the models that have been used for the vibration andradiation of the harp and the guitar.

All plucked string instruments consist, essentially, of two parts; the strings andthe body. It is common, and very simplifying, to study each of them separately,and subsequently their interaction.

The strings are plucked by the player. The plucks contain all the energy deliveredto the system. The strings store this energy in the form of vibration, which ispartially lost as heat, radiated to the air, and transferred to the instrument body.

The body, with its large surface area, is much more efficient than the strings intransferring the energy to the surrounding air. Vibration in the instrument body isalso transferred back to the strings, in a two-way exchange of energy. A portion ofthe energy is also lost internally in the body.

It is common to separate the instrument body into two parts; the soundboardand the sound box, under the assumption that only the soundboard’s vibrationcontributes considerably to the sound radiation.

The air transfers the vibrations of the instrument to the listeners, and is modeledas a linear medium obeying the three-dimensional wave equation.

7

Chapter 2

Plucked strings

Both of the instruments that have been studied during this project are plucked stringinstruments.

2.1 String models

Mathematical models of the string have been developed by investigating the me-chanical conditions within the string. A thorough description is given in [VAL].

2.1.1 Linear, first-order

The most basic, idealized model of a vibrating string, is the linearized, first-ordermodel. In this model, the string is regarded as totally one-dimensional and linear.The model only concerns the transverse motion of the string in one plane. Thetension is regarded as being constant throughout the string, and constant over time.The end supports in the guitar and harp are fairly rigid, and we idealize them in ourmodel to be completely immobile.

Under these assumptions, it can be developed (as in [FR91], among others) thatthe string displacement y as a function of time t and the position along the stringx follows the relation

�2y

�t2�

T

�

�2y

�x2 � �2�1�

the one-dimensional wave equation, where T is the string tension and � is the massof the string per unit length.

The termination of the string in the two ends can be modeled as a reflectionof the traveling waves. For an ideal termination, all energy is reflected at allfrequencies. This not the case in musical instruments, nor is it desirable, sincesome energy must propagate into the instrument body in order to obtain audiblesound. Along with various forms of energy losses within the string, the propagationof energy into the sound box is an important cause of the string’s decay.

Under a linear idealization, the amount of energy transported into the sound boxis at all times proportional to the amount of energy traveling towards the termination.This leads to a decay which follows an exponentially shaped envelope.

8

2.1. STRING MODELS 9

2.1.2 Model refinement

A number of additional phenomena affect the behavior of the string:

Transverse polarization: The string can support transverse motion in more thanone plane. Under a linear approximation, the string’s motion can be decomposedinto two orthogonal planes. The string’s transverse motion in each plane can beregarded as independent of the other.

The effect of polarization is more complicated at the end supports, though.

yF

ydx

Idle stringtermination point

dx

F

Figure 2.1: Displacement of a string due to forces exerted on the string termination

A force in a single direction, Fx, may well displace the string termination ina direction which has both x and y components. The two string polarizations willtherefore be coupled.

This gives the decay envelope an oscillatory component, in addition to thegenerally exponential decay described earlier.

String stiffness: Even a string which is not stretched will to some degree resistbending. This is referred to as stiffness, and causes dispersion in the string.

High frequencies have shorter distance between their nodes and antinodeson the string, and therefore bend the string more than low frequencies at the sameamplitude. The string’s resistance against this bending causes high frequency com-ponents to travel faster along the string than low frequency components. Therefore,the eigen frequencies of a string are not totally harmonic, but instead

�n � np

1 � Bn2� �2�2�

where B is a constant determined by the string’s physical parameters.

Varying tension: When a string is plucked vigorously, its tension will be largerthan at rest. This will increase the frequency of all the string’s transverse modes.The phenomenon can be heard as a descending pitch of the tone as it decays, andthe string’s average tension approaches its idle value. The varying tension alsocouples the two transverse polarizations.

Longitudinal waves: In addition to the transverse waves discussed so far, a stringcan propagate waves in the longitudinal direction as well. This is the same kindof wave propagation one finds in a long bar or rod, which is struck at its end. Thevarying compression of the material propagates at some speed cl along the string or

10 CHAPTER 2. PLUCKED STRINGS

bar. This propagation velocity is much higher than the propagation velocity of thetransverse waves. Therefore the longitudinal waves give rise to their own harmonicseries, with a much higher fundamental frequency than for the transverse series.The longitudinal waves have no dispersion, and are not affected by the tension inthe string.

Since there is energy transfer between the transverse and longitudinal modes atthe string termination point, the longitudinal mode is important in certain musicalinstruments, even though it is not deliberately excited [ASK].

Torsional waves: Yet another polarization of waves can propagate along thestring. A rotation of the string will, generally, propagate with another velocity thanboth the transverse and longitudinal waves. Torsional waves contribute very littleto the sound of plucked string instruments, since they are weakly excited, looselycoupled to the other modes, and poorly transferred to the soundboard.

2.2 String — soundboard interaction

For many stringed instruments, the strings are approximately parallel to the sound-board, and a bridge is fixed to the soundboard to obtain physical connection tothe string. The admittance (velocity per force) of the string connection point isnormally much higher in the direction normal to the soundboard than in the otherdirections. For a guitar, this means that a string plucked normal to the soundboardwill transfer its energy faster to the soundboard than if it had been plucked parallelto the soundboard. This gives a shorter, but stronger tone. A string plucked atanother angle will decay with two rates, one quick decay rate associated with thecomponent normal to the soundboard and a slower one for the other component.More details and references are found in [LC].

String

��

Pin

Knot��

��

T

F

Fp

n

α

��

��

beamReinforcing

Soundboard

��

(b)(a)

Figure 2.2: String — soundboard connection in a harp.

No detailed treatment has been found on the string — soundboard interactionof the harp. The assumption which is implicit in [F2] is that the admittance parallelto the soundboard is so much lower than that normal to it, that almost all energy

2.2. STRING — SOUNDBOARD INTERACTION 11

is transferred by the component of the string’s oscillating force normal to thesoundboard, Fn in figure 2.2. This is determined by the tension and deflection ofthe string:

Fn � T sinΔ� � cos� � TΔ� cos�� 2�3�

where Δ� is the deviation from the idle angle � between the string and the sound-board.

The vibration of the string causes its length to vary slightly, which in turnschanges the tension in the string. This also contributes to the force component nor-mal to the soundboard. This contribution only contains energy at even-numberedpartial frequencies, since the length of the string must vary symmetrically in thetwo half-cycles of its vibration.

Chapter 3

Vibrating instrument body

3.1 Modal Behavior

The concepts of mode shapes, natural frequencies, and damping are described inmany sources concerning vibration of elastic structures, including [GR93], [DJE]and [FR91]. The following description is adapted from [BKST].

An FRF (frequency response function) measurement made on any structure willshow its response to be a series of peaks. The individual peaks are often sharp, withidentifiable centre-frequencies, indicating that they are resonances, each typical ofthe response of a single-degree-of-freedom (SDOF) structure. A system consistingof one mass, a spring and a damper is the typical mechanical SDOF structure. Ifthe broader peaks in the FRF are analyzed with increased frequency resolution,two or more resonances are usually found close together. The implication is that astructure behaves as if it is a set of SDOF substructures. This is the basis of a modalanalysis, through which the behavior of a structure can be analyzed by identifyingand evaluating all the resonances, or modes, in its response.

Let us begin with a review of how structural response can be represented indifferent domains. Through this we will be able to see how the modal descriptionrelates to descriptions in the spatial, time and frequency domains.

As our example, we will take the response of a bell, which is a lightly dampedstructure. When the bell is struck, it produces an acoustical response containing alimited number of pure tones. The associated vibration response has exactly thesame pattern, and the bell seems to store the energy from the impact and dissipateit by vibrating at particular discrete frequencies.

In the physical domain, the shape of the bell continuously changes, since itis vibrating. The complicated deformation shapes can be represented by a set ofsimpler, independent deflection patterns, or mode shapes (see [FR91], chapter 21).

In the time domain, the vibration or (acoustic) response of the bell can be shownas a time history, which can be represented by a set of decaying sinusoids.

In the frequency domain, analysis of the time signal gives us a spectrum con-taining a series of peaks, each of which can be represented by a simple resonance.

12

3.2. MODAL TESTING AND ANALYSIS 13

In the modal domain, we regard the response of the bell as a modal modelconstructed from a set of SDOF models. Since a mode shape is the pattern ofmovement for all the points on the structure at a modal frequency1, a single modalcoordinate q can be used to represent the entire movement contribution of eachmode.

To sum up, each SDOF model is associated with a frequency, a damping and amode shape. These are the modal parameters:

� modal frequency

� modal damping

� mode shape

which together form a complete description of the inherent dynamic characteristicsof the bell, and are constant whether the bell is ringing or not.

Modal analysis is the process of determining the modal parameters of a structurefor all modes in the frequency range of interest. The ultimate goal is to use theseparameters to construct a modal model of the response. Two observations worthnoting here are that:

� Any forced dynamic deflection of a structure can be represented as a weightedsum of its mode shapes.

� Each mode can be represented by an SDOF model

3.2 Modal testing and analysis

There exists a number of different methods to perform modal testing. Two mea-surement approaches are commonly used:

� One point (or a few points) on the structure are subjected to a known forcegenerated by a shaker or an asymmetric rotor, and the response is measuredon a large number of points, preferably including the point or points ofexcitation.

� The response at one or a few points is measured while the instrument issequentially excited by a known force at many different points, normally byan impact hammer (see appendix B).

Both methods give transfer functions from force to displacement (or force toacceleration or velocity, which can be integrated to give the desired functions):

ars�� xs��

Fr�� 3�1�

1For heavily damped structures, the influence of neighboring modes will affect the deformationshape even at a mode’s resonancefrequency. The deformation shape at a mode’s resonance frequencyand the mode’s deformation shape are not identical.

14 CHAPTER 3. VIBRATING INSTRUMENT BODY

These functions are then analyzed in order to extract the essential information.Within the theory of vibration (see [GR93], chapter 3), it is developed that fordiscrete spring-mass systems with viscous damping, the transfer functions ars��can all be described as a superposition of second-order resonators:

ars�� NXk�1

�Akrs

j� � �k�

Akrs

�

j� � ��k

�� 3�2�

The set of eigenvalues, �, is the same for all ars��, but the coefficients used tocombine the distinct resonators are, in general, different from point to point. Themodal deformation shapes (eigenvectors) are identical to theArs coefficients, apartfrom a normalization factor.

The specific task of the modal analysis is to extract, from the transfer functions,the following:

� The natural frequencies of the vibrational modes.

� The damping of each mode.

� The coefficients needed to reconstruct each transfer function from a super-position of resonators. These coefficients can also be interpreted as modaldeformation shapes.

For lightly damped structures or small structures, where the modal density is small,inspection of the transfer functions will tell the mode frequencies, and a simplecurve-fitting step will give the damping factor for each mode. A number of methodsexist, and there have been made several commercial products for solving this task.One of them, the Star Modal System, was briefly tested during this project, but theresults that are presented in this report were obtained using the following method,which was implemented in Matlab.

The measured transfer functions were processed twice. First to estimate themode frequencies and damping of each mode, and then to extract the modaldeformation shapes.

Matrix pencil ESPRIT method: First, the impulse responses were calculatedfrom each transfer function, in order to use the matrix pencil method. This is ahigh-resolution spectral analysis method which is based on the assumption that theanalyzed signal is composed of damped sinusoids. Given a maximum number ofsinusoidal components, this methodfinds an optimum combination of exponentiallydamped sinusoids to match the analyzed signal. Since the transfer functions areassumed to be superpositions of resonator response curves, the impulse responsesmust be superpositions of exponentially damped sinusoids.

Clustering: The matrix pencil method is applied to each measurement point,which in turn gives a complete set of eigenvalue estimations. Due to noise andimperfections in the model, these estimations will not be identical for all transferfunctions. If we have n measurement points, we may have up to n estimations ofeach eigenvalue. An average of these is used in the subsequent analysis.

3.2. MODAL TESTING AND ANALYSIS 15

X3 (ω) (ω)F/

X2 (ω) (ω)F/

X1 (ω) (ω)F/Decayingsinusoids

Decayingsinusoids

DecayingsinusoidsTRANSFER

FUNCTIONS

MATRIXPENCIL

METHOD

CALCULATEBASIS FUNCTIONS

Dam

ping

h(t)

h(t)

h(t)

IFFTCLUSTERING

Eigen-values

Basisfunctions

DECOMPOSITION(CURVE FITTING)

Modaldeformationshapes

Linearcomb.coeff.

Frequency

12

3

4

Figure 3.1: The signal processing steps used in the modal analysis

Decomposition: After determining the set of eigenvalues to use, each responsefunction can be decomposed into the basic resonators. Equation 3.2 can be refor-mulated as follows:

ars�� NXk�1

��Akrs�

�1

j� � �k�

1j� � ��k

��

NXk�1

��Akrs�

�j

j� � �k� j

j� � ��k

��

�3�3�Finding the factors ��Ak

rs� and ��Akrs� can be done by decomposing ars into

the functions 1j��k

� 1j��

k

and jj��k

� jj��

k

. The Matlab script listed in

appendix D.5 does this, minimizing the energy in the residual.

Chapter 4

Instrument body radiation

One of the intentions of this project was to verify experimentally the validity ofcertain models for instruments’ radiation. A thorough description of the theoryinvolved can be found in the papers by Alexis Le Pichon and Jean Laroche, forinstance [PL95].

In short, the air is regarded as a linear, lossless medium obeying the three-dimensional wave equation.

4.1 Plane surfaces

Rayleigh’s integral can be used to calculate the sound field radiated by infiniteplanar sources. These are characterized by a specified distribution of normalvelocity everywhere. If we set this velocity to zero outside some finite region ofvibration, the physical equivalent is a vibrating source molded into an infinitelylarge and rigid baffle.

An iterative technique using FFT has been developed to evaluate the acousticpressure near unbaffled plates [WIL]. The boundary conditions are mixed; weknow that the pressure will be zero in the plane of the vibrating plate, except for atthe plate itself. On the plate itself, we do not initially know the sound pressure, butwe suppose that we know, by measurements or calculations, the normal velocity ofthe plate, which is identical to the normal velocity of the air.

The method will initially calculate the “baffled” pressure, that is, the pressurewhich would have been generated by the plate, if it had been molded into abaffle. Then, the velocity of the air across the plane of the baffle is approached byiteration, upto any required precision. Numerical topics like stability and speed ofconvergence are discussed in [PL95].

4.2 Three-dimensional instruments

As all musical instruments are three-dimensional, some adaptation must be donein order to apply the theory for plane radiation to them.

Let us consider the guitar as an example. Most of the sound of a guitar isradiated from the soundboard (including the rose) and the back plate. This ismostly because the other parts of the guitar (the neck and ribs) are much more rigid

16

4.2. THREE-DIMENSIONAL INSTRUMENTS 17

vf

vb

~ 0~ 0

Figure 4.1: Surface velocity on a guitar

and hence vibrate less, but also because their surface area is smaller than that of thetwo plates. We therefore neglect the vibration of the neck and ribs, see figure 4.1.

Since the soundboard and backplate are as good as plane, we can hope to applyour methods for plane radiators to them.

4.2.1 Baffle model

(c)(a) (b)

v =0p=0

v

-v

f

f

fv

fvfv

=0

Front plateBaffle Unbaffled Unbaffled with rigid back

Figure 4.2: Plane models for guitar radiation

The model of the guitar molded into a baffle has been used in previous studies ofguitar radiation [RW87]. The physical analogy has been explained, but there existsanother imaginable, physical system which also would recreate the same pressure:Two identical plates vibrating back to back, in opposite phase. The air velocitywill be zero in the plane of symmetry.

Figure 4.2a, illustrates the baffle model. The baffled plate model is best infront of the guitar, at high frequencies, when diffraction around the guitar isless important, and little sound from the back of the guitar diffracts around theinstrument and radiates forwards.

4.2.2 Baffleless model

As mentioned, there exists a fast, iterative method for calculating the sound pressuregenerated by a plate in air.

18 CHAPTER 4. INSTRUMENT BODY RADIATION

The baffleless model is illustrated in figure 4.2b. Since the backside of thesoundboard is not exposed to free air, the model is quite inaccurate, and its radiationis not calculated other than as an intermediary step which is necessary to evaluatethe rigid-back model, which will be discussed next.

At low frequencies, an unbaffled plate cannot radiate sound because of theacoustical short-circuit. At high frequencies, the unbaffled radiation is close to thebaffled radiation, apart from the plane of the plate.

4.2.3 Rigid-back model

The third model for one vibrating face of an instrument is that of a vibrating platewith a rigid back. That is, a plate vibrating identically to the plate in the instrument,and the rest of the instrument replaced by a super-stiff plate which covers exactlythe backside of the vibrating face.

The radiation from the rigid back model is simply half the sum of the radiationfrom the baffled and unbaffled models, respectively. To understand this, it isenough to imagine the addition of the curves on the bottom of figure 4.3a and b.The topsides are in phase and will sum up to a velocity which is twice that of eachplate, while the backsides in the two models are out of phase and will cancel eachother. Figure 4.2c illustrates the rigid-back model.

4.2.4 Several faces

vb

fv

Superposition of both plates

Figure 4.3: Superposition of several planes

To take into account several vibrating faces, simple superposition of rigid-backmodels for each vibrating face may be used, which brings us to figure 4.3. Thisneglects the influence of one plate on the other plate’s radiation. That is, the soundfrom the soundboard passes straight through the backplate and vice versa.

Part II

Preliminary experiments

As a first step in the process of acquiring the desired measurement data, it wasnecessary to find appropriate methods of measurement, and to test them.

The specific types of measurements that were tested, can be divided into threegroups:

� Measurement of string vibration

� Measurement of instrument body vibration

� Measurement of instrument radiation

19

Chapter 5

String excitation

A method of exciting a string instrument, which clearly suggests itself, is to exciteits strings. The strings form very sharp filters, and because of certain nonlineareffects, the coupling to an external oscillator is not trivial. The use of these methodswill be to investigate the vibrational response of the instrument, and its radiation,to a form of excitation which resembles the natural excitation as much as possible,but which is more reproducible and stable.

5.1 Electromagnetic excitation

This method for applying a magnetic force to a string is described in [HAM]. Thegeneral setup is displayed in figure 5.1.

Permanentmagnets

Metalstring

Lm

Suspension

To amplifier

Figure 5.1: Test setup for magnetic excitation of metal string

A current is passed through the string, thus generating a force on the portion ofthe string which is affected by the magnetic field between the two magnets. Thisforce is given by

F �

Z�B � �Jdl� �5�1�

where �B is the magnetic flux intensity, and �J is the current density. Supposing thatthe magnetic flux is constant, equal to B between the magnets, and zero outside,the force will be

F � I � Lm �B �5�2�

where I is the current in the string, Lm is the length of the magnets and B is themagnetic flux. The force is directed vertically.

20

5.1. ELECTROMAGNETIC EXCITATION 21

F

B

Figure 5.2: The magnetic force on the string

Measurements: The resistance of the guitar strings that were used in the prelim-inary experiments was in the range of 1.2 Ω to 12.3 Ω. A nylon string painted withsilver paint was also tested. The resistance was on the order of 100 Ω, but the stringwas possible to excite the same way as the metal and wound strings. The currentwas supplied by a conventional audio amplifier, driven by a sine-wave generator.

The string is suspended on heavy metal weights, so little energy is radiated assound. A current in the proximity of 50 mA RMS was enough to generate wellvisible oscillations on a guitar E-string tuned to 80 Hz. Currents up to 500 mARMS were applied. In those cases, several effects that are not covered by thefirst-order linear model were observed:

� As the 64 cm long string was deflected more than 1 cm at its antinode,the increased average tension changed its resonance frequency. This isdescribed, among others, in [HAM] and [FR91], chapter 5.

� The increased temperature in the string, caused by the current applied to it,leads to a decrease in resonance frequency.

These effects should not affect our experiments as long as we make sure to useonly gentle excitation of the string, using a current the range of 1–10 mA RMS.

One problem was to sustain steady sinusoidal excitation with a constant drivingfrequency. This was probably due to the fact that the string, regarded as a filter,has a very high Q factor, while its center frequency might float slightly due totemperature changes and possibly also other external conditions. This problemwill be discussed later.

Excitation of a radiating string: In the first experiments, very little energy wastransported away from the string, since it was suspended on a steel frame withheavy weights on each end. To see if this method of excitation was capable ofgenerating enough energy to do sound field measurements, it was applied to astring on a classical guitar. A current of 5 mA RMS generated a weak, but wellaudible sound, which will probably be sufficient for our purposes. At this intensity,nonlinearities are not prominent.

To check roughly for the presence of nonlinearities, the sound pressure near theguitar was analyzed with a spectrum analyzer. The drive current was intended tobe purely sinusoidal, but spectral analysis revealed a second. harmonic which was45 dB below the fundamental. The strongest harmonic component of the soundpressure, namely the second, was measured to be 40 dB below the fundamental.

22 CHAPTER 5. STRING EXCITATION

5.1.1 Chaotic oscillations

An oscillating string is known to be a nonlinear system. It is not obvious that itwill stabilize when coupled to an external oscillator. Under certain conditions,this nonlinearity might give rise to chaotic behavior. This would be severe if thepurpose is to obtain steady excitation, so it had to be checked.

Experimental setup: An excitation signal was synthesized on a computer, andrecorded on a digital tape recorder. This signal was amplified and applied to ashaker which was connected to a guitar. The measurements (string current, bridgevelocity measured with laser vibrometer, and microphone output), were recordedon another digital tape recorder and transferred back to the computers for analysis.

1Ω

Control boxOFV-2600 Laser vibrometer

OFV-352 sensor head

Current shunt

Guitar

Computerworkstation

Microphone

Preamp.

Amp.

AlesisADAT

DAT

Figure 5.3: Experimental setup

A guitar string was excited with a group of 8 sinusoids around its fundamentalresonance (at 110 Hz). The excitation signal was started abruptly, and lasted for20.0 s. The velocity of the bridge was measured. Figure 5.4 shows the responseat the frequencies of the imposed sinusoids. The data for the figure was extractedusing Fourier transformation with rectangular windows that are 2.0 s long. Thereis therefore some inaccuracy in the analysis of the first portion of the signal wherethe amplitudes vary rapidly. The main point with the figure is to show that thesystem seems to stabilize. Note that some of the phase of some of the frequencycomponents stabilize nearwhile others stabilize near�. These are the frequencycomponents that are below and above resonance, respectively.

5.2 Modeling the string’s resonance

The same eight frequency components, plus another eight around each of the nextfour partials, were imparted to a string, and the response was used to deduce theresonance frequency and the Q factor of different partials.

Figure 5.5a illustrates such an excitation signal in the frequency domain, andfigure 5.5b indicates the kind of response one could expect.

5.2. MODELING THE STRING’S RESONANCE 23

0

50

100

150

200

250

300

350

400

0 2 4 6 8 10 12 14 16 18 20P

ower

s

109.5 Hz110.0 Hz

-4

-3

-2

-1

0

1

2

3

4

0 2 4 6 8 10 12 14 16 18 20

Rad

s

109.5 Hz110.0 Hz

Figure 5.4: A string’s response to a cluster of simultaneous sinusoids (107.5–111.0 Hz) — envelope of amplitudes (a) and phase of each sinusoid (b)

5.2.1 Experimental results

Figure 5.6 shows amplitude and phase response at different points on the guitarbody, using string excitation and a signal as described.

The imposed signal was harmonic, with period 2.0 seconds. The measurementsare done over one period of 2.0 seconds, after some time of stabilization. Notethat the three curves indicate the same resonance frequency and phase evolution,but with a certain, constant offset. This offset is caused by the difference inmeasurement positions, and is what we wish to measure.

5.2.2 String variations

As mentioned earlier, the string’s behavior will not necessarily be the same from thetime we measure velocity A to the time when we measure velocity B somewhereelse on the sound box. However, the variation of f0 might possibly be so smallthat the instrument’s response can be assumed to be constant over the range ofvariation, so that measurements for different points on the guitar can be used as ifthey were measured under equal conditions.

Imposing several frequency components in a very narrow band around thestring’s resonance frequency, it would be possible to estimate the string’s resonancefrequency f0 and the string’s Q factor. The response of the instrument can then be


Pow

er

Frequency

Pow

er

Frequency

(a)

(b)

Figure 5.5: An excitation signal (a) and a surface velocity (b)

split into what is the response of the instrument body and what is the response ofthe string. This requires a model which describes the response of the string.

Model for string resonance: The normal velocity measured at one point willbe determined partly by the string’s response to the applied current and partlyto the body’s response to the force exerted by the string. By calculating thestring’s contribution, this may be eliminated, so that only the instrument body’sresponse remains. This requires a model for the response of the string and thebody, respectively. Within the narrow band that is studied, one may model thecontribution of the body to be of constant amplitude and phase. The model usedfor the string’s resonance is that of a damped spring-mass system, so the productof the two responses then becomes:

eH�s� �b1s

s2 � a1s� a0ej� �5�3�

The desired data is the value of H�s� at resonance (s � jpa0), that is

A � eH�jpa0� �

b1

a1ej� �5�4�

Estimation of model parameters: The model parameters can be estimated byusing the minimum square sum law. That is, if we have measured H�s� at a fewfrequencies, si, we wish to minimize the energy in the difference between eH�si�and H�si�:

�nXi�1

�H�si�� eH�si�

�2�5�5�

The error is minimized by solving the following set of equations:

�

�a0� 0 �5�6�

5.2. MODELING THE STRING’S RESONANCE 25

-30

-25

-20

-15

-10

-5

0

5

10

107.5 108 108.5 109 109.5 110 110.5 111dB

Hz

At bridgeOn box, pos. #2On box, pos. #1

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

107.5 108 108.5 109 109.5 110 110.5 111

rad

Hz

At bridgeOn box, pos. #2On box, pos. #1

Figure 5.6: The response of string and body of a guitar, at three different points ofthe guitar case (power and phase)

�

�a1� 0 �5�7�

�

�b1� 0 �5�8�

�

�� 0 �5�9�

Practical realization and results: The estimation of model parameters was donewith the MATLAB signal processing toolbox. The model described here was notavailable, but a normal pole-zero filter of higher order (3 poles, 2 zeros) matchedthe measured data well. Figure 5.7 shows the fitting in amplitude and phase,respectively.

A series of measurements was done in order to see if the method could be used.A guitar string was de-tuned slightly from measurement to measurement to movethe resonance frequency. This should not affect the bridge velocity at resonance,since the frequency changes were very small compared to the typical bandwidthsof the guitar body’s modes. The results are apparent in figure 5.8.

It is evident that the method would need significant improvement to be of anyuse. The amplitude of one of the peaks (a) was 5 dB above the others, and the Qfactor of another (b) was much higher than the others. Neither should theoretically


-25

-20

-15

-10

-5

0

5

10

15

107 107.5 108 108.5 109 109.5 110 110.5 111 111.5

dB

Hz

Measured pointsInterpolation

-1.5

-1

-0.5

0

0.5

1

1.5

107 107.5 108 108.5 109 109.5 110 110.5 111 111.5

Rad

Hz

Measured pointsInterpolation

Figure 5.7: Model-based interpolation of measured data

have changed significantly. Unacceptable variations in the detected phase werealso observed.

5.3 Photodetector

A photodetector can be used to measure a string’s displacement at one point alongits length. The method is described in [HAM]. The idea is that the (infra-red)light which is emitted by the LED will be stopped more or less, depending on thedisplacement of the string. The light which passes is detected by a photosensitivedevice. See figure 5.9.

The sensor proves to be sensitive to very small displacements. The majordisadvantage is that the linear range is quite small.

Rough measurements: The linear range and sensitivity of the detector weremeasured in four series of measurements, all using the same string and supplyingthe sensor with a stabilized voltage of 5 V.

The dynamic range of the sensor is from 18 mV when the light is stopped bya piece of cardboard between the LED and the phototransistor, to approximately4.2 V, when the light is allowed to pass.

Figure 5.10 is a plot of sensor output voltage vs. vertical displacement of thestring.

5.3. PHOTODETECTOR 27

-35

-30

-25

-20

-15

-10

-5

0

5

10

15

107.5 108 108.5 109 109.5 110 110.5 111 111.5dB

Hz

b

a

Figure 5.8: Model-based interpolation of measured data — 8 different stringtensions

String

device Light sensitive

Light emitting device

Figure 5.9: The use of photoelectronic devices to measure a string’s displacement

Note the very sharp response of the sensor, compared to the width of the string,which is 1 mm. This might indicate that the phototransistor is saturated, and that aweaker light intensity would increase the linear range of the sensor.

To check the sensitivity to the string’s horizontal position within the detector,two series of similar measurements were taken, one with the string as close aspossible to the detector, and the other with the string close to the emitter. Theresults are displayed in fig. 5.11.

The differences that are might be explained by the uncertainty which existsin the measurement setup. Since the horizontal displacement was in the order of3–4 mm, and the effect was almost unnoticeable, one can safely conclude that thesensitivity is much larger in the vertical direction.

Fine-scale measurements: In order to measure the linearity and fine-scale re-sponse of the device, the displacement detector was located close to one end ofa string, while the string was dislocated at the middle. The displacement at thephotodetector was then calculated, assuming the string took the shape of a straightline. The results are displayed in fig. 5.12.

Only one half of the response curve was measured, and shows that the sensitivityof the device is about 10 V/mm in the region of about 3.5 V output voltage bias.The linearity can be quantified by calculating the second-order term in a curvewhich is fitted to the measurement data. Using three consecutive measured points(at 3.650 V, 3.526 V, and 3.413 V), one finds that the magnitude of the second-


3

3.2

3.4

3.6

3.8

4

4.2

4.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Sen

sor

outp

ut /

V

String displacement / mm

Measured values

Figure 5.10: The sensor output voltage from the displacement sensor vs. stringdisplacement

3.5

3.6

3.7

3.8

3.9

4

4.1

4.2

4.3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Sen

sor

outp

ut /

V


Close to the sensorClose to emitter

Figure 5.11: The output voltage from the displacement sensor vs. displacement, attwo different horizontal positions of the string

order component is about 36 V/m2. This means, for instance, that a sinusoidaldisplacement with an amplitude of 10 �m will have a 2nd harmonic which is 80 dBbelow the fundamental. At 100 �m amplitude, the 2nd harmonic will be 34 dBbelow the fundamental.

A quick test with another string indicated that the thickness of the string affectsall these parameters.

5.4 Magnetic sensor

The use of a magnetic field to excite the string is discussed in section 5.1. Theintroduction of a magnetic field also gives rise to another measurement option.

A string which moves perpendicular to a magnetic field will induce a voltagewhich is proportional to its velocity, the field strength, and the length of string

5.4. MAGNETIC SENSOR 29

3

3.2

3.4

3.6

3.8

4

4.2

0.1 0.15 0.2 0.25 0.3 0.35 0.4S

enso

r ou

tput

/ V


Measured values2. order approximation

Figure 5.12: The output voltage from the displacement sensor vs. displacement,fine scale

which is subjected to the field:

us � �v ��l � �B

��5�10�

B

v

Figure 5.13: A string moving in a magnetic field

We assume that the string can be modeled as a circuit consisting of a voltagesource (the induced voltage) in series with a resistor. The velocity of the string canthus be measured simply by amplifying the voltage across the string.

There is a practical problem involved, though. The voltage induced was, fortypical velocities and field strengths, on the order of 1–2 mV. The string is 64 cmlong, and a normal, 2-pole connection to an amplifier will necessarily make a loopwhich functions as a one- turn coil, collecting noise from the ambient, magneticfields.

Input -

Input +

GND

Figure 5.14: Differential measurement setup

This can be overcome by using an instrumentational amplifier for differential


measurements. An HP-132A oscilloscope was used for this purpose, since it hasdifferential inputs, contains low-noise amplifiers and outputs from each channel.

Each input (the inverting, and the non-inverting) are connected to one end ofthe string. Theoretically, there should be no need to ground the system, since weare only interested in measuring the difference between the voltage at the two endsof the string. In practice, the amplifiers are saturated by noise if we do not groundthe system. Where and how it was grounded, did not seem to be critical.

Linearity: It would be surprising if the measurement method was severely non-linear, since all components involved are highly linear, in contrast to the photo-electronical sensor. A source of nonlinearity might be a vertical variation in themagnetical field strength. This would lead to a velocity sensitivity that varies withthe string’s displacement.

No linearity measurements were performed on this sensor, since there was noreference to compare it with at large amplitudes. However, to check the linearityroughly, the following experiment was performed: According to basic theory onthe vibration of strings, any excitation of a string should lead to a velocity functionat the center of the string, which is odd, that is, containing only the odd-numberedpartials of the string. Any even-order nonlinearity in the measurement systemwould generate these.

The magnets were placed on the middle of the string, and the output of theamplifier was measured with a B&K 2032 spectrum analyzer. The string wasplucked with a finger. Measurements showed that the 2nd harmonic was 60 dBbelow the fundamental, suggesting a very high degree of linearity.

Sensitivity: The sensitivity of the magnetic sensor (V�s/m) was measured roughlyusing the optical sensor. The string was plucked with a finger, while the outputsfrom both sensors were monitored. The optical sensor was biased to 3.5 V, corre-sponding to 10.1 V/mm sensitivity (ref. 5.3). The magnetic sensor was placed atthe middle of the 64-cm-long string, while the optical sensor was 0.75 cm from onesuspension. When the oscillation was sinusoidal with a frequency of 72.3 Hz, andthe optical sensor gave a 0.10 V amplitude signal, the output from the magneticalsensor was 1.4 mV in amplitude. This corresponds to a sensitivityof 7.4 �106V�s/m.

Noise: The noise produced in this method is almost exclusively the 50-Hz humand its harmonics, which are radiated from electrical wiring and the power suppliesof all the equipment in the laboratory. For the test setup, it amounted to approxi-mately 5.4 �V. This can probably be reduced by increasing the distance betweenthe string and the equipment and other electrical wiring.

5.5 Wiener bridge

A Wiener bridge gives the opportunity to magnetically excite the string whilesimultaneously measuring its velocity by the induced voltage. This might at firstthough seem difficult, since the voltage across the string is also the voltage outputfrom the amplifier.

5.5. WIENER BRIDGE 31

Recall that the magnetic force on the string is proportional to the current passingthrough the string, while the string’s velocity induces a proportional voltage. Bymeasuring current and voltage simultaneously, one might control the force, whilemeasuring the velocity. This is essentially what is done in the measurement bridgein fig. 5.15.

{String

Current shunt

+-

+

s

c

s

b

a

ogu

u

R

R R

R

uA+

Figure 5.15: Measurement bridge for simultaneous excitation and measurement

Let us imagine that the string is not moving, and that it is electrically equivalentto a resistor. Then us � 0. Let ug be a constant, DC voltage. The differentialvoltage input to the amplifier is then

ug

�Rs

Rc � Rs

� Rb

Ra �Rb

��5�11�

This voltage will be zero ifRc

Rs

�Ra

Rb

�5�12�

The output of the amplifier will then be zero, regardless of what ug is. Now,let the voltage ug � 0, while the string is moving at some, constant velocity. Thenegative input to the amplifier will be constant at zero, while the positive input willbe

ugRc

Rc � Rs

�5�13�

The output of the amplifier is then

uo � A � ug Rc

Rc �Rs

�5�14�

Since the system only consists of linear components, the two solutions maybe combined, and we see that with a balanced bridge, the output of the amplifiershould be proportional to the velocity of the string, regardless of the generatorvoltage, and hence the applied force.


Experimental verification: The circuit in figure 5.15 was connected, and mea-surements were performed. The system worked as predicted, with one exception:Even when the bridge was balanced, there was still some residual of the appliedsignal ug in the output uo from the amplifier. This residual was 90� out of phasewith ug. The cause for this is probably that the string’s impedanceRs is not simplyreal, but contains an imaginary part. This would be an inductance rather thana capacitance. The inductance of a straight wire with the string’s length shouldbe negligible, but since the string is made of a wound metal wire, it probablyhas a higher inductance. This could be compensated by introducing a similar(proportional) inductance in series with the resistor Rb.

5.6 Self-oscillation

Since it is somewhat problematic to couple an oscillator to the string, this sectiondemonstrates how the string itself can be turned into a part of an oscillator.

Any oscillator consists, in general, of a resonance circuit with feedback. Someactive amplification is necessary to compensate for the inevitable energy loss inthe system. In addition, an nonlinearity is necessary to limit the amplitude of theoscillation.

5.6.1 Linear feedback

An interesting experiment may easily be performed with the circuit from 5.5. Ifthe output of the instrumentation amplifier is connected to the input of the drivingamplifier, it should be possible to sustain an oscillation on the string, or to amplifyit. This would give the string an extra “puff” downwards while it is movingdownwards, and upwards on its way up. If the effect in these “puffs” exceedsthe effect which is lost, the mechanical energy stored in the string will steadilyincrease. The system is sketched in figure 5.16. A low-pass filter was used to avoidelectronic self-oscillation at high frequencies.

Inst. amp.

Stringmeasuredvelocityforce

Applied

FilterPow. amp.

Figure 5.16: Self-oscillating string

The experiment was performed, and indeed the string oscillated. The ampli-fication in the loop could be adjusted. Below one particular amplification, thestring could be plucked, and the oscillation would slowly decrease and die. Withincreased amplification, the plucked string would instead oscillate stronger and

5.6. SELF-OSCILLATION 33

stronger, until it hit the mechanical suspension of the magnets. Even if the stringwas not plucked, it would begin to oscillate at its fundamental frequency from aninitially idle state. By carefully adjusting the amplification in the loop, steady,sinusoidal oscillations were sustained. The amplitude was prone to vary over time,though.

By limiting the amplitude of the oscillation electronically, one could obtain anexcitation with a steady, controlled amplitude, even though the frequency of thestring might drift slightly. This moves the limiting nonlinearity in the system fromthe mechanical part to the electronic part.

Experiments were also performed in order to make the string vibrate in othermodes than its fundamental. The filter was made to cut off the fundamental fre-quency, and the string was plucked. The experiment was only partially successful.That is, the string could be made to decay more slowly than normally. Increasingthe amplification in the loop beyond a certain point caused high-frequency oscilla-tions due to the inability to perfectly balance the measurement bridge, as describedearlier.

5.6.2 Nonlinear feedback

One might imagine that a sinusoidal oscillation, as obtained with linear feedback,is ideal for studying the vibration of a musical instrument. On the contrary, sincewe assume that the instrument behaves linearly, a non-sinusoidal oscillation couldbe decomposed into a superposition of sinusoidal oscillations, and give informationabout the instrument’s vibration and radiation at each frequency component.

We wish, therefore, to excite as many string modes as possible. We also wishto maintain constant amplitude over time. A solution would be to excite the stringwith short-duration pulses, synchronized with the string’s own response. Thespectrum of such a pulse train contains practically equal energy at each of the firstharmonics, provided that the pulses are short compared to the distance betweenthem. The harmonics are not guaranteed to hit the string’s partials, though, sinceall strings are slightly inharmonic. However, the damping (and bandwidth) of thestring’s modes increases with the mode number, so we might hope to excite evenhigher-order string modes if we only slightly miss their resonance frequency.

outputTrigger

outputGate

Normal

Inverted

DC equalizer

inputSignal

Burst

PulseinputTrigger

displacementSignal

Gate

0.1 ms - 30 scircuitTrigger

Gate time0.1 ms - 30 s

Adj. delay

Figure 5.17: Bruel & Kjær type 2972 Tape Signal Gate (fig. 2 from [BK2972])

To synchronize the pulse train with the string’s fundamental, one might usea trigger circuit, which detects a certain phase of the string’s oscillation, a delay


circuit which determines the phase of the pulse and another delay circuit whichdetermines the duration of each pulse.

By an incredible stroke of luck, it proved unnecessary to build this circuit, sinceit already existed in the laboratory. Unit type 2972 from Bruel & Kjær contains,among others, the described components exactly. The unit is a “Tape Signal Gate,”and is intended to cut away weak portions of a signal. Its block diagram is shownin figure 5.17

Using the photoelectic sensor described in section 5.3, the tape signal gate, apower amplifier and magnetic excitation, steady oscillation was obtained. Becauseof the strong nonlinearities involved in the electronics, the amplitude was easilycontrolled. Some combinations of delay time and pulse width gave unstable oscil-lation, where the amplitude of the different partials changed continuously, often ina cyclic manner, repeating itself after 3–4 seconds. With other configurations, theoscillation was simply harmonic, with a period identical to the string’s fundamentalperiod. Adjusting the pulse width and height changed the spectral distribution. Thestring could easily be brought to oscillate at one of its harmonics. Stopping it at itsmidpoint effectively damped the odd-numbered modes, doubling the fundamentalfrequency and leaving the string to oscillate at its even-numbered modes.

5.7 External reference signal

If we for some reason cannot measure the force exerted by the string on the boxand we cannot keep it constant, we need another reference which can be usedto normalize the individual velocity measurements. This reference needs to belinearly related to the force exerted by the string on the instrument.

For instance, the velocity of the different points can be related to the velocityjust at the point where the string is fixed to the box. If the string de-tunes, thiswill change the velocity (amplitude and phase) at both points. Assuming that theinstrument body behaves linearly, the relative velocity (amplitude and phase) willnevertheless stay the same.

Since only one laser vibrometer is at our disposal for the experiments, onewould have to do the other velocity measurement by use of an accelerometer.Placing the accelerometer at the connection point between the string and the boxhas the disadvantage of loading the instrument mechanically, and possibly affectingits acoustical properties. Using a sufficiently light accelerometer would probablysolve the problem for most practical cases.

In order to avoid any mechanical connection with the instrument body, onemight use the recorded sound pressure from a microphone at some fixed pointas reference. If we can neglect the sound radiation from the string itself, andassume that the instrument body, the air and the room are all linear, the soundpressure at some fixed point around the instrument is linearly related (i.e. througha convolution) to the force exerted by the string on the instrument body.

The external reference signal can possibly also be used in a feedback loopdriving the string current, but this option was not tested.

5.7. EXTERNAL REFERENCE SIGNAL 35

MeasurementsTension Level Phase difference(f0 / Hz) (dB) (rad)

T1 �8�2307 2�6675(109.15) �8�1383 2�6806

T2 �9�0035 2�6552(109.11) �8�9961 2�6518

T3 �8�7069 2�6547(109.67) �8�7887 2�6575

T4 �9�2278 2�6571(110.02) �9�3018 2�6400

T5 �10�0823 2�6234(110.45) �10�0910 2�6225

T6 �10�2246 2�6126(110.48) �10�4564 2�6176

T7 �10�2015 2�6372(110.00) �10�1357 2�6153

T8 �9�7407 2�6337(109.47) �9�8117 2�6576

Table 5.1: Experimental results using an external reference signal

Experimental results: The same series of measurements, which were used tomake figure 5.8, were used to evaluate this method. A microphone was placedclose to the instrument, and the sound pressure was recorded. Then, the complexratios between surface velocity and sound pressure were evaluated at each of theimposed frequencies, for each tension. For each tension, the ratio at the differentfrequencies was then averaged (weighted according to the energy) to obtain arelative amplitude and phase.

The experiments were done in a normal, non-anechoic laboratory. Severalinstruments were making noise, which was picked up by the microphone. Themicrophone was placed just outside the hole in the guitar box, which means that itwas very near the strings of the instrument. This is not an ideal position, since weonly want the measured pressure to be dependent on box vibrations.

To get an idea of how much of the variation was caused by ambient noise,the ratios were evaluated twice, using data from two consecutive intervals of2.0 seconds. The results are summarized in table 5.2.

Sources of error: Very little of the error can be contributed to noise (0.04 dB /0.006 rad). Some of the variation can be explained by the varying response of thebox over the range of the string’s de-tuning, since there is a correlation betweenthe estimated resonance frequency and the measured amplitude and phase ratios.Subtracting values determined from a linear regression line from both amplitudeand phase still leaves approximately 0.5 dB and 0.01 radians of standard deviation.


Standard deviation(caused by tension change)Level Phase difference(dB) (rad)

0�7223 0�0198

Standard deviation after subtractionof regression lines

Level Phase difference(dB) (rad)

0�5046 0�0120

Average difference betweenmeasurements at same tension

Level Phase difference(dB) (rad)

0�0396 0�0055

Table 5.2: Error magnitudes when using an external reference signal

5.8 String — soundboard interaction

No detailed studies were performed on the interaction and energy transfer betweenstrings and soundboards. One quick experiment was performed, though. A stringwas tied to a rod and stretched vertically by a weight. A thin wooden plate restedon a knot on the string and functioned as a soundboard. See figure 5.18.

��

��

��

��

loadTension

Soundboard

Inertial massstring

Guitar

Impacthammer

��

��

��

��

Accelerometer

Soundboard Knot

Figure 5.18: Experimental setup

The angle between the string and the soundboard could be altered withoutchanging any other parameters. At angles near 90�, the sound emitted by thesoundboard was very weak and bright. The sound power increased considerablyas the angle was decreased, and the relative content of low frequency componentsincreased. Quantitative measurements were also done, but not analyzed.

5.9. CONCLUSIONS 37

5.9 Conclusions

Nonlinear feedback offers the possibility of exciting a string at its resonance fre-quencies without perturbing the musical instrument it might be attached to. Thespectral energy distribution can be controlled, for instance by introducing a filterin the feedback loop.

In cases where it is difficult or undesirable to measure the string’s displacementwith a photodetector, a Wiener bridge may be used to detect the string’s velocity.Problems might be encountered because the string’s resistance varies with tem-perature and some strings have a considerable inductance. An external referencesignal could possibly be used in the feedback loop, but this was not tested.

If feedback is not an option, strings may be excited with a predetermined signal,containing energy in narrow bands around each of the string’s resonances. Thishas the disadvantage of requiring a longer acquisition time, if the purpose of themeasurements is to compare the instrument’s response at different points to thesame excitation.

If the string’s vibration is unknown, measurements of the response at differentpoints of the instruments may still be compared by relating each measurement toa known, external reference. This might be the response at one fixed point onthe instrument or even the sound pressure at a point which is fixed relative to theinstrument.

Chapter 6

Velocity distribution

For this project, it is necessary to measure the velocity distribution on a vibrat-ing surface. This section describes some methods for performing the necessarymeasurements, and ways to extract the relevant data from the measurements.

Practical aspects of some of the measurement techniques described here arediscussed in section 9.2, in the chapter on the Laplane guitar.

6.1 Devices

6.1.1 Accelerometer

An accelerometer is a piezoelectric device which outputs a charge proportionalto the acceleration along its axis. It is used together with a charge amplifier,optionally with an integrator, which gives an output voltage proportional to eitherthe acceleration or velocity of the accelerometer. The useful frequency band islimited by the charge amplifier in the low end, and by mechanical resonances inaccelerometer in the upper end. Anyway, the frequency range is large enough forthe measurements that have been done in this project.

The use of an accelerometer obviously requires that the device is attached tothe instrument, which will affect its mechanical properties. The magnitude of thismight be sufficiently small for our purposes, since very light devices are available(approx. 2 g). The ideal size of the accelerometer to use is determined by itssensitivity and its mass, which demands a compromise.

6.1.2 Laser vibrometer

Another method for measuring the velocity of a surface is called laser vibrometry.A laser beam is directed at the vibrating surface, and the Doppler shift of thereflected beam, caused by the movement of the surface, is detected. This shift isproportional to the velocity of the surface, in the direction of the laser beam. Fortechnical details, refer to [OFV].

The laser, the optics, and the detection hardware are all located in one singleunit, which is mounted on a camera post. A few practical considerations have tobe taken when using the laser vibrometer:

38

6.1. DEVICES 39

� Reflective tabs must be attached to the instrument at all points that are to bemeasured.

� Any movement of the sensor head is detected with the same sensitivity asthe measurement on the instrument.

� The FM demodulation involved in the detection circuit sometimes generatesspikes in the signal, see figure 9.6.

� The laser beam should be perpendicular to the surface in order to give thetrue normal velocity.

6.1.3 Gramophone pickup

A gramophone pickup is an electromagnetic device. In most pickups, the vibrationson the needle are transferred to a little magnet which moves relative to a coil, thusinducing a voltage proportional to the velocity of the needle. In other pickups, themagnet remains still, while the coil moves.

The mechanical perturbation caused by a gramophone pickup would be smallerthan the one caused by an accelerometer. It has the practical disadvantage of beingmore difficult to attach properly to the instrument, since the device must remainstill, while only the needle is allowed to touch the instrument.

If the output of the pickup could be properly filtered, and the pickup is suffi-ciently linear, one could hold the pickup by hand while performing the measure-ments, and simply move it from point to point. This would be as easy as movingthe accelerometer around, without its mechanical load. Quick experiments thatwere done, measuring the output from a gramophone pickup which was held inone hand, pressed lightly against a guitar which was excited with a shaker. Thesemeasurements showed that it would be necessary to use a more stable mountingof the pickup in order to obtain relyable results. This would probably be a lesspractical arrangement than using the laser vibrometer and reflective tabs, so nofurther tests were done with pickups.

Chapter 7

Measurement of pressure fields

The interesting features of an instrument’s radiation are those that affect the soundperceived by the audience and the player of the instrument. The radiation efficiencyof an instrument should preferably be as high as possible, while more subtle criteriagovern other characteristics of the radiation.

Normally, the audience is located near a horizontal plane, and are in the far-field of the instrument (where kr �� 1). This plane therefore has first priority inthe measurements. The player is usually the only one who is situated well insidethe near-field, but since his or her perception of the sound is very important in theperformance of music, this listening position might deserve some special attention.Sound that is likely to be reflected by the floor and walls will also be heard by theaudience, and should be studied.

7.1 Direct approach

The most direct approach to measuring the interesting parameters of an instrument’ssound radiation would be to place microphones at various typical listener positionsand excite the instrument manually. Since this excitation is not reproducible, itwould be necessary to use one microphone for each listener position and to recordthe sound from all microphones simultaneously in order to compare the soundradiation in different directions.

7.2 Turntable setup

To measure the far-field sound radiation in one plane, it could be sufficient toplace a microphone far from the instrument, and record the sound pressure asthe instrument is rotated. The precondition for using a far-field model is that theinstrument’s dimensions are small compared with the distance to the microphone.The anechoic chamber is about 5 by 5 meters, which limits our approach to thisideal.

This measurement method is very convenient, but in practice it requires steadystate excitation of the instrument. For plucked string instruments, this excludes thenormal form of excitation. Using another form of excitation might give the same

40

7.2. TURNTABLE SETUP 41

information, as long as the measurement data can be used to predict the soundproduced by plucking.

Since the laboratory has equipment to record several channels simultaneously,and several identical microphones, the radiation can be measured in several differentplanes at once (fig. 7.1).

d

φ

Figure 7.1: Measurement setup for far-field sound pressure

Chargeamp.

Microphones

Turntable

Computerworkstation

AlesisADAT

DAT

Studiedinstrument

Preamp.

Amp.

Force output

Shaker drive current

Shaker w/ imp. head

.

.

.

Figure 7.2: Measurement setup for far-field sound pressure

The type 2931 turntable from Bruel & Kjær has two two-pole connectionsbetween its rotating and inert parts. These can be used, respectively, to feed theexcitation signal to the instrument, and to return the reference signal to the taperecorder. The reference signal might be the force output from an impedance head,the displacement output from a photo-optical sensor, or the pressure measured bya microphone which follows the instrument around. A reference signal is alwaysconvenient, and absolutely necessary if the amplitude of excitation might vary, aswith some types of string excitation. See figure 7.2 for the wiring schema.

42 CHAPTER 7. MEASUREMENT OF PRESSURE FIELDS

7.2.1 Choice of excitation signal

Of the investigated methods, only shaker and string excitation provide steady states.Using string excitation, the possibilities of excitation signal are limited, and mustfollow the string’s resonance frequencies.

As for the velocity measurements, a shaker with wide band noise providesthe largest amount of data in a minimum of time. Repeating pseudo-randomsequences give better signal-to-noise ratio in shorter time than true random noise.This is important, since the time spent on each acquisition determines the angularresolution.

7.2.2 Extraction of data

The measured signal is split into intervals that are treated separately. The variousfrequency components can be acquired from each block. If the acquisition time issufficiently short, we can assume that the signal is locally periodic, with a periodequal to the length of the pseudo-random sequence.

Rectangular time windows gives minimal cross-talk between neighboring fre-quencies, but on the other hand, the cross-talk which does occur will be spreadover a wider band than it would have been, using a smooth window. Therefore, itis favorable to use a smooth window if the spacing between measured frequencycomponents is sufficiently large.

A problem which arises when trying to extract information about certain fre-quencies, is that some of the imposed frequency components have so much weakerresponse than the others, that the signal-to-noise ratio of these may become unac-ceptably low. If all frequency components are required to be measured with thesame accuracy, one must do a preliminary measurement of the frequency responsein order to synthesize an excitation signal which will generate approximately equalamplitude response at all frequencies.

7.2.3 Linearity

A necessary assumption behind the measurement technique which has just beenpresented, is linearity in the system. This includes the playback and amplificationof the excitation signal, the shaker, the musical instrument, the microphones,amplifiers and recording equipment. Figure 7.3 shows the average spectrum ofthe measured sound pressure from a guitar excited by a shaker over a 25-secondperiod. The imposed frequencies are the six prominent peaks. The peaks at 770 Hzand 880 Hz are caused by nonlinearities. Since these are more than 40 dB belowthe weakest of the imposed frequencies, the assumption of linear behavior seemsto be verified.

7.2.4 Angular resolution

The obtained angular resolution is given by the measurement interval and rotationperiod of the turntable:

Δ� � 2 m r

� �7�1�


20

40

60

80

100

120

0 200 400 600 800 1000 1200 1400 1600 1800 2000dB

Hz

Measured response

Figure 7.3: Microphone output when imposing six frequency components, showingnonlinearity

where m is the measurement period and r is the rotation time of the turntable.High angular resolution must be weighed against the disadvantages of short

measurement periods, namely decreased frequency resolution and / or decreasedsignal-to-noise ratio On the other hand, increasing the length of the measurementperiod will also increase the amount of signal variation within that period, andthereby also the cross-talk between measured frequencies.

In our case, the rotation time m is 80.0 seconds and invariable. Using ameasurement period of 1.0 second gives a maximal frequency resolution of 1.0 Hz,and an angular resolution of 4.5�, which has proved to be sufficient for the frequencyranges that were be studied, which was up to 4 kHz for the guitar.

Even with several hundred frequency components, acceptable signal-to-noiseratios have been achieved for almost all frequency components, which indicatesthat 1 second acquisition time is sufficiently long.

7.2.5 Graphical presentation

In the literature, acoustical directivity patterns are most often plotted in polardiagrams, with dB as the radius unit, see for example fig. 7.4b. Usually, thesegraphs are normalized, so that maximum response approaches the limits of thefigure, and there is a 30–40 dB range between the center of the circle and itsperiphery. This can be an illustrative representation, for comparing levels indifferent directions or different signals with large dynamic difference, as infig. 9.2.

Figures 7.4a-c represent the same radiation pattern, with a 14 dB differencein the reference level of each graph. Everything that can be read off one of thegraphs, can be read off the others, but the appearance of the graphs are dramaticallydifferent. There is nothing inherent in the represented information that suggeststhat one is more correct than the others.

A linear scale, as in figure 7.5, has the disadvantage of giving a lower dynamicrange. For instance, if figure 9.2 had been made with a linear scale instead ofa logarithmic one, the inner graph would disappear. However, there are twoadvantages that apply to the fairly unidirectional patterns that are presented in this


5

10

15

20

25 dB

(a)

10

20

30

40 dB

(b)

20

40

60 dB

(c)

Figure 7.4: Same radiation pattern (unitless, relative pressure), represented on dBscales with different reference levels.

report:

� The visual appearance of the radiation pattern will stay the same, regardlessof choice of reference. An increased amplification in the system only changesthe size of the figure, not its shape.

� The shape of the curve has a physical significance, in that the sound pressurewill be constant along a path with the shape of the curve, if it is entirely inthe far field of the instrument, as will be shown.

To the extent that the far field approximation is valid at the two distances r andr0 from the instrument, the pressure at the two points are related:

jp�� r�j� jp�� r0�jr0

r�7�2�

This equation can be inverted to find the surfaces of equal pressure, i.e. the points�� r� that all have p�� r� � p0. Since we only measure p�� r� for certaindiscrete values of � and �, we can only find the intersection of this surface andthose planes �� . These intersections will be at

r � r0jp�� r0�j

jp0j � �7�3�

where r0 and p0 are constants that can be ignored as long as the absolute magnitudeis not necessary. This implies that a radius linearly related to jp�� r0�j, is anequi-potential curve under the far field approximation.

The radiation patterns presented in this report are on a linear scale, unlessotherwise noted.


Figure 7.5: Same radiation pattern as in fig. 7.4, (unitless, relative pressure)represented with unitless linear scale

Part III

The Laplane guitar

A particular guitar, no. 18 of type 161 by instrument maker Joel Laplane, was theobject of many of the measurement types described earlier.

This guitar comprises three unique innovations:

� The strings are not terminated on the bridge,but on a sturdy,wooden structurewhich is connected mechanically to the neck of the guitar. The strings reston the bridge, however, so it is only the static tension in the strings whichis releaved from the bridge and soundboard. The intention is then to use athinner plate for the soundboard, to improve the transfer of energy from thestrings.

� The guitar case has two openings; one in the soundboard as for most guitars,and an additional one on the upper side of the guitar, facing the player. Thismight change the near-field directivity at low frequencies, giving the playeran improved impression of the sound.

� There is a soundpost near the main rose, connecting the front plate andthe back plate. According to the instrument maker, this should rigidify thecontour of the rose, increasing the acoustical efficiency.

The fan struts on the inside of the soundboard are asymetrically placed. For moreinformation on this guitar, see [ROS], [LMC] and [CH96].

47

Chapter 8

First setup

Several series of measurements were done. Some of the results are discussedin two articles: [CHLB] and [CH96]. The first series of measurements wasdone in order to complete one paper [CH96]. That paper attempts to objectivelyquantify the observations that musicians have made concerning the Laplane guitar.Furthermore, each of the three modifications mentioned initially were done andundone individually, to study their isolated effects.

The purpose of the measurements to be described in the current chapter, wasonly to study the influence of the second soundhole in the Laplane guitar, and con-sisted in measuring the sound radiation to five different positions (typical listenerand player positions), in an anechoic chamber, using five microphones. The guitarwas excited by manually plucking the strings and by imparting band-filtered noisethrough a shaker at the bridge.

A curved piece of wood, similar to that in the ribs of the guitar, was used toclose the extra soundhole. Beeswax was used to connect this piece of wood to theguitar. All measurements were done with the extra soundhole both opened andclosed, to examine the differences. The soundpost was always kept in place, andthe strings were fixed to the interior arm. The five microphones were placed asshown in figure 8.1.

4

M

1M

3M

M2

M

5

0.3 m

1.3 m

0.5 m

1.4 m

0.3 m

Figure 8.1: Placement of microphones relative to the guitar

Appendix C sums up all the measurements that were done using manual exci-tation. These results are also discussed in [CH96]. We see that even though thestrength of each pluck might vary by several dB (column labeled ref. level), therelative strength picked up by each microphone only varies about 1-2 dB. It seems

48

49

like more energy is picked up by the microphones when the extra rose is open,particularly microphone no. 2 and 5, which are outside the extra hole, and morefor the low-pitched tones than for the high-pitched ones. The magnitude of thisdifference is about 3 dB. The excitation by shaker also showed that the differencescaused by opening the extra hole were most marked near the hole, and at lowfrequencies. At higher frequencies, there were also differences, but only in narrowbands. The response was not increased at all frequencies by opening the extra rose.

The differences can probably be explained by two factors. First of all, theHelmholz resonance frequency will be increased when the extra hole is opened,since this increases the total size of the opening of the cavity. Secondly, the twoopenings will radiate approximately as monopoles. In the far field, the location ofthese (on the soundboard or on the side of the guitar) is insignificant, but has someeffect in the near field. The radiation from the hole in the guitar is most importantat low frequencies, near the Helmholz resonance frequency.

Chapter 9

Second setup

In the weeks that followed, the laboratory setup was changed, and several newseries of measurements were performed.

One of the main objectives was to verify the model of the guitar described insection 4.2.4. Therefore, both the directivity pattern and the velocity distributionof the guitar had to be measured.

� Measurement of velocity distribution and sound radiation in three differentplanes:

– The shape of the soundboard was measured and transferred to thecomputer network.

– A regular mesh (5 by 5 cm) of reflective tabs was applied to the guitar’ssoundboard (see figure 9.4).

– The velocity of the soundboard was measured at the reflective tabs,using a laser vibrometer.

– The sound pressure was measured in three different planes, see fig-ure 9.3.

The guitar was excited with a shaker at the bridge, using various types ofexcitation signals, including single frequency, several, discrete frequencies,narrow- and wide-band noise.

Measurements were done with different configurations of the guitar:

� Both roses open.

� Extra rose closed, main rose open.

� Both roses closed.

� Glass-wool pressed against the backplate of the guitar, to mimic the effectof a player’s body on the vibration.

The results of the surface velocity measurements were used to predict the soundpressure field around the guitar, using the model for non-baffled vibrating plates

50

51

with rigid backs. These predictions were then compared with the measured soundpressures, in order to test that model’s validity.

In a first series of measurements, only the vibration of the soundboard of theguitar was measured and used for calculating the directivity patterns. The extra rosewas closed and the measurements were performed with the main rose both openand closed. The backplate of the guitar was damped with glass wool, which waspressed between the guitar and the microphone post supporting it. Measurementsshowed that this glass wool had very little effect on the radiation from the guitar.

Only taking into account the radiation from the soundboard, the discrepancieswere large at some frequencies, indicating that other sources of radiation existed.Subsequently, the back plate’s vibration was measured and taken into account.During these last measurements, the extra rose was closed and the front rose wasleft open.

52 CHAPTER 9. SECOND SETUP

9.1 Soundhole velocity

There exist several methods for measurement of air velocity (at audible frequen-cies). The most common approach is the use of an intensity probe. This deviceconsists of two microphones with a well-defined spacing. The air velocity betweenthem can be deduced from the pressure gradient [JFD]. Since the guitar had to bereturned after only a few weeks, it was necessary to limit the time spent on mea-surements, and there was no time to explore methods of measuring air velocity.Two possibilities remained:

� Close the soundhole.

� Attempt to calculate the velocity of the air in the soundhole, using othermeasurements.

Since guitars are never played with closed soundhole, it would be favorable to becapable of calculating the soundhole air velocity.

The extra soundhole (on the side of the guitar) was closed. Measurements ofboth vibration and radiation were done with the main hole both open and closed,in case the model for air velocity in the hole was not accurate. Since the resultsobtained using it were satisfactory, the results presented in this chapter are basedon measurements with open soundhole and the following model.

This problem is treated, among others in [CAL] and in [CHR82], which is morefrequently referred to. Christensen’s article does not only concern the calculation ofair velocity, but aims at explaining the whole guitar’s behavior at low frequencies,where the vibration is dominated by the lowest front- and backplate mode. Hisfirst approximation, the “two-oscillator model,” is illustrated in figure 9.1.

kp

Sp mpm h

Sh

Vb = 0

pV Vh

Top plate Soundhole

Figure 9.1: Simplified two-oscillator model for guitar vibration at low frequencies

In his article, Christensen models the air flowing through the soundhole as apiston with a certain mass. The model does not take into account wave-propagationphenomena in the air enclosed by the guitar. It is therefore best suited for lowfrequencies, which is the region where the soundhole contribution is important[STR].

The volume change caused by the vibration of the front and back plates, causesa change in the air pressure in the guitar. This, in turn, generates a force on theair piston, which accelerates. The system of equations is given and solved inChristensen’s article, and leads to the following expression for air hole velocity:

Vh � �Vp �2h

��2h � �2� � i��h

SpSh

�9�1�

9.1. SOUNDHOLE VELOCITY 53

where �h is the Helmholtz resonance frequency and �h the damping coefficientof the air piston. Vp is the average normal velocity of the soundboard and thebackplate.

The Helmholz resonance frequency was measured from a mobility plot to be atabout 130 Hz. At that frequency, the mobility shows a marked dip. The dampingcoefficient of the air piston was set to an average value given in Christensen’sarticle.


9.2 Experimental setup and measurements

Two types of measurements were performed in this series of measurements, surfacevelocity and sound-field pressure. Although a shaker might possibly perturb the di-rectivity patterns, the guitar was excited with a B&K type 4810 shaker, mounted atthe bridge. A string could have been excited instead, but that is a much more com-plicated procedure, and only gives information around the resonance frequenciesof the string (see section 5.1).

To find the order of magnitude of the noise that is introduced, the followingexperiment was conducted:

The shaker was connected to the guitar, in a setup similar to the one shownin figure 9.3. The radiation pattern was measured, supplying the shaker with pinknoise. Then the shaker was detached from the guitar. The mechanical arrangementstayed the same, the only difference being that the mounting surface of the shakerwas separated from the guitar by a millimeter or so. The radiation pattern was thenmeasured for the same excitation signal.

10

20

30 dB

30

210

60

240

90

270

120

300

150

330

180 0

Shaker connectedShaker disconnected

Figure 9.2: Magnitude of noise from shaker

The magnitudes found in this experiment are likely to be inaccurate, though.The force exerted by the shaker on the guitar also works back at the shaker, and willmake it vibrate more when connected to the guitar than when it is released fromit. The ratio of 20 dB, which appears in figure 9.2, must therefore be regarded asa lower bound estimate. The shaker was mounted on the rear of the guitar duringthese measurements.

The shaker was equipped with a B&K type 8001 impedance head. For the sur-face velocity measurements, an OFV-2600 laser vibrometer was used. The soundpressure measurements were done with three Shoeps type MK2G microphones,while the guitar was mounted vertically on a B&K type 3921 turntable. The stringswere damped, and the whole setup was done in an anechoic chamber, see figure 9.3.

9.2. EXPERIMENTAL SETUP AND MEASUREMENTS 55

=229.0 cm=13.5 cm

Rr

Microphones

d

=62.0 cm=100.0 cm

R

h

Shaker

Force transducer outputShaker current input

d

h

r

Turntable

Guitar

Figure 9.3: Setup for far-field sound pressure measurements

9.2.1 Excitation signal

A wide-band excitation signal was used, in order to get information about theguitar’s vibration and radiation at many frequencies, all in one measurement. Aperiodic (pseudo-random) excitation signal was used. Compared to random, non-periodic excitation, this has the advantage of higher speed and the disadvantageof not detecting nonlinearities. The latter will be detected as dips in the coher-ence function which is calculated along with transfer functions based on randomexcitation. An estimation of the noise level can also be done with pseudo-randomexcitation, but nonlinearities will not be detected.

The choices of period and spectral energy distribution were done by the fol-lowing criteria. A too short period would give a too large spacing between themeasured frequency components. This could lead to poor detection of narrow-banded resonances. Simply tapping the guitar with a finger and listening, indicatedthat all modes were sufficiently damped to fall back to rest within 1/3 second. Thismeans that the bandwidths of all modes were larger than 3 Hz. Thus, a 1/3 secondperiod was used. A too long period would give poor angular resolution (see thenext section) and would require longer measurement times for the surface velocity.

Next, the frequency span had to be chosen. From the mobility plot, it seemedthat the lowest mode’s resonance was well over 60 Hz, which was chosen as thelower limit. The bandwidth was traded against the signal- to noise ratio, and theupper frequency limit was set to 1 kHz, since the mode shapes become rathercomplicated at higher frequencies, and would require a very dense sampling of thesoundboard velocity distribution.

A white energy distribution was used, that is, equal amplitude of each frequencycomponent. The phase of the frequency components were randomized in order to


distribute energy more evenly over time.

9.2.2 Sound-field measurement

It was the objective to measure the radiation from the guitar in as many directionsas possible. An efficient way to do this, was to mount the guitar and shaker on aturntable, while the microphones remained fixed to a post. The sound radiated bythe guitar was measured along three circular orbits around the guitar, using threemicrophones simultaneously. The microphones were mounted one meter apart on atall, vertical beam (see Fig. 9.3). The turntable completes a rotation in 80 seconds.Since we need 1/3 second for each acquisition, we obtain an angular resolution of1.5�. The distance from the axis of the turntable to the microphones was 229 cm.For practical reasons, the center of the guitar could not be mounted on the axis ofthe turntable. The exact orbit of the microphones relative to the plates of the guitarwas taken into account when predicting the sound-field.

Signal processing

The extraction of the relevant data was done on computer workstations. The mi-crophone signals were Fourier transformed (using the FFT) in order to extract eachsingle frequency component. Since the signals were nearly periodic, a rectangulartime window with length equal to the period of the excitation signal was used topartition the signal.

9.2.3 Velocity measurement

The geometry of the guitar was measured and transferred to the computer network.A regular square grid was used as a basis for the points to be measured. On thefront plate, a few points had to be dislocated from the grid for practical reasons (seeFig. 9.4). The point of excitation and the point that fell in the soundhole, could notbe measured with the laser vibrometer, and were omitted.

On the back plate, the regular 5x5 cm grid was used without modification.

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Shadow behind shaker

Location of main rose

2

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1

5

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7

94

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46

47

48

49

50

51

52

53

5429

37

12

26

17

Figure 9.4: The mesh used for velocity measurements on the guitar soundboard


Hand-supported laser

When the laser vibrometer is fixed to a camera stand, it gives a fair signal to noiseratio over its frequency range, which is from practically DC to about 100 kHz.However, it would be very time-consuming to point the laser and fix the camerastand for every single point. Therefore, the possibility of not fixing the camerastand was tested. This means that the laser sensor is supported in one hand, andcan be tilted freely.

Two types of noise are then introduced. Firstly, since the laser shakes slightlyas the operator’s hand shakes, the relative velocity between the guitar and the sensoris perturbed, and low frequency noise is added. See figure 9.5a-b.

The excitation signal which makes the guitar vibrate, is a single sinusoid at220 Hz. The shaking of the hand can be seen as a slight deviation around thezero-line. This noise may be filtered away, and we appear to get about 60 dB signalto noise ratio (see fig. 9.5b), using an average over 0.25 seconds of the signal.

0 0.05 0.1 0.15 0.2 0.25s

20

30

40

50

60

70

80

90

100

0 200 400 600 800 1000 1200 1400 1600 1800 2000

dB

Hz

(a) (b)

Figure 9.5: Measured velocity of a point on the guitar case, using hand-supportedlaser vibrometer — some low-frequency noise due to shaking by the hand

The second source of noise is the FM demodulation involved in detecting thevelocity. This occasionally causes strong spikes in the signal, particularly whenthe measurement conditions are somewhat unstable, at large signal amplitudes, orwhen the laser beam does not hit a reflective tab properly. Figure 9.6a-b shows anexample of this, in the time and frequency domains.

Synchronization

The measured data is recorded as a continuous stream on a digital audio tape, andthere will be need for some sort of synchronization to associate the correct velocitywith the correct point.

The order of measurement must be determined. To be sure of the position ofthe measurement of each point on the tape, there are two possibilities:

� Synchronize the beginning of the measurement series, and spend a predeter-mined amount of time on measuring each point


0 0.05 0.1 0.15 0.2 0.25s

20

30

40

50

60

70

80

90

100

0 200 400 600 800 1000 1200 1400 1600 1800 2000

dB

Hz

(a) (b)

Figure 9.6: Measured velocity of a point on the guitar case, using hand-supportedlaser vibrometer — low-frequency + wide-band noise due to excessive shaking bythe hand

� Mark the measurement of each point as events on the tape.

The advantage of the first approach, is that it is the simplest. Event markingcould be done in many ways, using separate tracks, or manipulating the signal onthe tracks that are already used. It would be favorable to use only two tracks, sincemost recording equipment can handle two tracks. Both tracks must be used torecord essential measurement data, namely the reference (excitation) signal, andthe measured velocity. One practical possibility would be to insert a switch to cutone of the signals off while moving the laser from one point to another.

Practical test: In the first experiments, a stopwatch was used as a metronome,and only the beginning of the measurement series was synchronized, and this wasdone manually. The laser was directed at each point for a period of 2.4 seconds,and another 2.4 seconds were allowed for moving the laser between to points. Thedisadvantage of this approach, was that some points were just barely hit during the2.4 second measurement period, while others were found well before that periodstarted, and could have been finished in a shorter period. The 54 point measurementseries took 4 min 20 sec. This was not too exhausting, but a 200 point series wouldtake more than 15 minutes, and without the possibility of rest, it would certainlybe a great strain.

Therefore, a foot switch for synchronizing the measurements was included inthe setup, releaving and speeding up the process.

Signal processing

The cost of a quick and simple measurement procedure was a fairly complicateddata extraction process. This, however, was done automatically.

The most problematic of the two noise sources was obviously the wide-bandpeaks that occurred during the FM demodulation. The remedy to this problemwas to split the measurement signal from each point into a number of windows,


and extracting data only from the windows of highest “quality”. The quality ofeach window was evaluated by comparing the amount of energy at the excitationfrequencies to the overall amount of energy. For the good portions of the signal,most of the energy is at the frequencies of the excitation signal, while portions withnoise will have energy also at other frequencies.

The best windows were used to form an average velocity amplitude and phasefor each frequency and each point. For most points, 5–6 windows were used toform this average, depending on the time spent on measuring that particular point,and the “quality” of the signal.

The force exerted by the shaker was measured by the force transducer in theimpedance head. This signal was recorded along with the velocity signal. This wasused as a reference signal, making it possible to compare amplitudes and phasesacross measurement points (see section 5.7)

Recall that a wide-band excitation signal was used during the measurements.This way, each point had to be measured only once, during a short period of 2–3seconds.

The signal-to-noise ratio was estimated by regarding the variation between thewindows that were used to form the average. If n windows were used, and theamplitude for one particular frequency component was measured to bex1� x2 � � �xn,then the standard deviation in the average (x) is estimated to bevuut 1

n�n� 1�

nXi�1

(xi � x)2 �9�2�

The overall signal-to-noise ratio found this way was about 55 dB.

Interpolation

The next step in the processing of the measured data was interpolation and gridding.A cubic spline method which had already been implemented in Matlab was used[DTS]. Initially, the velocity was set to zero on the edge of the soundboard.This is only correct under the assumption that the ribs are stiff and immobile.Investigating the results clearly showed that the ribs were not at rest, particularlynot at low frequencies (below 80 Hz), where the whole guitar body was movingmore or less rigidly, probably caused by a mode of vibration where the neck isbending.

During later calculations, the ribs were assumed to be stiff but mobile. Theboundary points of the assumed were then supposed to be on a plane which touchesmeasurement points 47, 53 and the midpoint of the line between points 1 and 5(see figure 9.4). Not surprisingly, the effect on the calculated directivity patternwas unnoticeable. Even with the refined model of the rib’s movement, they stayalmost immobile at higher frequencies, and the velocity of the soundboard is lownear the edges anyway.

After interpolation, the velocity of the air piston in the rose of the guitar wascalculated (as described in section 9.1), and inserted into the grid, see figure 9.7b.The data could then, finally, be used by the numerical methods for calculation ofradiation.


(b)(a)

Bottom plate

Top plate

Figure 9.7: (a) Example of one measurement (at 441 Hz), (b) The output afterinterpolation

Measured vibration

The vibrational measurements consist of a real and an imaginary part, for 314 fre-quencies (60 Hz, 63 Hz � � � 999 Hz), for 106 points. It is impossible, and hardlyinteresting, to visualize all this in graphs and figures. Instead, interesting featurescan be extracted from this data base. An example will be given in the next section,concerning modal analysis based on this data set.

The shapes shown in figure 9.8 are samples of measured deformation, at someresonance frequencies. The gridpoints represent the actual measurements, whilethe lines between them are based on interpolation. The edge of the guitar, in itsidle position, is superimposed on the figure.


−9.315 dB63 Hz

−12.67 dB81 Hz

0 dB102 Hz

−17.54 dB165 Hz

−0.1902 dB198 Hz

−2.617 dB204 Hz

−7.557 dB240 Hz

−9.456 dB273 Hz

−1.565 dB342 Hz

−5.317 dB348 Hz

−14.28 dB432 Hz

−11.26 dB690 Hz

Figure 9.8: Simulated snapshots of guitar deformation at single frequencies, ex-tracted from wide-band excitation and measurements


9.3 Radiation field calculation

The methods for calculating the sound radiation from the guitar are describedbriefly in chapter 4.

9.3.1 Some results

The measured velocity distribution was used to predict the radiation pattern, usingthree different models, each of which can be compared with the measured radiationpattern. The models that are compared, are

1. Baffled front plate model.

2. Unbaffled front plate with rigid back.

3. Rigid-back model for both plates, superpositioning front and back plateradiation.

(a)

549 Hz219 Hz

(b)

Figure 9.9: Calculated and measured sound pressure. – – – Baffle model.— Rigid-back model. � � �Measured data.

Under normal playing conditions, the back plate vibration may be damped bythe body of the player, and much of the sound radiated by the back plate will beabsorbed by the player. In our setup, the guitar was hanging freely, so the backplate vibrated and radiated significantly.

� At some frequencies, the back plate vibration is much less important thanthe vibration of the front plate, including the air flowing through the rose. At219 Hz, for instance, the average amplitude of vibration of the front plate androse is approximately 20 times that of the back plate vibration. Therefore,the contribution of the back plate to the radiation pattern is feeble at thisfrequency (see Fig. 9.9a). The rigid-back model for the front plate gives

9.3. RADIATION FIELD CALCULATION 63

579 Hz

(a)

579 Hz

(b)

Figure 9.10: Calculated and measured sound pressure. (a) — Front plateradiation. – – – Back plate radiation. (b) — Superposition of the two.� � �Measured data.

approximately the same result as the rigid-back model for both plates. Notethat both these models were evaluated, and both are plotted with solid linesin the figure.

� At higher frequencies, the difference between the baffled plate model andthe rigid-back model diminishes. Figs. 9.9b and 9.11a are examples of this.Directly in front of the guitar, both models give fair results as long as theback plate vibration is weak.

� Figure 9.10 illustrates the significance of the back plate’s radiation. Theradiation from the front plate and the back plate was calculated individually,using the rigid-back model for both. Fig. 9.10a shows the calculated radiationfrom each of the plates individually. The two sound pressure functions werethen superimposed, and Fig. 9.10b shows the superposition of the two. Recallthat the sound pressure is complex-valued, so an addition of two radiationcurves does not necessarily increase the sound pressure in all directions.

9.3.2 Sources of error

The discrepancies that do exist between the theoretical results and measurements,may be due to:

The numerical model: The convergence of the FFT-based method is not guar-anteed at low frequencies. Details concerning this numerical problem aregiven in [PL95]. However, at these frequencies, the guitar’s radiation ismainly that of a monopole, and is therefore not too interesting to comparewith measurements.


(b)(a)

327 Hz501 Hz

Figure 9.11: Calculated and measured sound pressure. – – – Baffle model.— Rigid-back model. � � �Measured data.

The interpolation process: Different interpolation methods (linear interpolation,low-pass filtering, bicubic splines) were tested, and the choice of methodslightly affected the result. Different assumptions regarding the velocityaround the edge of the guitar were tested, but did not affect the result signif-icantly.

Air velocity in the rose: The air velocity in the soundhole was not measured,and the simple model which was used may be inaccurate, at least at highfrequencies, where wave propagation inside the sound box is important.

Noise from the shaker: The shaker itself vibrates and thereby radiates some sound.To estimate the amount of noise radiated by the shaker, it was disconnectedfrom the guitar, and the sound level was measured. On average, this levelwas 20 dB weaker than with the guitar connected to the shaker. However,the forces that make the box of the shaker vibrate include the forces exertedon the guitar, so the shaker makes more noise when connected to the guitar.

Diffraction around the shaker and suspensions: Probably insignificant since thedimensions of these structures are small compared with the acoustical wave-lengths in the studied frequency region.

Other sources of radiation: The contribution to the radiation from the side andneck of the guitar has been neglected. It is probably weak, since the amplitudeof vibration is small at these portions of the guitar, and their area is relativelysmall.

9.4. NEAR-FIELD INTENSITY 65

9.4 Near-field intensity

An instrument’s player is located in the near-field of its radiation. The near-fieldradiation therefore determines the sound that he or she perceives. Studies havebeen done on the near-field radiation of other instruments, using intensity probesand microphone arrays [STR].

Using the developed radiation model, it is possible to calculate the near-fieldintensity. The sound pressure can be calculated in a dense mesh on the plane ofinterest, and the velocity can be deduced from the Euler relation [JFD]:

�u � � 1j��

rp �9�3�

by using a first order finite difference approximation to the differentiation operator.Examples of results obtained this way is illustrated in figure 9.12. The figures showthe calculated active (real part) intensity. The guitar vibration is illustrated in eachfigure. On the left, the frequency of vibration is close to the Helmholz resonance.The air velocity in the soundhole is so much greater than the surface normal velocityon the soundboard that the latter seems negligible. Consequently, the radiation fromthe soundhole dominates the near-field intensity. Note the acoustical short-circuitabove the guitar at 576 Hz. No verification has been done of the accuracy of theradiation model in the near field, and one must expect a considerable error close tothe ribs of the guitar, which have been neglected completely in model. Near eachof the plates, away from the boundaries, the model should be accurate, since nocrude approximations have been done there.

f = 150 Hz f = 576 Hz

Figure 9.12: Near-field intensity at two different frequencies. The shape of theguitar’s vibration is also illustrated. Left: Intensity on a plane which intersects thesoundboard at the main rose. Right: On a plane normal to the soundboard withintersection at the lower boat of the guitar.


9.5 Conclusion

The measurement techniques that have been developed seem to work according totheir intention. The assumption that the backplate of the guitar does not contributesignificantly to the radiation, was shown to be inaccurate for this guitar. Althoughthere exists differences between the calculated and measured radiation, the accu-racy of the radiation model is probably good enough for the purposes of soundfield regeneration and for investigating and evaluating the quality and quantity ofradiation from non-existing, simulated instruments.

Chapter 10

Modal analysis

Using the procedures previously described, the measured transfer functions, fromforce on the bridge to displacement of the soundboard and back plate, were usedto do a modal analysis of the guitar as one single structure.

The measurements were only done in order to calculate the radiation fromthe guitar. The modal analysis was done afterwards, as an additional utilizationof measurement data. The most important changes that could have made themeasurements more suited for modal analysis would be:

� Place the shaker asymmetrically, to avoid antisymmetric modes’ modelines.

� Measure the response at the excitation point as well as all the other points,so that the modes can be normalized.

� Avoid coupling to other systems with resonances in the studied frequencyrange.

10.1 Extraction of eigenvalues

The matrix pencil method was used to estimate the natural frequencies and dampingof the modes in the range of 60 Hz – 900 Hz. The algorithm was run on all 106measured points. For each point, up to 45 decaying sinusoids were allowed. Thatmakes close to 4770 complex values, which are plotted in figure 10.1.

The natural frequency and damping of each mode must be the same all overthe guitar. One can clearly see how the estimations from different measurementpoints coincide and form clusters in figure 10.1a. A similar clustering is notfound in figure 10.1b, since the amplitude of the sinusoids varies from point topoint. However, one can clearly see that the points of high amplitude are very wellconcentrated in vertical bands around what is assumed to be the natural frequenciesof the various modes.

Figures similar to 10.1 were made for the soundboard and the back plateindividually. Both showed clustering of points around the same values.

The damping shows an increasing trend with frequency. Note that a large,negative value implies a quick recession, i.e. large damping. The unit used fordamping is s�1, and is the decay rate of the exponential envelope of the free regime

67

68 CHAPTER 10. MODAL ANALYSIS

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Figure 10.1: (a) Damping and (b) amplitude estimated for 106 measurement points

oscillation. The frequency is more conveniently presented in Hz than in angularfrequency, although the latter corresponds closer to the time constant chosen asdamping unit. Together, angular frequency and damping form the imaginary andreal part of each mode’s eigenvalue, respectively.

Using statistical clustering, the number of points (represented in fig. 10.1a)was progressively reduced to about 50. Finally, a manual selection of 38 pointswas done. These are drawn as circles on fig. 10.1a, and were used as eigenvaluesduring the extraction of the mode shapes.

10.2 Reconstruction of transfer functions

The next step in the modal analysis is to extract the eigenvectors. This was done asexplained in section 3.2. As explained, this is done by reconstructing the measuredtransfer functions as a superposition of basic resonator transfer functions, usingthe frequencies and dampings found in the first step. A typical example of sucha reconstruction is shown in figure 10.2. Observe that there is a particularitydiscrepancy between the reconstructed and measured function in three regions:

� At high frequencies (above 950 Hz)

� In the dips of the transfer functions

10.2. RECONSTRUCTION OF TRANSFER FUNCTIONS 69

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0 100 200 300 400 500 600 700 800 900 1000−4

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rad

Figure 10.2: Transfer function force – displacement at one point. — Measuredfunction – – – Reconstructed function

� In the range of approximately 490–570 Hz.

These errors occur in most of the reconstructed transfer functions. Figure 10.3shows the average error in the reconstructed amplitude, as a function of frequency.

0 100 200 300 400 500 600 700 800 900 10000

1

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5

6

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10

Frequency / Hz

Dev

iati

on /

dB

Figure 10.3: Average error at different frequencies in reconstruction of transferfunctions, – – – using 38 modes, — after adding another 2 modes

No modes above 1 kHz were taken into account. Approaching this frequency,the transfer functions will inevitably contain contributions from modes with reso-nance frequencies above 1 kHz. This would explain the rise in error above about950 Hz.

There are two factors that might explain the discrepancy in the reconstructionof the transfer functions in the “dips”, that is, where the amplitude of the transferfunction is low. The signal-to-noise ratio in the measurements is lower in theseregions. However, the errors are much too large to be explained by noise. Moreimportant is the fact that the curve-fitting which is used aims at fitting the peaksrather than the dips. The curve-fitting minimizes the error, which is measured bysumming up the energy of the difference over all frequencies. A 5 dB discrepancyin a dip will therefore count far less than 5 dB discrepancy on a peak, since the lattercontains much more energy. On top of this, the matrix pencil method can only take


into account viscous damping. Other forms of damping reduces the accuracy ofthe fitted curve.

The third region with particularly large errors, around 500 Hz, turned out tobe two vibrational modes that had been left out during the manual pruning ofeigenvalues. Another two eigenvalues, between number 22 and 23 (see fig. 10.1)were inserted. This effectively solved the problem, which can be seen fromfigure 10.3.

10.3 The phases of the modes

The mode shapes were complex valued. Thus, each point is assigned an amplitudeand an phase for each mode.

For lossless structures, each mode will be such that all points vibrate in phaseor antiphase. At the same time, their displacement must be in phase with theapplied force. Otherwise, there would be a net flow of energy. If the mode hadbeen lossless, this would have given rise to an ever increasing amplitude.

In reality, there is a finite damping. Therefore, not all points are in phase orantiphase, and they do not move in phase with the applied force. The energy flowis necessary to sustain vibration in the guitar.

Mode no. 39Mode no. 12 Mode no. 15

Figure 10.4: Amplitude and phase (compared to applied force) of the displacementfor the measured points on the guitar: � points on the soundboard, points on thebackplate.

As we can see from figure 10.4, all points on the soundboard and all points onthe backplate seem to be moving approximately in phase or antiphase. For modeno. 12, the back plate seems to be moving much more than the soundboard, whichwill be displayed more clearly in the next section. At least for mode no. 15, it isclear that there is a phase difference between the back plate and the soundboard.This indicates that the coupling between the plates is looser than within them,which is quite likely.

For most modes, the variation in phase within the measured points is largerthan for the three modes in figure 10.4.

10.4. MODE SHAPES 71

10.4 Mode shapes

The extracted mode shapes are displayed in figures 10.5–10.8, sorted by theirstrength relative to the strongest mode. The strength is defined as the square sumof displacement at all measurement points, divided by excitation force.

The three strongest modes all look like (0,0) modes. The splitting into threeresonance frequencies is in good accordance with the three-oscillator model of theguitar’s vibration. The resonance frequencies of this mode was found to be 102 Hz,197 Hz and 205 Hz. The difference in shape is mainly that the back plate, moves inopposite phase in thefirst resonance, compared with the two others. The movementof the back plate is small compared with that of the soundboard. Another differenceis that the deformation shape of the soundboard becomes somewhat unsymmetric atthe higher resonances, which could be caused by the different coupling conditionsto the cavity combined with the asymmetric fan strutting.

Mode no. 1 at 63 Hz seems to be a neck mode, but since the movement of theneck was not measured, this remains uncertain. Mode no. 2 at 80 Hz, which is13 dB weaker than no. 3, seems to have the same shape as the three (0,0) modes,but no mention of such a (0,0) resonance has been found in the literature on guitars.The low resonance frequency and prescence of strong rib movement in this mode,compared to top plate movement, might indicate that it is also a mode of neckvibration rather than a true top-plate mode.

The next top-plate mode, with (0,1) shape, is found as no. 15, 16 and 17, at315 Hz, 342 Hz and 348 Hz. The back plate is moving quite much in this mode,which indicates a strong coupling. The shape of backplate vibration is differentat each of the three resonances. The splitting of this mode might be due to thecoupling of the (0,1) modes of the soundboard and the back plate, and the secondair cavity resonance.

Next, the (1,0) soundboard mode is found at 239 Hz (mode no. 10). A mys-terious twin version of this mode is found at 164 Hz (mode no. 5), but this one is12 dB weaker. Along with its neighboring modes, no. 4 and 6, it lies between thetwo lowest (0,0) resonance frequencies. Figure 9.8 shows that the vibration in thisregion is more unsymmetric than at the (0,0) resonance frequencies. The modalanalysis interprets this as a superposition of almost symmetric (0,0) modes with avery unsymmetric mode (no. 5), which can not be excited alone by a sinusoidalforce at one point.

The (0,2) soundboard mode is found at 443 Hz (no. 20). No proper (1,1)mode can be found from the extracted mode shapes, which might be because theshaker was placed on the crossing point of this mode’s nodal lines. Mode no. 25 at624 Hz is a (0,3) soundboard mode, no. 22 at 528 Hz and no. 39 at 546 Hz can becharacterized as (2,0) modes. Mode no. 29 at 709 Hz is a (3,0) mode.

The backplate is harder than the soundboard, so the mode frequencies arehigher. Some of the recognizable mode shapes are: A (0,0) mode at 272 Hz(no. 12), a (0,1) mode at 315 Hz (no. 15), (0,2) modes at 342 Hz (no. 16), 348 Hz(no. 17), 423 Hz (no. 19) and 495 Hz (no. 21), a (2,4) mode at 951 Hz (no. 37),a (1,2) mode at 691 Hz (no. 28), a (1,1) mode for the whole backplate at 578 Hz(no. 23) and a (1,1) mode for the lower half at 709 Hz (no. 29), and a (0,3) modeat 600 Hz (no. 24).


A number of additional, weak modes were found. Since the matrix pencilmethod is very accurate in detecting decaying sinusoids, resonances in other me-chanical systems that are weakly coupled to the guitar will also be detected. Forinstance, all the modes of resonance of the neck of the guitar will be detected eventhough the vibration of the neck itself is not measured. Except for the simplestcases, the corresponding mode shapes of the soundboard and backplate will bedifficult to interpret, since they themselves are not the origin of the resonance.Other coupled resonators, like the microphone post that the guitar is hanging from,will also be detected, but will give low deformation amplitudes of unrecognizableshape.

10.5 Conclusion

The modal analysis has given valuable information about the guitar’s functionwithout requiring extra measurements. The shapes and frequencies of the modesare in accordance with known results on other guitars. The coupling between thetwo plates might be stronger than in other guitars due to the soundpost. For thelowest mode, where the soundboard and back plate modes in phase, the soundpostmight suppress the movement of the back plate.

The methods that have been chosen and their implementation work satisfactory,but care must be taken in the interpretation of the results. In particular, the boundaryconditions of the vibrating structure and its coupling to other resonant systems affectthe results by introducing modes that are not associated with the guitar itself.

10.5. CONCLUSION 73

0 dB 102.1 Hz, −13.96 /sMode no. 3

−1.814 dB 196.7 Hz, −25.66 /sMode no. 7

−5.648 dB 205.2 Hz, −34.59 /sMode no. 8

−9.802 dB 315.1 Hz, −15.66 /sMode no. 15

−10.54 dB 63.09 Hz, −8.042 /sMode no. 1

−11.8 dB 341.8 Hz, −16.62 /sMode no. 16

−11.99 dB 239.4 Hz, −21.77 /sMode no. 10

−12.51 dB 554.1 Hz, −20 /sMode no. 40

−13.1 dB 545.5 Hz, −21.79 /sMode no. 39

−13.45 dB 80.48 Hz, −13.72 /sMode no. 2

−14.43 dB 951.3 Hz, −36.11 /sMode no. 37

−16.33 dB 348.1 Hz, −29.44 /sMode no. 17

Figure 10.5: Some of the extracted mode shapes.


−16.41 dB 272 Hz, −18.26 /sMode no. 12

−17.01 dB 709.4 Hz, −37.8 /sMode no. 29

−17.1 dB 965 Hz, −45.82 /sMode no. 38

−17.4 dB 423.2 Hz, −29.9 /sMode no. 19

−18.55 dB 624.4 Hz, −34.53 /sMode no. 25

−18.92 dB 494.7 Hz, −24.61 /sMode no. 21

−19.46 dB 528.6 Hz, −32.58 /sMode no. 22

−20.27 dB 885.3 Hz, −42.14 /sMode no. 34

−20.88 dB 661.3 Hz, −36.47 /sMode no. 27

−22.32 dB 691.2 Hz, −29.11 /sMode no. 28

−22.72 dB 577.8 Hz, −22.93 /sMode no. 23

−23.03 dB 929.7 Hz, −44.31 /sMode no. 36


10.5. CONCLUSION 75

−23.54 dB 442.8 Hz, −23.93 /sMode no. 20

−23.66 dB 163.8 Hz, −21.46 /sMode no. 5

−23.77 dB 900.9 Hz, −40.36 /sMode no. 35

−25.24 dB 749.8 Hz, −31.18 /sMode no. 30

−28.42 dB 226.8 Hz, −17.07 /sMode no. 9

−28.75 dB 289.6 Hz, −20.28 /sMode no. 13

−29.25 dB 600.3 Hz, −23.47 /sMode no. 24

−29.91 dB 247.5 Hz, −18.42 /sMode no. 11

−30.14 dB 796.4 Hz, −36.37 /sMode no. 32

−30.43 dB 383.7 Hz, −30.86 /sMode no. 18

−30.52 dB 843.2 Hz, −39.1 /sMode no. 33

−31.04 dB 306 Hz, −14.84 /sMode no. 14



−31.62 dB 139.8 Hz, −11.34 /sMode no. 4

−32.3 dB 780.1 Hz, −25.97 /sMode no. 31

−32.52 dB 175.1 Hz, −10.6 /sMode no. 6

−32.78 dB 651 Hz, −23.56 /sMode no. 26


Part IV

The Celtic harp

A small Celtic harp from Aoyama Harp, Fukui, Japan was available for measure-ments. The harp is about 1 m tall, and has 34 strings. The deepest has a fundamentalfrequency of 65 Hz, and the brightest is at 1661 Hz. Each string’s pitch can beshifted up one semitone while playing by the use of a lever. The soundboard is25 cm wide at the lower end and only 5 cm at the top. On the visible side ofthe soundboard, the fibers run along the soundboard. Seen from the backside, thefibers run across the soundboard, which is evidently made of laminated wood. Thesound box has approximately the cross-section of a half circle which is joined withthe soundboard. There are four holes in the sound box, and another one in thesocket of the harp. These holes are necessary to gain access to the strings from theback side of the harp, in order to change them when they break.

77

Chapter 11

Measurements

We wish to find the vibrational modes of the harp, and to investigate the acousticalimportance of the sound box and the holes in it.

11.1 Vibrational measurements

Loosely based on preliminary measurements (see appendix A) and with goodsafety margins, the grid density was set to 4 cm. The harp’s geometry was fed intoa computer, and a mesh was constructed, giving approximately constant spacingbetween neighboring measurement points, while at the same time covering the mostimportant boundaries of the instrument. This process resulted in 199 measurementpoints on the soundboard, and 242 on the sound box. Another 12 points werespread evenly along the string suspension arm.

A suitable, plane projection of this mesh was then printed out on a laser printer,enlarged and glued to the instrument. Reflective tabs were placed on the instrumentthrough holes that had been cut in the paper at the appropriate locations.

The harp was then placed on its feet on a turntable together with a shaker. Theshaker was suspended by a steel beam, perpendicular to its direction of impact. Theother end of this steel beam was connected to the turntable through a joint, so as toavoid unmeasured forces to propagate to the harp. The shaker was connected to theharp through an impedance head, which was fastened to a point on the soundboardwith wax. All measurements were done using a pseudo-random excitation signalwith white energy distribution in the range of 60–3000 Hz, and with a period of1/3 s.

All strings on the harp where left on, at full tension, but thoroughly damped.The level of the excitation was augmented until unlinearities gave audible effects(rattling etc.), and then lowered until these sounds disappeared, with a margin.Each point was measured over a period of 2–3 seconds. The order which they weremeasured in was noted on a piece of paper. The harp had to be turned a little in orderto obtain a line-of-sight between the laser and each point on the soundboard. Formost points, this alteration of angle was fairly small, and was therefore overseen insubsequent data analysis. Only for a few points near the bottom of the soundboard,close to the arm, was the angle so sharp that special measures were taken. Theassumption was that the signal from the laser vibrometer was proportional to sin �,

78

11.2. INTENSITY MEASUREMENTS 79

where � is the angle between the laser beam and the surface. Measuring theseangles would be a tedious process, though. Instead, the few points that weremeasured at sharp angles were “normalized” by a factor determined by measuringa neighboring point at both angles. The point of excitation was measured byusing the accelerometer in the impedance head. To be on the safe side, this hadbeen calibrated against the laser vibrometer, by placing a reflective tab directly onthe impedance head and comparing the output of the two sensors on a spectrumanalyzer. The points just next to the excitation point were measured with anaccelerometer.

The harp was turned little by little when measuring the backside of the harp, inorder to keep the laser beam in the same plane as the surface’s normal vector. Theangle between the laser beam and the surface was then calculated for each point,both on the soundboard and behind the harp, and all measurements were scaledby 1� sin�. As mentioned, small horizontal displacement angles of the laser weredisregarded.

The quality of each measurement was evaluated, and those points that did notfulfill the quality requirements were re-measured. Of the 453 points, this amountedto about 20 points, mostly at boundaries, where the signal was weak. Of these,another 3 points had to be measured a third time. The quality requirement was thatthe average standard deviation in the transfer function between applied force andmeasured velocity should not exceed 5 % for any point in the frequency range of60–1500 Hz.

The order in which the points were measured was fed into the computer, soeach measured transfer function was related to a point with a known location. Thelocation of the points that were measured is shown in figure 11.1. The shaker wasmounted at point no. 120.

The average magnitudes of vibration on the sound box and the soundboard areshown in figures 11.2 and 11.3, respectively. These and other results are discussedlater.

11.2 Intensity measurements

The sound intensity in each of the four holes on the backside of the harp wasmeasured with a B&K 3519 sound intensity probe. For each of the holes in thesound box, the intensity was recorded at three locations; in the center, 2 cm fromthe bottom and 2 cm from the top. Only the component of the intensity in thedirection normal to the surface of the harp was measured.

11.3 Radiation measurements

The same setup as for the guitar, described in section 9.2.2, was used. Sevenmicrophones were spread out with 20-cm spacing along a vertical beam.

Because of the mechanical arrangement in the anechoic room, the turntablehad to be mounted on a wooden plate, which vibrated when the harp was excitedwith the shaker. Therefore, plates of hard foam rubber were placed between theharp and the turntable, which seemed to solve the problem.

80 CHAPTER 11. MEASUREMENTS

200201202203204

205 206207208209

210211 212213214215

216217 218219220221

222223 224225226227 228229230 231232233234 235236 237238 239240241242 243244 245246

247248249250 251252 253254

255256257258 259260 261262

263264265 266267 268269 270271

272273274275 276277 278279 280281

282283 284285 286287 288289 290291292293 294295 296297 298299 300301302303 304305 306307308 309 310311312313 314315316 317318 319320 321322323324 325326 327328 329 330331332 333334335336 337338 339340 341342 343344 345346

347348 349350 351352 353354 355356 357358359360 361362 363364 365366 367368 369370 371372 373374 375376 377378 379380 381382 383384 385386 387388 389390 391392 393394 395396 397398 399400 401402 403404 405406 407408 409410 411412 413414 415416 417418 419420 421422 423424425 426427 428429 430431 432433 434435 436437

438439440441

1 2 3 4

5

6 78 9101112 1314 151617 181920 212223 2425 26272829 3031 323334 3536 37 38 3940

4142 4344 45 46 474849 505152 53 54 555657 5859 60 61 626364 656667 68 69 707172 7374 75 76 777879 808182 83 84 8586 878889 90 91 929394 959697 98 99

100101

102

103104

105

106

107108

109

110111

112

113

114

115116

117118119120121

122123124125

126

127128

129

130

131

132

133

134

135

136137

138

139

140

141

142

143

144

145146147

148

149

150

151

152

153

154

155156

157

158

159

160

161

162

163

164165

166

167

168

169

170

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173174175

176

177

178

179

180

181

182

183184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

Backside of the harp Soundboard

Figure 11.1: Location of measurement points. The numbers were used for refer-encing while processing the measurements. The spots along the middle line on thesoundboard represent the string fixation points.

The directivity patterns were measured with the harp in its normal configuration,with all holes closed and with different combinations of holes closed, to check theirinfluence on the harp’s radiation.

Directivity patterns were also measured with the harp in its normal configura-tion, using electromagnetic excitation of one of the strings, namely the brightestof the metal strings, as described in section 5.6.2. The displacement of the stringclose to its termination point in the soundboard was measured and recorded duringthe experiment. The shaker was removed from the instrument during this measure-ment. It was done to allow a future assessment of the shaker’s influence on theradiation pattern, after an adequate radiation model has been developed.

Figure 11.5 shows the average pressure picked up by the seven microphonesover one rotation of the harp, an approximative measure of radiated power.

The peak at 150 Hz was found to be the second harmonic of the power supplyhum, picked up by a microphone preamplifier which was placed directly under thepower amplifier. Inspection of the directivity patterns at the neighboring frequencybands indicate that only the frequency band at 150 Hz is affected.

11.3. RADIATION MEASUREMENTS 81

0 500 1000 1500 2000 2500 3000−75

−70

−65

−60

−55

−50

−45

−40

−35

Frequency / Hz

Mea

n ve

loci

ty m

agni

tude

/ dB

447

72

774

753

732

657

687 81

092

794

5 1008

129

555

597

537

483

624

111

147

156

267

183

198

177

333

Figure 11.2: The measured velocity of all points on the sound box squared andaveraged. Reference level is 1 m/s�N. Numbers are peak frequencies, in Hz. Thefrequency resolution is 3 Hz.

0 500 1000 1500 2000 2500 3000−75

−70

−65

−60

−55

−50

−45

−40

−35

Frequency / Hz

Mea

n ve

loci

ty m

agni

tude

/ dB

105

111

177

183

195

270

333

366

447

555

537

597

684

705

807

927

945

1008

657

852

144

153

129

Figure 11.3: The measured velocity of all points on the soundboard squared andaveraged. Reference level is 1 m/s�N. Numbers are peak frequencies, in Hz. Thefrequency resolution is 3 Hz.

82 CHAPTER 11. MEASUREMENTS

0 500 1000 1500 2000 2500 3000−70

−65

−60

−55

−50

−45

−40

−35

−30

−25

Frequency / Hz

Mea

n ve

loci

ty m

agni

tude

/ dB

1005

228

174

336

363

432

495

437

594

654

549

612

678

705 85

281

3

915

Figure 11.4: The calculated normal air velocity of all four holes in the sound boxsquared and averaged. Reference level is 1 m/s�N. Numbers are peak frequencies,in Hz. The frequency resolution is 3 Hz.

0 500 1000 1500 2000 2500 3000−120

−110

−100

−90

−80

−70

−60

Frequency / Hz

Mea

n pr

essu

re /

dB

333

552

180 22

815

016

513

812

3

438

366

702

681

591

657

1005

948

915

861

810

537

Figure 11.5: The average sound pressure over all seven microphones, and allangles of rotation. Reference level is 1 Pa/N. Numbers are peak frequencies, in Hz.The frequency resolution is 3 Hz.

Chapter 12

Data analysis

12.1 Modal analysis

12.1.1 Eigenvalue extraction

As the total number of measurement points was very large (453), it was foundunnecessary to use all of them for the determination of resonance frequencies anddamping. One out of five points were used. Only the frequency range 60 Hz –1 kHz was studied, and gave the values shown in fig. 12.1.

Some of the clusters were less marked than for the guitar, and the dampingwas generally larger. These phenomena may be related. Recall that the impulseresponse of each point is calculated, and forms the basis for estimating the eigen-values. Highly damped modes lead to damped sinusoids that diminish and drownin noise very quickly. Therefore, heavily damped modes will be less accuratelydetected by the matrix pencil method.

Manual clustering

Another method of clustering than the one used for the guitar was tested. Since itseemed inevitable to do a manual selection, a method which relied more directlyon manual interaction was chosen.

A figure similar to 12.1 was drawn on the computer screen. Various regions ofthe figure could be zoomed in on, in order to get a good view of each cluster. Then,a rectangular region of the figure was marked. All points that fell within this regionwere then counted, to check how likely it is that these points were actually causedby the presence of a mode of vibration. At the same time, a check was done on howmany of the measurement points gave rise to more than one eigenvalue estimationinside the marked region. If most of the measurement points had two distinctestimations within the region, that indicated that the points within the region weremore likely to indicate two modes than one. If most of the measurement pointshave exactly one estimation within a region, it is likely that the average of theestimations within the region form a good estimation for an eigenvalue. Using thismethod, 33 eigenvalues were selected.

83

84 CHAPTER 12. DATA ANALYSIS

0 100 200 300 400 500 600 700 800 900 1000−200

−150

−100

−50

0

Frequency / Hz

Dam

ping

/ 1/

s

0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

Frequency / Hz

Rel

ativ

e am

plitu

de

Figure 12.1: Damping and amplitude estimated for 89 measurement points, selected eigenvalues

12.1.2 Eigenvector extraction

The subsequent decomposition and recomposition of transfer functions into simpleresonators gave acceptably good results in the frequency range 200 Hz – 1 kHz, ascan be seen from figure 12.2, which is the average error over all 441 measurementpoints on the soundboard and sound box.

There was some variation in the quality of the modal decomposition from pointto point. Particularly at and around the point of excitation, there were seriousproblems. Figure 12.3 shows this clearly, and the region around the excitationpoint stands out in many of the extracted modes’ shapes.

It is also evident that there are other points, especially on the backside of theharp (index � 199), where the reconstruction is fairly poor. These are generallypoints along the edges of the harp, where the measured signal was very weak, and afairly large error is neither unexpected, nor serious. The modal analysis still showsclearly the main property of these points, that they stay almost immobile.

The errors around the point of excitation are much more serious, since that isright on the middle of the soundboard, which is crucial to the sound productionof the instrument. The response at most of these points was measured with anaccelerometer, since they were in the shadow of the shaker, so the laser could notbe used. The sensitivity of the devices is different, and one might imagine that

12.1. MODAL ANALYSIS 85

0 200 400 600 800 10000

1

2

3

4

5

6

7

8

Frequency / Hz

Mea

n er

ror

/ dB

Figure 12.2: Average error in reconstruction of transfer functions

there could be some error in the scaling of the different measurements. However,the devices had been calibrated against each other, and were in perfect accordanceover the whole frequency range. An error in the scaling would give an equallylarge misfit at all frequencies, which is not the case. Another explanation might bethat the soundboard has significant movement in directions other than the normaldirection. The accelerometer only measures the surface normal component of themovement, while the laser beam was at a certain angle to the soundboard, anddetected other components as well. None of these possibilities can explain whythe modal reconstruction of the response curves was of lower quality in this region,though.

For the excitation point itself, the response was measured with the accelerom-eter in the impedance head. This measurement is affected by the non-rigidity ofthe force transducer and housing of the impedance head, which could explain theerrors at the excitation point, but similar errors are found at the neighboring points,which were measured with the laser vibrometer and an accelerometer.

A possible explanation might lie in nonlinear behavior of the mechanical struc-ture. It is obvious that the linearity assumptions that are inherent in the modalmodel are only first-order approximations, and that deviations from this model willoccur at large amplitudes of vibration. This would indicate that a too large forcehad been used to excite the instrument. This is a trade-off, though, since a lowerexcitation force would give larger measurement errors due to noise.


0 50 100 150 200 250 300 350 4000

0.5

1

1.5

2

2.5

3

3.5

4

Measurement point index

Mea

n er

ror

/ dB

pointExcitation

Figure 12.3: Average error in reconstruction of transfer function

12.1.3 Mode frequencies

In [FR91], section 11.2, the mode frequency is found to be related to the modenumber of the soundboard by the linear relation

fn � n � 103 Hz� �12�1�

This is for a free soundboard of a small harp, though with a central bar. For harp inthis study, with sound box and strings, a similar relationship has been found for thefrequency range below 1 kHz. Figure 12.4 shows the mode frequencies of someof the modes, related to the number of lobes along the soundboard. The averageincrease in mode frequency seems to be about 160 Hz from one mode to the next.

12.2 Input admittance

For the energy transfer between strings and soundboard, the ratio between thestring’s characteristic impedance and the soundboard’s input impedance is veryimportant. The string’s characteristic impedance can be calculated by measuring itslength, tension and fundamental frequency. The input impedance of the soundboardis less simple to measure.

In papers by Firth [F1][F2], such measurements are presented. Using a min-ishaker and a mechanical arrangement for moving it along the midline of thesoundboard, he could measure the input impedance at the stringholes in a sound-board of a harp. This did, however, require the removal of all the strings and thearm, which was not an option in this case.

Based on a modal analysis, the input impedance can be calculated anywhere onthe instrument, since the modal model predicts the movement of all points on the

12.2. INPUT ADMITTANCE 87

instrument due to any given distribution of forces everywhere on the instrument.For one single frequency, we have [PD95]:

bx�t�c �NXk�1

�bzkcbzkct

�j� � �k��k��bzkcbzkct��j� � ��k��k

�bscej�t� �12�2�

where bx�t�c is the displacement vector, bsc is the force vector, and bzkc are theeigenvectors. In our case, we wish to find the amplitude and phase of responsein point s due to a unitary, sinusoidal excitation at the same point, which is thecompliance of that point (displacement vs. force):

Cs�� NXk�1

�zk�s�zk�s�

�j� � �k��k�

�zk�s�zk�s��

�j� � ��k��k

�� 12�3�

The unknowns in this equation are zk�s�zk�s��k. These can be calculated from thecoefficients Ars calculated in the modal analysis (see section 3.2):

Akrs �

zk�r�zk�s�

�k� �12�4�

For the point of excitation r, we have:

Akrr �

zk�r�zk�r�

�k� �12�5�

Combining equations 12.4 and 12.5, we find

zk�s�zk�s�

�k�

AkrsA

krs

Akrr

�12�6�

The velocity was measured at points between each string hole in the harp,and the input admittance of the soundboard was calculated along this line, and ispresented in figure 12.5, in the same fashion as presented by Firth. Along one axisis the frequency, the other axis represents the string number along the midline ofthe soundboard.

Following Firth’s form of presentation, circles have been placed at the locationand fundamental frequency of each string. In addition, points have been placedat the first five harmonics of each string. Note that each string can be de-tunedby a semitone. Therefore, the location of each circle and point is somewhatambiguous. The exact location of the circles on the graph is that of a harp tuned toa logarithmically equally spaced scale with 7 intervals per octave, which makes asort of compromise.

The admittance is based on the modal analysis, and is normalized by theadmittance which was measured at the point of excitation. The quality of themodal analysis was poor below 200 Hz (see fig. 12.2, and for a broader region forthe point of excitation (fig. 12.3). Therefore, one might suspect that there is someinaccuracy in the normalization in the whole frequency range, and that the resultsare generally unreliable below 200 Hz.

To check the accuracy of this mehtod, two methods could be suitable:


� The shaker could be mounted at a point on the midline of the soundboard,to measure the admittance directly and compare with the calculated results.This would require the removal of a number of strings. Alternatively, theadmittance somewhere else on the soundboard or even on the sound boxcould be calculated, measured and compared.

� A string could be excited electromagnetically, and the displacement of thestring and the soundboard near the connection point could be measured.This method also contains some uncertainty, since the force exerted by thestring on the soundboard is not perpendicular to the soundboard, which isthe direction in which the admittance is calculated.

12.3 Discussion

The modal analysis gives, among others, a family of modes caused by standingwaves along the soundboard. We find a (0,0) mode at 332 Hz (no. 11) and anotherone at 367 Hz (no. 12). The difference between them is the location of the lobe, andthe vibration of the body of the harp. A similar double mode is found for the (1,0)shape, as both modes no. 17 and 18, at 534 Hz and 556 Hz, respectively, look like(1,0) modes. No good extraction was done of a (2,0) soundboard mode, probablybecause the shaker was placed too close to one of its modal lines. A (3,0) mode isfound at 705 Hz (no. 24), and a (4,0) mode at 849 Hz (no. 29). It seems like allthese modes contribute considerably to the radiation of the instrument, since theyall create peaks in the radiated pressure spectrum (see figure 11.5).

Other plate-like modes are also be found, but since the soundboard is so muchthinner at the top than at the bottom, there exist modes that can not be classifiedlike plate modes. For instance, mode no. 31 at 949 Hz seems to have 7 lobes, 6places symmetrically as in a (2,1) plate mode, plus one single lobe at the top ofthe soundboard. The same kind of modes are found for the sound box, which isequally much wider at the bottom than at the top. For instance, mode no. 23 at686 Hz contains 4+2+2 lobes. The vibration of the soundboard at the same moderesembles a (2,1) mode, but is quite unsymmetrical. Normal plate-like modes arealso found for the sound box. A (0,0) mode at 109 Hz (no. 2), a (0,1) mode at268 Hz (no. 8), a (0,2) mode at 331 Hz and at 556 Hz, in both cases connectedwith a strong soundboard mode. A (1,1) mode (no. 14) is found at 452 Hz. Themodes that mostly cause vibration of the sound box do not give marked peaks in theradiated pressure spectrum. This is probably because these modes are relativelyweakly excited when the shaker is placed on the soundboard.

There seems to be a strong resonance in the air cavity at 228 Hz. This can beseen from the peak in the hole air velocity (figure 11.4), while there is no peak inthe surface velocity at the same frequency. Additionally, the corresponding peak inthe radiated pressure disappears when the holes are closed (see fig. 12.6). The airmode is extracted as mode no. 9. The mode shape of mode no. 9 does not containmuch information, since the resonance is caused by movement of the air inside theharp, and not the harp itself.

Other air modes are found at 495 Hz and 612 Hz. The air flow through theholes is approximately in phase when the harp is vibrating at 228 Hz. At 495 Hz,

12.4. CONCLUSIONS 89

there is a phase shift of approximately 70� between the two upper and the two lowerholes. At 612 Hz, none of the wholes breathe in phase. The lowest resonance canpossibly be characterized as a Helmholz resonance, but whether it is or not, it doesnot contribute by far as much to the radiation of the instrument as the Helmholzresonance in a guitar.

12.4 Conclusions

The measurements suggest that the harp is a poor radiator of sound below the firstsoundboard mode at 330 Hz. Because of the relatively strong vibration of thebody of the harp, a radiation model which only takes into account the radiationfrom the soundboard will probably not be accurate. The modal analysis givesresults that compare well with those found elsewhere, but there were unresolvedproblems with the vibrational analysis at and near the excitation point. There arelarge variations in input mobility over the line where the strings are fixed. This isinevitable, and magnitude of this variation contributes in determining the qualityof the instrument. The air volume inside the harp has series of resonances, butthese affect the radiation and vibration much less than in more confined soundboxes, like that of a guitar. In particular, the air mode at 228 Hz, which might becharacterized as a Helmholz resonance, is below the lowest soundboard mode andwill therefore only be weakly excited when the instrument is played.


Mode no. 11, 0 dB Mode no. 2, –5 dB Mode no. 12, –5 dB

331.8 Hz, –25.6 s�1 109.0 Hz, –18.9 s�1 366.6 Hz, –54.3 s�1

Mode no. 6, –8 dB Mode no. 20, –10 dB Mode no. 29, –11 dB

183.9 Hz, –36.4 s�1 597.6 Hz, –53.0 s�1 849.2 Hz, –136.1 s�1


441.9 Hz, –30.8 s�1 155.6 Hz, –27.9 s�1 195.9 Hz, –28.0 s�1


452.0 Hz, –34.2 s�1 268.1 Hz, –23.2 s�1 556.4 Hz, –62.0 s�1



704.7 Hz, –91.6 s�1 810.7 Hz, –58.9 s�1 686.0 Hz, –32.3 s�1


534.1 Hz, –65.4 s�1 923.6 Hz, –74.1 s�1 305.4 Hz, –49.3 s�1


948.8 Hz, –65.0 s�1 126.9 Hz, –8.6 s�1 463.6 Hz, –51.2 s�1


623.1 Hz, –43.8 s�1 143.4 Hz, –1.81 s�1 227.1 Hz, –49.7 s�1



70.8 Hz, –14.3 s�1 655.8 Hz, –59.6 s�1 587.0 Hz, –48.6 s�1


401.6 Hz, –57.5 s�1 482.9 Hz, –37.0 s�1 732.6 Hz, –52.0 s�1


884.9 Hz, –49.4 s�1 752.2 Hz, –48.5 s�1 774.7 Hz, –38.2 s�1

Mode no. 34, –37 dB

497.0 Hz, –39.0 s�1


1 2 3 4 50

100

200

300

400

500

600

700

800

900

1000

Lobes along soundboard

Mod

e fr

eque

ncy

/ Hz

Figure 12.4: Relation between mode frequency and mode geometry. Soundboardof assembled and stringed Celtic harp.

100 200 300 400 500 600 700 800 900

5

10

15

20

25

30

-62

Frequency / Hz

-37 -37

-43

-43

-43

-49

-49

-49

-43

-56

-49

-49

-49

-49

-49

-49

-43

-31

Stri

ng n

umbe

r (f

rom

bas

s to

treb

le)

Figure 12.5: Input admittance calculated from modal analysis. Numbers on thegraph are in dB. Zero is 1 m/s�N. Contour lines are drawn for every 6 dB. Circlesare placed near the fundamental frequency and location of each string. Dots areplaced at the first four harmonics.


0 500 1000 1500 2000 2500 3000−120

−110

−100

−90

−80

−70

−60

Frequency / Hz

Mea

n pr

essu

re /

dB

138

180

321

378

432

528

858

942

150

Figure 12.6: The average sound pressure over all seven microphones, and allangles of rotation. Reference level is 1 Pa/N. Numbers are peak frequencies, in Hz.The frequency resolution is 3 Hz. All openings in the sound box have been closed

Part V

Appendices

This part contains various material which was not found to fit into the chronologicalpresentation in the previous parts. References to these appendices are made in theother parts.

95

Appendix A

Mesh grossness

Due to our measurement methods, only a few samples of the surface velocity canbe measured. We are naturally interested in the surface velocity over the wholesoundboard. We may justify saying something about the velocity between thepoints that are actually measured, if we can make some assumptions about thevelocity distribution.

The complexity of the deformation increases as the frequency increases. Thequestion raised in this section is how to determine the necessary measurementdensity, given a measurement frequency, or inversely, how to determine the highestmeasurement frequency, given a measurement density. Normally, the choice ofmesh grossness seems to be done fairly randomly. For instruments that havepreviously been studied, so the mode shapes are known, a good choice can easilybe made. If the wave propagation velocity in the structure is known, this can alsobe used, but otherwise the following approach could be an option.

A spatial frequency, or a wavenumber, is the reciprocal of a distance, namelya wavelength. For a homogeneous medium, a plane wave will propagate with awavenumber k � ��c, where � is the angular frequency [s�1], and c is the wavepropagation velocity [m/s]. In this case, the relationship between frequency (whichhas to do with a variation over time) and wavenumber (which represents a variationover space) is obvious.

Let us assume a similar relationship, even when the structure is not infinite, nothomogeneous, and there are a myriad of waves traveling across it simultaneously.

If this is the case, a certain excitation frequency �m will only cause deforma-tions of the structure that can be described with wavenumbers k � km. In thiscase, it is sufficient to sample the structure with an interval smaller than 1��2km�,according to the sampling theorem. The relationship between �m and km is nottrivial, though.

The following sections proposes a method to estimate experimentally thisrelationship.

A.1 Measuring k

The magnitude of deformation at different wavenumbers can be determined by per-forming a Fourier transformation of the deformation, a two-dimensional function

96

A.1. MEASURING K 97

f�x� y�. This gives the function F �k1� k2�, which should be small whenever k1 ork2 is greater than km.

One idea could then be to measure f�x� y� at one frequency for a portion of thestructure with a very fine grid, calculate F �k1� k2�, and estimate km. This processinvolves the measurement of many points, since the grid must be very dense, andmust be large enough to allow proper windowing of the measured data.

Another possibility would be to sample f�x� y� only along a line, thus reducingthe number of points to measure. This could work if one knew which directionwould give the largest k-values. This depends on the modes that happen to beexcited at the frequency of the excitation signal. For a wooden structure, thepropagation velocity is smallest across the lines of the wood, which, on average,gives the highest k-values in that direction. Even with a line, some extra lengthmust be added to allow windowing.

A third possibility, which also reduces the number of measurement points, isto measure a few points on the periphery of a circle.

rφ

1/k

x

y

Figure A.1: Sampling surface velocity around a circle periphery, incoming planewave

The major advantage of this configuration is that the proper wavenumber canbe found regardless of the direction of the wave.

Let us examine the case where

f�x� y� � e2�ikx� �A�1�

which is a plane wave propagating in positive x-direction. See figure A.1.The measured points represent samples of the function f�, where

f�� f�r cos �� r sin�� A�2�

For our example, we get

f�� exp�2ikr cos�� A�3�

This function is not easily Fourier transformed symbolically, but figure A.2ashows the most important features of F��f�, the power spectrum of f�, namelythat it only contains energy at discrete frequencies since f� is periodic, and thatalmost all energy is concentrated at frequencies below 2rk. Figure A.2b showshow the energy is distributed within F� for small values of k, for one plane wave.

98 APPENDIX A. MESH GROSSNESS

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

1

2

3

4

5

0 5 10 15 20 25 30 35 40 45 500

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

rk

(a) (b)

2π

Ene

rgy

Frequency

Rel

ativ

e en

ergy

rk

f=

f=

Figure A.2: (a) Power spectrum of f� for one choice of k and (b) Spectral energydistribution for small values of k

A.1.1 Measurements

The method was tested on the Celtic harp. The surface velocity was measured at32 points, 16 points on each of two circles, one on the soundboard and one on thebody of the instrument. The diameters of the circles were 70 mm.

The function F was calculated for each of the 981 measured frequencies in therange 60–3000 Hz. Figure A.3 shows how energy is distributed among the lowestcomponents of F at different excitation frequencies.

0 500 1000 1500 2000 2500 3000

0

1

2

3

4

5

0 500 1000 1500 2000 2500 3000

0

1

2

3

4

5

(b)(a)

Rel

ativ

e en

ergy

f=

f=

Rel

ativ

e en

ergy

Frequency / Hz Frequency / Hz

f=

f=

Soundboard Instrument body

Figure A.3: Measured energy distributionF . (a) Soundboard, (b) Body

Observe that for the front plate, almost all energy is contained in the lowesttwo frequency components up to about 2.8 kHz. For the body, the third frequencycomponent becomes generally important already at 2.0 kHz. This might indicatea difference of about 40 % in wave propagation velocity in the two parts of thestructure.

Thus, if we for instance wish to study the frequency region 0–2.0 kHz, we must

A.1. MEASURING K 99

choose a mesh which can capture k components such that rk 0�2, since thisis the wavenumber where higher frequency components in F becomes important(fig. A.2b). This gives k � 5�7 m�1, and a grid density of 1�2k � 8�75 cm. Forthe front plate, a 40 % less dense grid, 12.25 cm, could be used, according to thistheory. Since this supposes that the plates are homogeneous, which they are not, amuch denser grid was used according to thumb rules and gut feelings.

Appendix B

Instrument excitation

B.1 Shaker excitation

A shaker is an electromagnetic device which can impart an oscillating force on astructure. It can be compared with a loudspeaker where the membrane is replacedwith a piece of metal which can be connected to the instrument. The shaker can beequipped with an impedance head (fig. B.1), and give accurate information aboutthe excitation.

2 Piezo-

Seismic

Output SocketAcceleration

Force

Mounting Surface

Driving Point

Output Socket

2 Piezo-

Mass

Discselectric

electricDiscs

Figure B.1: Impedance head consisting of an accelerometer and a force transducer

A shaker gives accurate, reproducible results over a large frequency range.Previous experience in the acoustics group at ENST indicated that the use of ashaker might perturb the soundfield measurements, since the shaker itself makesnoise. In addition, the mass between the force transducer and the driving pointof the structure might affect the characteristics of the structure. Therefore, othermethods of excitation were searched for.

100

B.2. IMPACT HAMMER 101

B.2 Impact hammer

A light impact impact hammer, B&K 8203, and a heavier, hand-held model,B&K 8202 are available. The hammers are equipped with force transducers.One might suspect that such a mechanical arrangement would not yield goodreproducibility.

Preliminary experiments did, however, indicate good reproducibility for thelittle hammer, which is suspended by a wheel bearing. The velocity at a point ofthe surface of a violin was measured using the laser vibrometer. The light impacthammer was used to excite the violin at a point near the bridge. The output from thevibrometer was fed into a digital storage oscilloscope, and the traces of subsequentresponses were compared.

A disadvantage with this method, is that each impact only transfers a smallamount of energy to the system, which does not give an acceptable signal-to-noiseratio in the measurements. This problem can be overcome by averaging severalconsecutive measurements, which in turn has the disadvantage of being a tedious,time-consuming effort.

Other problems occur because of the extreme concentration in time of thetransferred energy. Nonlinear behavior of the structure will be provoked more thanwith coninuous excitation. For wooden structures, a small depression will be madeat each point of impact. This is a not modellable by a modal model, and may be asymptom of nonlinear behavior which affects the free vibration which follows theimpact.

Figure B.2: Impact hammers B&K 8203 and B&K 8202

Appendix C

Manual plucking

See chapter 8. The following table shows relative strength picked up by the fivemicrophones, plucking the different strings several times, with the extra soundholeopen and closed. Relative energy picked up by each microphone, measured in dB.For strings A-B, from 50 ms - 2050 ms after trig point. For low E string, 50 ms -3050 ms, for high E string, 50 ms - 1550 ms.

102

103

Side rose

open

open

open

open

open

open

open

open

open

open

closed

closed

closed

closed

closed

closed

Stringl. El. El. EAAADDDGGG

h. Eh. Eh. E

BBBGGGDDAAA

l. El. El. E

l. El. El. EAAADDDGGGBBB

h. Eh. Eh. E

Ref. level–12.6674–5.4305–4.4257–4.6324–3.6638–2.9858–8.8861–10.7894–9.6665–9.8062–13.1065–11.5287–15.9135–16.7120–16.7349–17.4169–12.6990–13.4876–6.4003–6.7670–15.5799–5.8559–6.3702–2.3250–1.2554

0–7.1897–5.3982–2.4378

–9.1168–9.8128–8.4962–9.8706–8.3656–7.9310–13.8897–13.2693–13.4508–15.6121–14.1250–14.2410–17.3278–21.5400–17.3181–21.4482–23.9257–19.7636

Mic 1 Mic 2 Mic 3 Mic 4 Mic 515.4708 9.1317 0 14.1512 –4.313316.4016 10.6030 0 16.5892 –2.776615.9162 9.6616 0 15.6488 –4.623816.3074 8.7820 0 14.8028 –4.626916.1237 9.1107 0 14.5144 –4.407216.1754 8.6869 0 14.5537 –5.072216.1992 8.6747 0 14.4904 –2.210216.7726 9.7945 0 15.2266 –2.583816.6743 9.7827 0 15.8768 –2.909417.0686 11.0673 0 14.7425 –1.974715.6551 10.7732 0 13.3653 –2.237715.6421 10.6836 0 13.5371 –2.307915.5158 11.5182 0 14.1448 –1.493615.3990 11.4480 0 13.7286 –0.618815.2851 11.2025 0 13.7337 –0.621316.1717 6.6336 0 16.7835 –2.726315.5533 6.4496 0 15.9775 –4.277815.4591 6.4042 0 16.1399 –4.463817.3605 11.8643 0 14.7865 –3.046517.5874 12.0878 0 15.0473 –2.912116.9543 11.8110 0 15.7364 –0.965016.9072 10.2087 0 15.6560 –4.784716.7303 10.1728 0 15.6495 –4.809516.1290 8.9998 0 14.3702 –5.814416.0136 8.8313 0 14.3028 –6.009315.8300 8.7526 0 14.0440 –5.982816.3871 10.1598 0 14.1119 –4.861116.2549 10.0519 0 14.1137 –5.266216.3776 9.9500 0 15.0401 –4.9154

14.2593 8.1127 0 14.0983 –7.381714.2462 8.0482 0 14.1537 –7.317314.2582 8.0639 0 13.8521 –6.802513.6391 6.7762 0 15.0959 –6.955613.6358 6.5210 0 14.6292 –7.168813.7920 6.8242 0 14.7340 –7.158813.9039 8.4409 0 16.2272 –5.658114.3556 8.6583 0 16.1506 –5.481114.0932 8.2565 0 15.7322 –6.366815.5134 10.5683 0 14.4654 –4.406015.8358 9.9161 0 14.6987 –5.129815.8261 9.9360 0 14.6857 –5.206214.1082 4.7913 0 15.2929 –6.093914.2554 6.0396 0 14.6397 –5.362913.8989 5.1275 0 14.3947 –6.225114.3951 11.5261 0 15.1680 –1.974413.9954 10.3531 0 15.7874 –3.728014.1504 11.9525 0 15.1972 –2.0092

Table C.1: Relative strength picked up by the five microphones

Appendix D

Matlab code

This appendix contains the listings of the Matlab scripts that were used to do modalanalysis. They can possibly be retrieved by FTP. Hopefully, they can be used inother, related projects. Included in this appendix are eight files. Six of these aremeant to be invoked by the user. For a complete modal analysis, the followingsteps must be followed:

1. Perform the necessary measurements and store the following variables in a.mat file:

� A matrix v containing one transfer function per column. Should bedisplacement vs. force.

� A vector freq containing the frequencies (in Hz) of the rows in v.

� A number expt containing the index of the point of excitation. Ifthe response in this point is not included, leave expt undefined. Theonly disadvantage is that the compliance of the structure can not becalculated.

2. Find the eigenvalue estimations by running Eigenvalues.m. Beforerunning this script, define the following variable:

� A number numeigval which limits the number of eigenvalues to extractfor each point. A slight overestimation is better than an underestima-tion. Base the estimation on counting peaks in mobility plots.

Optionally, the following variables may also be defined:

� A vector use which contains indexes of the columns of v to use forestimating the eigenvalues. If a very large number of points have beenmeasured, pick only a representative subset of these in order to savecomputing time.

� A vector frequse which contains the indexes of freq to use, i.e. thefrequency range to study. Must be a monotonically increasing sequenceof integers. If a large frequency range has been measured, it might bereasonable to split this and analyze each range individually. Make sureto allow the different bands to overlap much more than the bandwidth

104

105

of the modes. If the frequency resolution in the measured data is veryhigh, it might be sufficient to select only each second or third frequencycomponent.

� A vector timespan containing the start and stop times to use in studyingthe calculated impulse responses. Again, if the frequency resolution isunnecessarily high, this is another way to reduce computing time.

The script will generate the variables eigv, ampl, phas and errs, which mightbe convenient to save in a .mat file.

3. Run Mancluster.m. This is an interactive script which allows the userto pick what seems to be clusters of eigenvalue estimations. The script willreturn the variable eigvclust, which should be saved in a file. The choiceof eigenvalues is very important, and it might be necessary to return to thispoint several times.

4. Define the following variables before running Eigenvectors.m:

� The vector use now determines which points to include in the modalanalysis.

� The vector eigval contains the eigenvalues to use. These will normallybe the ones obtained by a manual clustering, so set eigval = eigvclust.

The script generates the matrix A which contains the eigenvectors.

5. Now, the quality of the curve-fitting can be evaluated using two scripts.Firstly, Reconstruct.m can be used to inspect the curve-fit for each ofthe transfer functions. The following may be defined before invokation:

� A vector recon containing the indexes of the points to inspect.

The scriptQuality.m automatically reconstructs all transfer functions andplots two graphs:

� A graph showing the average error, over all points, as a function offrequency.

� A graph showing the average error, over the selected frequency range,individually for all measurement points.

6. To visualize the mode shapes, it is convenient to define the geometry ofthe structure and use the graphics package supplied with Matlab, includingfunctions like mesh, contour and movie. The best suited visualizationmight vary from structure to structure.

7. Finally, one application of the modal model has been implemented. Thescript Compliance calculates the compliance at the points listed in thevector mobil.

106 APPENDIX D. MATLAB CODE

D.1 Eigenvalues.m

% This MATLAB script uses the ESPRIT algorithm (Matrix Pencil method)% as a tool to estimate eigenfrequencies, as a step in modal analysis.% Before calling this script, define the following variables:%% numeigval -- the maximum number of decaying sinusoids to use.% should be set higher than the estimated number of modes.% Too low setting gives poor results, too high setting% consumes time%% v -- a matrix containing the transfer functions to work on.% Must contain one transfer function per column.%% freq -- A vector containing the frequencies, in Hz, of the rows% in v.%% use -- A vector which lists the columns of v to use. May be% left undefined, in which case all points are used.%% frequse -- A vector which lists the rows of v to use. May be left% undefined, in which case all frequencies are used.%% timespan -- A two-element vector which defines the start and stop% times (in seconds) to use when selecting a portion of the% calculated impulse response. Too short setting gives poor% frequency resolution, too long setting consumes time.% May be left undefined. Half the impulse response it then used.%%%% Generates the variables%% eigv -- Imaginary part is angular frequency, real part is damping in s-1% ampl -- The initial amplitude of the corresponding sinusoids% phas -- The initial phase of the corresponding sinusoids% errs -- The error in trying to synthesize the signal, in dB%% For all three variables, elements that correspond to unused sinusoids are% set to NaN.%% Uses the functions fesprit and ampl written by Jean Laroche.% This script is written in 1996 by Svein Berge.

clear eigv;clear ampl;clear phas;clear errs;eigv=[];

var=0;tp=0;sa=0;

if (exist(’use’))use2=use;elseuse2=1:size(v,2);end

if (exist(’frequse’))frequse2=frequse;

D.2. FESPRIT.M 107

elsefrequse2=1:size(v,1);end

frequ=freq(frequse2);

maxfreq=frequ(length(frequ))+frequ(2)-frequ(1);minfreq=frequ(1)-(frequ(2)-frequ(1));

if (exist(’timespan’))startsample=timespan(1)*(maxfreq-minfreq)+1;stopsample=floor(timespan(2)*(maxfreq-minfreq))+1;elsestartsample=1;stopsample=length(frequ);end;

clghold onxlabel(’frequency / Hz’)ylabel(’damping / s-1’)

num=0;for i=use2,num=num+1;fprintf(’%d / %d (pt. %d) \n’,num,length(use2),i);x=real(ifft([0;v(frequse2,i);0;flipud(conj(v(frequse2,i)))]))’;

F=fesprit(x(startsample:length(x)),floor((stopsample-startsample)/2),...ceil(stopsample-startsample),numeigval);[amp pha freqs amor synth] = ampl(F, x(startsample:length(x)), ...ceil(stopsample-startsample),length(F)/2);

figure(2); hold offplot(startsample:stopsample,x(startsample:stopsample), ...

startsample:(startsample+length(synth)-1),synth)figure(1); hold off

freqs = freqs*2*(maxfreq-minfreq)+minfreq; % Frequency shiftamor = -amor*2*(maxfreq-minfreq); % Damping adjustmenterror = 20*log10(std(synth - x(startsample:startsample+...length(synth)-1))/std(synth));fprintf(’Error = %g dB\n’,error);

plot(imag(eigv)/2/pi,real(eigv),’.’); hold onplot(freqs,amor,’o’);

% sa=sa+sum(amp);% tp=tp+sum(amor.*amp);% axis([minfreq maxfreq 2*tp/sa 0])drawnow;

eigv(:,num)=[amor+2i*pi*freqs; zeros(numeigval-length(freqs),1)*nan];ampl(:,num)=[amp;zeros(numeigval-length(freqs),1)*nan];phas(:,num)=[pha;zeros(numeigval-length(freqs),1)*nan];errs=[errs;error];end

D.2 fesprit.m

function r = fesprit(x,p,N,L);% fesprit(x,p,N,L) retourne les racines signal d’apres la methode esprit.% L’implementation est plus rapide.


% x est le signal, N la longueur de la fenetre d’analyse, p habituellement N/2% et L le nombre de sinusoides.% ATTENTION, si N est grand (300) ca prend beaucoup de temps!%% Written by Jean Laroche.

X0 = toeplitz(x(p+1:N),x(p+1:-1:1));X1 = X0(:,(1:p));X0 = X0(:,(2:p+1));P = X0’*X0;

if(p>50) disp(’Multiplication...’); end[U,V] = eig(P);

if(p>50) disp(’Diagonalisation...’); endV = diag(V);[V m] = sort(V);V = V(p:-1:1);m=m(p:-1:1);V = diag([1 ./ V(1:2*L) ; zeros(p-2*L,1)]);M = U(:,m)*V*U(:,m)’ * X0’ * X1;

if(p>50) disp(’Calcul matrice...’); endr = sort(eig(M));

if(p>50) disp(’Vecteurs propres...’); endr = r(p-2*L+1:p);r = r(real(r) ˜= r);

D.3 ampl.m

function [ampl, phase, frequ, amort,synth] = ampl(a,x,N,L)% [ampl, phase, frequ, amort,synth] = ampl(a,x,N) travaille a partir d’un% vecteur a obtenu par toute methode (y compris esprit). Ca retourne% les amplitudes, phases, frequ et amort des sinusoides detectees dans le% signal x en utilisant N echantillons. De plus le signal synth est retourne.% Si a est complexe, ampl comprend que a contient les racines, et non% pas les coeff du polynome.% option = 1 permet de ne pas calculer l’erreur de reconstruction.%% Written by Jean Laroche.

if(real(a) == a)racines = roots(a);

elseracines = a;racines = racines(abs(racines) > eps);

endif(nargin == 3)

L = (length(racines))/2;endx = reshape(x(1:N),1,N);% mod = abs(abs(racines) - 1);% [n,m] = sort(mod);% racines = racines(m(1:2*L));arg = angle(racines);racines = racines(arg ˜= pi);arg = angle(racines);rac = log(racines);A = (0:N-1)’ * rac.’;A = exp(A);b = A\x(1:N)’;

synth = real(A * b)’;if(nargin ˜= 4)

erreur = 20*log10((std(synth - x(1:N))/ std(x(1:N))))end

ampl= 2*abs(b(arg>0 ));

D.4. MANCLUSTER.M 109

phase = angle(b(arg>0 ));frequ = arg(arg>0)/2/pi ;amort = - log(abs(racines(arg>0)));

D.4 Mancluster.m

% Script for manual clustering of modes%%% Uses the variables%% eigv -- contains eigenvalues% ampl -- contains amplitudes, used for weighting% errs -- contains errors, used for weighting (w=10ˆ-errs/20)%% Creates the variables%% eigvclust -- contains the clustered eigenvalues%% If the eigvclust variable is defined as the script is called,% clustering continues as if it was interrupted and restarted.

w=ampl;for i=1:size(w,2)su=w(:,i);w(:,i)=w(:,i).ˆ2/sum(abs(su(˜isnan(su))).ˆ2)*10ˆ(-errs(i)/20);end

figure(1)figure(2)

if ˜exist(’eigvclust’)eigvclust=[];amplclust=[];

eigvl=˜isnan(eigv);end

go=1;stay=0;

Ao=[min(min(imag(eigv(eigvl))))/2/pi,...max(max(imag(eigv(eigvl))))/2/pi,...min(min(real(eigv(eigvl)))),max(max(real(eigv(eigvl))))];

while(go)figure(1)clf;hold on;plot(imag(eigv(eigvl))/2/pi,real(eigv(eigvl)),’+’);plot(imag(eigv(˜eigvl))/2/pi,real(eigv(˜eigvl)),’.’);plot(imag(eigvclust)/2/pi,real(eigvclust),’o’);

if (˜stay)axis(Ao);

fprintf(’Zoom in on a region and press enter in text window.\n’)zoom on;pause;Ax=axis;zoom off;stay=1;elseaxis(Ax);end


fprintf(’Click on two corner points with left button or click\n’)fprintf(’right button to enter zoom mode.\n’)

figure(1)[x,y,b]=ginput(1);if (b==1)[x2,y2,b]=ginput(1);endif (b==1)x=[x x2];y=[y y2];

eigvc=imag(eigv)>min(x)*2*pi & imag(eigv)<max(x)*2*pi & ...real(eigv)>min(y) & real(eigv)<max(y) & eigvl;

line([x(1),x(1),x(2),x(2),x(1)],[y(1),y(2),y(2),y(1),y(1)]);

dist=abs(eigv-mean(x)*2i*pi-mean(y));dist(isnan(dist))=inf*ones(sum(sum(isnan(dist))),1);

[m,mi]=min(dist);av=0;we=0;

for i=1:size(eigv,2),if eigvc(mi(i),i)av=av+eigv(mi(i),i)*w(mi(i),i);we=we+w(mi(i),i);endend

plot(imag(av/we)/2/pi,real(av/we),’o’);

figure(2);clg;

s=sum(eigvc);h=[sum(s==0) sum(s==1) sum(s==2) sum(s==3) sum(s>3)]/size(eigv,2);bar([0,1,2,3,4],h);xlabel(’Number of eigenvalue estimations within area (4=more than 3)’);ylabel(’Percentage of measurement points’);axis([0,4,0,1])

a=input(’Extract a mode from this? [y/N] ’,’s’);if (length(a)==1)if a==’y’

for i=1:size(eigv,2),if eigvc(mi(i),i)eigvl(mi(i),i)=0;endend

eigvclust=[eigvclust av/we];amplclust=[amplclust sqrt(we)];

endend

a=input(’Extract another mode? [Y/n] ’,’s’);if (length(a)==1)if a==’n’ go=0;end

D.5. EIGENVECTORS 111

end

elsestay=0;end

end

D.5 Eigenvectors

% This MATLAB script estimates the eigenvectors, as a step in modal% analysis.%% Uses the following variables, which should be supplied before invocation:%% v -- a matrix containing the transfer functions to work on.% Must contain one transfer function per column.%% freq -- A vector containing the frequencies, in Hz, of the rows% in v.%% use -- A vector which lists the columns of v to use. May be% left undefined, in which case all points are used.%% frequse -- A vector which lists the rows of v to use. May be left% undefined, in which case all frequencies are used.%% eigval -- A vector containing the eigenvalues to use. Imaginary part% should be angular frequency, and real part should be damping.%% Will generate the variable%% A -- each row is an eigenvector and corresponds to an% eigenvalue.%% This code is written by Philippe Derogis, and brushed up by Svein Berge.


if (exist(’frequse’))frequse2=frequse;elsefrequse2=1:size(v,1);endfrequ=freq(frequse2);

N=length(frequse2);Nm=length(eigval);M=length(use2);

% Calculate resonance functions

f=zeros(N,Nm*2);for k=1:Nm,f(:,k)=(1./(2i*pi*frequ - eigval(k)*...ones(size(frequse2)))+1./(2i*pi*frequ -...conj(eigval(k))*ones(size(frequse2)))).’;end


for k=1:Nm,f(:,k+Nm)=(1i./(2i*pi*frequ - eigval(k)*...ones(size(frequse2)))-1i./(2i*pi*frequ -...conj(eigval(k))*ones(size(frequse2)))).’;end

A=zeros(Nm*2, M);%B=(f.’*conj(f))’;B=real(f.’*conj(f))’;Bi=inv(B);

for k=1:M,fprintf(’ %d / %d \r’,k,M);func=v(frequse2,use2(k));% p=(func.’*conj(f))’;p=real(func.’*conj(f))’;% A(:,k)=conj(Bi*p);A(:,k)=real(Bi*p);end

A=A(1:Nm,:)+1i*A(Nm+1:Nm*2,:);

fprintf(’\n’);

D.6 Reconstruct.m

% This MATLAB script reconstructs the transfer functions based on% a modal analysis. Used for assessing the quality of the modal% analysis, and to detect lacking or superfluous eigenvalues.%% Uses the following variables, which should be supplied before invocation:%% v -- a matrix containing the transfer functions to work on.% Must contain one transfer function per column.%% freq -- A vector containing the frequencies, in Hz, of the rows% in v.%% use -- A vector which lists the columns of v that exist in A. May be% left undefined, in which case all points are used.%% recon -- A vector which lists the columns of v to reconstruct. May be% left undefined, in which case all points are used.%% frequse -- A vector which lists the rows of v to use. May be left% undefined, in which case all frequencies are used.%% eigval -- A vector containing the eigenvalues to use. Imaginary part% should be angular frequency, and real part should be damping.%% A -- each row is an eigenvector and corresponds to an% eigenvalue.%

figure(1)clg


D.7. QUALITY.M 113

if (exist(’recon’))recon2=recon;elserecon2=use2;end





for k=1:Nm,f(:,k+Nm)=(1i./(2i*pi*frequ - eigval(k)*...ones(size(frequse2)))-1i./(2i*pi*frequ -...conj(eigval(k))*ones(size(frequse2)))).’;end

for k=1:length(recon2),r=zeros(N,1);mn=(use2==recon2(k));for l=1:Nm,r=r+f(:,l)*real(A(l,mn))+f(:,l+Nm)*imag(A(l,mn));end

subplot(211);plot(frequ,20*log10(abs(v(frequse2,recon2(k)))),’-’,...frequ,20*log10(abs(r)),’--’);title([’Point no. ’,num2str(recon2(k))]);subplot(212);plot(frequ,(angle(v(frequse2,recon2(k)))),’-’,frequ,(angle(r)),’--’);pauseend

D.7 Quality.m

% This MATLAB script reconstructs the transfer functions based on% a modal analysis. Used for assessing the quality of the modal% analysis, and to detect lacking or superfluous eigenvalues.%% Uses the following variables, which should be supplied before invocation:%% v -- a matrix containing the transfer functions to work on.% Must contain one transfer function per column.%% freq -- A vector containing the frequencies, in Hz, of the rows% in v.%


% use -- A vector which lists the columns of v to use. May be% left undefined, in which case all points are used.%% frequse -- A vector which lists the rows of v to use. May be left% undefined, in which case all frequencies are used.%% eigval -- A vector containing the eigenvalues to use. Imaginary part% should be angular frequency, and real part should be damping.%% A -- each row is an eigenvector and corresponds to an% eigenvalue.%

figure(1)clg






for k=1:Nm,f(:,k+Nm)=(1i./(2i*pi*frequ - eigval(k)*...ones(size(frequse2)))-1i./(2i*pi*frequ - ...conj(eigval(k))*ones(size(frequse2)))).’;end

e=zeros(N,1);ef=zeros(M,1);

for k=1:M,fprintf(’ %d / %d \r’,k,M);r=zeros(N,1);for l=1:Nm,r=r+f(:,l)*real(A(l,k))+f(:,l+Nm)*imag(A(l,k));endte=r./v(frequse2,use2(k));te=abs(20*log10(abs(te)));e=e+te;ef(k)=mean(te);end

plot(frequ,e/M);

D.8. COMPLIANCE.M 115

pauseplot(use2,ef,’o’);fprintf(’\n’);

D.8 Compliance.m

% This MATLAB script calculates the compliance% (displacement / force), based on a modal analysis.%% Uses the following variables, which should be supplied before invocation:%% A -- each row is an eigenvector and corresponds to an% eigenvalue.%% eigval -- A vector containing the eigenvalues to use. Imaginary part% should be angular frequency, and real part should be damping.%% expt -- Index of excitation point%% freq -- A vector containing the frequencies, in Hz, to evaluate%% use -- A vector which lists the indexes of the rows in A. May be% left undefined, in which case indexing follows row number.%% mobil -- A vector which lists the indexes for which to calculate% mobility. Leaving undefined calculates mobility for all points.%% frequse -- A vector which lists the elements of freq to use. May be left% undefined, in which case all frequencies are evaluated.%%% Generates the variable%% mob -- Each column corresponds to a point, indexed by mobil, each% row is a frequency.%% This script is written in 1996 by Svein Berge.


if (exist(’mobil’))mobil2=mobil;elsemobil2=use2;end




f=zeros(N,Nm*2);


for k=1:Nm,f(:,k)=(1./(2i*pi*frequ - eigval(k)*...ones(size(frequse2)))+1./(2i*pi*frequ - ...conj(eigval(k))*ones(size(frequse2)))).’;end

for k=1:Nm,f(:,k+Nm)=(1i./(2i*pi*frequ - eigval(k)*...ones(size(frequse2)))-1i./(2i*pi*frequ - ...conj(eigval(k))*ones(size(frequse2)))).’;end

mob=zeros(N,length(mobil2));ept=(use2==expt);

for k=1:length(mobil2),fprintf(’ %d / %d \r’,k,length(mobil2));mn=(use2==mobil2(k));for l=1:Nm,mob(:,k)=mob(:,k)+f(:,l)*real(A(l,mn)*A(l,mn)/A(l,ept))+ ...

f(:,l+Nm)*imag(A(l,mn)*A(l,mn)/A(l,ept));end

end

Bibliography

[ASK] Askenfelt, A. (1990), Five lectures on the acoustics of the piano, Almqvist& Wiksell Tryckeri, Uppsala.

[BEL] Belin, P. (1991), “Mesure de Reponses Impulsionelles par SequencesPseudo-Aleatoires.” Groupe Acoustique, Departement Signal, ENST, Paris

[BRO] Brooke, M. (1992), Numerical Simulation of Guitar Radiation Fields Usingthe Boundary Element Method, Dept. of Physics and Astronomy, University ofWales, College of Cardiff.

[CAL] Caldersmith, G. (1977), “Guitar as a reflex enclosure,” J. Acoust. Soc. Am.63(5), May 1978.

[CHR82] Christensen, O. (1982), “Quantitative models for low frequency guitarfunction,” Journal of Guitar Acoustics; The Chicago Papers, 6, pp. 10-25.

[CHR84] Christensen, O. (1984), “An Oscillator Model for Analysis of GuitarSound Pressure Response,” Acustica, vol. 54 pp. 289–295, 1984.

[JFD] Degeorges, J. F. (1988), Rayonnement acoustique des plaques en champproche, These, l’Universite du Maine 1988.

[PD95] Derogis, Ph. (1995), “Vibration de la table d’harmonie d’un piano droit.”IRCAM, Paris

[BKST] Døssing, O. (1988), Essais Structurels — 1ere partie: Mesures de mo-bilite, Bruel & Kjær, Denmark

[BKST] Døssing, O. (1988), Structural Testing — part II: Modal Analysis andSimulation, Bruel & Kjær, Denmark

[DJE] Ewins, D.J. (1984), Modal Testing: Theory and Practice, Research StudiesPress Ltd., Somerset, England.

[F2] Firth, I. M. (1977), “On the Acoustics of the Harp,” Acoustica, vol. 37 1977.

[F1] Firth, I. M. (1983), “On the Acoustics of the Concert Harp’s Soundboard andSoundbox,” Proceedings of the Stockholm Music Acoustics Conference 1983.

[F2] Firth, I. M. (1986), “Harps of the baroque period,” Dept. of physics andastronomy, University of St. Andrews, Scotland.

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[FR91] Fletcher, N. H., and Rossing, T. D. (1991), The Physics of Musical Instru-ments, Springer-Verlag New York Inc.

[GR93] Geradin, M and Rixen, D. (1993), Theorie des vibrations, Masson, Paris1983.

[HAM] Hanson, R. J., Anderson, J. M., and Macomber, H. K., “Measurements ofnonlinear effects in a vibrating wire”

[LC] Lambourg, C., Chaigne, A. (1993), “Measurements and Modeling of theAdmittance Matrix at the Bridge in Guitars,” Proceedings of the StockholmMusic Acoustics Conference 1993.

[LMC] Laplane, J., Minsen, E., Charron, S. (1995), “Different configurations ofthe same concert guitar characterized by means of modal analysis and noisetransfer,” Proceedings of the International Symposium on Musical Acoustics.

[JLR] Laroche, J. (1993), “The use of the matrix pencil method for the spectrumanalysis of musical signals,” J. Acoust. Soc. Am., October 1993 pp. 1958–1965.

[CHLB] Le Pichon, A., Berge, S., Chaigne, A. (1996), “Comparison betweenexperimental and simulated radiation of a guitar using an acoustical holographybased method,” submitted to Acta Acustica.

[PL95] Le Pichon, A. and Laroche, J. (1995), “Rayonnement Acoustique deSources Planes.” Groupe Acoustique, Departement Signal, ENST, Paris

[RW87] Richardson, B. E. and Walker, G. P. (1987), “Predictions of the soundpressure response of the guitar,” Proc. Inst. Acoust. 9, pp. 139–146.

[CH96] Rosen, M. and Chaigne, A. (1996), “Analyse de sons de guitare pourdifferentes configurations structurelles de l’instrument,” to appear in Acta Acus-tica.

[ROS] Rosen, M. (1995), “Guitar sounds analysis for various structural config-urations of the instrument,” Proceedings of the International Symposium onMusical Acoustics.

[DTS] Sandwell, D.T. (1987), “Biharmonic spline interpolation of GEOS-3 andSEASAT altimeter data,” Geophysical Research Letters, 2, pp. 139–142.

[STR] Strong, W. Y. Jr. (1982), “Studying a Guitar’s Radiation Properties withNearfield Holography,” Dept. of Physics and the Applied Research Laboratory,Pennsylvania State University.

[VAL] Valette, C., “The Mechanics of Vibrating Strings.” Laboratoired’Acoustique Musicale, CNRS, Universite Paris VI

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[WIL] Williams, E.G. (1983), “Numerical evaluation of the radiation from un-baffled, finite plates using the FFT”, J. Acoust. Soc. Am., Vol. 74, 1983, pp.343–347.

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