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==Ways to exite the string.==
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There are several possibilities to exite the string of the viotar. The only criteria is that is has to set in motion an Helmholz vibration, and that the Viotar has to remain playable with a normale guitar playing technique.
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From a list of possibilities four have been chosen that has to be further researched. We will look at the feasability of construction and if it is possible to meet the design requirements.
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The four subjects that will be researched are:
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'''Ways To Exite The String
'''
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<div style="position:relative; top:.25em; font-size:100%">'''<br/>William Schattevoet<br/>David Duwaer<br/>Eric Backx<br/> Arjan de Visser'''</div>
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'''Subpages:'''
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<h1 style="font-size:180%; border:none; margin:0; padding:0">
'''Overview:'''
</h1><br/>There are several possible ways to excite the string of the Viotar. The only two criteria are that a Helmholtz vibration is reached and that the Viotar remains playable as a normal guitar.<br/>
Four possible ways to excite the strings have been chosen for further research. The main focus of this research will be on feasibility of the construction, on whether Helmholtz can be achieved and if it’s possible to meet the design requirements.
The four possibilities that will be researched are:
* Electro-magnet
* Electro-magnet
* Piëzo-element
* Piëzo-element
* Bowing wheel
* Bowing wheel
* Bowing belt
* Bowing belt
The findings of these researches will be listed below, from the four possible ways the most feasible will be chosen for further research.<br/>
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==Ways to exite the string==
===Exitation by Electromagnet===
====Principle====
Another way to exite the string is by using an electromagnet force on the steel strings. This force is created by the flow of current through a coil wrapped around a magnetizable core, creating an electro magnet. The force can be controlled by changing the current that flows through the coil. Because current can be controlled by speeds in the order of several Gigahertz, the flow can be controlled very rapidly.
====Estimation for the maximum bow force on the string====
Figure x shows the string on the moment it's being exited by the violin bow. The displacement of the string is called 'u', creating two angles α and γ. The string will be exited on position βL.
[[File:snaex.PNG|frame|Border| Figure x: The string exited over a distance u]]
The force in y-direction is calculated with:
<math>F_y=sin(\gamma)A\sigma_{1}+sin(\alpha)A\sigma_{2}</math>
<math>\sigma=E\varepsilon</math>
<math>\varepsilon= {dL\over L_0 }</math>
<math>F_y = AE \left (sin(\gamma){\sqrt{\beta^2 L^2+u^2}-\beta L \over \beta L} + sin(\alpha){\sqrt{(1-\beta)^2 L^2+u^2}-(1-\beta) L \over (1-\beta) L} \right)</math>
The biggest static force acting on a string is the force when a short, thick string is exited. It is assumed that the length of this string is 20 cm and the diameter is 0.25mm.
The force acting on this particular string, exited over 0.5 cm, is about 40 N.
====Estimation for the maximum dynamical force acting on the string====
Since the string isn’t only exited statically but is being moved continuously, the static force calculated above isn’t the only force that has to be dealt with. The string should vibrate in a predefined pattern, and therefore big accelerations become important. The force needed to accelerate a certain mass m, is calculated using Newton’s second law of motion. This law states that this force is equal to acceleration times mass.
Since the acceleration of the string varies linearly over the length of the string, the acceleration (and thus force) should be calculated separately for every small segment of the string. This gives rise to an integral over the whole string. In order to do this, the string is divided into two pieces, being the part βL and the part (1-β)L.
This gives the following for the part βL:


The findings of the research will be made into small reports, which will be used to choose a the method of exitation we are going to research in depth.
<math>F = {ma}</math>


==Exitation by Elektromagnet==
<math>F = \sum_{i=1}^{n}{m_ia_i}</math>
===Principle===


Another way to exite the string is by using an electromagnet force on the steel strings. This force is created by the flow of current through a coil wrapped around a magnetizable core, creating an electro magnet. The force can be controlled by changing the current that flows through the coil. Because current can be controlled by speeds in the order of several Gigahertz, the flow can be controlled very rapidly.
<math>F_1 = \sum_{i=1}^{n}{m_i\left( {da \over dx} x_i \right)} = \sum_{i=1}^{n}{m_i\left( {a \over \beta L} x_i \right)}</math>
 
<math>m_i = {\rho {\beta L \over n} A_{cross.sect.}}</math>
 
<math>n = {x \over dx} = {\beta L \over dx}</math>
 
<math>m_i = \rho A_{cross.sect.} dx</math>
 
<math>F_1 = \int_0^{\beta L} {{a A_{cross.sect.} \rho} \over {\beta L}} x \;dx</math>
 
<math>F_1 = {{a A_{cross.sect.} \rho} \over {\beta L}} \int_0^{\beta L} x \;dx</math>
 
<math>F_1 = {{a A_{cross.sect.} \rho} \over {\beta L}} [ \frac{1}{2} x^2 ]_0^{\beta L} </math>




====Estimation for the maximum bow force on the string====
In figure x shows the string on the moment it's being exited by the violin bow. The displacement of the string is called 'u', creating two angles α and γ. The string will be exited on position βL.




[[File:snaex.PNG|frame|Border| Figure x: De snaar geëxciteerd over afstand u]]
And for the part (1-β)L:<br/><br/>


De kracht in y-richting kan berekend worden volgens:
<math>F_2 = {{a A_{cross.sect.} \rho} \over {(1-\beta) L}} [ \frac{1}{2} x^2 ]_0^{(1-\beta) L} </math>


[[File:forv.PNG]]
Where:


De grootste kracht die op een snaar komt is de kracht op een korte, dikke snaar. Er wordt aangenomen dat de lengte van deze snaar 20 cm is en de diameter 0.25 mm.
a =  The acceleration of the string on position βL<br/>
De kracht die op deze snaar komt te staan bij een uitwijking van 0.5 cm is ongeveer 40 N.
A<sub>cross.sect.</sub> =  The cross section area of the string <br/>
====Schatting voor de maximaal uitgeoefende dynamische kracht op de snaar====
ρ The density of the string <br/>


Aangezien de snaar niet statisch uitgerekt wordt maar continu in beweging is, is de statische kracht die hierboven staat uitgewerkt niet de enige kracht waar rekening mee moet worden gehouden. De snaar moet namelijk volgens een vooraf gedefinieerd patroon gaan trillen, waardoor grote snelheidsveranderingen een rol gaan spelen. Volgens de tweede wet van Newton kan de kracht die nodig is om een massa een versnelling te geven te berekenen volgens F=m*a.
Aangezien de versnelling van de snaar lineair varieert over de snaar, moet voor elk segment van de snaar afzonderlijk worden berekend wat de versnelling en dus de benodigde kracht is. Hiervoor wordt de snaar opgedeeld in twee stukken, namelijk het stuk βL en het stuk (1-β)L.
Dit levert voor het stuk βL:


[[File:forele.PNG]]
The total force needed to accelerate the string is given by:<br/>


Voor het stuk (1-β)L geldt:<br/><br/>
<math>F_{tot} = F_1 + F_2 </math>
[[File:forele1.PNG]]<br/>
Waarin:
a = de versnelling van de snaar op positie βL<br/>
Adoorsnede = Het doorsnedeoppervlak van de snaar<br/>
ρ = De dichtheid van de snaar<br/>


<math>F_{tot} = {{a A_{cross.sect.} \rho} \over {\beta L}} [ \frac{1}{2} x^2 ]_0^{\beta L} + {{a A_{cross.sect.} \rho} \over {(1-\beta) L}} [ \frac{1}{2} x^2 ]_0^{(1-\beta) L} </math>


De totale kracht die geleverd moet worden door de elektromagneet om de snaar te versnellen, wordt dus gegeven door:<br/>
[[File:forele2.PNG]]


Dit levert een kracht op ter grootte van 6,25 N voor de volgende parameters:
This gives a total acceleration force of 6,25N when the following parameters are used:


a = 500 m/s2<br/>
a = 500 m/s2<br/>
Adoorsnede = 5 ∙ 10-6 m2<br/>
A<sub>cross.sect.</sub> = 5 ∙ 10-6 m2<br/>
ρ = 8 ∙ 103 kg/m3<br/>
ρ = 8 ∙ 103 kg/m3<br/>


De maximale totale kracht, dus de kracht de geleverd moet worden om bij een grote uitwijking de snaar een grote versnelling te geven, is de som van Fy en Ftot.
The maximum total force, which is the force needed to give the string the maximum acceleration at the maximum position, is given by the sum of Fy and Ftot.
 
This means that the maximum magnetic force that is needed is about 50 N.
 
====Feasibility====
The calculation shows that a magnetic force of 50 N should be enough to exite the string in the desired pattern. Finding an electromagnet that generates this magnetic force is not a problem. However, the width of these electromagnets could be a problem, since there might not be enough space to place an electromagnet for each string separately. This problem could be solved by making the electromagnets on our own. The force that is generated in an electromagnet is only depending of the number of windings and the current through the coil. Since there is enough space underneath the strings, a long and tight electromagnet would work.
Another point is that the magnet of one string might interfere with another and disturb it’s magnetic field. However, this problem would be quite easy to deal with as will since a screen could be placed in between the strings or the control device a magnet could compensate for disturbances from the others.
 
====Drawbacks====
Despite the theoretical possibility to use electromagnets for actuation, there are two very important drawbacks in this concept. Firstly, the instrument will no longer be bowed although it sounds as such. In fact the instrument then will no longer be a bowing instrument, which was one of the main requirements. But probably the biggest disadvantage of this concept, is the quite illogical way of synthesizing the sound. To get the string in the desired vibration, this vibration should be known in advance. Getting this vibration would be no problem, since we can get it out of the model or simply record it from a real bowing instrument. But once the vibration is known, it has to be exited on the string using electromagnets, recorded using a sensor and then amplified before it is played. This is very strange since we already had the desired vibration when we started, and thus were already able to play it.
 
====Conclusion====
The conclusion about this concept is that it would work most likely and that every vibration pattern could be chosen. However, the big drawbacks that it is no longer a bowing instrument and that the synthesizing routine is very illogical, make the concept fail.
 
===Actuation by piëzo crystals===
====Principle====
 
A piezo actuator utilizes the phenomenon that crystals of certain substances expand when a voltage is applied on it or conversely, they give a voltage when pressure is exerted on the crystal. The name piezo is derived from the Greek word piezein, which means pressing. The concept for actuation by the piezo, is that the string is attached to a piezo element, which expands when a voltage is applied. The idea is that by varying the voltage, the movement of the string can be controlled, giving the desired vibration in the string
 
====Feasibility====
If the feasibility of this concept is considered, the combination of required force and deflection seems to be a problem. The reason is that piezo elements only enlarge a little, varying from a few micrometers to some tens of micrometers. There are piezo elements on the market that serve as a linear motor and have a maximum amplitude up to 100 mm, but again does not provide the required force. Another possibility is to use so-called piezo stacks. This a series of piezos allowing the maximum deflection to increase. However, this excursion is also just a little more than 1 mm for the required force, which is not enough.
 
====Drawbacks====
The main disadvantage of piezo actuation is that the deflection they give is not large enough. This problem is partly dealt with using a stack, but even then the deflection is not large enough to make the string vibrate in the desired shape. Furthermore, in this design the instrument will no longer be a bowing instrument


Dit betekent dat de totale benodigde magnetische kracht ongeveer 50 N bedraagt.
====Conclusion====
===Haalbaarheid===
From the foregoing it can be concluded that it is not possible to get the string in the desired vibration using piezoelectric actuators. Also the instrument would not be a bowing instrument anymore, which is an important factor in the design.
Volgens de berekeningen is een elektromagneet die een kracht levert van 50 N voldoende om de snaar in een gewenste trilling te krijgen. Elektromagneten die deze kracht leveren, zijn te koop. Een probleem zou echter kunnen zijn dat de magneet te breed is in verhouding tot de afstand tussen de snaren, omdat er dan niet genoeg ruimte is om voor iedere snaar een magneet te plaatsen. Dit zou echter opgelost kunnen worden door de elektromagneten zelf te maken. De kracht van de elektromagneet is namelijk alleen afhankelijk van het aantal windingen en de stroom die door de windingen gaat. Aangezien er voldoende ruimte onder de snaren is, zou een elektromagneet gemaakt kunnen worden die dun en langwerpig is.
Tevens zou de magneet van de ene snaar de andere snaar kunnen verstoren aangezien het magneetveld te dicht bij de naastliggende snaren kan liggen. Hier is echter ook wel een oplossing voor te vinden. Bijvoorbeeld door een scherm tussen de magneten te plaatsen, wat magnetische veldlijnen tegenhoudt. Een ander idee is dat iedere magneet compenseert voor de storing van de naastliggende magneten.


===Nadelen===
Ondanks dat het volgens de berekeningen en specificaties van bestaande magneten mogelijk zou moeten zijn om de snaar met een elektromagneet in de gewenste trillingsvorm te brengen, zijn er twee belangrijke nadelen aan dit concept. Het eerste nadeel is dat de snaar in dit ontwerp niet meer aangestreken wordt. Gevoelsmatig behoort het ontworpen instrument dus al niet meer tot de strijkinstrumenten, ondanks dat het wel hetzelfde geluid zal voortbrengen als een strijkinstrument. Waarschijnlijk het grootste minpunt aan dit concept is dat de synthese van het geluid erg onlogisch is. Om de snaar in de gewenste trillingsvorm te brengen, moet namelijk eerst deze trillingsvorm bekend zijn. Het verkrijgen van deze trilling zal geen problemen geven. Dit kan of uit een model komen, of opgenomen worden van een strijkinstrument. Maar als deze trilling eenmaal bekend is, is het erg onlogisch om dit via een elektromagneet, door de snaar, door een trillingsopnemer, en dan nog door een versterker te laten gaan, om het uiteindelijk af te spelen. Dit omdat het geluid eigenlijk al afgespeeld kan worden zodra de trillingsvorm bekend is.
===Conclusie===
De conclusie over dit concept is dat het hoogstwaarschijnlijk zal werken en dat de trillingsvorm vrij gekozen kan worden. Dit weegt echter niet op tegen het nadeel dat er niet meer gestreken wordt en met name tegen het nadeel dat er een erg onlogische route gevolgd moet worden terwijl dit totaal onnodig is.
==Actuatie door piëzo==
===Principe===
Een piëzo actuator maakt gebruik van het verschijnsel dat kristallen van bepaalde stoffen uitzetten als er een spanning op komt te staan of omgekeerd, ze geven een spanning af als er druk op het kristal wordt uitgeoefend. De naam piëzo is dan ook afgeleid van het Griekse woord piezein, wat drukken betekent.
Het concept voor actuatie door piëzo, is dat er aan de snaar een piëzo-element wordt gekoppeld, wat uitzet als er een spanning op wordt gezet. Het idee is dat door de spanning te variëren, de beweging van de snaar gecontroleerd kan worden, waardoor de gewenste trilling in de snaar kan worden gebracht.
===Haalbaarheid===
Als er naar de haalbaarheid van dit concept wordt gekeken, blijkt de combinatie van benodigde kracht en benodigde uitwijking een moeilijkheid te vormen. Piëzo’s zetten namelijk maar erg weinig uit, variërend van enkele micrometers tot soms tientallen micrometers. Er zijn wel piëzo-elementen op de markt die dienen als lineaire motor en een maximale uitwijking hebben tot soms wel 100 mm, maar deze leveren weer niet de benodigde kracht. Een andere mogelijkheid is zogenaamde piëzo stacks gebruiken. Dit is als het ware een serieschakeling van piëzo’s waardoor de maximale uitwijking toeneemt. Deze uitwijking is echter ook nauwelijks meer dan 1 mm.
===Nadelen===
Het belangrijkste nadeel van actuatie met piëzo’s is dat de uitwijking die ze geven niet groot genoeg is. Dit is deels op te vangen door een stack te gebruiken, maar zelfs dan is de uitwijking nog niet groot genoeg om de snaar in de gewenste trillingsvorm te brengen. Daarnaast is er in dit ontwerp geen sprake meer van een strijkinstrument.
===Conclusie===
Uit het voorgaande kan geconcludeerd worden dat het niet mogelijk is om de snaar in de gewenste trilling te brengen door middel van piëzo actuatoren. Tevens zou er dan geen sprake meer zijn van een strijkinstrument, wat ook een zwaar wegende factor is.
==The bowing wheel==
===The bowing wheel===
[[File:boww.PNG|frame|Border| Figure 1.1: Simple drawing of the bowing wheel with the string.]]
[[File:boww.PNG|frame|Border| Figure 1.1: Simple drawing of the bowing wheel with the string.]]
[[File:boww1.PNG|frame|Border| Figure 1.2: In both drawings the black dot is the cross section of the string and the bowing medium is red. Left: with a bow the bow hairs tilt slightly towards the string around the bowing point. Right: the surface of the bowing wheel tilt away from the string around the bowing point.]]
[[File:boww1.PNG|frame|Border| Figure 1.2: In both drawings the black dot is the cross section of the string and the bowing medium is red. Left: with a bow the bow hairs tilt slightly towards the string around the bowing point. Right: the surface of the bowing wheel tilt away from the string around the bowing point.]]

Latest revision as of 13:12, 5 April 2011

Ways To Exite The String


William Schattevoet
David Duwaer
Eric Backx
Arjan de Visser


Subpages:


Main page

Working principle of the violin and predicting it’s behavior

Ways to exite the string

Hardware Design

Software Design (Quantifying the signal we want to see)

Realisation and Proof of concept

Patent Research

Background information: Interview with Hendrick Zick

Recommendations

Overview:


There are several possible ways to excite the string of the Viotar. The only two criteria are that a Helmholtz vibration is reached and that the Viotar remains playable as a normal guitar.

Four possible ways to excite the strings have been chosen for further research. The main focus of this research will be on feasibility of the construction, on whether Helmholtz can be achieved and if it’s possible to meet the design requirements. The four possibilities that will be researched are:

  • Electro-magnet
  • Piëzo-element
  • Bowing wheel
  • Bowing belt

The findings of these researches will be listed below, from the four possible ways the most feasible will be chosen for further research.


Ways to exite the string

Exitation by Electromagnet

Principle

Another way to exite the string is by using an electromagnet force on the steel strings. This force is created by the flow of current through a coil wrapped around a magnetizable core, creating an electro magnet. The force can be controlled by changing the current that flows through the coil. Because current can be controlled by speeds in the order of several Gigahertz, the flow can be controlled very rapidly.


Estimation for the maximum bow force on the string

Figure x shows the string on the moment it's being exited by the violin bow. The displacement of the string is called 'u', creating two angles α and γ. The string will be exited on position βL.


Figure x: The string exited over a distance u

The force in y-direction is calculated with:

[math]\displaystyle{ F_y=sin(\gamma)A\sigma_{1}+sin(\alpha)A\sigma_{2} }[/math]

[math]\displaystyle{ \sigma=E\varepsilon }[/math]

[math]\displaystyle{ \varepsilon= {dL\over L_0 } }[/math]

[math]\displaystyle{ F_y = AE \left (sin(\gamma){\sqrt{\beta^2 L^2+u^2}-\beta L \over \beta L} + sin(\alpha){\sqrt{(1-\beta)^2 L^2+u^2}-(1-\beta) L \over (1-\beta) L} \right) }[/math]


The biggest static force acting on a string is the force when a short, thick string is exited. It is assumed that the length of this string is 20 cm and the diameter is 0.25mm. The force acting on this particular string, exited over 0.5 cm, is about 40 N.

Estimation for the maximum dynamical force acting on the string

Since the string isn’t only exited statically but is being moved continuously, the static force calculated above isn’t the only force that has to be dealt with. The string should vibrate in a predefined pattern, and therefore big accelerations become important. The force needed to accelerate a certain mass m, is calculated using Newton’s second law of motion. This law states that this force is equal to acceleration times mass. Since the acceleration of the string varies linearly over the length of the string, the acceleration (and thus force) should be calculated separately for every small segment of the string. This gives rise to an integral over the whole string. In order to do this, the string is divided into two pieces, being the part βL and the part (1-β)L. This gives the following for the part βL:

[math]\displaystyle{ F = {ma} }[/math]

[math]\displaystyle{ F = \sum_{i=1}^{n}{m_ia_i} }[/math]

[math]\displaystyle{ F_1 = \sum_{i=1}^{n}{m_i\left( {da \over dx} x_i \right)} = \sum_{i=1}^{n}{m_i\left( {a \over \beta L} x_i \right)} }[/math]

[math]\displaystyle{ m_i = {\rho {\beta L \over n} A_{cross.sect.}} }[/math]

[math]\displaystyle{ n = {x \over dx} = {\beta L \over dx} }[/math]

[math]\displaystyle{ m_i = \rho A_{cross.sect.} dx }[/math]

[math]\displaystyle{ F_1 = \int_0^{\beta L} {{a A_{cross.sect.} \rho} \over {\beta L}} x \;dx }[/math]

[math]\displaystyle{ F_1 = {{a A_{cross.sect.} \rho} \over {\beta L}} \int_0^{\beta L} x \;dx }[/math]

[math]\displaystyle{ F_1 = {{a A_{cross.sect.} \rho} \over {\beta L}} [ \frac{1}{2} x^2 ]_0^{\beta L} }[/math]



And for the part (1-β)L:

[math]\displaystyle{ F_2 = {{a A_{cross.sect.} \rho} \over {(1-\beta) L}} [ \frac{1}{2} x^2 ]_0^{(1-\beta) L} }[/math]

Where:

a = The acceleration of the string on position βL
Across.sect. = The cross section area of the string
ρ = The density of the string


The total force needed to accelerate the string is given by:

[math]\displaystyle{ F_{tot} = F_1 + F_2 }[/math]

[math]\displaystyle{ F_{tot} = {{a A_{cross.sect.} \rho} \over {\beta L}} [ \frac{1}{2} x^2 ]_0^{\beta L} + {{a A_{cross.sect.} \rho} \over {(1-\beta) L}} [ \frac{1}{2} x^2 ]_0^{(1-\beta) L} }[/math]


This gives a total acceleration force of 6,25N when the following parameters are used:

a = 500 m/s2
Across.sect. = 5 ∙ 10-6 m2
ρ = 8 ∙ 103 kg/m3

The maximum total force, which is the force needed to give the string the maximum acceleration at the maximum position, is given by the sum of Fy and Ftot.

This means that the maximum magnetic force that is needed is about 50 N.

Feasibility

The calculation shows that a magnetic force of 50 N should be enough to exite the string in the desired pattern. Finding an electromagnet that generates this magnetic force is not a problem. However, the width of these electromagnets could be a problem, since there might not be enough space to place an electromagnet for each string separately. This problem could be solved by making the electromagnets on our own. The force that is generated in an electromagnet is only depending of the number of windings and the current through the coil. Since there is enough space underneath the strings, a long and tight electromagnet would work. Another point is that the magnet of one string might interfere with another and disturb it’s magnetic field. However, this problem would be quite easy to deal with as will since a screen could be placed in between the strings or the control device a magnet could compensate for disturbances from the others.

Drawbacks

Despite the theoretical possibility to use electromagnets for actuation, there are two very important drawbacks in this concept. Firstly, the instrument will no longer be bowed although it sounds as such. In fact the instrument then will no longer be a bowing instrument, which was one of the main requirements. But probably the biggest disadvantage of this concept, is the quite illogical way of synthesizing the sound. To get the string in the desired vibration, this vibration should be known in advance. Getting this vibration would be no problem, since we can get it out of the model or simply record it from a real bowing instrument. But once the vibration is known, it has to be exited on the string using electromagnets, recorded using a sensor and then amplified before it is played. This is very strange since we already had the desired vibration when we started, and thus were already able to play it.

Conclusion

The conclusion about this concept is that it would work most likely and that every vibration pattern could be chosen. However, the big drawbacks that it is no longer a bowing instrument and that the synthesizing routine is very illogical, make the concept fail.

Actuation by piëzo crystals

Principle

A piezo actuator utilizes the phenomenon that crystals of certain substances expand when a voltage is applied on it or conversely, they give a voltage when pressure is exerted on the crystal. The name piezo is derived from the Greek word piezein, which means pressing. The concept for actuation by the piezo, is that the string is attached to a piezo element, which expands when a voltage is applied. The idea is that by varying the voltage, the movement of the string can be controlled, giving the desired vibration in the string

Feasibility

If the feasibility of this concept is considered, the combination of required force and deflection seems to be a problem. The reason is that piezo elements only enlarge a little, varying from a few micrometers to some tens of micrometers. There are piezo elements on the market that serve as a linear motor and have a maximum amplitude up to 100 mm, but again does not provide the required force. Another possibility is to use so-called piezo stacks. This a series of piezos allowing the maximum deflection to increase. However, this excursion is also just a little more than 1 mm for the required force, which is not enough.

Drawbacks

The main disadvantage of piezo actuation is that the deflection they give is not large enough. This problem is partly dealt with using a stack, but even then the deflection is not large enough to make the string vibrate in the desired shape. Furthermore, in this design the instrument will no longer be a bowing instrument

Conclusion

From the foregoing it can be concluded that it is not possible to get the string in the desired vibration using piezoelectric actuators. Also the instrument would not be a bowing instrument anymore, which is an important factor in the design.

The bowing wheel

Figure 1.1: Simple drawing of the bowing wheel with the string.
Figure 1.2: In both drawings the black dot is the cross section of the string and the bowing medium is red. Left: with a bow the bow hairs tilt slightly towards the string around the bowing point. Right: the surface of the bowing wheel tilt away from the string around the bowing point.
Figure 1.3: The bowing wheel rotating clockwise blocks the string from moving back when the string is deflected.

One straightforward way to excite the string electronically is by using a “bowing wheel”, which contacts the string as shown in Figure 1.1.



As we know from practice, exciting the string using stick-slip with a bow works. It is where the notion of a good-sounding bowing sound comes from in the first place, so this should be the reference of how the bow-string interaction should be. A good way to see how well a bowing wheel can produce a sound like this is to look into how far a bowing wheel can act like a bow, and determine what exactly marks the essential difference with a bow. Finally it should be determined if this difference will pose a problem. A set of properties of the bowing medium that influence the bow-string interaction has been assembled:

  • The stiffness of the medium in z-direction
  • The stiffness of the medium in y-direction
  • The impulse response of the medium in z-direction
  • The impulse response of the medium in y-direction
  • All the surface properties of the medium (including the presence of rosin)
  • The shape of the medium directly surrounding the point of contact with the string

The bow’s stiffness in z-direction can be simulated using a bowing wheel by suspending the wheel in some way in z-direction. The bow’s stiffness in y-direction, coming from the longitudal stiffness of the bow’s hairs, corresponds with the rotational stiffness of the wheel in ϕ direction. The wheel can be suspended in this direction as well, so the bow’s stiffness in y-direction can be accounted for. The impulse responses in both y- and z-directions can be simulated in the control of the rotation of the wheel and the z-translation of the wheel, respectively. The surface properties can then theoretically match those of a mat of bow hairs when a mat of bow hairs is somehow applied to the outside of the wheel. This leaves only the property of the shape of the wheel to be compensated for. The wheel, as it is round, doesn’t have the same shape around the string as a bow. This is important because when the string transverses in y-direction (which will be the direction in which it will ideally vibrate), it keeps contact with a bow because a bow is almost straight, with the bow hairs actually tilting a bit towards the string around the bowing point, as in the left side of Figure 1.2. With a bowing wheel this is different, because it tilts away from the string around the bowing point, as shown in the right side of Figure 1.2.



This does not necessarily mean that the string loses contact with the wheel. This is because the wheel is pushed against the string with a certain force, so the string will have a static positive deflection in z-direction and is pulled down by its terminations. Because of this, as the string transverses, the string will keep contact with the bowing wheel. However, the bowing wheel is not capable of moving in y-direction, so when the string is deflected in y-direction the situation will look like Figure 1.3. In this situation the string is actually pressing harder against the wheel because of its own urge to go back to a neutral position. This causes the friction force of the wheel to increase, probably resulting in a blockage of the system.



This effect could hypothetically be prevented by the controller for the z-translation of the wheel. If the current friction force and normal force between the wheel and the string could be determined at every moment, the controller could “make sure” that these two stay at “natural” values (i.e. the same as when using a bow). This requires a constant reference for these both forces over the course of one vibration period, and a construction together with an actuator capable of following this reference with a total delay time including the sampling of the force measurements t_d that is much smaller than one vibration period, t_d≪T. This in addition with the sensors that are needed to measure the forces, makes bowing wheel a very unrealistic concept.