The Violin Bow: Taper, Camber and Flexibility

Colin GoughSchool of Physics and Astronomy, University of Birmingham, B13 9SN,UKa)

(Dated: 11/22/10)

An analytic, small-deflection, simplified model of the modern violin bow is introduced to describethe bending profiles and related strengths of an initially straight, uniform cross-section, stick as afunction of bow tension. A number of illustrative bending profiles (cambers) of the bow are consid-ered, which demonstrate the strong dependence of bow strength on longitudinal forces across theends of the bent stick. Such forces are shown to be comparable in strength to critical buckling loadscausing excessive sideways buckling unless the stick is very straight. Non-linear, large deformation,finite element computations extend the analysis to bow tensions comparable with and above thecritical buckling strength of the straight stick. The geometric model assumes an expression forthe taper of Tourte bows introduced by Vuillaume, which we re-examine and generalise to describeviolin, viola and cello bows. Our results are then compared with recent measurements by Graebnerand Pickering of the taper and bending profiles of a particularly fine bow by Kittel.

PACS numbers: 43.75.DE

Many string players believe that the bow has a directinfluence on the sound of an instrument, in addition tothe more obvious properties that control its manipulationon and off the string. It is therefore surprising that untilrelatively recently little research has been published onthe elastic and dynamic properties of the bow that mightaffect the sound of a bowed instrument.

The present paper investigates the influence of the ta-per and bending profile on the elastic properties of thebow, which could influence the quality of sound producedby the bowed string via their influence on the slip-stickgeneration of Helmholtz kinks circulating around the vi-brating string. The influence of the taper and camber onthe flexibility of the bow will also affect the vibrationalmodes of the bow, which will be discussed in a subsequentpaper.

Two scientifically and historically important mono-graphs on the bow were written in the 19th century.The first by Francois-Joseph Fetis1 was commissionedand published in 1856 by J.B.Vuillaume (1798 – 1875),the leading French violin maker and dealer of the day.The monograph not only describes and celebrates the in-struments of the great Cremonese violin makers, but alsogives an account of Vuillaume’s own historical and scien-tific research on the “modern” violin bow. This had rela-tively recently been developed by Francois Xavier Tourte(1750-1835), who was already viewed as the Stradivariusof bow making.

In 1896, Henry Saint-George2 updated information onbow makers and described subsequent research on thebow. In addition to translating Vuillaume’s earlier re-search, he describes later research and measurementsby the eminent Victorian mathematician W.S.B. Wool-house, FRS (1809-1893) - the eponymous owner of the1720 Woolhouse Strad - who collaborated with the dis-tinguished English bow maker James Tubbs (1835-1921).

a)Electronic address: profgough@googlemail.com

Woolhouse3 reflected the views of many modern perform-ers in believing that the purity of the vibrations of thebow were as important as the vibrations of the violin it-self. Both monographs cite expressions for the taper offine Tourte bows, but neither indicate how such formulaewere derived.

There was then a lengthy hiatus in serious researchon the bow until the 1950’s. Two interesting papers onthe properties of the bow were then published in earlyCatgut Society Newsletters by Maxwell Kimball4 andOtto Reder5. This was followed in 1975 by a pioneer-ing theoretical and experimental paper on the dynamicproperties of the bow and stretched bow hair by RobertSchumacher6.

Subsequently. Anders Askenfelt7–9 at KTH in Stock-holm published several important papers on the bow,more recently in collaboration with the distinguisheddouble-bass virtuoso and teacher Knutt Guettler10,11.Their later publications are mainly devoted to the in-fluence of the bow-string interaction on the sound ofan instrument. George Bissinger12 has investigated themode shapes and frequencies of the bow plus tensionedbow hair and has highlighted the importance of thelow-frequency bouncing modes, especially for short bowstrokes. These modes have also been investigated byAskenfelt and Guettler10.

The present paper is largely devoted to models thatdescribe the static elastic properties of the bow, whichprovide the theoretical and computational framework todescribe their dynamic properties, to be described in asubsequent.

The paper is divided into three main sections followedby a discussion of the results and a final summary.

The first section introduces a small deflection, analyticmodel for the bending modes of a simplified constantcross-sectional area bow. This illustrates the dependenceof bow strength on the initial bending profile (camber)and bow tension. Three illustrative bow cambers are con-sidered equivalent to bending profiles generated by var-ious combinations of forces and couples across the ends

1

of the bent stick.The second section revisits and generalises Vuillaume’s

geometric model and related mathematical expression forthe tapered diameter of Tourte bows along their lengths.This is the taper chosen for the large deformation, non-linear, finite element analysis described in the third sec-tion. These computations extend the analysis to lon-gitudinal tensions between the ends of the bent stickapproaching and above the critical buckling load of thestraight stick ∼ 75N, comparable in size to typical play-ing tensions of ∼ 60N (Askenfelt).For small bending profiles, the finite-element compu-

tations reproduce the predictions of the simple analyticmodel remarkably well, justifying the most importantsimplifying assumption of the analytic model, but ex-tending the predictions to describe the asymmetries in-troduced by the tapered stick diameter and different ge-ometries of frog and head of the bow.

I. A SIMPLE BOW MODEL

A. Introduction to bending of bow stick

The 1-dimensional in-plane bending of a tapered bowstick is governed by the bending equation,

EI(s)d2y

ds2= M(s), (1)

where E is the elastic constant for stretching and com-pression above and below the neutral axis, relative towhich the perpendicular deflection y is measured. Dis-tances s are measured along the deformed neutral axis.The bending moment M(s) generated by couples andforces acting on the stick will, in general, vary with po-sition along the length of the stick.I(s) is the second moment of the cross-sectional

area∫area

y2dxdy, which for a constant radius a stick

is πa4/4. For a stick with octagonal cross-section, I =0.055w4, where w is the width across opposing flat sur-faces. For the short, quasi-elliptical, transitional cross-section between the stick and head of the bow, I =πab3/4, where b is the larger, in-plane, semi-axis.In general, the camber will be a function of both the

bending moment along the length of the stick and its lo-cal radius. When equal and opposite couples are appliedacross the ends of a constant cross-sectional radius stick,it will be bent into the arc of a circle with radius of cur-vature R(s) =

(d2y/ds2

)−1 ∼(d2y/dx2

)−1, where the

approximation for the curvature involves second-order,non-linear, corrections in (dy/dx)

2. The curvature of a

tapered stick will therefore vary along its length - in-versely proportional to the fourth power of its diameter.For a tapered Tourte bow, the diameter of the stick

changes from 8.6 mm at the frog end of the bow to 5.3mm at the tip end (Fetis1). When opposing couples areapplied across its ends, the curvature will therefore be al-most seven times larger at the upper end of the bow thannear the frog. Graebner and Pickering13 have suggestedthat the couple-induced curvature is the ideal camber for

a high quality bow. This suggestion will be assessed inthe light of the computations described in this paper.

In the following section, we initially consider the bend-ing of the bow based on a simplified model similar to thatused by Graebner and Pickering. The bow is of length ℓand constant cross-sectional radius a, with perpendicularrigid levers of height h at each end representing the frogand the head of the bow, between the ends of which thehair is tensioned.

For small bending angles θ, we can make the usualsmall deflection approximations with x = s measuredalong the length of the initially straight stick, θ ≃ tan θ ≃sin θ ≃ dy/dx and cos θ ≃ 1 and R(s) =

(d2y/dx2

)−1.

B. An analytic model

We adopt a linear analysis similar to that used by Tim-oshenko and Gere14 in Chapter 1 of their classic treat-ment of The Theory of Elastic Stability. This providesan invaluable introduction to the bending of straight andbent slender columns like the violin bow, under the com-bined influence of external moments and both lateral andlongitudinal forces.

First consider the bow stick bent by equal and oppo-site couples and forces of magnitudes C and F across itsends, promoting the upward deflection illustrated in fig.1.Imagine a virtual cut across the bent bow at a height y.In static equilibrium, the applied bending couples andforces acting on the separated left-hand section of thestick must be balanced by equal and opposite bendingmoments and shearing forces exerted on it by the adjoin-ing length of the stick. The bending moment along thelength of the stick must therefore vary as C+Fy, result-ing in a curvature along the length satisfying the bendingequation

EId2y

dx2= −(C + Fy), , (2)

which may be written as

d2y/dx2 + k2y = −C/EI, (3)

where

k2 = F/EI. (4)

These are the defining equation used throughout thispaper to describe various physically defined bending pro-files or cambers of the initially untensioned bow and theinfluence on such profiles of bow tension.

Because the displacement must be symmetrical andzero at both ends, the solutions can be written as

y = A sin kx sin k(ℓ− x) (5)

=C

F

[cos kℓ/2− cos k(ℓ/2− x)]

cos kℓ/2, (6)

2

y

C

M= C+Fy

F

F

FCx

FIG. 1. Equilibrium of bow bent by equal and opposite cou-ples and forces across its ends

which then satisfies the boundary condition d2y/dx2 =−C/EI at both ends (y = 0).The maximum deflection at the mid-point is given by

yℓ/2 =Cℓ2

8EI

2(1− cosu)

u2 cosu=

Cℓ2

8EIλ(u) (7)

where u = kℓ/2=(ℓ/2)√

F/EI.In the above expression, λ(u) is essentially a force-

dependent factor amplifying the deflection that wouldhave resulted from the bending moment alone, given bythe pre-factor in equ.7.The deflection becomes infinite when kℓ/2 = π/2. This

corresponds to a force FEuler = EIπ2/l2, which is themaximum compressive load that the straight stick, withhinged supports at both ends, can support without buck-ling.For compressive forces well below the critical limit,

λ(u) ≈ 1

1− T/TEuler, (8)

an approximation that differs from the exact expressionby less than 2% for compressive loads below 0.6 of theEuler critical value.In practice, the bending always remains finite, even for

forces well beyond the critical load. This is because, asthe bending increases, the two ends of the stick are forcedtowards each other. The external force then does addi-tional work introducing non-linear terms in the deflec-tion energy analysis, which limits the sideways deflection.This was already understood and mathematical solutionsobtained for deflections above the critical load by Eulerand Lagrange in the late 18th century (see Timoshenko15

History of Strength of Materials, pages 30-40 ).The neglect of the changes in length between the two

ends of the bent stick, limits the above linear analysis tocompressive forces significantly smaller than the criticalbuckling load. As we will show, the critical buckling ten-sion for a tapered violin stick is around 75 N for pernam-buco wood with along-grain elastic constant of 22 GPa.This is only slightly larger than typical bow tensions ofaround 60 N bow tensions (Askenfelt).Analytic expressions for large deflection bending were

published by Kirchhoff16 in 1859 based on a close anal-ogy between the bending of a slender column and thenon-linear deflections of a simple rigid pendulum. Theanalytic solutions of what is known as the elastica prob-lem involve complete elliptic integrals (see Timoshenko

and Gere14, art 2.7, pp 76-81). Alternatively, as in thispaper, large deflection, non-linear finite element compu-tations can be used to describe such bending, which alsoallows the influence of the tapered stick and geometry ofthe frog and head of the bow to be included.

A similar amplification factor applies to any initialsideways bending of the bow stick (Timoshenko andGere, art 1.12). This explains why bows with only aslight initial deviation from sideways straightness becomevirtually unplayable on tightening the bow because of ex-cessive sideways bending. In contrast, the downward cur-vature of the modern bow is in the opposite sense to thatproduced by the bow tension, which inhibits in-plane in-stabilities.

When the bending moments are reversed, the sense ofdeflection is simply reversed. However, reversing the di-rection of the longitudinal load, F → −F , has a moreprofound affect, with k2=F/EI now becoming nega-tive.Because cos ikx → cosh kx, the deflections now varyas

y(x) =C

F

[cosh kℓ/2− cosh k(ℓ/2− x)]

cosh kℓ/2. (9)

There is no longer a singularity in the denominator.Furthermore, the extensive load now tends to decreaseany bending initially present or induced by external cou-ples or forces.

C. Setting of initial camber of bow.

In practice, the bow maker adjusts the camber of abow by bending the stick under a gentle flame. Thislocks in internal strains equivalent to those that couldhave been provided by an equivalent combination of cou-ples and forces applied across the ends of the stick or bydistributed lateral forces along its length. Timoshenkoand Gere14, §1.12, provide several examples of the use ofsuch an approach to describe the bending of an initiallycurved bar.

Bow makers usually assume that a correctly camberedbow should pull uniformly straight along its length ontightening the bow hair. They then carefully adjust thebending profile to correct for changes in elastic propertiesand uneven graduations in taper along the length of thebow. We will show that the strength of the bow - theway the stick straightens on tightening the bow hair ordeflects on applying downward pressure on the string - isstrongly related to the camber of the bow.

The earliest suggestion for the most appropriate cam-ber of the modern bow stick was made by Woolhouse3:

Let a bow be made of the proper dimen-sions, but so as to be perfectly straight; thenby screwing it up in the ordinary way, it wouldshow, upside down, the exact curve to whichother bows should be set.

We will refer to such a profile as the mirror profile, asit is the mirror image of the upward bending profile of

3

the tensioned straight stick when inverted. A bow withsuch an initial camber would indeed straighten uniformlyalong its length on tightening.John Graebner and Norman Pickering13 have recently

proposed that the ideal camber would be one generatedby equal and opposite couples across the ends of the ini-tially straight stick. Their measurements provide per-suasive evidence that this might indeed be true for anumber of fine bows. We refer to this as the coupleprofile. Although such a stick with a uniform diame-ter would straighten uniformly along its length, we willdemonstrate that this is no longer true for a tapered stick.Moreover, such a stick must have a high elastic constant,if it is not to straighten prematurely before a satisfactoryplaying tensions could be achieved.In this paper, we also consider a bending profile equiv-

alent to that produced by an outward force between theends of the frog and head of the bow, in the oppositedirection to the hair tension. We refer to this as thestretch profile. Such a bow would also straighten uni-formly along it length on tightening, but would be signif-icantly stronger than bows withmirror or couple bendingprofiles.The equivalent couples and forces used to generate the

above three bending profiles are illustrated in figs.2a-c,while figs.2d-f illustrate the additional forces and couplesgenerated by the bow tension on the above cambers.

D. Influence of bow tension on bending profile.

The deflections of the stick are assumed to be wellwithin the elastic limits of the pernambuco wood used forthe stick. The net forces and couples acting across theends of the stick are therefore the sum of the equivalentforces ±F and ±C used to set the untensioned camberand the forces ±T and couples ±hT from the tensionof the hair stretched between the end levers representingthe frog and head of the bow. However, the deflectionsfrom the two sets of forces and couples are not necessarilyadditive because of the non-linear relationship betweenthe couple-induced deflections and the compressive or ex-tensive loads along the length of the stick.For the mirror and stretch bending profiles under ten-

sion, the longitudinal forces are compressive, so the de-flections under tension are given by equ.6 with C =h(F − T ) and longitudinal force F + T for the mirrorprofile, so that

ymirror(T ) =F − T

F + Th[cos kℓ/2− cos k(ℓ/2− x)]

cos kℓ/2, (10)

with k2 = (F + T )/EI, and

ycouple(T ) =M − hT

T

[cos kℓ/2− cos k(ℓ/2− x)]

cos kℓ/2, (11)

with k2 = T/EI for the couple profile.For the stretch profile, the net longitudinal forces (F >

T ) across the ends of the stick are extensive, so that

ystretch(T ) = h[cosh k(ℓ/2− x)− cosh kℓ/2]

cosh kℓ/2, (12)

with k2 = (F − T )/EI .The camber of the untensioned bow is given by letting

T → 0. The resulting cambers for above three bendingprofiles are shown in fig.3, where the equivalent couplesC and forces F setting the initial camber have been cho-sen to give the cited mid-point deflections. For relativelysmall deflections, the three bending profiles are almostindistinguishable. However, for all deflections, the couplebending profile has a slightly larger curvature towards theends than the mirror profile. As the mid-point deflectionincreases the stretch profile becomes progressively flatterin the middle of the bow, as the maximum deflectiongiven by equ.12 can never exceed the lever height h, re-quiring a much larger curvature towards the ends of thebow.

The solid lines in fig.4 plot the mid-point deflectionsas a function of the generating forces and couples used toset the mirror, couple and stretch profiles for our simplebow model. Note that, although the stresses and strainsin the stick are assumed to be elastic, the deflectionsof the mirror and stretch profiles are highly non-linear.This is a geometric effect arising from the non-linear am-plification of bending profiles by the longitudinal forcesas the bending changes (eq.2.

In plotting fig.4 a value of EI has been chosen to give acritical buckling load of 78N, at which force the analyticexpression for the mirror profile becomes infinite, un-physical and changes sign - indicated by the vertical line.Using this value as the only adjustable parameter, theanalytic small deflection predictions for all three bendingprofiles are almost indistinguishable from the open sym-bols plotting the maximum downward deflections of thetapered Tourte bow computed in section 3. Such compu-tations include the changes in geometry, hence bendingmoments, as the bending increases.

The excellent agreement between the analytic andour later non-linear finite element computations justifiesmany of the approximations made in the small deflectionanalysis and justifies its use to illustrate the most im-portant qualitative static and dynamic properties of thebow. Nevertheless, finite-element computations are stillrequired to account for the asymmetric properties asso-ciated with the taper and different dimensions of the frogand head of the bow.

Bows with the above bending profiles will straightenwhen the couples acting on the stick from the hair tensionacross the frog and tip of the bow are equal and oppositethose used to generate the initial untensioned camber. Abow with the mirror camber and a downward deflectionof 16mm would therefore straighten with a bow tension of30N, only bout half the normal playing tension of around60T (Askenfelt7). A Tourte bow with a couple bendingprofile would also straighten before a playing tension of60N could be achieved, unless its elastic constant wassignificantly higher than 22 GPa or it had a thicker di-ameter to increase the critical buckling load. In contrast,a bow with a pure stretch bending profile could support

4

T

C=M

h--

F

F

C=hF C= hF

F

(a) (b) (c)

F+T

C=h(F-T)

F

F

F

F

F

mirror couple stretch

F

C=h(F-T)F

F+T

T T F T T FT

C= M C=hF C=hF

(d) (e) (f)

C=M-hT C=M-hT C=h(F –T) C=h(F-T)

F-T F-T

h

h

h

h

h

h

FIG. 2. Figures a-c illustrate the mirror, couple and stretch bending profiles generated by equivalent couples and compressiveor extensive forces between the ends of the bent stick. Figures d-f illustrate the additional forces and couples generated by thebow tension between frog and tip of the bow, with both sets of forces and couples defining the net forces and couples acrossthe ends of the stick determining its bending profile.

-1

-0.8

-0.6

-0.4

-0.2

0

0 0.2 0.4 0.6 0.8 1

0.5 h mirror0.5 h couple0.5 h stretch0.7 h mirror0.7 h couple0.7 h stretch0.9 h mirror0.9 couple0.9 stretch

normalised length

defle

ction/h

FIG. 3. The mirror, couple and stretch bending profiles formid-point deflections of 0.5, 0.7 and 0.9 the end-lever heighth. The model assumes an initially straight bow stick of length70 cm, with uniform cross-sectional area and h = 2 cm.

very large bow tensions without straightening apprecia-bly. Bows of intermediate strength could be produced bysetting a camber equivalent to the bending profile pro-duced by a combination of a couple across the bow stickand a stretching force between the frog and head of thebow.Whatever profile is used by the maker to set the in-

plane rigidity, the critical buckling tension of the straightstick must always be significantly larger than the requiredplaying tension, to avoid major problems from out-ofplane buckling.The added strength of the stretch profile arises from

the increased curvature at the ends of the stick, which ina real bow compensates for the loss in bow strength fromthe reduction in diameter towards the tip. In practice,

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80 100

couple (cm)stretch (cm)mirror (cm)couple fit mirror fitstretch fit

do

wn

wa

rd d

eflectio

n

(cm

)

Force or Couple/lever height (N)

FIG. 4. The solid lines represent the bending couples C = Fhand forces F required to set the mirror, couple and stretchprofiles as a function of the mid-point deflection of the 70cm long, uniform diameter, initially straight stick with h=2cm levers on its end, for an assumed critical buckling loadof ∼ 78 N indicated by the vertical line. The open symbolslying almost exactly on top of the solid lines plot the maxi-mum deflection at around 30 cm from the end of the taperedTourte stick evaluated using non-linear finite-element softwareassuming a uniform elastic constant of 22 GPa.

the bow maker can choose any bending profile they wishand are not constrained to the specific examples we havechosen to consider here. However, for a tapered bow, onlythemirror or stretch bending profiles will pull uniformlystraight on tightening the bow hair.

5

II. THE TAPERED BOW

The taper used in our finite element computations willbe based on Vuillaume’s algebraic expression describinghis measurements on a number of fine Tourte bows. Vuil-laume noted that all such bows were tapered in much thesame way, with the stick diameter decreasing by a givenamount at almost exactly the same positions along theirlength. As cited by Fetis1, he therefore devised a geo-metric construction, so that these positions

might be found with certainty - by which,consequently, bows might be made whose con-dition should always be settled a priori.

Fetis1 gives the following mathematical expression,

d(x) = −6.22 + 5.14 log(x+ 175), (13)

derived by Vuillaume from his geometric model. Thisgives the tapered diameter d as a function of distancex from the end of the bow stick, with all dimensions inmm.Unfortunately, no explanation appears to have been

given for either the geometric model or mathematical ex-pression derived by Vuillaume and later by Woolhouse3

from an independent set of measurements on Tourte vi-olin, viola and cello bows.We therefore re-derive and justify the use of the Vuil-

laume geometric model and related mathematical expres-sion. We also show that the later Woolhouse expression,as originally correctly cited in mm, is virtually identicalto Vuillaume’s earlier expression.The geometric construction developed by Vuillaume

is illustrated in fig.5 together with a photograph of afine Tourte bow and a plot of the tapered diameter onwhich the geometric model and associated mathematicalexpression were based.The bendable length of stick from the frog end to the

head of the Tourte bow was measured by Vuillaume asℓ =700 mm, of which the first 110 mm had a constantdiameter of 8.6 mm. The remaining 590 mm was ta-pered towards the upper end with a diameter of 5.3 mm.Vuillaume noted that the tapered section could be di-vided into 11 sections marking the lengths over eachof which the diameter decreased by 0.3 mm, with thelength of each section decreasing by the same fractionβ, as illustrated schematically in fig.5. The decrease indiameter over consecutive sections is therefore describedby an arithmetic progression, while the decrease in sec-tion length is described by a geometric progression. Thisforms the basis of Vuillaume’s geometric model and re-lated mathematical expression.The model is generated by first drawing a base line of

length 700 mm to represent the length of the stick. Atthe frog end of the bow a vertical line is raised of heightA=110 mm equal in length to the initial constant radiussection of the bow. At the other end of the base-line asecond vertical line of height B = 22 mm is raised and astraight line drawn between their ends intercepting theaxis a distance 175 mm (ℓ/4) beyond the end of the stick.

A compass was then used to mark out 12 sections alongthe base line. With the point of the compass at the originof the base line, a point was first marked off along thebase line equal to the height of the first perpendicularrepresenting the initial 110 constant diameter section ofthe stick. At the end of this section, a new vertical linewas raised to intersect the sloping line. Using a compass,this length was added to the first section. The processwas then repeated until the remaining sections exactlyfitted into the length of the bow stick. The height ofthe 22 mm upright at the end was chosen so that thefractional decrease in length β = 1 − (A − B)/700 =0.874 between successive sections had the correct valueto allow this perfect fit. Apart from the initial, constant-diameter, section, each subsequent section marked thelengths over which the diameter decreased by 0.3mm (3.3mm in total).

Vuillaume presumably determined the length B bytrial and error. Mathematically, this requires the sum ofthe lengths A

∑11o βn= 700 mm, with a ratio of lengths

B/A =β12=22/110=1/5. This gives the above value for

β with A∑11

o βn= 700.5mm and B=Aβ12= 21.94 mm,in close agreement with Vuillaume’s choice of dimensionsfor his geometric model.

This simple geometric construction enabled Vuil-laume’s bow makers to graduate bows with the same ta-per as Tourte bows. However, it is most unlikely thatTourte would have used such a method. Like all greatbow makers, Tourte almost certainly had an innate feel-ing for the appropriate taper and camber of the bowbased on the flexibility of the stick and an aesthetic sensethat so often mirrors the elegance and beauty of simplemathematical constructs.

Vuillaume’s related mathematical expression for thetaper can be derived as follows. From the scaling of tri-angles with equal internal angles, the distance xn of then-th upright from the point of intersection of the slopingline with the base-line satisfies the recurrence formula,

xn = βxn−1. (14)

By iteration,

xn = xoβn, (15)

where xo is the distance of the far-end of the stick fromthe point of intersection of the sloping line. Each value ofn>0 corresponds to the position along the stick at whichthe diameter of the following section decreases by t= 0.3mm. The diameter dn at xnis therefore given by

dn = do − (n− 1) t, (16)

where d0 is the diameter of the initial section.To obtain the algebraic relationship between dn and

xn, the logarithm is taken of both sides of equ.15. Thevalue of n can then be replaced in equ.16 to give

dn = (do + t)− t [(log xn − log x0)/ log β] . (17)

6

r70 =

8.6 mmr0=

5.3 mm

110 mm

A =

110 mm B = 22 mm

L= 700 mm

xtapered section

L/4 = 175 mm

A A 2A3A

0x 1x 2x 3x 4x 5x 6x 7x 8x 9x 10x 11x 12x

d0=

d70=

FIG. 5. Vuillaume’s geometric model reproducing his measurements of the taper of Tourte bows, with a photograph of a bowon the same physical scale and a plot illustrating the functional form of the tapered stick diameter along its length.

This progression can be described by the continuous func-tion

d(x) = C −D log x, (18)

where x is still measured from the point of intersectionof Vuillaume’s sloping line, 175 mm beyond the end ofthe stick, D = t/ log β and C −D log x1 is the diameterat the end of the initial 110 mm constant cross-sectionlength (x1 = 175 + 700 − 110 = 765 mm). Insertion ofthe above parameters in equ.18 and redefining x as thedistance measured from the head-end of the tapered stickgives

d(x) = −6.22 + 5.14 log (x+ 175), (19)

with values of the constants essentially identical to thoseof the Vuillaume expression equ.13 cited by Fetis. Notethat this formula only applies to the upper 590 mm ta-pered section of the bow - the lower 110 mm has a con-stant diameter of 8.6 mm and includes a holding sectionthat is lapped with leather with a metal over-binding toprotect the stick from wear.The mathematical expression replaces the discrete

point values of the geometric model with a continuousexpression. It is clearly independent of the number ofsections over which the changes in diameter were origi-nally measured.It says much for French education at the time that

Vuillaume had the necessary mathematical skills to de-vise both the geometrical model and the equivalent math-ematical expression. It is, of course, likely that he col-laborated in their derivation with Felix Savart or someother scientifically trained researcher.Somewhat later, Woolhouse3 developed a similar for-

mula based on his independent measurements of thetaper of Tourte violin, viola and cello bows. Wool-house’s measurements are tabulated in Saint-George’smonograph2 along with an incorrect expression for the

5

6

7

8

9

10

0 100 200 300 400 500 600 700 800

Cello measVla measVln measVF celloVF violaVF fit to Woolhouse violinVF violin

distance from end of stick (mm)

dia

mete

r of bow

stick (

mm

)

FIG. 6. Measured tapers of violin, viola and cello bows byWoolhouse, with dashed lines representing fitted plots of thegeneralised Vuillaume-Ftis expression for the taper.

diameter expressed in inches. However, in his originalpublication, Woolhouse initially cites the diameter in mmas

d(x) = −6.17 + 5.08 log(x+ 184), (20)

before incorrectly transcribing it into inches. The aboveexpression differs from Vuillaume’s expression, equ.13,by less than 1% over almost the whole length of the bow,which is almost certainly within the accuracy of theirmeasurements.

7

It is not obvious why Tourte used the particular taperdescribed above, other than to reduce the weight at theend of the bow without over-reduction of its strength.Other smoothly varying tapers can be devised betweenfixed upper and lower diameters dU and dL of the generalform

d(x) = dU + (dL − dU )log[x/K + 1]

log[L/K + 1], (21)

where L is the length of the tapered section of the bowand K is a scaling distance equivalent to the distanceof the point of intersection of the sloping line beyondthe end of the bow in Vuillaume’s geometric model. Keffectively defines the length scale from the end of thebow over which most of the taper occurs. For Tourtebows, the Vuillaume taper parameterK ∼ ℓ/4 =175 mm,where ℓ is the total bow length.As K is increased, the taper is more uniformly dis-

tributed over the length of the stick, approaching that ofa simple truncated cone for large K-values, while smallervalues shift the major changes in tapering further to-wards the tip of the bow. The Woolhouse expressioncorresponds to a very slightly larger K−value than thatderived by Vuillaume.Figure 6 compares the tapers of fine violin, viola and

cello bows measured by Woolhouse3 and reproduced bySaint-George2. The dashed lines drawn through the mea-sured data points are generalised Vuillaume plots, withconstants chosen by hand to pass through the measuredvalues. The constants clearly have to be varied to de-scribe the larger diameters and slightly shorter lengthsof viola and cello bows, though the general form of thetaper remains much the same. Vuillaume derived a veryslightly larger thicker section for the bows he measuredillustrated by the solid line. The main difference betweenviolin, viola and cello bows is their slightly larger diam-eter and hence weight.

III. FINITE ELEMENT COMPUTATIONS

A. Geometric model and buckling

We now use the Vuillaume expression for the taperedTourte bow to describe the taper of our finite-elementbow model illustrated schematically in fig.7. A uniformelastic constant of E=22 GPa along the grain is assumed.The bow stick was divided into 14, equal-length, conical,sub-domains describing the tapered length of the stickwith a crudely modeled rigid bow head and frog at itsends. The bow hair was simply represented by the bowtension between its points of attachment on the undersideof the frog and head of the bow.Because the structure is relatively simple, at least in

comparison with the violin, we chose to perform a 3-dimensional analysis of the bending modes. Each subdo-main was divided into a medium-density mesh resultingin typically ∼10 K degrees of freedom. Deflections wereconfined to the plane of symmetry passing through the

FIG. 7. Finite element geometry of a Tourte-tapered violinbow stick with attached head and frog having effective leverheights from the neutral axis of 18 mm at the head of the bowand 24 mm at the frog.

-0.05

0

0.05

0.1

0.15

0.2

0 20 40 60 80 100

Force (N)

y(m

)

FIG. 8. Large-deformation, non-linear, finite-element compu-tations of the deflection of the mid-point of a Tourte-taperedbow stick as a function of bow tension, first applied in thenormal way across the ends of the frog and head of the bowand then only slightly offset from the central neutral axis ofthe stick by the stick radius. The associated bending profilesof the tapered stick are illustrated to exact scale.

stick, frog and head of the bows. The out-of-plane de-flections will be discussed in a later paper on vibrationalmodes.

The initially straight stick was bent by the same com-binations of forces and couples used to set the bendingprofile of the uniform diameter bow stick. Using COM-SOL linear FEA software on a modest PC, we were ableto compute bending profiles for small deflections in a fewseconds, while non-linear analysis, which takes into ac-count all changes in geometry on bending, typically tooka few tens of seconds. This increased to several minutesto compute the profile of a stick under the influence ofvery large bending couples and forces, when the two endsof the bow are forced to move close together.

8

As an initial example, fig.8 illustrates the bending pro-files of an initially straight bow as a function of bow ten-sion applied, first across the ends of the frog and headof the bow in the normal way and then slightly offsetfrom the central stick axis by the radius of the stick. Inboth cases, the point at which the tension is applied atthe frog-end is pinned allowing the bow to rotate, whilethe tip end of the hair is constrained to move in the di-rection of the applied tension. The plots illustrate therelative motion of the two ends of the tensioned hair to-wards each other, while the figures show the associatedbending profiles.For the slightly offset tension, there is very little bend-

ing or inward motion of the two ends until the Eulercritical load of TEuler = E ⟨I⟩π2/L2 ∼ 75N is reached.At this tension, there is a fairly sharp transition to thebuckled state, with the two ends of the stick forced tomove towards each other. ⟨I⟩ is the appropriately av-eraged second moment of the cross-sectional area of thetapered stick. The computations show that the criticalload of a straight stick with the Tourte taper is the sameas that of a constant radius stick of diameter 6.4 mm -somewhat larger than the 5.3 mm diameter at the tip endof the tapered bow but smaller than the 8.6 mm radiusat the frog-end.In contrast, when the tension is applied across the frog

and tip, the couples across the ends of the bow resultin a significant amount of bending (buckling) well be-low TEuler. There is then a continuous transition to thehighly buckled state with no sudden instability at TEuler.This is exactly what one would expect, as it correspondsto the drawing up of a simple hunting bow as the stringtension across its ends is increased. Because of the ad-ditional couple from the lever action of the bow tensionacting on the frog and head of the bow, the amount ofbending is always larger than the buckling produced bycompressive forces across the ends of an initially straightstick. The amount of bending at low bow tension isclearly proportional to the couple from the offset com-pressive load, which is determined by the frog and headlever heights.The non-linear finite element analysis extends the com-

putations to the whole range of possible bending forcesand bow tensions, whereas the analytic, small tensionanalysis is limited to compressive forces across the ends ofthe bow significantly smaller than TEuler, beyond whichthe solutions become unphysical.

B. Bending profiles of the tapered stick

We now consider the bending of an initially straightTourte-tapered bow by the same combinations of cou-ples and forces used to generate the couple, mirror andstretch bending profiles of the constant diameter bowstick described in the earlier analytic section. We havealso added a weight bending profile generated by addinga weight to the mid-point of the horizontal stick.Figure 9 illustrates the development of the bending

profiles as the couples and forces are increased in equalsteps over the ranges indicated. The solid lines show

the deflections computed using large-deformation, non-linear, finite-element software, while the dashed linesshow the computed profiles from linear, finite-elementanalysis. As a result of the taper, the maximum down-ward deflection of the stick is shifted to a position about5 cm beyond the mid-pint of the stick.

The difference between the two sets of computed bend-ing profiles occurs because the linear analysis neglects thechanging influence of the longitudinal forces on the stickas the bending changes (equ.2). This results in non-lineardeflections of the bow despite the strains within the stickitself remaining well within the linear elastic limit. Thelinear analysis only involves the contribution to the bend-ing of the initially straight stick from the couple actingon the stick and not from the longitudinal force, whichonly contributes to bending once bending has occurred.

The compressive forces across the ends of the bow gen-erating the mirror profile enhance any bending alreadypresent. This results in a non-linear softening of the bowas the bending is increased. In contrast, the extensiveforces generating the stretch profile tend to inhibit fur-ther increases in bending resulting in a non-linear in-crease in rigidity of the stick. Such non-linearities arereferred to as geometric non-linearities and occur eventhough the strains within the stick itself remain wellwithin the linear elastic limits.

For the couple and weight profiles, the deflections aresimply proportional to the bending forces, as there areno longitudinal forces giving rise to non-linearity.

The open symbols in figure 4 plot the computed maxi-mum downward deflections of the tapered Tourte for thebending forces and couples generating the mirror, coupleand stretch profiles. As remarked earlier, the computeddeflections are in excellent agreement with the mid-pointdeflections predicted by the simple analytic model. Theanalytic deflections are represented by the solid lines,with forces scaled to give the same critical buckling load.Such agreement is unsurprising, as the deflections - of or-der the frog and bow head heights - are small comparedto the length of the bow stick. The geometric approx-imations made in the analytic model are therefore welljustified. However, finite element computations are nec-essary to investigate the asymmetries produced by thetaper and the different geometries of the frog and headof the bow.

C. Influence of bow tension

Figure 10 illustrates the influence of bow tension onthe four bending profiles generated by external couplesand forces chosen to give a downward stick deflection of∼16 mm at around 30 cm from the end of the taperedstick. The plots also highlight the differences in initialbending profiles before the hair tension is applied. Forexample, the stretch profile is very much flatter in themiddle of the bow with much more curvature towards itsends than the mirror profile, confirming the results forthe symmetric, constant radius bow model in fig.3.

As expected, a bow with the mirror profile is theweakest and pulls “straight” with a bow tension of only

9

couple mirror

stretch weight

0 10 20 30 40 50 60 cm

0

-10

-2 0

mm

0 10 20 30 40 50 60 cm

0 10 20 30 40 50 60 cm 0 10 20 30 40 50 60 cm

0

-10

-2 0

0

-10

-2 0

0

-10

-2 0

mm

mm mm

C = 0 – 1.2 Nm F = 0 – 35 N

F = 0 – 100 NW = 0 – 12 N

couple mirror

stretch weight

0 10 20 30 40 50 60 cm

0

-10

-2 0

mm

0 10 20 30 40 50 60 cm

0 10 20 30 40 50 60 cm 0 10 20 30 40 50 60 cm

0

-10

-2 0

0

-10

-2 0

0

-10

-2 0

mm

mm mm

C = 0 – 1.2 Nm F = 0 – 35 N

F = 0 – 100 NW = 0 – 12 N

FIG. 9. The dependence of the bending profiles of an initially straight Tourte-tapered bow as a function of the forces andcouples used to generate them plotted for equal steps of the forces and couples involved. The dashed lines are the computedprofiles computed using linear finite element analysis and the solid lines show the profiles predicted by large-deformation,non-linear computations.

∼ 30 N. Bows with the couple and weight profiles areconsiderably stronger requiring a bow tension of around50N to straighten, while the bow with the stretch pro-file is very much stronger requiring a bow tension of150N to straighten. Only tapered bows with mirror andstretch untensioned cambers straighten uniformly alongtheir length.The computations confirm that a Tourte taper with a

couple bending profile would be unable to support a typ-ical bow tension of 60T unless its along-grain elastic con-stant was well in excess of the assumed value of 22GPa.Furthermore, Fig.10 shows that such a bow would notstraighten uniformly on tightening. The computationsalso confirm that increasing the curvature towards theupper end of the bow increases the strength or rigidityof the bow.

IV. DISCUSSION

Although our computations have focused on the bend-ing of violin bows, the results can easily be scaled toviola, cello and double-bass bows as demonstrated bythe fitting of the Vuillaume’s expression for the taper toWoolhouse’s measurements in Fig.6.The bending profiles and resultant flexibilities involve

forces and couples that scale with the critical Euler ten-sion EIπ2/ℓ2. Although the scaling clearly depends on

the length ℓ of the stick, by far the most important fac-tor is the r4 dependence of the second moment of thearea I. The larger diameter and slightly shorter violaand cello bows have a significantly larger Euler criticalload allowing them to support significantly larger bowtensions without straightening prematurely or bucklingsideways.

Until very recently, few measurements of both the ta-per and the bending profile had been made on the samebow, which would allow a direct comparison with thepredictions of this paper. However, Graebner and Pick-ering have13 recently published such data for a number ofbows of varying quality including measurements on a finebow of around 1850 by Kittel (the “German Tourte”),which was a favourite of Heifetz. Their measurementsare plotted in figs.11a and b. Superimposed on theirmeasurements in fig.11a is a smooth line representingthe Vuillaume-Tourte diameter and on 11b a line illus-trating the predicted curvature of the Vuillaume modelfor a pure couple across the ends of the stick. The cur-vature has been scaled to give a close fit to the measuredvalues.

Graebner and Pickering have suggested that the “purecouple” profile represents the optimum camber for a well-made bow, in qualitative agreement with their measure-ments on the Kittel bow. In contrast, our computationssuggest that a Tourte-tapered bow with a bending pro-

10

couple

weightstretch

mirror

0 10 20 30 40 50 60 cm 0 10 20 30 40 50 60 cm

0 10 20 30 40 50 60 cm 0 10 20 30 40 50 60 cm

0 – 50 N

0 – 50 N0 – 160 N

0 – 35 N

mm

0

- 2

- 4

- 6

- 8

-10

-12

-14

-16

mm

0

- 2

- 4

- 6

- 8

-10

-12

-14

-16

mm

0

- 2

- 4

- 6

- 8

-10

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-14

-16

mm

0

- 2

- 4

- 6

- 8

-10

-12

-14

-16

couple

weightstretch

mirror

0 10 20 30 40 50 60 cm 0 10 20 30 40 50 60 cm

0 10 20 30 40 50 60 cm 0 10 20 30 40 50 60 cm

0 – 50 N

0 – 50 N0 – 160 N

0 – 35 N

mm

0

- 2

- 4

- 6

- 8

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-12

-14

-16

mm

0

- 2

- 4

- 6

- 8

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-14

-16

mm

0

- 2

- 4

- 6

- 8

-10

-12

-14

-16

mm

0

- 2

- 4

- 6

- 8

-10

-12

-14

-16

FIG. 10. Straightening of the bow stick on increasing the bow hair tension in equal steps over the indicated range. In eachcase the forces and couples setting the initial camber of the bow were adjusted to give a typical maximum downward deflectionof ∼16 mm. For the couple profile C= 9.5 Nm, for the mirror profile F= 50 N, for the stretch profile F= 190 N and for theweight profile W=9 N.

0

0.3

0.6

0.9

1.2

1.5

0 100 200 300 400 500 600 700

distance along stick (mm)

curv

atu

re (

m-1

)

4

6

7

8

9

0 100 200 300 400 500 600 700

dia

mete

r (m

m)

distance along stick (mm)

5

FIG. 11. A comparison of (a) the taper of a bow by Kittel (∼1850) measured by Graebner and Pickering and the Vuillaume-Tourte model and (b) the measured radius of curvature of the camber with the solid line through the points calculated fromthe smoothed radius assuming a pure couple across the ends of the stick, with an additional solid line showing the curvatureexpected for the Vuillaume-Tourte bow scaled to the measured points.

files generated by couples alone across its ends would bemarginally too weak to support today’s playing tension of∼60 N without pulling straight prematurely - unless thelongitudinal elastic constant was well in excess of 22 GPa.Furthermore, a tapered bow with a couple bending pro-file would not pull straight uniformly along its length.A detailed comparison between the Graebner and Pick-

ering measurements and the Tourte-tapered bow suggests

a possible explanation in part for these apparently con-flicting conclusions. Fig.11 shows that, although the Kit-tel bow is slightly thinner than a Tourte bow at the lowerend, it has a longer constant diameter section and main-tains its thickness for a much greater length, with thethickness exceeding that of the Tourte bow over a dis-tance of ∼20 cm in the upper half of the bow. The radiusthen decreases significantly more rapidly than the Tourte

11

bow on approaching the tip. This accounts for the sig-nificant difference in predicted curvatures (proportionalto d−4) shown in fig.11b.It seems likely that, by maintaining the thickness of

the initially thinner bow stick over a longer length, Kit-tel was able to produce a significantly stronger bow stickthan a stick with a Tourte-taper. Leaving the thinningof the bow to further along the stick corresponds to a de-crease in the extrapolation length K in our generalisedVuillaume-Fetis expression for the taper (equ.13). Sig-nificantly larger couples are then required to achieve thesame downward deflection of the bow and correspond-ingly larger bow tensions are required to re-straightenthe stick. A stick with a couple-induced bending profilemight then be able to support normal bow tensions with-out straightening prematurely, as indeed appears to bethe case for the Kittel bow.

V. SUMMARY

A simple analytic model and large deformation, finiteelement computations have been used to demonstratehow the flexibility of the modern violin, viola and cellobow depends on both the taper and bending profile ofthe stick.A number of initial bow cambers bending profiles have

been considered generated by various combinations offorces and couples across the ends of the bent stick, bothin the untensioned state and with additional couples andforces from the tensioned bow hair.In all cases the bending profiles and rigidity of the bow

stick scale with the Euler critical buckling, so predictionscan be generalised to describe viola and cello bows. Fi-nite element computations demonstrate that the criticalbuckling force for a tapered violin bow stick is typicallyaround 75 N, which is not much larger than typical bowhair tensions of around 60 N. This explains why sidewaysbuckling is always a problem for sticks that deviate fromstraightness or have rather low elastic constants.The taper assumed for our finite element computations

is based on a geometrical model and related mathemati-cal expression originally introduced by Vuillaume to de-scribe his measurements on a number of Tourte bows.This model is re-derived and generalised to describe thetapers of viola and cello bows and shown to be equivalentto a similar expression cited by Woolhouse, after correc-tion for an arithmetic error in the original publicationreproduced in the later Fetis monograph.The predictions of our models are compared with re-

cent measurements by Graebner and Pickering13 on a fine∼ 1850 German bow by Kittel. The comparison suggeststhe Kittel bow achieves its strength by delaying the ma-jor taper of the bow to further along the bow stick thanexpected for a Tourte bow, which increases its effectiverigidity.The analytic and computation models developed in

this paper provide the theoretical framework for our sub-sequent analysis of the dynamic modes of the bow stickand their coupled modes of vibration with the stretchedbow hair.

VI. ACKNOWLEDGMENTS

I am grateful for very helpful correspondence and sug-gestions from John Graebner and Norman Pickering andfrom John Aniano and Fan Tao, who organised the 2010Oberlin one-day workshop on the Acoustics of Bows,which stimulated my initial interest in the elastic anddynamic properties of the bow. Robert Schumacher andJim Woodhouse also provided a number of extremelyhelpful comments on an earlier draft of the paper.

1 F. Fetis, D’analyses theoretiques sur l’archet, in AntoineStradivari-Luthiere celebre des Instruments a archet (Vuil-laume, Paris, 1856)Translated as A Theoretical Analy-sis of the Bow in Anthony Stadivari: the celebrated vi-olin maker by J. Bishop (Robert Cocks, London, 1864).http://books.google.co.uk. Last viewed 07/04/11 .

2 H. Saint-George, The Bow: Its history, manufac-ture and use. 3rd edition 1922 (S.A. Kellow, 1923),http://books.google.co.uk. Last viewed 07/04/11.

3 W. Woolhouse, “Note on the suitable proportions and di-mensions of a violin bow”, The Monthly Musical Record99–100 (1875).

4 M. Kimball, “On making a violin bow”, Catgut Acoust.Soc. Newsletter 11, 22–26 (1969).

5 O. Reder, “The search for the perfect bow”, CatgutAcoust. Soc. Newsletter 13, 21–23 (1970).

6 R. Schumacher, “Some aspects of the bow”, CatgutAcoust. Soc. Newsletter 24, 5–8 (1975).

7 A. Askenfelt, “Properties of violin bows”, Proc. Int. Symp.on Musical Acoustics (ISMA’92), Tokyo 27–30 (1992).

8 A. Askenfelt, “A look at violin bows”, STL-QPSR 34,41–48 (1993), http://www.speech.kth.se/qpsr, last viewed07/04/11.

9 A. Askenfelt, “Observations on the violin bow and the in-teraction with the string”, STL-QPSR 36, 23–42 (1995),http://www.speech.kth.se/qpsr, last viewed 07/04/11.

10 A. Askenfelt and K. Guettler, “The bouncing bow: someimportant parameters.”, TMH-QPSR 38, 53–57 (1997),http://www.speech.kth.se/qpsr, last viewed 07/04/11.

11 A. Askenfelt and K. Guettler, “Quality aspects of violinbows”, J. Acoust. Soc. Am. 105, 1216 (1999).

12 G. Bissinger, “Bounce tests, modal analysis, and playingqualitis of violin bow”, Catgut Acoust Soc. Jnl. (Series II)2, 17–22 (1995).

13 J. Graebner and N. Pickering, “Taper and camber ofbows”, J. Violin Soc. Am.: VSA Papers 22, 160–168(2010).

14 S. Timoshenko and J. Gere, Theory of elastic stability-2nd edition, pp 1–81 (McGraw Hill, New York, 1961),(reprinted Dover publications, 2009).

15 S. Timoshenko, History of Strength of Materials, pp 30–40,(McGraw Hill, New York, 1953).

16 G. Kirchhoff, J. Math (Crelle), 56 (1859).

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