The Violin Bow: Taper, Camber and Flexibility

Transcription

1 Te Violin Bow: Taper, Camber and lexibility Colin Goug Scool of Pysics and Astronomy, University of Birmingam, B13 9SN,UK a) (Dated: 11/22/1) An analytic, small-deflection, simplified model of te modern violin bow is introduced to describe te bending profiles and related strengts of an initially straigt, uniform cross-section, stick as a function of bow tension. A number of illustrative bending profiles (cambers) of te bow are considered, wic demonstrate te strong dependence of bow strengt on longitudinal forces across te ends of te bent stick. Suc forces are sown to be comparable in strengt to critical buckling loads causing excessive sideways buckling unless te stick is very straigt. Non-linear, large deformation, finite element computations extend te analysis to bow tensions comparable wit and above te critical buckling strengt of te straigt stick. Te geometric model assumes an expression for te taper of Tourte bows introduced by Vuillaume, wic we re-examine and generalise to describe violin, viola and cello bows. Our results are ten compared wit recent measurements by Graebner and Pickering of te taper and bending profiles of a particularly fine bow by Kittel. PACS numbers: DE Many string players believe tat te bow as a direct influence on te sound of an instrument, in addition to te more obvious properties tat control its manipulation on and off te string. It is terefore surprising tat until relatively recently little researc as been publised on te elastic and dynamic properties of te bow tat migt affect te sound of a bowed instrument. Te present paper investigates te influence of te taper and bending profile on te elastic properties of te bow, wic could influence te quality of sound produced by te bowed string via teir influence on te slip-stick generation of Helmoltz kinks circulating around te vibrating string. Te influence of te taper and camber on te flexibility of te bow will also affect te vibrational modes of te bow, wic will be discussed in a subsequent paper. Two scientifically and istorically important monograps on te bow were written in te 19t century. Te first by rançois-josep étis 1 was commissioned and publised in 1856 by J.B.Vuillaume ( ), te leading renc violin maker and dealer of te day. Te monograp not only describes and celebrates te instruments of te great Cremonese violin makers, but also gives an account of Vuillaume s own istorical and scientific researc on te modern violin bow. Tis ad relatively recently been developed by rancois Xavier Tourte ( ), wo was already viewed as te Stradivarius of bow making. In 1896, Henry Saint-George 2 updated information on bow makers and described subsequent researc on te bow. In addition to translating Vuillaume s earlier researc, e describes later researc and measurements by te eminent Victorian matematician W.S.B. Woolouse, RS ( ) - te eponymous owner of te 172 Woolouse Strad - wo collaborated wit te distinguised Englis bow maker James Tubbs ( ). a) Electronic address: profgoug@googl .com Woolouse 3 reflected te views of many modern performers in believing tat te purity of te vibrations of te bow were as important as te vibrations of te violin itself. Bot monograps cite expressions for te taper of fine Tourte bows, but neiter indicate ow suc formulae were derived. Tere was ten a lengty iatus in serious researc on te bow until te 195 s. Two interesting papers on te properties of te bow were ten publised in early Catgut Society Newsletters by Maxwell Kimball 4 and Otto Reder 5. Tis was followed in 1975 by a pioneering teoretical and experimental paper on te dynamic properties of te bow and stretced bow air by Robert Scumacer 6. Subsequently. Anders Askenfelt 7 9 at KTH in Stockolm publised several important papers on te bow, more recently in collaboration wit te distinguised double-bass virtuoso and teacer Knutt Guettler 1,11. Teir later publications are mainly devoted to te influence of te bow-string interaction on te sound of an instrument. George Bissinger 12 as investigated te mode sapes and frequencies of te bow plus tensioned bow air and as igligted te importance of te low-frequency bouncing modes, especially for sort bow strokes. Tese modes ave also been investigated by Askenfelt and Guettler 1. Te present paper is largely devoted to models tat describe te static elastic properties of te bow, wic provide te teoretical and computational framework to describe teir dynamic properties, to be described in a subsequent. Te paper is divided into tree main sections followed by a discussion of te results and a final summary. Te first section introduces a small deflection, analytic model for te bending modes of a simplified constant cross-sectional area bow. Tis illustrates te dependence of bow strengt on te initial bending profile (camber) and bow tension. Tree illustrative bow cambers are considered equivalent to bending profiles generated by various combinations of forces and couples across te ends 1

2 of te bent stick. Te second section revisits and generalises Vuillaume s geometric model and related matematical expression for te tapered diameter of Tourte bows along teir lengts. Tis is te taper cosen for te large deformation, nonlinear, finite element analysis described in te tird section. Tese computations extend te analysis to longitudinal tensions between te ends of te bent stick approacing and above te critical buckling load of te straigt stick 75N, comparable in size to typical playing tensions of 6N (Askenfelt). or small bending profiles, te finite-element computations reproduce te predictions of te simple analytic model remarkably well, justifying te most important simplifying assumption of te analytic model, but extending te predictions to describe te asymmetries introduced by te tapered stick diameter and different geometries of frog and ead of te bow. I. A SIMPLE BOW MODEL A. Introduction to bending of bow stick Te 1-dimensional in-plane bending of a tapered bow stick is governed by te bending equation, EI(s) d2 y = M(s), (1) ds2 were E is te elastic constant for stretcing and compression above and below te neutral axis, relative to wic te perpendicular deflection y is measured. Distances s are measured along te deformed neutral axis. Te bending moment M(s) generated by couples and forces acting on te stick will, in general, vary wit position along te lengt of te stick. I(s) is te second moment of te cross-sectional area area y2 dxdy, wic for a constant radius a stick is πa 4 /4. or a stick wit octagonal cross-section, I =.55 w 4, were w is te widt across opposing flat surfaces. or te sort, quasi-elliptical, transitional crosssection between te stick and ead of te bow, I = πab 3 /4, were b is te larger, in-plane, semi-axis. In general, te camber will be a function of bot te bending moment along te lengt of te stick and its local radius. Wen equal and opposite couples are applied across te ends of a constant cross-sectional radius stick, it will be bent into te arc of a circle wit radius of curvature R(s) = ( d 2 y/ds 2) 1 ( d 2 y/dx 2) 1, were te approximation for te curvature involves second-order, non-linear, corrections in (dy/dx) 2. Te curvature of a tapered stick will terefore vary along its lengt - inversely proportional to te fourt power of its diameter. or a tapered Tourte bow, te diameter of te stick canges from 8.6 mm at te frog end of te bow to 5.3 mm at te tip end (étis 1 ). Wen opposing couples are applied across its ends, te curvature will terefore be almost seven times larger at te upper end of te bow tan near te frog. Graebner and Pickering 13 ave suggested tat te couple-induced curvature is te ideal camber for a ig quality bow. Tis suggestion will be assessed in te ligt of te computations described in tis paper. In te following section, we initially consider te bending of te bow based on a simplified model similar to tat used by Graebner and Pickering. Te bow is of lengt l and constant cross-sectional radius a, wit perpendicular rigid levers of eigt at eac end representing te frog and te ead of te bow, between te ends of wic te air is tensioned. or small bending angles θ, we can make te usual small deflection approximations wit x = s measured along te lengt of te initially straigt stick, θ tan θ sin θ dy/dx and cos θ 1 and R(s) = ( d 2 y/dx 2) 1. B. An analytic model We adopt a linear analysis similar to tat used by Timosenko and Gere 14 in Capter 1 of teir classic treatment of Te Teory of Elastic Stability. Tis provides an invaluable introduction to te bending of straigt and bent slender columns like te violin bow, under te combined influence of external moments and bot lateral and longitudinal forces. irst consider te bow stick bent by equal and opposite couples and forces of magnitudes C and across its ends, promoting te upward deflection illustrated in fig.1. Imagine a virtual cut across te bent bow at a eigt y. In static equilibrium, te applied bending couples and forces acting on te separated left-and section of te stick must be balanced by equal and opposite bending moments and searing forces exerted on it by te adjoining lengt of te stick. Te bending moment along te lengt of te stick must terefore vary as C+y, resulting in a curvature along te lengt satisfying te bending equation wic may be written as were EI d2 y = (C + y),, (2) dx2 d 2 y/dx 2 + k 2 y = C/EI, (3) k 2 = /EI. (4) Tese are te defining equation used trougout tis paper to describe various pysically defined bending profiles or cambers of te initially untensioned bow and te influence on suc profiles of bow tension. Because te displacement must be symmetrical and zero at bot ends, te solutions can be written as y = A sin kx sin k(l x) (5) = C [cos kl/2 cos k(l/2 x)], cos kl/2 (6) 2

3 M= C+y y C x IG. 1. Equilibrium of bow bent by equal and opposite couples and forces across its ends wic ten satisfies te boundary condition d 2 y/dx 2 = C/EI at bot ends (y = ). Te maximum deflection at te mid-point is given by y l/2 = Cl2 8EI 2(1 cos u) u 2 cos u C = Cl2 λ(u) (7) 8EI were u = kl/2=(l/2) /EI. In te above expression, λ(u) is essentially a forcedependent factor amplifying te deflection tat would ave resulted from te bending moment alone, given by te pre-factor in equ.7. Te deflection becomes infinite wen kl/2 = π/2. Tis corresponds to a force Euler = EIπ 2 /l 2, wic is te maximum compressive load tat te straigt stick, wit inged supports at bot ends, can support witout buckling. or compressive forces well below te critical limit, λ(u) 1 1 T/T Euler, (8) an approximation tat differs from te exact expression by less tan 2% for compressive loads below.6 of te Euler critical value. In practice, te bending always remains finite, even for forces well beyond te critical load. Tis is because, as te bending increases, te two ends of te stick are forced towards eac oter. Te external force ten does additional work introducing non-linear terms in te deflection energy analysis, wic limits te sideways deflection. Tis was already understood and matematical solutions obtained for deflections above te critical load by Euler and Lagrange in te late 18t century (see Timosenko 15 History of Strengt of Materials, pages 3-4 ). Te neglect of te canges in lengt between te two ends of te bent stick, limits te above linear analysis to compressive forces significantly smaller tan te critical buckling load. As we will sow, te critical buckling tension for a tapered violin stick is around 75 N for pernambuco wood wit along-grain elastic constant of 22 GPa. Tis is only sligtly larger tan typical bow tensions of around 6 N bow tensions (Askenfelt). Analytic expressions for large deflection bending were publised by Kircoff 16 in 1859 based on a close analogy between te bending of a slender column and te non-linear deflections of a simple rigid pendulum. Te analytic solutions of wat is known as te elastica problem involve complete elliptic integrals (see Timosenko and Gere 14, art 2.7, pp 76-81). Alternatively, as in tis paper, large deflection, non-linear finite element computations can be used to describe suc bending, wic also allows te influence of te tapered stick and geometry of te frog and ead of te bow to be included. A similar amplification factor applies to any initial sideways bending of te bow stick (Timosenko and Gere, art 1.12). Tis explains wy bows wit only a sligt initial deviation from sideways straigtness become virtually unplayable on tigtening te bow because of excessive sideways bending. In contrast, te downward curvature of te modern bow is in te opposite sense to tat produced by te bow tension, wic inibits in-plane instabilities. Wen te bending moments are reversed, te sense of deflection is simply reversed. However, reversing te direction of te longitudinal load,, as a more profound affect, wit k 2 =/EI now becoming negative.because cos ikx cos kx, te deflections now vary as y(x) = C [cos kl/2 cos k(l/2 x)]. (9) cos kl/2 Tere is no longer a singularity in te denominator. urtermore, te extensive load now tends to decrease any bending initially present or induced by external couples or forces. C. Setting of initial camber of bow. In practice, te bow maker adjusts te camber of a bow by bending te stick under a gentle flame. Tis locks in internal strains equivalent to tose tat could ave been provided by an equivalent combination of couples and forces applied across te ends of te stick or by distributed lateral forces along its lengt. Timosenko and Gere 14, 1.12, provide several examples of te use of suc an approac to describe te bending of an initially curved bar. Bow makers usually assume tat a correctly cambered bow sould pull uniformly straigt along its lengt on tigtening te bow air. Tey ten carefully adjust te bending profile to correct for canges in elastic properties and uneven graduations in taper along te lengt of te bow. We will sow tat te strengt of te bow - te way te stick straigtens on tigtening te bow air or deflects on applying downward pressure on te string - is strongly related to te camber of te bow. Te earliest suggestion for te most appropriate camber of te modern bow stick was made by Woolouse 3 : Let a bow be made of te proper dimensions, but so as to be perfectly straigt; ten by screwing it up in te ordinary way, it would sow, upside down, te exact curve to wic oter bows sould be set. We will refer to suc a profile as te mirror profile, as it is te mirror image of te upward bending profile of 3

4 te tensioned straigt stick wen inverted. A bow wit suc an initial camber would indeed straigten uniformly along its lengt on tigtening. Jon Graebner and Norman Pickering 13 ave recently proposed tat te ideal camber would be one generated by equal and opposite couples across te ends of te initially straigt stick. Teir measurements provide persuasive evidence tat tis migt indeed be true for a number of fine bows. We refer to tis as te couple profile. Altoug suc a stick wit a uniform diameter would straigten uniformly along its lengt, we will demonstrate tat tis is no longer true for a tapered stick. Moreover, suc a stick must ave a ig elastic constant, if it is not to straigten prematurely before a satisfactory playing tensions could be acieved. In tis paper, we also consider a bending profile equivalent to tat produced by an outward force between te ends of te frog and ead of te bow, in te opposite direction to te air tension. We refer to tis as te stretc profile. Suc a bow would also straigten uniformly along it lengt on tigtening, but would be significantly stronger tan bows wit mirror or couple bending profiles. Te equivalent couples and forces used to generate te above tree bending profiles are illustrated in figs.2a-c, wile figs.2d-f illustrate te additional forces and couples generated by te bow tension on te above cambers. D. Influence of bow tension on bending profile. Te deflections of te stick are assumed to be well witin te elastic limits of te pernambuco wood used for te stick. Te net forces and couples acting across te ends of te stick are terefore te sum of te equivalent forces ± and ±C used to set te untensioned camber and te forces ±T and couples ±T from te tension of te air stretced between te end levers representing te frog and ead of te bow. However, te deflections from te two sets of forces and couples are not necessarily additive because of te non-linear relationsip between te couple-induced deflections and te compressive or extensive loads along te lengt of te stick. or te mirror and stretc bending profiles under tension, te longitudinal forces are compressive, so te deflections under tension are given by equ.6 wit C = ( T ) and longitudinal force + T for te mirror profile, so tat y mirror (T ) = T + T wit k 2 = ( + T )/EI, and y couple (T ) = M T T kl/2 cos k(l/2 x)] [cos, (1) cos kl/2 [cos kl/2 cos k(l/2 x)], (11) cos kl/2 wit k 2 = T/EI for te couple profile. or te stretc profile, te net longitudinal forces ( > T ) across te ends of te stick are extensive, so tat y stretc(t ) = [cos k(l/2 x) cos kl/2], (12) cos kl/2 wit k 2 = ( T )/EI. Te camber of te untensioned bow is given by letting T. Te resulting cambers for above tree bending profiles are sown in fig.3, were te equivalent couples C and forces setting te initial camber ave been cosen to give te cited mid-point deflections. or relatively small deflections, te tree bending profiles are almost indistinguisable. However, for all deflections, te couple bending profile as a sligtly larger curvature towards te ends tan te mirror profile. As te mid-point deflection increases te stretc profile becomes progressively flatter in te middle of te bow, as te maximum deflection given by equ.12 can never exceed te lever eigt, requiring a muc larger curvature towards te ends of te bow. Te solid lines in fig.4 plot te mid-point deflections as a function of te generating forces and couples used to set te mirror, couple and stretc profiles for our simple bow model. Note tat, altoug te stresses and strains in te stick are assumed to be elastic, te deflections of te mirror and stretc profiles are igly non-linear. Tis is a geometric effect arising from te non-linear amplification of bending profiles by te longitudinal forces as te bending canges (eq.2. In plotting fig.4 a value of EI as been cosen to give a critical buckling load of 78N, at wic force te analytic expression for te mirror profile becomes infinite, unpysical and canges sign - indicated by te vertical line. Using tis value as te only adjustable parameter, te analytic small deflection predictions for all tree bending profiles are almost indistinguisable from te open symbols plotting te maximum downward deflections of te tapered Tourte bow computed in section 3. Suc computations include te canges in geometry, ence bending moments, as te bending increases. Te excellent agreement between te analytic and our later non-linear finite element computations justifies many of te approximations made in te small deflection analysis and justifies its use to illustrate te most important qualitative static and dynamic properties of te bow. Neverteless, finite-element computations are still required to account for te asymmetric properties associated wit te taper and different dimensions of te frog and ead of te bow. Bows wit te above bending profiles will straigten wen te couples acting on te stick from te air tension across te frog and tip of te bow are equal and opposite tose used to generate te initial untensioned camber. A bow wit te mirror camber and a downward deflection of 16mm would terefore straigten wit a bow tension of 3N, only bout alf te normal playing tension of around 6T (Askenfelt 7 ). A Tourte bow wit a couple bending profile would also straigten before a playing tension of 6N could be acieved, unless its elastic constant was significantly iger tan 22 GPa or it ad a ticker diameter to increase te critical buckling load. In contrast, a bow wit a pure stretc bending profile could support 4

5 C= mirror couple stretc C= C=M C= M C= C= - (a) (b) (c) C=(-T) C=(-T) +T +T T T C=M-T C=M-T C=( T) C=(-T) -T T T T T (d) (e) (f) -T IG. 2. igures a-c illustrate te mirror, couple and stretc bending profiles generated by equivalent couples and compressive or extensive forces between te ends of te bent stick. igures d-f illustrate te additional forces and couples generated by te bow tension between frog and tip of te bow, wit bot sets of forces and couples defining te net forces and couples across te ends of te stick determining its bending profile. deflection/ mirror.5 couple.5 stretc.7 mirror.7 couple.7 stretc.9 mirror.9 couple.9 stretc downward deflection (cm) couple (cm) stretc (cm) mirror (cm) couple fit mirror fit stretc fit normalised lengt orce or Couple/lever eigt (N) IG. 3. Te mirror, couple and stretc bending profiles for mid-point deflections of.5,.7 and.9 te end-lever eigt. Te model assumes an initially straigt bow stick of lengt 7 cm, wit uniform cross-sectional area and = 2 cm. very large bow tensions witout straigtening appreciably. Bows of intermediate strengt could be produced by setting a camber equivalent to te bending profile produced by a combination of a couple across te bow stick and a stretcing force between te frog and ead of te bow. Watever profile is used by te maker to set te inplane rigidity, te critical buckling tension of te straigt stick must always be significantly larger tan te required playing tension, to avoid major problems from out-of plane buckling. Te added strengt of te stretc profile arises from te increased curvature at te ends of te stick, wic in a real bow compensates for te loss in bow strengt from te reduction in diameter towards te tip. In practice, IG. 4. Te solid lines represent te bending couples C = and forces required to set te mirror, couple and stretc profiles as a function of te mid-point deflection of te 7 cm long, uniform diameter, initially straigt stick wit = 2cm levers on its end, for an assumed critical buckling load of 78 N indicated by te vertical line. Te open symbols lying almost exactly on top of te solid lines plot te maximum deflection at around 3 cm from te end of te tapered Tourte stick evaluated using non-linear finite-element software assuming a uniform elastic constant of 22 GPa. te bow maker can coose any bending profile tey wis and are not constrained to te specific examples we ave cosen to consider ere. However, for a tapered bow, only te mirror or stretc bending profiles will pull uniformly straigt on tigtening te bow air. 5

6 II. THE TAPERED BOW Te taper used in our finite element computations will be based on Vuillaume s algebraic expression describing is measurements on a number of fine Tourte bows. Vuillaume noted tat all suc bows were tapered in muc te same way, wit te stick diameter decreasing by a given amount at almost exactly te same positions along teir lengt. As cited by étis 1, e terefore devised a geometric construction, so tat tese positions migt be found wit certainty - by wic, consequently, bows migt be made wose condition sould always be settled à priori. étis 1 gives te following matematical expression, d(x) = log(x + 175), (13) derived by Vuillaume from is geometric model. Tis gives te tapered diameter d as a function of distance x from te end of te bow stick, wit all dimensions in mm. Unfortunately, no explanation appears to ave been given for eiter te geometric model or matematical expression derived by Vuillaume and later by Woolouse 3 from an independent set of measurements on Tourte violin, viola and cello bows. We terefore re-derive and justify te use of te Vuillaume geometric model and related matematical expression. We also sow tat te later Woolouse expression, as originally correctly cited in mm, is virtually identical to Vuillaume s earlier expression. Te geometric construction developed by Vuillaume is illustrated in fig.5 togeter wit a potograp of a fine Tourte bow and a plot of te tapered diameter on wic te geometric model and associated matematical expression were based. Te bendable lengt of stick from te frog end to te ead of te Tourte bow was measured by Vuillaume as l =7 mm, of wic te first 11 mm ad a constant diameter of 8.6 mm. Te remaining 59 mm was tapered towards te upper end wit a diameter of 5.3 mm. Vuillaume noted tat te tapered section could be divided into 11 sections marking te lengts over eac of wic te diameter decreased by.3 mm, wit te lengt of eac section decreasing by te same fraction β, as illustrated scematically in fig.5. Te decrease in diameter over consecutive sections is terefore described by an aritmetic progression, wile te decrease in section lengt is described by a geometric progression. Tis forms te basis of Vuillaume s geometric model and related matematical expression. Te model is generated by first drawing a base line of lengt 7 mm to represent te lengt of te stick. At te frog end of te bow a vertical line is raised of eigt A=11 mm equal in lengt to te initial constant radius section of te bow. At te oter end of te base-line a second vertical line of eigt B = 22 mm is raised and a straigt line drawn between teir ends intercepting te axis a distance 175 mm (l/4) beyond te end of te stick. A compass was ten used to mark out 12 sections along te base line. Wit te point of te compass at te origin of te base line, a point was first marked off along te base line equal to te eigt of te first perpendicular representing te initial 11 constant diameter section of te stick. At te end of tis section, a new vertical line was raised to intersect te sloping line. Using a compass, tis lengt was added to te first section. Te process was ten repeated until te remaining sections exactly fitted into te lengt of te bow stick. Te eigt of te 22 mm uprigt at te end was cosen so tat te fractional decrease in lengt β = 1 (A B)/7 =.874 between successive sections ad te correct value to allow tis perfect fit. Apart from te initial, constantdiameter, section, eac subsequent section marked te lengts over wic te diameter decreased by.3 mm (3.3 mm in total). Vuillaume presumably determined te lengt B by trial and error. Matematically, tis requires te sum of te lengts A 11 o βn = 7 mm, wit a ratio of lengts B/A =β 12 =22/11=1/5. Tis gives te above value for β wit A 11 o βn = 7.5 mm and B=Aβ 12 = mm, in close agreement wit Vuillaume s coice of dimensions for is geometric model. Tis simple geometric construction enabled Vuillaume s bow makers to graduate bows wit te same taper as Tourte bows. However, it is most unlikely tat Tourte would ave used suc a metod. Like all great bow makers, Tourte almost certainly ad an innate feeling for te appropriate taper and camber of te bow based on te flexibility of te stick and an aestetic sense tat so often mirrors te elegance and beauty of simple matematical constructs. Vuillaume s related matematical expression for te taper can be derived as follows. rom te scaling of triangles wit equal internal angles, te distance x n of te n-t uprigt from te point of intersection of te sloping line wit te base-line satisfies te recurrence formula, By iteration, x n = βx n 1. (14) x n = x o β n, (15) were x o is te distance of te far-end of te stick from te point of intersection of te sloping line. Eac value of n> corresponds to te position along te stick at wic te diameter of te following section decreases by t=.3 mm. Te diameter d n at x n is terefore given by d n = d o (n 1) t, (16) were d is te diameter of te initial section. To obtain te algebraic relationsip between d n and x n, te logaritm is taken of bot sides of equ.15. Te value of n can ten be replaced in equ.16 to give d n = (d o + t) t [(log x n log x )/ log β]. (17) 6

7 A = 11 mm A A 2 A 3 A B = 22 mm x x1 x2 x3 x4 x5 x6 x7 x8 x9 x1 x11 x12 L= 7 mm L/4 = 175 mm dr 7 = = 11 mm d tapered section r 7 = 8.6 mm x 5.3 mm IG. 5. Vuillaume s geometric model reproducing is measurements of te taper of Tourte bows, wit a potograp of a bow on te same pysical scale and a plot illustrating te functional form of te tapered stick diameter along its lengt. Tis progression can be described by te continuous function d(x) = C D log x, (18) were x is still measured from te point of intersection of Vuillaume s sloping line, 175 mm beyond te end of te stick, D = t/ log β and C D log x 1 is te diameter at te end of te initial 11 mm constant cross-section lengt (x 1 = = 765 mm). Insertion of te above parameters in equ.18 and redefining x as te distance measured from te ead-end of te tapered stick gives d(x) = log (x + 175), (19) wit values of te constants essentially identical to tose of te Vuillaume expression equ.13 cited by étis. Note tat tis formula only applies to te upper 59 mm tapered section of te bow - te lower 11 mm as a constant diameter of 8.6 mm and includes a olding section tat is lapped wit leater wit a metal over-binding to protect te stick from wear. Te matematical expression replaces te discrete point values of te geometric model wit a continuous expression. It is clearly independent of te number of sections over wic te canges in diameter were originally measured. It says muc for renc education at te time tat Vuillaume ad te necessary matematical skills to devise bot te geometrical model and te equivalent matematical expression. It is, of course, likely tat e collaborated in teir derivation wit elix Savart or some oter scientifically trained researcer. Somewat later, Woolouse 3 developed a similar formula based on is independent measurements of te taper of Tourte violin, viola and cello bows. Woolouse s measurements are tabulated in Saint-George s monograp 2 along wit an incorrect expression for te diameter of bow stick (mm) Cello meas Vla meas Vln meas V cello V viola V fit to Woolouse violin V violin distance from end of stick (mm) IG. 6. Measured tapers of violin, viola and cello bows by Woolouse, wit dased lines representing fitted plots of te generalised Vuillaume-tis expression for te taper. diameter expressed in inces. However, in is original publication, Woolouse initially cites te diameter in mm as d(x) = log(x + 184), (2) before incorrectly transcribing it into inces. Te above expression differs from Vuillaume s expression, equ.13, by less tan 1% over almost te wole lengt of te bow, wic is almost certainly witin te accuracy of teir measurements. 7

8 It is not obvious wy Tourte used te particular taper described above, oter tan to reduce te weigt at te end of te bow witout over-reduction of its strengt. Oter smootly varying tapers can be devised between fixed upper and lower diameters d U and d L of te general form log[x/k + 1] d(x) = d U + (d L d U ) log[l/k + 1], (21) were L is te lengt of te tapered section of te bow and K is a scaling distance equivalent to te distance of te point of intersection of te sloping line beyond te end of te bow in Vuillaume s geometric model. K effectively defines te lengt scale from te end of te bow over wic most of te taper occurs. or Tourte bows, te Vuillaume taper parameter K l/4 =175 mm, were l is te total bow lengt. As K is increased, te taper is more uniformly distributed over te lengt of te stick, approacing tat of a simple truncated cone for large K-values, wile smaller values sift te major canges in tapering furter towards te tip of te bow. Te Woolouse expression corresponds to a very sligtly larger K value tan tat derived by Vuillaume. igure 6 compares te tapers of fine violin, viola and cello bows measured by Woolouse 3 and reproduced by Saint-George 2. Te dased lines drawn troug te measured data points are generalised Vuillaume plots, wit constants cosen by and to pass troug te measured values. Te constants clearly ave to be varied to describe te larger diameters and sligtly sorter lengts of viola and cello bows, toug te general form of te taper remains muc te same. Vuillaume derived a very sligtly larger ticker section for te bows e measured illustrated by te solid line. Te main difference between violin, viola and cello bows is teir sligtly larger diameter and ence weigt. III. INITE ELEMENT COMPUTATIONS A. Geometric model and buckling We now use te Vuillaume expression for te tapered Tourte bow to describe te taper of our finite-element bow model illustrated scematically in fig.7. A uniform elastic constant of E=22 GPa along te grain is assumed. Te bow stick was divided into 14, equal-lengt, conical, sub-domains describing te tapered lengt of te stick wit a crudely modeled rigid bow ead and frog at its ends. Te bow air was simply represented by te bow tension between its points of attacment on te underside of te frog and ead of te bow. Because te structure is relatively simple, at least in comparison wit te violin, we cose to perform a 3- dimensional analysis of te bending modes. Eac subdomain was divided into a medium-density mes resulting in typically 1 K degrees of freedom. Deflections were confined to te plane of symmetry passing troug te IG. 7. inite element geometry of a Tourte-tapered violin bow stick wit attaced ead and frog aving effective lever eigts from te neutral axis of 18 mm at te ead of te bow and 24 mm at te frog. y (m) orce (N) IG. 8. Large-deformation, non-linear, finite-element computations of te deflection of te mid-point of a Tourte-tapered bow stick as a function of bow tension, first applied in te normal way across te ends of te frog and ead of te bow and ten only sligtly offset from te central neutral axis of te stick by te stick radius. Te associated bending profiles of te tapered stick are illustrated to exact scale. stick, frog and ead of te bows. Te out-of-plane deflections will be discussed in a later paper on vibrational modes. Te initially straigt stick was bent by te same combinations of forces and couples used to set te bending profile of te uniform diameter bow stick. Using COM- SOL linear EA software on a modest PC, we were able to compute bending profiles for small deflections in a few seconds, wile non-linear analysis, wic takes into account all canges in geometry on bending, typically took a few tens of seconds. Tis increased to several minutes to compute te profile of a stick under te influence of very large bending couples and forces, wen te two ends of te bow are forced to move close togeter. 8

9 As an initial example, fig.8 illustrates te bending profiles of an initially straigt bow as a function of bow tension applied, first across te ends of te frog and ead of te bow in te normal way and ten sligtly offset from te central stick axis by te radius of te stick. In bot cases, te point at wic te tension is applied at te frog-end is pinned allowing te bow to rotate, wile te tip end of te air is constrained to move in te direction of te applied tension. Te plots illustrate te relative motion of te two ends of te tensioned air towards eac oter, wile te figures sow te associated bending profiles. or te sligtly offset tension, tere is very little bending or inward motion of te two ends until te Euler critical load of T Euler = E I π 2 /L 2 75 N is reaced. At tis tension, tere is a fairly sarp transition to te buckled state, wit te two ends of te stick forced to move towards eac oter. I is te appropriately averaged second moment of te cross-sectional area of te tapered stick. Te computations sow tat te critical load of a straigt stick wit te Tourte taper is te same as tat of a constant radius stick of diameter 6.4 mm - somewat larger tan te 5.3 mm diameter at te tip end of te tapered bow but smaller tan te 8.6 mm radius at te frog-end. In contrast, wen te tension is applied across te frog and tip, te couples across te ends of te bow result in a significant amount of bending (buckling) well below T Euler. Tere is ten a continuous transition to te igly buckled state wit no sudden instability at T Euler. Tis is exactly wat one would expect, as it corresponds to te drawing up of a simple unting bow as te string tension across its ends is increased. Because of te additional couple from te lever action of te bow tension acting on te frog and ead of te bow, te amount of bending is always larger tan te buckling produced by compressive forces across te ends of an initially straigt stick. Te amount of bending at low bow tension is clearly proportional to te couple from te offset compressive load, wic is determined by te frog and ead lever eigts. Te non-linear finite element analysis extends te computations to te wole range of possible bending forces and bow tensions, wereas te analytic, small tension analysis is limited to compressive forces across te ends of te bow significantly smaller tan T Euler, beyond wic te solutions become unpysical. B. Bending profiles of te tapered stick We now consider te bending of an initially straigt Tourte-tapered bow by te same combinations of couples and forces used to generate te couple, mirror and stretc bending profiles of te constant diameter bow stick described in te earlier analytic section. We ave also added a weigt bending profile generated by adding a weigt to te mid-point of te orizontal stick. igure 9 illustrates te development of te bending profiles as te couples and forces are increased in equal steps over te ranges indicated. Te solid lines sow te deflections computed using large-deformation, nonlinear, finite-element software, wile te dased lines sow te computed profiles from linear, finite-element analysis. As a result of te taper, te maximum downward deflection of te stick is sifted to a position about 5 cm beyond te mid-pint of te stick. Te difference between te two sets of computed bending profiles occurs because te linear analysis neglects te canging influence of te longitudinal forces on te stick as te bending canges (equ.2). Tis results in non-linear deflections of te bow despite te strains witin te stick itself remaining well witin te linear elastic limit. Te linear analysis only involves te contribution to te bending of te initially straigt stick from te couple acting on te stick and not from te longitudinal force, wic only contributes to bending once bending as occurred. Te compressive forces across te ends of te bow generating te mirror profile enance any bending already present. Tis results in a non-linear softening of te bow as te bending is increased. In contrast, te extensive forces generating te stretc profile tend to inibit furter increases in bending resulting in a non-linear increase in rigidity of te stick. Suc non-linearities are referred to as geometric non-linearities and occur even toug te strains witin te stick itself remain well witin te linear elastic limits. or te couple and weigt profiles, te deflections are simply proportional to te bending forces, as tere are no longitudinal forces giving rise to non-linearity. Te open symbols in figure 4 plot te computed maximum downward deflections of te tapered Tourte for te bending forces and couples generating te mirror, couple and stretc profiles. As remarked earlier, te computed deflections are in excellent agreement wit te mid-point deflections predicted by te simple analytic model. Te analytic deflections are represented by te solid lines, wit forces scaled to give te same critical buckling load. Suc agreement is unsurprising, as te deflections - of order te frog and bow ead eigts - are small compared to te lengt of te bow stick. Te geometric approximations made in te analytic model are terefore well justified. However, finite element computations are necessary to investigate te asymmetries produced by te taper and te different geometries of te frog and ead of te bow. C. Influence of bow tension igure 1 illustrates te influence of bow tension on te four bending profiles generated by external couples and forces cosen to give a downward stick deflection of 16 mm at around 3 cm from te end of te tapered stick. Te plots also igligt te differences in initial bending profiles before te air tension is applied. or example, te stretc profile is very muc flatter in te middle of te bow wit muc more curvature towards its ends tan te mirror profile, confirming te results for te symmetric, constant radius bow model in fig.3. As expected, a bow wit te mirror profile is te weakest and pulls straigt wit a bow tension of only 9

10 mm couple mm mirror C = 1.2 Nm -2 = 35 N cm cm mm stretc mm weigt = 1 N -2 W = 12 N cm cm IG. 9. Te dependence of te bending profiles of an initially straigt Tourte-tapered bow as a function of te forces and couples used to generate tem plotted for equal steps of te forces and couples involved. Te dased lines are te computed profiles computed using linear finite element analysis and te solid lines sow te profiles predicted by large-deformation, non-linear computations. 3 N. Bows wit te couple and weigt profiles are considerably stronger requiring a bow tension of around 5N to straigten, wile te bow wit te stretc profile is very muc stronger requiring a bow tension of 15N to straigten. Only tapered bows wit mirror and stretc untensioned cambers straigten uniformly along teir lengt. Te computations confirm tat a Tourte taper wit a couple bending profile would be unable to support a typical bow tension of 6T unless its along-grain elastic constant was well in excess of te assumed value of 22GPa. urtermore, ig.1 sows tat suc a bow would not straigten uniformly on tigtening. Te computations also confirm tat increasing te curvature towards te upper end of te bow increases te strengt or rigidity of te bow. IV. DISCUSSION Altoug our computations ave focused on te bending of violin bows, te results can easily be scaled to viola, cello and double-bass bows as demonstrated by te fitting of te Vuillaume s expression for te taper to Woolouse s measurements in ig.6. Te bending profiles and resultant flexibilities involve forces and couples tat scale wit te critical Euler tension EIπ 2 /l 2. Altoug te scaling clearly depends on te lengt l of te stick, by far te most important factor is te r 4 dependence of te second moment of te area I. Te larger diameter and sligtly sorter viola and cello bows ave a significantly larger Euler critical load allowing tem to support significantly larger bow tensions witout straigtening prematurely or buckling sideways. Until very recently, few measurements of bot te taper and te bending profile ad been made on te same bow, wic would allow a direct comparison wit te predictions of tis paper. However, Graebner and Pickering ave 13 recently publised suc data for a number of bows of varying quality including measurements on a fine bow of around 185 by Kittel (te German Tourte ), wic was a favourite of Heifetz. Teir measurements are plotted in figs.11a and b. Superimposed on teir measurements in fig.11a is a smoot line representing te Vuillaume-Tourte diameter and on 11b a line illustrating te predicted curvature of te Vuillaume model for a pure couple across te ends of te stick. Te curvature as been scaled to give a close fit to te measured values. Graebner and Pickering ave suggested tat te pure couple profile represents te optimum camber for a wellmade bow, in qualitative agreement wit teir measurements on te Kittel bow. In contrast, our computations suggest tat a Tourte-tapered bow wit a bending pro- 1

11 mm couple 5 N mm mirror 35 N cm cm mm stretc mm weigt N N cm cm IG. 1. Straigtening of te bow stick on increasing te bow air tension in equal steps over te indicated range. In eac case te forces and couples setting te initial camber of te bow were adjusted to give a typical maximum downward deflection of 16 mm. or te couple profile C= 9.5 Nm, for te mirror profile = 5 N, for te stretc profile = 19 N and for te weigt profile W =9 N diameter (mm) 7 6 curvature (m -1 ) distance along stick (mm) distance along stick (mm) IG. 11. A comparison of (a) te taper of a bow by Kittel ( 185) measured by Graebner and Pickering and te Vuillaume- Tourte model and (b) te measured radius of curvature of te camber wit te solid line troug te points calculated from te smooted radius assuming a pure couple across te ends of te stick, wit an additional solid line sowing te curvature expected for te Vuillaume-Tourte bow scaled to te measured points. files generated by couples alone across its ends would be marginally too weak to support today s playing tension of 6 N witout pulling straigt prematurely - unless te longitudinal elastic constant was well in excess of 22 GPa. urtermore, a tapered bow wit a couple bending profile would not pull straigt uniformly along its lengt. A detailed comparison between te Graebner and Pickering measurements and te Tourte-tapered bow suggests a possible explanation in part for tese apparently conflicting conclusions. ig.11 sows tat, altoug te Kittel bow is sligtly tinner tan a Tourte bow at te lower end, it as a longer constant diameter section and maintains its tickness for a muc greater lengt, wit te tickness exceeding tat of te Tourte bow over a distance of 2 cm in te upper alf of te bow. Te radius ten decreases significantly more rapidly tan te Tourte 11

12 bow on approacing te tip. Tis accounts for te significant difference in predicted curvatures (proportional to d 4 ) sown in fig.11b. It seems likely tat, by maintaining te tickness of te initially tinner bow stick over a longer lengt, Kittel was able to produce a significantly stronger bow stick tan a stick wit a Tourte-taper. Leaving te tinning of te bow to furter along te stick corresponds to a decrease in te extrapolation lengt K in our generalised Vuillaume-étis expression for te taper (equ.13). Significantly larger couples are ten required to acieve te same downward deflection of te bow and correspondingly larger bow tensions are required to re-straigten te stick. A stick wit a couple-induced bending profile migt ten be able to support normal bow tensions witout straigtening prematurely, as indeed appears to be te case for te Kittel bow. V. SUMMARY A simple analytic model and large deformation, finite element computations ave been used to demonstrate ow te flexibility of te modern violin, viola and cello bow depends on bot te taper and bending profile of te stick. A number of initial bow cambers bending profiles ave been considered generated by various combinations of forces and couples across te ends of te bent stick, bot in te untensioned state and wit additional couples and forces from te tensioned bow air. In all cases te bending profiles and rigidity of te bow stick scale wit te Euler critical buckling, so predictions can be generalised to describe viola and cello bows. inite element computations demonstrate tat te critical buckling force for a tapered violin bow stick is typically around 75 N, wic is not muc larger tan typical bow air tensions of around 6 N. Tis explains wy sideways buckling is always a problem for sticks tat deviate from straigtness or ave rater low elastic constants. Te taper assumed for our finite element computations is based on a geometrical model and related matematical expression originally introduced by Vuillaume to describe is measurements on a number of Tourte bows. Tis model is re-derived and generalised to describe te tapers of viola and cello bows and sown to be equivalent to a similar expression cited by Woolouse, after correction for an aritmetic error in te original publication reproduced in te later étis monograp. Te predictions of our models are compared wit recent measurements by Graebner and Pickering 13 on a fine 185 German bow by Kittel. Te comparison suggests te Kittel bow acieves its strengt by delaying te major taper of te bow to furter along te bow stick tan expected for a Tourte bow, wic increases its effective rigidity. Te analytic and computation models developed in tis paper provide te teoretical framework for our subsequent analysis of te dynamic modes of te bow stick and teir coupled modes of vibration wit te stretced bow air. VI. ACKNOWLEDGMENTS I am grateful for very elpful correspondence and suggestions from Jon Graebner and Norman Pickering and from Jon Aniano and an Tao, wo organised te 21 Oberlin one-day worksop on te Acoustics of Bows, wic stimulated my initial interest in te elastic and dynamic properties of te bow. Robert Scumacer and Jim Woodouse also provided a number of extremely elpful comments on an earlier draft of te paper. 1. étis, D analyses téoretiques sur l arcet, in Antoine Stradivari-Lutiere célèbre des Instruments a arcet (Vuillaume, Paris, 1856)Translated as A Teoretical Analysis of te Bow in Antony Stadivari: te celebrated violin maker by J. Bisop (Robert Cocks, London, 1864). ttp://books.google.co.uk. Last viewed 7/4/11. 2 H. Saint-George, Te Bow: Its istory, manufacture and use. 3rd edition 1922 (S.A. Kellow, 1923), ttp://books.google.co.uk. Last viewed 7/4/11. 3 W. Woolouse, Note on te suitable proportions and dimensions of a violin bow, Te Montly Musical Record 99 1 (1875). 4 M. Kimball, On making a violin bow, Catgut Acoust. Soc. Newsletter 11, (1969). 5 O. Reder, Te searc for te perfect bow, Catgut Acoust. Soc. Newsletter 13, (197). 6 R. Scumacer, Some aspects of te bow, Catgut Acoust. Soc. Newsletter 24, 5 8 (1975). 7 A. Askenfelt, Properties of violin bows, Proc. Int. Symp. on Musical Acoustics (ISMA 92), Tokyo 27 3 (1992). 8 A. Askenfelt, A look at violin bows, STL-QPSR 34, (1993), ttp:// last viewed 7/4/11. 9 A. Askenfelt, Observations on te violin bow and te interaction wit te string, STL-QPSR 36, (1995), ttp:// last viewed 7/4/11. 1 A. Askenfelt and K. Guettler, Te bouncing bow: some important parameters., TMH-QPSR 38, (1997), ttp:// last viewed 7/4/ A. Askenfelt and K. Guettler, Quality aspects of violin bows, J. Acoust. Soc. Am. 15, 1216 (1999). 12 G. Bissinger, Bounce tests, modal analysis, and playing qualitis of violin bow, Catgut Acoust Soc. Jnl. (Series II) 2, (1995). 13 J. Graebner and N. Pickering, Taper and camber of bows, J. Violin Soc. Am.: VSA Papers 22, (21). 14 S. Timosenko and J. Gere, Teory of elastic stability- 2nd edition, pp 1 81 (McGraw Hill, New York, 1961), (reprinted Dover publications, 29). 15 S. Timosenko, History of Strengt of Materials, pp 3 4, (McGraw Hill, New York, 1953). 16 G. Kircoff, J. Mat (Crelle), 56 (1859). 12