Dept. for Speech, Music and Hearing

Quarterly Progress andStatus Report

The bouncing bow: Someimportant parameters

Askenfelt, A. and Guettler, K.

journal: TMH-QPSRvolume: 38number: 2-3year: 1997pages: 053-057

http://www.speech.kth.se/qpsr

TMH-QPSR 2-3/1997

53

The bouncing bow: Some importantparameters.Anders Askenfelt & Knut Guettler*∗ Norwegian State Academy of Music, P.O. Box 5190 Majorstua, N-0302 Oslo, NORWAY.

AbstractThe bouncing bow, as used in rapid spiccato and ricochet bowing, has beenstudied. Dynamical tests were made by monitoring the force history when the bowwas played by a mechanical device against a force transducer as a substitute tothe string. The action of the bow stick was investigated by modal analysis. Bowsmade of wood, fibre glass, and carbon fibre composites were studied andcompared, as well as a bow which was modified from the normal (concave) shapeinto a straight stick.

IntroductionThe advanced bowing styles called rapidspiccato (sautill é) and ricochet, in which thebow bounces off the string between notes byitself, can be performed at high rates, between 8and 13 notes/s (sixteenth notes at M.M. ≈ 120-190). The dynamics of the bow plays an import-ant role in these bowing styles, and differences inthe action between bows are easily recognised byprofessional players. All i nstruments in the stringorchestra can perform these rapid bowingsdespite the large differences in scaling betweenthe instruments at the extremes, the violin and thedouble bass.

Bounce modeA low-frequent “bounce mode” of the bow is ofprimary importance for the rapid spiccato(Askenfelt, 1992a). With a light bow hold, whichis necessary for a rapid spiccato, the bow can beconsidered as pivoting around an axis roughlythrough the cut-out in the frog (thumb andmiddle/index finger at opposite sides). Themoment of inertia Jx of the bow with respect tothis axis and the restoring moment from thedeflected bow hair (and string) defines a bouncemode frequency which is dependent on thetension T of the bow hair and the distance rs

between the contact point with the string and thepivoting point (Figure 1). Assuming that the stickbehaves as a rigid body and that the bow hair oflength L stays in permanent contact with thestring, the bounce mode frequency fBNC against arigid support is given by

f

T rr L

JBNC

s

s

x=

−

1

2

1

π

Figure 1. Geometry of the bouncing bow.

Typical values of fBNC versus the distance rs

from the frog for a violin bow are shown inFigure 2. The calculated values range from 6 Hzclose to the frog to about 80 Hz at the very tip.Measured data for a real bow, given for three

0

20

40

60

80

0 100 200 300 400 500 600 700

DISTANCE FROM FROG rs mm

BOUNCE MODEFREQUENCY

Hz

BOW HAIRRIBBON

TYPICALSPICCATOPOSITION

fBNC

Figure 2. Bounce mode frequency fBNC against arigid support for a violin bow pivoting aroundan axis at the frog. Calculated values (dashedline) and measured (full li nes) for three valuesof hair tension (“ normal” = 55 N and ± 5 Ncorresponding to ± 1 turn of the frog screw).The hair was kept in permanent contact withthe support.

Askenfelt & Guettler: The bouncing bow: Some important parameters

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different hair tensions, follow the calculatedvalues closely up to about 2/3 of the bow length.For the remaining 1/3, the increase in fBNC ismuch less than the calculated case for a stiffstick, reaching only 40 Hz at the tip. The differ-ence indicates that a flexing of the thin outer partof the stick influences the transversal bowstiffness in this range (Pitteroff , 1995). The rapidspiccato is played well i nside this part, typicallyin a range between the midpoint and 2 -3 cmoutside the middle (towards the tip). In thisrange, fBNC will be between 13 - 15 Hz, and thestiff bow stick model predicts the bounce modefrequency accurately. The compliance of a violinstring will l ower this frequency by 1 Hzapproximately.

The player´s adjustment of the bow hairtension is precise. At a typical playing tension of55 - 60 N, one full turn of the frog screw changesthe tension ± 5 N, which seem to demarcate thelimits of the useable tension range in rapidspiccato.

The Q-value of the bounce mode is of theorder of 50 when measured with the bowpivoting freely around an axis at the frog as inFig. 1. When simulating playing conditions byletting a string player hold the bow while stillsupported by the axis, the Q-values were loweredto about 15.

Take-off and flightThe bow leaves the string when the contact forcehas decreased to zero. At that moment theangular acceleration is high enough to make the“ inertia moment” Jx d2ϕ1/dt2 match the “bowingmoment” MB supplied by gravity and the player.

The motion during the flight time is dependingon the character of the bowing moment MB. For ahorizontal bow, gravity will give a constantmoment independent of the bounce height and aflight time proportional to the angular velocity attake-off . The bow will bounce with an increasingrate when dropped against a support. At a typicalspiccato position on a violin bow, this occurs atan initial frequency of about 7 - 8 Hz for areasonable starting height (about 1 cm),approaching fBNC asymptotically as the bounceheight decreases. An additional restoring momentduring flight, supplied by the player, enables afaster spiccato than the gravity-controlled (muchlike a bouncing ball which not is allowed to reachits maximum height). Due to the compression offinger tissue, this additional moment is probablyof a “springy” character, increasing withdeflection angle ϕ1 . For the cello and doublebass, which are played with the strings nearlyupright, the contribution from gravity is much

reduced. In all , the two different restoringmoments acting during string contact (deflectedbow hair) and flight (the player´s bow hold),respectively, will roughly give a oscill atingsystem which switches between two slightlydifferent spring constants at take-off and landing.

Point of percussion and stickbendingThe “point of percussion” (PoP) is a parameterwhich often is referred to in discussion of bows.When hanging vertically, pivoted at the cut-out inthe frog, the bow will oscill ate with a lowfrequency, typically 0.72 - 0.75 Hz for a violinbow. This corresponds to the motion of a simplependulum (point mass) with length lPoP. For acompound pendulum consisting of a straight rodof uniform cross section, lPoP will be 2/3 of itslength. The violin bow is close to this case with atotal length between the pivoting axis at the frogand tip of typically 680 mm and lPoP between450 and 480 mm. A double bass bow will havePoP relatively closer to the tip.

An external force at PoP (such as when thebow lands on the string) will give no transversalreaction force at all at the player´s bow hold.Forces which are applied inside and outside PoP,respectively, will give reaction forces withopposite signs. Spiccato is always played wellinside PoP (about 10 cm), and the impact forceduring string contact and the correspondingreaction at the pivoting point will tend to bendthe bow so that the stick flexes upwards whenreferring to a horizontal bow (bending away fromthe hair). These forces will try to straighten outthe bow maker´s downward camber (concaveshape). The accompanying lengthening of thestick would increase the tension of the bow hair,by bringing the endpoints of the hair farther apart(Pitteroff , 1995). This effect is contradicted,however, by a decrease in the (upward) angle ofthe tip, and the net change in hair tension duringcontact can probably not be determined withoutdirect measurements.

In addition, a mode of the pivoted bow atabout 60 Hz will be excited at the impact atlanding. The shape of this mode is similar to thestick curvature (large motion at the middle andnodes at the frog and tip), including a rockingmotion of the tip. The 60-Hz mode can beobserved in the stick and tip during the entirespiccato cycle and will modulate the tension inthe bow hair, also during the contact with thestring. As will be seen in the following, a corre-sponding modulation is hardly observed in thecontact force, but the possibili ty of a periodic

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variation in the instantaneous bow velocityremains.

Mechanical spiccatoThe rapid spiccato can be reproduced in thelaboratory (Figure 3). The bow is supported byan axis in the cut-out in the frog and the hair isresting on a force transducer at the normalposition for rapid spiccato just outside the mid-point. A steady bow force of about 0.5 N issupplied by a spring, simulating the player´s“bowing moment.” The bow is driven by ashaker with a sinusoidal motion, which for eachcycle gives a short downward push on top of thestick at the normal position of the index finger.The motion of the stick is measured by aminiature accelerometer on top of the stick at theplaying position.

A typical registration of the motion of thestick and contact force and is shown in Figure 4,driven at a spiccato rate of 12.0 Hz and with thedriving adjusted to give a peak contact force of2 N. Contact force and stick displacement are inphase. After a flight tour, contact between thehair and support occurs with the stick movingdownward with a velocity of about 25 cm/s andthe stick displacement close to the equili briumposition. The motion of the stick is considerablewith an ampli tude of about 5 mm. A forcefulspiccato may drive the stick almost down to thehair, giving a stick amplitude of about 9 mm.

The bouncing bow in spiccato can be viewedas a mechanical series circuit with one restoringmoment during string contact due to the deflectedbow hair, and a second during flight (gravity andthe player´s bow hold), acting on the moment ofinertia Jx. The bow is driven close to the bouncemode frequency with the “ finger” of the shakergiving a down-ward push on the stick just afterit has passed its upper

turning point. The duration of the push is about aquarter of the spiccato period. This is similar tothe action of the player´s index finger (Guettler &Askenfelt, pp. 47-51, this issue). Depending onthe duration of the flight (determined by theplayer´s bow hold) in relation to the duration ofstring contact (set by the bounce mode), the bestspiccato driving can occur slightly below orabove the bounce mode frequency.

Contact forceThe contact with the string lasts a li ttle more thanhalf the spiccato period in a crisp, rapid spiccato(Guettler & Askenfelt, this issue, pp. ). Duringthis contact time (about 40 ms), the forceresembles a half period of a sine wave reachingpeak values of typically 1.5 - 2 N (Figure 4). Theforce waveform is somewhat peaked due to a“ ripple” component at about 150 Hz. Themagnitude of this ripple can change significantlyfor a slight change in driving frequency,depending on the relation to the spiccato rate(Figure 5). For this bow, a shift in drivingfrequency of 1 Hz boosted the 150-Hz ripplemarkedly.

A modal analysis of two bows showed thatthe 150-Hz component can be traced down totwo modes of the pivoted bow at approximately150 Hz (#3) and 170 Hz (#4) (Figure 6). Mode#3 resembles the second mode of a free-free

Figure 3. View of the experimental setup for amechanical spiccato. A hook fastened to theshaker contacts the upper side of stick via apiece of rubber, thus simulating the player´sindex finger. The bow force spring gives astatic bow force of about 0.5 N.

Figure 4. Registrations of bow motion in rapidspiccato when a violin bow is driven by ashaker at 12.0 Hz. The registrations show thecontact force against a rigid support 2 cmoutside the midpoint, and velocity anddisplacement of the stick above the position ofthe support. The vertical dashed lines indicatethe contact duration.

50 ms/DIV

2

1

0

2

1

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CONTACTFORCE

N

Askenfelt & Guettler: The bouncing bow: Some important parameters

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stick without hair (typically at 160 Hz(Askenfelt, 1992b), but with a marked rotationalmotion of the tip. The flexing of the stick in mode#4 occurs mainly in the outer part of the stick.By driving the bow under the hair at thesupporting point it was also observed that bothmodes include a strong motion of the hair oneach side of the support, cf. Bissinger (1995).Rather than being a doublet with the hair andstick moving in-phase and out-of-phase, respec-tively, the mode spli tting is probably due to theresonances of the hair on the long and short sideof the supporting point, respectively. The modefrequencies can be shifted by moving the support(spiccato point), and even brought to coincide infrequency. The question cannot be settledcompletely unless measurements are taken alsoon the bow hair.

The 150-Hz ripple seen in the contact force isprobably a combination of the two modes. Theboost of the ripple in Figure 5 (bottom) wasobserved when the 13th component of the drivingforce coincided with one of the mode frequencies.Interestingly, the bouncing motion of the bow canbe started by supporting the bow hair with ashaker at the spiccato point and driving with thefrequency of mode #3 or #4. This indicates anon-linear coupling between these modes and thebounce mode, which possibly has something todo with a certain quali ty of some bows “ to clingto the string” during long notes in legato anddetaché.

A crisp spiccatoA good bow will facili tate a rapid spiccato withclean, crisp attacks. In order to establish aprompt Helmholtz motion it seems essential thatthe contact force builds up fast to give the firstslip after shortest possible delay. Further, the

force should stay essentially constant during thefollowing periods in order to avoid multiple-slipping (Guettler, 1992). A box-shaped forceduring contact would then seem optimal for acrisp spiccato.

As seen in Figure 5, the 150-Hz componentcan increase the slope at force build-up (from0.12 to 0.23 N/ms), but on the other hand, theaccompanying stronger fluctuations in force giveincreased risk of multiple slipping. A histogramrepresentation of the force histories in Figure 5 isgiven in Figure 7. A distribution with few valuesat intermediate force values and a collectiontowards the maximum force percentile could beassumed to be advantageous according to thereasoning above. In this respect, the distributionfor 12 Hz driving frequency with only minor150-Hz activity would be more promising for acrisp spiccato. The damping due to the player´sholding of the bow will reduce the 150-Hz rippleslightly, but in no way cancel this activity in thebow. The bow hold in spiccato is light, andlocated close to a nodal point for most bowmodes (Askenfelt, 1992b).

Figure 7. Histogram distributions of the contact force histories in Fig. 5; 12-Hz case with littl e ripple(left) and 13-Hz case with pronounced ripple (right).

Figure 5. Comparison of the contactforces against a rigid support for aviolin bow driven in spiccato at12.0 Hz (top) and 13.0 Hz (bottom).The peak force is kept constant at 2N. Notice the difference in strengthof the 150-Hz ripple. The slopingdashed lines show the initial in-crease in force corresponding to0.12 and 0.23 N/ms.

Figure 6. Modal analysis of a violin bow showing mode #3 at 158 Hz (left) and mode #4 at 169 Hz(right). The bow is supported by an axis at the frog and a fixed support at a typical position forrapid spiccato 2 cm outside the midpoint.

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Comparisons between wooden bows of differentquali ty and bows of novel materials (fibre glass,carbon fibre composites) gives a picture which isfar from clear-cut. The strength of the 150-Hzripple in spiccato seems not to be related to thequali ty of the bow in a simple way. It is presentin poor bows as well as in excellent ones, theactual strength depending on the spiccatofrequency. Also a bow which had been bent to aconvex (“baroque”) camber which made itimpossible to tighten the hair to normal tension(25 N instead of 55 - 60 N) showed the rippleactivity (now at 110 Hz) in the contact force.This bow performed very poorly in spiccato,probably depending on a low bounce modefrequency (10 Hz instead of 13 -15 Hz) due tothe low tension of the hair. As noted by players,the abili ty to take a high tension of the bow hairwith only a minor straightening of the camber isone of the basic quality marks of a good bow.

ConclusionsThe rapid spiccato bowing is dependent on abounce mode of the pivoted bow at about 13 -15 Hz. This mode seems to be very similar for allbows. The contact force with the support showeda “ripple” component at about 150 Hz, whichwas traced to a combination of two modes withstrong activity in the outer part of the bow. Thestrength of this ripple, which is dependent on thespiccato rate, can reduce the force build-up timewhich would facili tate a fast and crisp attack, butthe accompanying variations in bow force may,on the other hand, increase the risk of multipleslipping. The net

effect of the ripple is not known yet, either beinga desired property or not.

In view of that bows perform very differentlyin rapid spiccato, it seems reasonable that thedifferences must be sought in the action of thestick, and not in the basic bounce mode whichcan described by a lumped mass-spring system.The differences in masses and moments of inertiabetween bows are small and the hair is taut tonearly the same tension. In view of thesesimilarities, the 150-Hz ripple, and possibly alsohigher modes, are interesting properties tocompare in a future study.

AcknowledgementsThis project was supported by the SwedishNatural Science Foundation (NFR) and theWenner-Gren Centre Foundation..

ReferencesAskenfelt A (1992a). Properties of violin bows,

Proc. of International Symposium on MusicalAcoustics (ISMA´92), Tokyo, August 1992; 27-30.

Askenfelt A (1992b). Observations on the dynamicproperties of violin bows. STL-QPSR, TMH,4/1992: 43-49.

Bissinger G (1995). Bounce tests, modal analysis,and the playing qualiti es of the violin bow. CatgutAcoust Soc J, 2/2 (Series II); 17-22.

Guettler K & Askenfelt A (1997). On the kinematicsof spiccato bowing. TMH-QPSR KTH, 2-3/1997:47-51 (this issue).

Guettler K (1992). The bowed string computersimulated - Some characteristic features of theattack, Catgut Acoust Soc J, 2/2 (Series II): 22-26.

Pitteroff R (1995). Contact mechanics of the bowedstring, PhD dissertation, University of Cambridge.

2 1.5 1 0.50

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