28-39 tlt feature 1-07 - university of cambridgejw12/jw pdfs/feature_1-07.pdf · the verge of a...

28 J A N U A R Y 2 0 0 7 T R I B O L O G Y & L U B R I C A T I O N T E C H N O L O G Y

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T R I B O L O G Y & L U B R I C A T I O N T E C H N O L O G Y J A N U A R Y 2 0 0 7 2 9

CONTINUED ON PAGE 30

Using—and sometimes inventing—

highly specific testing apparatus,

researchers continue to fiddle with the

bizarre lubrication properties of rosin.

By Linda Day

Analyzing thetribology of sound

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regime seems quirky or inscrutable. Violinists and researchers Jim Wood-

house of Cambridge University’s Engineer-ing Department in England and Bob Schu-macher, with the physics department ofCarnegie Mellon University in Pittsburgh,have been working for years to understandand simulate what goes on. If they succeed,computers may throw a little light on whythe violin is so hard to play.

At present, nearly everything about theway a violin and its strings vibrate to producesound is understood well enough for goodsimulations except for one thing: the interac-tion of bow and string. And here is where theWoodhouse-Schumacher team may be onthe verge of a simulation breakthrough. Ornot. To understand how far they’ve come, youhave to start with Helmholtz.

Strange cornersWhen you draw a rosined bow across a vio-lin string and create a sound that your violinteacher does not object to, something extra-ordinary happens to the string. Hermannvon Helmholtz discovered the resultingwaveform about 140 years ago, showing thatthe string moves in a V shape, two straight-line segments separated by a sharp “Helm-holtz corner.” What appears to be a smoothvibration is really just the envelope of amoving corner that circulates around oneway: When the bowing changes direction,the corner reverses and goes around theother way. For an animated version of this,see the Web article at http://plus.maths.org/issue31/features/woodhouse/index.html, since a moving picture is worth evenmore than the proverbial thousand words.

Sustaining Helmholtz motion dependson the bow’s rosin, which possesses uniquestick-slip tribological properties. While theHelmholtz corner travels on its long jour-

ney to the player’s finger and back, thebow is gripping the string so that bow

and string move together (sticking)until the Helmholtz corner arrivesback at the bow, at which point therosin gives and the string zips acrossthe bow hairs (sliding). The string

sticks again when the corner arrives backfrom the short journey to the bridge, andthe cycle repeats.

Good sound quality relies on achievingHelmholtz motion, which basically dependson three things: the downward force of thebow on the string, the bow position and thebow speed. If the bowing force is too light,the bow slips over the string with only asqueaky ‘surface sound’ to mark its passing.This is the result of insufficient sticking,which results in two or more episodes ofslipping per cycle of vibration. If the bowforce is too heavy, it creates chaotic vibra-tions that produce the raucous graunch of atortured string.

But it’s not enough to get the forceright—your bow also has to be moving atthe right speed and in the right place on thestring because Helmholtz motion also de-pends on the distance of the bow from thebridge as a fraction of the total length of thestring. J.C. Schelleng1 worked out the mathin the early ’70s, creating the Schelleng Dia-gram shown in Figure 2. It shows that if youplay near the fingerboard, you have a betterchance of sounding good.

OK, so now you can bow with the rightforce in the right place with the right speedto achieve Helmholtz motion—but howlong does it take you to do it? If you wantpeople to stay in the room, you’d better getpast the initial transients and intoHelmholtz in less than 50 ms. And here’swhere “playability” comes in.

Professional violinists judge an instru-ment not only on its sound but also on its

A violin’s beautiful tones depend on oscillations thatonly a slip-stick non-lubricant (the rosin on the bow) canprovide—and nearly everything related to this harmonic

CONTINUED FROM PAGE 29

Figure 1. Helmholtz motion of a bowed string.The dashed

line shows the envelope of the motion, and the solid lines

show three different snapshots of string position at dif-

ferent stages in the cycle. The “Helmholtz corner” circu-

lates in one direction as shown by the arrows. Circulation

reverses when the bow changes direction.

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playability, how quickly and easily the violinmoves past the initial transients to achieveHelmholtz motion. “That’s what motivatesour thread of research, playability,” saysWoodhouse. “When players are trying instru-ments, they say things like ‘I don’t like thesound, but it’s very easy to play.’ Sound qual-ity is a psychological thing, but playing qual-ities are more directly related to mechanicsand science, which we can measure andunderstand and hopefully learn to simulate.

“In order to make contact with the thingsmusicians really want to know about, anysimulation we develop has to be a reallygood model,” he continues. “It’s not enoughto get it roughly right. Everything has to dowith transients, and those are hard things toget right in theory.”

Woodhouse notes that for nearly allother types of frictional vibration likesquealing brakes and creaking hinges, thegoal is simply to get rid of the noise, andyou may not need to understand more thanthe basic science or common engineeringpractice. The musical application is uniquein its goal to understand and to simulate ona computer every component of Helmholtzmotion and related transients: the bow, theviolin body vibration, the string and espe-cially the rosin. With a reliable simulationmodel, many aspects of violin playabilitycould be probed by computer experiment.

Friction fact and fictionOne thing that Schumacher and Wood-house discovered from researching the tri-bology of rosin back in the 1980s was thatwhat everyone knew about slip-stick frictionwas wrong.

“People thought the friction depended

on sliding speed,” he says. “That followsnaturally from the measurements that peo-ple do. You’ve got a pin-on-disk tribometer,you measure the friction force during steadysliding, then you change the speed andmeasure the friction again. And becausethat was the data everyone has, peopleassumed the same thing would happen athigh frequencies. It’s a concept that is meas-urement-driven. There is no good reasonwhy that should be true, and as soon as wetested it, we found it wasn’t.”



About Jim WoodhouseJim Woodhouse received a mas-ter’s of arts degree in mathemat-ics from Cambridge University inEngland, in 1972. After receivinghis doctorate degree in 1977, hedid post-doctoral work on theacoustics of the violin in thedepartment of Applied Mathe-matics and Theoretical Physicsat Cambridge (this work beinginspired by a hobby interest inbuilding instruments). He then

worked for several years on a variety of structural vibration problemsfor a consultancy firm before joining the department of engineeringat Cambridge in 1985 as a lecturer, and then later as reader and pro-fessor. His research interests all involve vibration and musical instru-ments and have continued to play a major role in his career.

About Bob SchumacherBob Schumacher received hisundergraduate degree inphysics at the University ofNevada, Reno, and his doctorateat the University of Illinois, in1955. His thesis research was innuclear magnetic resonance un-der professor C.P. Slichter. Aftera brief instructorship at the Uni-versity of Washington, he joinedthe faculty of Carnegie Instituteof Technology (now CarnegieMellon University) in 1957. Inthe mid-’70s, he changed his

research interests to musical acoustics, a change that was solidifiedby a sabbatical at Cambridge University in 1977. His major interestshave been simulations of musical oscillators, with particular empha-sis on stringed instruments. He has been a professor of physics(emeritus) at Carnegie Mellon University since 1997.

Figure 2.The Schelleng Diagram shows the region of force

and position where Helmholtz motion can be achieved.




Figure 3 shows the debunked theorymatched up against experimental results. Inthe theory (the dashed line), friction forceincreases as sliding velocity decreases untilsticking occurs (the vertical line), at whichpoint the friction force can take a range ofvalues. In the experiments (the solid line),force follows a hysteresis loop.

“You can still see a faint shadow of theold curve,” says Woodhouse. “There’s avaguely vertical bit on the right, which issticking, and a slipping bit to the left. Butno part of that loop is anywhere near thedashed line. The original model was justwrong. The interaction with the dynamicsof the vibrating system changes the fric-tion behavior from the tribometer meas-urement.”

As you might imagine, the experimentaltechnology that produced the hysteresis

loop is quite a bit more complex than a pin-on-disk tribometer—dynamic friction forceis not so easy to measure directly. The firstdevice used a cantilever with a glued-onwedge of the material to be measured (seeFigure 4).

To excite the cantilever into stick-slipvibration, it was ‘bowed’ with a cylindricalrod coated with rosin and pressed againstthe cantilever by an arrangement of wiresand weights. The rod was driven at a steadyvelocity monitored by a displacement trans-ducer. Vibration of the cantilever was meas-ured using a small accelerometer placed asclose as possible to the driving point. Thevibration data covered only a few stick-slipcycles over a short time (~ 0.2 sec) and a fewmillimeters of travel, so that both the nor-mal load and the rod velocity could reason-ably be assumed constant.2

The cantilever produced interesting data,but it was still a very simple system, a far cryfrom the complexity of a violin string.

Starting from zeroMeanwhile, Knut Guettler took a leap for-ward in the study of playability in his disser-tation. Guettler taught himself string bass,became one of the world’s foremost players,and then, without even a secondary schooleducation, expanded his player’s knowledgeto a doctorate in music acoustics from theRoyal Institute of Technology in Sweden.

As a musician, Guettler realized the valueof the new experiments on slip-stick frictionbut recognized that steady-state experi-ments could not address playability. In thereal world, the bow must start with zerospeed (unless you’re using the “bouncingbow” technique, which no computer modelshave yet attempted to simulate). Guettler

Figure 3. The debunked theory of stick-slip friction

(dashed line) shows the coefficient of friction following a

smooth curve during sliding, while the vertical portion of

the dashed curve shows sticking. The solid line shows

what actually happens during a stick-slip vibration. The

arrows indicate the direction around the hysteresis loop.

Figure 4. Schematic diagram of the experimental rig. The cross-section is shown to scale.



sought to create a computer simulationbased on Woodhouse’s mathematics, recog-nizing that a meaningful simulation wouldnot only require the right equations for thehysteresis, it would also require initial tran-sients that were musically sensible.

As a start in addressing these issues,Guettler studied a family of bow gestureswith constant bow force and a velocity thatstarts at zero and accelerates at a constantrate2. Guettler’s bowed-string simulationsplotted the time taken to achieve Helmholtzmotion at a grid of points in the force/accel-eration plane. He also sought analyticalexpressions for the upper and lower boundsof a region containing “perfect transients,”combinations of force and acceleration thatproduce Helmholtz motion right from thestart of the note.

Guettler derived four necessary conditionsfor the production of a perfect transient; indoing so he simplified the real world signifi-cantly: He ignored the effects of wave disper-sion, torsional motion in the string and tem-perature, and he treated the ends of thestring as dashpots. He also assumed that thebow would contact the string only at onepoint, that the bow hair was rigid and thatthe bowing position would be an integer divi-sion of the string length.

Guettler’s four conditions relate to theway the initial stick-slip events interact witheach other—bogglingly complex! We shallsidestep nimbly around them here, but youcan get a good explanation in reference [4].3

Figure 5 presents Guettler’s diagrams for sixdifferent bow positions defined by the valueof β, the bow position as a fraction of thestring length. Like the Schelleng Diagram,the Guettler Diagrams show that it’s easierto achieve Helmholtz motion quickly (i.e.,the white wedge is bigger) when your bow isnearer the fingerboard (i.e., when β‚ is 1/7rather than 1/12), and when you use greaterbow velocity and force.

How well does Guettler’s theory matchup with fact? Woodhouse decided to createan experimentally measured Guettler dia-gram, which would reveal immediatelywhether the wedge-like region of perfecttransients in Figure 5 gives a useful approx-imation to real behavior. Such an experi-ment would require a versatile mechanicalbowing machine—no human has therequired control or patience—so he and a

graduate student, PaulGalluzzo, designed theone shown in Figure 6. Alinear motor carries abow, or a rosin-coatedPerspex rod, which pro-vides a match to simula-tion models with a sin-gle point of contact.

In the Woodhouse-Galluzzo bowing ma-chine, the bow positionis monitored by a digitalencoder that tracks thedesired trajectory, andthe bow force can be tai-lored to any desired pat-tern by another feedbackcontroller using a forcesignal from straingauges in the machine’s“wrist.” This computer-controlled robot canproduce a wide range ofbowing gestures withgood accuracy andrepeatability. Although itcan’t cope with “bounc-ing bow” gestures, forcontinuous contact itscapabilities meet or exceed those of ahuman player.4 The string motion inresponse to a given bowing gesture is mon-itored using a piezo-electric force sensorbuilt into the bridge of the test instrument,a cello.

At the same time he was testing the bow-ing machine, Woodhouse also was testinghis own simulation models that he hopedwould match the machine’s results. Figure 7shows how the results stack up.

Figure 7(c) shows simulation resultsbased on the debunked friction curvemodel—obviously no match for experimen-tal results. Figure 7(d)shows simulation resultsfor Woodhouse’s ther-mal plastic model. Inthis model, he treats therosin as a perfectly plas-tic solid, which will onlydeform (i.e., allow slip-ping) when the shearstress reaches the rosin’s Figure 6. Computer-controlled bowing machine fitted with

a rosin-coated Perspex rod to bow a cello.

Figure 5. Predicted and simulated regions in the

force/acceleration plane where “perfect transients” are

possible, reproduced from Reference 4. Right-hand col-

umn: the lines labeled ‘A’, ‘B’, ‘C’, ‘D’ correspond to the four

conditions described in that text. The white wedge shows

the region satisfying all four conditions. Left-hand col-

umn: results of simulations covering the same parameter

space, colored so that white pixels correspond to perfect

transients. The six rows show results for different values

of bow position, ß, expressed as the fractional distance of

the bow point from the bridge.




shear yield strength, which he assumes tobe temperature-dependent. Better, but stillno cigar.

And now Bob Schumacher enters the pic-ture with a different bowing machine.

Inverse calculations andmicroscopic wear tracksRecognizing the need for an experimentalsystem that would measure a wider range ofdynamical behavior than the Woodhouse-Galluzzo bowing machine could, Schumach-er invented a bowing machine designed tomeasure friction force and string velocity atthe contact point.

“The crucial thing is that you cannot putany kind of mass whatsoever on a string

without changing its properties,” Schu-macher says, “and if you want to know boththe force and velocity at the bowing point,that’s too much to measure at one place. SoI figured that we could get the informationwe needed by measuring forces at the ter-mination of the string—the bridge and thefinger—and then using inverse calculationto back out behavior under the ‘bow.’ Whenwe finally had this, we wondered why wehadn’t thought of it years ago.”

Figure 8 shows a schematic of Schu-macher’s bowing machine. The rosin-coatedrod is pressed against a violin E string (ahigh-tensile steel monofilament), which ismounted on a frame supported on microm-eter mounts that allow the normal force ofthe string on the rod to be adjusted. The rodis mounted on an aluminum plate, and elec-tric heaters on the plate allow the rosin-coated rod to be tested at varying tempera-tures. The rod and plate are moved by a con-stant velocity trolley that moves 0.2m at aconstant speed 0.2m/s. Acceleration anddeceleration occupy about 0.1s, so all runslast slightly longer than 1s. The tension inthe string (the note to be played) is adjustedto 650 Hz, giving many hundreds of periodsof oscillation in each run.

Piezoelectric force sensors mounted atthe string terminations measure the trans-verse component of the oscillating forceexerted by the string, without interferingwith the motion of the string. A reconstruc-tion algorithm then combines the forcesmeasured at the two terminations withparameters obtained from plucking experi-ments to calculate the velocity, v(t), of thecenter of the string, and the friction force,F(t), at the bowing point.5

This novel experimental approach offersimportant advantages: First, the reconstruc-

Figure 8. A sketch of the bowed-string apparatus

(not to scale). The side view shows rod pressing

on string to produce normal force.

Figure 7. Guettler diagrams for a cello D string. Top row:

measured using (a.) a conventional bow and (b.) with a

rosin-coated rod. Bottom row: simulated using (c.) the

(debunked) friction curve model and (d.) the thermal

plastic model, both assuming a single point of contact

between bow and string. All four plots correspond to the

same bow position ß = 0.0899 (about 1/12): for (a.), this

is the center of the bow-hair. (Note: This image is “pixi-

lated” because it is a computer calculation generated

from individual pixels, each representing one run of the

program).




tion algorithm takes into account all vibra-tion modes of the string so that useful datacan be collected over the full audio-frequen-cy bandwidth. And, second, the data gath-ered provides a sensitive test of which oscil-lation regime the string chooses undergiven conditions: A bowed string shows avery rich range of dynamical behavior, bothperiodic and nonperiodic, all of which canshed light on the underlying frictional con-stitutive law.6

Finally, a friction model that can predictthe correct sequence of oscillation regimesduring this kind of transient stick-slip testwould have a good claim to being convinc-ing. “It’s been shown that a skilled violinistcan control the length and nature of initialtransients with impressive consistency,”says Schumacher, “so the inherent variabili-ty of frictional interfaces can’t be used as anexcuse for poor models!”

Figure 9 shows sample data from Schu-macher’s experiments.

The top graph shows ideal Helmholtzmotion, where the velocity of the string rel-ative to the bow is close to zero during thelong sticking part of the cycle. The bottomgraph shows part of an initial transient withtwo incidents of sliding per cycle, which pro-duces a “surface” sound. As bowing contin-ues, the second smaller sliding event diesout, and the string attains the desiredHelmholtz regime.

The waviness of the flat part of the lineillustrates “Schelleng ripples:” “The ques-tions is, is the string really sticking, or doesit slip a little bit?” says Schumacher. “Theapparatus shows that the bow speed andthe string speed during the nominal stick-ing part are not necessarily the same. Butthis plot does not prove that sticking is nottaking place: the string may be rolling onthe bow, or creeping along the bow or a bitof both.”

Schumacher’s apparatus is capable ofextreme detail. In graphs of friction force vs.distance during the sticking phase (forwhich the debunked theory proposed thesmooth line shown in Figure 3), Schumach-er’s data leads to Figure 10.

When Schumacher plots the data as forcevs. velocity (the hysteresis curve), he getscurves that differ dramatically, dependingon the temperature of the rosin, but whichall show the sticking ‘scribble.’

To complement the work with the bowingmachine, Schumacher and Woodhouse alsoexamined scanning electron micrographs(SEM) of the wear scar created in the rosinedsurface after a single bow stroke. Figure 11(see page 38) shows the wear track for twoperiods of double-slip motion (e.g., Figure9(b)), plus a close-up view of location D. Thestring is vertical on the page, and therosined bow is moving left to right. Each

Figure 9. Helmholtz motion

(top) and “surface sound” (bot-

tom): The velocity of the string

relative to the bow is close to

zero during sticking (the

longest part of the cycle), and

then changes sharply during

sliding. The bottom graph

shows two incidents of sliding

per cycle.

Figure 10. The top graph

shows friction force as a func-

tion of position on the rod for

the Helmholtz motion of Fig-

ure 9(a). The bottom graph is

an extreme magnification of

the first slipping episode

shown in the top graph.




gouge in the rosin’s surface represents a sin-gle stick, while the mostly intact sections ofrosin in between the gouges reflect the slip.

“Each scar is a place where the string hasstuck to the bow,” Woodhouse explains.“And maybe while it was sticking, it wasrolling around a little, which is why it’schurned it up a little bit—that could relateto the Schelleng ripples. Then the string letsgo, and the rosin breaks in a brittle sort ofway, because the string pulls a chunk of itout and leaves a glassy looking fracture.

“It’s easy to believe that temperaturehas been varying a lot during this

cycle—the points where it’s been stick-ing and then let go, you can see itfailed in a brittle sort of way,” Wood-house adds. “Whereas in between

the scars, the little hori-zontal lines show whererosin has melted on thesurface and been pulledout into fingers. You’vegot to think treacle here.So during slipping, therosin gets really hot andmore or less melts, andduring sticking it’s hadtime to cool down a bitand get stronger, until itgets ripped off by brittlebreaking.”

Woodhouse had in-cluded some tempera-ture effects in his firstpaper (the Guettler Dia-grams shown in Figure 7),but this got him to think-ing about temperature ina deeper way. “It makesgeneric sense that tem-perature is the mostimportant thing,” hesays. “The most conspic-uous thing about rosin isthat it’s a glassy brittlesolid at room tempera-ture, but you only have tohold it in your fingers afew seconds and it goessticky. Its glass transitiontemperature is not farabove room temperature,which means that all of

its properties are varying rapidly with tem-perature—the effective viscosity changes byorders of magnitude.”

Woodhouse explains that in Helmholtzmotion, the string spends 90% of the cyclesticking. When it slides, it slides very fastand generates considerable heat. As therosin heats up, its viscosity goes down andthe friction force goes down. At the end ofthe sliding interval, the friction force is stilllow, but the string comes back to meet it asthe Helmholtz corner comes around andreattaches the string to the bow.

“Then the string sticks again for a rela-tively long time, during which the heat gen-erated during sliding leaks away into thethickness of the string or into the rod, andthe contact layer, which is just a few microns

Figure 11. SEM micrograph of wear track on the coated rod, showing approximately

two periods of a double-slip motion, and (b) enlarged view of the “D” area shown in

(a). (Reprinted with permission from Woodhouse, J., Schumacher, R.T. and Garoff, S.,

(2000), J. Acoust. Soc. Amer. 108, pp. 357–368. Figure 8(a). Copyright 2000, Acoustical

Society of America).




thick, cools down. As it cools, it goes backup the viscosity curve, creating the hystere-sis effect,” he adds.

“Our first model made a simple attemptto represent that, and it moved in the rightdirection, but it clearly wasn’t fully accurate,”Woodhouse says. “What I’m in the middle ofat the moment is going for something that’smuch more physics based. Rather thandoing a heat-flow calculation and guessing,I’m trying to do more careful measurementson the rosin as a material and a more thor-ough heat-flow model, where we actuallylook at temperature distribution throughoutthe rosin layer, not just an average tempera-ture. It’s work in progress.”

In particular, Woodhouse is interested inthe formation of shear bands within therosin. “Having shear bands in the back of mymind, it may be that something similar ishappening in our rosin layer,” he says. “Westart to slide, and we start to shear, and aswe shear it gets hot, and if the temperatureisn’t uniform—say it’s a bit warmer in themiddle than the edges, which could be heatleft over from the previous cycle—then theviscosity is already lower there, and maybethe rosin layer forms one of these shearbands in the middle, within the thickness ofthe rosin. Then the actual ‘break’ would takeplace at the shear band. That’s my mentalimage, that some or all of the rosin is essen-

tially melting and refreezing every cycle. Thetemperature may be fluctuating as much as20 degrees every cycle.

“People tend to think that sticking andslipping are two different states,” Wood-house notes, “so somewhere in the comput-er program should be a switch that says ‘Ifit’s sticking, do this, if it’s slipping, do that.’I think there won’t be two different states:There will be one law incorporating viscosi-ty and temperature that will link into thedynamics of the string to produce stick-slipmotion. There will be one model of thematerial, and the stick-slip friction will arisewhen you ask how that model behaves if youdrive the interface in a particular way.”

But for definitive answers, and the ulti-mate one-law simulation model, we’ll haveto wait.

“What’s endlessly intriguing about theviolin problem is the playability question inthe background,” Woodhouse says. “Oncewe’ve got the model, we’ll want to use it tolook much harder at detailed responses.That makes it a much more challengingproblem!” <<

Linda Day is a Houston-based free-lance writer.You can reach her at [email protected].

1. Schelleng, J.C. (1973),“The Bowed Stringand the Player,” J.Acoust. Soc. Amer., 53,pp. 26-41.

2. Smith, J.H. and Wood-house, J. (2000), “TheTribology of Rosin,” J.Mech. Phys. Solids, 48,pp. 1633-1681.

3. Guettler, K. (2002),“On the Creation ofthe Helmholtz Motionin Bowed Strings,”

Acustica—Acta Acustica,88, pp. 970-985.

4. Woodhouse, J. andGalluzzo, P.M. (2004),“The Bowed String asWe Know it Today,”Acustica—Acta Acus-tica, 90, pp. 579-589.

5. Galluzzo, P. (2004),“On the Playability ofStringed Instruments,”Doctoral dissertation,University of Cam-bridge.

6. Woodhouse, J., Schu-macher, R.T. and Gar-off, S. (2000), “Recon-struction of BowingPoint Friction Forcein a Bowed String,” J.Acoust. Soc. Amer., 108,pp. 357-368.

7. Schumacher, R.T., Gar-off, S. and Wood-house, J. (2005), “Prob-ing the Physics ofSlip-Stick Friction Us-ing a Bowed String,”J. of Adhesion, 81, pp.723-750.

References


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