Nylon strings

Nylon strings

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Initial Behavior of Nylon Guitar Strings MARK FRENCH1, DEBBIE FRENCH2, AND CRAIG ZEHRUNG 3 Abstract—Players have known about some of the quirks of nylon guitar strings since their introduction after WWII. In particular, new strings tend to decrease in frequency and need to be re-tuned repeatedly in the first few days of use as they stretch. This paper presents data on new strings in which the tension is held constant and the string length is allowed to vary – essentially the reverse of the condition on the guitar where the length is held fixed and the tension changes. The frequencies of monofilament nylon strings were observed to vary cyclically. This effect is not clearly explained by common expressions for resonant frequencies of stretched strings and may require a more sophisticated model for nylon strings under tension. I. INTRODUCTION Nylon (polyamide) guitar strings were introduced in the 1940s to replace gut strings [1]. One characteristic of nylon strings is that they creep when they are new so that the pitch decreases when they are installed on a guitar [2], [3]. This process typically continues over a period of hours or days and requires the instrument to be retuned regularly until the strings stop stretching. The effect is typically more pronounced in solid strings than stranded strings. The standard explanation of this effect is that the polymer chains forming the nylon are not straight when the string is first installed on the guitar. As tension is applied, the chains straighten out and the axial stiffness of the string increases. This time-dependent stiffness is welldocumented in plastics, including nylon [4]. Nylon strings are furnished in two forms. The higher pitch strings are made of monofilament and typically have diameters in the range of 0.025 in – 0.042in (0.635mm – 1.07mm). Lower pitch strings have stranded cores and are wound with silver plated copper wire or, less often, bronze wire. This winding increases the mass per unit length so that the wound strings can be brought to the correct pitch at approximately the same tension as the solid strings.

Photo 1 – A wound classical guitar string with a stranded core.

Stranded strings are less prone to pitch changes after being installed. There are at least two possible reasons. The first is that the strings are made of many fine strands of plastic and the polymer chains are necessarily forced to be straighter. The other is that the strands are placed in tension during the manufacturing process and may undergo some plastic deformation before shipment. Photo 1 shows a low E string partially unwound to reveal the stranded core. A guitar is tuned by stretching the strings using geared tuning machines. Photo 2 shows the headstock of a classical guitar with the universally-used worm gear tuning machines. The gear ratio is typically 14:1. The translucent strings on the right side of the headstock are the solid, higher pitch strings. The wound, low pitch, strings are on the left side of the headstock.

Photo 2 – Classical guitar tuners (Image is from Wikimedia Commons and is in the public domain).

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Mark French, Department of Mechanical Engineering Technology, Purdue University Debbie French, New Philadelphia High School 3 Craig Zehrung, Department of Mechanical Engineering Technology, Purdue University 2

Manuscript received November 11, 2012

Article published: January 08, 2013

url: http://SavartJournal.org/index.php/sj/article/view/17/pdf

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The guitar is tuned by changing the pitch of the strings with the geared tuners. The fundamental frequency of an ideal string is given by a simple relationship.

f =

1 T 2L ρ

(1)

Where L is the length of the string, T is the tension and ρ is the mass per unit length. Frequency is, thus, proportional to the square root of the tension [5]. Classical guitar strings are generally supplied in sets of six with three being monofilament and three being stranded, wound strings. Table 1 shows the frequencies and string types for classical guitars.

String 1 2 3 4 5 6

Note E B G D A E

Frequency 329.6 246.9 196.0 146.8 110.0 82.4

Type Plain Plain Plain Wound Wound Wound

Table 1 – String Notes, Frequencies and Types.

II. TEST SETUP It is a subtle, but important distinction that, when tuning, the player is not changing the tension directly, but rather stretching the string (applying a strain) and assuming that the tension will increase proportionally. This is true if the string is perfectly elastic. However, if there is any visco-elastic creep, the tension will decrease over time, even though the end points of the string haven’t moved. With this in mind, we performed a series of tests in which the strings were loaded with a weight and allowed to stretch as necessary. The general configuration is shown in Figure 1. The pulley had a small diameter in order to form an acceptable end condition and had very low friction to prevent binding. The saddle was cut from a piece of ¼” steel bar stock and had a distinct edge, also in order to form a good boundary condition. The distance from the saddle to the contact point on the pulley was 648mm (25.5 in). The same weight (69.0N, 15.5 lb.) was used for each test and the frequencies were recorded using a microphone placed very near the string. The fundamental frequency was determined from the sound produced when the string was plucked.

Figure 1 – The test fixture uses a weight to keep the string under a constant tension.

A. Test Fixture The fixture is of a very simple design as shown in Photo 3. The load-carrying beam is of hard maple and much stiffer than a guitar. The goal was for deformation of the fixture to be negligible. The string was attached to one end with a screw and washer and passed over a steel saddle as shown in Photo 4. There is a shallow groove cut into the top of the nut to prevent lateral motion of the string. As in conventional guitar nuts, the groove is at an angle to the horizontal so that the high end faces the pulley. This is to form a good end condition for the string. The point of the fixture was to load the string with a constant force rather than imposing a fixed strain, as with geared tuners. Thus the string was passed over a pulley and supported a weight. It is particularly important that the pulley turn freely, without any binding. Photo 5 shows the setup.

Article published: January 08, 2013

Photo 3 – Test fixture without string attached.

url: http://SavartJournal.org/index.php/sj/article/view/17/pdf

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The pulley was made from a nylon sleeve through which a ¼-20 bolt was passed. The diameter of the sleeve was kept small (around 8mm) to form a clear end of the string. A shallow groove was cut into the sleeve to keep the string aligned. The groove was wider than the strings to prevent binding. The portion of the bolt supporting the

Photo 4 – String anchor screw and steel saddle.

Photo 5 – Pulley over which the string passes.

spacer was smooth, with no threads. The spacer slipped easily over the bolt with no interference. The nylon resulted in a very low coefficient of friction and the pulley spun freely over the bolt. We could find no indication of any binding of the pulley. Photo 6 shows the string under load.

Photo 6 – String installed in the fixture and under load.

Photo 7 – Microphone used for frequency measurement.

It is worth noting that the fixed weight means that the strings were not tuned to any specific frequency. Since the string was free to stretch and the tension was constant, the simple expression in Equation (1) suggests that the frequency should be approximately constant, even in the presence of visco-elastic creep. Since the length of the string increases slightly with time and its mass doesn’t change, the mass per unit length could be assumed to decrease slightly. The effect would be a very slight increase in the frequency of the string over time. The observed behavior, however, did not conform to these assumptions. B. Frequency and Strain Measurements Because the frequency and strain measurements were made by high school students as part of a class, the procedure was simple and, we think, robust. The string was plucked by hand to induce vibration. String motion was

Article published: January 08, 2013

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observed using a dynamic microphone placed very close to the string. At very short distances, a microphone essentially measures velocity of the radiating surface. Photo 7 shows the microphone near the string. The microphone was connected to a PC sound card and the fundamental frequencies of the strings were identified using a program called Soundcard Scope (http://www.zeitnitz.de/Christian/scope_en). Measuring strain over short distances can be difficult, particularly on a light structure like a string. To avoid this problem, two widely separated reference marks were put on the strings so that distance between them could be easily measured with a steel ruler mounted on the fixture. Using this method, changes in distance between in the reference marks were relatively easy to measure and strain was then calculated using the definition of strain, ε=ΔL/L. C. Argon Testing We tested some of the strings in an argon atmosphere to see whether eliminating oxygen changed the observed behavior. To do this, the fixture was placed in a large cardboard box that had been coated to reduce permeability. Corners and edges were taped and the box was slightly pressurized so that any leakage was outward. A rubber glove was mounted in the side of the box so that the string could be plucked without having to open the box. III. TEST RESULTS Figure 2 shows frequencies measured from a monofilament nylon string as a function of time. For the first 20 minutes, the fundamental frequency is essentially unchanged. After that, however, there are two long cycles during which the frequency exhibited a sharp jump followed by an approximately exponential decay. It is important to note that they are not accompanied by a proportional change in strain. Strain increases monotonically after the first 20 minutes. 1.06

1.02 Normalized Frequency Normalized Length

1.04 1.01 Normalized Frequency

Normalized Frequency and Length

1.05

1.03

1.02

1.01

1.00

1 0.99

0.99 0

50

100

150

Time (Minutes) Figure 2 – Normalized frequency, length vs. time for a monofilament string.

200

0

50

100

150

200

Time (Minutes) Figure 3 – Monofilament flouropolymer string showed only limited changes in frequency over time.

Successive tests on different brands and different types of strings established that this behavior is not anomalous. It was found in strings from different manufacturers and in strings that had been ground by the manufacturer to reduce diameter variation. We observed some frequency cycling in monofilament strings made of a flouropolymer and sold as carbon strings, though the effect was muted as shown in Figure 3. The strings are not made of carbon fiber as the name would suggest. Rather, the manufacturers use the term because the polymer from which they are made contains carbon.

Article published: January 08, 2013

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Cyclic behavior was not observed in wound, stranded strings. Figure 4 shows the frequency vs. time behavior for a wound low E string. It does not display the cyclic behavior that the monofilament string showed. This was typical of all the wound, stranded strings tested.

1

0.995 Normalized Frequency

While there is no cyclic effect as with the solid strings, there is a steady decrease in frequency with time. This suggests some time dependent mechanical change within the string. If the ideal string equation above is applicable, there is a bit of a problem. The test fixture is designed to ensure that tension and vibrating string length are constant, so the only remaining parameter is mass per unit length. There is no obvious mechanism that would explain such a change. The observed behavior would require a reduction in mass per unit length of about 1.1%. Equation (2) relates change in fundamental frequency to change in mass per unit length.

1.005

0.99

0.985

0.98

0.975

f2 f1

=

1

T2

2 L2

ρ2

1

T1

2 L1

ρ1

0

50

100

150

200

Time (Minutes)

=

ρ1 ρ2

(2)

Figure 4 – Wound strings with stranded nylon cores (this data is from a low E string) did not exhibit the frequency cycling behavior seen in monofilament nylon strings.

Solid strings are generally extruded and there it is possible for diameter to change slightly along the length of the string. At least one manufacturer used an optical inspection method to reduce the diameter variation in strings. Another approach is to precision grind the extruded strings to reduce diameter variation. Figure 5 shows that 1.02 this grinding process does not eliminate the frequency cycling.

The majority of solid strings exhibited cyclic behavior. We saw variation in typical parameters like time of the initial frequency jump, cyclic period and amplitude. However, strings showing cyclic behavior exhibited some broadly common behaviors shown in Figure 6. The first is that the fundamental frequency increases in a stepwise fashion and then decreases in a manner analogous to relaxation. The three periods in Figure 6 during which the frequency cycles are roughly equal in width, so the period is increasing approximately logarithmically. This is also broadly typical of the strings tested so far. Finally, on a semi-log scale, the frequency tends to decrease in a pattern closely approximated by two straight lines (superimposed on the data).

Article published: January 08, 2013

1.01

Normalized Frequency

There are some general patterns in the frequency data, though no simple relationships suggest themselves. The first is evident when the data is viewed on a semi-log scale. This is at least qualitatively consistent with viscoelastic constitutive models that include a velocity dependent damping term. Figure 6 shows data from the previous figure on a semi-log axis.

1.00

0.99

0.98

0.97 0

20

40

60 80 Time (Minutes)

100

120

Figure 5 – Precision ground strings also display frequency cycling behavior.

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IV. POSSIBLE MECHANISMS The data suggest that some micromechanical mechanism is at work. It is well-known that polymer chains straighten under load until they fracture. After fracture, the new ends of the chain bind with oxygen. If this effect is present in the stretched strings and is at least partially responsible for the observed behavior, testing them in an inert gas should affect the behavior. Figure 7 shows results from a test in which the string was in an argon atmosphere.

Another facet to consider is that the behavior of real strings varies slightly from that predicted by the ideal string equation presented above. The ideal equation makes several simplifying assumptions, including small deformations (constant tension) and zero bending stiffness. On most guitars, the saddle (the end support on the body of the guitar) is typically offset, in part, to account for these effects.

1.02

1.01

Normalized Frequency

The cyclic behavior is significantly attenuated, but there is a significant frequency decrease over the first 20 minutes of the test. This result is qualitatively typical of those from a larger set of measurements. Since removing oxygen from the environment around the string significantly affects the cyclic behavior, it is possible that oxidation of severed polymer chains contributes to the observed frequency cycling behavior.

1.00

0.99

0.98

0.97

Only approximate expressions are available for the frequencies of real strings [6]. One of the most widely used is

1

10 Time (Minutes)

100

Figure 6 – Periodic frequency cycling.

fn =

n T n π Er 1+ 2L ρ 4 L2T 2

3

4

(3)

where E is the elastic modulus that relates stress and strain, r is the string radius and n is the frequency number. The fundamental frequency results when n=1.

1 0.99 0.98 Normalized Frequency

If micro-mechanical phenomena are at work, this equation suggests that either the radius or elastic modulus may be increasing. An increase in the radius is essentially ruled out by a positive Poisson ratio, η, for nylon; η≈0.35 for nylon, depending on processing and formulation. By elimination, this suggests that the elastic modulus may be changing. A change in elastic modulus might be accompanied by a change in the strain rate, depending on the nature of the visco-elastic behavior of the material.

1.01

0.97 0.96 0.95 0.94 0.93

V. CONCLUSIONS

0.92

Frequencies from a pool of classical guitar strings were measured while the strings were subjected to a constant tension. The observed behavior was surprising and did not conform to expectations based on the ideal string equation and the known visco-elastic properties of nylon. Rather, the solid strings exhibited a frequency change that varied cyclically with time.

0.91

Article published: January 08, 2013

0

50

100

150

200

Time (Minutes) Figure 7 – Testing in argon appears to eliminate frequency cycling.

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A. Future Work This testing series was originally conceived as an experiment for a high school physics class wanting to learn research methods. Since the results were so unexpected, we are now working to repeat the results using more sophisticated equipment. The fixture will be made of milled aluminum and the pulley will be a high-grade ball bearing with a v-groove outer face. Frequencies will be recorded with an audio data acquisition system and an instrumentation grade microphone. VI. ACKNOWLEDGMENTS We wish to acknowledge the contribution of a group of Debbie French’s physics students at New Philadelphia High School in New Philadelphia OH. In Alphabetical order, they are: Joel Borton, Zakk Boyd, Joey Clark, Jacob Heslop, Thad Marshall, Ali Maus, Scott Mizer, Natalie Neidig, Alyssa Norman, Chris Perrine, Ben Potts, Nick Roth, Courtney Spears, Austin Smith, Zach Troyer, Kelsey Willoughby, Luke Yurich, Joe Zalesky We also wish to thank Fan Tao at D’Addario Strings for providing test samples. VII. BIBLIOGRAPHY 1. D.Martin, “Innovation and the Development of the Modern Six-String Guitar”, Galpin Soc. J. 51, 86-109 (1998). 2. P.M.Vilela, R.M. Moscosco and D. Thompson, “What Every Musician Knows About Viscoelastic Behavior”, Am. J. Phys. 65, 1000-1003 (1997). 3. W. Browstow, “Mechanical Properties”, in Physical Properties of Polymers Handbook 2nd Edition, edited by J.E. Mark (Springer 2006), Chap. 24, pp. 423-446. 4. H.F. Brinson and L. C. Brinson, Polymer Engineering Science and Viscoelasticity – An Introduction, Springer (2008). 5. M. French, Engineering the Guitar, Springer (2009) 6. R.W. Young, “Inharmonicity of Plain Wire Piano Strings”, J. Acoust. Soc. of Am. 24, No. 3 pp 267- 273 (1952).

Article published: January 08, 2013

url: http://SavartJournal.org/index.php/sj/article/view/17/pdf