15th International Conference on Experimental Mechanics

PAPER REF: 3009

DYNAMICAL ANALYSIS OF PORTUGUESE GUITAR STRINGS

Elisa Costa, Pedro Serrão*, A. M. R. Ribeiro, Virgínia Infante Department of Mechanical Engineering, Instituto Superior Técnico, Lisboa, Portugal (*)Email: [email protected]

ABSTRACT This work gives details on the measurement and analysis of the dynamical behavior of Portuguese guitar strings. An experimental string set-up (monochord) was assembled and the corresponding modal parameters identified. The string dynamical testing included frequency response, inharmonicity and damping phenomena. Relaxation tests were also performed where the string was repeatedly plucked by a mechanical device. Two types of strings were tested: hard draw steel (music wire) and stainless steel. Test results will be incorporated on the instrument string-body coupled dynamics model.

INTRODUCTION The Portuguese guitar has 12 strings, 6 double courses, typically with a 44 cm or 47 cm scale respectively for the Lisboa and Coimbra type. The first 3 courses are plain strings in high carbon spring wire or in stainless steel spring wire. The others are wound strings made of silver plated copper wrap wire in a steel hexagonal core. Musicians mention noticeable differences in sound between string materials with similar acoustic properties like steel and stainless steel. Why do they sound and feel different from each other? Could we correlate acoustical characteristics of a string to the quality of tone? Frequent retuning over time and playing will deteriorate the properties of the string. How will it affect the tone quality of the string? Damping in vibrating strings can be attributed to different loss mechanisms- aerodynamic, viscoelastic, thermoelastic and transfer of energy to other vibrating systems. Musicians mention a progressive brightening of the string sound in the period following replacement, the string sounds gradually less dull until it becomes typically brilliant. As shown in Fig.1 string testing was performed in a monochord with adjustable string length assembled with a Portuguese guitar tuner and tail piece to support the string and adjust the tension. String was plucked by an artificial nail actuated by an electrical motor. Excitation mechanism is similar enough to actual playing, and after each initial plucking doesn’t interact with the string dynamics ensuring reproducibility of the experiments.

- Fig. 1 – Monochord – string test jig

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Two force transducers, Fig 2 on the right, measure string vibration motion in the vertical and horizontal directions.

Fig. 2 – Force Transducers

Modal testing, Fig. 3 on the right, was performed on the monochord to assess how it could affect the string modal parameters. The monochord structure modal parameters were obtained using 4 accelerometers AC1-4 and a force transducer which was placed alternatively in the direction of the four accelerometers respectively. The graphics of frequency response function, FRF, and phase are shown Fig. 3 - Modal testing below, for the force transducer in position 3 with accelerometer AC3.

Fig. 4 - FRF function For this analysis, it was important to identify if any of these modes would be nearby the modes of the strings, and therefore, if the analysis and results of the tests to the strings would be affected.

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INHARMONICITY To study inharmonicity, tests were performed in the monochord. The strings were tuned and Fourier Spectrum curves were obtained from the force transducers signals. The frequencies of the modes were then identified. Inharmonicity depends on the tension T of the strings. Tension at given pitch was determined from the linear density of the string and this was obtained by weighting the strings in a precision scale. The results are in table 1.

Stainless Frequency Diameter Linear density Tension Steel (Hz) (mm) (Kg/m) (N) B 493.88 0.24 0.000378 71.40 A 440 0.25 0.000394 59.07 E 329.63 0.32 0.00062 52.17 Frequency Diameter Linear density Tension Steel (Hz) (mm) (Kg/m) (N) B 493.88 0.23 0.000318 60.07 A 440 0.25 0.00038 56.97 E 329.63 0.33 0.00064 53.85 Table 1 - Strings tension In the ideal flexible string the partials are whole-number multiples of the fundamental. The flexural stiffness of the real strings cause the natural frequencies to departure from the harmonic series as per (Fletcher 1998)

(1)

where E is typically 210GPa for steel and 193GPa for stainless steel strings. Inharmonicity tests results are shown for the first 3 strings B, A and E, steel (S) and stainless steel (SS) strings. In the table the data used for calculations is shown. Normalized frequency difference to fundamental (fn - nf0) / f0 is plotted in Fig.5.

The tested plain strings exhibit a low inharmonicity. The 3rd string E is the largest diameter and more prone to intonation problems as reported by musicians. The E strings experimental results are in good agreement with theoretical curve. Inharmonicity can be described by the flexural stiffness in the string model.

DAMPING To calculate damping, free vibration tests were made. The strings were tuned in the monochord and a single pluck was applied. The strings were then left to vibrate freely. These tests were made to the strings B, A and E for stainless steel and steel, to compare the effect of the diameter and material in damping. Results are shown in fig. 6 for the vertical force transducer.

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B (SS) B (S) 0,02 0,02

0,015 0,015 0,01 0,01 0,005 0,005 0 -0,005 0 1 2 3 4 5 6 7 8 9 10 11 0 0 5 10 -0,01

Partial number Partial Number

Frequency differenceFrequencynormalized Frequency differenceFrequencynormalized

experiment theorical experiment theorical

A (SS) A (S) 0,14 0,05 0,12 0,04 0,1 0,08 0,03 0,06 0,02 0,04 0,02 0,01 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 -0,01 0 5 10 15

Frequency differenceFrequencynormalized Partial number Partial Number Frequency differenceFrequencynormalized

experiment theorical experiment theorical

E (SS) E (S) 0,4 0,4

0,3 0,3

0,2 0,2

0,1 0,1

0 0 0 5 10 15 20 0 5 10 15 20

Frequency differenceFrequencynormalized Partial Number Partial Number Frequency differenceFrequencynormalized

experiment theorical experiment theorical

Fig. 5 - Frequency difference normalized

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B (SS) B (S) 6,0E-01 1,00E+00

4,0E-01 5,00E-01

2,0E-01 0,00E+00 0 2 4 0,0E+00 -5,00E-01 0 1 2 3 4 -2,0E-01 (N)Amplitude Amplitude (N)Amplitude -1,00E+00 -4,0E-01 -1,50E+00 -6,0E-01 Time (s) Time (s)

A (SS) A (S) 3,0E-01 3,00E-01

2,0E-01 2,00E-01

1,0E-01 1,00E-01

0,0E+00 0,00E+00 0 1 2 3 4 0 1 2 3 4

-1,0E-01 -1,00E-01

Amplitude (N)Amplitude Amplitude (N)Amplitude -2,0E-01 -2,00E-01

-3,0E-01 -3,00E-01 Time (s) Time (s)

E (SS) E (S) 1,0E+00 2,00E+00

1,50E+00

5,0E-01 1,00E+00 5,00E-01 0,0E+00 0 1 2 3 4 0,00E+00

0 1 2 3 4 Amplitude (N)Amplitude Amplitude (N)Amplitude -5,0E-01 -5,00E-01 -1,00E+00 -1,0E+00 -1,50E+00 Time (s) Time (s)

Fig. 6 - Amplitude - Time graphics

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The damping ratio, , for the first mode was obtained to the strings mentioned according to Maia [3],

where the logarithmic decrement, for the n cycles is calculate by

which relates displacements n cycles apart. Since the data presented discrete values to calculate the damping coefficient peaks were chosen and compared to the peak of the 500th cycle after each. Moreover, it was decided that the peaks should be chosen not from the beginning of the sample but from the values indicated in table 2, in order to avoid errors in calculations due to the influence of higher order modes .

String Ti=1 (s) String Ti=1 (s) B (SS) 1.5 B (S) 1.5 A (SS) 1.5 A (S) 1.9 E (SS) 1 E (S) 1.3 Table 2 - Time selection The graphics obtained for the damping ratio are shown in Fig. 7. Note that the shape of the curve should be a line. The irregular shape appears due to the point selections: although they are the higher value indicated in the time function to ith cycle, they doesn’t correspond to the higher points of the curve, since that the acquisition of points doesn’t match with the peaks of the curves (quantization error). The damping ratio was obtained by calculating an average from 100 decays. The values obtained are in table 3. 1st Mode Damping frequency (Hz) ratio B (SS) 493.5 0.000294 A(SS) 439.5 0.000339 E (SS) 329.5 0.000436 B (S) 493.25 0.000566 A (S) 440.25 0.000301 E (S) 329 0.000551 Table 3 - Damping ratio results Since the values obtain are considered low, the approximation in the formula might be used, according to Maia,

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B (SS) damping ratio B (S) damping ratio

0,0008 0,0008

0,0006 0,0006

0,0004 0,0004 Dampingratio Dampingratio 0,0002 0,0002

0 0 0 20 40 60 80 100 0 20 40 60 80 100 Cycle i Cycle i

A (SS) damping ratio A (S) damping ratio

0,0008 0,0008

0,0006 0,0006

0,0004 0,0004 Dampingratio Dampingratio 0,0002 0,0002

0 0 0 20 40 60 80 100 0 50 100 Cycle i Cycle i

E (SS) damping ratio E (S) damping ratio

0,0008 0,0008

0,0006 0,0006

0,0004 0,0004

Dampingratio 0,0002 0,0002 Dampingcoefficient 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Cycle i Cycle i

Fig. 7 - Damping ratio

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In table 4 the data and the results for the viscous and hysteretic damping are presented. Stainless Steel Steel B A E B A E Mass (g) 0.159241 0.172788 0.283095 0.142591 0.168468 0.293538 Natural frequency (Hz) 493.5 439.5 329.5 493.25 440.25 329 Viscous damping ratio, 0.000294 0.000339 0.000436 0.000556 0.000301 0.000551 Histeretic damping ratio, η 0.000587 0.000678 0.000873 0.001132 0.000602 0.001102 Table 4 - Viscous and hysteretic damping ratio

CONCLUSION The proposed tests were successfully performed and the data required for the next steps in the research line obtained. This data will be used to model the complete instrument, allowing for a more complete study of the Portuguese Guitar. To perform the tests, a universal test platform for guitar strings was built and characterized. Called monochord, it allows for the testing of the strings without interference of the guitar body. The data collected included the quantification of inharmonicity and damping, two essential parameters for the modeling of the complete guitar. The results presented in this paper concern 3 of the guitar strings with 2 different materials each, but the method can be generalized to any of the remaining strings and, likely, to strings from other musical instruments.

ACKNOWLEDGMENTS The authors gratefully acknowledge the funding by Ministério da Ciência, Tecnologia e Ensino Superior, FCT, Portugal, project reference PTDC/FIS/103306/2008.

REFERENCES Fletcher N.H. and Rossing T. D., The Physics of Musical Instruments, 2nd ed. (Springer- Verlag, New York, 1998) Maia N, Silva J, Theoretical and Experimental Modal Analysis, (Research Studies Press LTD., Taunton, Somerset, England) Maia N, Vibrações e Ruído (Associação de Estudantes do Instituto Superior Técnico) Vallete C. and Cuesta C, Mécanique de la Corde Vibrante, (Hermés, 1993)

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