DYNAMICAL ANALYSIS OF PORTUGUESE GUITAR STRINGS … · DYNAMICAL ANALYSIS OF PORTUGUESE GUITAR STRINGS Elisa Costa, ... (monochord) was assembled and ... The results are in table

15th International Conference on Experimental Mechanics

ICEM15 1

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


ICEM15 2

Two force transducers, Fig 2 on the

right, measure string vibration motion

in the vertical and horizontal

directions.

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

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.

Fig. 2 – Force Transducers

Fig. 3 - Modal testing


ICEM15 3

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 Steel

Frequency (Hz)

Diameter (mm)

Linear density (Kg/m)

Tension (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

Steel Frequency

(Hz) Diameter

(mm) Linear density

(Kg/m) Tension

(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.


ICEM15 4

Fig. 5 - Frequency difference normalized

-0,01

-0,005

0

0,005

0,01

0,015

0,02

0 1 2 3 4 5 6 7 8 9 10 11

Fre

qu

en

cy d

iffe

ren

ce n

orm

aliz

ed

Partial number

B (SS)

experiment theorical

0

0,005

0,01

0,015

0,02

0 5 10

Fre

qu

en

cy d

iffe

ren

ce n

orm

aliz

ed

Partial Number

B (S)


0 0,02 0,04 0,06 0,08

0,1 0,12 0,14

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Fre

qu

en

cy d

iffe

ren

ce n

orm

aliz

ed

Partial number

A (SS)


-0,01

0

0,01

0,02

0,03

0,04

0,05

0 5 10 15

Fre

qu

en

cy d

iffe

ren

ce n

orm

aliz

ed

Partial Number

A (S)


0

0,1

0,2

0,3

0,4

0 5 10 15 20

Fre

qu

en

cy d

iffe

ren

ce n

orm

aliz

ed

Partial Number

E (SS)


0

0,1

0,2

0,3

0,4

0 5 10 15 20

Fre

qu

en

cy d

iffe

ren

ce n

orm

aliz

ed

Partial Number

E (S)



ICEM15 5

Fig. 6 - Amplitude - Time graphics

-6,0E-01

-4,0E-01

-2,0E-01

0,0E+00

2,0E-01

4,0E-01

6,0E-01

0 1 2 3 4

Am

plit

ud

e (

N)

Time (s)

B (SS)

-1,50E+00

-1,00E+00

-5,00E-01

0,00E+00

5,00E-01

1,00E+00

0 2 4

Am

plit

ud

e (

N)

Time (s)

B (S)

-3,0E-01

-2,0E-01

-1,0E-01

0,0E+00

1,0E-01

2,0E-01

3,0E-01

0 1 2 3 4

Am

plit

ud

e (

N)

Time (s)

A (SS)

-3,00E-01

-2,00E-01

-1,00E-01

0,00E+00

1,00E-01

2,00E-01

3,00E-01

0 1 2 3 4

Am

plit

ud

e (

N)

Time (s)

A (S)

-1,0E+00

-5,0E-01

0,0E+00

5,0E-01

1,0E+00

0 1 2 3 4

Am

plit

ud

e (

N)

Time (s)

E (SS)

-1,50E+00

-1,00E+00

-5,00E-01

0,00E+00

5,00E-01

1,00E+00

1,50E+00

2,00E+00

0 1 2 3 4 Am

plit

ud

e (

N)

Time (s)

E (S)


ICEM15 6

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 frequency (Hz)

Damping 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,


ICEM15 7

Fig. 7 - Damping ratio

0

0,0002

0,0004

0,0006

0,0008

0 20 40 60 80 100

Dam

pin

g ra

tio

Cycle i

B (SS) damping ratio

0

0,0002

0,0004

0,0006

0,0008

0 20 40 60 80 100

Dam

pin

g ra

tio

Cycle i

B (S) damping ratio

0

0,0002

0,0004

0,0006

0,0008

0 20 40 60 80 100

Dam

pin

g ra

tio

Cycle i

A (SS) damping ratio

0

0,0002

0,0004

0,0006

0,0008

0 50 100

Dam

pin

g ra

tio

Cycle i

A (S) damping ratio

0

0,0002

0,0004

0,0006

0,0008

0 20 40 60 80 100

Dam

pin

g ra

tio

Cycle i

E (SS) damping ratio

0

0,0002

0,0004

0,0006

0,0008

0 20 40 60 80 100

Dam

pin

g co

eff

icie

nt

Cycle i

E (S) damping ratio


ICEM15 8

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)

Documents

DYNAMICAL ANALYSIS OF PORTUGUESE GUITAR STRINGS … · DYNAMICAL ANALYSIS OF PORTUGUESE GUITAR STRINGS Elisa Costa, ... (monochord) was assembled and ... The results are in table