Complex modes of vibration due to small–scale damping in a guitar top–plate

 

J. A. Torres, P. L. Rendón, R. R. Boullosa

 

Centro de Ciencias Aplicadas y Desarrollo Tecnológico, Universidad Nacional Autónoma de México, México D.F. 04510 jesusalejandrott@yahoo.com.mx, pablo.rendon@ccadet.unam.mx, ricardo.ruiz@ccadet.unam.mx

 

ABSTRACT

Modal analysis is one of the preeminent methods used by scientists and engineers to study vibrating structures. The frequency response functions obtained through this method, are, in general, complex–valued. There is, however, no agreed–upon interpretation given to the real and imaginary parts of these functions, even though it is acknowledged that their relative magnitude for different frequencies is related to the behaviour of the corresponding modes. A simple model is deduced to describe the shape of the spectrum associated with a finite–length time–signal. There is very good agreement between results obtained using this model and numerical results obtained for, in this case, the vibration of a guitar top–plate using finite element methods. One interpretation of the relative magnitudes of the real and imaginary parts of the frequency response functions is advanced. It is found that stationary–wave behaviour is associated with the dominance of the real or imaginary part; traveling–wave behaviour, on the other hand, occurs when the real and imaginary parts are of the same order of magnitude, as long as the scale of damping is large enough and resonance peaks in the spectrum are close enough.

Keywords: Complex modes, mode shapes, damped plates, guitar.

 

RESUMEN

El análisis modal es uno de los métodos utilizados con mayor frecuencia por científicos e ingenieros para estudiar estructuras vibrantes. Las funciones de respuesta de frecuencia obtenidas mediante este método tienen, en general, valores complejos. No existe, sin embargo, una interpretación universalmente aceptada asociada a las partes real e imaginaria de estas funciones, aun cuando se sabe que la magnitud relativa de estas cantidades para diferentes frecuencias esta relacionada con el comportamiento de los distintos modos. Se obtiene un modelo sencillo que describe la forma del espectro asociado a una señal temporal de duración finita. Hay una muy buena concordancia entre los resultados obtenidos utilizando este modelo y aquellos obtenidos para, en este caso, la vibración de la tapa de una guitarra simulada a través de elementos finitos. Se propone una interpretación de las magnitudes relativas de las partes real e imaginaria de la función de respuesta de frecuencia. El comportamiento de onda estacionaria se asocia a la dominancia de la parte real o imaginaria; por otra parte, el comportamiento de onda viajera ocurre cuando las partes real e imaginaria son del mismo orden de magnitud, y siempre y cuando la escala del amortiguamiento sea lo suficientemente grande y los picos de resonancia en el espectro sean suficientemente cercanos.

 

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Acknowledgements

The authors would like to thank the Universidad Nacional Autónoma de México, which, through Coordinación de Estudios de Posgrado and project PAPIIT IN120008, has awarded financial support towards the production of this paper.

 

References

[1] Fletcher, N.H., and Rossing T.D., The Physics of Musical Instruments. Springer, New York, 1998.         [ Links ]

[2] Ahluwalia, D.S., Kriegsmann, G.A., and Reiss, E.L., Scattering of low–frequency acoustic waves by baffled membranes and plates. J. Acoust. Soc. Am., 78(2):682–687, 1985.         [ Links ]

[3] Schelleng, J.C., On the polarity of resonance. Catgut Acoust. Soc. Newsletter, (10):14–18, 1968.         [ Links ]

[4] Bissinger, G., Some mechanical and acoustical consequences of the violin soundpost. J. Acoust. Soc. Am., 97(5):3154–3164, 1995.         [ Links ]

[5] Spiekermann, C. E., and Radcliffe, C. J, Decomposing one–dimensional acoustic pressure response into propagating and standing waves. J. Acoust. Soc. Am., 84(4):1536–1541, 1988.         [ Links ]

[6] Spiekermann, C. E., and Radcliffe, C. J, Stripping one–dimensional acoustic pressure response into propagating – and standing – wave components. J. Acoust. Soc. Am., 84(4):1542–1548, 1988.         [ Links ]

[7] Inman, D. J., Engineering Vibration. Prentice Hall, London, 1994.         [ Links ]

[8] Hatch, M. R., Vibration simulation using Matlab and Ansys. Chapman and Hall, London, 2001.         [ Links ]

[9] Rossing, T. D., Modal analysis. In Thomas D. Rossing, editor, Springer Handbook of Acoustics, pages 1127–1138. Springer, New York, 2007.         [ Links ]

[10] Newland, D.E., Mechanical vibration analysis and computation. Longman Scientific & Technical, Harlow, England, 1989.         [ Links ]

[11] Marshall, K.D., Modal analysis of a violin. J. Acoust. Soc. Am., 77(2):695–709, 1985.         [ Links ]

[12] Feeny, B.F., Acomplexorthogonal decomposition forwave motion analysis. J. Sound Vibrat., 310:77–90, 2008.         [ Links ]

[13] Kammler, D.W., A First Course in Fourier Analysis. Cambridge University Press, 2007.         [ Links ]

[14] Torres, J.A., and Boullosa, R. R., Influence of the bridge on the vibrations of the top plate of a classical guitar. Applied Acoustics, 70(11–12):1371–1377, 2009.         [ Links ]