Abstract
We study the characteristics of stiff strings vibrating against a curved boundary obstacle which is a situation encountered in stringed musical instruments like sitar, veena and tanpura. The wrapping and unwrapping of the string over the curved obstacle introduces a moving boundary which upon appropriate spatial scaling reveals the resulting nonlinearity. We find dominant quadratic nonlinearities which facilitate modal interactions leading to mode-locked solutions in the absence of damping. The analytical results for the phase-locked solutions are obtained using the method of multiple scales (MMS). We have further found a mode-locked state which tends to an equipartition of energy among the various vibrational modes with an increase in the number of modes considered for the analysis. \(\dots \)
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Deva, B.C.: Musical Instruments of India: Their History and Development, 2nd edn. Munshiram Manoharlal, New Delhi (1987)
Raman, C.V.: On some Indian stringed instruments. Proc. Indian Assoc. Cultiv. Sci. 7, 29–33 (1921)
Burridge, R., Kappraff, J., Morshedi, C.: The sitar string, a vibrating string with a one sided inelastic constraint. SIAM J. Appl. Math. 42(6), 1231–1251 (1982)
Taguti, T.: Dynamics of simple string subject to unilateral constraint: a model analysis of sawari mechanism: acous. Sci. Tech. 29(3), 203–214 (2008)
Vyasarayani, C.P., Birkett, S., McPhee, J.: Modeling the dynamics of a vibrating string with finite distributed unilateral constraint: application to the sitar. J. Acous. Soc. Am. 125(6), 3673–3682 (2009)
Alsahlani, A., Mukherjee, R.: Vibration of a string wrapping and unwrapping around an obstacle. J. Sound Vib. 329, 2707–2715 (2010)
Mandal, A.K., Wahi, P.: Natural frequencies, mode shapes and modal interactions for strings vibrating against an obstacle: relevance to sitar and veena. J. Sound Vib. 338, 42–59 (2015)
Mandal A.K.: Vibration characteristics of strings in the presence of a smooth curved boundary obstacle: relevance to stringed musical instruments. Ph.D. thesis, IIT Kanpur (2016)
Mandal, A.K., Wahi, P.: Mode-locking and improved harmonicity for real strings vibrating in the presence of a curved obstacle. Nonlinear Dyn. 88(3), 2203–2224 (2017)
Benade A.H., Messenger W.G.: Sitar spectrum properties. In: The 103rd Meeting of Acoustical Society of America, Chicago, Illinois; Abstract available at J. Acous. Soc. Am. 71(S1), S83–S83 (1982)
Donkin, W.F.: Acoustics. Clarendon Press, Oxford (1870)
Rayleigh, J.W.S.: The Theory of Sound. Macmillan and Co. London (1877)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Mandal, A.K., Wahi, P. (2019). Equipartition of Modal Energy in a Stiff Vibrating String Due to a Finite Curved Boundary Obstacle. In: Gutschmidt, S., Hewett, J., Sellier, M. (eds) IUTAM Symposium on Recent Advances in Moving Boundary Problems in Mechanics. IUTAM Bookseries, vol 34. Springer, Cham. https://doi.org/10.1007/978-3-030-13720-5_21
Download citation
DOI: https://doi.org/10.1007/978-3-030-13720-5_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-13719-9
Online ISBN: 978-3-030-13720-5
eBook Packages: EngineeringEngineering (R0)