The vibration of strings
Experimental results from D.W. Jordan & P. Smith also show the effect of large driving force. As shown in the diagram, new trajectories start to emerge as the driving force increases. In other words, higher order corrections with larger periods in the approximated solution become more evident which eventually leads to chaos. However, in this paper, we only consider small nonlinearities which theoretically will not generate such effect. From experimental results, it will be seen that the damping coefficient b cannot be considered a constant. In the first experiment, the lowering and widening of the resonance curves as current increases, before the rightward bending becomes appreciable, implies a damping factor that depends upon signal amplitude, i.e., a nonlinear damping. Further complexity can be added by considering damping constant models in fluid dynamics. In the second experiment, further study can be carried out on period-doubling bifurcation which generalizes the nonlinear behaviour of oscillations with larger nonlinearities.
Conclusion
This paper shows that the motion of a driven guitar string can be modelled by the duffing equation. The equation describes well the resonance amplitude of the string when driving forces are applied. With larger nonlinearities, more realistic model for damping should be considered.
References
[1] D.W. Jordan & P. Smith (2007) Nonlinear ordinary differential equations – An introduction for scientists and engineers (4th ed.), Oxford University Press, pp. 453 – 462. [2] Ali H. Nayfeh and Dean T. Mook (1979) Nonlinear Oscillations. New York. [3] Nicholas B. Tufillaro (1989) ‘Nonlinear and chaotic string vibrations’, Am. J. Phys . 57(5), 408 – 414. [4] John W. Miles (1965) ‘Stability of forced oscillations of a vibrating string’, J. Acoust. Soc. Am. 38(5), 855 – 861. [5] David Politzer (2015) ‘The plucked string: an example of non -normal dynamics ’, Am. J. Phys . 83, 395 – 402. [6] G. F. Carrier (1945) ‘ On the nonlinear vibration problem of the elastic string ’, Quart. Appl. Math.
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