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Nonlinearities

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Acoustics of Musical Instruments

Part of the book series: Modern Acoustics and Signal Processing ((MASP))

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Abstract

In the previous chapters, it was assumed that the amplitude of both air and structural oscillations in musical instruments were sufficiently small so that the assumption of linearity for their underlying models was fulfilled. However, this assumption is no longer valid in a number of situations encountered in musical acoustics, and a nonlinear approach becomes necessary for describing the observed phenomena. This chapter starts with the presentation of a simple example of nonlinear oscillator, the interrupted pendulum, whose aim is to introduce some fundamental properties of nonlinear systems, such as the dependence of resonance frequency on amplitude. The generic Duffing equation, which is found in many areas of nonlinear physics, is then examined. Musical applications are found first in piano strings, where the transverse-longitudinal coupling and the presence of additional partials in the spectrum (or “phantom” partials) are the consequence of nonlinearity due to high amplitude motion (geometric nonlinearity). In brass instruments, high values of the acoustic pressure induce nonlinear propagation which, in turn, might give rise to shock waves. In gongs and cymbals, a strong excitation produces the so-called bifurcations materialized by the emergence of new frequencies in the spectrum, which ultimately can lead to chaos. Specific methods are used for characterizing chaotic signals, such as the Lyapunov exponents. New emerging tools, such as the nonlinear normal modes (NNMs), appear to be very efficient for describing the dynamics of nonlinear systems with a reduced number of degrees of freedom. Self-sustained oscillations of reed, flute-like and bowed string instruments are treated in the three following chapters.

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Notes

  1. 1.

    The tambura is an Indian plucked stringed instrument.

  2. 2.

    The origin of this denomination is due to the fact that these terms were highlighted as first in the field of celestial mechanics, where the time scales are of the order of centuries rather than of milliseconds!

  3. 3.

    Let us recall (see Chap. 6) that the boundary conditions at the bridge are another cause of coupling between y and z in the linear case. In the general complex case of a real stringed instrument during normal playing, these two factors coexist and it is often difficult to separate them.

  4. 4.

    Strictly speaking, the model should contain an additional horizontal component. In fact, it is observed experimentally that the polarization of most piano strings changes with time. It is often almost vertical during the initial transient, due to the action of the hammer, and is then evolving progressively towards a horizontal motion, even for small amplitudes. It can be demonstrated mathematically that such a polarization change can only occur if some asymmetry exists in the system that allows a transfer of energy from one component to the other. If the string is assumed to be homogeneous and perfectly rectilinear, with ideal boundary conditions (assuming a vertical motion of the bridge, for example), then the string will keep the initial polarization induced by the hammer during its motion. In [13], the authors made such assumptions, with a vertical initial motion of the hammer, and this is the reason why only one transverse component of the string is considered here. Revisiting the bridge model would be necessary for allowing a horizontal component.

  5. 5.

    For a more detailed study of fundamental nonlinear acoustics, the reader may refer, for example, to the following authors: [18, 34, 54, 57].

  6. 6.

    A shock wave is a pressure field which has an abrupt, and almost discontinuous, transition.

  7. 7.

    This method is detailed in Sect. 8.5 devoted to nonlinear vibrations of gongs and cymbals.

  8. 8.

    So far, the volume visco-thermal losses, proportional to ω 2, were ignored because in a wind instrument they are very low compared to losses in the boundary layers, proportional to \(\sqrt{\omega }\). However, volume losses are essential in free space. Equation (8.74) including only the volume loss term (B = 0) is the equation called “Burgers equation,” referring to the similar nonlinear equation used by the Dutch physicist J.M. Burgers in his work on turbulence.

  9. 9.

    The shock zone is the zone where sudden changes of pressure, acoustic velocity, and temperature may occur.

  10. 10.

    For a pure traveling plane wave, a particle velocity of 1 m/s corresponds to a level of 400 Pa, or 146 dB SPL.

  11. 11.

    The definition of chaos, as well as the method used for analyzing and quantifying such oscillations, will be presented throughout this chapter (see Sect. 8.6, in particular). Here, we can see some first properties of chaotic oscillations: irregularity in the time-domain, and a broadband spectrum where it is not possible to discriminate individual spectral peaks anymore. Sensitivity to initial conditions is another essential feature of chaotic oscillations, which will be discussed later in detail.

  12. 12.

    An important part of the topics presented in this section are the results of new insights on the nonlinear vibrations of thin structures which were published during the last 15 years. See, in particular, the Ph.D. thesis by O. Thomas, and his joint work with colleagues [8, 11, 60, 61, 64, 68].

  13. 13.

    In this example, the terms in y 2 and y 3, respectively, are comparable as long as α o is not supposed to be small. We will see later that the quadratic terms are predominant in thin shells .

  14. 14.

    For the sake of clarity, the overlinings are now removed from the equations.

  15. 15.

    Details on numerical methods for solving the von Kármán plate equations can be found in [8].

  16. 16.

    As shown in Chap. 3 for the circular membrane (see Fig. 3.32), a mode (n, m) of a structure with a circular geometry is characterized by n nodal diameters and m nodal circles. However, in contrast to the case of stretched membranes described in Chap. 3, the outer edge of the spherical shells considered here are free, so that the lowest number of nodal circles is zero.

  17. 17.

    The choice of T F determines the accuracy of the estimated Lyapunov exponents. These very technical considerations will not be detailed here, and we invite the interested reader to refer to the specialized literature [40, 43]. We simply recommend to choose T∕2 ≤ T F  < T, where T is the time interval chosen for the construction of the vectors y [see Eq. (8.113)].

  18. 18.

    New results on the use of normal form theory can be found in [41].

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Authors and Affiliations

  1. Institute of Music Acoustics, University of Music and Performing Arts Vienna (MDW), Anton-von-Webern-Platz 1, 1030, Vienna, Austria

    Antoine Chaigne

  2. Laboratoire d’Acoustique de l’Université du Maine (LAUM), Avenue Olivier Messiaen, 72085, Le Mans Cedex 09, France

    Joël Gilbert & Jean-Pierre Dalmont

  3. Institute of Mechanical Sciences and Industrial Applications (IMSIA), ENSTA ParisTech, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762, Palaiseau Cedex, France

    Cyril Touzé

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  1. Antoine Chaigne
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Chaigne, A., Gilbert, J., Dalmont, JP., Touzé, C. (2016). Nonlinearities. In: Acoustics of Musical Instruments. Modern Acoustics and Signal Processing. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3679-3_8

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