Calculation of the spectrum of self-sustained oscillators using a variable truncation method: Application to cylindrical reed instruments

摘要

Two limit cases are well known concerning the spectrum of cylindrical instruments excited by a reed: the small oscillations case, for which the nth odd harmonic has an amplitude proportional to a nth power of the first harmonic, and the non dissipation case, where the spectrum is that of a square signal. The present paper investigates the transition between these two cases, and proposes approximate formulae for the spectrum with respect to the mouth pressure and dissipation parameter. The method, called the "variable truncation method", is an intermediate one between the series expansion method, valid for small oscillations, and the general, numerical "harmonic balance" method for solving a system of nonlinear equations. The models used are classical, the nonlinear function describing the mouthpiece being first a polynomial of the third order, then a more complicated one based upon the Bernoulli law. It is first established that the operating frequency and the spectrum are in practice almost independent of the shape of the nonlinearity. Resonators with harmonically related resonance frequencies are first studied in order to justify as far as possible the calculation method. Rather simple formulae exhibit a two-slopes behaviour for the diagrams plotting the amplitude of the odd harmonics versus that of the first one. Then inharmonicity due to visco-thermal effects is considered, and its effect is found to be very important, independently on the aspect ratio of the cylindrical tube. It is shown to reduce the amplitudes of the odd harmonics of the external pressure compared to these of the even harmonics. However the amplitudes Of the even harmonics of the pressure inside the mouthpiece are found to be very small, allowing great simplifications in the approximate calculations. Qualitative comparison with experiments published by some authors is given. A short investigation is also presented concerning the effect of changes in the shape of the resonator, the approximate method remaining useful. [References: 20]