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

Two limit cases are well known concerning the spectrum of cylindrical instr uments excited by a reed: the small oscillations case, for which the nth od d harmonic has an amplitude proportional to a nth power of the first harmon ic, and the non dissipation case, where the spectrum is that of a square si gnal. The present paper investigates the transition between these two cases , and proposes approximate formulae for the spectrum with respect to the mo uth pressure and dissipation parameter. The method, called the "variable tr uncation method", is an intermediate one between the series expansion metho d, valid for small oscillations, and the general, numerical "harmonic balan ce" 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 B ernoulli law. It is first established that the operating frequency and the spectrum are in practice almost independent of the shape of the nonlinearit y. Resonators with harmonically related resonance frequencies are first stu died in order to justify as far as possible the calculation method. Rather simple formulae exhibit a two-slopes behaviour for the diagrams plotting th e amplitude of the odd harmonics versus that of the first one. Then inharmo nicity 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 exte rnal pressure compared to these of the even harmonics. However the amplitud es Of the even harmonics of the pressure inside the mouthpiece are found to be very small, allowing great simplifications in the approximate calculati ons. Qualitative comparison with experiments published by some authors is g iven. A short investigation is also presented concerning the effect of chan ges in the shape of the resonator, the approximate method remaining useful.