Convergence improvement of the harmonic balance method to obtain periodic solutions for self-sustained musical instruments Snorre Farner 1 , Christophe Vergez 2 , Jean Kergomard 2 1 Department of Electronics and Telecommunications, NTNU, Trondheim, Norway 2 Laboratoire de M´ ecanique et d’Acoustique, CNRS, Marseille, France farner@tele.ntnu.no,{vergez|kergomard}@lma.cnrs-mrs.fr Abstract The harmonic balance method was originally developed for solving periodic solutions of forced-oscillation elec- tronic circuits but has later been adapted to self-sustained musical instruments. A computer program for solving general problems involving nonlinearly coupled exciter and resonator has been developed using this method, and the convergence has been improved as well as continua- tion. We briefly present the harmonic balance method be- fore we address one specific problem of convergence due to sampling and describe the backtracking algorithm used to efficiently reduce the problem. This improvement fa- cilitates continuation and we demonstrate the method on a simple model of a clarinet and compare with analytical results. 1. Introduction A self-sustained musical instrument may be described by an exciter coupled to a resonator [5]. For most of them, both the exciter and the resonator can be rather well de- scribed by a linear model, the coupling being the only nonlinearity in the system. Although time-domain cal- culations have grown popular for sound synthesis, calcu- lation in the frequency domain offer a simple manner to study periodic solutions of the model under study. Fifteen years ago, Jo¨ el Gilbert and colleagues popu- larised the use of the harmonic balance method (HBM) to study the spectrum of the clarinet [3], based on the work of Nakhla and Vlach [6]. The HBM is a numerical method to calculate the steady-state spectrum of periodic solutions of nonlinear dynamical systems. The principle is that by guessing a variable characteristic of the state of the model, the system of equations will respond by a deviation from zero if the guessed value is not a solution. The HBM searches to minimise this deviation and in an iterative manner end up with the solution. It works in the frequency domain, and the variables are expressed as a sum of their Fourier harmonics. Furthermore, by apply- ing the discrete Fourier transform (DFT) and its inverse to the variables, nonlinear time-domain equations are eas- ily supported by the method. This makes the HBM very p Lips m p Reed Pipe u y+H Figure 1: Diagram of the mouthpiece of a clarinet suitable for studying the steady-state regimes of a self- sustained instrument as well as the control of the play- ing parameters. We have developed a free, open-source computer program called Harmbal [1] (under the GNU General Public License) for applying the HBM to such models of self-sustained instruments. In this paper we present the HBM applied to a sim- ple model of the clarinet, discuss a convergence problem that was encountered, and propose the backtracking al- gorithm as a remedy. This is all included in the current version of Harmbal as well as a script program (hbmap) for continuation. 2. Equations for the clarinet Following McIntyre and Schumacher, most self- sustained musical instrument can be modelled by a lin- ear exciter with output impedance H(ω) coupled through a nonlinear time-domain function f to a linear resonator with input impedance Z (ω): H(ω)Y (ω)= P (ω), Z (ω)U (ω)= P (ω), and (1) u(t)= f (p(t),y(t)). The capitalised quantities P , Y , and U denote the Fourier transform of the time-domain variables p, y, and u. In the case of the clarinet, illustrated in Figure 1, the player sets the (constant) pressure p m in the mouth to be higher than the pressure ˜ p(t) inside the mouthpiece, the tilde being used on physical variables to distinguish them from their dimensionless counterparts. The air flow ˜ u(t) through the reed opening makes the reed close the opening. The reed displacement from equilibrium is ˜ y(t). Tu5.C1.4 II - 1429