Search

The lowest fingerings of the saxophone can lead to several different regimes, depending on the musician’s control and the characteristics of the instrument. This is explored in this paper through a physical model of saxophone. The harmonic balance method shows that for many combinations of musician control parameters, several regimes are stable. Time-domain synthesis is used to show how different regimes can be selected through initial conditions and the initial evolution (rising time) of the blowing pressure, which is explained by studying the attraction basin of each stable regime. These considerations are then applied to study how the produced regimes are affected by properties of the resonator. The inharmonicity between the first two resonances is varied in order to find the value leading to the best suppression of unwanted overblowing. Overlooking multistability in this description can lead to biased conclusions. Results for all the lowest fingerings show that a slightly positive inharmonicity, close to that measured on a saxophone, leads to first register oscillations for the greatest range of control parameters. A perfect harmonicity (integer ratio between the first two resonances) decreases first register production, which adds nuance to one of Benade’s guidelines for understanding sound production. Thus, this study provides some a posteriori insight into empirical design choices relative to the saxophone.

The continuation of quasi-periodic solutions for autonomous or forced nonlinear systems is presented in this paper. The association of the Asymptotic Numerical Method, a robust continuation method, and a two-frequency Harmonic Balance Method, is performed thanks to a quadratic formalism. There is no need for a priori knowledge of the solution: the two pulsations can be unknown and can vary along the solution branch, and the double Fourier series are computed without needing a harmonic selection. A norm criterion on Fourier coefficients can confirm a posteriori the accuracy of the solution branch. On a forced system, frequency-locking regions are approximated, without blocking the continuation process. The continuation of these periodic solutions can be done independently. On an autonomous system an example of solution is shown where the number of Fourier coefficients is increased to improve the accuracy of the solution. [ABSTRACT FROM AUTHOR]

• The Hill's method of stability analysis is proved to be a by-product of the Harmonic Balance Method. • A nonlinear quadratic recast of the dynamical system is used for an automatic generation of the Hill's matrix. • The issue of eigenvalue sorting is solved by a robust criterion on the eigenvectors. • The method is validated by the stability computation of several canonical systems. • The complex topology of the backbone curves of a hinged-hinged beam is presented in details. Since the founding theory established by G. Floquet more than a hundred years ago, computing the stability of periodic solutions has given rise to various numerical methods, mostly depending on the way the periodic solutions are themselves determined, either in the time domain or in the frequency domain. In this paper, we address the stability analysis of branches of periodic solutions that are computed by combining a pure Harmonic Balance Method (HBM) with an Asymptotic Numerical Method (ANM). HBM is a frequency domain method for determining periodic solutions under the form of Fourier series and ANM is continuation technique that relies on high order Taylor series expansion of the solutions branches with respect to a path parameter. It is well established now that this HBM-ANM combination is efficient and reliable, provided that the system of ODE is first of all recasted with quadratic nonlinearities, allowing an easy manipulation of both the Taylor and the Fourier series. In this context, Hill's method, a frequency domain version of Floquet theory, is revisited so as to become a by-product of the HBM applied to a quadratic system, allowing the stability analysis to be implemented in an elegant way and with good computing performances. The different types of stability changes of periodic solutions are all explored and illustrated through several academic examples, including systems that are autonomous or not, conservative or not, free or forced. [ABSTRACT FROM AUTHOR]

• A1: Alternating Frequency-Time scheme, Predictor-Corrector Continuation (AFT-PreCo). • A2: classical Harmonic Balance, Asymptotic Numerical Method (cHB-ANM). • Benefits cHB-ANM: continuous repr. of solution path, robust step length control. • Benefits AFT-PreCo: suitable for non-smooth problems, computationally efficient. In this work we apply and compare two algorithms for setting up Harmonic Balance equations and numerical continuation of the solution path for harmonically driven mechanical systems. The first algorithm relies on a predictor-corrector scheme and an Alternating Frequency-Time approach (AFT-PreCo). The second algorithm relies on a high-order Taylor series expansion of the solution path (asymptotic numerical method) and classical Harmonic Balance formulated entirely in the frequency domain (cHB-ANM). We conclude that the cHB-ANM is suited for smooth nonlinearities, for instance geometrically nonlinear finite element models. Here, cHB-ANM avoids aliasing errors and convinces with a numerically robust adjustment of the continuation step length and a continuous representation of the solution path. For non-smooth nonlinearities such as stick-slip friction or unilateral constraints, AFT-PreCo is better suited, and convinces with high numerical robustness and efficiency. [ABSTRACT FROM AUTHOR]