长笛类乐器：一个玩具模型的数值延拓研究
Flute-like musical instruments: a toy model investigated through numerical continuation
摘要：
自持乐器（弓弦乐器、木管乐器和铜管乐器）可以用非线性集总动力系统来建模。在这些乐器中，长笛和烟道管风琴表现出被建模为延迟动力系统的特殊性。在本文中，这样一个系统，玩具模型的长笛般的文书，研究使用数值延拓。平衡和周期性的解决方案进行了探索，相对于吹制压力，重点是振幅和频率的演变沿不同的解决方案分支，以及“跳跃”之间的周期性解决方案分支。第二个模型参数（即非谐性）对系统的行为的影响。结果表明，谐波起着关键作用，在滞后或准周期制度的存在。在整个文件中，一个真正的仪器上的实验结果来说明各种现象，并允许一些定性的比较与数值结果。
英文摘要：
Self-sustained musical instruments (bowed string, woodwind and brass instruments) can be modeled by nonlinear lumped dynamical systems. Among these instruments, flutes and flue organ pipes present the particularity to be modeled as a delay dynamical system. In this paper, such a system, a toy model of flute-like instruments, is studied using numerical continuation. Equilibrium and periodic solutions are explored with respect to the blowing pressure, with focus on amplitude and frequency evolutions along the different solution branches, as well as "jumps" between periodic solution branches. The influence of a second model parameter (namely the inharmonicity) on the behaviour of the system is addressed. It is shown that harmonicity plays a key role in the presence of hysteresis or quasi-periodic regime. Throughout the paper, experimental results on a real instrument are presented to illustrate various phenomena, and allow some qualitative comparisons with numerical results.
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