Debut V, Antunes J, Inácio O
Instituto de Etnomusicologia, Música e Dança, Faculdade de Ciências Sociais e Humanas, Universidade Nova de Lisboa, 1069-061 Lisboa, Portugal.
Centro de Ciências e Tecnologias Nucleares, Instituto Superior Técnico, Universidade de Lisboa, Estrada Nacional 10, Km 139.7, 2695-066 Bobadela LRS, Portugal.
J Acoust Soc Am. 2017 Mar;141(3):2107. doi: 10.1121/1.4976092.
Linearised models are often invoked as a starting point to study complex dynamical systems. Besides their attractive mathematical simplicity, they have a central role for determining the stability properties of static or dynamical states, and can often shed light on the influence of the control parameters on the system dynamical behaviour. While the bowed string dynamics has been thoroughly studied from a number of points of view, mainly by time-domain computer simulations, this paper proposes to explore its dynamical behaviour adopting a linear framework, linearising the friction force near an equilibrium state in steady sliding conditions, and using a modal representation of the string dynamics. Starting from the simplest idealisation of the friction force given by Coulomb's law with a velocity-dependent friction coefficient, the linearised modal equations of the bowed string are presented, and the dynamical changes of the system as a function of the bowing parameters are studied using linear stability analysis. From the computed complex eigenvalues and eigenvectors, several plots of the evolution of the modal frequencies, damping values, and modeshapes with the bowing parameters are produced, as well as stability charts for each system mode. By systematically exploring the influence of the parameters, this approach appears as a preliminary numerical characterisation of the bifurcations of the bowed string dynamics, with the advantage of being very simple compared to sophisticated numerical approaches which demand the regularisation of the nonlinear interaction force. To fix the idea about the potential of the proposed approach, the classic one-degree-of-freedom friction-excited oscillator is first considered, and then the case of the bowed string. Even if the actual stick-slip behaviour is rather far from the linear description adopted here, the results show that essential musical features of bowed string vibrations can be interpreted from this simple approach, at least qualitatively. Notably, the technique provides an instructive and original picture of bowed motions, in terms of groups of well-defined unstable modes, which is physically intuitive to discuss tonal changes observed in real bowed string.
线性化模型常常被用作研究复杂动力系统的起点。除了其具有吸引力的数学简洁性外，它们在确定静态或动态状态的稳定性特性方面具有核心作用，并且常常能够揭示控制参数对系统动态行为的影响。虽然从多个角度对弓弦动力学进行了深入研究，主要是通过时域计算机模拟，但本文建议采用线性框架来探索其动态行为，在稳定滑动条件下对平衡状态附近的摩擦力进行线性化，并使用弦动力学的模态表示。从库仑定律给出的最简单摩擦力理想化模型出发，考虑速度相关的摩擦系数，给出了弓弦的线性化模态方程，并使用线性稳定性分析研究了系统随弓法参数的动态变化。根据计算得到的复特征值和特征向量，绘制了模态频率、阻尼值和模态形状随弓法参数的演变图，以及每个系统模态的稳定性图。通过系统地探索参数的影响，这种方法似乎是对弓弦动力学分岔的初步数值表征，与需要对非线性相互作用力进行正则化的复杂数值方法相比，具有非常简单的优点。为了明确所提出方法的潜力，首先考虑经典的单自由度摩擦激励振荡器，然后是弓弦的情况。即使实际的粘滑行为与这里采用的线性描述相差甚远，但结果表明，至少在定性方面，可以从这种简单方法解释弓弦振动的基本音乐特征。值得注意的是，该技术从定义明确的不稳定模态组的角度，提供了一幅关于弓法运动的具有启发性和原创性的图景，从物理角度直观地讨论了在实际弓弦中观察到的音调变化。
