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Vibration of strings with nonlinear supports

机译：带有非线性支撑的琴弦振动

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The dynamic string motion, which displacement is unilaterally constrained by the rigid termination condition of an arbitrary geometry has been simulated and analyzed. The treble strings of a grand piano usually terminate at a capo bar, which is situated above the strings. The apex of a V-shaped section of the capo bar defines the end of the speaking length of the strings. A numerical calculation based on the traveling wave solution is proposed for modeling the nonlinearity inducing interactions between the vibrating string and the contact condition at the point of string termination. It was shown that the lossless string vibrates in two distinct vibration regimes. In the beginning the string starts to interact in a nonlinear fashion with the rigid terminator, and the resulting string motion is aperiodic. Consequently, the spectrum of the string motion depends on the amplitude of string vibrations, and its spectral structure changes continuously with the passage of time. The duration of that vibration regime depends on the geometry of the terminator. After some time of aperiodic vibration, the string vibrations settle in a periodic regime where the resulting spectrum remains constant.
机译：仿真和分析了动态弦运动，该运动单向位移受任意几何形状的刚性终止条件约束。三角钢琴的高音弦通常终止于变调夹，该变调夹位于琴弦上方。变调夹的V形截面的顶点定义弦的说话长度的终点。提出了一种基于行波解的数值计算方法，用于对振动弦与弦终止点接触条件之间的非线性相互作用进行建模。结果表明，无损弦在两种不同的振动状态下振动。在开始时，琴弦开始以非线性方式与刚性终止子相互作用，并且由此产生的琴弦运动是非周期性的。因此，弦运动的频谱取决于弦振动的幅度，并且其频谱结构随着时间的流逝而连续变化。该振动状态的持续时间取决于端接器的几何形状。在经过一段时间的非周期性振动后，琴弦振动会以周期性状态稳定下来，在该状态下，所得频谱保持恒定。

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