Phase-locked loops are commonly used for shaker-based backbone tracking of nonlinear structures. The state of the art is to tune the control parameters by trial and error. In the present work, an approach is proposed to make backbone tracking much more robust and faster. A simple PI controller is proposed, and closed-form expressions for the gains are provided that lead to an optimal settling of the phase transient. The required input parameters are obtained from a conventional shaker-based linear modal test, and an open-loop sine test at a single frequency and level. For phase detection, an adaptive filter based on the LMS algorithm is used, which is shown to be superior to the synchronous demodulation commonly used. Once the phase has locked, one can directly take the next step along the backbone, eliminating the hold times. The latter are currently used for recording the steady state, and to estimate Fourier coefficients in the post-process, which becomes unnecessary since the adaptive filter yields a highly accurate estimation at runtime. The excellent performance of the proposed approach is demonstrated for a doubly clamped beam undergoing bending–stretching coupling leading to a 20 percent shift of the lowest modal frequency. Even for fixed control parameters, designed for the linear regime, only about 100 periods are needed per backbone point, also in the nonlinear regime. This is much faster than what has been reported in the literature so far.

中文翻译：

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通过锁相环实现稳健、快速的骨干跟踪


锁相环通常用于非线性结构的基于振动台的骨干跟踪。最先进的技术是通过反复试验来调整控制参数。在目前的工作中，提出了一种使骨干跟踪更加稳健和更快的方法。提出了一种简单的 PI 控制器，并提供了增益的封闭式表达式，从而实现相位瞬态的最佳稳定。所需的输入参数是通过传统的基于振动台的线性模态测试以及单一频率和电平的开环正弦测试获得的。对于相位检测，使用基于LMS算法的自适应滤波器，该算法优于常用的同步解调。一旦相位锁定，就可以直接沿着骨干网进行下一步，从而消除了保持时间。后者目前用于记录稳态，并在后处理中估计傅立叶系数，这变得不必要，因为自适应滤波器在运行时产生高度准确的估计。该方法的优异性能在双夹紧梁经历弯曲-拉伸耦合时得到了证明，导致最低模态频率发生 20% 的偏移。即使对于为线性状态设计的固定控制参数，每个骨干点也只需要大约 100 个周期，在非线性状态下也是如此。这比迄今为止文献报道的要快得多。