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Computation of Nonlinear Normal Modes through Shooting and Pseudo-Arclength Computation

机译：通过射击和伪弧长计算计算非线性法线模

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Most existing constructive techniques for computing NNMs are based on asymptotic approaches and rely on fairly involved mathematical developments. In this context, algorithms for the numerical continuation of periodic solutions are really quite sophisticated and advanced (see, e.g., (1; 2), and the AUTO and MATCONT softwares). These algorithms have been extensively used for computing the forced response and limit cycles of nonlinear dynamical systems. Interestingly, there have been very few attempts to compute the periodic solutions of conservative mechanical structures (i.e., NNM motions) using numerical continuation techniques. One of the first approaches was proposed by Slater in (3) who combined a shooting method with sequential continuation to solve the nonlinear boundary value problem that defines a family of NNM motions. Similar approaches were considered in Lee et al. (4) and Bajaj et al. (5). A more sophisticated continuation method is the so-called asymptotic-numerical method. It is a semi-analytical technique that is based on a power series expansion of the unknowns parameterized by a control parameter. It is described in the next chapter. In this study, a shooting procedure is combined with the so-called pseudo-arclength continuation method for the computation of NNM motions. We show that the NNM computation is possible with limited implementation effort, which holds promise for a practical and accurate method for determining the NNMs of nonlinear vibrating structures.
机译：大多数现有的用于计算NNM的构造技术都基于渐近方法，并且依赖于相当复杂的数学发展。在这种情况下，用于周期解的数值连续的算法实际上是相当复杂和先进的（例如参见（1； 2），以及AUTO和MATCONT软件）。这些算法已被广泛用于计算非线性动力系统的强制响应和极限环。有趣的是，很少有人尝试使用数值连续技术来计算保守机械结构（即NNM运动）的周期解。 Slater在（3）中提出了第一种方法，他将射击方法与顺序连续相结合，以解决定义一系列NNM运动的非线性边界值问题。 Lee等人考虑了类似的方法。 （4）和Bajaj等。 （5）。一种更复杂的延续方法是所谓的渐近数值方法。这是一种半分析技术，基于通过控制参数设置参数的未知数的幂级数展开。下一章将对其进行描述。在这项研究中，将射击程序与所谓的伪弧长连续方法相结合，用于计算NNM运动。我们表明，在有限的实现工作量下，NNM的计算是可能的，这为确定非线性振动结构的NNM的实用而准确的方法提供了希望。

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