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An upper bound theory to approximate the natural frequencies and parameters identification of composite beams

机译：组合梁固有频率和参数识别的上限理论

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For the free vibration of composite beams and non-uniform beams we propose a new upper bound theory to approximate the first few natural frequencies. The Rayleigh quotient is expressed in terms of boundary functions, instead of that in terms of eigenfunctions. The boundary function satisfies all boundary conditions of the given beam, and is at least the fourth-order polynomial with leading coefficient to be one. We prove that the maximality of the Rayleigh quotient in the space of the kth order boundary functions is equivalent to the orthogonality of the kth order boundary function to lower order optimal boundary functions. Hence, we can easily find the kth order natural frequency through an orthogonalization technique provided. When the first three natural frequencies are compared with the exact or numerically found ones, good results are obtained, which confirm the applicability of the present upper bound theory. We address the inverse problems of composite beam equations, where we use the orthogonal system of boundary functions as bases to expand the unknown functions and derive linear algebraic equations to determine the expansion coefficients. As a consequence, we can fast and accurately estimate the unknown rigidity function and planar inertial function with the help of the first three natural frequencies, and the supplemented measured data of recovered function on two boundaries. The robustness of the present inversion methods is demonstrated by numerical examples. (C) 2017 Elsevier Ltd. All rights reserved.
机译：对于复合梁和非均匀梁的自由振动，我们提出了一种新的上限理论来近似前几个固有频率。瑞利商用边界函数表示，而不是用特征​​函数表示。边界函数满足给定光束的所有边界条件，并且至少是前导系数为1的四阶多项式。我们证明了在第k阶边界函数空间中瑞利商的最大值等于第k阶边界函数与低阶最优边界函数的正交性。因此，我们可以通过提供的正交化技术轻松找到第k阶固有频率。将前三个固有频率与精确的或在数值上找到的固有频率进行比较，可以获得良好的结果，这证实了本上限理论的适用性。我们解决了复合梁方程的反问题，其中我们使用边界函数的正交系统作为基础来展开未知函数，并导出线性代数方程来确定展开系数。结果，我们可以借助前三个固有频率以及两个边界上恢复函数的补充测量数据，快速准确地估计未知刚度函数和平面惯性函数。通过数值实例证明了当前反演方法的鲁棒性。 （C）2017 Elsevier Ltd.保留所有权利。

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