Instead of the traditional method of utilizing the extremal values of Rayleigh quotient, in this research we develop an upper bound theory to determine the natural frequencies in a linear space of boundary functions. The boundary function will have the following restrictions: boundary conditions are satisfied, fourth-order polynomial at least with unit leading coefficient. It proves that the maximum value of Rayleigh quotient in terms of kth-order boundary function builds a good upper bound when the \((k - 3)\)th-order natural frequency is approximated. There are four boundary conditions of the beam, so the fourth-order boundary function is unique without having any parameter. On the contrary, the kth-order boundary function has \(k-4\) free parameters, which leave us a chance to optimize them. We employ the orthogonality to derive the higher-order optimal boundary functions from the lower-order optimal boundary functions exactly, which are constructed sequentially by starting from the fourth-order boundary function. We examine the upper bound theory for uniform beams under three types of supports, and the single- and double- tapered beams under cantilevered support. Comparison is made between the first four natural frequencies with the exact/numerical ones, which proves the usefulness of the upper bound theory.

中文翻译：

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边界函数线性空间中的瑞利商和正交性，找到非均匀波束固有频率的准确上限

代替使用瑞利商极值的传统方法，在本研究中，我们开发了一种上限理论来确定边界函数线性空间中的固有频率。边界函数将具有以下限制：满足边界条件，至少具有单位前导系数的四阶多项式。证明了当近似于（（k-3）\）阶自然频率时，以k阶边界函数表示的瑞利商的最大值建立了良好的上限。光束有四个边界条件，因此四阶边界函数是唯一的，没有任何参数。相反，第k阶边界函数具有\（k-4 \）个免费参数，这使我们有机会对其进行优化。我们利用正交性从低阶最优边界函数中精确推导出高阶最优边界函数，这些函数是从四阶边界函数开始依次构建的。我们研究了三种支撑下均匀梁的上限理论，以及悬臂支撑下的单锥形和双锥形梁的上限理论。通过对前四个固有频率与精确/数字频率之间的比较，证明了上限理论的有用性。