Based on the principle of energy variation, an improved Fourier series is introduced as an allowable displacement function. This paper constructs a calculation model that can study the in-plane and out-of-plane free and forced vibrations of curved beam structures under different boundary conditions. Firstly, based on the generalized shell theory, considering the shear and inertial effects of curved beam structures, as well as the coupling effects of displacement components, the kinetic energy and strain potential energy of the curved beam are obtained. Subsequently, an artificial spring system is introduced to satisfy the constraint condition of the displacement at the boundary of the curved beam, obtain its elastic potential energy, and add it to the system energy functional. Any concentrated mass point or concentrated external load can also be added to the energy function of the entire system with a corresponding energy term. In various situations including classical boundary conditions, the accuracy and efficiency of the method in this paper are proved by comparing with the calculation results of FEM. Besides, by accurately calculating the vibration characteristics of common engineering structures like slow curvature (whirl line), the wide application prospects of this method are shown.

中文翻译：

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弯曲梁平面内和平面外振动分析的一般傅立叶公式

根据能量变化原理，引入改进的傅立叶级数作为容许位移函数。本文构建了一个计算模型，可以研究不同边界条件下弯曲梁结构的平面内和平面外自由振动和受迫振动。首先，基于广义壳理论，考虑曲梁结构的剪切和惯性效应，以及位移分量的耦合作用，得到曲梁的动能和应变势能。随后，引入人工弹簧系统，满足曲梁边界位移的约束条件，获得其弹性势能，并将其加入系统能量泛函中。任何集中的质点或集中的外部载荷也可以用相应的能量项添加到整个系统的能量函数中。在包括经典边界条件在内的各种情况下，通过与有限元计算结果的对比，证明了本文方法的准确性和有效性。此外，通过准确计算慢曲率（回旋线）等常见工程结构的振动特性，显示了该方法的广泛应用前景。