The paper proposes a closed-form dynamic stiffness (DS) formulation for exact transverse free vibration analysis of tapered and/or functionally graded beams based on Euler–Bernoulli theory. The novelties lie in both the DS formulation and the solution technique. For the formulation, the developed DS is applicable to a wide range of non-uniform beams whose bending stiffness and linear density are assumed to be polynomial functions of position. This fills a gap of existing closed-form DS element library which is generally limited to linearly tapered/functionally graded beams. For the solution technique, an elegant and efficient $J_0$ count of tapered element is proposed to apply the Wittrick-Williams (WW) algorithm most effectively. The investigation sheds lights on the so-called $J_0$ count challenge of the algorithm for other DS elements. The above two novelties make exact and highly efficient modal analysis possible for a wide range of tapered and/or functionally graded beams, without resorting to series solution, numerical integrations or refined mesh discretization. Results for a particular case show excellent agreement with published results. Moreover, we investigate the effects of the taper/functional gradient rate/index and boundary conditions on the free vibration behaviour. Benchmark solutions are provided for individual beams as well as beam assemblies.

本文提出了一种基于欧拉-伯努利理论的封闭形式动态刚度 (DS) 公式，用于锥形和/或功能梯度梁的精确横向自由振动分析。新颖之处在于 DS 公式和解决方案技术。对于公式，开发的 DS 适用于范围广泛的非均匀梁，其弯曲刚度和线密度被假定为位置的多项式函数。这填补了现有封闭形式 DS 元素库的空白，该库通常仅限于线性锥形/功能渐变梁。对于求解技术，一个优雅而高效的$J_0$提出了锥形元素的计数以最有效地应用 Wittrick-Williams (WW) 算法。调查揭示了所谓的$J_0$计算其他 DS 元素的算法挑战。上述两个新颖之处使得对各种锥形和/或功能梯度梁进行精确和高效的模态分析成为可能，而无需求助于系列求解、数值积分或细化网格离散化。特定案例的结果与公布的结果非常吻合。此外，我们研究了锥度/功能梯度率/指数和边界条件对自由振动行为的影响。为单个梁以及梁组件提供基准解决方案。