Abstract In this paper we estimate an unknown space-time dependent force being exerted on the vibrating Euler–Bernoulli beam under different boundary supports, which is obtained with the help of measured boundary forces as additional conditions. A sequence of spatial boundary functions is derived, and all the boundary functions and the zero element constitute a linear space. A work boundary functional is coined in the linear space, of which the work is approximately preserved for each work boundary function. The linear system used to recover the unknown force with the work boundary functions as the bases is derived and the iterative algorithm is developed, which converges very fast at each time step. The accuracy and robustness of the boundary functional method (BFM) are confirmed by comparing the estimated forces under large noise with the exact forces. We also recover the unknown force on the damped vibrating Euler–Bernoulli beam equation.

中文翻译：

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用边界函数法识别振动欧拉-伯努利梁上的时空相关力

摘要 在本文中，我们估计了在不同边界支撑下施加在振动欧拉-伯努利梁上的未知时空相关力，该力是在测量的边界力作为附加条件的帮助下获得的。推导出一系列空间边界函数，所有边界函数和零元素构成一个线性空间。工作边界函数是在线性空间中创造的，其中的工作对于每个工作边界函数都近似保留。用于恢复未知力的线性系统以工作边界函数为基础，并开发了迭代算法，该算法在每个时间步都收敛得非常快。通过将大噪声下的估计力与精确力进行比较，确认了边界泛函法 (BFM) 的准确性和稳健性。我们还恢复了阻尼振动欧拉-伯努利梁方程的未知力。