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 引用本文: 周俊, 饶柱石, 塔娜. 欧拉-伯努利梁运动场的正交性及其能量传导特性分析[J]. 力学学报, 2015, 47(1): 135-146.
Zhou Jun, Rao Zhushiy, Ta Na. THE ORTHOGONALITY AND ENERGY TRANSMITION CHARACTERISTICS OF EULER-BERNOULLI BEAM DYNAMIC MOTION FIELD[J]. Chinese Journal of Theoretical and Applied Mechanics, 2015, 47(1): 135-146.
 Citation: Zhou Jun, Rao Zhushiy, Ta Na. THE ORTHOGONALITY AND ENERGY TRANSMITION CHARACTERISTICS OF EULER-BERNOULLI BEAM DYNAMIC MOTION FIELD[J]. Chinese Journal of Theoretical and Applied Mechanics, 2015, 47(1): 135-146.

## THE ORTHOGONALITY AND ENERGY TRANSMITION CHARACTERISTICS OF EULER-BERNOULLI BEAM DYNAMIC MOTION FIELD

• 摘要: 从无阻尼欧拉—伯努利梁振动方程解析解出发, 推导了有限长梁的关于谱系数的时间—空间平均能量和功率流表达式. 在此基础上, 从泛函分析观点, 探讨了弯曲运动场: 衰减振动、行波模式分解关于能量、功率泛函的正交性. 结果表明: 弯曲衰减振动模式和行波模式关于功率流、机械能时间—空间平均是相互独立的, 即关于场能和场功率互不干涉, 满足叠加原理; 衰减振动场导能与行波场导能的重要区别在于功率流关于右、左衰振动模式分解不满足叠加原理, 即弯曲衰减振动场间的相互"干涉"是使其具有能量传导能力的内在原因. 通过右端集中阻尼器支撑的梁的稳态功率流仿真分析计算, 表明低频区振动导能不可忽略, 同时, 衰减振动场和行波场间存在一定的能量交换现象, 但随着频率升高, 振动场传导能量不断下降, 同时能量传导效率也不断下降.

Abstract: Based on analytical solution of undamped Euler-Bernoulli beam's equation of motion, the temporal & spatial average of mechanical energy and power flow's calculation formula are derived. The formula is related to spectral coefficients and based on finite length beam. From the view of functional analysis, the bending motion field, which can be decomposed into evanescent vibration and travelling wave mode, is investigated and the discussion focuses on orthogonality. The result shows that, the evanescent vibration mode is mutual independent of travelling wave mode with regard to energy and power flow functional. In another word, there is no interference between these two modes and the energy superposition principle is satisfied. The important difference between evanescent vibration and wave energy transfer is that the superposition principle is unsatisfied in the evanescent vibration energy transmit mode. That means the interference of two local vibration mode is the reason why the whole vibration field having the capability of energy conduction. The simulation result of a beam with damping at right end and harmonic exciting at middle length is given. The result exhibits that evanescent vibration energy transmit mode shouldn't be neglected in low-frequency range and it is existed that energy exchange between the two modes. But with the increase of frequency, evanescent vibration transmission mode decreases and the total efficiency of energy conducting also falls down.

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