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摘要:
运用铁木辛柯梁理论和K-V阻尼理论，研究了非比例阻尼梁在冲击载荷作用下的频域振动求解方法。推导采用了传统拉普拉斯正变换和基于Durbin公式的拉普拉斯反变换策略（统称拉普拉斯法），发展了阻尼梁系统的动力学方程解法。拉普拉斯法的推演同时涵盖了3种典型的梁边界条件，具有广泛的适用性。数值法的验证采用了特殊构造的比例阻尼点条件，并与基于模态叠加法的求解结果进行了对比分析，且数值算例充分考虑了数值参数和系统参数的影响。计算结果表明：在不同边界条件和受载状态下，拉普拉斯法与模态叠加法均能合理地计算出基本阻尼梁系统的动响应曲线，且两者的求解精度保持在同一量级；同时，捕捉到拉普拉斯法的求解精度会受到系统长细比等参数的影响。拉普拉斯法具有比传统实、复模态叠加法更易操作的特性，但其精度受到了算法固有参数和阶跃外载型式的影响，稳定性仍需进一步提高。
Abstract:Based on Timoshenko's beam theory and K-V damping model, the method for the frequency domain vibration solution of the non-proportionally damped beam under a stationary impact load is studied. The dynamic response of the damped beam is derived by introducing traditional Laplace transformation and Durbin's Laplace inverse transformation (Laplace method). Three typical beam boundaries are taken into consideration in the derivation of Laplace method to demonstrate its applicability. Thereafter, the numerical method is validated under a special proportional damping condition and compared with the modal superposition method. The numerical experiments fully investigate the impact of algorithmic parameters and system parameters. The calculation results indicate that the dynamic responses of the fundamental damped beam system can be reasonably computed by the Laplace method under various boundary and loading conditions, showing comparable accuracy with the modal superposition method. However, the Laplace method is slightly affected by the slenderness ratio of the system. Although Laplace method is easier to manipulate than traditional modal superposition method, its accuracy is affected by its inherent numerical parameters and step external load type, and thus the algorithm stability needs further improvement.
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Key words:
- structural mechanics /
- Laplace transformation /
- Durbin method /
- K-V damped beam /
- dynamic response
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图 2 d*=0.5，Lr=10，X=0.2状态下系统总响应
Figure 2. Curves of total dynamic responses of system under d*=0.5, Lr=10 and X=0.2 conditions
图 3 d*=0.5，Lr=10，X=0.8状态下系统总响应
Figure 3. Curves of total dynamic responses of system under d*=0.5, Lr=10 and X=0.8 conditions
图 4 d*=0.5，Lr=100，X=0.2状态下系统总响应
Figure 4. Curves of dynamic responses in d*=0.5, Lr=100 and X=0.2 conditions
图 5 d*=0.5，Lr=100，X=0.8状态下系统总响应
Figure 5. Curves of total dynamic responses of system under d*=0.5, Lr=100 and X=0.8 conditions
图 6 d*=0.25，Lr=10，X=0.2状态下系统总响应
Figure 6. Curves of total dynamic responses of system under d*=0.25, Lr=10 and X=0.2 conditions
图 7 d*=0.25，Lr=10，X=0.8状态下系统总响应
Figure 7. Curves of total dynamic responses of system under d*=0.25, Lr=10 and X=0.8 conditions
图 8 d*=0.25，Lr=100，X=0.2状态下系统总响应
Figure 8. Curves of total dynamic responses of system under d*=0.25, Lr=100 and X=0.2 conditions
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