﻿ 基于K-V阻尼模型的铁木辛柯梁振动响应分析<sup>*</sup>
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Response analysis of Timoshenko beam based on K-V damping model
ZHANG Xiayang, ZHU Ming, WU Zhe
School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2017-04-05; Accepted: 2017-07-12; Published online: 2017-09-22 10:12
Foundation item: National Key R&D Program of China (2016YFB1200100)
Corresponding author. ZHU Ming, E-mail:zhumingbuaa@163.com
Abstract: Based on Timoshenko beam theory, this paper has analyzed the dynamic properties when a clamped beam subjected to step load and moving load respectively. In addition, K-V damping model is considered to study the influence of damping on dynamic performance of the system. To acquire the theoretical solution, proportional damping utilization condition is derived, the real modal superposition method is applied, and eventually obtain the analytical responses when beam subjected to external loads. The numerical analysis results indicate that the solving process is accurate and reliable, providing a measurement reference to other methods, like Laplace transformation. The results of damping cases demonstrate that the high modes inherit over damping property, while in low modes present oscillation convergent characteristic. Sometimes, the damping can have significantly impact on the whole system, and for large slender ratios, the amplitude under moving load is even enlarged. Furthermore, the dynamic response subjected to step load is dominated by the low modes.
Key words: Timoshenko beam     K-V damping     real mode theory     step load     moving load

1 建模与求解 1.1 建模

 (1)

 (2)

 (3)

 (4)

 (5)

 (6)

1.2 求解

 (7)

 (8)
 (9)

1.2.1 无阻尼振动求解

 (10)

 (11)

 (12)

 (13)

1) 当Ωλ0Lr2时，式(10)由一对实根和一对共轭复根构成，分别记为±η和±α。其频率方程和特征向量分别为

 (14)
 (15)

2) 当Ω=λ0Lr2时，式(10)由2个0重根和一对虚根±α组成。其频率方程和特征向量分别为

 (16)
 (17)

3) 如果Ωλ0Lr2，式(10)的解由一对虚根±α和另一对虚根±β构成，其频率方程和特征向量分别为

 (18)
 (19)

1.2.2 阻尼振动求解

 (20)

 (21)

1) 阻尼比满足κm=0时,

 (22)

2) 阻尼比满足0＜κm＜1时,

 (23)

3) 阻尼比满足κm=1时,

 (24)

4) 阻尼比满足κm＞1时,

 (25)

 (26)

 (27)
2 算例分析

 图 1 Durbin数值反拉普拉斯求解效果示意图 Fig. 1 Schematic of solving accuracy of Durbin's inverse Laplace
2.1 模态求解

 图 2 模态频率和阻尼比变化趋势图 Fig. 2 Changing tendency of modal frequencies and damping ratios

 图 3 归一化模态振型 Fig. 3 Normalized modal shapes
2.2 阶跃载荷

 图 4 Lr=10模态响应 Fig. 4 Modal response when Lr=10
 图 5 Lr=100模态响应 Fig. 5 Modal response when Lr=100
 图 6 d*=0.5时无量纲位移总响应 Fig. 6 Non-dimensionalized response of total displacement when d*=0.5
 图 7 d*=0.25时无量纲位移总响应 Fig. 7 Non-dimensionalized response of total displacement when d*=0.25

2.3 移动载荷

 图 8 Lr=10无量纲位移总响应 Fig. 8 Non-dimensionalized response of total displacement when Lr=10
 图 9 Lr=100无量纲位移总响应 Fig. 9 Non-dimensionalized response of total displacement when Lr=100
3 结论

1) 铁木辛柯梁模型可以预测更复杂的模态振型形状，因而其频率求解更准确。

2) 系统在承受阶跃载荷时，系统的响应主要由低阶频率主导，对于小长细比，阻尼振动可以很快收敛，而对于大长细比，振动收敛很慢，这是因为当长细比较大时，各阶模态的阻尼比明显减小。

3) 系统在承受低速移动载荷时，无阻尼振动和有阻尼振动的结果表明，小长细比情况下，有阻尼振动由于能量很快被耗散掉，所以响应没有呈现出波动特性，而在大长细比下，有阻尼振动波动十分明显，而且小速度移动的载荷引起了有阻尼振动的共振，使得振幅远远超过了无阻尼的响应。

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#### 文章信息

ZHANG Xiayang, ZHU Ming, WU Zhe

Response analysis of Timoshenko beam based on K-V damping model

Journal of Beijing University of Aeronautics and Astronsutics, 2018, 44(3): 500-507
http://dx.doi.org/10.13700/j.bh.1001-5965.2017.0196