﻿ 舰载设备在基础运动激励下的动力响应分析方法
 舰船科学技术  2024, Vol. 46 Issue (3): 178-183    DOI: 10.3404/j.issn.1672-7649.2024.03.033 PDF

1. 中国船舶集团有限公司第七一三研究所 河南 郑州 450015;
2. 上海交通大学 机械与动力工程学院，上海 200240

Dynamic response analysis method of shipborne equipment underfoundation motion excitation
LU Feng1, HE Chao-xun1,2
1. The 713 Research Institute of CSSC, Zhengzhou 450015, China;
2. School of Mechanical Engineering, Shanghai Jiaotong University, Shanghai 200240, China
Abstract: In order to solve the dynamic response problem of shipborne equipment under the excitation of foundation motion, by comparing the dynamic equations of inertial load method, large mass method, large stiffness method and direct excitation method, this paper discusses the characteristics and application conditions of several methods, and gives the corresponding calculation examples and result analysis, so as to provide theoretical basis and methods for the dynamic response calculation of shipborne equipment.
Key words: shipborne equipment     dynamic response     inertia load method     large mass method     direct excitation method
0 引　言

1 惯性载荷法、大质量法、大刚度法、直接激励法对比分析 1.1 惯性载荷法原理

 $\left\{\boldsymbol{U}\right\}=\left\{\boldsymbol{\delta }\boldsymbol{U}\right\}+\left[{\boldsymbol{D}}_{\boldsymbol{R}}\right]\left\{{\boldsymbol{U}}_{0}\right\} ，$ (1)
 $\left[\boldsymbol{K}\right]\left[{\boldsymbol{D}}_{\boldsymbol{R}}\right]=0 \text{，} \left[\boldsymbol{C}\right]\left[{\boldsymbol{D}}_{\boldsymbol{R}}\right]=0。$ (2)

 $\left[\boldsymbol{M}\right]\left\{\ddot{\boldsymbol{U}}\right\}+\left[\boldsymbol{C}\right]\left\{\dot{\boldsymbol{U}}\right\}+\left[\boldsymbol{K}\right]\left\{\boldsymbol{U}\right\}=\boldsymbol{P}\left(\boldsymbol{t}\right) 。$ (3)

 $[{\boldsymbol{M}}]\{ \delta {\boldsymbol{\ddot U}}\} + [{\boldsymbol{C}}]\{ \delta {\boldsymbol{\dot U}}\} + [{\boldsymbol{K}}]\{ \delta {\boldsymbol{U}}\} = {\boldsymbol{P}}({\boldsymbol{t}}) - [{\boldsymbol{M}}][{{\boldsymbol{D}}_{\boldsymbol{R}}}]\{ {{\boldsymbol{\ddot U}}_0}\} 。$ (4)

1）由于释放了结构的刚体运动，在求解时可避免积分过程中因刚体漂移造成的计算误差；

2）获取的结构位移、速度及加速度$\{ \delta U\} $$\{ \delta \dot U\}$$\{ \delta \ddot U\}$是相对于活动坐标系的向量，去除了刚体运动部分，与在工程中的测量结果不同，需进行相应修正；

3）因结构的弹性变形与刚体运动无关，由位移推导的应力场也未变化；

4）从等效转换过程可知，仅适用于基础载荷为加速度的工况。

1.2 大质量法原理

 $\begin{split} & \left[ {\begin{array}{*{20}{c}} {{\boldsymbol{M }}+ {\boldsymbol{\tilde M}}}&0 \\ 0&{{\boldsymbol{J}} + {\boldsymbol{\tilde J}}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {\ddot \delta (t)} \\ {\ddot \theta (t)} \end{array}} \right\} + C\left\{ {\begin{array}{*{20}{c}} {\dot \delta (t)} \\ {\dot \theta (t)} \end{array}} \right\}+ \\ & \left[ {\begin{array}{*{20}{c}} {[{{\boldsymbol{K}}_{xx}}]}&{[{{\boldsymbol{K}}_{x\theta }}]} \\ {[{{\boldsymbol{K}}_{\theta x}}]}&{[{{\boldsymbol{K}}_{\theta \theta }}]} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {\delta (t)} \\ {\theta (t)} \end{array}} \right\} = {\boldsymbol{P}}({\boldsymbol{t}})。\end{split}$ (5)

1）其本质是将基础加速度载荷转化为力和力矩，增加了基础的大质量和大转动惯量，其载荷输入误差为${\boldsymbol{M}}/({\boldsymbol{\tilde M}} + {\boldsymbol{M}})$

2）因去除了激励方向的自由度，所以增加了激励方向上的刚体模态，但其弹性模态未发生变化；

3）由大质量法求解获得的位移可以是相对位移也可以是绝对位移，只要将基础位移去除即可获得相对位移；

4）大质量法可用于基础激励为位移、速度、加速度的工况。

1.3 大刚度法原理

 $\begin{split} & \left[ {\begin{array}{*{20}{c}} {\boldsymbol{M}} &0 \\ 0 &{\boldsymbol{J}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{\boldsymbol{\ddot \delta}} ({\boldsymbol{t}})} \\ {{\boldsymbol{\ddot \theta}} ({\boldsymbol{t}})} \end{array}} \right\} + {\boldsymbol{C}}\left\{ {\begin{array}{*{20}{c}} {{\boldsymbol{\dot \delta}} ({\boldsymbol{t}})} \\ {{\boldsymbol{\dot \theta}} ({\boldsymbol{t}})} \end{array}} \right\} + \\ & \left[ {\begin{array}{*{20}{c}} {[{{\boldsymbol{K}}_{\delta \delta }} + {{{\boldsymbol{\tilde K}}}_{{\delta _0}{\delta _0}}}]}&{[{{\boldsymbol{K}}_{\delta \theta }} + {{{\boldsymbol{\tilde K}}}_{{{\boldsymbol{\delta}} _0}{{{\boldsymbol\delta}} _0}}}]} \\ {[{\boldsymbol{K}}_{\theta \delta } + {{{\boldsymbol{\tilde K}}}_{{\delta _0}{\delta _0}}}]}&{[{{\boldsymbol{K}}_{\theta \theta }} + {{{\boldsymbol{\tilde K}}}_{{\delta _0}{\delta _0}}}]} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{\boldsymbol{\delta}} (t)} \\ {{\boldsymbol{\theta}} (t)} \end{array}} \right\}= \\ & \left[ {\begin{array}{*{20}{c}} {[{{{\boldsymbol{\tilde K}}}_{{\delta _0}{\delta _0}}}]}&{[{{{\boldsymbol{\tilde K}}}_{{\delta _0}{\theta _0}}}]} \\ {[{\boldsymbol{\tilde K}}{}_{{\theta _0}{\delta _0}}]}&{[{{{\boldsymbol{\tilde K}}}_{{\theta _0}{\theta _0}}}]} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{{\boldsymbol{\delta}} _0}({\boldsymbol{t}})} \\ {{{\boldsymbol{\theta}} _0}({\boldsymbol{t}})} \end{array}} \right\}。\end{split}$ (8)

1）大刚度法会造成结构的高阶模态，而采用模态叠加法求解时往往忽略高阶模态，所以大刚度法不适宜用模态叠加法进行动力响应的求解；

2）大刚度法适用于基础激励为位移的工况，载荷施加方便，无需进行其他转换，避免了两次微分造成的舍入误差，也可避免刚体漂移问题；

3）大刚度法与大质量法相比，其求解精度和求解效率比大质量法要差，因此除基础激励为位移以外，建议用大质量法求解。

1.4 直接激励法原理

 $\begin{split} & \left[ {\begin{array}{*{20}{c}} {\boldsymbol{M}} &0 \\ 0&{\boldsymbol{J}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{\boldsymbol{\ddot \delta}} ({\boldsymbol{t}})} \\ {{\boldsymbol{\ddot \theta}} ({\boldsymbol{t}})} \end{array}} \right\} + {\boldsymbol{C}}\left\{ {\begin{array}{*{20}{c}} {{\boldsymbol{\dot \delta}} ({\boldsymbol{t}})} \\ {{\boldsymbol{\dot \theta}} ({\boldsymbol{t}})} \end{array}} \right\} + \\ & \left[ {\begin{array}{*{20}{c}} {[{{\boldsymbol{K}}_{xx}}]}&{[{{\boldsymbol{K}}_{x\theta }}]} \\ {[{\boldsymbol{K}}_{\theta x}]}&{[{{\boldsymbol{K}}_{\theta \theta }}]} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{\boldsymbol{\delta}} ({\boldsymbol{t}})} \\ {{\boldsymbol{\theta}} ({\boldsymbol{t}})} \end{array}} \right\} = {\boldsymbol{P}}({\boldsymbol{t}})。\end{split}$ (9)

1）未进行等效变换，只将运动激励作为动力学方程的边界条件施加到结构与基础的连接处；

2）适用范围广，能解决结构在任意位置受到激励时的结构动力响应问题；

3）当结构组成较复杂时，由于计算所需最小时间步长与划分单元的最小尺度有关，当单元网格划分较密时，计算花费时间较长，受计算机硬件影响较大。

1.5 工程应用

2 算例与结果分析

 图 1 箱体结构示意图 Fig. 1 Schematic diagram of box structure

 图 2 基础加速度曲线 Fig. 2 Foundation acceleration curve

 图 3 大质量法的有限元模型 Fig. 3 Finite element model of large mass method

 图 4 惯性载荷法有限元模型 Fig. 4 Finite element model of inertial load method

 图 5 直接激励法有限元模型 Fig. 5 Finite element model of direct excitation method

 图 7 大质量法节点191与节点1000速度比较 Fig. 7 Comparison of speed between node 191 and node 1000 in the large mass method

 图 8 大质量法中加速度比较 Fig. 8 Comparison of the acceleration of the large mass method

 图 9 大质量法相对位移与惯性载荷法位移 Fig. 9 Large mass method relative displacement and inertial load method displacement

 图 10 大质量法相对速度与惯性载荷法速度 Fig. 10 The relative velocity of the large mass method and the inertial load method

 图 11 大质量法相对加速度与惯性载荷法加速度 Fig. 11 The relative acceleration of the large mass method and the inertial load method

 图 12 直接激励法相对位移与惯性载荷法比较 Fig. 12 Comparison of relative displacement between direct excitation method and inertial load method

 图 13 直接激励法相对速度与惯性载荷法比较 Fig. 13 Comparison of the relative velocity of direct excitation method and inertial load method

 图 14 直接激励法相对加速度与惯性载荷法比较 Fig. 14 Comparison of relative acceleration between direct excitation method and inertial load method

 图 15 大质量法、惯性载荷法与直接激励法应力值比较 Fig. 15 Comparison of stress values of large mass method, inertial load method and direct excitation method

 图 16 大质量法与直接激励法位移 Fig. 16 Displacement by large mass method and direct excitation method

 图 17 大质量法与直接激励法速度 Fig. 17 Velocity of large mass method and direct excitation method

 图 6 大质量法节点191与节点1000位移比较 Fig. 6 Comparison of displacement between node 191 and node 1000 in the large mass method

 图 18 大质量法与直接激励法加速度 Fig. 18 Acceleration by large mass method and direct excitation method

3 某舰载设备在基础运动激励下的动力响应分析

 图 19 某舰载设备在5种工况下冯·米赛斯最大应力值比较 Fig. 19 Comparison of von Mises maximum stress values of a shipborne equipment under five working conditions
4 结　语

 [1] 王波, 张海龙, 武修雄, 等. 基于大质量的高墩大跨连续刚构桥地震时程反应分析[J]. 桥梁建设, 2006, 36(5): 17-20. WANG B, ZHANG H L, WU X X, et al. Analysis of seismic time-history response of high-rise pier and long span continuous rigid-frame bridge based on great mass method[J]. Bridge Construction, 2006, 36(5): 17-20. DOI:10.3969/j.issn.1003-4722.2006.05.005 [2] 杨德庆, 王德禹, 汪庠宝. 考虑星-箭动力耦合作用结构瞬态响应分析[J]. 宇航学报, 2003(2): 213-216. YANG D, WANG D Y, WANG X B. Structural transient response analysis considering satellite-rocket dynamic coupling effect[J]. Journal of Astronautics, 2003(2): 213-216. [3] 周国良, 李小军, 刘必灯, 等. 大质量法在多点激励分析中的应用、误差分析与改进[J]. 工程力学, 2011, 28(1): 48-54. ZHOU G L, LI X J, LIU B D, et al. Error analysis and improvements of large mass method used in multi-support seismic excitation analysis[J]. Engineering Mechanics, 2011, 28(1): 48-54. [4] 楼梦麟, 田仲业. 基于大质量法的结构零频率振型的动力特性[J]. 力学季刊, 2020, 41(2): 240-248. LOU M L, TIAN Z Y. Dynamic characteristics of structural modes with zero frequency based on large mass method[J]. Chinese Quarterly of Mechanics, 2020, 41(2): 240-248. DOI:10.15959/j.cnki.0254-0053.2020.02.004 [5] 雷虎军, 李小珍. 车-轨-桥耦合系统中的非一致地震输入方法研究[J]. 铁道科学与工程学报, 2015, 12(4): 769−777. LEI H J, LI X Z. Input methods of non-uniform seismic excitation in coupling system of vehicle- track- bridge[J]Journal of Railway Science and Engineering, 2015, 12(4): 769−777.