﻿ 弹性支撑对多跨梁结构振动特性的影响
 舰船科学技术  2020, Vol. 42 Issue (10): 17-22    DOI: 10.3404/j.issn.1672-7649.2020.10.004 PDF

1. 海军装备部装备项目管理中心，北京 100071;
2. 中国舰船研究院，北京 100101;
3. 哈尔滨工程大学，黑龙江 哈尔滨 150001

Research on the effect of elastic support on vibration characteristics of multi-span beam structure
SONG Chao1, ZHAO Yan1, Liu Jiang-tao2, ZHANG Hang3, GAO Cong3
1. Equipment Project Management Centre of Navy Equipment Department, Beijing 100071, China;
2. China Ship Research and Development Academy, Beijing 100101, China;
3. Harbin Engineering University, Harbin 150001, China
Abstract: Aiming at the insufficient disclosure of the vibration mechanism of the ship's multi-span beam structure, the effect of elastic support on the vibration characteristics of the multi-span beam structure was studied based on the transfer matrix method. The vibration analysis model of the typical multi-span beam structure of the ship is established, the inherent modal and vibration transfer characteristics of the multi-span beam structure are explored, and the vibration mechanism of the multi-span beam structure of the ship is revealed; The structure has explored the influence of support stiffness, span, etc. on the vibration characteristics of ship plate frame structure. The research shows that the natural frequency of the multi-span beam structure decreases with the increase of the span and the stiffness; under certain local support stiffness, the vibration of the ship slab structure can be simplified to the vibration of the multi-span beam structure, and its vibration characteristics are similar to those of the simply supported beam.
Key words: elastic support     multi-span beam     transfer matrix method     vibration characteristics
0 引　言

1 多跨梁结构振动特性分析 1.1 梁系结构振动方程

 图 1 简单梁结构模型示意图 Fig. 1 Schematic diagram of simple beam model

 ${{Z}} = {\left\{ {y\;\theta \;M\;S} \right\}^{\rm{T}}}\text{。}$ (1)

 $\left\{ \begin{array}{l} \left\{ Z \right\}_i^R = {{{C}}_{ip}}\left\{ Z \right\}_i^L \text{，}\\ \left\{ Z \right\}_{i + 1}^L = {{{C}}_{if}}\left\{ Z \right\}_i^R \text{。}\end{array} \right.$ (2)

 图 2 梁结构计算模型 Fig. 2 Calculation model of beam structure

 $\left\{ {\begin{array}{*{20}{c}} {S_{i - 1}^R - S_i^L = 0} \text{，}\\ { - M_{i - 1}^R + M_i^L - S_{i - 1}^R{l_i} = 0} \text{，}\end{array}} \right.$ (3)

 \left\{ {\begin{aligned} & {\theta _i^L = \theta _{i - 1}^R + \frac{{{l_i}M_{i - 1}^R}}{{EI}} + \frac{{l_i^2S_{i - 1}^R}}{{2EI}}} \text{，}\\ & {y_i^L = y_{i - 1}^R + {l_i}\theta _{i - 1}^R + \frac{{l_i^2M_{i - 1}^R}}{{2EI}} + \frac{{l_i^3S_{i - 1}^R}}{{6EI}}} \text{。}\end{aligned}} \right. (4)

 $\left\{ \!\!{\begin{array}{*{20}{c}} y \\ \theta \\ M \\ S \end{array}} \!\!\right\}_i^L = \left\{\!\! {\begin{array}{*{20}{c}} 1&{{l_i}}&{\dfrac{{l_i^2}}{{2EI}}}&{\dfrac{{l_i^3}}{{6EI}}} \\ 0&1&{\dfrac{{{l_i}}}{{EI}}}&{\dfrac{{l_i^2}}{{2EI}}} \\ 0&0&1&{{l_i}} \\ 0&0&0&1 \end{array}} \!\!\right\}\left\{\!\! {\begin{array}{*{20}{c}} y \\ \theta \\ M \\ S \end{array}} \!\!\right\}_{i - 1}^R \equiv {C_{if}}Z_{i - 1}^R\text{，}$ (5)

 $\left\{ {\begin{array}{*{20}{c}} y \\ \theta \\ M \\ S \end{array}} \right\}_i^R = \left\{ {\begin{array}{*{20}{c}} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ {{m_i}{\omega ^2}}&0&0&1 \end{array}} \right\}\left\{ {\begin{array}{*{20}{c}} y \\ \theta \\ M \\ S \end{array}} \right\}_i^L \equiv {C_{ip}}Z_i^L \text{，}$ (6)

 $Z_i^R = {C_{ip}}{C_{if}}Z_{i - 1}^R \equiv {C_i}Z_{i - 1}^R\text{。}$ (7)

 ${{{C}}_i} = \left\{ {\begin{array}{*{20}{c}} 1&{{l_i}}&{\dfrac{{l_i^2}}{{2EI}}}&{\dfrac{{l_i^3}}{{6EI}}} \\ 0&1&{\dfrac{{{l_i}}}{{EI}}}&{\dfrac{{l_i^2}}{{2EI}}} \\ 0&0&1&{{l_i}} \\ {{\omega ^2}{m_i}}&{{\omega ^2}{m_i}{l_i}}&{\dfrac{{{\omega ^2}ml_i^2}}{{2EI}}}&{1 + \dfrac{{{\omega ^2}{m_i}l_i^3}}{{6EI}}} \end{array}} \right\}\text{，}$ (8)

 $Z_n^R = {C_n}{C_{n - 1}} \cdots {C_1}{Z_0} \equiv {{C}}{Z_0}\text{。}$ (9)

1.2 梁系结构振动特性分析

 图 3 平面横向振动弹性梁段 Fig. 3 Elastic beam section with transverse vibration

 ${{Z}} = {[0,\;\theta ,\;0,\;S]^{\rm{T}}}\text{。}$ (10)

 图 4 梁结构振动模态振型 Fig. 4 Vibration mode of beam structure

 图 5 振动响应频谱图 Fig. 5 Vibration response spectrum

 图 6 梁结构振动响应分布 Fig. 6 Vibration response of beam structure
1.3 船舶结构多跨梁简化模型

 图 7 多跨梁计算模型 Fig. 7 Calculation model of multi span beam

 $\left\{\!\! {\begin{array}{*{20}{c}} y \\ \theta \\ M \\ S \end{array}}\!\! \right\}_i^R = \left\{\!\! {\begin{array}{*{20}{c}} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ {{m_i}{\omega ^2} - k}&0&0&1 \end{array}}\!\! \right\}\left\{\!\! {\begin{array}{*{20}{c}} y \\ \theta \\ M \\ S \end{array}}\!\! \right\}_i^L \equiv {C_{ip}}Z_i^L\text{。}$ (11)

2 弹性支撑跨度对振动特性的影响 2.1 结构跨度对多跨梁振动特性的影响

 图 8 不同跨度系统首阶固有频率变化 Fig. 8 First order natural frequency variation of different span system
2.2 支撑跨度对多跨梁振动特性的影响

 图 9 系统自由振动特性变化 Fig. 9 Change of free vibration characteristics of the system
3 弹性支撑刚度对结构振动特性的影响 3.1 整体刚度对结构振动特性的影响

 图 10 多跨梁计算模型 Fig. 10 Calculation model of multi span beam

 图 11 系统自由振动特性变化 Fig. 11 Change of free vibration characteristics of the system

 图 12 振动速度响应频谱图 Fig. 12 Frequency spectrum of vibration speed response
3.2 局部刚度对结构振动特性的影响

 图 13 系统自由振动特性变化 Fig. 13 Change of free vibration characteristics of the system

 图 14 振动速度响应频谱图 Fig. 14 Frequency spectrum of vibration speed response
4 结　语

1）对弹性支撑多跨梁结构，系统固有频率随结构跨度增加而下降，随弹性支撑跨度增大而下降，且均逐渐趋于平缓。

2）对弹性支撑多跨梁结构，当支撑刚度整体增加时，结构的固有频率随即增大，增大幅度随刚度的增加趋于平缓。同时结构振动响应曲线右移，右移幅度随频率的升高而减小。

3）选取适当的局部支撑刚度，船体板架结构振动可简化为多跨梁振动。当多跨梁上局部支撑刚度增大时，结构一阶振型最大位移点逐渐由中部向两侧移动，与两端简支的低阶梁结构一阶阵型趋于一致。

 [1] 金咸定, 夏利娟. 船体振动学[M]. 上海: 上海交通大学出版社, 2011. [2] 王杰德, 杨永谦. 船体强度与结构设计[M]. 北京: 国防工业出版社, 1995. [3] 李俊, 金咸定. Timoshenko薄壁梁弯扭耦合振动的动态传递矩阵法[J]. 振动与冲击, 2001(4): 59-61. [4] 汤华涛, 吴新跃. 基于有限元的空间变截面梁传递矩阵[J]. 南京理工大学学报(自然科学版), 2014(1): 78-82. [5] NANDAKUMAR P, SHANKAR K. Structural parameter identification using damped transfer matrix and state vectors[J]. International Journal of Structural Stability & Dynamics, 2013, 13(4): 1250076. [6] 王献忠, 江晨半, 计方, 等. 有限长加筋圆柱壳水下声辐射的精细传递矩阵法[J]. 船舶力学, 2017, 21(4): 503-511. [7]