﻿ 浮筏动力特性分析时瑞利阻尼系数的计算方法
 舰船科学技术  2023, Vol. 45 Issue (19): 55-59    DOI: 10.3404/j.issn.1672-7649.2023.19.010 PDF

Calculation method of Rayleigh damping coefficient in dynamic analysis of floating raft
LI Hai-feng, FU Jian, ZENG Fan
Naval Submarine Academy, Qingdao 266100, China
Abstract: In order to reduce the influence of Rayleigh damping parameters on the calculation accuracy of floating raft dynamic characteristics, based on the experimental data of mode frequency and damping ratio, the method of weighted least square is used to calculate coefficient of rayleigh damping which can insure that the sum of square of relative error of mode damping ratio is smallest, we can get satisfying result compared with the common method. The number of mode which participate in calculation can effect the relative errors of mode damping ratio, combined with the method of mode analysis method based on effective mode mass and mode participation factor ,The mode selection principle is determined that accumulative effective mass participation factor ought to be bigger than 95% in the main freedom degree. The principle is used in determination of rayleigh damping coefficient for dynamic analysis of floating raft. the rayleigh damping coefficient determined by this principle can satisfy the need for dynamic analysis of floating raft in engineering.
Key words: Rayleigh damping     floating raft     method of weighted least square     effective mass participation factor     modal selection principle
0 引　言

1 瑞利阻尼介绍

 $f = c\dot u = j{\omega _n}cu 。$ (1)

 $\xi = \frac{c}{{{c_{cr}}}} = \frac{c}{{2m{\omega _0}}} = \frac{c}{{2\sqrt {km} }} 。$ (2)

 ${{F_d}} = {\boldsymbol{C}}{\dot u} 。$ (3)

 ${\boldsymbol{C}} = \alpha {\boldsymbol{M}} + \beta {\boldsymbol{K}}。$ (4)

 ${\boldsymbol{M}}{\ddot u(t)} + {\boldsymbol{C}} {\dot u(t)} + {\boldsymbol{K}} {u(t)} = {f(t)}。$ (5)

 ${ {\boldsymbol{\phi}} ^{\rm{T}}}{\boldsymbol{M}} {\boldsymbol{\phi}}{\ddot \xi } + { {\boldsymbol{\phi}} ^T}{\boldsymbol{C}} {\boldsymbol{\phi }} {\dot \xi } +{{\boldsymbol{\phi}} ^{\rm{T}}}K {\boldsymbol{\phi }} \xi = { {\boldsymbol{\phi}} ^{\rm{T}}}f(t)。$ (6)

 $\left\{ {{{\ddot \xi }_j}} \right\} + 2{\zeta _j}{\omega _j}\left\{ {{{\dot \xi }_j}} \right\} + \omega _j^2\left\{ {{\xi _j}} \right\} = \left\{ {{f_j}(t)} \right\}。$ (7)

 ${{\boldsymbol{\phi}} ^{\rm{T}}}{\boldsymbol{C}} {\boldsymbol{\phi}} = \left( {\begin{array}{*{20}{c}} {\alpha + \beta \omega _1^2}& \cdots &0 \\ \vdots & \ddots & \vdots \\ 0& \cdots &{\alpha + \beta \omega _n^2} \end{array}} \right)。$ (8)

 $2{\zeta _i}{\omega _i} = \alpha + \beta \omega _i^2。$ (9)
2 瑞利阻尼系数的确定

 图 1 瑞利阻尼对隔振效果的影响 Fig. 1 Influence of Rayleigh damping on vibration isolation effect

 $a = \frac{{2\left( {{\zeta _j}{\omega _i} - {\zeta _i}{\omega _j}} \right){\omega _i}{\omega _j}}}{{\omega _i^2 - \omega _j^2}}，$ (10)
 $\beta = \frac{{2\left( {{\zeta _i}{\omega _i} - {\zeta _j}{\omega _j}} \right)}}{{\omega _i^2 - \omega _j^2}} 。$ (11)

2.1 加权最小二乘法计算瑞利阻尼系数

 ${w_i} = 1/\sigma _i^2 。$ (12)

 ${e_i} = \left| {\frac{{{\zeta _{\rm{iinput}}} - {\zeta _{\rm{irequest}}}}}{{{\zeta _{\rm{irequest}}}}}} \right|。$ (13)

 $E = \sum\limits_{i = 1}^n {{w_i}} e_i^2。$ (14)

 $E = \sum\limits_{i = 1}^n {{w_i}} \left(\frac{{\alpha + \beta \omega _i^2}}{{2{\omega _i}{\zeta _i}}} - 1\right)。$ (15)

 $a = \frac{{2\left[ {\displaystyle\sum\limits_{i = 1}^n {\frac{{{w_i}}}{{{\omega _i}{\zeta _i}}}\displaystyle\sum\limits_{i = 1}^n {\frac{{{w_i}\omega _i^2}}{{\zeta _i^2}}} } - \displaystyle\sum\limits_{i = 1}^n {\frac{{{w_i}\omega {}_i}}{{{\zeta _i}}}\displaystyle\sum\limits_{i = 1}^n {\frac{{{w_i}}}{{\zeta _i^2}}} } } \right]}}{{\displaystyle\sum\limits_{i = 1}^n {\frac{{{w_i}\omega _i^2}}{{\zeta _i^2}}\displaystyle\sum\limits_{i = 1}^n {\frac{{{w_i}}}{{\omega _i^2\zeta _i^2}}} } - \displaystyle\sum\limits_{i = 1}^n {\frac{{{w_i}}}{{\zeta _i^2}}\displaystyle\sum\limits_{i = 1}^n {\frac{{{w_i}}}{{\zeta _i^2}}} } }}，$ (16)
 $\beta = \frac{{2\left[ {\displaystyle\sum\limits_{i = 1}^n {\frac{{{w_i}{\omega _i}}}{{{\zeta _i}}}\displaystyle\sum\limits_{i = 1}^n {\frac{{{w_i}}}{{\omega _i^2\zeta _i^2}}} } - \displaystyle\sum\limits_{i = 1}^n {\frac{{{w_i}}}{{\omega {}_i{\zeta _i}}}\displaystyle\sum\limits_{i = 1}^n {\frac{{{w_i}}}{{\zeta _i^2}}} } } \right]}}{{\displaystyle\sum\limits_{i = 1}^n {\frac{{{w_i}\omega _i^2}}{{\zeta _i^2}}\displaystyle\sum\limits_{i = 1}^n {\frac{{{w_i}}}{{\omega _i^2\zeta _i^2}}} } - \displaystyle\sum\limits_{i = 1}^n {\frac{{{w_i}}}{{\zeta _i^2}}\displaystyle\sum\limits_{i = 1}^n {\frac{{{w_i}}}{{\zeta _i^2}}} } }}。$ (17)
2.2 不同方法的相对误差比较

3 浮筏瑞利阻尼系数的确定

 图 2 浮筏有限元模型 Fig. 2 Finite element model of floating raft

 图 3 前15阶模态阻尼比的相对误差 Fig. 3 Relative errors of the first 15 modal damping ratios

3.1 模态选择方法

 $({\boldsymbol{K}} -{\omega ^{\text{2}}}{\boldsymbol{M}})\phi = \phi。$ (18)

 $\phi _j^{\rm{T}}{\boldsymbol{M}}{ \phi _j} = 1。$ (19)

 ${F_{ji}} = \phi _{ji}^{\rm{T}}{\boldsymbol{M}}{ {\boldsymbol{D}} _j}。$ (20)

i阶模态第j方向的有效模态质量定义为：

 ${M_{ji}} = \frac{{F_{ji}^2}}{{\phi _j^{\rm{T}}{\boldsymbol{M}}{{ \phi }_j}}}。$ (21)

 ${r_j} = \frac{{\displaystyle\sum\limits_{i = 1}^m {{M_{ji}}} }}{{{M_{\rm{total}}}}}。$ (22)

3.2 浮筏瑞利阻尼系数的确定

 图 4 模态阻尼比相对误差 Fig. 4 Relative error of modal damping ratio

 图 5 不同阻尼系数对应的隔振效果 Fig. 5 Vibration isolation effect corresponding to different damping coefficients

4 结　语

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