﻿ 基于梁变形微分方程与奇异函数的轴系校中计算研究
 舰船科学技术  2019, Vol. 41 Issue (6): 71-75 PDF

Rsearch on the shafting alignment calculation based on differential equations for beam deformation and singularity functions
WANG Jian-wu, LOU Jing-jun, LI Xin-yi, YANG Qing-chao
College of Power Engineering, Naval University of Engineering, Wuhan 430033, China
Abstract: The alignment calculation of shafting is theoretical basis for the design, manufacture, installtion and inspection of vessel’s propulsion shafting, which has important influence on alignment quality and operation performance of shafting. Based on the research of differential equations for beam deformation and singularity functions, the expressions of shear fore, bending moment, sectional angle and deflection under different alignment proposals are deduced in the paper, then through theoretical modeling, the alignment calculation and conclusion analysis of ship shafting under straight line alignment and load alignment are carried out. The results show that the method is more quick and concise, and can meet the needs of practical engineering. Thus it may provide a theoretical reference for the selection and evaluation of alignment proposals of ship shafting.
Key words: alignment calculation of shafting     differential equations for beam deformation     singularity functions
0 引　言

1 校中计算方法研究 1.1 梁变形微分方程

 $\frac{1}{\rho } = \frac{{{{\rm d}^2}w/{\rm d}{x^2}}}{{{{\left[ {1 + {{\left( {\rm d}w/{\rm d}x \right)}^2}} \right]}^{3/2}}}} {\text{。}}$ (1)

 $\frac{1}{\rho } = {{\rm d}^2}w/{\rm d}{x^2}{\text{。}}$ (2)

 $\frac{{{{\rm d}^2}w}}{{{\rm d}{x^2}}} = \frac{M}{{EI}} {\text{，}}$ (3)

 $\frac{{{{\rm d}^2}M}}{{{\rm d}{x^2}}} = \frac{{{\rm d}Q}}{{{\rm d}x}} = q {\text{。}}$ (4)

1.2 奇异函数

 $F(x) = < x - a{ > ^n} {\text{。}}$ (5)

 $\left\{ \begin{array}{l} {\text{当}n \text{≥} 0\text{时}, < x - a{ > ^n} = \left\{ {\begin{array}{*{20}{c}} {{{\left( {x - a} \right)}^n}{\rm{ }}x \text{≥} a}{\text{，}}\\ {0{\rm{ }}x < a}{\text{，}} \end{array}} \right.}\\ {\text{当}n < 0\text{时}, < x - a{ > ^n} = \left\{ {\begin{array}{*{20}{c}} {\infty {\rm{ }}x = a}{\text{，}}\\ {0{\rm{ }}x \ne a}{\text{。}} \end{array}} \right.} \end{array} \right.$ (6)

 $\left\{ \begin{array}{l} {\displaystyle\frac{{{\rm d} < x - a{ > ^n}}}{{{\rm d}x}} = \left\{ {\begin{array}{*{20}{c}} {n < x - a{ > ^{n - 1}}}{\text{，}}\\ { < x - a{ > ^{n - 1}}}{\text{，}} \end{array}} \right.{\rm{ }}\begin{array}{*{20}{c}} {n > 0}{\text{，}}\\ {n \text{≤} 0}{\text{，}} \end{array}}\\ {\int { < x - a{ > ^n}{\rm d}x = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{{n + 1}} < x - a{ > ^{n + 1}}}{\text{，}}\\ { < x - a{ > ^{n + 1}}}{\text{，}} \end{array}{\rm{ }}\begin{array}{*{20}{c}} {n > 0}{\text{，}}\\ {n \text{≤} 0}{\text{。}} \end{array}} \right.} } \end{array} \right.$ (7)

1.3 轴系校中计算方法的推导

 图 1 简化轴段载荷分布图 Fig. 1 Load distribution of simplified shafting

1）将作用于[aibi]区域的均布载荷qi转化为载荷集度函数为：

 ${q_i}(x) = {q_i}( < x - {a_i}{ > ^0} - < x - {b_i}{ > ^0}) {\text{；}}$ (8)

2）将作用于ah点的集中载荷转化为载荷集度函数为：

 ${q_h}(x) = {F_h} < x - {a_h}{ > ^{ - 1}} {\text{；}}$ (9)

3）作用在ak处的轴承支反力的载荷集度载荷函数为：

 ${q_k}(x) = {R_k} < x - {a_k}{ > ^{ - 1}} {\text{；}}$ (10)

4）将作用于aj点的外力偶转化为载荷集度函数为：

 ${q_j}(x) = {M_j} < x - {a_j}{ > ^{ - 2}} {\text{。}}$ (11)

 $\begin{split} q(x) =& \displaystyle\sum\limits_{i = 1}^n {{q_i}( < x - {a_i}{ > ^0} - < x - {b_i}{ > ^0}) + } \displaystyle\sum\limits_{h = 1}^m {{F_h} < x - {a_h}{ > ^{ - 1}}}+ \\ &\displaystyle\sum\limits_{k = 1}^l {{R_k} < x - {a_k}{ > ^{ - 1}} + } \displaystyle\sum\limits_{j = 1}^o {{M_j} < x - {a_j}{ > ^{ - 2}}}{\text{。}} \quad\quad\;\;\,\,(12) \end{split}$

 $\begin{array}{l} Q(x) = \displaystyle\sum\limits_{i = 1}^n {{q_i}( < x - {a_i}{ > ^1} - < x - {b_i}{ > ^1}) + } \displaystyle\sum\limits_{h = 1}^m {{F_h} < x - {a_h}{ > ^0}} +\\ \qquad\;\;\;\displaystyle\sum\limits_{k = 1}^l {{R_k} < x - {a_k}{ > ^0} + } \displaystyle\sum\limits_{j = 1}^o {{M_j} < x - {a_j}{ > ^{ - 1}}} {\text{；}}\quad\quad\;\;\,\,(13) \end{array}$

 $\begin{array}{l} M(x) = \displaystyle\sum\limits_{i = 1}^n {\frac{{{q_i}}}{2}( < x - {a_i}{ > ^2} - < x - {b_i}{ > ^2}) + } \displaystyle\sum\limits_{h = 1}^m {{F_h} < x - {a_h}{ > ^1}} + \\ \qquad\;\;\;\;\displaystyle\sum\limits_{k = 1}^l {{R_k} < x - {a_k}{ > ^1} + } \displaystyle\sum\limits_{j = 1}^o {{M_j} < x - {a_j}{ > ^0}} {\text{；}}\quad\quad\;\;\;\;(14) \end{array}$

 $\begin{array}{l} \displaystyle\frac{{\theta (x)}}{{EI}} = \displaystyle\sum\limits_{i = 1}^n {\frac{{{q_i}}}{6}( < x - {a_i}{ > ^3} - < x - {b_i}{ > ^3}) + } \displaystyle\sum\limits_{h = 1}^m {\frac{{{F_h}}}{2} < x - {a_h}{ > ^2}}+ \\ \qquad\;\;\displaystyle\sum\limits_{k = 1}^l {\frac{{{R_k}}}{2} < x - {a_k}{ > ^2} + } \displaystyle\sum\limits_{j = 1}^o {{M_j} < x - {a_j}{ > ^1} + C} {\text{；}}\quad\;\;(15) \end{array}$

 $\begin{array}{l} \displaystyle\frac{{Y(x)}}{{EI}} \!=\! \displaystyle\sum\limits_{i = 1}^n {\frac{{{q_i}}}{{24}}( < x \!-\! {a_i}{ > ^4} - < x - {b_i}{ > ^4}) \!+\! } \displaystyle\sum\limits_{h = 1}^m {\frac{{{F_h}}}{6} < x \!-\! {a_h}{ > ^3}}\!+\! \\ \qquad\;\;\displaystyle\sum\limits_{k = 1}^l {\frac{{{R_k}}}{6} < x \!-\! {a_k}{ > ^3} \!+\! }\displaystyle \sum\limits_{j = 1}^o {\frac{{{M_j}}}{2} < x \!-\! {a_j}{ > ^2} \!+\! C} x \!+\! D{\text{。}}\;(16) \end{array}$

2 推进轴系校中模型

 图 2 某船舶推进轴系结构示意图 Fig. 2 The schematic diagram of ship propulsion shafting
2.1 轴系的简化

1）轴段自重的简化

 图 3 某船舶推进轴系载荷分布图 Fig. 3 Load distribution of ship propulsion shafting

2）载荷的简化

3）轴承支点的简化

 $\left\{ \begin{array}{l} \text{对铁梨木轴承}:S = (1/4\sim1/3)B\text{或}S = (1\sim1.4)D\text{，}\\ \text{对白合金轴承}:S = (1/7\sim1/3)B\text{或}S = (0.3\sim0.7)D\text{。} \end{array} \right.$

4）轴的简化

①轴系中每个轴承都视为梁的一个实支座；

②轴系尾端悬伸于尾轴后轴承外，作自由端处理；

③轴系通过轮胎离合器与主机主轴相连，计算模型首端取到轮胎离合器从动部分，且作自由端处理；

④阶梯轴系各段刚度差小于30%时，轴系抗弯刚度可视为常数，故只要在以上轴系模型简化时对轴段自重进行合理简化，可将整个轴系看作等刚度均匀轴。

2.2 校中计算模型的构建

3 校中计算结果

 图 4 轴系剪力与弯矩图 Fig. 4 Shear force and bending moment of shafting

 图 5 轴系截面转角与挠曲度图 Fig. 5 Sectional angle deflection of shafting
4 结　语

 [1] 刘玉君, 张生俊, 汪骥. 考虑工艺要求的船舶推进轴系校中改进算法研究[J]. 中国造船, 2017, 58(2): 138-144. LIU Yu-jun, ZHANG Sheng-jun, WANG Ji. Research on improved algorithm of ship alignment with consideration of technical requirements[J]. Shipbuiding of China, 2017, 58(2): 138-144. DOI:10.3969/j.issn.1000-4882.2017.02.015 [2] 周瑞平. 超大型船舶推进轴系校中理论研究[D]. 武汉: 武汉理工大学, 2005. ZHOU Rui-ping. The theoretic studies on the proplusion shafting alignment of ultra-large vessels[D]. Wuhan:Wuhan University of Technology, 2005. [3] 李冬梅. 船舶轴系动态校中三弯矩方程的应用[J]. 舰船科学技术, 2017, 39(8): 73-75. LI Dong-mei. The application of three moment equation in dynamic alignment of ship shafting[J]. Ship Science and Technology, 2017, 39(8): 73-75. [4] YANG Yong, TANG Wen-yong, CHE Chi-dong. Shafting alignment based on improved three-moment method with hydrodynamic simulation for twin propulsion systems[J]. Journal of Ship Mechanics, 2013, 17(9): 1038-1052. [5] 张辉, 陈嘉伟, 封海宝. 船舶轴系校中计算中优化算法的应用[J]. 船舶工程, 2017, 39: 41-42+87. ZHANG Hui, CHEN Jia-wei, FENG Hai-bao. Optimization algorithm application on marine shafting alignment calculation[J]. Ship Engineering[J], 2017, 39: 41-42+87. [6] 尤国英, 杜尚林, 顾忠明, 等. 有限元法用于船舶轴系校中计算[J]. 舰船科学技术, 2009, 31(8): 60-62+66. YOU Guo-ying, DU Shang-ming, GUI Xiao-chun, et al. Application of FEM in ship shafting alignment[J]. Ship Science and Technology, 2009, 31(8): 60-62+66. [7] 冷坳坳. 舰船推进轴系校中技术研究[D]. 武汉:武汉理工大学, 2015. LENG Ao-ao. Research on the proplusion shaft alignment technology for warships[D]. Wuhan:Wuhan University of Technology, 2015. [8] 金蓉. 工程力学[M]. 大连:大连海事大学出版社, 2012. [9] 徐斌. 高层建筑分析奇异函数法[M]. 北京: 科学出版社, 2009. [10] FALSONE G. The use of generalised function in the discontinuous beam bending differential equations[J]. Int.J.Engng Ed, 2002, 18(3): 337-344. [11] CB/Z 338-2005, 船舶推进轴系校中[S].