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 哈尔滨工程大学学报  2020, Vol. 41 Issue (2): 184-193  DOI: 10.11990/jheu.201902007 0

### 引用本文

CHEN Dongyang, XIAO Qing, XIE Junchao, et al. Vibration characteristics study of a hydroplane system based on the transfer matrix method for multibody systems[J]. Journal of Harbin Engineering University, 2020, 41(2): 184-193. DOI: 10.11990/jheu.201902007.

### 文章历史

1. 扬州大学 电气与能源动力工程学院, 江苏 扬州 225100;
2. 中国舰船研究设计中心, 湖北 武汉 430064;
3. 南京理工大学 发射动力学研究所, 江苏 南京 210094

Vibration characteristics study of a hydroplane system based on the transfer matrix method for multibody systems
CHEN Dongyang 1, XIAO Qing 2, XIE Junchao 2, ZHU Weijun 1, RUI Xiaoting 3
1. College of Electrical, Energy and Power Engineering, Yangzhou University, Yangzhou 225127, China;
2. China Ship Development and Design Center, Wuhan 430064, China;
3. Institute of Launch Dynamics, Nanjing University of Science and Technology, Nanjing 210094, China
Abstract: Efficient and accurate simulation of the vibration characteristics of the hydroplane system of an underwater vehicle is the key to the study of the hydroelasticity and vibration control of the hydroplane system. To analyze the influence of the structural parameters of the hydroplane system's key components on its vibration characteristics, an efficient dynamic simulation model of the hydroplane system established on the basis of the ransfer matrix method for multibody systems (MSTMM). The simulation results of the proposed model compared with those of the commercial software ANSYS, and the accuracy of the model verified. The simulation results show that the calculation efficiency of the hydroplane system's vibration characteristics based on the MSTMM is higher than that of commercial software ANSYS. The first and third orders of the hydroplane system are symmetric modes. Meanwhile, the second and fourth orders are antisymmetric and local modes. Increasing the stiffness of the hydraulic spring can increase the frequency of the first and third orders of the hydroplane system, which is equivalent to the increase in the equivalent torsional stiffness of the steering system, and restrain the pitching motion of the hydroplanes.
Keywords: hydroplane system    vibration characteristics    transfer matrix method for multibody systems    dynamic modeling    efficient model    finite element    multi-rigid-flexible coupling system    modal

1 水下航行器舵系统动力学建模 1.1 动力学模型

 Download: 图 1 水下航行器舵系统 Fig. 1 The hydroplanes system of an underwater vehicle

 Download: 图 2 舵系统结构示意及动力学模型 Fig. 2 The hydroplanes system and its dynamic model

1.2 传递方程推导

 $\begin{array}{l} {Z_{19,0}} = U_{{\rm{pla}}}^{\rm{R}}{Z_{12,13}} = U_{{\rm{pla}}}^{\rm{R}}\left( {{U_{12,{I_1}}}{Z_{12,11}} + {U_{12,{I_2}}}{U_{30}}{U_{31}}{Z_{31,0}}} \right) = \\ U_{{\rm{pla}}}^{\rm{R}}\left( {{U_{12,{I_1}}}{U_{11}}{Z_{10,11}} + {U_{12,{I_2}}}{U_{30}}{U_{31}}{Z_{31,0}}} \right) = \\ U_{{\rm{pla}}}^{\rm{R}}\left( {{U_{12,{I_1}}}{U_{11}}\left( {{U_{10,{I_1}}}{Z_{10,9}} + {U_{10,{I_2}}}{Z_{10,20}} + {U_{10,{I_3}}}{Z_{10,28}}} \right) + } \right.\\ \left. {{U_{12,{I_2}}}{U_{30}}{U_{31}}{Z_{31,0}}} \right) = \\ U_{{\rm{pla}}}^{\rm{R}}\left( {{U_{12,{I_1}}}{U_{11}}\left( {{U_{10,{I_1}}}{U_9}{Z_{8,9}} + {U_{10,{I_2}}}{U^{col}}{Z_{25,0}} + } \right.} \right.\\ \left. {\left. {{U_{10,{1_3}}}{U_{28}}{U_{29}}{Z_{2,0}}} \right) + {U_{12,{l_2}}}{U_{30}}{U_{31}}{Z_{31,0}}} \right) = \\ U_{{\rm{pla}}}^{\rm{R}}\left( {{U_{12,{I_1}}}{U_{11}}\left( {{U_{10,{I_1}}}{U_9}\left( {{U_{8,{I_1}}}{Z_{8,7}} + {U_{8,{I_2}}}{U_{26}}{U_{27}}{Z_{27,0}}} \right) + } \right.} \right.\\ \left. {\left. {{U_{10,{I_2}}}{U^{{\rm{col}}}}{Z_{25,0}} + {U_{10,{I_3}}}{U_{28}}{U_{29}}{Z_{29,0}}} \right) + {U_{12,{I_2}}}{U_{30}}{U_{31}}{Z_{31,0}}} \right) = \\ U_{{\rm{pla}}}^{\rm{R}}\left( {{U_{12,{I_1}}}{U_{11}}\left( {{U_{10,{I_1}}}{U_9}\left( {{U_{8,{I_1}}}U_{{\rm{pla}}}^{\rm{L}}{Z_{1,0}} + {U_{8,{I_2}}}{U_{26}}{U_{27}}{Z_{27,0}}} \right) + } \right.} \right.\\ \left. {{U_{10,{I_2}}}{U^{{\rm{col}}}}{Z_{25,0}} + {U_{10,{I_3}}}{U_{28}}{U_{29}}{Z_{29,0}}} \right) + {U_{12,{I_2}}}{U_{30}}{U_{31}}{Z_{31,0}} = \\ U_{{\rm{pla}}}^{\rm{R}}{U_{12,{I_1}}}{U_{11}}{U_{10,{I_1}}}{U_9}{U_{8,{I_1}}}U_{{\rm{pla}}}^{\rm{L}}{Z_{1,0}} + \\ U_{{\rm{pla}}}^{\rm{R}}{U_{12,{I_1}}}{U_{11}}{U_{10,{I_1}}}{U_9}{U_{8,{I_2}}}{U_{26}}{U_{27}}{Z_{27,0}} + \\ U_{{\rm{pla}}}^{\rm{R}}{U_{12,{I_1}}}{U_{11}}{U_{10,{I_3}}}{U_{28}}{U_{29}}{Z_{29,0}} + \\ U_{{\rm{pla}}}^{\rm{R}}{U_{12,{I_1}}}{U_{11}}{U_{10,{I_2}}}{U^{{\rm{col}}}}{Z_{25,0}} + U_{{\rm{pla}}}^{\rm{R}}{U_{12,{I_2}}}{U_{30}}{U_{31}}{Z_{31,0}} \end{array}$ (1)

 $\left\{ {\begin{array}{*{20}{l}} {U_{{\rm{pla}}}^{\rm{R}} = {U_{19}}{U_{18}}{U_{17}}{U_{16}}{U_{15}}{U_{14}}{U_{13}}}\\ {U_{{\rm{pla}}}^{\rm{L}} = {U_7}{U_6}{U_5}{U_4}{U_3}{U_2}{U_1}}\\ {{U^{{\rm{col}}}} = {U_{20}}{U_{21}}{U_{22}}{U_{23}}{U_{24}}{U_{25}}} \end{array}} \right.$ (2)

 $\left\{ {\begin{array}{*{20}{l}} {{T_{1 - 19}} = U_{{\rm{pla}}}^{\rm{R}}{U_{12,{I_1}}}{U_{11}}{U_{10,{I_1}}}{U_9}{U_{8,{I_1}}}U_{{\rm{pla}}}^{\rm{L}}}\\ {{T_{27 - 19}} = U_{{\rm{pla}}}^{\rm{R}}{U_{12,{I_1}}}{U_{11}}{U_{10,{I_1}}}{U_9}{U_{8,{I_2}}}{U_{26}}{U_{27}}}\\ {{T_{29 - 19}} = U_{{\rm{pla}}}^{\rm{R}}{U_{12,{I_1}}}{U_{11}}{U_{10,{I_3}}}{U_{28}}{U_{29}}}\\ {{T_{25 - 19}} = U_{{\rm{pla}}}^{\rm{R}}{U_{12,{I_1}}}{U_{11}}{U_{10,{I_2}}}{U^{{\rm{col}}}}}\\ {{T_{31 - 19}} = U_{{\rm{pla}}}^{\rm{R}}{U_{12,{I_2}}}{U_{30}}{U_{31}}} \end{array}} \right.$ (3)

 $\begin{array}{*{20}{c}} { - {Z_{19,0}} + {T_{1 - 19}}{Z_{1,0}} + {T_{27 - 19}}{Z_{27,0}} + }\\ {{T_{29 - 19}}{Z_{29,0}} + {T_{25 - 19}}{Z_{25,0}} + {T_{31 - 19}}{Z_{31,0}} = 0} \end{array}$ (4)

 $\left\{ {\begin{array}{*{20}{l}} {{H_{8,{I_1}}}{Z_{8,7}} = {H_{8,{I_2}}}{U_{26}}{U_{27}}{Z_{27,0}}}\\ {{H_{8,{I_1}}}U_{{\rm{pla}}}^L{Z_{1,0}} = {H_{8,{I_2}}}{U_{26}}{U_{27}}{Z_{27,0}}}\\ {{G_{1 - 8}}{Z_{1,0}} + {G_{27 - 8}}{Z_{27,0}} = 0}\\ {{G_{1 - 8}} = - {H_{8,{I_1}}}U_{{\rm{pla}}}^L}\\ {{G_{27 - 8}} = {H_{8,{I_2}}}{U_{26}}{U_{27}}} \end{array}} \right.$ (5)
 $\left\{ \begin{array}{l} {H_{10,{I_1}}}{Z_{10,9}} = {H_{10,{I_2}}}{Z_{10,20}}\\ {H_{10,{I_1}}}{U_9}{Z_{8,9}} = {H_{10,{I_2}}}{U^{{\rm{col}}}}{Z_{25,0}}\\ {H_{10,{I_1}}}{U_9}\left( {{U_{8,{I_1}}}U_{{\rm{pla}}}^L{Z_{1,0}} + {U_{8,{I_2}}}{U_{26}}{U_{27}}{Z_{27,0}}} \right) = \\ \;\;\;\;\;\;\;{H_{10,{I_2}}}{U^{{\rm{col}}}}{Z_{25,0}}\\ {G_{1 - 10}}{Z_{1,0}} + {G_{27 - 10}}{Z_{27,0}} + {G_{25 - 10}}{Z_{25,0}} = 0\\ {G_{1 - 10}} = - {H_{10,{I_1}}}{U_9}{U_{8,{I_1}}}U_{{\rm{pla}}}^L\\ {G_{27 - 10}} = - {H_{10,{I_1}}}{U_9}{U_{8,{I_2}}}{U_{26}}{U_{27}}\\ {G_{25 - 10}} = {H_{10,{I_2}}}{U^{{\rm{col}}}} \end{array} \right.$ (6)
 $\left\{ {\begin{array}{*{20}{l}} {{H_{10,{{I'}_1}}}{Z_{10,9}} = {H_{10,{I_3}}}{U_{28}}{U_{29}}{Z_{29,0}}}\\ {{H_{10,{{I'}_1}}}{U_9}{Z_{8,9}} = {H_{10,{I_3}}}{U_{28}}{U_{29}}{Z_{29,0}}}\\ {{H_{10,{{I'}_1}}}{U_9}\left( {{U_{8,{I_1}}}U_{{\rm{pla}}}^L{Z_{1,0}} + {U_{8,{I_2}}}{U_{26}}{U_{27}}{Z_{27,0}}} \right) = }\\ {\;\;\;\;\;\;\;{H_{10,{I_3}}}{U_{28}}{U_{29}}{Z_{29,0}}}\\ {G_{1 - 10}^\prime {Z_{1,0}} + G_{27 - 10}^\prime {Z_{27,0}} + G_{29 - 10}^\prime {Z_{29,0}} = 0}\\ {G_{1 - 10}^\prime = - {H_{10,{{I'}_1}}}{U_9}{U_{8,{I_1}}}U_{{\rm{pla}}}^L}\\ {G_{27 - 10}^\prime = - {H_{10,{{I'}_1}}}{U_9}{U_{8,{I_2}}}{U_{26}}{U_{27}}}\\ {G_{29 - 10}^\prime = {H_{10,{I_3}}}{U_{28}}{U_{29}}} \end{array}} \right.$ (7)
 $\left\{ \begin{array}{l} {H_{12,{I_1}}}{Z_{12,11}} = {H_{12,{I_2}}}{U_{30}}{U_{31}}{Z_{31,0}}\\ {H_{12,{I_1}}}{U_{12,{I_1}}}{U_{11}}{Z_{10,11}} = {H_{12,{I_2}}}{U_{30}}{U_{31}}{Z_{31,0}}\\ {H_{12,{I_1}}}{U_{12,{I_1}}}{U_{11}}\left( {{U_{10,{I_1}}}{Z_{10,9}} + {U_{10,{I_2}}}{Z_{10,20}} + {U_{10,{I_3}}}{U_{28}}{U_{29}}{Z_{29,0}}} \right) = {H_{12,{I_2}}}{U_{30}}{U_{31}}{Z_{31,0}}\\ {H_{12,{I_1}}}{U_{12,{I_1}}}{U_{11}}\left( {{U_{10,{I_1}}}{U_9}{Z_{8,9}} + {U_{10,{I_2}}}{U^{{\rm{col}}}}{Z_{25,0}} + {U_{10,{I_3}}}{U_{28}}{U_{29}}{Z_{29,0}}} \right) = {H_{12,{I_2}}}{U_{30}}{U_{31}}{Z_{31,0}}\\ - {H_{12,{I_1}}}{U_{12,{I_1}}}{U_{11}}\left( {{U_{10,{I_1}}}{U_9}\left( {{U_{8,{I_1}}}{Z_{8,7}} + {U_{8,{I_2}}}{U_{26}}{U_{27}}{Z_{27,0}}} \right) + {U_{10,{I_2}}}{U^{{\rm{col}}}}{Z_{25,0}} + {U_{10,{I_3}}}{U_{28}}{U_{29}}{Z_{29,0}}} \right) = {H_{12,{I_2}}}{U_{30}}{U_{31}}{Z_{31,0}}\\ - {H_{12,{I_1}}}{U_{12,{I_1}}}{U_{11}}\left( {{U_{10,{I_1}}}{U_9}\left( {{U_{8,{I_1}}}U_{{\rm{pla}}}^L{Z_{1,0}} + {U_{8,{I_2}}}{U_{26}}{U_{27}}{Z_{27,0}}} \right) + {U_{10,{I_2}}}{U^{{\rm{col}}}}{Z_{25,0}} + {U_{10,{I_3}}}{U_{28}}{U_{29}}{Z_{29,0}}} \right) = {H_{12,{I_2}}}{U_{30}}{U_{31}}{Z_{31,0}}\\ {G_{1 - 12}}{Z_{1,0}} + {G_{27 - 12}}{Z_{27,0}} + {G_{25 - 12}}{Z_{25,0}} + {G_{29 - 12}}{Z_{29,0}} + {G_{31 - 12}}{Z_{31,0}} = 0\\ {G_{1 - 12}} = - {H_{12,{I_1}}}{U_{12,{I_1}}}{U_{11}}{U_{10,{I_1}}}{U_9}{U_{8,{I_1}}}U_{{\rm{pla}}}^L\\ {G_{27 - 12}} = - {H_{12,{I_1}}}{U_{12,{I_1}}}{U_{11}}{U_{10,{I_1}}}{U_9}{U_{8,{I_2}}}{U_{26}}{U_{27}}\\ {G_{25 - 12}} = - {H_{12,{I_1}}}{U_{12,{I_1}}}{U_{11}}{U_{10,{I_2}}}{U^{{\rm{col}}}}\\ {G_{29 - 12}} = - {H_{12,{I_1}}}{U_{12,{I_1}}}{U_{11}}{U_{10,{I_3}}}{U_{28}}{U_{29}}\\ {G_{31 - 12}} = {H_{12,{I_2}}}{U_{30}}{U_{31}} \end{array} \right.$ (8)

1.3 舵系统各部件的传递矩阵

 ${\mathit{\boldsymbol{U}}^{CB}} = \left[ {\begin{array}{*{20}{c}} {\cos \left( {{\beta _r}l} \right)}&0&0&0&{ - \sin \left( {{\beta _r}l} \right)/\left( {{\beta _r}EA} \right)}&0&0&0\\ 0&{D_{11}^{CB}}&{D_{12}^{CB}}&{D_{13}^{CB}}&0&{D_{14}^{CB}}&{D_{15}^{CB}}&{D_{16}^{CB}}\\ 0&{D_{21}^{CB}}&{D_{22}^{CB}}&{D_{23}^{CB}}&0&{D_{24}^{CB}}&{D_{25}^{CB}}&{D_{26}^{CB}}\\ 0&{D_{31}^{CB}}&{D_{32}^{CB}}&{D_{33}^{CB}}&0&{D_{34}^{CB}}&{D_{35}^{CB}}&{D_{36}^{CB}}\\ {{\beta _r}EA\sin \left( {{\beta _r}l} \right)}&0&0&0&{\cos \left( {{\beta _r}l} \right)}&0&0&0\\ 0&{D_{41}^{CB}}&{D_{42}^{CB}}&{D_{43}^{CB}}&0&{D_{44}^{CB}}&{D_{45}^{CB}}&{D_{46}^{CB}}\\ 0&{D_{51}^{CB}}&{D_{52}^{CB}}&{D_{53}^{CB}}&0&{D_{54}^{CB}}&{D_{55}^{CB}}&{D_{56}^{CB}}\\ 0&{D_{61}^{CB}}&{D_{62}^{CB}}&{D_{63}^{CB}}&0&{D_{64}^{CB}}&{D_{65}^{CB}}&{D_{66}^{CB}} \end{array}} \right]$ (9)

 ${\mathit{\boldsymbol{U}}_{10,{I_1}}} = \left[ {\begin{array}{*{20}{c}} 1&0&0&0&0&0&0&0\\ 0&1&{{b_1}}&0&0&0&{ - {b_3}}&0\\ 0&0&1&0&0&0&0&0\\ 0&{m{\omega ^2}\left( {{b_1} - {c_1}} \right)}&{ - m{\omega ^2}\left( { - {b_2}{c_2} - {b_1}{c_1}} \right) - {\omega ^2}\left( {{J_{zz}} + mc_3^2} \right)}&1&0&{{b_1}}&{ - m{\omega ^2}{b_1}{c_3} + {\omega ^2}{J_{xz}}}&0\\ {m{\omega ^2}}&0&0&0&1&0&0&0\\ 0&{m{\omega ^2}}&{m{\omega ^2}{c_1}}&0&0&1&{ - m{\omega ^2}{c_3}}&0\\ 0&0&0&0&0&0&1&0\\ 0&{m{\omega ^2}\left( { - {b_3} + {c_3}} \right)}&{ - m{\omega ^2}{b_3}{c_1} + {\omega ^2}{J_{xz}}}&0&0&{ - b3}&{ - m{\omega ^2}\left( { - {b_3}{c_3} - {b_2}{c_2}} \right) - {\omega ^2}\left( {{J_{xx}} + mc_1^2} \right)}&1 \end{array}} \right]$ (10)
 ${\mathit{\boldsymbol{U}}_{10,{I_2}}} = \left[ {\begin{array}{*{20}{c}} 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&1&0&{\left( {{b_1} - {a_1}} \right)}&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&{\left( { - {b_3} + {a_3}} \right)}&0&1 \end{array}} \right]$ (11)
 ${\mathit{\boldsymbol{U}}_{10,{I_3}}} = \left[ {\begin{array}{*{20}{c}} 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&1&0&{\left( {{b_1} - {d_1}} \right)}&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&{\left( { - {b_3} + {d_3}} \right)}&0&1 \end{array}} \right]$ (12)

 ${\mathit{\boldsymbol{U}}_{8,{I_1}}} = {\mathit{\boldsymbol{U}}_{12,{I_1}}} = \left[ {\begin{array}{*{20}{c}} 1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1 \end{array}} \right]$ (13)
 ${\mathit{\boldsymbol{U}}_{8,{I_2}}} = {\mathit{\boldsymbol{U}}_{12,{I_2}}} = \left[ {\begin{array}{*{20}{c}} 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&1 \end{array}} \right]$ (14)

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{U}}_{27}} = {\mathit{\boldsymbol{U}}_{28}} = {\mathit{\boldsymbol{U}}_{29}} = {\mathit{\boldsymbol{U}}_{31}} = }\\ {\left[ {\begin{array}{*{20}{c}} 1&0&0&0&0&0&0&0\\ 0&1&0&0&0&{ - 1/{K_y}}&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1 \end{array}} \right]} \end{array}$ (15)
 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{U}}_{25}} = {\mathit{\boldsymbol{U}}_{26}} = {\mathit{\boldsymbol{U}}_{30}} = }\\ {\left[ {\begin{array}{*{20}{c}} 1&0&0&0&{ - 1/{K_x}}&0&0&0\\ 0&1&0&0&0&{ - 1/{K_y}}&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1 \end{array}} \right]} \end{array}$ (16)
 ${\mathit{\boldsymbol{U}}_{24}} = \left[ {\begin{array}{*{20}{c}} 1&0&0&0&{ - 1/{K_x}}&0&0&0\\ 0&1&0&0&0&{ - 1/{K_h}}&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1 \end{array}} \right]$ (17)

 ${\mathit{\boldsymbol{U}}_{20}} = \left[ {\begin{array}{*{20}{c}} 1&0&0&0&0&0&0&0\\ 0&1&0&0&0&{ - 1/{K_y}}&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&1&{1/{{K'}_x}}\\ 0&0&0&0&0&0&0&1 \end{array}} \right]$ (18)

 ${\mathit{\boldsymbol{U}}_{22}} = \left[ {\begin{array}{*{20}{c}} 1&0&0&0&{ - 1/{K_x}}&0&0&0\\ 0&1&0&0&0&{ - 1/{K_y}}&0&0\\ 0&0&1&{1/{{K'}_z}}&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&1&{1/{{K'}_x}}\\ 0&0&0&0&0&0&0&1 \end{array}} \right]$ (19)

 $\begin{array}{l} {\mathit{\boldsymbol{U}}_1} = {\mathit{\boldsymbol{U}}_2} = {\mathit{\boldsymbol{U}}_3} = {\mathit{\boldsymbol{U}}_4} = \\ {\mathit{\boldsymbol{U}}_5} = {\mathit{\boldsymbol{U}}_6} = {\mathit{\boldsymbol{U}}_7} = \\ {\mathit{\boldsymbol{U}}_{13}} = {\mathit{\boldsymbol{U}}_{14}} = \\ {\mathit{\boldsymbol{U}}_{15}} = {\mathit{\boldsymbol{U}}_{16}} = {\mathit{\boldsymbol{U}}_{17}} = {\mathit{\boldsymbol{U}}_{18}} = {\mathit{\boldsymbol{U}}_{19}} = {\mathit{\boldsymbol{U}}^{CB}} \end{array}$ (20)

 ${\mathit{\boldsymbol{U}}^{UB1}} = \left[ {\begin{array}{*{20}{c}} {\cos \left( {{\beta _r}l} \right)}&0&0&0&{ - \sin \left( {{\beta _r}l} \right)/\left( {{\beta _r}EA} \right)}&0&0&0\\ 0&{S(\lambda l)}&{T(\lambda l)/\lambda }&{U(\lambda l)/\left( {EI{\lambda ^2}} \right)}&0&{V(\lambda l)/\left( {EI{\lambda ^3}} \right)}&0&0\\ 0&{\lambda V(\lambda l)}&{S(\lambda l)}&{T(\lambda l)/(EI\lambda )}&0&{U(\lambda l)/\left( {EI{\lambda ^2}} \right)}&0&0\\ 0&{EI{\lambda ^2}U(\lambda l)}&{EI\lambda V(\lambda l)}&{S(\lambda l)}&0&{T(\lambda l)/\lambda }&0&0\\ {\beta EA\sin \left( {{\beta _r}l} \right)}&0&0&0&{\cos \left( {{\beta _r}l} \right)}&0&0&0\\ 0&{EI{\lambda ^3}T(\lambda l)}&{EI{\lambda ^2}U(\lambda l)}&{\lambda V(\lambda l)}&0&{S(\lambda l)}&0&0\\ 0&0&0&0&0&0&{\cos (\gamma l)}&{\sin (\gamma l)}\\ 0&0&0&0&0&0&{ - \gamma Gl\sin \left( {\gamma Ll} \right)}&{\cos (\gamma l)} \end{array}} \right]$ (21)

 $\left\{ {\begin{array}{*{20}{l}} {\gamma = \sqrt {\rho {\omega ^2}/G} ,\lambda = \sqrt[4]{{m{\omega ^2}/EI}},S = \frac{{ch + c}}{2},}\\ {T = \frac{{{\rm{sh}} + s}}{2},U = \frac{{{\rm{ch}} - c}}{2},V = \frac{{{\rm{sh}} - s}}{2},}\\ {{\rm{ch}} = \cosh (\lambda l),sh = \sinh (\lambda l),}\\ {c = \cos (\lambda l),s = \sin (\lambda l)} \end{array}} \right.$ (22)

lA分别是梁的长度和截面积。

 ${\mathit{\boldsymbol{U}}_9} = {\mathit{\boldsymbol{U}}_{11}} = {\mathit{\boldsymbol{U}}^{{\rm{UB1}}}}$ (23)

 ${\mathit{\boldsymbol{U}}^{\rm{1}}} = \left[ {\begin{array}{*{20}{c}} {\cos \left( {{\beta _r}l} \right)}&0&0&0&{ - \sin \left( {{\beta _r}l} \right)/\left( {{\beta _r}EA} \right)}&0\\ 0&{S(\lambda l)}&{T(\lambda l)/\lambda }&{U(\lambda l)/\left( {EI{\lambda ^2}} \right)}&0&{V(\lambda l)/\left( {EI{\lambda ^3}} \right)}\\ 0&{\lambda V(\lambda l)}&{S(\lambda l)}&{T(\lambda l)/(EI\lambda )}&0&{U(\lambda l)/\left( {EI{\lambda ^2}} \right)}\\ 0&{EI{\lambda ^2}U(\lambda l)}&{EI\lambda V(\lambda l)}&{S(\lambda l)}&0&{T(\lambda l)/\lambda }\\ {{\beta _r}EA\sin \left( {{\beta _r}l} \right)}&0&0&0&{\cos \left( {{\beta _r}l} \right)}&0\\ 0&{EI{\lambda ^3}T(\lambda l)}&{EI{\lambda ^2}U(\lambda l)}&{\lambda V(\lambda l)}&0&{S(\lambda l)} \end{array}} \right]$ (24)

 $\mathit{\boldsymbol{R}} = \left[ {\begin{array}{*{20}{c}} 0&{ - 1}&0&0&0&0\\ 1&0&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&0&{ - 1}\\ 0&0&0&0&1&0 \end{array}} \right]$ (25)

 ${\mathit{\boldsymbol{U}}^2} = {\mathit{\boldsymbol{R}}^T}{\mathit{\boldsymbol{U}}^1}\mathit{\boldsymbol{R}}$ (26)

 ${\mathit{\boldsymbol{U}}^{\mathit{\boldsymbol{UB}}{\rm{2}}}} = \left[ {\begin{array}{*{20}{c}} {U_{11}^2}&{U_{12}^2}&{U_{13}^2}&{U_{14}^2}&{U_{15}^2}&{U_{16}^2}&0&0\\ {U_{21}^2}&{U_{22}^2}&{U_{23}^2}&{U_{24}^2}&{U_{25}^2}&{U_{26}^2}&0&0\\ {U_{31}^2}&{U_{32}^2}&{U_{33}^2}&{U_{34}^2}&{U_{35}^2}&{U_{36}^2}&0&0\\ {U_{41}^2}&{U_{42}^2}&{U_{43}^2}&{U_{44}^2}&{U_{45}^2}&{U_{46}^2}&0&0\\ {U_{51}^2}&{U_{52}^2}&{U_{53}^2}&{U_{54}^2}&{U_{55}^2}&{U_{56}^2}&0&0\\ {U_{61}^2}&{U_{62}^2}&{U_{63}^2}&{U_{64}^2}&{U_{65}^2}&{U_{66}^2}&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1 \end{array}} \right]$ (27)

 ${\mathit{\boldsymbol{U}}_{21}} = {\mathit{\boldsymbol{U}}^{{\rm{UB2}}}}$ (28)

 ${\mathit{\boldsymbol{U}}^{{\rm{rod}}}} = \left[ {\begin{array}{*{20}{c}} 1&0&0&0&0&0&0&0\\ 0&{\cos \left( {{\beta _r}l} \right)}&0&0&0&{ - \sin \left( {{\beta _r}l} \right)/\left( {{\beta _r}EA} \right)}&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&{{\beta _r}EA\sin \left( {{\beta _r}l} \right)}&0&0&0&{\cos \left( {{\beta _r}l} \right)}&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1 \end{array}} \right]$ (29)

 ${\mathit{\boldsymbol{U}}_{23}} = {\mathit{\boldsymbol{U}}^{{\rm{rod}}}}$ (30)

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{H}}_{8,{I_1}}} = {\mathit{\boldsymbol{H}}_{8,{I_2}}} = {\mathit{\boldsymbol{H}}_{12,{I_1}}} = {\mathit{\boldsymbol{H}}_{12,{I_2}}} = }\\ {\left[ {\begin{array}{*{20}{l}} 1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&0&0&0&1&0 \end{array}} \right]} \end{array}$ (31)
 ${\mathit{\boldsymbol{H}}_{10,{I_1}}} = \left[ {\begin{array}{*{20}{c}} 1&0&0&0&0&0&0&0\\ 0&1&{{a_1}}&0&0&0&{ - {a_3}}&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&0&0&0&1&0 \end{array}} \right]$ (32)
 ${\mathit{\boldsymbol{H}}_{10,{I_2}}} = {\mathit{\boldsymbol{H}}_{10,{I_3}}} = \left[ {\begin{array}{*{20}{c}} 1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&0&0&0&1&0 \end{array}} \right]$ (33)
 ${\mathit{{H}}_{10,{I_1}}} = \left[ {\begin{array}{*{20}{c}} 1&0&0&0&0&0&0&0\\ 0&1&{{d_1}}&0&0&0&{ - {d_3}}&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&0&0&0&1&0 \end{array}} \right]$ (34)
1.4 舵系统特征值求解

 ${\left. {{{\left. {{\mathit{\boldsymbol{U}}_{{\rm{all}}}}} \right|}_{24 \times 48}}{\mathit{\boldsymbol{Z}}_{{\rm{all}}}}} \right|_{48 \times 1}} = 0$ (35)

 $\left\{ \begin{array}{l} {\mathit{\boldsymbol{U}}_{{\rm{all}}}} = \left[ {\begin{array}{*{20}{c}} {{{\left. {{\mathit{\boldsymbol{T}}_{1 - 19}}} \right|}_{8 \times 8}}}&{{{\left. {{\mathit{\boldsymbol{T}}_{27 - 19}}} \right|}_{8 \times 8}}}&{{{\left. {{\mathit{\boldsymbol{T}}_{29 - 19}}} \right|}_{8 \times 8}}}&{{{\left. {{\mathit{\boldsymbol{T}}_{25 - 19}}} \right|}_{8 \times 8}}}&{{{\left. {{\mathit{\boldsymbol{T}}_{31 - 19}}} \right|}_{8 \times 8}}}&{ - {{\left. I \right|}_{8 \times 8}}}\\ {{{\left. {{\mathit{\boldsymbol{G}}_{1 - 8}}} \right|}_{4 \times 8}}}&{{{\left. {{\mathit{\boldsymbol{G}}_{27 - 8}}} \right|}_{4 \times 8}}}&{{{\left. {\bf{O}} \right|}_{4 \times 8}}}&{{{\left. {\bf{O}} \right|}_{4 \times 8}}}&{{{\left. {\bf{O}} \right|}_{4 \times 8}}}&{{{\left. {\bf{O}} \right|}_{4 \times 8}}}\\ {{{\left. {{\mathit{\boldsymbol{G}}_{1 - 10}}} \right|}_{4 \times 8}}}&{{{\left. {{\mathit{\boldsymbol{G}}_{27 - 10}}} \right|}_{4 \times 8}}}&{{{\left. {\bf{O}} \right|}_{4 \times 8}}}&{{\mathit{\boldsymbol{G}}_{25 - 10}}}&{{{\left. {\bf{O}} \right|}_{4 \times 8}}}&{{{\left. {\bf{O}} \right|}_{4 \times 8}}}\\ {{{\left. {\mathit{\boldsymbol{G}}_{1 - 10}^\prime } \right|}_{4 \times 8}}}&{{{\left. {\mathit{\boldsymbol{G}}_{27 - 10}^\prime } \right|}_{4 \times 8}}}&{{{\left. {\mathit{\boldsymbol{G}}_{29 - 10}^\prime } \right|}_{4 \times 8}}}&{{{\left. {\bf{O}} \right|}_{4 \times 8}}}&{{{\left. {\bf{O}} \right|}_{4 \times 8}}}&{{{\left. {\bf{O}} \right|}_{4 \times 8}}}\\ {{{\left. {{\mathit{\boldsymbol{G}}_{1 - 12}}} \right|}_{4 \times 8}}}&{{{\left. {{\mathit{\boldsymbol{G}}_{27 - 12}}} \right|}_{4 \times 8}}}&{{{\left. {{\mathit{\boldsymbol{G}}_{25 - 12}}} \right|}_{4 \times 8}}}&{{{\left. {{\mathit{\boldsymbol{G}}_{29 - 12}}} \right|}_{4 \times 8}}}&{{{\left. {{\mathit{\boldsymbol{G}}_{31 - 12}}} \right|}_{4 \times 8}}}&{{{\left. {\bf{O}} \right|}_{4 \times 8}}} \end{array}} \right]\\ \mathit{\boldsymbol{Z}}_{{\rm{all}}}^{\rm{T}} = {\left[ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{Z}}_{1,0}^{\rm{T}}}&{\mathit{\boldsymbol{Z}}_{27,0}^{\rm{T}}}&{\mathit{\boldsymbol{Z}}_{29,0}^{\rm{T}}}&{\mathit{\boldsymbol{Z}}_{25,0}^{\rm{T}}}&{\mathit{\boldsymbol{Z}}_{31,0}^{\rm{T}}}&{\mathit{\boldsymbol{Z}}_{19,0}^{\rm{T}}} \end{array}} \right]^{\rm{T}}} \end{array} \right.$ (36)

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{Z}}_{1,0}} = {{\left[ {X,Y,{\mathit{\Theta }_z},0,0,0,{\mathit{\Theta }_x},0} \right]}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{Z}}_{27,0}} = {{\left[ {0,0,0,{M_z},{Q_x},{Q_y},{\mathit{\Theta }_x},0} \right]}^{\rm{T}}}}\\ {{Z_{29,0}} = {{\left[ {X,0,{\mathit{\Theta }_z},0,0,{Q_y},{\mathit{\Theta }_x},0} \right]}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{Z}}_{25,0}} = {{\left[ {0,0,0,{M_z},{Q_x},{Q_y},0,{M_x}} \right]}^{\rm{T}}}}\\ {{Z_{31,0}} = {{\left[ {0,0,0,{M_z},{Q_x},{Q_y},{\mathit{\Theta }_x},0} \right]}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{Z}}_{19,0}} = {{\left[ {X,Y,{\mathit{\Theta }_z},0,0,0,{\mathit{\Theta }_x},0} \right]}^{\rm{T}}}} \end{array}} \right.$ (37)

2 水下航行器舵系统振动特性分析

 Download: 图 3 圆频率计算结果 Fig. 3 Calculation results of circular frequency

 Download: 图 4 舵系统模态仿真结果(ANSYS) Fig. 4 Modal simulation results of the hydroplane system(ANSYS)

 Download: 图 5 舵系统前四阶X方向振动(MSTMM) Fig. 5 The first fourth x-direction modes of the hydroplane system (MSTMM)
 Download: 图 6 舵系统前四阶Y, Θx方向振型(MSTMM) Fig. 6 The first fourth Y, Θx-direction modes of the hydroplane system (MSTMM)

 Download: 图 7 不同液压刚度情况下的舵系统振动频率 Fig. 7 The hydroplanes system natural frequencies vs. hydraulic stiffness
3 结论

1) 基于MSTMM可以快速建立水下航行器舵系统的动力学模型，且计算效率高。

2) 基于MSTMM可以方便的考虑系统各部件的结构参数和连接刚度对系统振动特性的影响，且可以直观的计算出各振动方向上的振型。

3) 舵系统的1、3阶模态为对称模态，2、4阶模态为反对称模态，且为局部模态。增加液压弹簧刚度可以增加系统第1、3阶的频率，相当于增加了操纵系统的等效扭转刚度，可以抑制舵叶俯仰运动。

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