﻿ 轴系偏斜因素对于横向振动特性的影响研究
 舰船科学技术  2016, Vol. 38 Issue (3): 85-91 PDF

Research on effects of shafting deflection factor for lareral vibration characteristics
LI Yong-zhe, ZHOU Qi-zheng, WANG De-shi
Department of Weaponry Engineering, Naval University of Engineering, Wuhan 430033, China
Abstract: Research the problem of propulsion shafting lareral vibration response problems caused by angle misalignment in uniform motion process. Considering as the basis of shaft angle misalignment and shaft eccentricity factors, establish the angle misalignment propulsion shafting lareral vibration dynamic model, use the numerical algorithm for solving the equation, and analysis the form of shafting vibration response and the effect of misalignment angle quantitymass ratio of active shaft and dimensionless shaft speed on the lareral vibration characteristics of the shafting with the angle misalignment in uniform process. The results indicate that shafting with angle misalignment can the dynamic behavior. The research work makes sense in providing a theoretical basis for warships to improve the acoustic performance from reducing shafting vibration.
Key words: angle misalignment    shafting    transverse vibration    uniform motion
0 引言

1 动力学模型

 图 1 偏角不对中轴系动力学模型 Fig. 1 The dynamical model of shafting with angular misalignment

 图 2 轴系坐标系 Fig. 2 Coordinate systems of the shafting

 \left\{ \begin{align} & {{x}_{c2}}=\left( {{x}_{2}}+{{e}_{2}} \right)\cos \alpha \\ & {{y}_{c2}}=\left( {{y}_{2}}+{{e}_{2}} \right)\cos \alpha \\ \end{align} \right. (1)

 \begin{align} & f({{x}_{1}},{{y}_{1}},{{x}_{2}},{{y}_{2}})= \\ & {{({{x}_{1}}-{{x}_{2}})}^{2}}+{{({{y}_{1}}-{{y}_{2}})}^{2}}-{{(l\sin \alpha )}^{2}}=0 \\ \end{align} (2)

 \left\{ \begin{align} & {{m}_{1}}{{{\ddot{x}}}_{1}}+{{K}_{1}}{{x}_{1}}+{{D}_{1}}{{{\dot{x}}}_{1}}= \\ & {{m}_{1}}{{e}_{1}}\left( {{\left( \varphi {{'}_{1}} \right)}^{2}}\cos {{\phi }_{1}}+\varphi _{_{1}}^{,,}\sin {{\varphi }_{1}} \right)+\lambda ({{x}_{1}}-{{x}_{2}}) \\ & {{m}_{1}}{{{\ddot{y}}}_{1}}+{{K}_{1}}{{y}_{1}}+{{D}_{1}}{{{\dot{y}}}_{1}}= \\ & {{m}_{1}}{{e}_{1}}\left( {{\left( \varphi {{'}_{1}} \right)}^{2}}\sin {{\phi }_{1}}-\varphi _{_{1}}^{,,}\cos {{\varphi }_{1}} \right)-{{m}_{1}}g+\lambda ({{y}_{1}}-{{y}_{2}}) \\ & {{m}_{2}}{{{\ddot{x}}}_{2}}+{{K}_{2}}{{x}_{2}}+{{D}_{2}}{{{\dot{x}}}_{2}}= \\ & {{m}_{2}}{{e}_{2}}\cos \alpha \left( {{\left( \varphi {{'}_{2}} \right)}^{2}}\cos {{\varphi }_{2}}+\varphi _{_{2}}^{,,}\sin {{\varphi }_{2}} \right)+\lambda ({{x}_{2}}-{{x}_{1}}) \\ & {{m}_{2}}{{{\ddot{y}}}_{2}}+{{K}_{2}}{{y}_{2}}+{{D}_{2}}{{{\dot{y}}}_{2}}= \\ & {{m}_{2}}{{e}_{2}}\cos \left( \alpha {{\left( \varphi {{'}_{2}} \right)}^{2}}\sin {{\varphi }_{2}}-\varphi _{_{2}}^{,,}\cos {{\varphi }_{2}} \right)-{{m}_{2}}g+\lambda ({{y}_{2}}-{{y}_{1}}) \\ \end{align} \right. (3)

 ${\varphi _1} = {\varphi _0'}t$ (4)

 ${\varphi _2} = {\varphi' _0}t + {\varphi _0}$ (5)

 ${\omega _{n1}} = \sqrt {\frac{{{K_1}}}{{{m_1}}}} ,\tau = {\omega _{n1}}t{\text{,}}{\omega _{n2}} = \sqrt {\frac{{{K_2}}}{{{m_1}}}} {\text{,}}\eta = \frac{{{\omega _{n2}}}}{{{\omega _{n1}}}}$

 $\omega = {\text{ }}\frac{{{\varphi'' _1}}}{{{\omega _{n1}}}},m' = \frac{{{m_2}}}{{{m_1}}}{\text{,}}\varepsilon = \frac{{{e_1}}}{{{c_z}}}{\text{,}}{e'_2} = \frac{{{e_2}}}{{{c_z}}}{\text{,}}\bar l = \frac{l}{{{c_z}}}$

 ${c_1} = \frac{{{D_1}}}{{\sqrt {{K_1}{m_1}} }},{c_2} = \frac{{{D_2}}}{{\sqrt {{K_1}{m_1}} }},{X_1} = \frac{{{x_1}}}{{{c_z}}},{Y_1} = \frac{{{y_1}}}{{{c_z}}},$

 ${X_2} = \frac{{{x_2}}}{{{c_z}}},{Y_2} = \frac{{{y_2}}}{{{c_z}}},\gamma = \frac{\lambda }{{{m_1}\omega _{n1}^2}},\bar g = \frac{g}{{\omega _{n1}^2{c_z}}},$

 $\left\{ \begin{gathered} X_{_1}^{,,} + {X_1} + {\zeta _1}X{'_1} = \hfill \\ {\varepsilon _1}{\omega ^2}\cos {\varphi _1} + \gamma ({X_1} - {X_2}),\hfill \\ Y_{_1}^{,,} + {Y_1} + {\zeta _1}{Y_1} = \hfill \\ {\varepsilon _1}{\omega ^2}\sin {\varphi _1} - \bar g + \gamma ({Y_1} - {Y_2}),\hfill \\ m'X_{_2}^{,,} + {\eta ^2}{X_2} + {c_1}X{'_2} = \hfill \\ m'{\varepsilon _2}{\omega ^2}\cos {\varphi _2}\cos \alpha + \gamma ({X_2} - {X_1}),\hfill \\ m'Y_{_2}^{,,} + {\eta ^2}{Y_2} + {c_1}Y{'_2} = \hfill \\ m'{\varepsilon _2}{\omega ^2}\sin {\varphi _2}\cos \alpha - m'\bar g + \gamma ({Y_2} - {Y_1}) \hfill \\ \end{gathered} \right.$ (6)

 ${({X_1} - {X_2})^2} + {({Y_1} - {Y_2})^2} = {(\bar l\sin \alpha )^2}$ (7)

 $\left\{ \begin{gathered} {X_2} = {X_1} + \bar l\sin \alpha \cos \theta ,\hfill \\ {Y_2} = {Y_1} + \bar l\sin \alpha \sin \theta ,\hfill \\ \end{gathered} \right.$ (8)

 $M{u^{,,}} + Du' + Ku = {F_0} + {F_\varepsilon } + {F_\varphi } + {F_m},$ (9)

 ${u^{,,}} = {M^{ - 1}}({F_0} + {F_e} + {F_\phi } + {F_m} - Du{,^,} - Ku)$ (10)

 $K = \left[{\begin{array}{*{20}{c}} {1 + {\eta ^2}}&0&0 \\ 0&{1 + {\eta ^2}}&0 \\ { - {\eta ^2}\sin \theta }&{{\eta ^2}\cos \theta }&0 \end{array}} \right],$

 $D = \left[{\begin{array}{*{20}{c}} {{c_1} + {c_2}} & 0 & { - {c_2}\bar l\sin \alpha \sin \theta }\\ 0 & {{c_1} + {c_2}} & {{c_2}\bar l\sin \alpha \cos \theta }\\ { - {c_2}\sin \theta } & {{c_2}\cos \theta } & 0 \end{array}} \right] ,$

 ${F_m} = \left[\begin{gathered} - 0 \hfill \\ (1 + m')\bar g \hfill \\ m{\text{'}}\bar g\cos \theta ,\hfill \\ \end{gathered} \right]$

 ${F_\varepsilon } = \left[{\begin{array}{*{20}{c}} {{\varepsilon _1}{\omega ^2}\cos {\varphi _1} + \bar m{\varepsilon _2}\cos \alpha {\omega ^2}\cos {\varphi _2}} \\ {{\varepsilon _1}{\omega ^2}\sin {\varphi _1} + \bar m{\varepsilon _2}\cos \alpha {\omega ^2}\sin {\varphi _2}} \\ {\bar m{\varepsilon _2}\cos \alpha ({\omega ^2}\sin {\varphi _2}\cos \theta - {\omega ^2}\cos {\varphi _2}\sin \theta )} \end{array}} \right]$

2 算例分析

1)偏角不对中量 α 对系统振动特性的影响

 图 3 ω = 0.8， m' = 0.8时，无量纲位移X1 的分岔图 Fig. 3 The bifurcation diagrams of X1 for ω= 0.8, $\bar m$= 0.8

 图 4 ω=0.8、α= 5×10-4时系统振动响应 Fig. 4 vibration responses of the system for ω=0.8 and α= 5×10-4

 图 5 α = 0时系统振动响应 Fig. 5 The vibration responses of the system for α = 0

2)转轴质量比m'对系统振动特性的影响

 图 6 ω = 2，α = 5 × 10-4时，无量纲位移X1 的分岔图 Fig. 6 The bifurcation diagrams of X1 for ω = 2， α = 5 × 10-4

 图 7 ω=0.8, m' = 0.3时系统振动响应 Fig. 7 The vibration responses of the system for ω=0.8, m'=0.3

3)转轴转速 ω对系统振动特性的影响

 图 8 以转轴速度ω为参量的分岔图 Fig. 8 The bifurcation diagrams with rotation speed for the parameter

 图 9 α=5×10-4, ω=1.4时轴系振动响应 Fig. 9 The vibration responses of the system for α=5×10-4, ω=1.4

θ的时间历程图 10(a)和频谱图 10(b)知，θ是时间的函数，表现为混沌运动，且振动幅度大，说明θ不能被视为常量或是小量，因此不能采用级数展开的近似法求解系统的解析解。

 图 10 α=5×10-4, ω=1.4时，广义坐标θ的时间历程图和频谱图 Fig. 10 The time history chart and spectrum chart of generalized coordinates θ for α=5×10-4 and ω=1.4
3 结语

1)由于存在偏角不对中，广义坐标θ为具有复杂变化的时间函数，使偏角不对中轴系成为强非线性、非定常的三自由度耦合系统，导致难以求出系统的解析解；偏角不对中与转轴偏心量相结合改变了轴系横向振动的质量矩阵、阻尼矩阵和力矩阵，使轴系的固有频率发生变化进而影响轴系的振动特性。

2)偏斜因素导致的非线性使得系统呈现出复杂多变的振动特征，并且这些特征的变化难以从理论上进行控制，表现为：①当存在偏角不对中时，系统运动响应从简单规律的周期运动变为复杂的混沌运动，且振动幅度增大；给定转速情况下，随着不对中偏角量的变化，系统振动幅值较大且随机变化，系统响应存在概周期运动和混沌运动等复杂形式，表现为系统产生复杂的轴心轨迹图、时间历程图和功率谱图。②给定不对中偏角量情况下，随着转轴质量比和主动轴转速的变化，系统响应同样存在概周期运动和混沌运动等复杂形式，并且系统振动幅值也是随机变化。③适当减小转轴质量比或者降低转轴转速，系统振动能够得到一定抑制。④较小的转轴偏心量变化对轴系的振动响应形式影响较小。

3)由于存在偏角不对中，系统产生非线性和运动耦合，形成了复杂的非线性动力学行为，表现为概周期运动和混沌运动。初始条件的微小变化会使系统振动响应产生较大误差，导致更为复杂的运动响应变化。这种振动响应的复杂性、初态敏感性、自发随机性和长期预测的不可能性，决定了偏角不对中轴系非线性振动理论研究的难度。

4)为了避开理论研究难题，可以利用本文算列所示方法进行大量数值仿真计算，确定抑制轴系振动的偏角量阈值，从而制定最大容许的轴系不对中工艺误差以更高效地指导舰船推进轴系制造加工的工程实践。

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