推进轴系回旋振动及其支撑结构振动特性研究
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摘要
推进轴系是船舶动力装置的重要组成部分，其与壳体耦合结构的振动与噪声预报及控制是船舶工程领域的重要研究课题。面对悬臂推进器在旋转过程中所产生的回旋力矩以及艉轴承的等效支撑刚度及支撑位置，准确建立其动力学模型，并研究其对轴系及耦合结构的影响均具有重要的理论意义及应用前景。本文围绕推进轴系的回旋效应、艉轴承的动力学特性、悬臂推进器对轴系的影响以及轴系与壳体的耦合结构振动问题展开了相关研究，具体的研究内容简述如下：
采用傅里叶级数展开方法建立了基于欧拉-伯努利梁（Euler-Bernoulli beam）理论的轴系回旋振动动力学分析模型。通过与文献及有限元模型所得结果的对比验证了傅里叶级数展开方法应用的正确性，并引入狄拉克（Dirac）delta函数，将轴系推进器考虑为集中质量点，并分析其回旋力矩，推导得到了带集中质量点及考虑回旋效应的轴系控制微分方程，利用三角级数的正交性推导得到了轴系回旋振动的标准特征方程，并利用数值逐步积分方法Newmark法求解简谐激励或任意激励下的回旋振动时域响应，通过与已出版文献，国标及有限元模型得到的结果进行对比分析验证了本方法具有良好的收敛性及正确性。本方法基于连续体理论，可以求解弹性支撑边界、变轴颈、多跨距轴系回旋振动固有特性及强迫响应。
基于Reynolds方程，利用倾斜轴颈状态下的油膜厚度公式，通过阻抗关系推导得到了无限短及无限长轴承的刚度和阻尼共8个动力特性参数的计算方法，并根据适用于有限长轴承的结合了无限长短轴承阻抗表达式的一种经验阻抗关系，得到了有限长轴承的刚度及阻尼计算方法，并由此获得有限长轴承的等效支撑刚度及支撑位置。通过与已出版文献不同倾斜角度下最大油膜压力的对比验证了本方法的正确性，并分析了艉轴承润滑特性、等效支撑刚度及支撑位置与艉轴承末端下沉量及轴系转速之间的关系，进一步研究了其对轴系回旋振动特性的影响。
采用二维的傅里叶级数展开方法建立了弹性支撑及连接边界的多跨曲梁面内自由振动模型。从能量原理出发，将曲梁面内径向和切向位移函数表示成傅里叶级数形式，并引入辅助多项式函数用以解决弹性支撑及连接边界的不连续性，采用Rayleigh-Ritz方法求解哈密顿方程，得到关于未知位移幅值系数的标准特征值问题，求解得到多跨曲梁的固有频率和振型。同时，考虑了其他分布或者集中特征量对整体矩阵的影响。通过两跨的自由、简支、固支等经典边界及弹性边界的曲梁模型结果与有限元模型所得结果的对比验证了本方法的正确性，并对曲率对曲梁振动特性的影响及两跨曲梁中间连接刚度对其固有频率的影响作了一定的研究。
引入波传播方法解决了圆柱壳弹性支撑边界自由振动的求解问题。首先采用波传播方法研究了经典边界圆柱壳的振动特性，研究了使用相同边界梁单元波数求解圆柱壳振动波数的方法，并另辟蹊径直接通过控制方程求解波数，并应用到弹性边界条件，通过设定弹性支撑弹簧刚度为无穷大或者零来得到与经典边界相对应的结果，并与文献及有限元模型所得结果进行了对比，并研究了各个方向弹性支撑对圆柱壳振动特性的影响。
采用机械阻抗综合法研究梁及其支撑圆柱壳体的耦合结构振动特性。首先对机械阻抗的基本思路进行了说明，求解了基于机械阻抗综合法的结构在简谐激励下的响应。其次，推导得到了多跨梁及简支边界下圆柱壳体在点激励下频域内的阻抗表达形式，求解计算了在简谐激励下的谐响应，并与有限元模型所得结果进行了对比，结果吻合良好。对双层梁、梁-壳耦合结构进行了分析，研究了质量点质量及悬臂长度对质量点-梁-壳耦合结构响应的影响。
设计了实验台架，定量分析弹性挠曲对轴系回旋振动的影响规律。通过推进器与整个轴系的质量比相似设计、推进器当量半径与轴系主要半径的惯量比相似设计、以及轴系长度与轴系主要半径的长径比相似设计，最后设置轴承位置来实现轴系的动力学相似性，使实验台架最大限度的体现船舶轴系的特性。通过精心的垂直及水平方向位置调心保证挠曲前的轴系状态，之后通过调整艉轴承的倾斜量分析轴系产生弹性挠曲之后其回旋振动响应特性的变化，并定量研究弹性挠曲对轴系回旋振动的影响规律。
The propulsion shafting is one of the most important components of a ship’s powerdevice, and the noise and vibration prediction and control of the shafting and its coupledstructure with cylindrical shell is a very important research project in ship engineering. Forthe gyroscopic moment because of the rotation cantilever propeller and the effectivesupported stiffness and location of the stern bearing, developing the precise dynamic models,and studying the influence to the shafting and the coupled structure will be of great theoreticalsignificance and applied value. Surrounding the gyroscopic effect of the propulsion shafting,the dynamic characteristics of the stern bearing, and the influence of the cantilever propellerto the shafting, the coupled vibration problem of the shafting and cylindrical shell, thedetailed research work has been carried out in this thesis as follows:
The dynamic analysis model of the shafting’s gyroscopic vibration is established usingthe Fourier Series Expanded Method based on the Euler-Bernoulli beam theory. Thecorrectness of the application of the method is validated by comparing with the resultsobtained by published papers and FEA model. Through the introduction of Dirac deltafunction, considering the propeller as a lumped mass and analyzing its gyroscopic moment,the governing differential equation of the shafting with lumped mass and gyroscopic effect isobtained and a standard matrix eigenvalue problem for gyroscopic vibration can be developedusing the orthogonal of trigonometric functions. The gyroscopic dynamic response in timedomain under harmonic or arbitrary load is obtained using the Newmark numericalstep-by-step integration method. The accurate and efficient of the method is validated bycomparing with the results obtained by published papers, National Standard and FEA model.The method which is based on theory of continuous system can solve natural characteristicsand forced responses of gyroscopic vibration of shafting with elastic-surpport boundaryconditions, various section areas and multi-span.
Based on the Reynolds equation, the calculating approach of eight dynamic parametersabout stiffness and damp of infinite long and short bearing is deduced through the impedancerelationship using the oil film thickness of misaligned journal bearing. Combining theimpedance relationships of the infinite long and short bearing, the calculating formulas ofstiffness and damp of finite long bearing are derived using an experiential impedancerelationship for the finite long bearing. And the effective supported stiffness and location areobtained. The correctness of the method is validated by comparing with the results ofmaximal oil film pressure obtained by published papers with difference misalignments. Andthe influences of the misalignment to the lubricating characteristics, the effective supported stiffness and location of the journal bearing and to the gyroscopic vibration of shafting areanalyzed.
The in-plane free vibration model of multi-span curved beam system with elasticllysupported and connected boundary condition is established using the two-dimentional FourierSeries Expanded Method. According to the enegy principle, the in-plane vibrationdisplacements along radial and tangent directions are both expressed as the superposition of adouble Fourier cosine series and four supplementary functions in the form of the product of apolynomial fuction. The use of these supplementary functions is to overcome thediscontinuity problems of the elastic boundary conditions. And a standard matrix eigenvalueproblem about the unknow displacement amplitude coefficients is derived through solving theHamilton’s equation using the Rayleigh-Ritz Method and the natural frequencies and modeshapes of multi-span curved beams can be solved. At the same time, the contribution of otherdistributed and lumped parameters to the whole mass and stiffness matrix is considered. Theresults of two-span curved beams with free, simple supported, clamped and elasticllysupported boundary conditions are obtained and compared with the results got from the FEMmodel to validate the correctness of the present method. And the effect of curvature andconnecting stiffnesses between two-span curved beams on the vibration frequencies isdescribed.
The vibration of cylindrical shell with elasticlly supported boundary conditions is solvedthrough introducing the wave propagation method. The method is applied to the cylindricalshell with traditional boundary conditions. And the wave number along the axial direction issolved using the governing equation directly rather than solved from beam structures havingthe same boundary conditions with the shell used by the most papers. Applying to the elasticboundary conditions, the accurate of the method is validated by comparing with the resultsobtained by published papers and FEA model through considering the classical homogeneousboundary conditions as the special cases when the stiffness for each set of springs is equal toeither infinity or zero. The influence of the elasticlly supported along various directions to thevibration of the cylindrical shell is studied.
The vibration characteristic of coupled structure of the beam and its supportedcylindrical shell is worked out using the Mechanical Impedance Synthesis Method. The basicconcept the mechanical impedance are introduced firstly, and the methods used formulti-coordinate, coupled substructures under harmonic load are analyzed. The impedanceexpressions in frequency domain of the multi-span beam system with elastic boundaryconditions and the cylindrical shell with shear diaphragm-shear diaphragm (SD-SD) boundary conditions under harmonic load are obtained, and the results of the harmonic responses arevalidated accurate compared with the results calculated from FEA models. The approach isfirst used to determine the modal properties of a double-deck beam system consisting of twobeams coupled in parallel, which serves to both validate the substructure synthesis techniqueand demonstrate the versatility of the beam substructure in contrasting to its seeminglylimited face value. The vibrational responses of a coupled beam-cylindrical shell system arethen used to validate the current solution technique and to study the effects of modifying somemodel parameters.
Finally, an experimental system is designed to quantitative analyze the influence ofelastic deformation to gyroscopic vibration of the shafting. Through mass ratio of thepropeller to the whole shafting design, inertia ratio of the equivalent radius of propeller to themaster radius of shafting design and length-radius ratio of length to master radius of shaftingdesign, and the dynamic design setting the location of the bearing, the system could make ananalogy to the shafting. Various experimental measurement works are performed throughresizing the vertical location of the stern bearing in order to describe the influence of elasticdeformation to gyroscopic vibration of the shafting.
采用傅里叶级数展开方法建立了基于欧拉-伯努利梁（Euler-Bernoulli beam）理论的轴系回旋振动动力学分析模型。通过与文献及有限元模型所得结果的对比验证了傅里叶级数展开方法应用的正确性，并引入狄拉克（Dirac）delta函数，将轴系推进器考虑为集中质量点，并分析其回旋力矩，推导得到了带集中质量点及考虑回旋效应的轴系控制微分方程，利用三角级数的正交性推导得到了轴系回旋振动的标准特征方程，并利用数值逐步积分方法Newmark法求解简谐激励或任意激励下的回旋振动时域响应，通过与已出版文献，国标及有限元模型得到的结果进行对比分析验证了本方法具有良好的收敛性及正确性。本方法基于连续体理论，可以求解弹性支撑边界、变轴颈、多跨距轴系回旋振动固有特性及强迫响应。
基于Reynolds方程，利用倾斜轴颈状态下的油膜厚度公式，通过阻抗关系推导得到了无限短及无限长轴承的刚度和阻尼共8个动力特性参数的计算方法，并根据适用于有限长轴承的结合了无限长短轴承阻抗表达式的一种经验阻抗关系，得到了有限长轴承的刚度及阻尼计算方法，并由此获得有限长轴承的等效支撑刚度及支撑位置。通过与已出版文献不同倾斜角度下最大油膜压力的对比验证了本方法的正确性，并分析了艉轴承润滑特性、等效支撑刚度及支撑位置与艉轴承末端下沉量及轴系转速之间的关系，进一步研究了其对轴系回旋振动特性的影响。
采用二维的傅里叶级数展开方法建立了弹性支撑及连接边界的多跨曲梁面内自由振动模型。从能量原理出发，将曲梁面内径向和切向位移函数表示成傅里叶级数形式，并引入辅助多项式函数用以解决弹性支撑及连接边界的不连续性，采用Rayleigh-Ritz方法求解哈密顿方程，得到关于未知位移幅值系数的标准特征值问题，求解得到多跨曲梁的固有频率和振型。同时，考虑了其他分布或者集中特征量对整体矩阵的影响。通过两跨的自由、简支、固支等经典边界及弹性边界的曲梁模型结果与有限元模型所得结果的对比验证了本方法的正确性，并对曲率对曲梁振动特性的影响及两跨曲梁中间连接刚度对其固有频率的影响作了一定的研究。
引入波传播方法解决了圆柱壳弹性支撑边界自由振动的求解问题。首先采用波传播方法研究了经典边界圆柱壳的振动特性，研究了使用相同边界梁单元波数求解圆柱壳振动波数的方法，并另辟蹊径直接通过控制方程求解波数，并应用到弹性边界条件，通过设定弹性支撑弹簧刚度为无穷大或者零来得到与经典边界相对应的结果，并与文献及有限元模型所得结果进行了对比，并研究了各个方向弹性支撑对圆柱壳振动特性的影响。
采用机械阻抗综合法研究梁及其支撑圆柱壳体的耦合结构振动特性。首先对机械阻抗的基本思路进行了说明，求解了基于机械阻抗综合法的结构在简谐激励下的响应。其次，推导得到了多跨梁及简支边界下圆柱壳体在点激励下频域内的阻抗表达形式，求解计算了在简谐激励下的谐响应，并与有限元模型所得结果进行了对比，结果吻合良好。对双层梁、梁-壳耦合结构进行了分析，研究了质量点质量及悬臂长度对质量点-梁-壳耦合结构响应的影响。
设计了实验台架，定量分析弹性挠曲对轴系回旋振动的影响规律。通过推进器与整个轴系的质量比相似设计、推进器当量半径与轴系主要半径的惯量比相似设计、以及轴系长度与轴系主要半径的长径比相似设计，最后设置轴承位置来实现轴系的动力学相似性，使实验台架最大限度的体现船舶轴系的特性。通过精心的垂直及水平方向位置调心保证挠曲前的轴系状态，之后通过调整艉轴承的倾斜量分析轴系产生弹性挠曲之后其回旋振动响应特性的变化，并定量研究弹性挠曲对轴系回旋振动的影响规律。
The propulsion shafting is one of the most important components of a ship’s powerdevice, and the noise and vibration prediction and control of the shafting and its coupledstructure with cylindrical shell is a very important research project in ship engineering. Forthe gyroscopic moment because of the rotation cantilever propeller and the effectivesupported stiffness and location of the stern bearing, developing the precise dynamic models,and studying the influence to the shafting and the coupled structure will be of great theoreticalsignificance and applied value. Surrounding the gyroscopic effect of the propulsion shafting,the dynamic characteristics of the stern bearing, and the influence of the cantilever propellerto the shafting, the coupled vibration problem of the shafting and cylindrical shell, thedetailed research work has been carried out in this thesis as follows:
The dynamic analysis model of the shafting’s gyroscopic vibration is established usingthe Fourier Series Expanded Method based on the Euler-Bernoulli beam theory. Thecorrectness of the application of the method is validated by comparing with the resultsobtained by published papers and FEA model. Through the introduction of Dirac deltafunction, considering the propeller as a lumped mass and analyzing its gyroscopic moment,the governing differential equation of the shafting with lumped mass and gyroscopic effect isobtained and a standard matrix eigenvalue problem for gyroscopic vibration can be developedusing the orthogonal of trigonometric functions. The gyroscopic dynamic response in timedomain under harmonic or arbitrary load is obtained using the Newmark numericalstep-by-step integration method. The accurate and efficient of the method is validated bycomparing with the results obtained by published papers, National Standard and FEA model.The method which is based on theory of continuous system can solve natural characteristicsand forced responses of gyroscopic vibration of shafting with elastic-surpport boundaryconditions, various section areas and multi-span.
Based on the Reynolds equation, the calculating approach of eight dynamic parametersabout stiffness and damp of infinite long and short bearing is deduced through the impedancerelationship using the oil film thickness of misaligned journal bearing. Combining theimpedance relationships of the infinite long and short bearing, the calculating formulas ofstiffness and damp of finite long bearing are derived using an experiential impedancerelationship for the finite long bearing. And the effective supported stiffness and location areobtained. The correctness of the method is validated by comparing with the results ofmaximal oil film pressure obtained by published papers with difference misalignments. Andthe influences of the misalignment to the lubricating characteristics, the effective supported stiffness and location of the journal bearing and to the gyroscopic vibration of shafting areanalyzed.
The in-plane free vibration model of multi-span curved beam system with elasticllysupported and connected boundary condition is established using the two-dimentional FourierSeries Expanded Method. According to the enegy principle, the in-plane vibrationdisplacements along radial and tangent directions are both expressed as the superposition of adouble Fourier cosine series and four supplementary functions in the form of the product of apolynomial fuction. The use of these supplementary functions is to overcome thediscontinuity problems of the elastic boundary conditions. And a standard matrix eigenvalueproblem about the unknow displacement amplitude coefficients is derived through solving theHamilton’s equation using the Rayleigh-Ritz Method and the natural frequencies and modeshapes of multi-span curved beams can be solved. At the same time, the contribution of otherdistributed and lumped parameters to the whole mass and stiffness matrix is considered. Theresults of two-span curved beams with free, simple supported, clamped and elasticllysupported boundary conditions are obtained and compared with the results got from the FEMmodel to validate the correctness of the present method. And the effect of curvature andconnecting stiffnesses between two-span curved beams on the vibration frequencies isdescribed.
The vibration of cylindrical shell with elasticlly supported boundary conditions is solvedthrough introducing the wave propagation method. The method is applied to the cylindricalshell with traditional boundary conditions. And the wave number along the axial direction issolved using the governing equation directly rather than solved from beam structures havingthe same boundary conditions with the shell used by the most papers. Applying to the elasticboundary conditions, the accurate of the method is validated by comparing with the resultsobtained by published papers and FEA model through considering the classical homogeneousboundary conditions as the special cases when the stiffness for each set of springs is equal toeither infinity or zero. The influence of the elasticlly supported along various directions to thevibration of the cylindrical shell is studied.
The vibration characteristic of coupled structure of the beam and its supportedcylindrical shell is worked out using the Mechanical Impedance Synthesis Method. The basicconcept the mechanical impedance are introduced firstly, and the methods used formulti-coordinate, coupled substructures under harmonic load are analyzed. The impedanceexpressions in frequency domain of the multi-span beam system with elastic boundaryconditions and the cylindrical shell with shear diaphragm-shear diaphragm (SD-SD) boundary conditions under harmonic load are obtained, and the results of the harmonic responses arevalidated accurate compared with the results calculated from FEA models. The approach isfirst used to determine the modal properties of a double-deck beam system consisting of twobeams coupled in parallel, which serves to both validate the substructure synthesis techniqueand demonstrate the versatility of the beam substructure in contrasting to its seeminglylimited face value. The vibrational responses of a coupled beam-cylindrical shell system arethen used to validate the current solution technique and to study the effects of modifying somemodel parameters.
Finally, an experimental system is designed to quantitative analyze the influence ofelastic deformation to gyroscopic vibration of the shafting. Through mass ratio of thepropeller to the whole shafting design, inertia ratio of the equivalent radius of propeller to themaster radius of shafting design and length-radius ratio of length to master radius of shaftingdesign, and the dynamic design setting the location of the bearing, the system could make ananalogy to the shafting. Various experimental measurement works are performed throughresizing the vertical location of the stern bearing in order to describe the influence of elasticdeformation to gyroscopic vibration of the shafting.
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