动压径向轴承形状泛函与优化设计理论研究
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摘要
动压径向轴承被广泛应用于各种旋转机械中,随着现代机械向大型化、高速化、高性能方向发展,动压径向轴承形状设计方法研究成为改善或提高机组性能非常重要的发展方向。然而目前国内外的轴承形状研究局限于固有型线的改进,修型和参数优化,在己有基本结构基础上,提出新的形状设计思想或基本结构形式的难度也越来越大。为了克服这一障碍,本文重点开展了对动压轴承形状泛函及优化设计理论的研究,主要研究内容和成果包括:
①分析轴承动压润滑原理,建立轴承产生动压最基本的控制方程--雷诺方程。为了求解控制方程,分析比较求解非线性偏微分方程的方法,采用基于MATLAB的PDETOOL工具箱的有限元方法求解控制方程,使原本复杂的求解过程简单化。计算结果证明基于MATLAB的PDETOOL工具箱可以简单、准确、有效地求解动压轴承优化模型中的控制方程。
②分析了轴承产生动压必须具有周期性,非定常性和正定性,提出能够满足条件的四种通用型线方程表示轴承型线:基于点、基于泰勒级数、基于贝塞尔函数和基于付里叶级数的通用型线方程。从表示的复杂性、准确性和计算的难易度对四种方法进行比较,证明了基于付里叶级数的通用型线方程最适合动压轴承形状优化。通过分析基于付里叶级数的通用型线对常用轴承型线的集成性,并以型线的连续性为前提条件,从基于付里叶级数的通用型线中推导出径向轴承的三种典型型线,验证了以付里叶级数为基础的通用型线不仅可以集成现有轴承型线,而且可以表征出不受固有型线限制的新型轴承型线。
③利用表征动压轴承型线函数特征的泛函集成形式表示动压轴承的膜厚方程建立形状控制方程,以轴承型线的泛函系数为变量,最大承载力为目标,用遗传算法进行形状优化,优化结果摆脱了原始形状的限制,并且不产生拟合误差,得到具有最大承载力大于现有文献报道的新型轴承型线。在形状优化中,通用膜厚方程的无穷级数应用逐步逼近法,用有限项级数值表示了无限项的最佳近似值。优化结果显示,当级数为2阶时出现收敛趋势,该结果进一步论证了具有一个动压收敛腔的轴承型线对应的轴承承载力最大。
④针对动压径向轴承的多目标要求,以最小化侧漏流速和最小化功耗为目标函数,以宽径比,粘度和通用轴承型线方程系数为设计变量,以最小油膜厚度和最小承载力为限制条件建立了动压径向轴承的多目标优化设计数学模型。采用NSGA-Ⅱ的多目标优化方法,避免了权重因子的确定,利用惩罚因子处理非线性约束条件,实现了动压轴承形状多目标优化,得到Pareto前沿解集。实例证明,多目标优化结果比单目标得到的结果更优,同时从物理本质上论证了优化后的设计变量与性能参数的正确关系。
⑤轴承动压润滑理论本身具有近似性,为了验证理论的正确性,进行了实验研究。分析了动压轴承的材料、加工方法和实验设备和实验方法等,根据得到的理论优化结果加工了实验样件,完成了对比实验,实验结果验证了动压径向轴承形状泛函与优化设计理论的可行性和正确性。
本论文的研究为设计出新的具有更佳性能的轴承形状提供了新的思路,提出了一种动压轴承型线设计的新理论和新方法,对动压轴承的设计具有重要的理论意义,同时对以动压润滑为工作原理的机械和其它利用微小间隙进行润滑的传动装置的廓线理论研究的深入发展也具有重要的推动作用。
There has always been a huge demand for efficient journal bearings in machines like steam turbines, centrifugal compressors, pumps, and motors. Ideally the bearing achieves good damping, durability, negligible friction, and zero wear in the smallest possible space. However, the starting point for most bearing shape development is still limited to classical geometrical bearing shapes. The radial clearance, the eccentricity ratio, and the attitude angle are all parameters that describe the location between the journal and bush for a given bearing shape. The shape optimized result is therefore limited to cylindrical or elliptical bearings. Using this approach the shape of the bush itself is not modified. In this work a fully generalized film thickness profile was used as the basis for shape optimization. An investigation into the geometrical theory required for hydrodynamic journal bearings was started, the mathematic model was built then optimization methods for shape optimization were used. The main work and results are as follows:
①The general Reynolds equation for the shape optimization model of hydrodynamic journal bearings was deduced from lubrication theory with some hypothesis. The Reynolds equation, which is the most importand government formula to express the hydrodynamic pressure generated by bearings, is a non-linear elliptic second order partial differential equation, which does not have a known analytical solution. In this work numerical solutions were found by employing the PDE Toolbox in MATLAB using a finite element method.
②Three necessary geometric conditions to generate hydrodynamic pressure were analyzed which were periodicity, non-constant and positive definiteness. Four kinds of general bearing profiles satisfying geometrical conditions were proposed which were the general profile based on points, the general profile based on Tylor series, the general profile based on Bazier functions and the general profile based on Fourier serises. Comparison was made under complexity and accuracy in expression and difficulty in computing, the general bearing profile based on a Fourier series was taken as the most suitable to shape optimization for hydrodynamic journal bearings. Some classical bearing profiles were shown to be a subset of this general bearing profile. Under continuity, three typicall profiles were deduced from the gernal profile based on a Fourier series, which testified that this kind of profiles can not only integrated current shapes but also can create new shapes for hydrodynamic bearings with optimal performance. The new approach based on the general film thickness breaked the limitation in current bearing shapes, which was convenient to analyze directly and adjust reversely. Consequently, arbitrary shapes of bearings can be obtained using optimizing method. The general profile and its parameterization make it possible to investigate the characteristics of bearing profiles. This could potentially be used to improve designs for better performance in industrial applications.
③A uniform shape optimization model was created using this general profile. The objective of the optimization had been to maximize the load capacity using the coefficients in the Fourier series as design variables. The lubrication equations had been solved based using the PDE Toolbox in MATLAB. This new method for optimizing the hydrodynamic journal bearings shapes outperformed that of previous publications. The optimization process was carried out using increasing orders of Fourier series to describe the bearing profile. As the order increased the load capacity increased. However the optimized result was almost fully achieved at a profile of order 2. This optimal shape was a single wedge bearing, which proved that a hydrodynamic journal bearing with the largest load capcity had a single wedge.
④The necessarity of multi-objective optimization was studied. This paper dealt with multi-objective shape optimization (minimization of side leakage rate, minimization of power loss) and design variables (lubricant viscosity, coefficients of general oil film thickness, slenderness ratio) related to the profile design of a hydrodynamic journal bearing. The aim was to show optimized shapes with better performance proposed here breaks the limitation in existing bearing profiles. The modified Non-Dominant Sort in Genetic Algorithm (NSGA-Ⅱ) can be used as the optimization tool with a penalty to those solutions that violate constraints for bearing shape optimization. The difficulty of selecting the weighting factor was removed in multi-objective journal bearing optimization problems. The Pareto optimal set obtained in this paper was a set of solutions close to the optimal solutions which were diverse enough to represent the true spread of optimal solutions. The multi-objective optimization result was better than single objective optimization one. The relation between designed variables and performance paremters was proved.
⑤Experiment was done required by product design and improvement to theory model. The material, processing method, experimental equipment and operation method were studied before the modal was manufactured. Correctless of the shape optimization based on the general profile was testified.
This new shape design method provides a new thought for designers in order to design some completely new shape for hydrodynamic bearings with optimal performance. It also plays an important role in mechanism working in hydrodynamic theory and profile theory of transmission devices which make use of micro-clearance to lubricate.
①分析轴承动压润滑原理,建立轴承产生动压最基本的控制方程--雷诺方程。为了求解控制方程,分析比较求解非线性偏微分方程的方法,采用基于MATLAB的PDETOOL工具箱的有限元方法求解控制方程,使原本复杂的求解过程简单化。计算结果证明基于MATLAB的PDETOOL工具箱可以简单、准确、有效地求解动压轴承优化模型中的控制方程。
②分析了轴承产生动压必须具有周期性,非定常性和正定性,提出能够满足条件的四种通用型线方程表示轴承型线:基于点、基于泰勒级数、基于贝塞尔函数和基于付里叶级数的通用型线方程。从表示的复杂性、准确性和计算的难易度对四种方法进行比较,证明了基于付里叶级数的通用型线方程最适合动压轴承形状优化。通过分析基于付里叶级数的通用型线对常用轴承型线的集成性,并以型线的连续性为前提条件,从基于付里叶级数的通用型线中推导出径向轴承的三种典型型线,验证了以付里叶级数为基础的通用型线不仅可以集成现有轴承型线,而且可以表征出不受固有型线限制的新型轴承型线。
③利用表征动压轴承型线函数特征的泛函集成形式表示动压轴承的膜厚方程建立形状控制方程,以轴承型线的泛函系数为变量,最大承载力为目标,用遗传算法进行形状优化,优化结果摆脱了原始形状的限制,并且不产生拟合误差,得到具有最大承载力大于现有文献报道的新型轴承型线。在形状优化中,通用膜厚方程的无穷级数应用逐步逼近法,用有限项级数值表示了无限项的最佳近似值。优化结果显示,当级数为2阶时出现收敛趋势,该结果进一步论证了具有一个动压收敛腔的轴承型线对应的轴承承载力最大。
④针对动压径向轴承的多目标要求,以最小化侧漏流速和最小化功耗为目标函数,以宽径比,粘度和通用轴承型线方程系数为设计变量,以最小油膜厚度和最小承载力为限制条件建立了动压径向轴承的多目标优化设计数学模型。采用NSGA-Ⅱ的多目标优化方法,避免了权重因子的确定,利用惩罚因子处理非线性约束条件,实现了动压轴承形状多目标优化,得到Pareto前沿解集。实例证明,多目标优化结果比单目标得到的结果更优,同时从物理本质上论证了优化后的设计变量与性能参数的正确关系。
⑤轴承动压润滑理论本身具有近似性,为了验证理论的正确性,进行了实验研究。分析了动压轴承的材料、加工方法和实验设备和实验方法等,根据得到的理论优化结果加工了实验样件,完成了对比实验,实验结果验证了动压径向轴承形状泛函与优化设计理论的可行性和正确性。
本论文的研究为设计出新的具有更佳性能的轴承形状提供了新的思路,提出了一种动压轴承型线设计的新理论和新方法,对动压轴承的设计具有重要的理论意义,同时对以动压润滑为工作原理的机械和其它利用微小间隙进行润滑的传动装置的廓线理论研究的深入发展也具有重要的推动作用。
There has always been a huge demand for efficient journal bearings in machines like steam turbines, centrifugal compressors, pumps, and motors. Ideally the bearing achieves good damping, durability, negligible friction, and zero wear in the smallest possible space. However, the starting point for most bearing shape development is still limited to classical geometrical bearing shapes. The radial clearance, the eccentricity ratio, and the attitude angle are all parameters that describe the location between the journal and bush for a given bearing shape. The shape optimized result is therefore limited to cylindrical or elliptical bearings. Using this approach the shape of the bush itself is not modified. In this work a fully generalized film thickness profile was used as the basis for shape optimization. An investigation into the geometrical theory required for hydrodynamic journal bearings was started, the mathematic model was built then optimization methods for shape optimization were used. The main work and results are as follows:
①The general Reynolds equation for the shape optimization model of hydrodynamic journal bearings was deduced from lubrication theory with some hypothesis. The Reynolds equation, which is the most importand government formula to express the hydrodynamic pressure generated by bearings, is a non-linear elliptic second order partial differential equation, which does not have a known analytical solution. In this work numerical solutions were found by employing the PDE Toolbox in MATLAB using a finite element method.
②Three necessary geometric conditions to generate hydrodynamic pressure were analyzed which were periodicity, non-constant and positive definiteness. Four kinds of general bearing profiles satisfying geometrical conditions were proposed which were the general profile based on points, the general profile based on Tylor series, the general profile based on Bazier functions and the general profile based on Fourier serises. Comparison was made under complexity and accuracy in expression and difficulty in computing, the general bearing profile based on a Fourier series was taken as the most suitable to shape optimization for hydrodynamic journal bearings. Some classical bearing profiles were shown to be a subset of this general bearing profile. Under continuity, three typicall profiles were deduced from the gernal profile based on a Fourier series, which testified that this kind of profiles can not only integrated current shapes but also can create new shapes for hydrodynamic bearings with optimal performance. The new approach based on the general film thickness breaked the limitation in current bearing shapes, which was convenient to analyze directly and adjust reversely. Consequently, arbitrary shapes of bearings can be obtained using optimizing method. The general profile and its parameterization make it possible to investigate the characteristics of bearing profiles. This could potentially be used to improve designs for better performance in industrial applications.
③A uniform shape optimization model was created using this general profile. The objective of the optimization had been to maximize the load capacity using the coefficients in the Fourier series as design variables. The lubrication equations had been solved based using the PDE Toolbox in MATLAB. This new method for optimizing the hydrodynamic journal bearings shapes outperformed that of previous publications. The optimization process was carried out using increasing orders of Fourier series to describe the bearing profile. As the order increased the load capacity increased. However the optimized result was almost fully achieved at a profile of order 2. This optimal shape was a single wedge bearing, which proved that a hydrodynamic journal bearing with the largest load capcity had a single wedge.
④The necessarity of multi-objective optimization was studied. This paper dealt with multi-objective shape optimization (minimization of side leakage rate, minimization of power loss) and design variables (lubricant viscosity, coefficients of general oil film thickness, slenderness ratio) related to the profile design of a hydrodynamic journal bearing. The aim was to show optimized shapes with better performance proposed here breaks the limitation in existing bearing profiles. The modified Non-Dominant Sort in Genetic Algorithm (NSGA-Ⅱ) can be used as the optimization tool with a penalty to those solutions that violate constraints for bearing shape optimization. The difficulty of selecting the weighting factor was removed in multi-objective journal bearing optimization problems. The Pareto optimal set obtained in this paper was a set of solutions close to the optimal solutions which were diverse enough to represent the true spread of optimal solutions. The multi-objective optimization result was better than single objective optimization one. The relation between designed variables and performance paremters was proved.
⑤Experiment was done required by product design and improvement to theory model. The material, processing method, experimental equipment and operation method were studied before the modal was manufactured. Correctless of the shape optimization based on the general profile was testified.
This new shape design method provides a new thought for designers in order to design some completely new shape for hydrodynamic bearings with optimal performance. It also plays an important role in mechanism working in hydrodynamic theory and profile theory of transmission devices which make use of micro-clearance to lubricate.
引文
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