复杂环境下V带和齿轮传动设计优化理论与方法研究
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摘要
V带和齿轮传动是两类最基本和重要的机械传动形式,应用极为广泛。对它们进行全局优化设计,以及考虑它们在复杂不确定的实际应用背景下的优化设计,不仅具有重要的理论价值,而且具有广阔的应用前景。本文根据工程实际中这两类传动面临的复杂设计和工作环境特点,以现代优化理论与方法和现代不确定数学理论与方法为研究工具,研究了这两类传动设计的确定性全局优化方法,首次提出了处理一类多态不确定性非线性优化问题的方法并应用到V带传动设计优化中。主要研究工作如下：
1、分别建立了以V带承载能力最大化和V带疲劳寿命最长为目标的V带传动设计优化问题的数学模型,研究了这两类模型目标函数的凹性、单调性等特点,证明了它们的可行域是有界闭凸集。以此为基础,提出了寻求其全局最优设计方案的最优值线段算法。克服了已有研究成果的不足,即以往设计中要么依赖于没有收敛性理论保证的启发式算法,要么只能得到可行设计方案或局部最优设计方案。
2、在包含随机、模糊、区间等多种形态不确定参数的环境下,分别建立了V带传动承载能力最大化和疲劳寿命最大化问题的多态不确定非线性优化(PUNP)模型。在给定模糊隶属度和随机变量的置信水平的条件下,推导了这些不确定性优化模型的确定型等价式。
3、基于一种建立在描述区间不等式的最大和最小范围不等式定义基础上的区间规划理论,对具有多态不确定性的V带传动承载能力最大化和疲劳寿命最大化问题,分别提出了寻求其区间最优解的两步抽样算法,为实际工程设计提供了灵活的设计方案。
4、分别基于比较区间序的一种满意度理论和一种可能度理论,提出了求解多态不确定非线性优化问题的两种交互式抽样算法,案例研究表明这两种算法能够有效地得到稳健的V带传动设计优化方案。
5、针对一类载荷系数确定的直齿圆柱齿轮传动的体积优化设计问题,提出了三种全局优化方法。第一种方法通过变量变换,将原非线性模型转换成了含混合变量的线性规划问题。再通过设计该类规划的全局优化方法,使得原问题在连续变量空间和混合变量空间中均能求出所有全局最优解；第二种全局优化方法针对一类优化模型中增加了重合度系数的软齿面直齿轮传动设计,以第一种方法为基础,通过对模型非线性特点的分析,求得该问题在连续变量空间中的全局最优解；而第三种是基于离散变量枚举与模型单调性分析相结合的方法,无论优化模型考虑重合度系数与否,这种方法都能在混合变量空间中求得所有全局最优解。
V-belt drive and gear drive are two classes of fundamental and important mechanical transmissions with wide application in mechanism. It has outstanding significance in theory and application to find a globally optimal design scheme and make an optimal decision under a complex environment of uncertainty for the design of V-belt drive and gear drive. In this dissertation, on the basis of the analysis on the real-world complex design and working conditions, some deterministic global optimization methods for the two classes of mechanical transmissions are investigated and some polymorphic uncertain nonlinear programming methods are first proposed to solve the optimal design problem of V-belt under a complex environment of uncertainty from the viewpoint of the theory and the methods in modern optimization and uncertain mathematics. The main contribution in this dissertation is as follows.
1. Two deterministic optimal design models for maximizing the transmission capacity of V-belt drive and maximizing the fatigue life of V-belt are constructed, respectively. The concavity, the monotonicity and the global optimality condition are investigated for the objective functions in the constructed models. It is proved that the feasible regions in the models are bounded, closed and convex under some design conditions. On the basis of these analyses, a solution method, called an optimal segment algorithm, is developed to find the global maximizer of the optimization models. The proposed method remedied the shortage in the existent researches. For example, for the employed heuristic algorithms in the optimal drive design, there is no convergence theory to be established, while all of the classical local optimization algorithms can only obtain local optimal design scheme for a noncovex problem.
2. Polymorphic uncertain nonlinear programming (PUNP) models are respectively constructed to formulate the problem of maximizing the power transmission capacity and the fatigue life of V-belt under a complex environment of uncertainty which are involved with some uncertain parameters such as stochastic, fuzzy and interval ones,etc.. Then, some deterministic equivalent formulations for the PUNP models are obtained with given degrees of membership and levels of confidence.
3. Based on the interval programming theory where the maximal and minimal range inequalities are obtained for an interval inequality, a two-step based sampling algorithm is developed to find interval optimal solutions for the problems of maximizing the transmission capacity of V-belt drive and maximizing the fatigue life of V-belt under polymorphic uncertainties. Both of them provide flexible design schemes for the practical engineering problems.
4. Two types of sampling-based interactive algorithms are developed to find out robust optoimal solutions for the constructed nonlinear uncertain models, respectively. The concepts of satisfaction level and possibility degree are introduced respectively to describe an interval inequality. These algorithms are applied into case study in the optimal design of V-belt drive under polymorphic uncertain environment.
5. Three global optimization metthods are presented for the problem of minimizing the gear's volume for a type of spur gear drives with fixed load coefficient. In the first approach, the original nonlinear optimization model is converted into a linear program with mixed discrete variables by a suitable variable transformation. By means of developing a class of global optimization methods for linear programming with mixed discrete variables, all global optimal solutions are found for the original problem both in continuous variable space and in mixed variables space. Furthermore, a special global optimization approach is provided to optimize the spur gear drive with soft tooth flank in continuous variable space by taking into account modification of the contact ratio factors in addition. The third global optimization method is based on enumerating the discrete variables and analyzing the monotonicity for the original problem. The effectiveness of the algorithm does not depend upon taking an account on the modification of the contact ration factors in the optimization model.
1、分别建立了以V带承载能力最大化和V带疲劳寿命最长为目标的V带传动设计优化问题的数学模型,研究了这两类模型目标函数的凹性、单调性等特点,证明了它们的可行域是有界闭凸集。以此为基础,提出了寻求其全局最优设计方案的最优值线段算法。克服了已有研究成果的不足,即以往设计中要么依赖于没有收敛性理论保证的启发式算法,要么只能得到可行设计方案或局部最优设计方案。
2、在包含随机、模糊、区间等多种形态不确定参数的环境下,分别建立了V带传动承载能力最大化和疲劳寿命最大化问题的多态不确定非线性优化(PUNP)模型。在给定模糊隶属度和随机变量的置信水平的条件下,推导了这些不确定性优化模型的确定型等价式。
3、基于一种建立在描述区间不等式的最大和最小范围不等式定义基础上的区间规划理论,对具有多态不确定性的V带传动承载能力最大化和疲劳寿命最大化问题,分别提出了寻求其区间最优解的两步抽样算法,为实际工程设计提供了灵活的设计方案。
4、分别基于比较区间序的一种满意度理论和一种可能度理论,提出了求解多态不确定非线性优化问题的两种交互式抽样算法,案例研究表明这两种算法能够有效地得到稳健的V带传动设计优化方案。
5、针对一类载荷系数确定的直齿圆柱齿轮传动的体积优化设计问题,提出了三种全局优化方法。第一种方法通过变量变换,将原非线性模型转换成了含混合变量的线性规划问题。再通过设计该类规划的全局优化方法,使得原问题在连续变量空间和混合变量空间中均能求出所有全局最优解；第二种全局优化方法针对一类优化模型中增加了重合度系数的软齿面直齿轮传动设计,以第一种方法为基础,通过对模型非线性特点的分析,求得该问题在连续变量空间中的全局最优解；而第三种是基于离散变量枚举与模型单调性分析相结合的方法,无论优化模型考虑重合度系数与否,这种方法都能在混合变量空间中求得所有全局最优解。
V-belt drive and gear drive are two classes of fundamental and important mechanical transmissions with wide application in mechanism. It has outstanding significance in theory and application to find a globally optimal design scheme and make an optimal decision under a complex environment of uncertainty for the design of V-belt drive and gear drive. In this dissertation, on the basis of the analysis on the real-world complex design and working conditions, some deterministic global optimization methods for the two classes of mechanical transmissions are investigated and some polymorphic uncertain nonlinear programming methods are first proposed to solve the optimal design problem of V-belt under a complex environment of uncertainty from the viewpoint of the theory and the methods in modern optimization and uncertain mathematics. The main contribution in this dissertation is as follows.
1. Two deterministic optimal design models for maximizing the transmission capacity of V-belt drive and maximizing the fatigue life of V-belt are constructed, respectively. The concavity, the monotonicity and the global optimality condition are investigated for the objective functions in the constructed models. It is proved that the feasible regions in the models are bounded, closed and convex under some design conditions. On the basis of these analyses, a solution method, called an optimal segment algorithm, is developed to find the global maximizer of the optimization models. The proposed method remedied the shortage in the existent researches. For example, for the employed heuristic algorithms in the optimal drive design, there is no convergence theory to be established, while all of the classical local optimization algorithms can only obtain local optimal design scheme for a noncovex problem.
2. Polymorphic uncertain nonlinear programming (PUNP) models are respectively constructed to formulate the problem of maximizing the power transmission capacity and the fatigue life of V-belt under a complex environment of uncertainty which are involved with some uncertain parameters such as stochastic, fuzzy and interval ones,etc.. Then, some deterministic equivalent formulations for the PUNP models are obtained with given degrees of membership and levels of confidence.
3. Based on the interval programming theory where the maximal and minimal range inequalities are obtained for an interval inequality, a two-step based sampling algorithm is developed to find interval optimal solutions for the problems of maximizing the transmission capacity of V-belt drive and maximizing the fatigue life of V-belt under polymorphic uncertainties. Both of them provide flexible design schemes for the practical engineering problems.
4. Two types of sampling-based interactive algorithms are developed to find out robust optoimal solutions for the constructed nonlinear uncertain models, respectively. The concepts of satisfaction level and possibility degree are introduced respectively to describe an interval inequality. These algorithms are applied into case study in the optimal design of V-belt drive under polymorphic uncertain environment.
5. Three global optimization metthods are presented for the problem of minimizing the gear's volume for a type of spur gear drives with fixed load coefficient. In the first approach, the original nonlinear optimization model is converted into a linear program with mixed discrete variables by a suitable variable transformation. By means of developing a class of global optimization methods for linear programming with mixed discrete variables, all global optimal solutions are found for the original problem both in continuous variable space and in mixed variables space. Furthermore, a special global optimization approach is provided to optimize the spur gear drive with soft tooth flank in continuous variable space by taking into account modification of the contact ratio factors in addition. The third global optimization method is based on enumerating the discrete variables and analyzing the monotonicity for the original problem. The effectiveness of the algorithm does not depend upon taking an account on the modification of the contact ration factors in the optimization model.
引文
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