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附录：英文翻译无级变速链的优化介绍相对于传统的手动或自动变速箱的普通齿轮，无级变速传动是一个行之有效的替代品。他们可以在发动机最有效操作的范围内工作，在此范围内能够同时提高性能和燃油经济性。如图1所示，无级变速链的传动基本上由链条本身和两对锥盘完成，分别具有一个固定的和可移动的滑轮轴。滑轮由链耦合，而链是由链接在板块上的铰链组成的。锥盘和铰链末端的摩擦力转化为动力。液压执行器向可移动的滑轮施加力。因此，在滑轮链的齿轮传动比半径可以调整和持续改变。 如今无级变速器必须满足与普通齿轮高要求的竞争，因此无级变速器的某些动态性能的改善是巨大的。一个重要的方面是它可以通过减少轴承在滑轮处的振幅进而在给定频域取得更小的齿轮噪音。另一个目标是尽量减少链环节中的最大拉伸力，增加链条寿命。另外，齿轮需要有更高的效率。链条的这些改进是由最佳参数的计算实现的，例如长度和链接的刚性，或摇臂脚关节的几何形状。用于数值模拟的齿轮模型在第4章中进行了描述。对下一步的优化问题进行了表述和分析，以找到一个合适的解决方法。第6部分涵盖了优化过程与实际执行情况的说明。最后，通过对尽量减少无级变速器启动噪声问题的解决，讨论了所引进的优化环境的高度潜能。CVT的建模与仿真一个涵盖所有无级变速器相关效果的精密的动力齿轮是最优化齿轮的基础。CVT的建模为一个多体系统，如图2所示。它包含三个基本组件，滑轮组，链，和描述链和滑轮之间的连接的连接模型。每个滑轮设置有两个自由度：一个围绕滑轮轴转动的自由度，和一个滑轮移动的转换自由度。在这里，滑轮被认为是刚性的，模型的建立需要考虑弹性变形。驱动滑轮A是通过角速度驱动的，同时，外部扭矩M2作用于驱动滑轮B。传动比是通过力Fp,1和Fp,2作用于移动滑轮来调整的。在这项工作中，链的平面模型就足够了。为了考虑引起多角度激振的离散结构链条，每个单独的链接都注意到了。每个链接由一个无质量弹簧阻尼单元组成，代表连结板和连接两个相邻关节（刚度c，阻尼系数d）。一个质量为m的物体位于每个关节，是该摇杆引脚对质量以及对相邻板块上下半场各环节的质量组成。该仿真模型的最后部分指摇臂之间的针脚和滑轮的两端摩擦接触。忽视其自己的动力，对摇杆引脚对可以建模为一个单一的，无质量弹簧作用完全垂直的模型飞机。图3显示了螺栓模型和力量的接触面作用。为了推导接触力，对螺栓弹簧力FB进行量化是有必要的。这取决于长度lB和刚度cB,以及齿轮的表面距离s（如图4）：这些力的静态平衡垂直于模型飞机，提供了一个正常的力条件方程。考虑到公式（1）取决于链条链接和径向摩擦力FRr的最小配合。要确定作为正常的力函数的剩余摩擦力，一库仑的摩擦法连续近似（图4）用于其中，为在接触的相对速度矢量：在文献1和2中可以找到模型的具体解说。为进一步优化讨论的一个重要任务是对运动所产生的方程和数值模拟的属性。如上所述，该模型包括与打开和关闭在模拟过程中摩擦力的接触。此外，微分方程是刚性的因为代表的弹簧板连接刚度。公式（4）的间断式和刚度都造成高度的仿真倍数。制定和优化问题分析在本节中的数学优化问题是指定的，即当出现一个无级变速器进行优化时。在此基础上制定的数学算法的优化选择，这能够有效地解决问题。一个目标功能一个目标函数必须在给定优化参数p的依赖定义，即量化的优化目标。完整的解决方案的初始值问题，这是在动力系统（4）基础的一套，需要f的每一个评估。该数据被计算在一个等距网格，至于目标函数评价中使用的一些程序需要它，例如快速傅立叶变换。为了提供更灵活地选择最优化的目标，任意复杂的目标函数f必须为评估仿真数据所允许。如最小，最大，最小平均数甚至FFT（快速傅立叶变换）的操作均可使用，以及它们的组合。我们的目标是通过找到最优参数p调整为每个组件的上限和下限而尽量减少f的目标函数。这导致了约束的优化问题当寻找一个方便的优化算法，实际的问题必须根据以下两个方面考虑分析：首先是目标函数的性质，其次在参数空间的限制。就手头上的问题，约束是简单的约束限制，这可以用梯度投影方法处理，见文献3。事实上，困难在于目标函数。作为模拟耗时要为每个评估执行，所需的方法，必须依靠尽可能少的功能评价研究。因此，遗传算法是毫无疑问的。前调查表明，即使最优化问题有关，而简单的机械系统可以导致非光滑，有很多局部极小的目标函数，并可能取决于优化参数。一个机械问题可能的优化表面可能形如图5之一。这种最坏的情况已经可以预料的。另一个问题是来自一个事实，一个相当大的仿真模型在综合目标函数f的对数值积分过程无保留的输出。如前所述模拟任意的数据处理方法都必须接受。对于这样一个问题的优化算法IFFCO似乎非常合适。这是一个隐式过滤与绑定约束问题的方法实现。隐式过滤的基本思想是利用近似的梯度递增的序列差异，当迭代的收益减少时具有相当大的价值，而不是一开始非常小的增量。这提供了不卡在第一个最低位置，但当增量减少时，能够达到最低位置的机会。在文献3中对隐式过滤进行了详细的介绍。这个实现的一个重要特征就是，由中央差异梯度近似平行的手段为整个优化过程的计算时间显着减少。优化过程在本节中的无级变速器和优化算法的仿真模型相结合，构成了无级变速器所需的优化工具。对其实际执行方面进行讨论。首先，一些额外的组件和功能都必须提供。对于一个复杂的优化目标更顺利的实施，目标是开发函数库。它包含几个目标函数的原型（表1），可直接使用或混合使用，以评估目标函数。一流的目标函数计算目标函数值f（p）直接从仿真结果，从而维护之间的模拟和优化算法接口。模拟程序的一个扩展是必要的，即该参数p，其中叙述了经过优化的属性，可以从外部变化的模拟。进一步的功能返回所需的值，例如在某些力的组成部分，必须建立。数值模拟无级变速的结合，优化算法和目标函数库是优化无级变速器的有力工具。图6说明了该工具的组件的交互。主进程是优化算法。每当评价必要时它就调用无级变速器模拟并传递函数参数p。用这个参数设置一个完全仿真的CVT并执行操作。优化器所要求的目标函数值，是使用目标函数库从模拟结果中计算的。这个过程，直到迭代优化算法终止和最佳参数集被发现。优化结果该无级变速器推出优化工具的巨大潜力，表现出解决问题的一个例子。噪音是一个CVT的重大问题，应尽可能小。产生的噪音主要是由多边形效应，发生这家连锁店是进入滑轮，由于链的离散结构。减少这些效果与不同长度的做法或链的联系与优化摇杆链脚关节的几何形状。这里有一个同样长度的所有链接链进行了研究。一个最佳的链路长度，因此应计算的参数向量是齿轮的声学性能是根据滑轮轴承B处的力的振幅测量的。其目的是减少在频域的力峰，指的是多边形的频率和它的倍数。选择下面的目标函数是指在平面上是垂直于轴滑轮滑轮的支持力的两个组成部分。FFT（s）是用来作为缩写的变量s的快速傅里叶变换。f0和f1的上下频域，其中以上的力组成的幅度应尽量减少约束。目标函数f数值计算，包括两个步骤：首先，整个无级变速传动仿真用其中大部分的CPU时间得到的力，第二，在方程式（9）中自动进行目标函数分类的FFT和积分计算成本。对于所提出的优化的出发点是与链路长度为alink = 9:85毫米的PIV链。在这篇文章中传动比i= 1:0，滑轮A的角速度= 104:7 rad/s，外部扭矩M2 = 150 nm。用这个介绍的优化工具，目标函数降低到了50%。图7显示了链长度alink和目标函数f的相关性。很明显，开始成形浓度的优化准则的最优度不够。相对于以上图8中提及的初始浓度，链长度的最优模拟结果为alink = 9:93毫米。每个支持力的两个主要组成部分的振幅峰可减少到其中的一个的三分之一。通过改变0.08毫米的连接长度来显着改善齿轮的声学性能。结论优化无级变速器的工具已经推出。它包括了一个优化无级变速器算法和目标函数库详细的模拟模型。目的是为了找到一个合适的优化算法，对现在的优化问题进行了分析和隐式过滤原来是一个很好的选择。优化环境的能力是通过实例说明：齿轮噪声的排放减少了一链的结构优化。复杂的优化目标可以顺利实现，都取得了很好的效果。通过利用这个工具，整个无级变速器将得到优化。附录：原文6thWorld Congresses of Structural and Multidisciplinary OptimizationRio de Janeiro, 30 May - 03 June 2005, BrazilOptimisation of a CVT-ChainLutz Neumann, Heinz UlbrichDepartment of Mechanical Engineering, Technical University of MunichBoltzmannstrasse 15, 85478 Garching, GermanyEmail: neumannamm.mw.tum.de, ulbrichamm.mw.tum.de1. AbstractIn order to account for current and future requirements regarding the further improvement of the dynam-ical behaviour of a chain CVT (Continuous Variable Transmission) optimisation methods are substantial.The primary focus lies on the chain, which is the central part of the gear. In this work a concept to dealwith this challenging task is introduced, leading to an optimisation tool for CVTs. First of all, a de-tailed dynamical model of the CVT is necessary to estimate the performance of the gear using numericalsimulation techniques. The optimisation problem arising from the aim to optimise CVTs is identifiedin the next step. This includes the formulation of a target function quantifying the property which hasto be optimised. For its evaluation simulated data is used. The optimisation problem is analysed inorder to identify the requirements for a suitable optimisation algorithm. Implicit filtering will turn outto be the method of choice. For the implementation of complicated optimisation goals, a class of targetfunctions is introduced. Therewith a tool for optimising CVTs is developed, which basically consists ofthe numerical simulation, the optimisation algorithm and the class of target functions. Its capability isshown by reducing the noise emission of a CVT which is achieved by obtaining optimal values for certaingeometry parameters of the chain.2. Keywords: CVT, optimisation, multibody dynamics, multibody simulation3. IntroductionContinuously variable transmissions are a well-established alternative to common gears such as manualor conventional stepped automatic transmissions. They allow to drive in the most efficient operatingrange of the engine, which can enhance both performance and fuel economy.As illustrated in Fig. 1, a continuous variable chain drive basically consists of the chain itself andplaterocker pinrocker pin chainFigure 1: CVT-drive (left), rocker pin chain (right)two pairs of cone discs, each with a fixed and an axial moveable sheave. The pulleys are coupled bythe chain, which is composed of rocker pins connected by plates (Fig. 1). The power is transmitted byfrictional forces between the discs and the ends of the rocker pins. Hydraulic actuators apply forces ontothe moveable sheaves. Thus the radius of the chain in the pulleys and as a consequence the transmissionratio of the gear can be adjusted and changed continuously.1Nowadays CVTs have to satisfy high requirements to compete with common gears and therefore animprovement of certain dynamic properties of a CVT is substantial. An important aspect is a preferablysmall noise emission of the gear which can be achieved by a reduction of the force amplitudes in thebearings of the pulleys in a given frequency domain. Another goal is to minimise the maximum tensileforces in the chain links to increase the lifetime of the chain. Furthermore a higher efficiency of the gearis desired. These improvements are realised by calculating optimal parameters of the chain, which arefor example the length and the stiffness of the links or the geometry of the rocker pin joints.The model of the gear, which is used for the numerical simulation, is described in chapter 4. Next theoptimisation problem is stated and analysed to find an appropriate algorithm for its solution. Section 6covers a description of the optimisation process and its practical implementation. Finally, by solvingthe starting problem to minimise the noise emission of the CVT, the high potential of the introducedoptimisation environment is disscussed.4. Simulation Model of the CVTA detailed dynamical model of the gear that covers all relevant effects of the CVT is the basis forthe optimisation of the gear. The CVT is modelled as a multibody system which is shown in Fig. 2.It contains three essential components, the pulley sets, the chain and the contact model, describingthe connection between chain and the pulleys. Each of the pulley sets has two degrees of freedom: onePulley APulley BM2FP,1FP,2Figure 2: CVT-drive: Simulation modelrotational degree of freedom around the pulley axis and one translational degree of freedom of its movablesheave. Here the sheaves are assumed to be rigid, models which take into account elastic deformationsare described in Sedlmayr 2. Pulley A the driving pulley is kinematically excited by the angularvelocity whereas an external torque M2acts on the driven pulley B. The transmission ratio is adjustedby the forces FP,1and FP,2which act on the movable sheaves.In this work, a planar model of the chain is sufficient. In order to consider the discrete structureof the chain which causes the polygonal excitation, every single link is regarded. Each link consists ofa massless spring damper element, representing the link plates and connecting two neighbouring joints(stiffness c, damping coefficient d). A mass m is located in each joint and is composed of the mass ofthe rocker pin pair as well as the half mass of each of the adjacent link plates.The last component of the simulation model represents the frictional contact between the ends ofthe rocker pins and the sheaves. Neglecting its own dynamics, the pair of rocker pins can be modeledas one single, massless spring acting exclusively perpendicular to the model plane. Figure 3 shows themodel of the bolt and the forces acting in the contact plane. For the derivation of the contact forces itis necessary to quantify the spring force FBof the bolt. It depends on the length lBand stiffness cBas2xByBzBcBFBFBrFRrFRrFRtFRtFNFNFigure 3: Model of a boltwell as on the local distance of the surfaces s of the sheaves (Fig. 4):FB=?cB(lB s)s lB0s lB.(1)The static equilibrium of forces perpendicular to the model plane provides a conditional equation forthe normal force. Taking into account Eq. (1) it depends on the minimal coordinates of the accordingchain link and the contact force FRrin radial direction:FN= FRrtan +FBcos= FRrtan +cB(lB s)cos(2)To determine the remaining frictional forces as a function of the normal force, a continuous approximationof Coulombs friction law (Fig. 4) is used where g is the vector of the relative velocity in the contact:FR= FN g| g|; = 0(1 e| g| gh)(3)A detailed exposition of the model can be found in Srnik 1 and Sedlmayr 2.ssCoulombcontinuousapproximation| g|lBFB gh0Figure 4: Bolt force and friction characteristicImportant for the further discussion of the optimisation task are the properties of the resultingequations of motion and the numerical simulation. As mentioned above, the model includes contactswith friction that may open and close during the simulation. This results in nonlinear equations ofmotion with a discontinuous right hand side in the formM q = h(q, q,t).(4)Furthermore, the differential equations are stiffbecause of the stiffness of the springs representing thelink plates. Both the discontinuities and the stiffness of Eq. (4) cause high simulation times.35. Formulation and Analysis of the Optimisation ProblemIn this section the mathematical optimisation problem, that arises when a CVT is to be optimised,is specified. Based on this mathematical formulation an optimisation algorithm is selected, which iscapable to solve the problem efficiently. A goal functionf(p) = f(q(p), q(p)(5)has to be defined in dependence of given optimisation parameters p, that quantifies the optimisationtarget. The complete solution set(q(p), q(p) := (q(p,ti), q(p,ti),i = 0,.,k(6)of the initial value problem, which is based on the dynamical system (4), is needed for each evaluationof f. The data has to be computed on an equidistant grid, as some routines used for the evaluation ofthe target function require it, e.g. the fast fourier transformation. In order to provide more flexibilityto choose optimisation goals, arbitrarily complex target functions f must be allowed for evaluating thesimulation data. Operations like min, max, least squares or even a FFT (Fast Fourier Transformation)may be used, as well as combinations of them. The objective is to minimise the target function f byfinding optimal parameters p subject to upper and lower bounds for each component of p. This leadsto a constrained optimisation problemminpBf(q(p), q(p),B = p IRm: li pi uii = 1,.,m.(7)When looking for a convenient optimisation algorithm, the actual problem has to be analysed underconsideration of the following two aspects: firstly the properties of the objective function and secondlythe constraints in the parameter space. For the problem at hand, the constraints are simple bound con-straints, which can be handled with a gradient projection method, see Kelley 3. Indeed the difficultieslie in the objective function. As a time consuming simulation has to be performed for each evaluation,the desired method has to rely on as few function evaluations as possible. Therefore genetic algorithmsare out of question. Former investigations have shown, that even optimisation problems concerningrather simple mechanical systems can result in target functions that are nonsmooth, have many localminima and may depend on the optimisation parameters chaotically. A possible optimisation surfacefor a mechanical problem may be shaped like the one in Fig. 5.fp1p2Figure 5: Possible optimisation surfaceSuch a worst case has to be expected. Another problem arises from the fact, that a quite large sim-ulation model is implicitely integrated in the goal function f by the output of the numerical integration4process. As stated before arbitray methods processing the simulation data have to be accepted. Hence itcomes clear that it is not possible to calculate analytical gradients nor hessians. For such a problem thethe optimisation algorithm IFFCO (Choi, et. al. 4) seems to be very suitable. It is an implementationof the implicit filtering method for problems with bound contraints. The basic idea of implicit filteringis to utilise a sequence of difference increments for gradient approximation, starting with a rather bigvalue which is reduced while the iteration proceeds, instead of using very small increments right at thebeginning. This provides the opportunity not to get stuck in the first local minimum, but later when theincrement gets small, a local minimum can be reached. A comprehensive description of implicit filteringis provided in Kelley 3. An important feature of this implementation is, that the approximation ofgradients by central differences is parallelised, what means a significant reduction of computation timefor the whole optimisation process.6. Optimisation ProcessIn this section the simulation model of the CVT and the optimisation algorithm are combined to form thedesired optimisation tool for CVTs. Practical aspects for its implementation are to be discussed. Firstsome additional components and features have to be provided. For a more comfortable implementationof complicated optimisation goals, a library of target functions is developed. It contains several targetfunction prototypes (Tab. 1), which can be used directly or in combination to evaluate the objectivefunction.Table 1: Target function library.Target functionsmaxminfast fourier transformationleast squares.The class of target functions computes the goal function value f(p) directly from the simulationresults and thus maintains an interface between the simulation and the optimisation algorithm. Anextension of the simulation program is necessary, namely that the parameters p, which describe theproperty that is optimised, can be changed from outside of the simulation. Further the functionality toreturn the required values, e.g. forces in a certain component, has to be established. The combinationof the numerical simulation of the CVT, the optimisation algorithm and the library for target functionsresults in a powerful tool to optimise CVTs. Fig. 6 illustrates the interaction between the componentsof the tool. The master process is the optimisation algorithm. It calls the CVT simulation and passesf(p)IFFCOsimulation resultstarget function librarypCVT simulationFigure 6: Interaction of the components of the optimisation toolthe parameter p whenever a function evaluation is needed. With this parameter set a complete simula-5tion of the CVT is executed. The target function value, requested by the optimiser, is calculated fromthe simulation results by using the target function library. This process iterates until the optimisationalgorithm terminates and an optimal parameter set pis found.8. Optimisation ResultsThe high potential of the introduced tool for optimising CVTs is shown by solving an example problem.Noise emission is a significant problem of a CVT and it should be as small as possible. The noise ismainly generated by the polygonal effect, which occurs where the chain is entering the pulleys, due tothe discrete structure of the chain. Approaches to reduce these effects are chains with links with differentlengths or chains with an optimised rocker pin joint geometry. Here a chain with links having all thesame length is investigated. An optimal link length shall be calculated and therefore the parametervector isq = (alink).(8)A measure for the accoustical performance of the gear are the amplitudes of the forces in the bearingsof pulley B. The aim is to reduce the force peaks in the frequency domain, which refer to the polygonalfrequency and its multiples. The following objective function is chosenf =Xy,zf1Zf0FFT(F)d.(9)F, y,z refers to the two components of the support forces of the pulleys in the plane whichis perpendicular to the axes of the pulleys.FFT(s) is used as an abbreviation for the fast fouriertransformation of the variable s. f0and f1are the lower and upper bound of the frequency domainin which the integral over the amplitudes of the forces shall be minimised. The target function f isevaluated numerically and includes two steps: firstly, the simulation of the entire CVT drive to get theforces F, which costs most of CPU-time and secondly, the calculation of the FFT and integration inEq. (9), which is carried out automatically by the target function class.The starting point for the presented optimisation is a PIV chain with the link length alink= 9.85 mm.In the scope of this article only a special loadcase with the transmission ratio i = 1.0, the angular ve-locity = 104.7 rad/s of pulley A and the external torque M2= 150 Nm is considered. Using of thealinkmmtarget function f -startoptimum20004000600080009.69.810.0Figure 7: Influence of the link length alinkon the target function roduced optimisation tool, the target function was reduced to 50%. Figure 7 shows the correlationbetween the link length alinkand the value of the target function f. It is obvious, that the startingconfiguration was not optimal with respect to the optimisation criterion at hand. The simulation resultsfor the optimal link length alink= 9.93 mm and for the special loading case as mentioned abov