8字形变桩距无碳小车及其转向机构设计10张CAD图
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附录 1中文翻译高速钻床的动力学分析摘要通常情况下，术语“高速钻床”就是指具有较高切削速率的钻床。高速钻床(HSDM)也是指具有非常快的和正确的点到点运动的钻床。新的 HSDM 是由带有两个直线电动机的平面并联机构组成。本文主要就是对并联机器(PKM)的动力学分析。运动合成是为了介绍一种新方法，它能够完善钻孔操作和点到点定位的准确性。动态合成旨在减少因使用弹簧机械时 PKM 的输入功率。关键词: 并联运动机床; 高速钻床; 动力学的合成1.介绍在最近的几年里，研究所和工业协会介绍了各式各样的 PKM。其中大部分（但不是所有）,以众所周知的斯图尔特月台1为基础结构。这一做法的好处是高公称的负载重量比，良好的位置精度和结构刚性2。斯图尔特式 PKM 的主要缺点是相对小的工作空间和相对慢的操作速度 3,4。机床刀具的工作空间是指刀尖能够移动和切削材料所需要的容积。平面的斯图尔特月台的设计在5中被提到，像是对无 CNC 机器作翻新改进的方法需要塑料的铸模机制一样。PKM5 的设计允许可以调整几何学已经被规定了的最佳的再配置的任何路径。 一般的，改变一根或较多连杆的长度是以 PKM 受约束的顺序来做几何学的调整。在机床设计中，“定长度连杆”的 PKM 应用比“不定长度连杆”的共同点要少的多。一个优秀“定长度连杆”型的机器例子被显示在6。Renault- Automation Comau 已经建造叫做“Urane SX”的机器。在此 HSDM 被描述成是一个采用“定长度连杆”组成的并联机械装置。钻床操作在文学7中被很好的介绍了。汽车工业中，一项关于高速钻孔的操作的广泛的实验研究在8中被报告。数据从数百个钻床控制实验上收集起来，是为了具体指定钻床质量所必须的参数。理想的钻床运动和制造高质量钻床的指导方针通过理论和实验的研究被呈现在9中。在被建议的 PKM 综合中，我们遵循9中的结论。新推出的 PKM 的详细机械结构在10,11被介绍，机器的大致结构显示在图 1 中；它有很大的工作空间，点到点的高速运动和非常高的钻速。并联的机械装置提供给了 Y 和 Z 轴的动作，X 轴动作是由工作台提供的。为了达成高速的运转，用了两个线性马达来驱驶机械装置和用一个高速的主轴来钻孔。这篇文章的目的就是描述新的运动学的和动力学合成的方法的发展，为了改良机器的运转。通过输入运动，规划钻井和点对点定位，机器的误差将会被减少，而且完成孔的质量能被极大的提高。通过增加一个弹簧机械要素到 PKM，输入动力就能被最小，以便机器的尺寸和能量损耗降低。数字模拟的正确查证和热交换率的方法呈现在这篇文章中。1. PKM 模型的运动学和动力学的运动方程式PKM 模型的概要线图在图 2 中被显示。由于机床刀具库的一致，Z 轴是沿着工具运动的方向的。PKM 模型有部分 1 和部分 6 二个输入指示(二个线性电机)，和一个刀具的输出动作。在 PKM 模型应用中，定位和钻孔运动分别通过( y 轴动作相对点到点的定位)和 (z 轴动作相对钻孔)表示。刚体和柔性体的 PKM 模型运动方程式都被发展了。刚体方程式被用于合成输入钻床的动作计划和输入力量还原。柔性体方程式被用来在刀具点到点定位之后的剩余振动控制。1.1. 刚性连杆的 PKM 模型的运动方程式机械装置12的特点是使用了数字集成,刀具设备(含工作台，主轴和刀具 3 部份)。它的运动学方程式的发展依下列各项。刀具的变位是且其中 b 是点 B 和点 C 之间的距离,r 是连杆 AB 的长度(连杆 AB、CD 和 CE 的长度是相等的)。刀具的速度是其中刀具的加速度是其中PKM 模型的动力学方程式的发展如方程（7）所示，使用了拉格朗日的第二个类型的方程式13。其中 t 是系统的总动能； 和是总坐标值和速度值;是总力对应到的的值。k 是坐标系中总的独立数目。在这里,k=2，q1= y1 和 q2=y6，引出之后， 公式(7)可被表达成其中 n 是移动连杆的数目；是连杆 i 的大量惯性矩；是连杆 i的质量中心坐标； 是 PKM 模型中连杆 i 的旋转角。总力 的值通过（9）决定其中 V 是势能， 是没有势能的力。为了对 PKM 模型的钻孔操作，我们有其中是切削力, F1 和 F6 是线性马达在 PKM 上输入的力。情绪商数。公式(1)到公式(10)构成了刚性连杆 PKM 模型的运动学和动力学方程式。1.2. 柔性连杆的 PKM 模型的动作方程式顺从的机械装置的动微分方程式能用有限的机械要素方法和以下的公式得到其中M、C和K分别是系统质量，阻尼和刚性母体；D是在全球同等坐标系中的每个机械要素平移和旋转变形表现的总坐标值；R是总外力值，与D 保持一致；n 是坐标的总数目值(机械装置的柔性自由度)。在我们的 FEA 模型中，我们使用在图 3 中被显示的机械要素结构，其中 EIe 是弯曲刚性(E 是材料的柔性系数,Ie 是惯性矩),是物质的密度,le 是机械要素的最初长度。是(x,y)坐标系统中表现的结点变位。机械要素的大众基地和刚性基地将会是 66 个对称的矩阵，能从动能和应变能中得到，表达在公式（12）和（13）中其中 t 是动能，U 是机械要素的应变能；是机械要素基本坐标系中线性的 123456 和角变形节。详细的推论能在14被发现。典型地，在有限的机械要素分析中，一个顺从的机械装置是被离散成许多个机械要素的。每个机械要素与一个质量和一个刚性母体有关。每个机械要素有它自己的基本坐标系。我们结合机械要素质量和所有机械要素的刚性矩阵运行坐标转换时，必须把机械要素的基本坐标系转换成世界坐标系，这就提供了系统质量M和刚性K矩阵。在一个顺从的系统中捕获阻尼特性不是这么顺利的。即使, 在许多应用中，阻尼可能很小，但是它能作用在系统安全性和动力的频率响应中，尤其在共振区域中,可能是重要的。阻尼基地C能被写做一种质量和刚性矩阵15的线性结合，构成比例阻尼C如下式表达所示其中和是二个通常由实验决定的正系数。一个表现阻尼基地的交互方法16 表达成C如下机械要素C被定义为,其中, 和是K和M的机械要素, 是材料的阻尼比。机械要素结构中的总力被定义为其中 和 是 的外力和力矩，包括在 上动作的机械要素的惯性力和力矩,m 是在机械要素上动作的外力数目。机械要素的总力,组合构成了系统总力R。系统动作的第二次序普通微分方程式，如公式(11), 用一个数字能直接被整合的方法,就像是Runge- Kutta 的方法那样。对于我们研究的 PKM,每个连杆被分离成 15 个机械要素结构。Matlab 和 ADAMS 软件都被用来规划和解决这些方程式。2. 为钻床输入动作计划假如我们知道钻床理想的动作功能。高质量钻床的关键是如何决定输入电动机动作以便刀具的理想动作能被了解。创建明白的输入动作功能时也为机器控制提供了必需的数据。依照研究在9中所做的,钻孔的过程能分为三个时期: 入口期,中间期和出口期。为了增加生产能力和钻孔的质量，许多操作限制，例如最小刀具的寿命限制，孔位置误差限制,退出毛边限制,钻头扭转破坏限制等等，一定要考虑而且要满意。在这些条件之下，刀具的补给速度在入口期应该是慢的，以减少孔位置的误差。刀具的速度在出口期也应该是慢的，以减少出口毛边。在中央期,刀具的钻速应该很快速并且保持持续。刀具在完成钻孔之后的退回应该被做的尽可能的快，以增加生产能力。基于这些考虑, 我们采取了公式（17）中得到的理想钻床和刀具的退回速度。其中是最大的钻孔速度，T1、T2 和 T3 是分别对应入口期，中间期和出口期的时间。vT2 是退回的最大速度。T4 、T5 和 T6 对应的分别是加速的，持续的速度,和缩回操作时减速的时间。是一个单一钻孔的周期。用一个数字为例,我们打算利用钻一个 25.4mm(1 在)深的孔，0.3s 用来钻孔，0.1s 用来刀具退回。设定 T1=T3=0.06s,T4=T6=0.03s。在这些条件下， 。图 4 显示了理想刀具运动的图解式。如果 PKM 中连杆长度 r=500mm，在钻孔出发点时的角=53 ，与理想刀具动作相关的对应输入电动机的速度显示在图 5 中。一般 的，曲线装配方法能用来产生输入运动的函数，但是依照图 5 中显示的曲线形状，我们创建的线性马达速度函数详尽的显示在公式(18)中其中。当按公式（18）计画速度曲线时,没有不同的曲线能被发现，通过图（5）中显示的曲线。公式(18)由四个旋轮线的函数和两个线性函数共六个函数组成。假如我们像公式（18）中描述的那样控制两个线性电动机就会有相同的动作,那么刀具钻孔和退回的速度将几乎是与在图 4 中显示的相同。 在理想的和真正的刀具速度之间的绝对误差在图 6 中被显示,图中最大的误差不足 8mm/s,相对误差不足 1.5%，在钻孔的开始和结束的位置,误差是等于零的。这些小的绝对和相对的误差说明了输入动作的产生并且容易接受。这些已知的函数能非常简单被整合进 PKM 的控制运算法则里。3. 输入点到点的定位动作计划为了在整个的钻孔过程中达到快速的和正确的定位运动,应该适当地计划输入动作，以便刀具尖端的剩余振动能被最小化。照惯例加速度运动函数在机床中能被普遍用来驱动轴的运动。虽然这种动作函数很容易被控制, 但是由于它在加速度中的突然变化可能引起系统的柔性振动。举个早先使用相同的 PKM 例子来说。 一个 FEA 模型是通过有机械要素结构的 ADMAS 建造起来的。定位动作是 Y 轴的动作, 也就是在同一方向上通过两个线性电动机的运动实现的。假如在二个孔之间的定位距离是 75mm,等加速度是 3g(接近 30m/s)。等加速度和减速度的线性电动机的输入动作在图 7 中被显示,其中最大的速度是1500mm/s，定位时间为 0.1s。 假定材料的阻尼率为 0.01,则刀具尖端的剩余振动显示在图 8 中。为了要减少剩余振动和定位动作的平稳,建了一个输入动作的六次多元函数如(19)所示其中系数 Ci 必须是由刀具尖端的最小剩余振动决定的设计变数。选择接口条件为 ， 时，其中是点到点的定位时间，就产生了最初六个系数如下：合乎逻辑地,设立最佳目的如下其中 C6 是独立的设计变数，是刀具尖端在点到点定位之后的剩余振动的最大变动。设定并从 C6=0 开始计算，最佳导致 C6=-10mm/s。因此。可以看见最佳化计算使得变数 C6 的设计到了极限。如果给 c6 深层的释放极限,那么目的将会在价值中连续减少,但是输入动作的加速度的最大价值将会变成太大。最佳化后的最佳输入动作在图 9 中被显示。对应的刀具尖端的剩余振动在图 10 中被显示。比较图 8 和图 10，可以看到，在最佳化之后，振幅和刀具尖端的剩余振动被减少到了 30 次。较小的剩余振动将会对增加定位精度非常有用。这里应当注意，只有柔性连杆被包含在上述的计算之中。剩余振动在最佳化后将会仍然非常小，如果柔度是来自其他的来源，如压力和驱动系统，会比在图 10 中显示的结果高的 10 倍。4. 通过增加弹簧机械要素减少输入动力减少输入动力是机床刀具设计中的众多考虑之一。对于我们研究的 PKM,两个线性马达是使 PKM 模型做钻孔运动和定位运动的输入设备。在选择一个线性马达时要考虑的一个因数就是它需要的最大动力。PKM 模型的输入动力是由输入力乘以二个线性的电动机输入速度决定的。省略接触处的磨擦, 输入力是通过平衡钻削力和连杆与主轴设备的惯性力决定的。增加一个能量储存的机械要素，例如加一个弹簧到 PKM 上，如果弹簧的刚性和最初的(自由的) 长度被适当地选择，或许能够减少输入动力。减小最大输入动力导致用比较小的线性电动机驱动 PKM 模型。这将会依次减少能量的损失和机床的结构尺寸。一个线性的弹簧可以被把加到二个连杆的中央如图 11(a)所示，或者在 B 点和 C 点加入两个减震弹簧如图 11（b）所示。我们将会像举例子一样讨论线性弹簧来说明设计程序。公式（10）中的总力有以下形式：其中和 k 是线性弹簧的初始长度和弹性模量。 线性马达的输入动力取决于为了要减少输入动力，我们依下列各项设定最佳数值:其中 v 是一个设计变数的矢量，包括弹簧长度和弹性模量。对于我们研究的 PKM 模型,大量的数值在表 1 中被列出。 设计变数的初始数值被设定为。设计变数的范围被设定为， 。PKM 模型是通过公式（18）描述的输入动作函数驱动的。经过数值(24)的最小化，最佳的弹簧参数和 k=14.99N/mm 被得到。有优化弹簧的线性电动机和没有优化弹簧的线性电动机的输入动力如图 12 所示,图中实线表示没有弹簧的输入动力，虚线表示用了优化弹簧的输入动力。从结果中可以看出，右边线性马达的最大输入动力从 122.37 降到了 70.43W，减少量达到了 42.45%。对于左边的线性马达,最大的输入动力从 114.44 降到了 62.72W，减少量达到了45.19%。通过增加一个弹簧机械要素来减少机器输入动力，实现热交换的方法被证实了。减震弹簧可能被用来减少惯性作用和弹簧附属件的尺寸。5. 结论本文展现了一个以平面的 PKM 模型为基础的新型高速钻床。介绍了获得 PKM 需要的动作函数的方法。这种方法适用于钻孔操作中点到点的运动和上下运动。结果已经显示，它能够通过优化多元的动作函数，从实质上减少刀尖的震动。对于 HSDM 中，在几微米范围内定位刀具时，减少剩余振动是具有决定意义的。通过把“好的-调谐的”最佳的弹簧加入到结构中，能够降低驱动线性电动机的输入动力。前面已经模拟演示了在需要输入的动力减少到 40%时，就实现了无弹簧的结构。 必需输入动力的减少，就可能允许选择较小的电动机，那么就会减少硬件和操作的成本。为了更好的了解 HSDM 的性能并且完成它的设计，就需要更深入的去研究。它将包含机器的误差分析，同时还有控制策略和系统的控制设计。6. 感激作者非常的感谢密西根大学的 NSF 工程研究中心对于机制系统改造的资金支持，以及在中心投资的股东们。附录 2英文附录：Kinematic and dynamic synthesis of a parallel kinematic high speed drilling machineAbstractTypically, the termhigh speed drilling is related to spindle capability of high cutting speeds. The suggested high speed drilling machine (HSDM) extends this term to include very fast and accurate point-to-point motions. The new HSDM is composed of a planar parallel mechanism with two linear motors as the inputs. The paper is focused on the kinematic and dynamic synthesis of this parallel kinematic machine (PKM). The kinematic synthesis introduces a new methodology of input motion planning for ideal drilling operation and accurate point-to-point positioning. The dynamic synthesis aims at reducing the input power of the PKM using a spring element.Keywords: Parallel kinematic machine; High speed drilling; Kinematic and dynamic synthesis1. IntroductionDuring the recent years, a large variety of PKMs were introduced by research institutes and by industries. Most, but not all, of these machines were based on the well-known Stewart platform 1 configuration. The advantages of these parallel structures are high nominal load to weight ratio, good positional accuracy and a rigid structure 2. The main disadvantages of Stewart type PKMs are the small workspace relative to the overall size of the machine and relatively slow operation speed 3,4. Workspace of a machine tool is defined as the volume where the tip of the tool can move and cut material. The design of a planar Stewart platform was mentioned in 5as an affordable way of retrofitting non-CNC machines required for plastic moulds machining. The design of the PKM 5 allowed adjustable geometry that could have been optimally reconfigured for any prescribed path. Typically, changing the length of one or more links in a controlled sequence does the adjustment of PKM geometry.The application of the PKMs with constant-length links for the design of machine tools is less common than the type with varying-length links. An excellent example of a constant- length links type of machine is shown in 6. Renault-Automation Comau has built the machine named Urane SX. The HSDM described herein utilizes a parallel mechanism with constant-length links.Drilling operations are well introduced in the literature 7. An extensive experimental study of highspeed drilling operations for the automotive industry is reported in 8. Data was collected fromhundreds controlled drilling experiments in order to specify the parameters required for quality drilling. Ideal drilling motions and guidelines for performing high quality drilling were presented in 9 through theoretical and experimental studies. In the synthesis of the suggested PKM, we follow the suggestions in 9.The detailed mechanical structures of the proposed new PKM were introduced in 10,11. One possible configuration of the machine is shown in Fig. 1; it has large workspace, highspeed point-to-point motion and very high drilling speed. The parallel mechanism provides Y, and Z axes motions. The X axis motion is provided by the table.For achieving highspeed performance, two linear motors are used for drivingthe mechanism and a highspeed spindle is used for drilling. The purpose of this paper is to describe new kinematic and dynamic synthesis methods that are developed for improving the performance of the machine. Through input motion planning for drilling and point-to- point positioning, the machining error will be reduced and the quality of the finished holes can be greatly improved. By adding a well-tuned spring element to the PKM, the input power can be minimized so that the size the machine and the energy consumption can be reduced. Numerical simulations verify the correctness and effectiveness of the methods presented in this paper.2. Kinematic and dynamic equations of motion of the PKM moduleThe schematic diagram of the PKM module is shown in Fig. 2. In consistent with the machine tool conventions, the z-axis is along the direction of tool movement. The PKM module has two inputs (two linear motors) indicated as part 1 and part 6, and one output motion of the tool. The positioning and drilling motion of the PKM module in thisapplication is characterized by (y axis motion for point-to- point positioning) and (z axis motion for drilling).Motion equations for both rigid body and elastic body PKM module aredeveloped. The rigid body equations are used for the synthesis of input motion planning of drilling and input power reduction. Theelastic body equations are used for residual vibration control after point-to-point positioning of the tool.2.1. Equations of motion of the PKM module with rigid links Using complex-number representation of mechanisms 12, thekinematic equations of the tool unit (indicated as part 3 which includes the platform, the spindleand the tool) are developed as follows. The displacement of the tool isandwhere b is the distance between point B and point C, r is the length of link AB (the lengths of link AB, CD and CE are equal). The velocity of the tool iswhereThe acceleration of the tool iswhereThe dynamic equations of the PKM module are developed using Lagranges equation of the second kind 13 as shown in Eq. (7).where T is the total kinetic energy of the system; andare thegeneralized coordinates and velocities; is the generalized forcecorresponding to . k is the number of the independent generalized coordinates of the system. Here, k=2, q1=y1 and q2=y6. After derivation, Eq. (7) can be expressed aswhere n is the number of the moving links; are mass and mass moment of inertia of link i; are the coordinates of thecenter of mass of link i; hi is the rotation angle of link i in the PKM module. The generalized force can be determined bywhere V is the potential energy and Fi are the nonpotential forces. For the drilling operation of the PKM module, we havewhere Fcut is the cutting force, F1 and F6 are the input forces exerted on the PKM by the linear motors. Eqs. (1) to (10) form the kinematic and dynamic equations of the PKM module with rigid links.2.2. Equations of motion of the PKM module with elastic linksThe dynamic differential equations of a compliant mechanism can be derived using the finite element method and take the form ofwhere M, C and K are system mass, damping and stiffness matrix, respectively; D is the set of generalized coordinates representing the translation and rotation deformations at each element node in global coordinate system; R is the set of generalized external forces corresponding to D; n is the number of the generalized coordinates (elastic degrees of freedom of the mechanism). In our FEA model, we use frame element shown in Fig. 3 in which EIe is the bending stiffness (E is the modulus of elasticity of the material, Ie is the moment of inertia), q is the material density, le isthe original length of the element.are nodaldisplacements expressed in local coordinate system(x, y). The mass matrix and stiffness matrix for the frame element will be 66 symmetric matrices which can be derived fromthe kinetic energy and strain energy expressions as Eqs. (12) and (13)where T is the kinetic energy and U is the strain energy of the element; are the linear 1 2 3 4 5 6 and angular deformations of the node at the element local coordinatesystem. Detailed derivations can be found in 14. Typically, a compliant mechanism is discretized into many elements as in finite element analysis. Each element is associated with a mass and a stiffness matrix. Each element has its own local coordinate system. We combine the element mass and stiffness matrices of all elements and perform coordinate transformations necessary to transform the element local coordinate systemto global coordinate system. This gives the systemmass M and stiffness K matrices. Capturing thedamping characteristics in a compliant systemis not so straightforward. Even though, in many applications, damping may be small but its effect on the systemstability and dynamic response, especially in the resonance region, can be significant. The damping matrix C can be written as a linear combination of the mass and stiffness matrices 15 to form the proportional damping C which is expressed aswhere a and b are two positive coefficients which are usually determined by experiment. An alternate method 16 of representing the damping matrix is expressing CasThe element of C is defined as,where signKij=(Kij/|Kij|), Kij and Mij are the elements of K and M, is the damping ratio of the material.The generalized force in a frame element is defined aswhere Fj and Mj are the jth external force and moment including the inertia force and moment on the element acting at (xj ,yj), and m is the number of the externalforces acting on the element. The element generalized forces,are then combined to formthe systemgeneralized force R. The second order ordinary differential equations of motion of the system, Eq. (11), can be directlyintegrated with a numerical method such as Runge-Kutta method. Forthe PKM we studied, each link was discreted as 15 frame elements.Both Matlab and ADAMS software are used for programming and solving these equations.3. Input motion planning for drillingSuppose we know the ideal motion function of the drilling tool. How to determine the input motor motion so that the ideal tool motion can be realized is critical for high quality drillings. The created explicit input motion function also provides the necessary information for machine controls. According to the study done in 9,the drilling process can be divided into three phases: entrance phase, middle phase, and exit phase. In order to increase the productivity and quality of the drilling, many operation constraints such as minimum tool life constraint, hole location error constraint, exit burr constraint, drill torsion breakage constraint, etc. must be considered and satisfied. Under these conditions, the feed velocityof the tool should be slow at the entrance phase to reduce the hole location errors. The tool velocity should also be slow at the exit phase to reduce the exit burr. At the middle phase, the tool drilling velocity should be fast and kept constant. The retraction of the tool after finishing the drilling should be done as quickly as possible to increase the productivity. Based on these considerations, we assume that the ideal drilling and retracting velocities of the tool are given by Eq. (17).where vT1 is the maximum drilling velocity, T1, T2,and T3 are the times corresponding to the entrance phase, the middle phase and the exit phase. vT2 is the maximum retracting velocity. T4, T5, and T6 are corresponding to accelerating, constant velocity, anddecelerating times for retracting operation. is the cycle time for a single drilling. As a numerical example, suppose we drill a 25.4 mm (1 in) deep hole with Tc=0.4s, 0.3s for drilling, 0.1s for retracting. Set T1=T3 0.06s, T4=T6=0.03s. Under these con- ditions, vT1=106(mm/s), vT2=-363(mm/s). The graphical expression of the ideal tool motion is shown in Fig. 4. If the link length in PKM r=500 mm, the angle=53 at the starting point of drilling, the corresponding input motor velocity relative to the idealtool motion is shown in Fig. 5. Generally, the curve fitting method can be used to create the input motion function. But according to the shape of the curve shown in Fig. 5, we create the linear motor velocity function manually section by section as shown in Eq. (18).where vB=143.48mm/s, vC=165.77mm/s, vE=-557.36mm/s, vF=-499.44mm/s. When plotting the velocity curve with Eq. (18), no visual difference can be found with the curve shown in Fig. 5. Eq. (18) is composed of six parts with four cycloidal functions and two linear functions. If we control the two linear motors to have the same motion as described in Eq. (18), the drilling and retracting velocity of the tool will bealmost the same as shown in Fig. 4. The absolute errors between the ideal and real tool velocity are shown in Fig. 6, in which the maximum error is less than 8 mm/s, the relative error is less than 1.5%. At the start and the end positions of the drilling, theerrors are zero. These small absolute and relative errors illustrate the created input motion and are quite acceptable. The derived function is simple enough to be integrated into the control algorithmof the PKM.4. Input motion planning for point-to-point positioningIn order to achieve fast and accurate positioning operation in the whole drilling process, the input motion should be appropriately planned so that the residual vibration of the tool tip can be minimized. Conventionally the constant acceleration motion function is commonly used for driving the axes motions in machine tools.Although this kind of motion function is simple to be controlled, it may excite the elastic vibration of the systemdue to the sudden changes in acceleration. Take the same PKM module used in previous for example. A FEA model is built using ADMAS with frame elements.The positioning motion is the y-axis motion, which isrealized by the two linear motors moving in the same direction. Suppose the positioning distance between the two holes is 75mm, the constant acceleration is 3g(approximated as 30m/s here). The input motion of the linear motors with constant acceleration and deceleration is shown in Fig. 7, in which the maximum velocity is 1500 mm/s, the positioning time is 0.1 s. Assuming the material damping ratio as 0.01, the residual vibration of the tool tip is shown in Fig. 8. In order to reduce the residual vibration and make the positioning motion smoother, a six order polynomial input motion function is built as Eq. (19)where the coeffcients ci are the design variables which have to be determined by minimizing the residual vibration of the tool tip.Selecting the boundary conditions as that when t=0, sin=0, vin=0, ain=0;and when t=Tp, sin=h, vin=0, ain=0, where Tp is the point-to-point positioning time, the first six coeffcients are resulted:Logically, set the optimization objective aswhere c6 is the independent design variable;is the maximum fluctuation of residual vibrations of the tool tip after the point-to-point positioning. Set and start thecalculation from c6=0. The optimization results in c6=-10mm/s .Consequently, c5=7.510mm/s , c4 =-1.42510mm/s , c3=8.510mm/s , c2=c1=c0=0. It can be seen that the optimization calculation brought the design variable c6 to the boundary. If further loosing the limit for c6, the objective will continue reduce in value, but the maximum value of acceleration of the input motion will become too big. The optimal input motions after the optimization are shown in Fig. 9. The corresponding residual vibration of the tool tip is shown in Fig. 10. It is seen from comparing Fig. 8 and Fig. 10 that the amplitude and tool tip residual vibration was reduced by 30 times after optimization. Smaller residual vibration will be very useful for increasing the positioning accuracy. It should be mentioned that only link elasticity is included in above calculation. The residual vibration after optimization will still be very small if thecompliance from other sources such as bearings and drive systems caused it 10 times higher than the result shown in Fig. 10.5. Input power reduction by adding spring elementsReducing the input power is one of many considerations in machine tool design. For the PKM we studied, two linear motors are the input units which drive the PKM module to perform drilling and positioning operations. One factor to be considered in selecting a linear motoris its maximum required power. The input power of the PKM module is determined by the input forces multiplying the input velocities of the two linear motors. Omitting the friction in the joints, the input forces are determined frombalancing the drilling force and inertia forces of the links and the spindle unit. Adding an energy storage element such as a spring to the PKM may be possible to reduce the input power if the stiffness and the initial (free) length of the spring are selected properly.The reduction of the maximum input power results in smaller linear motors to drive the PKM module. This will in turn reduce the energy consumption and the size of the machine structure. A linear spring can be added in the middle of the two links as shown in Fig. 11(a). Or two torsional springs can be added at points B and C as shown in Fig. 11(b). The synthesis process is the same for the linear or torsional springs. We will take the linear spring as an example to illustrate the design process. The generalized force in Eq. (10) has the form ofwhere l0 and k are the initial length and the stiffness of the linear spring. The input power of the linear motors is determined byIn order to reduce the input power, we set the optimization objective as follows:where v is a vector of design variables including the length and the stiffness of thespring, . For the PKM module westudied, the mass properties are listed in Table 1. The initial value
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