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湖南科技大学本科生毕业设计（论文）Virtual Design and Optimization of Machine Tool SpindlesY. Altintas(l), Y. CaoManufacturing Automation Laboratory, Department of Mechanical Engineering University of British Columbia, Vancouver, Canadah t t p : / / w .mech. ubc.ca/-ma1AbstractAn integrated digital model of spindle, tool holder, tool and cutting process is presented. The spindle is modeled using an in-house developed Finite Element system. The preload on the bearings and the influence of gyroscopic and centrifugal forces from all rotating parts due to speed are considered. The bearing stiffness, mode shapes, Frequency Response Function at any point on the spindle can be predicted. The static and dynamic deflections along the spindle shaft as well as contact forces on the bearings can be predicted with simulated cutting forces before physically building and testing the spindles. The spacing of the bearings are optimized to achieve either maximum dynamics stiffness or maximum chatter free depth of cut at the desired speed region for a given cutter geometry and work-piece material. It is possible to add constraints to model mounting of the spindle on the machine tool, as well as defining local springs and damping elements at any nodal point on the spindle. The model is verified experimentally.Keywords:Spindle, Cutting, Vibration1INTRODUCTIONHigh-speed machining is widely used in industry due to increased manufacturing efficiency. However, high speed spindles have smaller shaft diameter and bearings which lead to chatter unless the spindle is designed to operate at the desired cutting conditions. Chatter leads to poor surface finish and overloads the bearings which shorten the spindle life I . The dimensions of the spindle shaft, and the stiffness, preload, and spacing of the bearings, tool geometry and holder, and work material affect the overall performance of the spindle during machining. The aim of the modeling study is to simulate the performance of the spindle and optimize its dimensions to achieve maximum dynamic stiffness and increased material removal rate.Angular contact ball bearings are most commonly used in high-speed spindles due to their low-friction properties and ability to withstand external loads in both axial and radial directions 2. The stiffness of the bearings is dependent on the contact angle, which in turn depends on the speed, contact loads between the balls and rings. Jones developed a general theory for the load-deflection analysis of bearings, including centrifugal and gyroscopic loading of the rolling elements under high-speed operation 3 which is used in this paper. The rotating shafts and stationary housing have been commonly modeled by Finite Element techniques 4, 5. Most past research did not consider the nonlinear behaviour of the bearing stiffness. For example, Nelson 6 employed Timoshenko beam theory to establish the system matrices for analyzing the dynamics of rotor systems with the effects of rotary inertia, gyroscopic moments, shear deformation, and axial load, but the bearings are modeled as linear springs. As presented by Abele 7, the structural dynamics of spindles change at high speeds, which affect the location and shape of stability pockets 8.This paper presents a general Finite Element model which can predict the stiffness of the bearings, contact forces on bearing balls, natural frequencies and mode shapes, frequency response functions and time history response under cutting loads. The model includes the bearing preload, rotating effects from both bearings and the spindle shaft. Henceforth, the paper is organized as follows.The nonlinear finite element model of the spindle shaft and bearings, which considers the bearing preload, gyroscopic and centrifugal speed affects, are presented. The Model of a spindle is experimentally verified in section 3. A Bearing spacing optimization method to obtain either maximum dynamics stiffness or maximum chatter free depth of cut for multiple flute cutters is presented in section 4.The paper is concluded with a summary of contributions.2FINITE ELEMENT MODEL OF SPINDLE SYSTEMSFigure 1 shows the experimental spindle instrumented with non-contact displacement sensors along its shaft. The spindle has a standard CAT 40 tool holder interface with maximum 15000 rev/min speed, and driven by a 15kW motor connected to the shaft with a pulley-belt system.Figure 1: Spindle systemFigure 2: Finite element model for spindle bearing systemThe spindle is modeled by an in house developed Finite Element system dedicated for spindles as shown in Figure2. The Timoshenko beam theory is used to model the spindle shaft and housing. The black dots represent nodes, where each node has three translational and two rotational degrees of freedom. The pulley is modeled as a rigid disk. The spindle has two front bearings (BI and B2) in tandem and three rear bearings (B3, B4 and B5) in tandem. The preload is applied hydraulically on the outer ring defined as node A3, which can move along the spindle housing with nodes A4 and A5. The forces are transferred to inner rings B3 to B5 through bearing balls, then to the spindle shaft through inner ring B5. The forces are transmitted to front bearings by inner ring B I , which is also fixed to the spindle shaft, then to the housing by outer ring A2, which is fixed to the housing. The whole spindle is self-balanced in the axial direction under the preload. An initial preload is applied during the assembly and can be adjusted through the hydraulic unit. The tool is assumed to be rigidly connected to the tool holder which is fixed to the spindle shaft through springs with stiffness in both translation and rotation. Depending on the rigidity of the machine tool, the spindle housing can be rigidly fixed or elastically supported on the spindle head. The inner and outer rings are related by nonlinear equations from which bearing stiffness is obtained by solving equations of the system.2.1 Equations of motion for the spindle shaft with rotating effectsThe following discrete equations in matrix forms for the beam can be obtained using the finite element method: where M is the mass matrix, Mc is the mass matrix used for computing the centrifugal forces,Gis the gyroscopic matrix which is skew-symmetric, K is the stiffness matrix, KP is the stiffness matrix due to the axial force P ,is the spindle speed, q is the displacement vector and F is the force vector that includes distributed and concentrated forces. The damping matrix is not included here and is estimated from experimentally identified modal damping ratios.2.2 Nonlinear bearing modelThe Hertzian contact theory is used to predict the bearing contact force and elastic ball displacements.Figure 3: Bearing modeThe force acting on the bearing ring is:where , and , are contact displacements between bearing balls and rings; i, and o are bearing contact angles;represent the displacement vectors for the nodes on the spindle shaft, inner ring, outer ring and spindle housing, respectively; are functions of ,respectively, depending on the configuration of bearing rings; Qi, and Q0, are contact forces; Fc, and Mg, are centrifugal force and gyroscopic moment depending on the spindle speed .The derivative of force with respect to the displacement is the bearing stiffness matrix:where KI, and KO are 5 by 5 matrices. The bearing stiffness matrix depends on the displacements which are in turn affected by the stiffness of bearings, hence the system dynamics is nonlinear. By assembling all matrices of spindle shaft/housing, disk and bearings, the following general non-linear dynamic equations for the spindlebearing system can be obtained:where M is the total mass matrix; C is the equivalent damping matrix including gyroscopic matrix; F(t) is the external force and R(x) is the internal force of the system which depends on the displacement x. The Newton-Raphson method is used to solve Eq.(4).3EXPERIMENTAL VERIFICATIONThe nonlinear Finite Element model of the spindles is experimentally verified using an instrumented spindle. Arrays of non-contact displacement sensors are installed in the spindle housing in two radial directions along the shaft, and two axial displacement sensors are mounted close to the spindle nose. First, the spindle is hung using elastic strings as a free-free system as shown in Figure 4. The frequency response functions under different preloads are measured by performing the impact modal tests. Figure 4: Experimental setup.3.1 Frequency response function (FRF)An impact force which is measured from a real impact blow test is applied at the spindle nose in the radial direction while the bearings are preloaded with a 500N force which changes the bearing contact angle as well as the bearing stiffness. Experimentally identified modal damping ratios 4% and 3% are used for the two dominate modes (506 Hz, 2685 Hz) respectively, and 3% is used for the rest of the modes. The FRF at the spindle nose is measured and also predicted by applying the same measured impact force to the nonlinear Finite Element model presented here. The FRF is calculated by using Fourier transforms of the simulated acceleration and input force. The measured and predicted FRF are shown in Figure 5, which is in good agreement. The two modes are most dominant at the spindle nose, which influence the machining stability most.The proposed model is able to predict the influence of preload accurately, which is quite important in designing and operating the spindle shafts at chatter vibration free spindle speeds.Figure 5: FRF in the radial direction at the spindle nose3.2 Effects of preload and speed on spindle dynamicsThe bearing stiffness increases with the increasing preload, but decreases as the spindle speed increases. Figure 6 shows the relation among radial bearing stiffness, preload and spindle speeds for bearing number 1. The speed effects are more obvious at lower preloads. Since the bearing stiffness is difficult to measure experimentally, the validity of the mathematical model is measured from the accuracy of FRF prediction which agrees quite well here with measurements. In general, the natural frequencies of all modes increase with the preload due to increased bearing stiffness, but decrease with the spindle speed due to decreasing stiffness caused by centrifugal forces. The lower modes are most affected by the spindle speed. A sample relationship between the speed, preload and the first natural frequency is shown in Figure 7.Figure 7: Natural frequency vs preload and spindle speed3.3 Prediction of FRF with the toolThe tool-spindle connection is the main source of flexibility in practice, and it is also difficult to model due to unknown contact stiffness and damping at the tool holder joints 9. As an example, two scenarios are tested: The elastic tool is rigidly connected, or connected via distributed springs at the spindle taper. The end mill has a diameter of 19.05 mm with a stick out of 55 mm, and is attached to the tool holder with a mechanical collet. The FRF at the tool tip for two interface connections is shown in Figure 8. The first mode is less affected, but the rigid tool connection leads to a higher second natural frequency than the spring connection. The first two modes match experimental results better with the spring connection since they are from the whole spindle system. However, the added spring to the tool interface brings a third mode which is not visible from the experiments. The results indicate that the tool-spindle interface mechanics and dynamics require more research since it has a strong influence on the dynamics of spindles during high speed machining.Figure 8: FRF with the tool in radial direction at the tool tip.3.4 Bearing Force Prediction under Cutting LoadsThe developed finite element system permits virtual cutting with the spindle. The milling forces whose peak value is about 1000N have been simulated for a 4 fluted end mill cutting AL7050 and are applied to the tool tip in three directions. A preload of 1500 N is first applied, followed by the cutting forces acting on the end mill after transient response due to the preload diminishes. The corresponding changes in the bearing stiffness as well as the contact forces experienced by the bearings are shown in Figures 9 and 10. The front bearings are most affected by the cutting forces which are periodic at tooth passing intervals. Since the axial force is opposite to the preload force when a right handed helix angle is used on the end mill, the front bearing stiffness decreases and the bearing contact forces increase under the cutting periodic load. If the axial force is larger than the bearing preload, the bearing stiffness can be lost momentarily. Due to periodicity of milling forces at tooth passing frequency, the bearing stiffness and contact forces change, which is a major nonlinearity in analyzing the dynamic behaviour of spindles during milling.Figure 9: Bearing stiffness.Figure 10: Contact forces on bearings.4 SPINDLE DESIGN OPTIMIZATIONThe spindles should be designed either to achieve maximum dynamic stiffness at all speeds for general operation, or remove maximum axial depth of cut at the specified speed with a designated cutter for a specific machining application. Although both criteria are implemented, the objective of cutting maximum material at the desired speed is presented here. The spindle modes are automatically tuned in such a way that chatter free pockets of stability is created at the desired spindle speed and depth of cut by optimizing the locations of bearings and the integral motor. The objective function is defined as follows:where Wi and(aclim)i are the weight and critical depth of cut for the cutter respectively, which is evaluated by the stability theory developed by Altintas 10; Nf is the total number of cutters with different flutes.A motorized spindle with the tool is shown in Figure 11 where six design variables are defined. The required cutting conditions are listed in Table 1.Figure 11: Initial design and design variablesFigure 12: Stability lobes before and after optimizationThe chatter stability lobes for a four fluted cutter computed from the three initial design trials and the final optimized design are shown in Figure 12. The desired spindle speed is 9,000 rpm, and the minimum depth of cut is 3 mm. The cutting is not stable for all three initial designs, but it becomes stable after automatic optimization. The physics behind the optimization is to locate the natural frequency of the spindle at the desired tooth passing frequency and satisfy the dynamic stiffness requirement, which is done by automatic adjusting of the bearing spacing ( X l , X 2 , . . , X 6 ) .The algorithm allows the optimization of multiple cutters with different flutes at the desired speeds as well.5 SUMMARYA general finite element method, which can predict the static and dynamic behavior of spindle systems, is presented. The spindle and housing are modeled by Timoshenko beam elements. The gyroscopic and centrifugal effects from both spindle shaft and bearings are included in constructing the dynamic model of the spindle system. The nonlinear stiffness matrix for the angular contact bearings are established through the analysis of load deflection of bearings. Hertzian theory is used to determine the relationship between the contact force and displacement of bearing balls and rings which are considered as elastic elements. The stiffness matrix of the bearing, the contact angle, preload and deflection of spindle shaft and housing are all coupled in the Finite Element model of the spindle assembly. The simulated results are compared favorably well against experimental measurements conducted on an instrumented, industrial size spindle. The simulation shows that the rotational speed of the spindle shaft has a bigger influence on the lower natural frequencies. The proposed optimization method is used to achieve maximum depth of cut or dynamic stiffness by tuning of the spindle modes through optimizing the locations of bearings and the motor for motorized spindles. The overall design and analysis model allows virtual testing of spindles under simulated cutting forces. The dynamic behaviour of the spindle, contact loads experienced by the bearings, the displacements of the shaft at any point can be predicted under simulated cutting conditions. The proposed model can be used to improve the design of spindles for targeted machining applications.6 REFERENCES1 Altintas, Y., Weck, M., 2004, Chatter Stability of Metal Cutting and Grinding, Annals of CIRP, vol. 53/2, pp. 619-642.2 Weck, M., Koch, A, 1993, Spindle Bearing Systems for High Speed Applications in Machine Tools, Annals of CIRP, vol. 42/1, pp. 445-448.3 Jones, A. B., 1960, A General Theory for Elastically Constrained Ball and Radial Roller Bearings Under Arbitrary Load and Speed Conditions, ASME J. Basic Eng., pp. 309-320.4 Jedrzejewski, J., Kowal, Z., Kwasny, W., Modrzycki, W., 2004, Hybrid Model of High Speed machining Centre Headstock, Annals of CIRP, vo1.53/1. pp. 285-288.5 Zeljkovic, M., Gatalo, R., 1999, Experimental and Computer Aided analysis of High-speed Spindle Assembly Behaviour, Annals of CIRP, vol. 48/1, pp. 325-329.6 Nelson, H. D., 1980, A finite rotating shaft element using Timoshenko beam theory, ASME J. Mech. Des V0l.102, pp.793-803.7 Abele, E., Fiedler, U., 2004, Creating Stability Lobe Diagrams during Milling, Annals of CIRP, vol. 53/1, pp. 309-312.8 Smith, S., Snyder, J., 2001, A Cutting Performance Based Template for Spindle Dynamics, Annals of CIRP, VOI. 50/1, p.259-262.9 Rivin, E., 2000, Tooling Structure - Interface Between Cutting Edge and Machine Tool, Annals of CIRP, VOI. VO1.49/2, pp. 591-643.10 Altintas Y., Budak E., 1995, Analytical Prediction of Stability Lobes in Milling, Annals of CIRP, 44/1, pp.357-366- 23 -虚拟机床主轴的设计和优化英属哥伦比亚大学机械工程系与加拿大温哥华大学制造自动化实验室Y. Altintas(1), Y. 曹h t t p : / / w .mech. ubc.ca/-mal摘要 本文呈现了一个主轴, 工具架、工具和切削过程一体化的数字模型。主轴是在一个内部开发的有限元建模系统建模的。本文涵盖了轴承预负荷，以及旋转部件的转速给陀螺和离心力所造成的影响。主轴上的每一点的轴承刚度、模态、频率响应函数可以预测出来。沿主轴轴的静态和动态变形量以及接触力轴承可以用模拟预测切削力在身体上构建和测试纺锤波。对轴承的间距进行优化可以使动态刚度最大或最大震颤免费深度削减速度所需的地区对于一个给定的刀具几何形状和工件材料。添加约束模型在机床主轴的安装,以及定义本地弹簧和阻尼元素在主轴上的任何节点是可行的。模型验证实验。关键词：主轴、切割、振动1介绍 由于生产效率的增加，高速切削广泛应用于工业。然而, 高速运转的主轴要配置引起震颤的直径较小的轴和轴承,除非该主轴是专为在理想的切削条件下运行而设计的。震颤导致不良表面光洁度，使轴承超载，从而会缩短主轴的寿命1。在机械加工中，影响主轴的总体性能的因素有主轴的尺寸、刚度、预加载,各轴承的间距,刀具几何形状，工具架以及加工材料。建模研究的目的是模拟主轴的性能和优化其尺寸,以达到最大程度的动态刚度和增加材料去除率。 角接触球轴承最常用在高速主轴中，由于其低摩擦性能和承受工作载荷的能力为轴向和径向2。轴承的刚度取决于接触角,反过来依赖于速度、接触球和环之间的负载。琼斯发展了挠度曲线分析的一般理论。用来分析轴承,包括高速运转的轴承滚子的离心和旋转运动3，这些在本文中有使用。转动轴和固定箱体通常通过有限元建模技术来建模4, 5。过去的绝大多数研究没有考虑轴承刚度的非线性行为。例如，纳尔逊6采用的一种Timoshenko梁理论建立了系统矩阵分析转子系统的动力学，其受到转动惯量，陀螺力矩,剪切变形、轴承和轴向负荷的影响,但被建模为线性弹簧。正如Abele7提出的主轴结构动力在速度较高的情况下发生改变，这将影响稳定容器的位置和形状8。 本文提出一种通用有限元模型，它可以预测轴承的刚度,接触力轴球承、固有频率和振型、频率响应函数和在切削负载下为响应次数计数。模型包括轴承预负荷、轴承和轴的旋转带来的影响。此后,本文组织如下文：本文展示了轴的和轴承的非线性有限元模型,考虑了轴承预负荷,陀螺和离心速度的影响。第三节实验验证了主轴模型。第四节提出了轴承间距优化方法为获得最大动力刚度或最大振颤槽刀具。本文还包括对贡献的总结。2主轴系统的有限元模型 图1是装备非接触式位移传感器的实验主轴。主轴有标准的CAT40刀架接口，最大转速为15000转速/分钟,由15千瓦电机驱动，它与轴和传动皮带系统息息相关。主轴根据内部开发的转为主轴设计的有限元建模系统开模,如图2所示。Timoshenko 梁理论是用来模拟心轴（锭杆）和外壳。黑色圆点代表节点,每个节点有三个平移和两个旋转自由度。皮带轮被建模为一个刚性圆盘。轴有两个前轴承(BI和B2)串联和三个后轮轴的(B3,B4和B5)。外环上的预加载应用液压A3定义为节点,可以沿着轴住房节点A4和A5。部队转移到内部环B3通过轴承球B5,然后通过内圈B5主轴轴。部队传送到前线轴承内圈的我,这也是固定在主轴,然后外环A2的住房,这是固定的住房。整个主轴平衡轴向方向的预加载。初始预加载在组装和应用可以通过液压调节单元。工具被认为是刚性连接的的工具架固定在主轴轴平移和旋转通过弹簧刚度。根据机床的刚度,可以严格固定轴住房或弹性支承主轴头。相关的内环和外环的非线性方程组,得到轴承刚度通过求解方程的系统。Figure 1: Spindle system图一：主轴系统Spindle nose：主轴端部 tool：刀具 tool-holder:刀架 housing：外罩 shaft：主轴 Hydraulic fluid：液压流体 bearing：轴承 pulley：齿轮Figure 2: Finite element model for spindle bearing system图二：主轴系统的有限元模型2.1 运动方程的主轴轴旋转的影响下面的离散方程的矩阵形式的梁可以获得使用有限元方法： 质量矩阵是M,Mc是质量矩阵用于计算离心力,G是陀螺矩阵是反对称的,刚度矩阵K,KP是由于轴向力刚度矩阵P,是主轴转速,q是位移矢量， F 是力向量,包括分布和集中力量。这里不包括阻尼矩阵和估计实验确定模态阻尼比。2.2 非线性轴承模型赫兹接触理论用于预测弹性球轴承接触力和位移。图三：轴承模型力作用于轴承套圈大小是:和是轴承球和环之间的接触位移;i,o轴承接触角,分别代表节点在主轴,内环,外环和轴外圈的位移向量;分别是的功能,取决于轴承的配置;Qi、Q0代表接触强度;Fc和Mg离心力和旋转力矩瞬时值取决于主轴转速。力对位移的导数的轴承刚度矩阵为:KI和K0是5X5矩阵。轴承刚度矩阵取决于位移,进而影响轴承的刚度,此系统动力学是非线性的。通过装配所有矩阵的主轴/外圈、磁盘和轴承。由一般主轴轴承系统非线性动力学方程可得到:M是总质量矩阵;C相当于阻尼矩阵包括旋转矩阵; F(t)是外力, R(x)是取决于系统位移 x 的内力。牛顿迭代法用于解答此方程。3 实验验证 主轴的非线性有限元模型是使用一个已被检测的主轴实验验证的。一系列的非接触式位移传感器沿着轴安装在主轴外圈在两个径向方向，两个轴向位移传感器安装接近主轴的突出部分。首先,主轴是悬挂使用弹力带作为完全自由系统如图4所示。通过模型试验的影响测定在不同预紧载力作用下的频率响应函数。 图片四：实验装配Accelerometer：加速度计 hammer：锤3.1 频率响应函数 冲击力由一个真正的冲击试验测量施加在主轴径向方向的力，轴承预装500N力改变轴承的接触角以及轴承刚度。实验分别确定了用于模态阻尼比4%和3%两种主导模式(506 Hz,2685 Hz)和3%用于其它模式。在主轴突出部分的频率响应函数的测量与预测，采用相同的测量冲击力的非线性有限元模型。频率响应函数的计算通过使用傅里叶变换的模拟输入加速度和力。测量和预测的频率响应函数如图5所示,这结论完全一致。轴突出部分的两种模式最主要影响加工稳定性。该模型能够准确预测预载力的影响,这在设计和操作的主轴自由振动和主轴速度是很重要的。图五：主轴突出部分径向的频率响应函数3.2预紧力和速度对主轴动力学的影响随着预紧力增加而增加轴承刚度,但随着主轴转速的增加而轴承刚度减少。图6显示了关于编号1的径向轴承刚度,预紧力和主轴速度之间的关系。在预紧力较小时，速度的影响更明显。由于轴承刚度是很难实验测量,数学模型的有效性从频率响应函数的预测中准确测量，这种测量方法在这里非常的好。一般来说,所有模型的自然频率随着预紧力的增加而增加，这是由于轴承刚度的增加,但随着主轴转速降低而减小，这是由于离心力引起的刚度减小。较低能量的模型是最易受到主轴转速的影响。如图7所示为一模型的速度,预紧力和第一固有频率之间的关系。图六：预紧力和主轴速度对轴承刚度的影响图七：固有频率与预紧力和主轴转速的对比3.3 频率响应函数预测的工具 机床主轴的连接