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湖南科技大學(xué)本科生畢業(yè)設(shè)計(jì)（論文） Virtual Design and Optimization of Machine Tool Spindles Y. Altintas(l), Y. Cao Manufacturing Automation Laboratory, Department of Mechanical Engineering University of British Columbia, Vancouver, Canada h t t p : / / w .mech. ubc.ca/-ma1 Abstract An 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, Vibration 1 INTRODUCTION High-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. 2 FINITE ELEMENT MODEL OF SPINDLE SYSTEMS Figure 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 system Figure 2: Finite element model for spindle bearing system The 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 effects The following discrete equations in matrix forms for the beam can be obtained using the finite element method: where [M] is the mass matrix, [M]c is the mass matrix used for computing the centrifugal forces,[G]is the gyroscopic matrix which is skew-symmetric, [K] is the stiffness matrix, [K]P 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 model The Hertzian contact theory is used to predict the bearing contact force and elastic ball displacements. Figure 3: Bearing mode The force acting on the bearing ring is: where , and , are contact displacements between bearing balls and rings; θi, and θo 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). 3 EXPERIMENTAL VERIFICATION The 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 nose 3.2 Effects of preload and speed on spindle dynamics The 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 speed 3.3 Prediction of FRF with the tool The 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 Loads The 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 OPTIMIZATION The 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 variables Figure 12: Stability lobes before and after optimization The 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 SUMMARY A 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 REFERENCES [1] 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 - 虛擬機(jī)床主軸的設(shè)計(jì)和優(yōu)化 英屬哥倫比亞大學(xué)機(jī)械工程系與加拿大溫哥華大學(xué)制造自動(dòng)化實(shí)驗(yàn)室 Y. Altintas(1), Y. 曹 h t t p : / / w .mech. ubc.ca/-mal 摘要 本文呈現(xiàn)了一個(gè)主軸, 工具架、工具和切削過(guò)程一體化的數(shù)字模型。主軸是在一個(gè)內(nèi)部開(kāi)發(fā)的有限元建模系統(tǒng)建模的。本文涵蓋了軸承預(yù)負(fù)荷，以及旋轉(zhuǎn)部件的轉(zhuǎn)速給陀螺和離心力所造成的影響。主軸上的每一點(diǎn)的軸承剛度、模態(tài)、頻率響應(yīng)函數(shù)可以預(yù)測(cè)出來(lái)。沿主軸軸的靜態(tài)和動(dòng)態(tài)變形量以及接觸力軸承可以用模擬預(yù)測(cè)切削力在身體上構(gòu)建和測(cè)試紡錘波。對(duì)軸承的間距進(jìn)行優(yōu)化可以使動(dòng)態(tài)剛度最大或最大震顫免費(fèi)深度削減速度所需的地區(qū)對(duì)于一個(gè)給定的刀具幾何形狀和工件材料。添加約束模型在機(jī)床主軸的安裝,以及定義本地彈簧和阻尼元素在主軸上的任何節(jié)點(diǎn)是可行的。模型驗(yàn)證實(shí)驗(yàn)。 關(guān)鍵詞：主軸、切割、振動(dòng) 1介紹 由于生產(chǎn)效率的增加，高速切削廣泛應(yīng)用于工業(yè)。然而, 高速運(yùn)轉(zhuǎn)的主軸要配置引起震顫的直徑較小的軸和軸承,除非該主軸是專為在理想的切削條件下運(yùn)行而設(shè)計(jì)的。震顫導(dǎo)致不良表面光潔度，使軸承超載，從而會(huì)縮短主軸的壽命[1]。在機(jī)械加工中，影響主軸的總體性能的因素有主軸的尺寸、剛度、預(yù)加載,各軸承的間距,刀具幾何形狀，工具架以及加工材料。建模研究的目的是模擬主軸的性能和優(yōu)化其尺寸,以達(dá)到最大程度的動(dòng)態(tài)剛度和增加材料去除率。 角接觸球軸承最常用在高速主軸中，由于其低摩擦性能和承受工作載荷的能力為軸向和徑向[2]。軸承的剛度取決于接觸角,反過(guò)來(lái)依賴于速度、接觸球和環(huán)之間的負(fù)載。瓊斯發(fā)展了撓度曲線分析的一般理論。用來(lái)分析軸承,包括高速運(yùn)轉(zhuǎn)的軸承滾子的離心和旋轉(zhuǎn)運(yùn)動(dòng)[3]，這些在本文中有使用。轉(zhuǎn)動(dòng)軸和固定箱體通常通過(guò)有限元建模技術(shù)來(lái)建模[4, 5]。過(guò)去的絕大多數(shù)研究沒(méi)有考慮軸承剛度的非線性行為。例如，納爾遜[6]采用的一種Timoshenko梁理論建立了系統(tǒng)矩陣分析轉(zhuǎn)子系統(tǒng)的動(dòng)力學(xué)，其受到轉(zhuǎn)動(dòng)慣量，陀螺力矩,剪切變形、軸承和軸向負(fù)荷的影響,但被建模為線性彈簧。正如Abele[7]提出的主軸結(jié)構(gòu)動(dòng)力在速度較高的情況下發(fā)生改變，這將影響穩(wěn)定容器的位置和形狀[8]。 本文提出一種通用有限元模型，它可以預(yù)測(cè)軸承的剛度,接觸力軸球承、固有頻率和振型、頻率響應(yīng)函數(shù)和在切削負(fù)載下為響應(yīng)次數(shù)計(jì)數(shù)。模型包括軸承預(yù)負(fù)荷、軸承和軸的旋轉(zhuǎn)帶來(lái)的影響。此后,本文組織如下文：本文展示了軸的和軸承的非線性有限元模型,考慮了軸承預(yù)負(fù)荷,陀螺和離心速度的影響。第三節(jié)實(shí)驗(yàn)驗(yàn)證了主軸模型。第四節(jié)提出了軸承間距優(yōu)化方法為獲得最大動(dòng)力剛度或最大振顫槽刀具。本文還包括對(duì)貢獻(xiàn)的總結(jié)。 2主軸系統(tǒng)的有限元模型 圖1是裝備非接觸式位移傳感器的實(shí)驗(yàn)主軸。主軸有標(biāo)準(zhǔn)的CAT40刀架接口，最大轉(zhuǎn)速為15000轉(zhuǎn)速/分鐘,由15千瓦電機(jī)驅(qū)動(dòng)，它與軸和傳動(dòng)皮帶系統(tǒng)息息相關(guān)。主軸根據(jù)內(nèi)部開(kāi)發(fā)的轉(zhuǎn)為主軸設(shè)計(jì)的有限元建模系統(tǒng)開(kāi)模,如圖2所示。Timoshenko 梁理論是用來(lái)模擬心軸（錠桿）和外殼。黑色圓點(diǎn)代表節(jié)點(diǎn),每個(gè)節(jié)點(diǎn)有三個(gè)平移和兩個(gè)旋轉(zhuǎn)自由度。皮帶輪被建模為一個(gè)剛性圓盤(pán)。軸有兩個(gè)前軸承(BI和B2)串聯(lián)和三個(gè)后輪軸的(B3,B4和B5)。外環(huán)上的預(yù)加載應(yīng)用液壓A3定義為節(jié)點(diǎn),可以沿著軸住房節(jié)點(diǎn)A4和A5。部隊(duì)轉(zhuǎn)移到內(nèi)部環(huán)B3通過(guò)軸承球B5,然后通過(guò)內(nèi)圈B5主軸軸。部隊(duì)傳送到前線軸承內(nèi)圈的我,這也是固定在主軸,然后外環(huán)A2的住房,這是固定的住房。整個(gè)主軸平衡軸向方向的預(yù)加載。初始預(yù)加載在組裝和應(yīng)用可以通過(guò)液壓調(diào)節(jié)單元。工具被認(rèn)為是剛性連接的的工具架固定在主軸軸平移和旋轉(zhuǎn)通過(guò)彈簧剛度。根據(jù)機(jī)床的剛度,可以嚴(yán)格固定軸住房或彈性支承主軸頭。相關(guān)的內(nèi)環(huán)和外環(huán)的非線性方程組,得到軸承剛度通過(guò)求解方程的系統(tǒng)。 Figure 1: Spindle system 圖一：主軸系統(tǒng) Spindle nose：主軸端部 tool：刀具 tool-holder:刀架 housing：外罩 shaft：主軸 Hydraulic fluid：液壓流體 bearing：軸承 pulley：齒輪