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, 51. 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.

fluid

Figure 1: Spindle system

Tool and Tool-Holder

0 Node 9 Bearing 1 Rigidly Connected Movable

0-0 Spacer 5 Spring (for both Translation and Rotation)

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 Figure 2. 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 [MI is the mass matrix, [MI, is the mass matrix used for computing the centrifugal forces, [GI is the gyroscopic matrix which is skew-symmetric, [K] is the stiffness matrix, [KIP is the stiffness matrix due to the axial force P , L2 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 model

The force acting on the bearing ring is:

k=l

where 6, and 6, are contact displacements between bearing balls and rings; S, and So are bearing contact

angles; 6(') , 6(2) ,6(3) and 6(4) represent the displacement vectors for the nodes on the spindle shaft, inner ring, outer ring and spindle housing, respectively; 6(2) and 6(3) are functions of 6(') , 6(4) , respectively, depending on the configuration of bearing rings; Q, and Q, are contact

forces; F, and M, 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:

(3)

where K, 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 spindle- bearing system can be obtained:

(4)

where [MI 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 interface connections is shown in Figure 8. The first mode preload accurately, which is quite important in designing is less affected, but the rigid tool connection leads to a and operating the spindle shafts at chatter vibration free higher second natural frequency than the spring spindle speeds. 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 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

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.

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

Figure 10: Contact forces on bearings.

and depth of cut by optimizing the locations of bearings and the integral motor. The objective function is defined as follows:

N

Minimize: fob = - ~ W , ( u C b m ) , (i =1,2, ..., Nf ( 5 ) 1=1

where W, and (uCbm), are the weight and critical depth of

cut for the i fh cutter respectively, which is evaluated by the stability theory developed by Altintas [ lo] ; 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 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

Figure 12: Stability lobes before and after optimization

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.

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