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UBC Theses and Dissertations

Modeling of high-speed machine-tool spindle systems Cao, Yuzhong

Abstract

The successful application of high-speed machining technology is highly dependent on spindles operating free of chatter vibration without overloading the angular contact ball bearings. Unless avoided, vibration instability in the metal-cutting process leads to premature failure of the spindles, which have small shafts and bearing diameters and rotate at high speeds. This thesis presents a general structural model of the machine-tool spindle assembly, consisting of the rotating shaft, tool holder, angular contact ball bearings, stationary housing, and the machine-tool mounting. The model enables performance testing and analysis of the spindles by specifying cutting conditions, tool geometry, and the work-piece material in a virtual machining environment that avoids the present, costly trial design and prototyping of the spindles. A generalized finite element (FE) model of the spindle system is developed using Timeshenko’s beam theory. The effects of bearing preload, axial cutting forces, gyroscopic and centrifugal forces are considered in the formulation. Bearing stiffness, which varies as a function of the preload, cutting force, and spindle speed, is the fundamental nonlinearity in the dynamics of high-speed spindles. A nonlinear finite element model of the angular contact ball bearing is formulated by considering both centrifugal forces and gyroscopic moments from the rolling elements of the bearing. The elastic deformation of the shaft, rolling elements, housing, and the inner and outer races of the bearings are considered in the bearing model. The proposed model predicts stiffness changes in the bearing as a function of axial preloads. The model is experimentally verified on an instrumented spindle by comparing the predicted and measured static displacements in axial direction at the spindle nose as the preload is varied. The general model of the spindle predicts the mode shapes, frequency response function (FRF), vibrations along the spindle shaft, and contact loads on the bearings, which are essential for assessing the performance of the spindle during high-speed machining. While the spindle dynamics is considered to be approximately linear at rest because of the constant preload of the bearing, it becomes nonlinear in machining due to the dependency of the bearing stiffness on cutting forces and spindle speeds. The linear dynamic model is experimentally verified with impact modal tests on a spindle in free-free boundary conditions. The nonlinear model is verified by comparing the measured impact response of the spindle against predicted displacements under different preloads, which constitutes the fundamental nonlinearity in the spindles. It is shown that high preloads increase the bearing stiffness and the natural frequencies of the spindle system. However, excessive preloads also decrease modal damping, causing reduced dynamic stiffness of the spindles, which is not desirable in avoiding chatter vibrations. Chatter vibration can be avoided either by having high dynamic stiffness on all structural modes, or by creating a stability pocket at a desired spindle speed where the tooth-passing frequency resonates the mode, thereby allowing large depths of cut. An optimization method, which identifies the most optimal bearing spacing along the spindle shaft, is presented by using either the dynamic stiffness or chatter-free depth of cut at a desired speed range as the objective. The number of teeth on the cutter and the influence of the work-piece material properties are considered in optimizing the bearing locations. This thesis demonstrates that the dynamics of machine-tool structures may significantly affect the overall structural dynamics of spindle systems. A method of identifying the dynamics of the machine tool without the spindle is presented, and it is incorporated into the spindle model, which allows the simulation of the spindle systems under working conditions. The proposed numerical model allows designers to optimize spindles and test the performance of the spindles in a virtual metal-cutting environment before they are manufactured. The static and dynamic deflections along the cutter and 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 proposed mathematical models are experimentally proven on a research spindle instrumented with non-contact displacement sensors along its shaft. The virtual testing of a complete, industrial-sized spindle is shown to agree well with the experimental results of the same spindle tested in actual metal-cutting operations.

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