Shock and Vibration

Volume 2016, Article ID 4390185, 9 pages

http://dx.doi.org/10.1155/2016/4390185

## Effects of Machine Tool Spindle Decay on the Stability Lobe Diagram: An Analytical-Experimental Study

Department of Aerospace Engineering, Ryerson University, 350 Victoria Street, Toronto, ON, Canada M5B 2K3

Received 3 July 2015; Revised 4 October 2015; Accepted 21 October 2015

Academic Editor: Evgeny Petrov

Copyright © 2016 Omar Gaber and Seyed M. Hashemi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An analytical-experimental investigation of machine tool spindle decay and its effects of the system’s stability lobe diagram (SLD) is presented. A dynamic stiffness matrix (DSM) model for the vibration analysis of the OKADA VM500 machine spindle is developed and is validated against Finite Element Analysis (FEA). The model is then refined to incorporate flexibility of the system’s bearings, originally modeled as simply supported boundary conditions, where the bearings are modeled as linear spring elements. The system fundamental frequency obtained from the modal analysis carried on an experimental setup is then used to calibrate the DSM model by tuning the springs’ constants. The resulting natural frequency is also used to determine the 2D stability lobes diagram (SLD) for said spindle. Exploiting the presented approach and calibrated DSM model it is shown that a hypothetical 10% change in the natural frequency would result in a significant shift in the SLD of the spindle system, which should be taken into consideration to ensure chatter-free machining over the spindle’s life cycle.

#### 1. Introduction

A great number of airframe structural components are manufactured by high speed milling, where problems can arise related to the instability in the process, dimensional errors in the work pieces, and even breakage of the tools. The instability of the process, a vibration phenomenon known as chatter, appears in the high removal rate roughing, as well as in the finishing of low rigidity airframe sections. Machine chatter has been researched since the early 20th century [1]. It has been well established that chatter is dependent on the dynamic behavior of machine tool structure and is directly linked to the machine vibrational characteristics [2], which is often expressed in terms of the frequency response function (FRF) of the system at the tool point. As the machine vibrational characteristics change, the machine stability behavior changes.

In the mid-1990s, Altintas and Budak [3] presented an analytical form of the so-called two-dimensional (2D) stability lobe theory for milling. The two-dimensional stability lobe theory deals with the stability of solutions for dynamical cutting systems, which usually stands for the spindle speed and axial depth of cut. As a function of these two cutting parameters, the border between a stable cut (i.e., chatter-free) and an unstable one (i.e., with chatter) can be visualized in a chart known as stability lobes diagram (SLD). A stable cut is defined as cut where the tool tooth displacement decreases from one pass to the next dampening the effect of the initial deflection. If the displacement increases or stays the same causing a wave pattern on the part, the cut is unstable. Numerous effective experimental and analytical techniques have been developed to establish SLD and to predict stable processes in recent decades (see, e.g., Altintas and Weck [4]). A more recent review of methods of obtaining stability lobe diagram in high speed milling operation is presented by Palpandian et al. [5]. 3D stability lobes were also later established considering radial depth of cut as another parameter [6]. Both of these stability lobe theories can help to select the appropriate cutting parameters of the spindle speed and axial depth to avoid chatter in machining processes. Attempts have also been made to integrate the dynamical behavior variation of the part with respect to the tool position in 3D lobes construction, with application to thin-walled structure milling, in order to determine optimal cutting conditions during the machining process [7].

The stability lobes calculation requires the dynamic parameters of the system, namely, stiffness, natural frequency, and damping ratio of the workpiece for each natural mode. Thevenot et al. [7] used FEM-based numerical models to determine the system’s stiffness terms, but the damping ratio cannot be easily calculated numerically, and therefore it is usually evaluated through experimental measurements. Ertürk et al. [8] modeled the machine tool spindle with a set of springs and dampers to simulate bearing behavior and to predict tool tip FRF. Cao and Altintas [9] investigated the effect of preload applied to a bearing on the overall system natural frequency and showed that the greater the preload, the higher the system’s natural frequency.

Many researchers have studied the stability through machine behavior, assuming a rigid work piece. The tool tip transfer function is then elaborated through models or experimental approaches. In addition, most of the previous models reported in the open literature have been developed assuming that spindle-tool set dynamics do not change over the full spindle speed range. However, this assumption needs to be reconsidered in high speed machining, where gyroscopic moments and centrifugal forces on both bearings and spindle shaft induce spindle speed dependent dynamics changes. Furthermore, the change of spindle system dynamics has not been accounted for in most existing stability studies. Few studies were also reported considering the flexibility of work piece [10].

In summary, the methods of obtaining stability lobe diagram (SLD) can be divided in three main categories, namely, experimental, semianalytical, and analytical approaches.

The aim of experimental methods is to obtain SLD by conducting a series of experiments on work piece by machining it using a milling machine tool; while machining at a certain depth of cut along the tool path, forced vibrations turn into self-excited vibrations, causing the milling process to become unstable, that is, chatter onset. This procedure is used in various experiments and is repeated for various depth of cut and spindle speed combinations.

In semianalytical methods, most of the parameters required to obtain stability lobes are calculated analytically. The modal parameters of spindle/tool-holder/tool systems, however, are obtained experimentally (e.g., using tap test), from the resulting FRF of the system. The system’s parameters can then be used to calculate its SLD.

The analytical approaches (see, e.g., [3]) aim to obtain the transfer functions of structure, which is required in the stability model, and to eliminate the need for series of costly experimental impact testing at various points on a work piece. The dynamic characteristics of the entire (spindle/tool-holder/tool) system, contributing in the transfer functions of structure, have been shown to depend on a large number of factors, including holder characteristics, spindle shaft geometry and drawbar force, and the stiffness and damping provided by the bearings. Thus the analytical approaches, complemented with numerical methods such as FEM, eliminate the need for experimentation and save the time and cost involved in determining the stability lobes.

The aim of this paper is to present a semianalytical stability technique, developed to incorporate the spindle’s dynamic behavior variations in the stability lobes diagram (SLD). The change in the spindle’s dynamic behavior, also referred to as aging, is generally caused by system’s bearings wear, translated through a reduction in the system’s natural frequencies. Exploiting and adapting the dynamic stiffness matrix (DSM) method [11], the model of the spindle system is generated, where the boundary conditions (BC) at the bearings are originally enforced using simple supports. The spindle DSM model is also validated against conventional FEM-based simulations generated in the educational version of commercial software ANSYS V13. Once the correctness of the DSM method has been established, the simple supports BC are replaced with linear spring elements to incorporate the inherent flexibility of the bearings. The experimental (working) fundamental frequency of the spindle system, determined from FRF data obtained from tap testing, is subsequently used to adjust the spring stiffness constants, leading to a calibrated dynamic stiffness matrix (CDSM) [12] model of the system. Using the system’s CDSM model and the experimental FRF data, the SLD can then be determined. Exploiting the proposed semianalytical method, it will then be possible to evaluate updated SLD and optimized machining parameters, should one know the variation of system’s stiffness (i.e., changes in the fundamental frequency) over the spindle life cycle.

The application of the proposed model is demonstrated through an OKADA VM500 machine spindle, where the shift in the SLD resulting from a simulated 10% reduction of the system’s fundamental frequency and its effects on the machining parameters are investigated. The spindle, initially examined while mounted on the original machine tool, was then installed on a bench top fixture to carry out further experimentations.

#### 2. Theory and Governing Equations

In what follows, the differential equations governing the bending-bending (BB) vibrations of a spinning beam segment are briefly discussed. Following mainly the theory presented by Banerjee and Su [11], the effect of torsional stiffness is assumed to be large enough so that the torsional vibrations can be ignored (see, e.g., [12]). Figure 1 shows the beam in a right-handed rectangular Cartesian coordinates system. The beam length is , the mass per unit length is , the polar mass moment of inertia per unit length is , the principal axes bending rigidities are for both planes, and the torsional rigidity is .