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.

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 [

In the mid-1990s, Altintas and Budak [

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. [

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 [

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., [

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 [

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.

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 [

Degrees of freedom of a spinning beam segment.

The differential equations governing the bending-bending (BB) vibrations of a spinning beam segment can be written as [

The DSM formulation was used to model an OKADA VM500 vertical milling machine spindle [

Schematic diagram of an OKADA VM500 spindle [

Spindle model.

The spindle was modeled using the DSM method presented in the previous section, where only one element per uniform segment (i.e., a total of 26 elements) was used and simply supported boundary conditions were applied at the bearings locations. The material properties of tool steel, elastic modulus of ^{3}, were used. Modal analysis was carried out and the fundamental flexural natural frequency of the spindle system was found to be 2303 Hz. To validate the spindle DSM model, frequency data were compared with those obtained from a 154-element FEM model created in ANSYS V13 software [

The experimental modal analysis results, obtained using tap testing, were used to determine the system’s nominal fundamental frequency. Experimental Apparatus (Figure

352A21 (Light, 0.8 g) accelerometer,

086C04 (5000 N Hammer) Hammer,

SIM3 Module Photon+ Data Acquisition,

MetalMax Software (TXF).

Experimental equipment.

Spindle experimental frequency response function (FRF) (real part).

Spindle experimental frequency response function (FRF) (imaginary part).

The system’s fundamental frequencies were found to be much higher than that of the experimentally measured fundamental natural frequency, that is, 2303 Hz (from DSM) and 2367 Hz (from FEM) versus 1722 evaluated experimentally. The large difference (34%) between numerical and experimental results can be associated with the bearings flexibility. As mentioned earlier in this paper, the spindle DSM and FEM models were both initially equipped with simply supported (frictionless pin) boundary conditions at the bearing locations, that is, infinite lateral stiffness. A calibrated dynamic stiffness matrix (CDSM) model of the system was created by implementing spring boundary conditions (i.e., finite lateral stiffness) in the spindle’s DSM model (Figure

Spindle with bearings replaced with springs.

System fundamental frequency versus bearing equivalent spring constant for a low speed spindle (in log scale).

To achieve the system’s experimental natural frequency of 1722 Hz, the required bearing stiffness was found to be

Spindle natural frequency versus spindle RPM; high torque spindle.

The stability lobes were generated using the experimental FRF data obtained. The peak picking method was used to determine the equivalent stiffness at the tool tip [

Stability chart an OKADA VM500 spindle using 2 different frequencies.

Furthermore, exploiting the calibrated CDSM method, it is 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. This reduction can correspond to a combination of bearing degradation and loss of bearing preload due to age and/or damage. For the spindle system at hand, however, the authors were not able to find any data publicly available on the time history and reduction of the system’s fundamental frequency versus service time. However, to demonstrate the application of the presented CDSM and the process, let us consider a hypothetical 10% reduction in the system’s fundamental frequency.

As presented in [

Exploiting the graph given in Figure

Let us consider a very small natural frequency,

Finally, the reason for not using the FRF directly and entirely is the extremely large amount of data points in the FRF file. The goal of the present study was to use a quick approach to generate stability lobes that would benefit someone working as a machine operator in a manufacturing facility, with limited computation capabilities, that is, the CNC machine programmers. The stability lobes generated in this research, however, were confirmed against ones directly generated from FRF graphs. Additionally, as also suggested in [

An experimentally calibrated dynamic stiffness matrix (DSM) formulation was presented and used to model an OKADA VM500 spindle. Unlike conventional FEM, the modeling method commonly used in this field, the presented element DSM is exact within the limits of the theory. As a result, in comparison with FEM, the DSM model requires much fewer elements to achieve the desired precision, that is, one element per uniform segment. The system’s natural frequency found experimentally was used to calibrate the spindle model, equipped with bearings represented as linear spring elements, and to generate spindle’s stability lobes. This natural frequency was then reduced by 10% to simulate a bearing degradation, or loss of preload due to spindle aging and/or damage. Incorporating the reduced frequency into the calibrated DSM, it was utilized to calculate the spindle’s equivalent stiffness value, used to generate updated stability lobe diagram. It was found that a 10% reduction in frequency leads to a significant shift in the stability lobes, which should be taken into consideration to ensure that a machine spindle runs chatter-free over its entire life cycle.

The authors declare that this paper presents the results of a recent research, conducted by the first author, under the supervision of the second (corresponding) author. The authors also declare that the present paper in its present form has not been published, nor has it been submitted for publication, in any other journals.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors wish to acknowledge the supports provided by Natural Sciences and Engineering Research Council of Canada (NSERC) and Ryerson University.