**1. Introduction**

**Author details**

, F. Impinna, N. Amati and A. Tonoli

Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Turin, Italy

[1] Post, R. F., & Ryutov, D. D. (1998). Ambient-temperature passive magnetic bearings: Theory and design equations. Massachusetts, USA. *Proceedings of the 6th International*

[2] Filatov, A., & Maslen, E. H. (2001). Passive magnetic bearing for flywheel energy

[3] Tonoli, A., Amati, N., Impinna, F., & Detoni, J. G. (2011). A solution for the stabiliza‐ tion of electrodynamic bearings: modeling and experimental validation. *ASME Jour‐*

[4] Lembke, T. A. (2005). Design and analysis of a novel low loss homopolar electrody‐ namic bearing. *PhD thesis*, Royal Institute of Technology, Stockholm, Sweden.

[5] Kluyskens, V., & Dehez, B. (2009). Parameterized electromechanical model for mag‐ netic bearings with induced currents. *Journal of System Design and Dynamics*, 3(4).

[8] Tonoli, A., Amati, N., Bonfitto, A., Silvagni, M., Staples, B., & Karpenko, E. (2010). Design of Electromagnetic Dampers for Aero-Engine Applications. *Journal of Engi‐*

[9] Detoni, J. G., Impinna, F., Tonoli, A., & Amati, N. (2012). Unified Modelling of Pas‐ sive Homopolar and Heteropolar Electrodynamic Bearings. *Journal of Sound and Vi‐*

\*Address all correspondence to: joaquimd@gmail.com

66 Advances in Vibration Engineering and Structural Dynamics

*Symposium on Magnetic Bearings*, Cambrige.

*neering for Gas Turbines and Power*, 132(11).

*bration*, 331(19), 4219-4232.

*nal of Vibration and Acoustics*, 133.

storage systems. *IEEE Transactions on Magnetics*, 37(6).

[6] Genta, G. (2005). *Dynamics of rotating systems*, Springer, New York.

[7] Dorf, R. C., & Bishop, R. H. (2010). *Modern Control Systems*, Prentice Hall.

J. G. Detoni\*

**References**

The booming aerospace industry and high levels of competition has forced companies to constantly look for ways to optimize their machining processes. Cycle time, which is the time it takes to machine a certain part, has been a major concern at various Industries deal‐ ing with manufacturing of airframe parts and subassemblies. When trying to machine a part as quick as possible, spindle speed or metal removal rates are no longer the limiting factor; it is the chatter that occurs during the machining process. Chatter is defined as self-excited vi‐ brations between the tool and the work piece. A tight surface tolerance is usually required of a machined part. These self-excited vibrations leave wave patterns inscribed on the part and threaten to ruin it, as its surface tolerances are not met. Money lost due to the destructive nature of chatter, ruining the tools, parts and possibly the machine, has driven a lot of re‐ search into determining mathematical equations for the modeling and prediction of chatter. It is well established that chatter is directly linked to the natural frequency of the cutting system, which includes the spindle, shaft, tool and hold combination.

The first mention of chatter can be credited to Taylor [18], but it wasn't until 1946 that Ar‐ nold [3] conducted the first comprehensive investigation into it. His experiments were con‐ ducted on the turning process. He theorized that the machine could be modelled as a simple oscillator, and that the force on the tool decreased as the speed of the tool increased with relation to the work piece. In his equations, the proportionality constant of the speed of the tool to the force was subtracted from the damping value of the machine; when the propor‐ tionality constant increases beyond the damping value of the machine, negative damping occurs causing chatter. This was later challenged by Gurney and Tobias who theorized the now widely accepted belief that chatter is caused by wave patterns traced onto the surface

© 2012 Hashemi and Gaber; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Hashemi and Gaber; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

of the work piece by preceding tool passes [9]. The phase shift of the preceding wave to the wave currently being traced determines whether there is any amplification in the tool head movement. If there exists a phase shift between the two tool passes, then the uncut chip cross-sectional area is varied over the pass. The cutting force is dependent on the chips cross sectional area, and so, a varying cutting force is produced [19]. To perform calculations on this system, they modelled a grinding machine as a mass-spring system as opposed to an oscillating system. It had a single degree of freedom, making its calculations quite simple. The spring-mass system is also the widely used modelling theory for how a vibrating tool should be characterized today.

adding the behaviour of the tool after the onset of chatter [12]. The paper discusses the ef‐ fects preceding passes of the tool have on the current state. It was generally accepted that wave patterns left on the work piece from a previous pass greatly effects the current pass, however, it is demonstrated that tool passes two or more turns prior to the current also have an effect. The phase difference and frequency of the waves etched into the surface of the pri‐ or turns interact with one another, and if the conditions are correct, interact in a critical way that produces increasing amplitude vibrations [10]. While Tlusty was able to theorize that chatter stabilizes at a certain point due to the tool leaving the work piece, [12] set off to prove this theory. They had the novel idea to turn the machine-work piece system into a cir‐ cuit. Current was passed through the machine and into the work piece while turning. When chatter occurred, they noticed drops in current at the machine-tool contact point. This proves that an open circuit was being created, proving that the tool was losing contact with the work piece. They also sought out to prove why cutting becomes more stable at lower speeds. They believed that there was a resistive force caused by the tool moving forward along the cut. They found this resistive force to be inversely proportional to the cutting speed, and directly proportional to the relative velocity of the tool to the work piece. When this force was taken into account in their equations, it produces a wider region of stability while the spindle is at lower speeds. This resistive force was proven to be responsible for the large regions of stability at low spindle speeds, and is what diminishes at higher spindle speeds resulting in less stability. The majority of papers published prior to the 80's exam‐ ined chatter with reference to the turning and boring processes. Milling is plagued with the same issues of chatter, but its modelling becomes more complicated. Tlusty and Ismail [21] characterized the chatter in the milling process by examining the periodicity of the forces that occur at the tool that are not present in other processes. During the milling process, cut‐ ter teeth come into and out of contact with the work piece. It is on the surface of these teeth that the force is applied. The same number of teeth are not always in contact with the work piece, and each tooth may be removing a different amount of material at a time. This leads to widely varying forces at the tool tip, creating a more challenging system to model. Forced vibrations can be attributed to periodic forces that the machine is subjected to. This can in‐ clude an imbalance on rotating parts, or forces the machine transfers to the tool while mov‐ ing. Chatter must be isolated from this in experiments so that the observations and

Free Vibration Analysis of Spinning Spindles: A Calibrated Dynamic Stiffness Matrix Method

http://dx.doi.org/10.5772/51174

69

calculations can be kept specific to the chatter phenomenon.

Once an accurate model of the milling process had been created, a reliable stability lobe can be constructed. Stability lobes plot the axial depth of cut vs. spindle speed. The resulting graph has a series of lobes that intersect each other at certain points. The area that is formed underneath the intersection of these lobes represents conditions that will produce stable ma‐ chining. The area above these intersections represent unstable machining conditions. The concept of the stability lobe was first proposed by Tobias [24] As the mathematical model‐ ling of the machining systems improved, so did the accuracy of the stability lobes. Prior to the paper by Tlusty *et al.* [22], most stability lobe calculations contained many simplifying assumptions, and therefore, were not very accurate; all teeth on the cutter were assumed to be oriented in the same direction, and also had a uniform pitch. They eliminated all of these assumptions and proved their math represented reality more accurately. A quarter of a cen‐

Prior to 1961, the research papers published on the machining processes regarded them as steady state, discrete processes [8]. This erroneous idea led to the creation of machines that were overly heavy and thick walled. It was believed that this provided high damping to the forces on the tool tips that were thought to be static. To properly predict chatter, one must realize that machining is a continuous, dynamic process with tooltip forces that are in constant fluctuation. When performing calculations, the specific characteristics of each machine must also be taken into consideration. If one takes two identical tools, placed into two identical machines, and perform the same machining process on two identical parts, the lifespan of the tools will not be the same. The dynamics and response of each of the machines differs slightly due to structural imperfections, imbalances, etc. Therefore, calcu‐ lations must always take the machine-tool dynamics into consideration [23]. The modes of the machines structure determine the frequency and the direction that the tool is going to vibrate at [11]. Rather than the previously used machine design philosophy of "where there's vibration, add mass", it was then stated that designers must investigate the modeforms, weak points, bearing clearances, and self-inducing vibratory components of their machine design to try to reduce chatter [8, 15]. Certain researchers even further investiga‐ tedthe required number of structural modes to produce accurate results [6]. Since it is impractical to investigate an unlimited bandwidth of a signal, restrictions must be made. This has generally been restrained to one or two modes of vibration of the machine. Their study proves that using low order models, that only incorporate two modes, are sufficient‐ ly accurate to model the machines.

In 1981, one of the first papers documenting the non-linearity of the vibratory system occur‐ ring during chatterwas published [20]. Self-excited chatter is a phenomenon that grows, but does not grow indefinitely. There is a point in time where the vibrations stabilize because of the tool jumping out of the cut. As the vibrations amplify, the tool head displacement in‐ creases. The displacement of the tool is not linear, but occurs in all three dimensions. When the force on the tool due to chatter causes displacement away from the work piece that ex‐ ceeds the depth of cut, the tool will lose contact with the work piece. When this occurs, the work piece exerted forces on the tool all go to zero. The only forces acting on it now are the structural forces that want to keep the tool on its planned route. It is impossible for chatter to amplify further past this point, and so, this is where it stabilizes. Previous reports do not account for this stabilization. Their results are accurate up to this point, but then diverge from the experimentally obtained results. Tlusty's investigation was then complemented by adding the behaviour of the tool after the onset of chatter [12]. The paper discusses the ef‐ fects preceding passes of the tool have on the current state. It was generally accepted that wave patterns left on the work piece from a previous pass greatly effects the current pass, however, it is demonstrated that tool passes two or more turns prior to the current also have an effect. The phase difference and frequency of the waves etched into the surface of the pri‐ or turns interact with one another, and if the conditions are correct, interact in a critical way that produces increasing amplitude vibrations [10]. While Tlusty was able to theorize that chatter stabilizes at a certain point due to the tool leaving the work piece, [12] set off to prove this theory. They had the novel idea to turn the machine-work piece system into a cir‐ cuit. Current was passed through the machine and into the work piece while turning. When chatter occurred, they noticed drops in current at the machine-tool contact point. This proves that an open circuit was being created, proving that the tool was losing contact with the work piece. They also sought out to prove why cutting becomes more stable at lower speeds. They believed that there was a resistive force caused by the tool moving forward along the cut. They found this resistive force to be inversely proportional to the cutting speed, and directly proportional to the relative velocity of the tool to the work piece. When this force was taken into account in their equations, it produces a wider region of stability while the spindle is at lower speeds. This resistive force was proven to be responsible for the large regions of stability at low spindle speeds, and is what diminishes at higher spindle speeds resulting in less stability. The majority of papers published prior to the 80's exam‐ ined chatter with reference to the turning and boring processes. Milling is plagued with the same issues of chatter, but its modelling becomes more complicated. Tlusty and Ismail [21] characterized the chatter in the milling process by examining the periodicity of the forces that occur at the tool that are not present in other processes. During the milling process, cut‐ ter teeth come into and out of contact with the work piece. It is on the surface of these teeth that the force is applied. The same number of teeth are not always in contact with the work piece, and each tooth may be removing a different amount of material at a time. This leads to widely varying forces at the tool tip, creating a more challenging system to model. Forced vibrations can be attributed to periodic forces that the machine is subjected to. This can in‐ clude an imbalance on rotating parts, or forces the machine transfers to the tool while mov‐ ing. Chatter must be isolated from this in experiments so that the observations and calculations can be kept specific to the chatter phenomenon.

of the work piece by preceding tool passes [9]. The phase shift of the preceding wave to the wave currently being traced determines whether there is any amplification in the tool head movement. If there exists a phase shift between the two tool passes, then the uncut chip cross-sectional area is varied over the pass. The cutting force is dependent on the chips cross sectional area, and so, a varying cutting force is produced [19]. To perform calculations on this system, they modelled a grinding machine as a mass-spring system as opposed to an oscillating system. It had a single degree of freedom, making its calculations quite simple. The spring-mass system is also the widely used modelling theory for how a vibrating tool

Prior to 1961, the research papers published on the machining processes regarded them as steady state, discrete processes [8]. This erroneous idea led to the creation of machines that were overly heavy and thick walled. It was believed that this provided high damping to the forces on the tool tips that were thought to be static. To properly predict chatter, one must realize that machining is a continuous, dynamic process with tooltip forces that are in constant fluctuation. When performing calculations, the specific characteristics of each machine must also be taken into consideration. If one takes two identical tools, placed into two identical machines, and perform the same machining process on two identical parts, the lifespan of the tools will not be the same. The dynamics and response of each of the machines differs slightly due to structural imperfections, imbalances, etc. Therefore, calcu‐ lations must always take the machine-tool dynamics into consideration [23]. The modes of the machines structure determine the frequency and the direction that the tool is going to vibrate at [11]. Rather than the previously used machine design philosophy of "where there's vibration, add mass", it was then stated that designers must investigate the modeforms, weak points, bearing clearances, and self-inducing vibratory components of their machine design to try to reduce chatter [8, 15]. Certain researchers even further investiga‐ tedthe required number of structural modes to produce accurate results [6]. Since it is impractical to investigate an unlimited bandwidth of a signal, restrictions must be made. This has generally been restrained to one or two modes of vibration of the machine. Their study proves that using low order models, that only incorporate two modes, are sufficient‐

In 1981, one of the first papers documenting the non-linearity of the vibratory system occur‐ ring during chatterwas published [20]. Self-excited chatter is a phenomenon that grows, but does not grow indefinitely. There is a point in time where the vibrations stabilize because of the tool jumping out of the cut. As the vibrations amplify, the tool head displacement in‐ creases. The displacement of the tool is not linear, but occurs in all three dimensions. When the force on the tool due to chatter causes displacement away from the work piece that ex‐ ceeds the depth of cut, the tool will lose contact with the work piece. When this occurs, the work piece exerted forces on the tool all go to zero. The only forces acting on it now are the structural forces that want to keep the tool on its planned route. It is impossible for chatter to amplify further past this point, and so, this is where it stabilizes. Previous reports do not account for this stabilization. Their results are accurate up to this point, but then diverge from the experimentally obtained results. Tlusty's investigation was then complemented by

should be characterized today.

68 Advances in Vibration Engineering and Structural Dynamics

ly accurate to model the machines.

Once an accurate model of the milling process had been created, a reliable stability lobe can be constructed. Stability lobes plot the axial depth of cut vs. spindle speed. The resulting graph has a series of lobes that intersect each other at certain points. The area that is formed underneath the intersection of these lobes represents conditions that will produce stable ma‐ chining. The area above these intersections represent unstable machining conditions. The concept of the stability lobe was first proposed by Tobias [24] As the mathematical model‐ ling of the machining systems improved, so did the accuracy of the stability lobes. Prior to the paper by Tlusty *et al.* [22], most stability lobe calculations contained many simplifying assumptions, and therefore, were not very accurate; all teeth on the cutter were assumed to be oriented in the same direction, and also had a uniform pitch. They eliminated all of these assumptions and proved their math represented reality more accurately. A quarter of a cen‐ turylater, Mann *et al.* [14] discovered unstable regions in a stability lobe graph that existed underneath the stability boundary for the milling operation. They resemble islands in the fact that they are ovular areas contained within the stable regions, complicating the previ‐ ously thought simple stability lobe model. It was found that stability lobes taken from mo‐ dal parameters of the machine at rest (static) were not as accurate as the stability lobes produced from the dynamic modal properties. Zaghbani & Songmene [25] obtained these dynamic properties using operational modal analysis (OMA). OMA uses the autoregressive moving average method and least square complex exponential method to obtain these val‐ ues, producing a dynamic stability lobe that more accurately represents stable cutting condi‐ tions. These stability lobes have proven to be an invaluable asset to machinists and machine programmers. They provide a quick and easy reference to choose machining parameters that should produce a chatter free cut [2].

dramatically. Previous research papers assume a constant FRF throughout the whole proc‐ ess for the sake of reducing computations. However, a constantly updated FRF would allow

Free Vibration Analysis of Spinning Spindles: A Calibrated Dynamic Stiffness Matrix Method

http://dx.doi.org/10.5772/51174

71

To the authors' best knowledge, the spindle decay/bearings wear over the service time and their effects on the system natural frequencies, and consequently change of the stability lobes, have not been investigated.The objective of the present study is to determine the natu‐ ral frequencies/vibration characteristics ofmachine tool spindle systems by developing its Dynamic Stiffness Matrix (DSM) [4] and applying the proper boundary conditions. These re‐ sults would then be compared to the experimental results obtained from testing a common cutting system to validate/tune the model developed. The Hamilton's Principle is used to derive the differential equations governing the coupled Bending-Bending (B-B) vibration of a spinning beam, which are solved for harmonic oscillations. A MATLAB® code is devel‐ oped to assemble the DSM element matrices for multiple components and applying the boundary conditions (BC). The machine spindles usually contain bearings, simulated by ap‐ plying spring elements at said locations. The bearings are first modeled as Simply-Support‐ ed (S-S) frictionless pins. The S-Sboundary conditions are then replaced by linear spring elements to incorporate the flexibility of bearings into the DSM model. In comparison with the manufactures' data on the spindle's fundamental frequency, the bearing stiffness coeffi‐ cients, *K <sup>S</sup>*, are then varied to achieve a Calibrated Dynamics Stiffness Matrix (CDSM) vibra‐ tional model. Once the non-spinning results are confirmed and the spindle model tuned to represent the real system, the formulation could then be extended to include varying rota‐ tional speeds and torsional degree-of-freedom (DOF) for further modeling purposes. The re‐ search outcomepresented in this Chapter is to be used in the next phase of the authors' ongoing research to establish the relationship between the tool/system characteristics (incor‐ porating spindle's service time/age), and intended machining process, through the develop‐

Computer Numeric Control (CNC) machines are quite often found in industries where a great deal of machining occurs. These machines are generally 3-, 4- or 5-axis, depending on the number of degrees of freedom the device has. Having the tool translate in the *X*, *Y* and Z direction accounts for the first three degrees-of-freedom (DOF). Rotation about the spindle axes account for any further DOF. The spindle contains the motors that rotate the tools and all the mechanisms that hold the tool in place. Figure 1 displays a sample spindle configura‐

for accurate, real time stability calculations.

ment of relevant Stability Lobes, to achieve the best results.

tion and a typical tool/holder configuration is shown in Figure 2.

**2. Mathematical model**

Tool wear is an often-overlooked factor that contributes to chatter. With the aid of more powerful computers this variable can now be included in simulations. The cutting tool is not indestructible and will change its shape while machining, and consequently affects the sta‐ bility of the system and stability lobes [7]. As the tool becomes worn, its limits of stability increase. Therefore, the axial depth of cut can be increased while maintaining the same spin‐ dle speed that would have previously created chatter. The rate of wear was incorporated in‐ to the stability lobe calculations for the tools so that it was now also a function of wear. To verify their calculations, the tools were ground to certain stages of wear and then tested ex‐ perimentally. They were found to be in strong agreement. Tool wear, however, is not some‐ thing that machine shops want increased. Chatter increases the rates of tool wear, shortening their lifespan, and increasing the amount of money the shops must spend on new tools. Li *et al.* [13] determined that the coherence function of two crossed accelerations in the bending vibration of the tool shank approaches unity at the onset of chatter. A thresh‐ old needs to be set [16] and then detected using simple mechanism to alert the operator to change the machining conditions and avoid increased tool wear.

In most of the previous stability prediction methods, a Frequency Response Function (FRF) is required to perform the calculations. FRF refers to how the machine's structure reacts to vibration. It is required to do an impact test to acquire the system's FRF [17]. In this case, an accelerometer is placed at the end of the top of the tool, and a hammer is used to strike the tool. The accelerometer will measure the displacement of the tool, telling the engineer how the machine reacts to vibration. This test gives crucial information about the machine, such as the damping of the structure and its natural frequencies. This method of obtaining infor‐ mation is impractical; because the FRF of the machine is always changing, it would require the impact test to be performed at all the different stages of machining. Also, having to do this interrupts the manufacturing processing and having machines sitting idle costs the com‐ pany money. An offline method of obtaining this information could greatly benefit machin‐ ing companies by eliminating the need for the impact test. Adetoro *et al.* [1] proposed that the machine, tool and work piece could be modelled using finite element analysis.A com‐ puter simulation would be able to predict the FRF during all phases of the machining proc‐ ess. As the part is machined and becomes thinner, its response to vibration changes dramatically. Previous research papers assume a constant FRF throughout the whole proc‐ ess for the sake of reducing computations. However, a constantly updated FRF would allow for accurate, real time stability calculations.

To the authors' best knowledge, the spindle decay/bearings wear over the service time and their effects on the system natural frequencies, and consequently change of the stability lobes, have not been investigated.The objective of the present study is to determine the natu‐ ral frequencies/vibration characteristics ofmachine tool spindle systems by developing its Dynamic Stiffness Matrix (DSM) [4] and applying the proper boundary conditions. These re‐ sults would then be compared to the experimental results obtained from testing a common cutting system to validate/tune the model developed. The Hamilton's Principle is used to derive the differential equations governing the coupled Bending-Bending (B-B) vibration of a spinning beam, which are solved for harmonic oscillations. A MATLAB® code is devel‐ oped to assemble the DSM element matrices for multiple components and applying the boundary conditions (BC). The machine spindles usually contain bearings, simulated by ap‐ plying spring elements at said locations. The bearings are first modeled as Simply-Support‐ ed (S-S) frictionless pins. The S-Sboundary conditions are then replaced by linear spring elements to incorporate the flexibility of bearings into the DSM model. In comparison with the manufactures' data on the spindle's fundamental frequency, the bearing stiffness coeffi‐ cients, *K <sup>S</sup>*, are then varied to achieve a Calibrated Dynamics Stiffness Matrix (CDSM) vibra‐ tional model. Once the non-spinning results are confirmed and the spindle model tuned to represent the real system, the formulation could then be extended to include varying rota‐ tional speeds and torsional degree-of-freedom (DOF) for further modeling purposes. The re‐ search outcomepresented in this Chapter is to be used in the next phase of the authors' ongoing research to establish the relationship between the tool/system characteristics (incor‐ porating spindle's service time/age), and intended machining process, through the develop‐ ment of relevant Stability Lobes, to achieve the best results.
