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ARTICLE IN PRESSG ModelRE-6213; No. of Pages 15

Precision Engineering xxx (2015) xxx–xxx

Contents lists available at ScienceDirect

Precision Engineering

jo ur nal ho me p age: www.elsev ier .com/ locate /prec is ion

orm error in diamond turning

. Tauhiduzzaman, A. Yip, S.C. Veldhuis ∗

cMaster Manufacturing Research Institute, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4L7, Canada

r t i c l e i n f o

rticle history:eceived 20 November 2014eceived in revised form 3 February 2015ccepted 16 March 2015vailable online xxx

eywords:mbalanceotordynamicsampbell’s waterfall diagramir-hammeringpindle starynchronousiamond turning

a b s t r a c t

The main feature of ultraprecision, single point diamond turning (SPDT) is its ability to produce highquality surface finishes on the order of nanometers while meeting tight form tolerances on the order ofmicrometers. This capability allows for the production of optical devices with minimum post-processingoperations. The issue of form error is critical since it may severely compromise the performance of thedesigned optical system. Tool-workpiece relative vibration is a major cause of this error. In this article, theauthors have demonstrated that imbalance of the spindle is a major cause of form error which eventuallyleads to a structured error called “spindle star” that appears as straight concentric spokes radiating outfrom the center of the part. The formation of a spindle star pattern can be explained using Campbell’srotordynamics analysis. This analysis explains how assisted air-hammering instability and cross-couplingeffect of the air-bearing spindle can contribute to spindle star. This experimental approach used force andaccelerometer data with the help of modal analysis to conclude that spindle star is a synchronous errorand is a function of rotational frequency of the spindle and its harmonics. The integer harmonic from the
Campbell’s waterfall diagram predicts the number of spokes in the spindle star. It was also observed thatthe height of the spindle star undulations increases with higher rotational speed. It was also observedthat cutting material and tool geometry has no influence on the spindle star formation; rather this is aninherent characteristic of air-bearing journals. Finally, the analysis was successfully validated by changingthe natural frequency of the spindle by adding mass.
© 2015 Elsevier Inc. All rights reserved.

. Introduction

Ultra precision, single point diamond turning (SPDT) is a toolased machining technology to produce high quality surface fin-

shes on the order of nanometers while meeting tight formolerances on the order of micrometers. The purpose of improv-ng the surface finish in diamond turned surfaces is to reduce theost of rotationally-symmetric optical components by reducingr eliminating costly manual polishing and lapping. In the dia-ond turning process, the proper combination of good workpiece

uality, sharp and controlled waviness tools, and a high qualityositioning system will create a part with high accuracy and opti-al surface quality. It is generally agreed that surface finish in SPDTs primarily affected by four factors [1,2] which are: tool geometrynose radius, machining parameters), relative vibrations between

Please cite this article in press as: Tauhiduzzaman M, et

http://dx.doi.org/10.1016/j.precisioneng.2015.03.006

ool-workpiece, material properties and microstructure, and toolutting edge quality. Among these factors, form error is generatedrom the undulations between the “relative vibrations between

∗ Corresponding author. Tel.: +1 905 525 9140x27044; fax: +1 905 572 7944.E-mail address: [email protected] (S.C. Veldhuis).

ttp://dx.doi.org/10.1016/j.precisioneng.2015.03.006141-6359/© 2015 Elsevier Inc. All rights reserved.

tool-workpiece” and may become a part of the surface rough-ness measurement. This issue is extremely critical if the error isstructured and regular in pattern as it severely compromises thedesigned optical system performance.

Many articles have reported on the relative vibration betweenthe tool and workpiece and their effect on surface roughness.Off line and in process metrologies have also been performed todevelop new technologies to compensate for such errors; how-ever, the source and characteristics of these errors have not beenfully developed. In this article, the authors have summarized theirobservations on spindle star formation on a diamond turned flatworkpiece surface using rotordynamic analysis and assisted air-hammer instability of the air-bearing spindle. The observationsexplain the formation of structured undulations which have beenconsistently observed on the machined surface of parts. This arti-cle also explains how to predict the expected number of undulationspokes from a given air-bearing spindle system and provides rec-ommendations for minimizing their formation.

al. Form error in diamond turning. Precis Eng (2015),

One of the pioneering works on relative tool-workpiece vibra-tion was done by Takasu et al. [3] who assumed that a simpleharmonic motion with small amplitude and low frequency existsbetween the tool and the workpiece. He presented equations

dx.doi.org/10.1016/j.precisioneng.2015.03.006


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mailto:[email protected]


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escribing the resulting pattern in the feed direction by consid-ring the effect of vibration on surface roughness, specifically as itoncerns the interplay between the tool nose radiuses. His analysisor fly-cutting was further adopted by Fawcett [4] who also mea-ured the relative tool/workpiece vibrations (not during cutting) inhe in-feed direction with a capacitive gage to develop a fast-low-mplitude tool servo to compensate for such errors during turning.hese equations [3] are in turn used by Cheung and Lee [5–7] with

capacitive displacement probe between the tool and the spin-le in predicting values of peak–valley surface roughness, Rt, andean arithmetic roughness, Ra, and on comparing predicted valuesith those measured on ultraprecision machined workpieces. In

heir works [6,7] the surface roughness evaluation length used was.5 mm. Since such vibrations can be large enough to be considereds part of the form error, the authors of this article believe that thismall evaluation length may result in a misrepresentation of theurface finish. In their work, Cheung and Lee [6] also concluded thataterial crystallography induced vibration is another major factor

ontributing to the generation of surface roughness, in addition tohe tool feed rate, the spindle rotational speed, the tool geometry,he material properties, as well as the relative tool-work vibration,hich have been previously studied. The possibility of chatter iniamond turning has been ruled out by them [7] since the depth ofut in ultraprecision machining is usually in the range of a few toeveral tens of micrometers and the chatter vibration induced byhe regenerative effect seldom occurs as it is generally developednder heavy-duty cutting. However, some high frequency marks12 kHz to 14 kHz) on the cutting force signal and surface rough-ess profile have recently been reported in the cutting directionhich is believed to be from the tool-tip vibration [8] during SPDT.evertheless, none of these articles explain the source of the errorotion, except in the case of tool-tip vibration which is not a focus

f this article.Bittner [9] simulated all possible sources of error while dia-

ond turning diffractive optical elements. His simulations coveredot only physical errors from decentering of the tool with respecto the center of rotation and tilting of the machine slide relativeo the spindle, but also periodic errors like variable shift betweenool and spindle, periodic structures formed due to thermal effects,nd tool wear and vibration of the axis of rotation which produces

spindle star pattern on the part. Kim and Kim [10] analyzednd compensated the thermal growth spindle error in real timesing a fast tool servo along with a capacitive displacement sen-or and were able to bring the form error down to 0.1 �m in a00 mm diameter specimen. Finally, vibrations of the spindle per-endicular to the workpiece can result in azimuthal ripples on theurface forming the spindle star pattern. Again, according to Juer-ens et al. [11], spindle star can be caused by imbalance of theorkpiece on the machine and a properly designed spindle gen-

rally produces little or no spindle star. The size and location of their bearing orifice can cause axial oscillations of the spindle; how-ver, newer spindles all have continuous air slots based on the usef porous bearing materials. The use of these designs has greatlyeduced this source of error but they remain a significant sourcef workpiece quality issues as product specifications have tight-ned. It is difficult to see further reductions in the spindle star formrror through spindle balancing as it is impossible to make a partithout error and compensation for imbalance is often done out

f the error plane on the body of the vacuum chuck. Torque rip-le from the motor can also cause vibration. It has been observedhat parts produced from current study and off-the-shelf compo-ents produced from other machines all suffer from spindle star



rror. Fig. 1 shows optical metrology of a diamond turned copperorkpiece showing a star pattern radially spreading outward from

he center as observed in this study. Fig. 1 has a field of view of0 mm × 24 mm.

Fig. 1. Spindle star formation on a diamond turned copper workpiece.

Khanfir et al. [12] in 2005 has also observed the spindle star andstated that the error is due to resultant facial and radial motionof the machine spindle. By using a spindle with active magneticbearings, Khanfir et al. was able to filter out the fifth harmonicvibration of the spindle, eliminating the five lobed spindle star, andleaving a surface with a seven lobed spindle star pattern instead.Although the vibration amplitude decreased when the active mag-netic bearings were applied, the pattern continues to exist. A similarobservation on spindle star was also made by Zhang et al. [13]in 2012. They developed a mathematical model with five degreesof freedom of an aerostatic spindle and analyzed how its move-ments affect the surface topography in spindle star formation ofa machined surface in SPDT. Zhang et al. [13] also observed thatthe operating speed has a significant role on how that imbalancewill act and the moving point contact of the tool-workpiece canalso trigger an imbalance. It is very important to know what maycause the spindle star form error since each machine may oper-ate at different spindle speeds. However, both these articles [9,11]addressed the source of such error as coming from the spindle axialmotion due to the imbalance in the air-bearing spindle system.

The spindle star error is a form error that results in a wavy sur-face in which the peaks and valleys form straight, radial spokes onthe surface of the workpiece. These straight, radial spokes are con-stant across the surface and are not dependent on machine position.Although the spindle star pattern is not visible to the naked eye, thestructured form error can be amplified in an optical system whichcauses imaging problems.

In air bearing spindles, motion error is often characterized by3 features: eccentricity, higher order synchronous motion, andasynchronous motion errors. Synchronous vibration is defined asa vibration that is equal to an integer multiple of the rotating fre-quency. Asynchronous vibration is defined as a vibration which isat a frequency that is other than an integer multiple of the rota-tional frequency. In other words, when the disturbance frequencyis a harmonic of spindle rotating frequency, it is termed as syn-chronous error motion and when the disturbance frequency isindependent of spindle rotating frequency it is termed as asyn-chronous error motion. Basic eccentricities in motion error canbe addressed by proper bearing alignment and balancing; how-ever, reducing synchronous and asynchronous motion requires acombination of attention to air flows and bearing geometry. Oneof the major factors in ensuring low asynchronous motion erroris an air supply that is free from pressure pulses and contami-nants [14]. Improvements in manufacturing methods have had animportant effect on motion error and have enabled errors to be


reduced from about 200 nm down to 50 nm [14,15] over the lasttwenty years. Further improvements are needed as tolerances con-tinue to tighten. To meet these challenges, our understanding of the


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nderlying mechanisms which drive this problem must bemproved. By applying rotordynamics and assisted air-hammernstability theory these motion errors can be explained and in theuture action can be taken to further minimize their impact onuality.

. Rotordynamic and assisted air-hammer instability inir-bearing spindle dynamics

The type of bearing typically used in the diamond turningpindle is the externally pressurized aerostatic bearing. Externallyompressed air is supplied through ports inside the spindle to causet to ‘float’ on a layer of compressed air. At higher pressures, theylso have a high load capacity and can boast a stiffness of several/�m with limited cross-coupling effects [16] and with very low

otational friction and heat generation. Due to their characteristics,ir bearings are an obvious choice for ultra precision machining.ompared to hydrostatic oil bearings, the machine tool is simplifieds it no longer needs a system of seals, filters, and heat exchang-rs. These air bearing spindles can be analyzed in a similar fashions other rotating machinery. Extensive rotordynamics analysis haseen performed on turbines and centrifugal pumps [17] where aommon cause of vibration is imbalance in mass distribution. Theres an extensive amount of literature and books available on rotor-ynamics with examples of successfully applying this knowledgeo predict chatter stability in machining processes [18,19].

Imbalance will always be a cause of vibration as it is impos-ible to produce a perfectly balanced rotor in the physical world.ven very small rotary imbalances and misalignments can createynchronous vibration, typically occurring one or two times theotating frequency [17]. To show the dynamic properties of theotating system a Campbell plot can be constructed. The Campbelllot is a diagram that shows the vibration amplitude of a rotatingystem as a function of frequency and rotational speed [20]. Theibration amplitude can be shown either by color intensity or bysing a waterfall plot. Several trends can be observed from a Camp-ell diagram of a system, such as frequencies where synchronousnd asynchronous vibration occurs, as well as problem frequen-ies which are close to the system’s dominant natural frequency orrequencies.



There are several modes of vibration that can occur in the rotor-earing system in the spindle. This can be visualized by modelinghe rotor-bearing system as an inflexible shaft with two bearingupports and a mass at its end. The spindle shaft can vibrate as

rdynamic body in air-bearings.

it is rotating like a beam with two pivoting supports, with differ-ent vibration modes [17]. If we assume the shaft to be perfectlyrigid and the bearings to be flexible, the rotating spindle shaft maytrace an axial, cylindrical and conical orbit as shown in Fig. 2(a–c).The first motion is vibration of the spindle in the axial direction.The second motion is a cylindrical orbit where the front and rearbearing both vibrate in phase with each other. The third motionis a conical orbit where the vibration of the front bearing is outof phase with the rear bearing, creating a conical pattern whenin motion. The difference between static structural vibration androtating vibration is in addition to this orbital movement [21]. Dueto the machine configuration used when face turning flat work-pieces, the cylindrical orbit is the only type of movement that willnot affect the workpiece’s form error. However, this may not be truewhen the features are not concentric, flat or round [22]. All the othertypes of shaft bending modes and soft bearing orbits can cause thedepth of cut to change, creating a change in cutting force and formerror.

To understand the whole phenomenon, assume a simple rotat-ing shaft with a point unbalanced mass, m of angular velocity ωrad/s with an eccentricity of r and a vertical downward force, Facting on one end of the shaft as shown in Fig. 2(d). The point ofsupport is assumed to be on the other end of the shaft. First, con-sider the conservative angular momentum of a system which can beexpressed as L = mωr2 = Iω where the direction is determined by theright hand rule. I is the moment of inertia of the rotating unbalancedmass, m. If the air-journal beating is assumed to be soft bearing [21],then this angular momentum will give an axial movement alongthe rotational axis in either direction along the Z-axis dependingon the rotational direction. In this example, the axial movementshould be in the +Z direction along the angular momentum, L. Thisalso means a higher mass should increase the axial movement;however, this will decrease the natural frequency of the system.Secondly, due to the unbalanced mass, the rotating shaft will alsohave a cylindrical orbital forward whirl movement around the rota-tional axis with a rotational frequency as shown in Fig. 2(b). TheJeffcott-Laval rotor system is often used to explain shaft whirling.A combination of these motions will create a cylindrical error vol-ume. Finally, the force F shall create a bending moment aroundthe X-axis on the system which will create a counter clockwise


precession of the Z–X plane around the Y axis with the same magni-tude of angular momentum due to momentum conservation. Thisprecession is a gyroscopic effect that will result in an unwantedconical orbit as it is spinning. The force F here is analogues to the



4 M. Tauhiduzzaman et al. / Precision Engineering xxx (2015) xxx–xxx

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Fig. 3. Rotordynamic error motions and air-hamme

esultant force between the overhang weight (W) of the vacuumhuck and the workpiece, and cutting force in diamond turning.his effect, combined with the centripetal force of the workpiece,ill cause the workpiece to wobble and vibrate in a conical orbit.

he amount of precession due to the gyroscopic effect will dependn the cross-coupling stiffness difference between the axial andadial direction of the air-bearing journals. To understand the inter-lay between conical error movement and the dynamic air-bearingpindle response, assume an air-bearing spindle cross section ashown in Fig. 3(a) and consider the last factor which is air-hammernstability.

An externally pressurized aerostatic bearing consists of amooth shaft with a flange, and an outer ring with a system ofrifices in which the air is pumped through. Fig. 3(a) shows a hypo-hetical cutaway of the air-bearing spindle system in a diamondurning machine. In Fig. 3(a), authors represent an imaginary topiew of an air bearing spindle in action at its ideal state where theorkpiece is placed with the help of a vacuum chuck mounted on

he same axis of the spindle rotor shaft. The rotation is usually pow-red by a DC brushless induction motor. The DC brushless motoran be a source of error due to magnetic residual unbalance [12]hich has not been considered in this study. In this case the RPM

s assumed to be in the clockwise direction and the chuck has annbalanced mass as shown in Fig. 3. The source of this unbalancedass can be inherent or from the mounting error of the workpiece

nd is too unpredictable to quantify. The locations of the axial andadial air bearings are shown with the air inlet having an uneventiffness of kaxial and kradial respectively. The imaginary geometricenter for the bearing systems are at point “O” on the shaft. Any con-ideration of rotation and translation shall be referenced from thisoint. Applying the understanding of Fig. 2 to Fig. 3(a) will result

n a situation as shown in Fig. 3(b) where the axes-systems beforend after are expanded to aid understanding and the new origin isepresented by O′.

There are three motions involved between O–O′:

. Translation in Z-axis for angular momentum;

. Circular orbit due to unbalance mass;



. A precession motion in the X–Z plane around the Y-axis which isdue to the moment generated around the X-axis from the grav-itational force of the chuck and the workpiece (W) and cuttingforce (Fc).

n a rotating air-bearing spindle system (TOP view).

These movements will create one high-point and one low-pointat two extreme ends of the X-axis due to the X–Z plane preces-sion movement. As a result, the axis of rotation will have a conicalshape and air inlet volume and compression volume shall havean energy difference as shown by the colored arrows of differ-ent sizes in Fig. 3(b). This unequal air-gap must try to reach anequilibrium (Fig. 3a) given a similar orifice configuration and airsupply pressure. When this happens, the shaft extracts energy toreach an equilibrium position from the external air supply andthus vibrates at its natural frequency. This natural frequency isdependent on the apparent stiffness, damping, and mass of theshaft. This phenomenon is called air hammer instability. Air ham-mer instability is highly dependent on the orifice configuration,supply pressure and, of course the natural frequency of the sys-tem, and can be present even when no rotation is present [23–26].However, air-hammer instability can be exacerbated by the gyro-scopic precession or whirling phenomena discussed earlier. It canalso occur due to external perturbations. When a vibration at a syn-chronous frequency is sustained over time by extracting energyfrom the external air supply, it is known as assisted air hammerinstability. Any displacement of the shaft in this case is correctedby the shaft exciting itself to one of the closest harmonics of thesystem’s rotational damped natural frequency to conserve energy,in an attempt to return to its equilibrium position. A compressiblefluid like air may not be displaced at all for extremely fast motions. Ifthat is the case, no damping occurs; instead a proportional reactionforce is exerted from the displacement due to the compression ofthe gas according to Boyle’s law [27]. The air film begins to behaveas if it was a spring element and less like a damper element [28]and a system with little or no damping where resonances are easilyexcited [23].

The rotating spindle and gyroscopic effects creates a forcedvibration problem. Due to the characteristics of air bearings andBoyle’s Law, the spring rate (cross-coupling stiffness) is non-linearand the hysteresis effects may be introduced as the stiffness is notuniform for tension and compression cycles in the air-bearings. Thisresults in a different natural frequency for the first and second half-cycle of vibration representing the tension and compression phase


of a single cycle of vibration. The resulting motion is no longer sim-ple harmonic motion, but can be represented by a Fourier serieswith infinite number of natural frequencies as multiples of the cir-cular frequency of a full cycle ω, 2ω, 3ω. . . [29]. In forced vibration,




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performed at a 2 �m depth of cut, 5 �m/rev feed rate with varyingRPMs. The detailed cutting matrix is presented in Table 2. The cop-

Fig. 4. Experimental setup: (a) Machine axes and configuration; (b) SP90 sp

uch as that of a rotating imbalance, the circular frequency willatch the rotational frequency. In a physical sense, a shaft with

rotating imbalance will cause it to vibrate at the rotational fre-uency as well as its harmonic frequencies. This forces the numberf spindle star lobes to be an integer value.

. Experimental setup and procedure

A series of face turning operations were performed at differentPMs to understand and characterize the spindle star formationuring SPDT. In the following sections, the equipment used androcedures performed in this study will be described.

.1. Machine tool

The experiments were conducted on a Precitech Freeform 700Gltra precision CNC machine tool. This machine is suitable for theroduction of optical and mechanical components. The machineool has three linear axes (X, Y, Z) as well as one rotational axis (B).he axes configuration of the machine is shown in Fig. 4(a). Feednd depth of cut is provided by the X- and Z-axis slides respectivelyhere the Y-axis is used to set the tool height adjustment to the

enter of part rotation. The work holding SP-90 air-bearing spin-le is positioned on the rotary B-axis on top of the Z slide with an0 psi supply air pressure. The rotary B axis was adjusted to hold theorkpiece parallel to the X–Y plane and held to this fixed angularosition to make a flat surface. The overall flatness of the surfaceas confirmed by inspecting the machined workpiece on a ZygoPI laser interferometer instrument. The linear axes are guidedy hydrostatic oil slide ways and controlled by multi-phase linearotors. The home position of these linear motors is sensed by a Lin-

ar Variable Differential Transformer (LVDT) transducer. Positionaleedback is given by laser holographic linear encoders which arethermally mounted to the slideways. The X- and Y-axis encodersave a resolution of 8.6 nm, while the Z-axis encoder has a resolu-ion of 1.4 nm [15]. The straightness of these X, Y, and Z slideways isithin 0.5 �m per 150 mm length [15]. The machine is controlled

y the use of the Precitech UltrapathTM digital signal processing-ased control system. The machine is mounted to a sealed natural



ranite base and is leveled and isolated from the ground using pressurized air system [15] that isolates the machine fromigh frequency ground vibrations. An SP-90 air-bearing spindleith a 100 mm diameter vacuum chuck holds the workpiece. This

able 1recitech machine stiffness [15].

Axis Direction Stiffness (N/�m)

X (Linear) Vertical/Horizontal 438 N/�m/438 N/�mY (Linear) Vertical/Horizontal 263 N/�m/263 N/�mZ (Linear) Z-Direction/X-Direction 438 N/�m/438 N/�mB (Rotary) Axial/Radial 1750 N/�m/525 N/�mSpindle (RPM) Axial/Radial 70 N/�m/26 N/�m

on Z-B axis and (c) dynamometer and tool holder showing force directions.

spindle has a maximum speed of 10,000 RPM. The maximum spin-dle speed used in this study did not exceed 3600 RPM. The spindle isshown in Fig. 4(b). The machine enclosure is connected to a stand-alone heater/cooler in order to control the machine temperature.The machine axes are stiff enough to keep deflection within nano-meter scale when cutting with sub-Newton forces and appropriatefor diamond turning. The stiffness values of the machine providedby the manufacturer are presented in Table 1 [15].

3.2. Diamond tools, workpiece and cutting parameters

Single crystal diamond (SCD) tools were used to perform a faceturning operation and the tool was positioned on a dynamometeron the X–Y slides of the machine as shown in Fig. 4(a) and (c). Fourdiamond tools were used in this study. The tools with a 12 mm noseradius were produced by DiamTech SCD Tool Technology, locatedin Quebec, Canada. Two more tools with a 1.5 mm and a 0.5 mmnose radius were supplied by K&Y Diamond, also located in Quebec.The four tools and conditions used in this study are outlined inmore detail in Table 2. The tools with sharp edges are commerciallyavailable but the tool with honed edge was specially ordered by B-Con Engineering in Ontario, Canada. The tools were inspected withan optical microscope with at least 800× magnification to ensurethat the tool’s cutting edges were sharp and chip-free, a standardindustrial practice.

The workpiece used for the majority of the cutting tests wasaluminum 6061 made of Rapidly Solidified Aluminum (RSA-6061).It is 60 mm in diameter and 15 mm thick. It was mounted to aworkpiece-holding aluminum disk in order for it to be held by thevacuum chuck of the machine tool with the help of a dowel pinfollowed by precision dialing to be within ±1 �m run out. The RSA-6061 workpiece was also placed into a large steel holder weighing4 kg. This steel workpiece holder was used to test the dynamics ofthe machine with a heavier workpiece without changing the mate-rial being cut. A 25.4 mm diameter copper rod was also diamondturned for comparison purpose. Odorless mineral spirit (OMS) mistwas used for all the cutting tests. All of the cutting parameters were


per material was machined at 1000 RPM with 0.5 mm nose radiustool.

Table 2Cutting conditions.

2 �m depth of cut, 5 �m/rev feed rate, RSA 6061 aluminum andCopper, OMS mist coolant

Diamond tools (i) 12 mm nose radius, (ii) 12 mm nose radiushoned edge, (iii) 1.5 mm nose radius and (iv)0.5 mm nose radius

RPMs 400, 500, 600, 800, 1000, 1021, 1100, 1200,1400, 1600, 1800, 2000, 2200, 2400, 2600,2800, 3000, 3200, 3310, 3400, 3429 and 3600


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Table 3Instrument list.

Instrument type Model

3 component Force Dynamometer Kistler 9256B1Accelerometer Kistler 8730A500M1

Kistler 8640A5Charge Amplifier Kistler Type 5010Voltage Amplifier PCB Piezotronics Model 480D06Data Acquisition Modules National Instruments NI 9234

National Instruments NI 9215Data Acquisition to USB Interface National Instruments NI cDAQ 9172Data Acquisition to PCI Interface National Instruments PCI-6023E

National Instruments BNC-2110



.3. Spindle balancing procedure

The Precitech’s SP-90 work-holding spindle must be balancedefore use. The chuck allows for single-plane balancing using 12et screws placed around the vacuum chuck with the help of a ded-cated spindle balancing procedure implemented on the machine.he balancing procedure is limited by the arrangement of thealancing screws so only a single plane of balancing can be per-ormed. The true location and magnitude of the shaft imbalance isnknown and cannot be fully compensated for using the balanc-

ng screws alone. The SP-90 spindle is equipped with a 1000-lineptical encoder with an index pulse that can be used as part of thealancing procedure. Several instruments need to be mounted tohe machine to allow for single-plane balancing. The index pulserom the SP-90 optical encoder is attached to an opto-isolatorreakout board. The one used here is a SparkFun Model BOB-09118quipped with ILD213T opto-isolator chip. A Kistler Type 8640A5ccelerometer was mounted to the SP-90 spindle in the radial direc-ion near the chuck to measure vibration. The PCB Piezotronics

odel 480D06 power unit was used to amplify the signal from theccelerometer. These two devices were connected to a data acquisi-ion system consisting of a National Instruments NI PCI-6023E andNC-2110. A key feature of this data acquisition system is that itupports hardware triggering. LabView 7.1 VI was used to capturehe accelerometer data using the signal from the opto-isolator torigger the data recording. The hardware trigger ensures that theata acquisition system begins recording at the index pulse of thepindle rotation each time. 40,960 samples were collected at a ratef 40.960 kS/s for each measurement.

.4. Data acquisition system

The forces encountered by a cutting tool are important ashey reflect the quality and condition of the tool, machine, fix-ure, and sometimes even the finished surface. The use of a forceynamometer is critical and allows for easy monitoring of theutting process. The data acquisition system (DAQ) consists of

MiniDyn 3-component dynamometer (Kistler 9256B1), a dual-ode three-channel charge amplifier (Kistler Type 5010), and a USB

ata acquisition system with four input channels (NI compactDAQ-172, NI 9215). A PC using National Instruments’ (NI) LabViewoftware is used to collect and analyze the force data. The cuttingorces and their directions are shown in Fig. 4(c).

Kistler type 8730A500M1 [30] and 8640A5 [31] accelerometersere used in this study to measure vibration. The NI 9234 module

s used to record accelerometer data [32]. These accelerometersere attached to various parts of the machine tool using hot glue.

he PCB Piezotronics Model 086B01 [33] is a small impulse ham-er used to strike the machine structure for modal analysis. The

ard tip is used to create as sharp of an excitation force as possibleo generate a wide band of excitation. The Kistler 9726A5000 is aarger impulse hammer used to excite larger structures. The soft-

are used for the modal tests is produced by the Manufacturingutomation Laboratories Corporation in British Columbia, Canada.his program was used to measure natural frequencies, dampingatios, modal mass and modal stiffness of dominant modes in theachine structure [34]. A list of instrumentations used in this study

s presented in Table 3.

.5. Measurement of surface topography

ZYGO NewView 5000 White Light Interferometer (WLI) was



xtensively used to characterize the surface form error. The spin-le star pattern was resolved with the stitching application built

nto the ZYGO MetroPro software with a 1X Michelson objective at.5× magnification with a lateral resolution of 22.7 �m and a FOV of

Optical isolator SparkFun Electronics BOB-09118Force hammer PCB Piezotronics Model 085B01

Kistler 9726A5000

14.5 mm × 10.9 mm with 15% overlap. This reduced spatial resolu-tion allowed the machine to essentially ‘ignore’ surface roughnessand measure larger undulations and waves in the finished surface.

4. Results and discussion

In the following subsections, the results and discussions on anumber of observations made in this study will be presented.

4.1. Balancing: major source of error

Balancing of the rotating shaft is extremely important sinceunbalanced mass can lead to cylindrical orbital movement, axialtranslation and eventually a gyroscopic precession movement asdiscussed before. The spindle balance procedure was performedto reduce the vibration amplitude as much as mechanically possi-ble before considering the form error. Then a systematic imbalancewas introduced on the spindle by adjusting the balancing screws.Machining was done before and after the balancing to demonstratehow precession due to imbalance influences the process and tohighlight the importance of balancing. Balancing was monitoredby an accelerometer in the radial direction of the spindle. Bandpass filtering of the accelerometer signal was done to only observethe rotational frequency (1st harmonic frequency). Fig. 5(a) showsthe acceleration signature before and after balancing at 1000 RPMfor 1st harmonic excitation for one revolution of the spindle. Thepeak vibration amplitude after balancing decreased from 0.034 g to0.001 g, where 1 g = 9.81 m/s2. Machining was done with a spindlespeed of 3000 RPM, feed rate of 5 �m/rev, and a depth of cut of 2 �mwith a 1.50 mm nose radius tool in balanced and unbalanced condi-tions. Fig. 5(b), (c) show the WLI measurement of the whole surfacewith circular line profiles at a radial distance of 10 mm from the cen-ter of the workpiece showing that peak to valley (PV) undulationon the surfaces are 174 nm and 92 nm respectively for a slightlyunbalanced and balanced spindle. This observation suggests thatit is not possible to remove all imbalances from the spindle usingonly the off plane set screws; however, careful set up can greatlyminimize the error.

Before proceeding toward the reason for formation of spindlestar which is distinctively visible near the center of the machinedpieces, the authors would like to introduce another observationon the spread of spindle star on the generated surface. To explainthis observation, Fig. 5(d) is introduced where the circular line pro-files are drawn at 27 mm, 17 mm and 5 mm radial distances fromthe center of the workpiece showing a gradual decrease in theirPV values and presence of a similar number of radial spindle star


spokes over the entire surface. It is reasonable to argue that at thispoint the amplitude of such precession movement will be higher ata higher radial distance due to Abbe’s principle of error as describedwith Fig. 3. Only the axial movements near the center will be more




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whereas Fig. 5 captured the whole machined surface. The reasonthe thrust force component was used to draw the conclusions willbe discussed in the next section.

Table 4Expected and actual number of spokes in the star pattern.

Obs. no Spindle speed(RPM)

Actual disturbancefrequency (Hz)

Actual numberof lobes (N)

a 500 400.00 48b 1000 400.00 24c 1021 400.80 24d 2000 400.00 12

ig. 5. Spindle balancing and its effect on form error: (a) Accelerometer signature

nbalanced condition; (c) surface topography when machined under a balanced ceing machined under a balanced condition.

ominant for the same reason. More observation on the spread ofpindle star is presented later.

.2. Spindle star: not an asynchronous error

From the results presented in Fig. 5(b), (c) and (d), the 8 spin-le star lobes are prominent on the surfaces when machined at000 RPM. The star pattern is very clear near the center; however,

t is obscured near the periphery. Thus it can also be concludedhat the star pattern is a result of some systematic disturbance fre-uency in the axial direction normal to the workpiece face. Creating

lobes in the star pattern also means that there should be an axialisturbance frequency of 400 Hz when the spindle is rotating at aub synchronous frequency of 3000 RPM (50 rps) according to theollowing equation:

d = [(RPM/60) × N] (1)

here fd is the disturbance frequency and N is a positive integerepresenting a harmonic of the rotational frequency. Thus, if theisturbance frequency is asynchronous at 400 Hz, the number of

obes should always have a simple mathematical relationship withPM according to Eq. (1). Experiments were performed to create a



hase lag of vibration between rotations of the spindle in order toink or scramble the lobes of the spindle star pattern on the dia-ond turned surface by slightly changing the RPM. It was observed

hat the number of spindle star lobes stayed constant despite the

lanced and unbalanced spindle; (b) surface topography when machined under anon; (d) circular profile map of the surface at different radial locations taken after

small change in spindle speed. Moreover, it has been observed thatthe number of spokes in a spindle star created by the disturbancefrequency will change depending on the spindle RPM. The disturb-ance frequency was detected using the Kistler force dynamometerin the tool thrust direction. A summary of the force dynamome-ter measurements is shown in Table 4. To confirm the results, thefreshly cut surfaces were scanned on the Zygo WLI, shown in Fig. 6.The numbers of spokes are observable using 360◦ surface profiletaken from these figures. Fig. 6 is the WLI measurement of thesurface near the center with a stitched FOV of 30.2 mm × 24.4 mm


e 3000 400.00 8f 3200 426.67 8g 3310 441.33 8h 3429 457.20 8




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dtfnLlatigsd

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tioapdrCfiwfpt

ig. 6. Observation on spindle star: (a) 500 RPM (48 spokes); (b) 1000 RPM (24 spo200 RPM (8 spokes); (g) 3310 RPM (8 spokes) and (h) 3429 RPM (8 spokes). 10× m

The observations shown in Table 4 and Fig. 6 will be used toevelop the understanding of form error. The assumption thathe disturbance frequency is consistently exciting the naturalrequency around 400 Hz is incorrect and the spindle star phe-omenon does not appear to be an asynchronous error problem.ikewise, small changes in the RPM do not change the number ofobes or disturb their structure; for example between 1000 RPMnd 1021 RPM the number of lobes stays the same. Rather, the dis-urbance frequency (Eq. (1)) shifts to produce an equal number ofnteger patterns (lobes) depending on the RPM range. It also sug-ests that the PV of the undulations is dominated by the spindletar pattern at higher RPMs. A thorough investigation of the rotor-ynamic properties of the air bearing spindle is therefore required.

.3. Rotordynamic analysis: observation on spindle star

A rotordynamic analysis was performed to further understandhe dynamic behavior of the air bearing spindle in SPDT. Analyt-cal and numerical simulations were not performed due to a lackf information on the geometry of the spindle. An experimentalpproach was taken instead. Since gyroscopic effects cross cou-led with direction stiffness of the air bearings are responsible for aependency of the natural frequencies on the spin velocity, it can beepresented by the Campbell waterfall plots [20]. To generate theampbell plot for rotordynamic analysis, cutting tests were per-

ormed with spindle speeds ranging from 400 RPM to 3600 RPMn 200 RPM intervals. A sharp SCD tool with 12 mm nose radius



as used to create Campbell plots as shown in Fig. 7. In the water-all plots, a line can be drawn on the frequency-spindle speedlane using Eq. (1) described earlier. The accelerometer attachedo the spindle in the radial direction represents all the frequencies

(c) 1021 RPM (24 spokes); (d) 2000 RPM (12 spokes); (e) 3000 RPM (8 spokes); (f)cation with field of view of 30 × 24 mm.

generated by the spindle during operation, including electricalnoise and turbulent air in the air bearing as shown in Fig. 7(a).Despite the presence of noise, Fig. 7(a) shows a clear trend as a1st harmonic excitation in the graph, as well as a range of largepeaks ranging from 300 Hz to 500 Hz in the radial direction. Again,an accelerometer was placed on the tool in the thrust direction.This accelerometer and the tool’s force dynamometer representthe mechanical forces transmitted from the spindle, through themachine structure, and to the cutting tool itself. In Fig. 7(b) for theaccelerometer on the tool thrust direction one can see that mostof the frequencies that make up the spindle star pattern fall within300 Hz to 500 Hz. The 1st harmonic frequencies detected by theKistler 8730A500M1 500 g accelerometer (Fig. 7a) was not detectedby the Kistler 8640A5 5 g accelerometer (Fig. 7b). If this region is iso-lated for spindles speed below 2000 RPM (Fig. 7c), the peak frequen-cies are observed to stay near 400 Hz. Diamond turning is typicallyperformed below 2000 RPM, leading to the incorrect assumptionthat the spindle star phenomenon is an asynchronous spindle error.However, at all of these spindle speeds, the peak frequency creat-ing the spindle star was always a harmonic of the spindle frequencyand was in fact a synchronous error. This trend is also observed inthe cutting force data where a change in the depth of cut will imparta large change in the thrust force. Fig. 7(d) shows the waterfall plotgenerated using the frequency component of the thrust force.

1st harmonic and 2nd harmonic are obvious from the trendsin the waterfall diagram of the thrust force as shown in Fig. 7(d).There are also a series of peaks at the 8th harmonic between 300 Hz


and 500 Hz when the spindle speed is over 2800 RPM. Interestingly,this 8th harmonic appears on the surface as a spindle star with 8lobes at 3000 RPM and above as shown in Fig. 6(e), (f), (g) and (h).This is also true for Fig. 6(a), (b), (c) and (d); however, the spikes


Please cite this article in press as: Tauhiduzzaman M, et al. Form error in diamond turning. Precis Eng (2015),http://dx.doi.org/10.1016/j.precisioneng.2015.03.006



Fig. 7. Waterfall plot for: (a) 500 g accelerometer mounted on spindle body in radial direction; (b) 5 g accelerometer mounted on tool holder in axial direction; (c) after (b)for spindle speed under 2000 RPM and (d) Thrust force.




F surfac2

olhiddlo

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fltdgmrer

Fa

ig. 8. Influence of harmonic vibrations in forming waviness and spindle star on a

nd and 48th harmonic vibration.

f the disturbance frequencies are unclear at lower RPM due toess strength in the signal when compared to higher a RPM. 1starmonic, 2nd harmonic and 8 lobe undulations are clearly visible

n Fig. 5(b), (c) and (d) by two high points and two low points and theominant star pattern near the center. By analyzing the frequencyomain of a force or accelerometer measurement, the spindle star

obe pattern can be easily determined and can be a useful tool fornline monitoring of the spindle star error.

.4. Causes of spindle star error

There are three frequencies that have the most influence on theatness and form error of the part. These errors can be seen asrends in the waterfall plot for the force dynamometer in the thrustirection (Fig. 7d) and in the interferometer scans for all tests. Thereatest contributors to the form errors are the 1st and 2nd har-



onics of the rotational frequency, creating 1 or 2 lobes on theesulting surface. The next contributor is the spindle star error, anrror that ranges from the 8th through to the 48th harmonic of theotational frequency in this study.

ig. 9. Vibration Data Overlaid on FRF of Spindle, Axial Direction: Frequency response funir-hammering at no RPM in radial and axial directions.

e: (a) 3000 RPM showing 1st and 8th harmonic vibration and (b) 500 RPM showing

Using a scan from the Zygo WLI, the frequency that affects thesurface can be clearly seen by taking a circular profile centeredon the part. In Fig. 8(a), the circular profile is bowed, showing anexample where both the first harmonic and the eighth harmoniccontribute to the flatness of the part when turned at 3000 RPM.The same part turned at a 500 RPM was also scanned using theZygo WLI and shown in Fig. 8(b). It showed less magnitude in spin-dle star error (48th harmonic) and more error in its 2nd harmonicerror. The slower speed reduced the spindle star error but did notimprove the 2nd harmonic error. This can be seen from the PV ofthe surface profile of both of these parts. The 1st and 2nd harmonicerrors are caused by spindle imbalance. Van Osch et al. published avibration troubleshooting chart [17], stating that a common fault inrotordynamic system is the misaligned, bent shafts with imbalancetypically cause 1st and 2nd harmonic frequencies. These vibrationswere detected using accelerometers and force dynamometers and


the spindle balance procedure can make an improvement in vibra-tion by adjusting the balancing set screws around the spindle chuck.

The disturbance frequency associated with a spindle star errorover the entire spindle speeds used in this study was between

ction (100 times) and FFT of thrust forces for all RPMs. Inset is showing spontaneous




Table 5Summary of Modal Tests.

Test condition Test location Dominant frequency (Hz) Machine features eliminated Modal stiffness (N/�m) Damping ratio (%)

Machine online Spindle FaceTool Fixture

384 None 66.201 11.503

411 80.488 4.670

Y-Axis disabled Tool Fixture 408 Y-axis active position control 80.018 5.640

Emergency Stop Spindle FaceTool Fixture

385 Active PositionControl, All Axes

41.173 14.151

419 90.372 4.198

3ostiatFasIbiiwosFpeabs

sbniaa

pateatds3umtm

TT

Air supply shutdown Spindle Face, TwoPeaks

384629

00 Hz to 500 Hz. This region is also close to the natural frequencyf the spindle in both the axial and radial direction. The modal testhowed that the natural frequency of the spindle in the axial direc-ion is 385 Hz. The frequency response function (FRF) of the spindlen the radial direction shows two peaks, one at 392 Hz and anothert 465 Hz. The vibration data from the thrust forces are overlaid onhe FRF of the spindle in the axial direction and are presented inig. 9. When performing the modal test, the spindle is struck withn instrumented force hammer. The accelerometer attached to thepindle measures the spindle’s response to the hammer strike.n this case, the spindle is responding by vibrating across the airearings which absorb the blow and undergo assisted air hammer

nstability. Air hammer instability is also present when the spindles not moving. A measurement was made with the accelerometers

ithout the use of the force hammer to determine the frequencyf the air hammering effect. The FFT of the accelerometer mea-urements are shown in the inset of Fig. 9. It can be observed fromig. 9 that the disturbance frequencies from the thrust force com-onent are shifting to a higher value with increasing RPM. Kozanekt al. [28] in 2009 performed a study on the dynamic properties oferostatic journal bearings and concluded that that for aerostaticearings, the bearing stiffness coefficient increases with spindlepeed.

When the spindle is not rotating, the accelerometer signals aretill not zero. This is because the air hammer instability of the airearings causes the spindle to vibrate. An FFT analysis on these sig-als reveals that they are vibrating below 600 Hz due to air hammer

nstability. Once the spindle begins rotating, the spindle imbal-nce results in gyroscopic precession which eventually leads to thessisted air hammer instability.

An investigation of the individual machine components wereerformed to isolate the source for spindle star error by performing

series of modal tests. The first set of modal tests was performed onhe machine with all its components online to detect the source ofrror more accurately. A natural frequency of 384 Hz was detectedt the spindle, and 411 Hz was detected at the tool fixture. An extraest was performed with the accelerometer attached to the outeriameter of the spindle head to find the natural frequency of thepindle in the radial direction. Two frequencies were detected at92 Hz and 465 Hz as described earlier. The second test required the



se of the Precitech’s cylinder balancing utility to disable the linearotion control of the Y-axis. By doing this and repeating the modal

est, the disturbance from the controller compensating for move-ent in the Y-axis is eliminated. A modal test was not performed

able 6ools used in spindle star investigation.

Tool Nose radius Edge preparation

1 12 mm None

2 12 mm Honed cutting edg3 0.5 mm None

Spindle Air Bearing 348.28 9.55469.181 24.638

on the spindle as the disabling of the Y-axis does not affect theZ-axis and spindle. The detected natural frequency was identical tothe previous test at 408 Hz, eliminating the Y-axis controller as thecause of asynchronous motion. The third test required the machineto be in “Emergency Stop” mode. When the emergency stop buttonis pressed, mechanical brakes are engaged on the Y-axis while thecontroller releases control to the X- and Z-axes, allowing themto freely float on the machine’s hydrostatic bearings. A modaltest was performed on the spindle face with a detected naturalfrequency of 385 Hz. The modal test for the tool holding fixturedetected a dominant frequency at 419 Hz. The mechanical brakesincreased the stiffness of the Y-axis, causing the natural frequencyto shift upwards slightly. The Z-axis slide was also tested with anaccelerometer mounted to the base of the Z-axis. A frequency of396 Hz was detected. For the fourth series of tests, the air supplyto the machine tool was shut off, disabling the air bearings in thespindle. The “Emergency Stop” mode continued on the machine,locking the Y-axis using its mechanical brakes and disabling thecontrol of the linear motion of this axis on the machine. No changeswere detected to the frequencies in the tool-holding fixture ascompared to the previous tests. While testing the spindle face, twodominant frequencies were detected, at 384 Hz and 629 Hz. Themodal stiffness associated with 384 Hz is much higher without anair supply as the rushing air will no longer cushion the structureinside the spindle, the bearing surfaces inside the spindle are nowin direct contact. When the air supply is engaged, the rushing of airinto the spindle body will excite a structure within the air bearingspindle at approximately 385 Hz. Once the air supply has beendisabled, the structure is no longer excited by the air, allowing thevibration of other components in the spindle to be detected. There isalso no possibility of air hammer instability without an external airsupply. The results of these modal tests are summarized in Table 5.

4.5. Factors influencing spindle star error

4.5.1. Tool geometry: process dampingCutting tests were performed with different tools to determine

the effects of process damping in an attempt to decrease the ampli-tude of the waviness in spindle star. Three SCD tools were used witha spindle speed of 1000 RPM, feed rate of 5 �m/rev and a depth of


cut of 2 �m. All three tools produced the same 24-lobe pattern, cor-responding with a disturbance frequency of 400 Hz at 1000 RPM.These surfaces were scanned using the white-light interferometer.The annular Peak-To-Valley (PV) reading was taken from a circular

Contact length PV (nm)

0.21909 mm 80.693e 0.21909 mm 87.061

0.04474 mm 82.137




F RPM ds s too

pa

PlSc

lw0acc

ig. 10. Tool geometry in spindle star formation: (a) thrust force signature at 1000ignature at 3000 RPM during air-cutting and during cutting with 12 mm nose radiu

rofile 10 mm in diameter from the center of the part. These resultsre summarized below in Table 6.

The result was that these tools do not show significant change inV in the amplitude of the spindle star error. With a small contactength and low cutting forces, the damping effect of the differentCD tools is insignificant when compared to other factors in theutting process.

The vibration frequency that causes the spindle star error isargely independent of the tool-workpiece contact. An experiment

as performed where the RSA 6061 workpiece was cut using a



.5 mm nose radius tool at 1000 RPM with a feed rate of 5 �m/revnd a depth of cut of 2 �m. The machine tool was then set to run theutting tool about 2 �m away from the freshly cut surface in a pro-ess referred to as air cutting. A second test was performed with the

uring air-cutting and during cutting with 0.5 mm nose radius tool; (b) thrust forcel and (c) waterfall plot of thrust direction during air cutting.

same cutting parameters, but with a 12 mm nose radius tool andat a spindle speed of 3000 RPM. The FFT of the force dynamome-ter signals were extracted from LabVIEW and shown in Fig. 10(a)and (b). Both tests showed similar frequencies at 400 Hz but withdifferent force amplitudes. This means, the changes in tool noseradius and tool contact do not affect the disturbance frequencythat causes spindle star error. Therefore, the tool-workpiece con-tact is not a major factor in determining the disturbance frequencyof the spindle star phenomenon. These two results also suggestthat the disturbance frequencies and its characteristics are inde-


pendent of the cutting process and are in fact a part of the machine’sdynamic behavior. The amplitude of such vibration increaseswith higher RPM due to less damping properties of air at higherspeeds [23,28].




310 H

td2

Fig. 11. Effect on adding 4 kg mass: (a) natural frequency went down to

During rotordynamic analysis, the force dynamometer gave



he most complete picture of vibration in SPDT when cutting. Itetected the low frequency vibrations associated with the 1st andnd harmonic, but also the higher frequency vibrations associated

Fig. 12. Force Measurement near outer diameter

z at 1000 RPM and (b) WLI measurement at 1000 RPM with 4 kg mass.

with the spindle star pattern. Similar rotordynamic analysis was


done during air-cutting. From the results shown in Fig. 10(c), theforce dynamometer only detected the 8th, 10th, and higher har-monics of the spindle frequency. This further confirms that the

, middle and inner diameter at 3000 RPM.




F ce; (b

dbaco

4

mw4a1matb1s

utiptTsTaeon

4

ooetsdaewv

4

twdiWt

i

v

ig. 13. Material and its effect on spindle star: (a) Diamond turned copper workpie

isturbance frequency associated with spindle star is not affectedy the tool-workpiece contact. Two asynchronous frequencies atround 550 Hz and 675 Hz were not observed during cutting, asan be seen from Fig. 7(d) and had no effect in forming the spokesf the spindle star pattern.

.5.2. Adding a mass: changing natural frequencyIt has also been observed that the spindle star error can be

anipulated by adding mass to the spindle. The same RSA 6061orkpiece was mounted to a steel workpiece holder weighing

kg. This decreased the natural frequency of the spindle to 300 Hz,s determined by a modal test. A cutting test was performed at000 RPM, 5 �m/rev feed rate, with a 2 �m depth of cut. The addedass changed the disturbance frequency from 400 Hz to 300 Hz,

s seen by the force dynamometer in Fig. 11(a). This change inhe disturbance frequency from 400 Hz to 300 Hz revised the num-er of spindle star lobes from 24 to 18 at a rotational frequency of6.667 Hz according to Eq. (1). This was confirmed with a workpiececan using the Zygo WLI (Fig. 11b).

This result indicated that the spindle star pattern can be manip-lated by adding mass to change the natural frequency. Howeverhis increased mass, and consequently, increased inertia, did notmprove the PV of the surface. Using a 10 mm diameter circularrofile on the part, the PV was 193 nm. From Fig. 5(b), we can seehat the average PV is typically falls around 80 nm at 1000 RPM.his higher value of PV is due to the higher gyroscopic preces-ion motion resulting from higher gravitational force W (Fig. 3b).he remaining method to reduce vibration amplitude is thus theddition of damping. Process damping due to the cutting tool wasxplored as mentioned earlier but did not provide sufficient levelsf damping to impact the form error. Thus other sources of dampingeed to be considered in the design of the spindle.

.5.3. Spread of a spindle star signatureFor a cutting test at 3000 RPM, feed rate of 5 �m/rev and a depth

f cut of 2 �m, three force measurements were taken: near theuter diameter, in the center of the part, and near the inner diam-ter of the part. The FFT data is shown below in Fig. 12. Again,he 60 Hz reading is not considered as it corresponds with thewitching frequency of the charge amplifier used with the forceynamometer. These three measurements show the same disturb-nce frequency of 400 Hz and 500 Hz is present in generating thentire surface that can also be seen as a spindle star on the finishedorkpiece with optical measurements as shown in Fig. 5(d). These

ibrations are consistent, even at the center of the workpiece.

.5.4. Material and its effect on spindle starIt was also observed that these vibrations are independent of

he workpiece material. A copper workpiece, 25.4 mm in diameter,as diamond turned at 1000 RPM with a feed rate of 5 �m/rev and a



epth of cut of 2 �m. An optical-quality finish was achieved, shownn Fig. 13(a). This workpiece was scanned using the Zygo WLI. The

LI scan in Fig. 13(b) shows a spindle star pattern that extends tohe very center of the workpiece with 24 lobes, a pattern identical

) Spindle star in copper workpiece, 1000 RPM, 24 Spokes, 100× magnifications.

to that of the aluminum workpiece. Thus, the spindle star error isindependent of workpiece material.

5. Conclusions

This analysis leads us to the following conclusions:

i. In an air-bearing spindle system, residual imbalance is the majorcause of form error. Imbalance is practically impossible to elim-inate; however, should be minimum.

ii. Due to residual imbalance, overhang mass, and soft air bear-ings, the spindle may have axial, cylindrical and conical orbitalmovements. Axial movement is generated by angular momen-tum where cylindrical movement is generated by imbalance.The overhung mass causes a bending moment of the shaft dueto gravity which results in a gyroscopic precession and conicalorbital movement.

ii. Axial and conical movement causes the depth of cut to bechanged, which creates undulations on the surface at a levelthat are critical for most diamond turned components.

iv. Once the conical movement has been initiated, the air-bearingwill try to reach equilibrium by exciting the spindle at somehigher order harmonic of its rotational natural frequency. Thisis called assisted air hammering. Air-hammering can be inher-ent to an air-bearing system due to imbalance. Star formationdue to air hammering is more visible near the center where thegyroscopic precession has minimum influence.

v. The rotating spindle and gyroscopic effects creates a forcedvibration problem. Due to the characteristics of air bearingsand Boyle’s Law, the spring rate (cross-coupling stiffness) isnon-linear and the hysteresis effects may be introduced as thestiffness is not uniform for tension and compression cycles in theair-bearings. This results in a different natural frequency for thefirst and second half-cycle of vibration representing the tensionand compression phase of a single cycle of vibration. The systemthus have an infinite number of natural frequencies, multiplesof the circular frequency of a full cycle ω, 2ω, 3ω. . . Since, gyro-scopic effects cross coupled with direction stiffness of the airbearings are responsible for a dependency of the natural fre-quencies on the spin velocity, it can represent by the Campbellwaterfall plots.

vi. Due to the inherent rotordynamic nature of a system, it is possi-ble to characterize a rotating air-bearing spindle to extrapolateits behavior during cutting. The nature of rotordynamic and air-hammering can be monitored and predicted by: thrust forcemeasurements and with the use of accelerometers.

ii. Spindle star is a synchronous error. It was found that the naturalfrequency of the spindle lies in between 300 Hz to 500 Hz. Fromthe Campbell diagram, it was concluded that the disturbancefrequency needed to create a given number of spokes in a spindle


star pattern lies in that range. The expected numbers of spokesare then a simple integer value according to Eq. (1) when takenin relation to the rotational frequency of the spindle. However,in this study an even number of spokes in the star pattern was


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always observed while the reason for this is still unknown atthis stage.

ii. It was observed that the use of different diamond tool geome-tries did not change the star pattern. This is probably because ofthe low cutting force associated with diamond turning.

x. A copper workpiece was machined to compare with RSA 6061aluminum and no difference was observed in the star patternsuggesting that the phenomenon is independent of the work-piece material.

x. To validate our understanding, the mass of the workpiece wasincreased to change the natural frequency of the system. Byadding a 4 kg mass, the natural frequency of the spindle wasdropped down to 300 Hz from 400 Hz. The number of spokesin the star pattern dropped down to 18 from 24 with addedmass as expected; however, the undulation due to precessionincreased due to more gyroscopic effect from a heavier massthus the magnitude of the undulations increased.

i. For air-bearing spindle, the gyroscopic effect (and thus the spin-dle star error) can be minimized by orifice modification, reducedcross-coupling stiffness, and decreased natural frequency or byadding damping to the rotating system.

cknowledgments

The authors would like to thank the Natural Sciences and Engi-eering Research Council of Canada (NSERC), the Ontario Centers ofxcellence (OCE) and B-Con Engineering for their financial supportf this research. Authors would like to thank Ms. Heather Morrisonor proof reading the document.

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