Non-contact measurements and modelling of milling machine tool vibrations

6:5 LICENTIATE T H E S I S Non-contact measurements and modeing of miing machine too vibrations Matti Rantatao Lueå University of Technoogy Department of Human Work Sciences Division of Sound & Vibration 6:5 ISSN: 14-1757 ISRN: LTU-ic -- 6 5-- SE

Non-contact measurements and modeing of miing machine too vibrations Matti Rantatao Division of Sound & Vibration Lueå University of Technoogy SE-971 87 Lueå Sweden Key words: machine too spinde, centrifuga, gyroscopic, non-contact measurement, anguar contact ba bearings, machine too chatter, aser Dopper Vibrometry, specke noise, rotor bearing systems, FEM, magnetic excitation, inductive dispacement sensors.

ACKNOWLEDGEMENTS First of a I woud ike to thank my oving and encouraging famiy; Christina, Uno, Kerstin and Marianne, my parents Magda and Hugo and my parents in-aws Mona and Bengt - you a made it possibe for me to combine a oving famiy ife with the effort that must be invested in writing the thesis. Secondy, I woud ike to thank my supervisor, Professor Anders Ågren, for giving me this once in a ifetime opportunity to participate in a PhD study. A his hep during my studies and his being a great boss made a the difference. I woud aso ike to express my gratitude to Dr. Per Gren and Dr. Jan-Oof Aidanpää of Lueå University of Technoogy (LTU) and Bo Göransson of SKF for their hep, dedication and genuine interest in my work. I woud aso ike to thank my coeagues Peter Norman and Kourosh Tatar for their work and for making the beginning of my PhD studies a joyfu and memorabe time in my ife. I woud aso ike to express my gratitude to the rest of the peope invoved in LTU s research project in this genera area for interesting discussions, ideas and their interest: Tore Siver, Dr. Aes Svoboda, Prof. Inge Svennigson, Tommy Gunnarsson and Dr. Mikae Bäckström. Finay I woud ike to thank my current and previous coeagues at the Department of Sound and Vibration for sharing their knowedge and being great friends. The work conducted was financed by the Swedish Agency for Innovation Systems (Vinnova). The funds to pay for the aser Dopper Vibrometer (LDV) used in this research came from the Kempe Foundation.

ABSTRACT This thesis concerns the deveopment of non-contact measurement methods and anaysis of rotors. The methods have been verified and appied to miing machine spindes in order to investigate the speed dependency in the miing machine spinde dynamic. The research was financed by the Swedish Agency for Innovation Systems (Vinnova). Turning operations ike miing are common in the automotive and aerospace industries where arge meta work pieces are reduced to a fraction of their origina weight when creating compex thin structures. During these operations it is important that unwanted behaviours such as excessive too vibrations be avoided (this is normay caed chatter ). Chatter causes poor surface finish and/or materia damage and can expose machine operators to annoying and/or dangerous noise eves. In order to predict processes parameters for a chatter free miing operation, knowedge of the properties of the dynamic system are essentia. Normay the system dynamics are measured during no rotation; in order to incude the infuence of the spin speed the system must be anaysed for a spinde speeds intended for the miing operation. This can be done either by measurement or modeing. Non-contact measurement techniques are however, often based on dispacement sensors which do not have the same sensitivity as veocity or acceeration based methods. To improve the sensitivity in non-contact measurements of rotors a aser Dopper Vibrometry (LDV) based method has been deveoped. The deveoped LDV method is based on the reduction of the rotor surface structure and makes it possibe to use singe beam LDV measurements of rotors. These types of measurements were previousy considered inaccurate but now have become feasibe through the use of the method described in this research. Furthermore the dynamic properties of a high-speed-miing machine spinde were studied by a contactess dynamic spinde tester (CDST) deveoped by SKF. The measurements were substantiated by simuations using a finite eement mode (FEM) which confirmed the measurement resuts. The CDST measurements coud be performed without vioating safety reguations regarding human interaction with high speed spindes through the use of a magnetic excitation method. In the measurements conducted by the CDST a speed dependency in the spinde dynamic coud be detected. By performing FEM simuations the major source of this dependency coud be identified. The centrifuga force of the bas in the anguar contact ba bearings was shown to have the argest infuence on the overa dynamics compared to the gyroscopic moment of the rotor. The study performed indicates that predictions of high-speed-miing stabiity must incude consideration of the speed dependency in the dynamic.

CONTENTS 1 THESIS 1 INTRODUCTION 5 3 LDV MEASUREMENT METHOD 8 4 SPINDLE MEASUREMENTS 16 5 MODELLING 6 DISCUSSION 8 7 CONCLUSIONS 3 8 FUTURE WORK 33 9 NOMENCLATURE 33 1 REFERENCES 34 APPENDIX A: SPINDLE DRAWING APPENDED PAPERS PAPER I PAPER II PAPER III PAPER IV

1 THESIS This thesis begins by introducing readers to the research probems of miing machine vibrations and spinde measurements. It then continues on by describing different non-contact measurement methods and, in particuar, the theoretica background and probems associated with aser Dopper Vibrometry (LDV) when appied to rotating targets such as miing machine spindes. A method for soving these LDV probems is presented together with a method for miing machine spinde anaysis using inductive dispacement sensors and eectromagnetic excitation substantiated by FEM simuations. Lasty, the findings are discussed aong with suggestions to future work. This thesis covers work described in greater detai in the four attached papers. 1.1 Paper I M. Rantatao, P. Norman, K. Tatar, Non-contact measurements of too vibrations in a miing machine, SVIB vibrations Nytt, () (4) -9 This paper presents a pre-study of two types of non-contact measurement methods for miing machine spinde vibrations. LDV on a rotating spinde and the use of an active magnetic bearing (AMB) for spinde dynamic measurements were tested. The AMB study was conducted on a ow speed (-7 rpm) miing machine spinde. The methods and findings in this work were further studied in Papers II, III and IV. Matti Rantatao prepared the too, outined the work and performed the AMB measurements with the assistance of Peter Norman. The LDV measurements were performed by Kourosh Tatar and Matti Rantatao and the miing machine was operated by Peter Norman. Matti Rantatao performed the post processing and anaysis of the data. Matti Rantatao wrote the paper with the assistance of Peter Norman and presented the work at the SVIB 4 Conference in Stockhom. Pubished in SVIB s member journa 4-. The paper was not subject to a review procedure. 1. Paper II M. Rantatao, K. Tatar, P. Norman, Laser Dopper Vibrometry measurements of a rotating miing machine spinde, in: Proc of the Eighth Internationa Conference on Vibrations in Rotating Machinery, Swansea UK, (4) () 31-4. The paper describes a method for specke noise remova in LDV on rotating targets. The study is an extension of the LDV study presented in Paper I. The tite of the paper is a bit miseading in that a more genera work is suggested than was actuay 1

presented. A more representative tite woud be. A method for specke noise remova by surface structure reduction - an LDV appication to rotating targets. Matti Rantatao, Kourosh Tatar and Peter Norman outined the work. Matti Rantatao performed the post processing and anaysis of the data with the assistance of Kourosh Tatar. Measurement data ogged in Paper I was used in this paper. Matti Rantatao wrote most of the paper with the assistance of Kourosh Tatar and Peter Norman. Correspondence with the editor of the conference proceedings and the reviewers was conducted by Matti Rantatao Pubished in conference proceedings. The paper was subjected to a review procedure. 1.3 Paper III K. Tatar, M. Rantatao, P. Gren, Laser vibrometry measurements of an opticay smooth rotating spinde, submitted to: Mechanica Systems and Signa Processing, (6). The work incudes an investigation in the presence of crosstak between radia vibration components after using the method for specke noise remova presented in Papers I and II. Kourosh Tatar and Matti Rantatao outined the work with the assistance of Per Gren. Kourosh Tatar prepared and verified the dummy too quaity and performed together with Matti Rantatao the experiments. Matti Rantatao conducted the AMB measurements and Kourosh Tatar the LDV measurements. Kourosh Tatar performed the post processing of the data and together with Matti Rantatao and Per Gren the data was anaysed. Kourosh Tatar and Matti Rantatao wrote most of the paper with the assistance of Per Gren. Kourosh Tatar was the corresponding author. The paper has been accepted for pubication by Mechanica Systems and Signa Processing. 1.4 Paper IV M. Rantatao, J-O. Aidanpää, B. Göransson, P. Norman, Miing machine spinde anaysis using FEM and contactess spinde excitation and response measurement. Manuscript submitted for pubication. This paper demonstrates a method for spinde anaysis incuding FEM, contactess excitation and response measurement. The method was appied to a high-speedmiing machine with a spinde speed capacity of up to 4 rpm. The experimenta part in this paper, regarding the inductive measurement method, is a seque to the study presented in Paper I where a ow speed spinde was studied.

Matti Rantatao outined the work and performed the CDST measurements and tap tests with the assistance of Peter Norman. Matti Rantatao performed the post processing and the anaysis of data. Matti Rantatao impemented a FEM representation of the rotor bearing system and performed the simuations with the assistance of Jan-Oov Aidanpää. Measurement of the physica spinde dimensions was performed by the SKF spinde service and Bo Göransson at SKF performed the bearing stiffness cacuations and wrote parts about the physics behind the speed dependent bearings. Peter Norman outined the pre-oad test together with Bo Göransson and wrote about procedure and performed the measurements with the assistance of Matti Rantatao. Matti Rantatao performed the LDV measurements of the mode shapes. Matti Rantatao wrote most of the paper with the assistance of Jan- Oov Aidanpää and Bo Göransson. The paper has been submitted for pubication. 3

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INTRODUCTION Turning operations ike miing are common in the automotive and aerospace industries where arge meta work pieces are reduced to a fraction of their origina weight in the creation of compex thin structures. In Fig 1 a picture of the high speed (4 rpm) state-of-the-art 5-axes research miing machine used in this thesis can be seen. As with most operations, the deveopment of more efficient processes resuts in increased productivity. This is the case in miing where the process can be made more efficient by optimizing performance in such a way that it better fufis various demands such as fexibiity, machine too endurance, production time, operator work environment and production quaity. Fig 1. Photo of the Liechti Turbomi ST1 miing machine used in studies described in this thesis. A property highy associated with these optimization parameters is machine too vibration. Vibration can expose machine operators to annoying and/or dangerous noise eves. Regenerative machine too vibration (chatter) wi reduce the quaity of a surface finish and can damage a workpiece (down to the moecuar eve). Chatter can aso resut in increased too wear/faiure and/or compex and costy machine too faiures. Therefore it is important that the vibration eve be kept under contro. Chatter is sef-excited machine too vibration which can be caused by various physica parameters [1]. Chatter caused by friction, thermodynamics or mode couping in the cutting process is caed primary chatter. Secondary chatter is caused 5

by the waviness of a machined surface and is considered to be one of the most important causes of instabiity in the cutting process. The secondary chatter phenomenon is a significant issue and has been addressed and modeed by numerous authors over the past decades e.g. [-5]. The deveoped modes predict a specific chatter free depth of cut for a specific spinde speed. The predictions are based on the spinde speed and the frequency response function (FRF) of the too tip; assuming a rigid workpiece. The chatter free depth of cut is cacuated for different spinde speeds which can be potted as stabiity obe charts. Fig shows a typica stabiity obe chart where the area above the curve represents production parameters which wi resut in an instabe process. The area beow represents a stabe process. Increasing the depth of cut to the wrong side of the curve wi generate chatter. 16 14 Stabiity obes Depth of cut [mm] 1 1 8 6 4 Instabe zone Stabe zone 5 1 15 Spinde speed [rpm] Fig. Stabiity obes: The area above the stabiity obe curve represents a depth of cut which wi generate chatter vibrations. Vaues beow the curve wi render in a stabie miing process..1 Research probems The FRF of the too tip, which is used in the stabiity obe cacuations, is normay measured manuay by tap-tests of a non-rotating (rpm=) spinde/hoder/too system where the too tip is excited by an impuse hammer and the response is measured by a vibration transducer. Stabiity obes based on FRF measurements made at rpm are cacuated for a spinde speeds. The assumption in this procedure is that the dynamics of the spinde/hoder/too system is independent of the spin speed. This is 6

however not true for the case of high-speed-miing operations where the effect of gyroscopic moments and centrifuga forces wi change the FRF. [6-8]. The vaue of the depth of cut, in the rpm based stabiity obes, is ony vaid for the speed rpm and not for other spinde speeds uness the FRF is speed independent. In order to achieve spinde speed dependent stabiity obes the FRF s for each spinde speed must be inserted into the cacuations. Tap-tests to measure the high speed FRF woud add an extra risk to work environments of the operator or engineers who conducts such measurements. An aternative, safer excitation method for spinde anaysis is therefore desirabe. Knowedge of the origin of the speed dependency in the system dynamics can aso give vauabe information for spinde design. The physica properties of a fuy operating miing machine can either be anaysed by various measurement methods or by modeing. To accuratey mode spinde dynamics speed dependency the modes must incude a parts which can contribute significanty. Present measurement methods use dispacement based response sensors and various types of excitation methods which are a in some way in contact with the rotor during the measurement procedure. Dispacement sensors have a ower sensitivity to higher vibration frequencies than acceeration and veocity based methods. No substantia research using dispacement sensors for measurement of high frequencies was found during a survey of major databases. As an aternative, a veocity based measurement method woud increase the sensitivity to higher frequencies. LDV is such a method and it is aready a common instrument for vibration measurement. However, the method possesses imitations when appied to rotating targets because specke noise and crosstak between radia vibration components occurs. A method for LDV measurement of rotating rotors has been deveoped [9] but is based on a mutipe set of continuousy measuring LDV s. The possibiity of using a singe beam aser woud reduce the investment in measurement equipment and make it possibe to measure spinning rotors in, for exampe, situations where measurement from two directions is not practica.. Research questions In the previous section the research probems, which are the subject of this thesis, were outined. When anaysing the presented research probem the foowing research questions emerged: Is it possibe to measure radia vibrations of a spinning rotor using a singe beam LDV? How can a spinde be measured without vioating safety reguations regarding human interaction with high-speed-miing machines? What contributes the most to the speed dependent spinde dynamic? 7

3 LDV MEASUREMENT METHOD Vibration measurements of spinning rotors can be carried out using inductive/capacitive dispacement sensors or aser sensors based on trianguation. These methods are however, imited regarding detectabe frequency and ampitude and must be positioned reativey cose to the measurement surface. Laser Dopper Vibrometry (LDV) offers a more sensitive veocity based vibration measurement technique than other non-contact measurements methods. A 64 Hz B Frame mode 1,8 m String mode 1 Snowmobie frame,6 m C R = 18 mm D b = 4 mm String mode Speakers Tennis racket Fig 3. Exampe of scanning LDV measurements on vibrating objects. A: Large area LDV scan of a snowmobie frame. B: An LDV scan of the frame and strings of a tennis racket. C: LDV scan of a sma thin speaker membrane. D: LDV scan of a sma computer component. LDV is now a commony used method for vibration measurements. This technique has many advantages compared to traditiona vibration measurements. For exampe, it is easier to use than acceerometers pus it is often faster to use. Fig 3 shows an exampe of LDV measurements of four objects of different size and shape. The exampe shows a scan of the supporting frame of a scooter reveaing an eigen-mode at 64 Hz ocated at the bet tunne. The other scans show a racket, a speaker membrane and a sma computer component which demonstrates measurement of sma, thin and ightweight structures. The nature of the LDV system permits measurement without additiona mass oading and aows a wide range of distances between the sensor head and the object (from miimetres up to severa meters). Vibration measurement of hot objects can be performed as we as measurements of sma and ight weight structures as e.g. the tympanic membrane in a human ear [1]. 8

The LDV technoogy can aso be appied in other medica appications e.g. for teeth vibration measurements during driing [11]. Another appication of aser Dopper Vibrometry is the measurement of sound wave propagation in transparent medias ike gases [1, 13]. When a sound wave is propagating through a medium it changes the pressure. The changing pressure wi affect the refraction index of the medium and hence moduate the aser frequency. This moduation is then interpreted as a vibration veocity by the LDV system. 3.1 LDV principa A Poytech 1 PSV 3 scanning LDV system was used in this thesis. The LDV was equipped with a scanning aser head with a scanning ange of about ±. The LDV used for this work has a heium neon (He-Ne) aser with a waveength of 63.8 nm. The LDV measures the veocity component of the object aong the direction of the aser beam and is based on the detection of the Dopper shift in the aser ight refected from the surface. (f B =4 MHz) Mirror Osciator Object ωb = πf B Ω Ω + ω B Laser Bragg-ce Ω v v(t) Veocity decoder Beam spitter x(t) Dispacement decoder Detector x(t) Fig 4. Sketch of an LDV. In an LDV the emitted ight with the frequency Ω + ωb is aowed to interfere with the ight refected from the measured object. 1 Poytec GmbH www.poytec.com 9

To determine the target s veocity the aser ight is demoduated by an interferometer where the refected ight is aowed to interfere with the origina transmitted ight (reference ight) on a photodetector. To be abe to determine the direction of the target veocity a virtua constant veocity is added by a Bragg-ce which frequency moduates the reference ight with a 4 MHz signa ( ω B Bragg-ce frequency), see Fig 4. The registered Dopper shift is then used to cacuate the veocity of the object by using the foowing equation; λ v = f D (1) where f D is the Dopper frequency and λ is the aser waveength. 3. Specke When iuminating an opticay rough surface with a aser, a phenomenon caed specke wi occur. Speckes are bright and dark spots that can be seen in a refected aser ight (see Fig 5). These spots are the resut of superimposing ight waveets which are caused by differences in traveed path ength, from ight source to detector, due to the rough surface. A surface is considered opticay rough if the surface structure exceeds the eve of approximatey haf the waveength of the ight source, in this case 316 nm. The specke phenomena is sometimes considered a disturbing noise but in other cases, ike in TV hoography [14], the speckes are used as the information carrier. Fig 5. Speckes on a detector. Speckes are formed at a detector due to superimposing refected ight waveets with different traveed path engths. 1

For a non moving target the summarized waveets are ony seen as an added DC eve to the detector signa. If the target starts to move this DC eve wi start to vary according to the specke popuation present on the detector. This resuts in specke noise. The noise wi moduate the phase of the aser ight and add a noise to the detected Dopper signa. This noise eve can be reduced by averaging due to its random behaviour. 3.3 Specke noise on rotating targets When measuring mutipe revoutions of a rotor using LDV the specke noise can not be averaged out from the signa and must be considered separatey. In a static LDV measurement of a rotor the same specke noise occurs each revoution. If the measurement spans a number of rotor revoutions the specke noise wi be apparent as peaks in the frequency domain at mutipe integers of the rotationa speed. If the rotor speed is constant and the measurement sequence contains many revoutions the specke peaks can easiy be distinguished from ordinary structura vibrations due to their narrow appearance. The specke noise has a simiar appearance to ordinary outof-roundness components which aso are ocated in the frequency spectra at mutipe integers of the rotationa speed. The main difference between specke peaks in the frequency domain and typica peaks originating from the out-of-roundness components is the decaying appearance with increasing frequency that the out-ofroundness components show (see Fig 6). Magnitude A Structura vibrations B Miss aignment & Roundess C x Specke noise f n = 1 3 4 5 6 7 8 9 f n = 1 3 4 5 6 7 8 9 f D n=1 n= n=3 n=4 x Fig 6. Spectra components from LDV measurements. A: Structura vibrations. B: Rotor centre miss aignment (n=1) roundness components (n=, n=3, ). C: Repeated specke noise. D: n=1 iustrates the rotor miss aignment and n=, n=3, iustrate the out-of-roundness components of the rotor surface structure. 11

3.4 Specke noise remova As described here, it has been shown by others that the specke noise eve in LDV can be reduced or removed by different methods. By optimizing the detector size and position within a vibrometer the noise eve can be reduced but not competey removed [15]. It has aso been shown that the specke noise in aser torsiona vibrometry measurements can be removed by randomising the path that the aser ight is undertaking during the revoutions. This is by either by moving the aser aong the shaft [16] or simpy by adding a new surface structure. The atter strategy can be achieved by continuousy appying oi or some other substances to the surface during measurement [17]. In theory, these techniques shoud aso work for LDV measurements. Another approach to specke noise remova is by reducing the target surface structure to beow haf the aser waveength [18, 19] (Papers I and II). Fig 7 shows a comparison between LDV measurements performed on a rotor with a rough (ower graph) and a smooth surface structure (upper graph). The ower graph shows specke peaks at mutipe integers of the rotationa speed (1 Hz). The upper graph shows a spectra with no peaks due to specke. The out-of-roundness components in the upper eft graph coud be extracted with no disturbing specke noise and the resut was verified by a mechanica roundness tester (see Fig 8). Magnitude [mm/s] 1.5 1.5.5 1 1.5 Veocity FFT, 6 [rpm] n=1 Poished n=5 n=3 n= n=6 n=8 n=4 n=7 n=9 Rough 1 3 4 5 6 7 8 9 1 Frequency [Hz] Veocity FFT, 6 [rpm].6.4.. n=14 n=16.4 n11 n=1 n=13 11 115 1 15 13 135 14 145 15 Frequency [Hz] Fig 7. Spectra of the poished and the rough surface at a spinde speed of 6 rpm. The spectrum of the rough rotor has been mirrored aong the frequency axis down to the negative side to simpify comparison between the two. Mutipe harmonics of n*1 Hz (marked by back dots) can be seen in the spectrum of the sprayed surface (n = 1,,3 ). 1

Dummy too circumference profie [μm] 1º 9º 6 6º Mechanica LDV 15º 4 3º 18º º 1º 33º 4º 7º 3º Fig 8. Out-of-roundness. Out-of-roundness components extracted from the LDV measurements compared to a separate mechanica roundness measurement. 3.5 Crosstak When appying LDV to rotating opticay rough targets two probems occur; the presence of specke noise which has been described in the previous section and crosstak between vibration veocity components. The crosstak probem is caused by different veocity components which affect the Dopper shift in the refected ight. In this thesis (from Paper III) the crosstak between the two radia vibration components are studied. Due to an opticay rough surface, backscattered ight from the tangentia veocity component moduated by the in-pane dispacement wi be added to the outof-pane veocity. [9, -]. The measured veocities of a rotating opticay rough shaft in the two orthogona directions, v x, v y, can be expressed as [9]: v y and v x = y& + Ω( x x ) () = x& Ω( y y ), (3) where x& and y& are vibration veocities, x and y are the vibration dispacements, x and y are the distances to the spin axis due to aignment errors and Ω is the tota anguar veocity incuding torsiona vibrations. x& and y& are the desired veocity components for each direction. The methods for specke noise reduction/remova 13

described by previous authors cope with the specific specke noise probem but are not abe to neutraize the effect of crosstak in a singe beam LDV measurement. Consequenty; the signa obtained during measurements under these circumstances wi be a mix of the vibration components in both directions. A method for resoving the true vibrations in the two x- and y-directions using a setup of two simutaneousy measuring asers in both directions and an accurate measurement of the rotationa anguar veocity has been deveoped by Hakon and Rothberg [9]. The method does require a setup of two simutaneousy measuring LDV systems in an orthogona arrangement. In Paper III the cross sensitivity in the deveoped method presented in Papers I and II is investigate experimentay. The crosstak in LDV appied to a poished rotor is compared with the crosstak present when measuring an opticay rough rotor. As a reference, a set of inductive dispacement sensors (DS) measured the position of the rotor in the x and y-direction (see Fig 9). The excitation of the rotor was carried out by eectromagnets. y (t) DS x y x (t) DS Ω F (4 Hz) v& y (t) LDV Fig 9. Crosstak test. Sketch of setup used to examine the crosstak in LDV on rotating targets. 14

Simuated crosstak based on vaues from the reference sensors y and x (DS) LDV on a rough surface LDV on a smooth surface Dispacement sensors (y) for a smooth and rough surface Fig 1. Effect of crosstak in LDV: Veocities at 4 Hz for different spinde speeds 7, 14, 8, 56 and 7 rpm. Fig 1 iustrates the effect of crosstak on LDV measurements of a rotating rough surface for different spinde speeds and an excitation of 4 Hz orthogona to the aser beam. The measured vibration veocity of the dummy too after being sprayed with paint (triange up) shows a spinde speed dependent crosstak as expected from Eq. ( and 3), whie the same measurements on the smooth surface (trianges down) do not. The outputs from the dispacement sensor (DS) in the y-direction for both sets of measurements (smooth and rough measurement surfaces - square and pentagram) showed no physica crosstak during the excitation. Inserting the signas from the dispacement sensors y and x into Eq. () gave the same veocity (circes) as from the vibrometer when the surface was rough (triange up). This cacuation confirms the measured crosstak detected by the LDV when measuring the rough surface. 15

4 SPINDLE MEASUREMENTS During a miing process the cutting force and the frequency response function (FRF) at the too tip are two important parameters. The cutting force can be measured indirecty by the use of a force pate. The force pate measures the force on the work piece by measuring the force transferred by the work piece to the machine tabe. This procedure assumes that the work piece is rigid with no interfering moda properties. A Response Response Ba Bearings B Command votage response Too F Rotor F AMB 1 Rotor AMB AMB 3 Response Radia Bearings Axia Bearing C Response D Response Actuator Rotor Rotor F F Sensor Fig 11. Force and FRF measurement methods. A: Indirect cutting force measurement using inductive/capacitive dispacement sensors. B: Indirect cutting force measurement using the AMB command votage response. C and D: Speed dependent FRF measurement methods. An aternate approach is to measure the force that the work piece has on the too. In order to do this, non-contact sensors measuring the rotor vibrations are used. Noncontact measurement of rotating miing machine spindes customariy use inductive and capacitive dispacement sensors together with aser based dispacements probes [1]. Abrecht et a. [3] describe an indirect method of force measurement when miing that uses capacitive or inductive dispacement sensors. In this method, the transfer function between the force appied on the too tip and the dispacement from capacitive sensors mounted on the spinde cose to the housing is measured for different spinde speeds. Different Kaman fiters were then cacuated for each speed and appied to the dispacement sensor signa to produce rea time cutting force measurement data. Tap tests on a ba bearing mounted on the too tip were used to excite the structure (see Fig 11A). Spiewak [4] presented an aternative acceerometer based cutting force measurement method where a miing cutter was www.kister.com 16

instrumented with a 3-axia acceerometer inside the too cose to the tip. This method requires speciay manufactured toos. For spindes equipped with active magnetic bearings (AMB), Auchet et a. [5] have outined a another method for indirect cutting force measurement based on command votage of the AMB. The method used the reationship between cutting force and an increasing command votage in the magnetic bearings in order to keep the rotor in pace (see Fig 11B). The reationship was estabished by measuring the FRF between a force (tap test) appied to the too tip and the command votage of the AMB. The FRF measured at rpm as a predictor of the cutting force at high speed machining wi, in the view of this author, be a source of error. Using AMB s for measurement and chatter contro purposes has been investigated. Knospe [6] ooked at active chatter suppression through the use of AMB. Chen and Knospe [7] estimated cutting dynamics by both exciting the system and increasing the damping of the athe too using an AMB. Simiar to expoiting the reationship between the AMB command votage and the cutting force, methods which use the current in motorised spindes have been studied. For exampe, Jeong and Cho [8] deveoped a method where they improved the frequency rage from earier methods by a factor of two up to 13 Hz. 4.1 Speed dependent stabiity obes Knowedge about the too tip FRF is important when cacuating the stabiity obes and optimizing the maximum depth of cut. When deaing with rotating dynamic systems ike a miing machine spinde, especiay during high speed machining, the FRF depends on the spinde speed. To achieve the spinde speed dependent FRF, the machine too must be anaysed in a rotating state that spans the whoe range of spinde speeds intended for the operation. An experimenta method for the prediction of stabe cutting regions was presented by Schmitz et a. [9] which took into account the dynamic change that a rotating spinde undergoes. The method is based on impuse hammer excitation and capacitive probe response measurement of a too rotating during different spinde speeds (see Fig 11C). Stabiity obes for a discrete number of spinde speeds are cacuated and the imit vaue corresponding to the actua spinde speed used during the measurement is picked out to form a spinde speed dependent stabiity obe chart. Experimenta tests found that there is a changing stabe imit of cut above 16 rpm. Sims et a. [3] demonstrated a method for too tip FRF prediction based on piezoeectric actuators and sensors mounted near the base of the too (see Fig 11D). The predicted FRF was compared with ordinary impuse hammer tap tests at the too tip. 4. Spinde measurements For this thesis, a contactess dynamic spinde testing (CDST) instrument for measuring the speed dependency was used. The CDST uses inductive response measurement and eectromagnetic excitation of the tip of a dummy too (Papers I and 17

IV) for spinde testing. The too tip FRF can be measured in the radia directions x and y without vioating safety reguations regarding human interaction with high speed rotating spindes. Fig 1. Setup of the CDST measurement performed on the high-speed-miing machine. The rotor excitation was caused by eectromagnets which were fed by a frequency step vice sine sweep coi current which generated a magnetic force that acted on the rotor. The rotor consisted of a speciay manufactured dummy too with a aminated rotor part designed to reduce the energy osses caused by eddy current effects. In each radia direction two eectromagnets (on opposite sides of the rotor) worked together in attracting the rotor to cause the excitation. Two types of miing machine spindes were tested using this measurement method. Fig 1 shows the measurement setup of the CDST measurement of the high speed spinde. During the measurement procedure the spinde was owered, thereby inserting the dummy too into the CDST measurement unit. 4.3 Low speed spinde testing In an initia study presented in Paper I a 3-axes Dynamite miing machine with a spinde speed capacity up to 7 rpm was measured (see Fig 13). 18

7 Ex W13, Magnitude W13 Magnitude Position Position response Response [µm/a] [µm/a] 56 Spinde speed [rpm] 4 8 Rotor unbaance Mode Mode 3 14 Mode 1 5 91 166 3 55 1 Frequency Log(Hz) [Hz] Fig 13. Measured FRF s for different spinde speeds. Bright coours correspond to high magnitudes. Arrows point out detected structura modes and the rotor unbaance. In Fig 13 the infuence of the rotor unbaance can be seen in the upper eft corner of the figure. Three structura modes of the rotating spinde can be seen in the figure between and 6 Hz. These modes do not seem to posses any visibe speed dependency except for the step between and 35 rpm. This change coud be due to the different dynamic properties of non-rotating and rotating bearings. 4.4 High speed spinde testing In a more extensive study presented in Paper IV, a Liechti Turbomi ST1 stateof-the-art machining centre (capabe of mutipe movement up to 5-axes) equipped with a Fischer spinde (MFWS-35/4/8) capabe of speeds of up to 4 rpm was studied. The FRF measured in this study were recacuated from [m/a] to [m/f] and the measurements were preformed in the intervas,, 4,,4 rpm. The detected eigen-modes in the measurements showed a cear speed dependency especiay above 1 rpm (see Fig 13). 19

Hyy [og(m/n)] -1.5 16 Mode 4-13 -13.5 Frequency [Hz] 1-14 -14.5 Mode 3 8 Mode Mode 1 4 8 16 4 Spinde speed [rpm] -15-15.5-16 Fig 14. Measured FRF s for different spinde speeds. Bright coours represent arger magnitudes. 4.5 Stabiity obes for different spinde speeds By using the speed dependent FRFs, stabiity obes coud be cacuated for each spinde speed. When appying the method presented by [9] the depth of cut of a certain spinde speed can be picked out. Fig 15 shows stabiity obe cacuations 3 based on different FRFs measured during different spinde speeds. The thick red curve represents the stabiity imit cacuated when based on the FRF measured at rpm. Back thin curves are based on FRFs measured during speeds in the intervas,, 4,,4 rpm. Green dots mark the speed dependent depth of cut for the speeds 1, 1, 14,,4 rpm which were picked out from each curve. For miing, based on rpm stabiity oad predictions, green dots ocated above the red curve indicate that the depth of cut coud be increased for that speed whie dots beow indicate that chatter vibrations wi occur at that speed. The exampe in Fig 15 iustrates the importance of investigating the speed dependency in FRFs when performing predictions of high-speed-miing machine stabiity. Correct predictions can mean avoidance of chatter vibrations or identification of speeds where the depth of cut can be increased. 3 The stabiity obes were cacuated by Sandvik Coromant using CutPro.

16 Stabiity obes 14 1 Depth of cut [mm] 1 8 6 4 4 8 1 16 4 Spinde speed [rpm] Fig 15. Stabiity obes cacuated for different spinde speeds. The red curve is based on rpm FRF. Back curves are based on FRFs measured at the speed intervas of, 4, 6,,4. Green dots represent the speed dependent depth of cut picked out from stabiity obes based on FRFs measured at that specific speed. 1

5 MODELLING The speed dependency shown in the high speed spinde measurements wi ater the stabiity obe predictor criterion. Paper IV describes investigations into the spinde speed dependency described in the previous section through the use of numerica simuations. The simuation examined the infuence of the gyroscopic moment of the rotor and the centrifuga effects in the ba bearings on the eigen-frequencies of the spinde. In their anaysis of a miing machine spinde, Wang and Chang [31] presented a modeing method based on FEM. The mode did not incude rotation and therefore centrifuga forces and gyroscopic moments were not considered. In 1976 Neson and McVaugh [3] presented a FEM formuation of a rotor bearing system based on the Euer Bernoui beam theory where the effect of gyroscopic moments and centrifuga forces is incuded. Zorzi and Neson [33] ater added interna damping and in 198 Neson [34] presented another formuation based on the Timoshenko beam theory which factored in shear deformation effects. Xiong et a. [7] presented a way of combining this FEM representation and the miing cutting force mode formuated by Atintas [3]. The mode, which ony consisted of the rotor, predicted that the gyroscopic moment woud not affect the stabiity regions in miing but woud increase the rea part of the eigen-frequency and thus reduce the axia depth of cut. The mode aso predicted a change in spinde resonance frequencies of about ± 1 Hz. Chi-Wei Lin et a. [35] integrated a thermo-mechanica-mode into the Timoshenko FEM description. Numerica and practica experiments verified an increase in bearing stiffness with increasing bearing preoad. The work aso predicted a softening of the spinde shaft with increasing spinde speed. It was suggested that the softening of the bearing radia stiffness due to speed coud be compensated for by the thermay-induced preoad. Cao and Atintas [8] presented a genera method for the modeing of a spinde bearing system which incuded the axia coordinate pus a corresponding spinde speed and preoad dependent five degree of freedom bearing stiffness matrix. In the spinde mode a rotor reated centrifuga force was modeed by subtracting a Ω term 4 from the stiffness matrix. The numerica simuations presented in Paper IV were based on the FEM formuation described by Neson [34] and were extended by incuding a separate mass and stiffness radius together with a stiffness radius dependent shear deformation factor [36]. A second order homogenous differentia equation was used to describe the dynamica system; [ M ] q& + Ω[ G] q& + [ K ] q = {} (4) 4 Ω mutipied by a radia version of the transationa mass matrix.

where [ M ], [ G ], and [ K ] are the system matrixes of a shaft eement (see the Appendix in Paper IV). q is the generaized coordinates of the FEM assemby. The assembed second order homogenous differentia equation was transformed into a first order differentia equation, using the state vector notation described in [37]. The equation of motion coud then be rewritten as: Ω[ G] [ M ] [ ] [ ] {} [ K ] [ ] {} = {} h& + [] [ ] h M M, (5) where {} h {} { &} = q q (6) 5.1 Spinde mode In the simuation of the miing machine spinde a reduced mode without the machine foundation and spinde housing was used. The mode was based on the drawings shown in Appendix A. The mode consisted of the rotor and a pair of supporting ba bearings. +y +z +x Front bearing Motor Back bearing Dummy too Fig 16. Spinde mode incuding the rotor and the bearings. The rotor incuded the spinde shaft, motor package, inner bearing rings, too hoder and dummy too. The rotor was divided into FEM eements and speed dependent 3

bearing stiffness was added to nodes corresponding to the bearing positions. Fig 16 shows the rotor and the bearings modeed as springs. The figure is a simpified iustration of modeed bearing stiffness. Apart from the iustrated radia stiffness of the bearings, an anguar stiffness component and a cross couping term are incuded in the mode. 5. Preoad measurement To be abe to cacuate the bearing stiffness the preoad force of the bearings needs to be known. However, when deaing with a rea miing machine spinde the bearing preoad is normay not known to the operator and must be measured. On-ine measurements which give continuous information about the preoad status woud be preferabe. Studies of on-ine measurements and contro of the bearing oad have been described by Chen and Chen [38]. Most spindes now used for production work do not have and can not be retrofitted for on-ine measuring. The preoad must therefore be measured or be provided by the spinde designers. The spinde used in this study was designed to use spring-oaded bearings as shown in Fig 17. It consists of a rotor with two hybrid ange-contact bearings paced back to back. The preoad is measured by puing the spinde towards z whie the dispacement is measured using a dia indicator. The force needed to unoad the front bearing, hence the preoad, was estimated as 145N. Front bearing F Motor Preoad spring Housing Rotor Z+ Dia indicator Front bearing Back bearing Fig 17. Sketch of spinde showing the preoad measurement setup 5.3 Bearing stiffness cacuations Using the resut from the preoad measurement the bearing stiffness within the speed interva,, 4,, 4 rpm were cacuated. The bearing stiffness cacuations were performed by SKF with their in-house deveoped software Bearing Beacon. The bearing stiffness for different spinde speeds in reation to the stiffness at rpm are potted in Fig 18. The pot shows that the stiffness in the radia direction of the back bearing decreased to a eve of 6% of its origina ( rpm) vaue 4

when the speed increased to 4 rpm. The corresponding vaue for the front bearing was 38%. This bearing softening was caused by the centrifuga force which acted on the bas by dispacing them axiay and radiay. In doing this, an extra spring is added in series with the norma Hertzian contact spring. The shape of the stiffness variation of the bearings can be seen in the eigen-frequency variation of the bearing reated modes measured by the CDST (see Fig 14). 11 1 Bearing stiffness 4 rpm (Preoad 145 N) Back bearing Fx/x Percentage of stiffness at rpm [%] 9 8 7 6 5 4 Front bearing 3.5 1 1.5 Spinde speed [rpm] x 1 4 Fig 18. Changes in bearing stiffness in the x direction with increasing spinde speed. 1% represents the bearing stiffness at rpm. 5.4 Simuations The eigen-frequencies of the modeed spinde can be seen in Fig 19. In Graph A the gyroscopic effect of the rotor can be seen; each mode spits up into a backward and a forward mode. Graph B, shows the same simuation with speed dependent bearing stiffness. The speed dependency in the bearing stiffness originates from the centrifuga force that acts on the bas in the bearing. In this case, the effect of the centrifuga force acting on the bas in the bearings had a more significant infuence on the eigen-frequencies than the gyroscopic moment. With the speed dependent bearing stiffness a reduction of the eigen-frequencies of the first and second modes of about 4% and 37% coud be detected. In comparison, the reduction was approximatey 1% when ony the gyroscopic effect was present. When incuding the speed dependent bearing stiffness, the same shape - as seen in the bearing stiffness variation (Fig 18) - coud be seen in the variation of the eigen-frequencies of bearing reated modes. 5

A 18 16 Mode 4 Frequency [Hz] 14 1 1 Mode 3 8 6 Mode Mode 1 4 8 16 4 Spinde speed [rpm] B 18 16 Mode 4 Frequency [Hz] 14 1 1 Mode 3 8 6 Mode Mode 1 Intersection area 4 8 16 4 Spinde speed [rpm] Fig 19. Simuated eigen-frequency for different spinde speeds. A: Infuence of the gyroscopic moment of the rotor. B: Infuence of both the gyroscopic moment of the rotor and the centrifuga effect in the bearings. 6

5.5 Mode shape and eigen-frequency identification The rigid body modes of the rotor are governed by the mass distribution and the bearing stiffness [39]. The rigid body mode governed by the front bearing has a node point at the back bearing position which causes a arger dispacement at the front bearing position. This mode wi form a conica shape and vice versa for the back bearing. The radia bearing stiffness mainy contributes to the rigid body modes whie the stiffness of the rotor shaft dictates the fexura modes. In reaity, the modes often appear as a mix of rigid body modes and fexura modes. The mode shape anaysis of the simuation reveaed that the first mode shape was governed by the back bearing and the second mode by the front bearing. The third mode was a pure fexura mode and the forth was a mix of a fexura mode and the front bearing. Due to the reduced bearing stiffness, the first and second mode shapes intersected (see Fig 19B) at approximatey 14 rpm. The simuated mode shapes were verified by an LDV ine scan of the visibe part of the rotor at rpm (see Fig ). Mode 1 3 4 Simuation LDV LDV measured mode shapes Simuated mode shape Bearing position Fig. Mode shapes of the spinde at rpm. Simuations compared to LDV measurements. 7

6 DISCUSSION In this thesis a method for singe beam LDV measurements on rotating spindes has been presented together with a spinde anaysis method based on inductive dispacement measurement, eectromagnetic excitation and FEM. The LDV method removed the specke noise and the cross sensitivity in the measurements and the miss aignment and out-of-roundness components coud be extracted from the measurements signa. By appying this method, singe beam LDV measurements can be conducted on rotating spindes. The second method for spinde anaysis identified the bearing stiffness as the weakest ink in the spinde design regarding the speed dependent spinde dynamics. A simpified mode of the spinde system showed good agreement with the measurements. 6.1 LDV on rotating targets The deveoped singe beam LDV method for radia rotor vibration measurements requires that the measurement surface be poished. When using this method the measurement surface must be kept smooth and cean during the measurement procedure. When choosing materia for the dummy too, stainess stee is preferabe. This type of materia is non-corrosive and thus avoids surface probems which woud ead to inaccurate measurement. The aignment of the aser is aso a parameter that must be considered when using this method. The aignment was however, found to be an easy procedure. The biggest issue in aignment is that too arge a dispacement during the measurement coud refect the returning ight away from the path of the emitted ight. This effect was however, not detected in the measurements performed on the miing machines studied. 6. Modeing Excuded parts: The mode, used in Paper IV, is a reduced miing machine mode. The machine foundation, spinde housing and parts beonging to the pre-oad mechanism were not incuded. Consequenty the bearing support was modeed as a rigid support. Asymmetry in the spinde housing or in the mounting of the spinde was not incuded in the mode. Rotor unbaance was aso not incuded in the simuation. Couping: The couping between the hoder and the spinde shaft and the couping between the hoder and the too was not considered. The couping was simpy modeed as a rigid connection. This simpification resuted in a sighty stiffer spinde being simuated. Drawbar and springs: Inside the spinde shaft, mechanica components ike the drawbar and springs are mounted. These parts are used to pu the too and connect it to the spinde shaft. The content of the hoow spinde shaft coud not be determined 8

with a high degree of confidence and therefore the inner mass radius of the hoow spinde shaft was set to zero. The eigen-frequencies and gyroscopic moments coud be affected by an inaccurate mode of the mass distribution inside the spinde. Motor: The part of the integrated motor which is mounted on the spinde shaft was modeed as a mass without any stiffness properties. In reaity, it may be that the motor part adds some stiffness to the shaft; meaning that this assumption woud resut in a softer mode rotor. FEM description: A FEM formuation based on the Euer Bernoui beam theory presented by Neson and Mc Vaugh [3] was compared to the formuation based on the Timoshenko beam theory [34]. Eigen-vaues simuated by the use of the Timoshenko formuation are marked by circes and simuations made by the Euer Bernoui formuations are marked by asterisks in Fig 1. The absence of the shear deformation in the Euer Bernoui formuation means that a stiffer spinde with higher eigen-frequencies is depicted. The difference is greater when the frequency increases. Mode 1-6 showed a change in eigen-frequency at rpm of about 4%, 3%, 4%, 4%, 8% and 13%. It can be argued that the impact of the difference between the two beam theories coud be ignored and that the shear deformation is of itte importance. In this case, the difference for the 1 st and 6 th mode corresponded to 7 Hz and 419 Hz respectivey. The amount of difference is dependent on the actua geometry of the rotor and must be considered for each case. Damping, gravity and the centrifuga effects of the rotor described by [8] were not incuded in the mode. The axia oad during free run was considered negigibe and therefore excuded from the simuation. Shear deformation factor: The shear deformation factor incuded in the shear effect in the Timoshenko beam theory is normay determined experimentay and a typica vaue of this variabe is.9 for a soid circuar shaft. In the study presented in Paper IV an anaytica method [36] to achieve the shear deformation factor of a circuar hoow shaft was used. The use of the anaytica vaue resuted in ower eigenfrequencies of the modes (see Fig ). The difference between the two methods increased for higher frequencies but the infuence of the shear deformation factor was not as evident as the infuence of the actua shear deformation effect. Mode 1-6 shows a change in eigen-frequency at rpm of about 1%, 1%, 1%, 1%, % and 3%. For modes 1 and 6 the percentage corresponds to 8 Hz and 15 Hz respectivey. 9

4 Eigen-frequency 35 3 Frequency [Hz] 5 15 1 5 8 16 4 Spinde speed [rpm] Fig 1. Simuated eigen-vaues. The FEM formuations based on Euer Bernoui (asterisks) and Timoshenko (circes) beam theory are compared. 4 Eigen-frequency 35 3 Frequency [Hz] 5 15 1 5 8 16 4 Spinde speed [rpm] Fig. Simuated eigen-frequencies regarding the shear deformation coefficient effect. The effect of the anayticay obtained shear deformation coefficient of a hoow shaft (circes) compared to the soid circuar shaft vaue.9 (asterisks) 3

6.3 Centrifuga effects of the shaft According to Cao and Atintas [8] the eigen-frequency of a rotating spinde is affected by the centrifuga effects of the rotor which woud then ead to a softening of the rotor shaft. This softening was introduced by a Ω term added to the stiffness matrix. This term wi indeed weaken the stiffness, if present. This term is however not incuded in the origina FEM formuation presented by Neson [34] in 198. A comparison between two simuations with and without this term can be seen in Fig 3. Looking at Mode 3 the softening caused by the Ω term reduces the eigenfrequencies which is not the case in the absence of that term. The softening can not be seen in the CDST measurements (see Fig 14) which supports the FEM formuation without the Ω term in the stiffness matrix. Eigen-frequency 18 Mode 1 Frequency at 4 rpm = 3 & 334 Hz 16 Mode 4 Frequency [Hz] 14 1 1 Mode 3 8 6 Mode Mode 1 Intersection area 4 8 16 4 Spinde speed [rpm] Fig 3. Comparison between FEM simuations. Circes denote simuated eigen-frequencies with no Ω term in the stiffness matrix. Asterisks denote eigen-frequencies of a simuation with the Ω term. 31

7 CONCLUSIONS LDV method: A method for singe beam LDV as presented in Papers I and II is based on the reduction of the rotor surface structure. The specke noise in the measurement signa coud be removed and the out-of-roundness components (form error) coud be extracted. A crosstak sensitivity study (Paper III) confirmed that the theory of crosstak in LDV on rotors and verified this method s abiity to eiminate crosstak from in-pane vibrations. The eimination aso impies that the sensitivity to torsiona vibrations is eiminated. The deveoped singe beam LDV vibration measurement method makes it possibe to carry out LDV measurement of spinning rotors without specke noise and speed dependent crosstak from in-pane vibrations. This method can, in contrast to other dispacement-based methods, be used when a ong distance between the measurement object and the sensors is necessary. The method can, as compared to other LDV methods, be appied where ony one radia direction of the rotor can be accessed. The capacity of the LDV system in this method makes it possibe to measure vibration of rotors up to frequencies of 1 MHz. The method requires that the measurement surface be poished to an optica smoothness; a eve ess than haf the aser waveength. CDST and FEM anaysis: The method presented in Paper IV describes a measurement method for rotors that functions in combination with speed dependent FEM simuations. By using the CDST measurement method, spinde anaysis can be performed without any vioation of appicabe safety reguations regarding human interaction with rotating miing machine spindes. The centrifuga force of the ba bearings was shown to have a more significant effect than the gyroscopic moment of the rotor on the speed dependent dynamic of the studied miing machine spinde. It must however, be noted that this concusion is based on data obtained from the specific spinde used in the study. Other resuts may be possibe with other spinde designs. The studies reported on in this thesis do indicate that predictions of highspeed-miing stabiity based on a rpm tap-test can be difficut due to the speed dependency of the system dynamics. A rotor softening reported by previous authors coud not be detected in the CDST measurements. The measurement supports the simuation method chosen for Paper IV. This disagreement between the presented resuts in this thesis and the resuts presented by previous authors must be investigated further. Research questions: According to the concusions drawn, the research questions that initiated the presented studies coud be answered. Radia vibrations of a spinning rotor can be measured using a singe beam LDV setup. When modeing a miing machine spinde the suggested method in this thesis is based on the FEM description initiay presented by Neson [34] together with a spinde speed sensitive bearing stiffness, a separate mass and stiffness radius, and a stiffness radius sensitive shear deformation factor. The most significant part contributing to the speed dependency 3

of the spinde was found to be the bearing stiffness sensitivity to centrifuga forces acting on the bas in the bearings. To verify the simuated spinde behaviour a method for spinde anaysis was used which incuded inductive dispacement sensors and magnetic excitation. The eectromagnetic excitation of the spinde simpified and removed hazardous eements in the measurement procedure normay performed by an impuse hammer. 8 FUTURE WORK If ony one radia direction of the rotor is accessibe, the deveoped singe beam LDV method is suitabe. The deveopment of a method for measuring the hidden radia direction is a possibe avenue for future work. If such a method coud be deveoped, both radia directions coud be measured without moving the aser. An investigation into crosstak caused by miss aignment and out-of-roundness components together with high frequency measurements of miing machines aso suggests itsef for future work. The absence of rotor softening in the measurements presented in Paper IV shoud be investigated further. Further studies can incude the conversion of the CDST measurements from a dummy too setup to a rea cutting too setup. The impact of the forward and the backward modes on the miing stabiity coud aso be investigated. 9 NOMENCLATURE v Veocity λ Laser waveength f D Dopper frequency ω B Brag-ce frequency Ω Laser frequency x, y, z Spatia coordinates Ω M K G q h [ ] [ ] [ ] Rotor spin speed Mass matrix Stiffness matrix Gyroscopic matrix Generaized coordinates Transformed generaized coordinates 33

1 REFERENCES [1] Faassen, R.P.H., et a., Prediction of regenerative chatter by modeing and anaysis of high-speed miing. Internationa Journa of Machine Toos and Manufacture, 3. 43(14): p. 1437. [] Tobias, W.F., A Theory of Regenerative Chatter, The Engineer. 1958, London. [3] Atintas, Y. and E. Budak, Anaytica prediction of stabiity obes in miing. CIRP Annas - Manufacturing Technoogy, 1995. 44(1): p. 357-36. [4] J. Tusty and M. Poacek. The Stabiity of the Machine-Too against Sef- Excited Vibration in Machining. in The Internationa Research in Production Engineering Conference. 1963. Pittsburgh, PA: ASME: New York. [5] Meritt, H.E., Theory of sef-excited machine too chatter. ASME J. Eng. Ind., 1965(87): p. 447. [6] Tang, W.X., et a. Dynamic Modeing for High-speed Miing System with Centrifuga Force and Gyroscopic Effect. 4. Kunming, China: Trans Tech Pubications Ltd. [7] Xiong, G.L., et a., Study of the gyroscopic effect of the spinde on the stabiity characteristics of the miing system. Journa of Materias Processing Technoogy, 3. 138(1-3): p. 379. [8] Cao, Y. and Y. Atintas, A genera method for the modeing of spinde-bearing systems. Journa of Mechanica Design, Transactions of the ASME, 4. 16(6): p. 189. [9] Hacon, B. and S. Rothberg. Automatic post-processing of aser vibrometry data for rotor vibration measurements. in Eighth Internationa Conference on Vibrations in Rotating Machinery. 4. University of Waes, Swansea, UK: Professiona Engineering Pubishing. [1] Whittemore, J., Kenneth R., et a., A normative study of tympanic membrane motion in humans using a aser Dopper vibrometer (LDV). Hearing Research, 4. 187(1-): p. 85-14. [11] Casteini P, S.L., Tomasini EP, Teeth mobiity measurement: a aser vibrometry approach. Cin Laser Med Surg, 1998. 16((5)). [1] Moin NE, Z.L., Optica methods of today for visuaizing sound fieds in musica acoustics. ACTA ACUSTICA UNITED WITH ACUSTICA, 4. 9((4)): p. 618-68. [13] Gren, P., et a., Laser vibrometry measurements of vibration and sound fieds of a bowed vioin. Measurement Science and Technoogy, 6. 17(4): p. 635. [14] Gren, P., Bending wave propagation in rotating objects measured by pused TV hoography. Appied Optics,. 41(34): p. 737-74. [15] Denman, M.H., Nei A.; Rothberg, Steve. Specke noise reduction in aser vibrometry: experimenta and numerica optimisation. in Second Internationa 34

Conference on Vibration Measurements by Laser Techniques: Advances and Appications. 1996. Washington, DC, USA. [16] Haiwe N. A., THE LASER TORSIONAL VIBROMETER: A STEP FORWARD IN ROTATING MACHINERY DIAGNOSTICS. Journa of Sound and Vibration, 1996. 19(3): p. 399-418. [17] Drew, S.J. and B.J. Stone, REMOVAL OF SPECKLE HARMONICS IN LASER TORSIONAL VIBROMETRY. Mechanica Systems and Signa Processing, 1997. 11(5): p. 773-776. [18] Rantatao, M., K. Tatar, and P. Norman. Laser dopper vibrometry measurements of a rotating miing machine spinde. in Eighth Internationa Conference on Vibrations in Rotating Machinery. 4. University of Waes, Swansea UK. [19] Rantatao, M., P. Norman, and K. Tatar. NON-CONTACT MEASUREMENTS OF TOOL VIBRATIONS IN A MILLING MACHINE. in Nordic Vibration Research 4. 4. Stockhom, KTH campus. [] BELL, J.R. and S.J. ROTHBERG, LASER VIBROMETERS AND CONTACTING TRANSDUCERS, TARGET ROTATION AND SIX DEGREE- OF-FREEDOM VIBRATION: WHAT DO WE REALLY MEASURE? Journa of Sound and Vibration,. 37(): p. 45-61. [1] BELL, J.R. and S.J. ROTHBERG, ROTATIONAL VIBRATION MEASUREMENTS USING LASER DOPPLER VIBROMETRY: COMPREHENSIVE THEORY AND PRACTICAL APPLICATION. Journa of Sound and Vibration,. 38(4): p. 673-69. [] Rothberg, S. and J. Be, On the appication of aser vibrometry to transationa and rotationa vibration measurements on rotating shafts. Measurement SU -, 4. [3] Abrecht, A., et a., High frequency bandwidth cutting force measurement in miing using capacitance dispacement sensors. Internationa Journa of Machine Toos and Manufacture, 5. 45(9): p. 993. [4] Spiewak, S.A., Acceeration based indirect force measurement in meta cutting processes. Internationa Journa of Machine Toos and Manufacture, 1995. 35(1): p. 1-17. [5] Auchet, S., et a., A new method of cutting force measurement based on command votages of active eectro-magnetic bearings. Internationa Journa of Machine Toos and Manufacture, 4. 44(14): p. 1441-1449. [6] Knospe, C.R., Active magnetic bearings for machining appications. Contro Engineering Practice, 5. In Press, Corrected Proof. [7] Chen, M. and C.R. Knospe, A new approach to the estimation of cutting dynamics using active magnetic bearings. Transactions of the ASME. Journa of Manufacturing Science and Engineering, 5. 17(4): p. 773. [8] Jeong, Y.-H. and D.-W. Cho, Estimating cutting force from rotating and stationary feed motor currents on a miing machine. Internationa Journa of Machine Toos and Manufacture,. 4(14): p. 1559-1566. 35

[9] Schmitz, T.L., J.C. Ziegert, and C. Stanisaus. A method for predicting chatter stabiity for systems with speed-dependent spinde dynamics. 4. Charotte, NC, United States: Society of Manufacturing Engineers, Dearborn, 4811-93, United States. [3] Sims, N.D., P.V. Bayy, and K.A. Young, Piezoeectric sensors and actuators for miing too stabiity obes. Journa of Sound and Vibration. In Press, Corrected Proof. [31] W.R. Wang, C.N.C., Dynamic anaysis and design of a machine too spindebearing system. Journa of Vibration and Acoustics, Transactions of the ASME 116, 1994: p. 8-85. [3] H.D. Neson, J.M.M., and., The dynamics of rotor-bearing systems using finite eements. Journa of Engineering for Industry, 1976(Transactions of the ASME 93 ()): p. 593-6. [33] E.S. Zorzi, H.D.N. Finite eement simuation of rotor-bearing systems with interna damping. Transactions of the ASME Journa of Engineering for Power, 1977: p. 71-76. [34] H.D. Neson., A finite rotating shaft eement using Timoshenko Beam Theory. Journa of Mechanica Design, 198(Transactions of the ASME 1): p. 793-83. [35] Lin, C.-W., J.F. Tu, and J. Kamman, An integrated thermo-mechanicadynamic mode to characterize motorized machine too spindes during very high speed rotation. Internationa Journa of Machine Toos and Manufacture, 3. 43(1): p. 135. [36] Hutchinson, J.R., Shear coefficients for timoshenko beam theory. Transactions of the ASME. Journa of Appied Mechanics, 1. 68(1): p. 87. [37] Ewins, D.J., Moda Testing: Theory, Practice and Apications. ed., Phiadephia: Research studies press Ltd. [38] Chen, J.-S. and K.-W. Chen, Bearing oad anaysis and contro of a motorized high speed spinde. Internationa Journa of Machine Toos and Manufacture, 5. 45(1-13): p. 1487. [39] Genta, G., Dynamics of rotating systems. Mechanica Engineering, ed. F. F.Ling. 5: Springer. 36

APPENDIX A: SPINDLE DRAWING

Paper I

NON-CONTACT MEASUREMENTS OF TOOL VIBRATIONS IN A MILLING MACHINE Matti Rantatao 1 Peter Norman Kourosh Tatar 3 Lueå University of Technoogy 1 Div. of Sound and Vibrations Div. of Manufacturing Systems Engineering 3 Div. of Experimenta Mechanics Lueå Sweden Abstract Empirica knowedge of the machining process is now being compemented by more scientific methods to determine machine too vibration properties that are associated with stabe machining. Current methods used to predict the behaviour of a miing process (e.g. impact-test ) use measurements made on non-rotating systems. However, such measurements do not detect the effects of gyroscopic and centrifuga forces present in a rotating system. To be abe to fuy investigate the behaviour of a high speed rotating system one need to use non-contact measurement methods e.g. the Laser Dopper Vibrometry (LDV) method, which is becoming common for vibration measurements; athough the method is not without its probems when appied to rotating objects. The work presented here consists of an investigation into the use of LDV and inductive position sensors with magnetic excitation to measure vibrations of a too in a miing machine. The abiity to perform non intrusive measurements shoud make it possibe to anayse changes in machine too dynamics during the cutting process and in the ong term find ways to stabe machining. 1 INTRODUCTION Manufacturers of modern machine toos are increasingy impementing advanced process monitoring and supervisory process contro [1] to compement the basic functionaity of the machine too contro system. At their simpest, process monitoring systems are used to hep prevent or imit the effects of catastrophic events such as too breakage [] or spinde faiure. Such events can be detected by monitoring the current drawn by axis drives and spinde motor [3], or by more advanced techniques such as 1

cutting force monitoring or measurements of vibrations using acceerometers or acoustic emission using sensitive transducers and signa conditioning software[4-8]. By setting safe imits for the monitored parameter(s) based on experience or trias, unusua or unexpected events which may indicate a catastrophic faiure can be used as a trigger to stop the machine. To be abe to fuy investigate the behaviour of a high-speed rotating system, such as a machine too spinde, it is necessary to use non-contact measurement methods. Severa approaches to the non-contact measurement of rotating objects have been deveoped. These incude optica techniques such as Pused Laser TV-Hoography [9] and Laser Dopper Vibrometer techniques (LDV) [1] which a have their difficuties when appied to rotating objects. [11] Another way of measuring vibrations is with the use of inductive position sensors which is commony used for dispacement measurement. E.g. in SKF s (www.skf.se) magnetic bearing product; position sensors are used in order to keep track of the axe position whie it is being controed by magnets keeping it hovering inside the bearing free of mechanica contact. This paper describes an investigation in the possibiities of measuring vibrations and Frequency Response Functions (FRF) of a too in a miing machine using LDV or inductive position sensors. The two methods have been tested separatey and no comparison between the two has been made. LDV MEASUREMENTS OF A POLISHED DUMMY TOOL A dummy too with a radius of 1 mm and a ength of 1 mm was manufactured from a soid stainess stee too bank. The shaft was mounted in a athe and poished using emery paper, diamond paste with partices ranging from 9 µm to.5 µm and with a chemica poishing fuid. In the actua experiment a spray which is normay used for crack detection was used to create a removabe diffuse (opticay rough) surface on the poished dummy too. Both the poished too surface and the sprayed surface were measured by the optica profier, and a representative area of the too (of 34 x 199 µm) was samped in steps of 414 nm. The measurements showed a normay distributed surface structure with Ra = 11,9 nm, impying that the poished surface was opticay smooth compared to the aser waveength of 633 nm. The sprayed surface showed substantiay higher vaues, Ra = 1.µm, giving an opticay raw surface. For the vibration measurements, a PSV 3 LDV system from Poytec GmbH (www.poytec.com) incuding a dispacement decoder was used. The LDV scanning head was mounted on a sturdy tripod and paced approximatey m from the tip of the dummy too on a soft damped materia to reduce the infuence of structura foor vibrations. The maximum detectabe frequency was set to 16 khz. The LDV was used to measure the vibrations at the tip of the poished dummy too in the radia direction at a spinde speed of 6 rpm. After this the same measurement was carried out on the too after being sprayed to give an opticay raw finish.

3 FRF MEASUREMENTS USING INDUCTIVE SENSORS AND MAGNETC EXCITATION FRF measurements using inductive sensors and magnetic excitation were carried out by the use of a standard magnetic bearing manufactured by SKF Revove and a speciay made dummy too with magnetic properties. (The inductive sensors in the bearing have 1% inearity, a sensitivity of 5µm/V, a range of 1.5 mm, a resoution of 3 nm and are immune to shifts in temperature). The bearing was mounted in a miing machine on the machine tabe with its W axis aong the y direction of the machine (Figure 1). A moda test of the mounted bearing was performed which reveaed no significant moda properties in the frequency band of interest, impying that the FRF measurements of the spinde coud be performed without unwanted infuences from the mounted bearing structure. A dummy too with magnetic properties was mounted in the machine spinde and owered down into the bearing. The FRF of the mounted dummy too was then cacuated by measuring the response from the inductive position sensors whie exciting the structure with the magnets. The magnetic bearing contro software MBscope was set up to perform an anaysis of the system in the interva 5-1 Hz with 6 points/decade and 3 s/decade using a sine sweep with an ampitude of 1A. Measurements were made whie exciting the structure in the W13 direction for 1 different spinde speeds from to 7 rpm in steps of 35 rpm. The measurements started with rpm and with a machine structure temperature of approximatey 18º C. Figure 1 Miing machine with a magnetic bearing mounted on the machine tabe for spinde FRF measurements. The bearing W direction is aong the norma y direction for the miing machine. 3

4 RESULTS 4.1 LDV The resuts from the LDV measurements at 6 rpm are presented in a veocity spectrum chart were the poished and rough dummy too measurements are dispayed together Figure. The spectrum of the rough surface has been fipped down to the negative side in the chart to simpify comparison of the two spectra. In the charts it can be seen that the spectrum of the rough dummy too, in opposite to the poished one contains peaks at f * n Hz where f is the rotationa speed of 1 Hz (6 rpm) and n = 1,, 3 These peaks are expected due to the presence of a specke noise repeated for each dummy too revoution in the samped data. 3 Veocity FFT, 6 rpm Poished (upper) Rough (ower) Magnitude [mm/s] 1-1 - 1 3 4 5 6 7 8 9 1 Frequency [Hz] Figure Spectra of the poished and the sprayed surface at a spinde speed of 6 rpm. The spectrum of the rough dummy too has been mirrored aong the frequency axis down to the negative side to simpify comparison between the two. Mutipe harmonics of n*1 Hz where n= 1,,3 can be seen in the spectrum of the sprayed surface. 4

5. Inductive sensors The resuts from the FRF measurements with the magnetic bearing are presented in Figure 3 and Figure 4. Figure 3 dispays the magnitude response and the phase response for a non rotating spinde. Figure 4 shows the measured magnitude response functions for a spinde speeds where the response functions are dispayed aong the x axis from 5-1 Hz. (High response vaues are represented by bright pixes in the image). A resonance frequency can be seen at 5 Hz in Figure 3 and Figure 4 for the non rotating case. The difference between the FRF of the non rotating spinde and the rotating ones is obvious especiay regarding the resonance frequency at 5 Hz. On the upper eft hand side in Figure 4 the spinde speed can be seen as a disturbance in the measurement. 5 Spinde speed rpm w13 Response µm/a 4 3 1 1 3 4 5 6 7 8 9 1 Frequency Hz w13 Phase (degre) 1-1 - 1 3 4 5 6 7 8 9 1 Frequency Hz Figure 3 Frequency response function for rpm. 5

7 56 W13 Magnitude Position Response [µm/a] 4 35 3 Spinde speed [rpm] 4 8 14 5 15 1 5 5 91 166 3 55 1 Frequency [Hz] -5-1 Figure 4 Pot of the frequency response functions in µm per ampere in to the magnets [µm/a] with the spinde speeds aong the y axis and the frequency between 5-1Hz on the x axis (og scae). 5 DISCUSSION AND CONCLUDING REMARKS By poishing the too to a eve where the surface coud be considered as opticay smooth; the speckes noise is removed and the measurements of axia vibrations can be performed. If this method is to be used for vibration measurements of a rotating too when it is cutting, severa other difficuties must be tacked. The aser beam must have a cear path to the too with no interfering meta chip, oi or cooing fuid from the cutting process. The poished part of the too surface, where the measurement wi be conducted, has to be free of partices such as dust or process fuids. Any disturbances on the surface wi cause specke noise and unwanted peaks in the spectrum. The test with the magnetic bearing showed that it is possibe to measure the FRF during different spinde speeds with a magnetic bearing, and therefore making it possibe to detect e.g. speed dependent components in the FRF. The difference detected in the measurements of the non rotating spinde and the rotating ones can have its origin in temperature or speed dependent components in the spinde structure. More tests wi be conducted with other test signas and during other machine conditions. The infuence of the unknown excitation force from the machine itsef must aso be considered in future measurements. Comparisons between LDV and inductive position sensors wi aso be carried out. 6

6 ACKNOWLEDGEMENTS The financia and technica support for this work was provided by the Swedish Agency of Innovation Systems (Vinnova) and SKF Nova, respectivey. The LDV system and auxiiary equipment have been financed by the Kempe Foundations. 7 REFERENCES [1] M. Bäckström, "On monitoring and contro of machining processes," in Department of Appied Physics and Mechanica Engineering. Lueå: Lueå university of technoogy, 1999, pp. 148. [] P. T. Huang and J. C. Chen, "Fuzzy ogic-base too breakage detecting system in end miing operations," Computers & Industria Engineering, vo. 35, pp. 37-4, 1998. [3] Y.-H. Jeong and D.-W. Cho, "Estimating cutting force from rotating and stationary feed motor currents on a miing machine," Internationa Journa of Machine Toos and Manufacture, vo. 4, pp. 1559-1566,. [4] G. D. Byrne, D.;Inasaki, I.;Ketteer, G.;Konig, W.;Teti, R., "Too condition monitoring (TCM) - the status of research and industria appication," CIRP Annas - Manufacturing Technoogy, vo. 44, pp. p 541-567, 1995. [5] D. A. Dornfed, "Monitoring technoogies for inteigent machining," presented at CIRP/VDI Conference on Monitoring of Machining and Forming Processes, Dussedorf, Germany, 1995. [6] P. W. Prickett and C. Johns, "An overview of approaches to end miing too monitoring," Internationa Journa of Machine Toos and Manufacture, vo. 39, pp. 15-1, 1999. [7] D. Yan, T. I. E-Wardany, and M. A. Ebestawi, "A muti-sensor strategy for too faiure detection in miing," Internationa Journa of Machine Toos and Manufacture, vo. 35, pp. 383-398, 1995. [8] D. E. DimaSnr., "Sensor signas for too-wear monitoring in meta cutting operations--a review of methods," Internationa Journa of Machine Toos and Manufacture, vo. 4, pp. 173-198,. [9] P. Gren, "Bending wave propagation in rotating objects measured by pused TV hoography," Appied Optics, vo. 41, pp. 737-74,. [1] D. C. Wiiams, Optica Methods in Engineering Metroogy. London: Chapman and Ha, 1993. [11] S. J. Rothberg, J. R. Baker, and N. A. Haiwe, "Laser vibrometry: Pseudovibrations," Journa of Sound and Vibration, vo. 135, pp. 516-5, 1989. 7

Paper II

Laser dopper vibrometry measurements of a rotating miing machine spinde Matti Rantatao 1 Kourosh Tatar Peter Norman 3 Lueå University of Technoogy 1 Div. of Sound and Vibrations Div. of Experimenta Mechanics 3 Div. of Manufacturing Systems Engineering Lueå Sweden ABSTRACT Finding an optimum process window to avoid vibrations during machining is of great importance; especiay when manufacturing parts with high accuracy and/or high productivity demands. In order to make more accurate predictions of the dynamic moda properties of a machining system in use, a non-contact method of measuring vibrations in the rotating spinde is required. Laser Dopper Vibrometry (LDV) is a non-contact method, which is commony used for vibration measurements. The work presented consists of an investigation into the use of LDV to measure vibrations of a rotating too in a miing machine, and the effects of specke noise on measurement quaity. The work demonstrates how the axia misaignment and the roundness of a poished shaft can be evauated from LDV measurements. 1 INTRODUCTION Manufacturers of modern machine toos are increasingy impementing advanced process monitoring and supervisory process contro (1) to compement the basic functionaity of the machine too contro system. At there simpest, process monitoring systems are used to hep prevent or imit the effects of catastrophic events such as too breakage () or spinde faiure. Such events can be detected by monitoring the current drawn by axis drives and spinde motor (3, 4), or by more advanced techniques such as cutting force monitoring or measurements of vibrations using acceerometers or acoustic emission using sensitive transducers and signa conditioning software (5-9). By setting safe imits for the monitored parameter(s) based on experience or trias, unusua or unexpected events which may indicate a catastrophic faiure can be used as a trigger to stop the machine. 1

Since vibrations are the resut of reative movement between the cutter and work piece, the dynamic behaviour of both the machine structure and rotating spinde/cutter together with the behaviour of the component being machined has to be considered. In most situations, the work piece can be considered a soid part fixed to the machine tabe with no significant moda properties of its own. This assumption tends to weaken, however, when machining components with reativey thin was (1). Regenerative machine too chatter is a fundamenta type of vibration that can occur during miing. These vibrations have their origin in the cosed oop nature of the cutting process and are dependent on the structura vibration modes, described by the frequency response function (FRF) of the machine too. The FRF is normay measured on a non rotating/static system from which the imits for chatter free machining can be cacuated (11). In modern machine toos, spinde speeds of, rpm and upwards are not uncommon, since the dynamic characteristics of the spinde such as damping change, this causes the FRF of the system as a whoe to change. To be abe to fuy investigate the behaviour of a high-speed rotating system, such as a machine too spinde, it is necessary to use non-contact measurement methods. Severa approaches to the non-contact measurement of rotating objects have been deveoped. These incude optica techniques such as Pused Laser TV-Hoography (1) and Laser Dopper Vibrometer techniques (LDV) (13). LDV is a we-estabished technique for measuring the veocity of a moving object. It is based on the Dopper effect, which expains the fact that ight changes its frequency when detected by a stationary observer after being refected from a moving object. The vibrating object scatters or refects ight from the aser beam and the Dopper frequency shift is used to measure the component of veocity which ies aong the axis of the aser beam. As the aser ight has a very high frequency, direct demoduation of the ight is not possibe and optica interferometry is therefore used. When a coherent ight source iuminates a surface that is opticay rough, i.e. the surface roughness is arge on the scae of the aser waveength, a granuar pattern caed specke which has random ampitude and phase is seen. This is due to interference between the components of backscattered ight. The intensity of a specke pattern obeys negative exponentia statistics and their phases are uniformy distributed over a vaues between -π and π (14). If the specke pattern changes during LDV measurement the rate of change in the resuting phase wi be nonzero, and the frequency spectrum wi contain peaks. These kinds of specke fuctuations are induced by non-norma target motions, such as tit, in-pane motions or rotation (15). Specke fuctuations due to target rotation are periodic and wi repeat for each revoution. This eads to peaks in the spectrum at the fundamenta rotation frequency and higher order harmonics. These moduations are difficut to distinguish from the true vibrations and in the worst case, can amost competey mask the vibration pattern. It is therefore important that the target to be measured has a surface smooth enough so that the specke noise is avoided.

EXPERIMENTAL SET-UP AND PROCEDURE.1 Preparation of the dummy too A dummy too with a radius of 1 mm and a ength of 1 mm was manufactured from a soid stainess stee too bank. The shaft was mounted in a athe and poished using emery paper with grades ranging from 4 (grains/mm) to 1. The shaft was finay poished using diamond paste with partices ranging from 9 µm to.5 µm and with a chemica poishing fuid. Quaity contro of the poished surface was performed using non-contact optica surface profie measurement (www.veeco.com). In the actua experiment a spray which is normay used for crack detection was used to create a removabe diffuse (opticay rough) surface on the poished dummy too. Both the poished too surface and the sprayed surface were measured by the optica profier, and a representative area of the too of 34 x 199 µm was samped in steps of 414 nm. The measurements showed a normay distributed surface structure with Ra = 11,9 nm, impying that the poished surface is opticay smooth compared to the aser waveength of 633 nm. The sprayed surface showed substantiay higher vaues, Ra = 1.µm, giving an opticay raw surface.. The miing machine The LDV measurements were made on a Liechti Turbomi ST1 state-of-the-art machining centre offering mutipe (5-axis) movement and a spinde capabe of speeds of up to 4, rpm. The poished dummy too was mounted in a Corogrip hoder with an HSK shank which was in turn mounted in the machine and was not removed unti a the measurements had been made..3 Setting up the LDV For the measurements, a PSV 3 LDV system from Poytec GmbH (www.poytec.com) incuding a dispacement decoder was used. The LDV scanning head was mounted on a sturdy tripod and paced approximatey m from the tip of the dummy too on a soft damped materia to reduce the infuence of structura foor vibrations. Care was taken to aign the aser beam so that it s centre ine passed through the centre ine of the shaft and was perpendicuar to the shaft s axis of rotation. This was necessary to ensure that the true veocity vector associated with the vibrations was aong the incident direction of the aser beam. The LDV system was set up to perform samping with a frequency of 4,96 khz. The maximum detectabe frequency was set by the system to 16 khz. The LDV system produced frequency spectra with a standard FFT agorithm using a compex averaging method with 1 averages of 8 ms each giving a frequency resoution of 1.5 Hz and a tota measuring time of 1.8 s..4 LDV measurements A series of experiments were carried out to estabish whether vibrations of a rotating too coud be measured using the LDV system. Four different spinde speeds 7, 4, 6 and 7 rpm were studied. The LDV was used to measure the vibrations at the tip of the poished dummy too in the radia direction at these speeds. The same set of measurements was carried out on the too after being sprayed to give an opticay raw finish. 3

Figure 1. LDV measurement of the rotating opticay raw (sprayed) dummy too. Logged data was exported from the LDV as ASCII fies and then imported into Matab 6. where more detaied anaysis and fitering of the data was carried out. The arge-scae profie around the circumference of the dummy too was measured using a mechanica roundness tester from C E Johansson (www.cej.se), with an accuracy of +.3µm. This was performed after that the LDV measurements were carried out. 3 RESULTS In this section the resuts from the measurements at 6 rpm are presented. The veocity spectrum of the poished and rough dummy too measurements are dispayed in the same chart for different frequency bands, Figure -5. The spectrum of the rough surface has been fipped down to the negative side in the charts to simpify comparison of the two spectra. In the charts it can be seen that the spectrum of the rough dummy too contains peaks at f * n Hz where f is the rotationa speed of 1 Hz (6 rpm) and n = 1,, 3 These peaks are expected due to the presence of a specke noise repeated for each dummy too revoution in the samped data. A zoomed part of the spectrum covering the frequency band 8.8-1 khz shows ceary the specke noise in the form of peaks at integer mutipes of the rotationa speed of 1 Hz. These are marked with circes aong the frequency axis, Figure 3. These peaks coud not be seen in the graph of the poished too. Between 1.1-1.5 khz contains both specke noise peaks 4

and ordinary vibrations, Figure 4. Note that the vibrations are present in both curves but the peaks are ony present in the spectrum of the rough surface. The frequency band covering -1 khz shows harmonic peaks in both FFT graphs, see Figure 5. However; the first peak at 1 Hz in the poished measurement spectrum was detected as the dummy too axia misaignment and the other six harmonics as the roundness profie. Figure 6 shows the signa from the dispacement decoder. This signa is band-pass fitered between.15-.75 khz, thus fitering out the roundness profie. The resut is shown in Figure 7, where the fitered time signa for one revoution is presented in a poar pot (dashed ine), together with an independent mechanica measurement of the roundness made by the roundness tester (soid ine). The difference between the curves is ess than the error given by the manufacturer of the roundness tester (+.3 µm). For the sprayed dummy too the roundness coud not be measured propery due to specke noise caused by the rough surface. Simiar resuts where achieved for measurements made at spinde speeds of 7, 4, and 7 rpm. 3 Veocity FFT, 6 rpm Poished (upper) Rough (ower) Magnitude [mm/s] 1-1 - 1 3 4 5 6 7 8 9 1 Frequency [Hz] Figure. Spectra of the poished and the sprayed surface at a spinde speed of 6 rpm. The spectrum of the rough dummy too has been mirrored aong the frequency axis down to the negative side to simpify comparison between the two. Mutipe harmonics of n*1 Hz where n= 1,,3 can be seen in the spectrum of the sprayed surface. 5

.1.5 Veocity FFT, 6 rpm Poished (upper) Rough (ower) Magnitude [mm/s] -.5 -.1 88 9 9 94 96 98 1 Frequency [Hz] Figure 3. Zoomed part of the spectrum, 8.8-1 khz. Frequencies where peaks are expected due to specke noise are marked with a ring on the frequency axis..6.4 Veocity FFT, 6 rpm Poished (upper) Rough (ower) Magnitude [mm/s]. -. -.4 11 115 1 15 13 135 14 145 15 Frequency [Hz] Figure 4. Zoomed part of the spectrum, 1.1-1.5 khz. Frequencies where peaks are expected due to specke noise are marked with a ring on the frequency axis. It can ceary be seen that no peaks is present in the spectrum of the poished surface at the marked positions. Note that the vibration signa is present in both measurements. 6

1.5 1 Veocity FFT, 6 rpm Poished (upper) Rough (ower) Magnitude [mm/s].5 -.5-1 -1.5 1 3 4 5 6 7 8 9 1 Frequency [Hz] Figure 5. Zoomed part of the spectrum, -1 khz. Frequencies where peaks are expected due to specke noise are marked with a ring on the frequency axis. In this graph peaks in both spectra are present at the marked frequency positions. For the poished case the peaks are identified as too axia misaignment and roundness. 8 Dispacement decoder, 6 rpm 6 Dispacement [µm] 4 - -4-6 -8.5.1.15..5.3.35 Time [s] Figure 6. Dispacement measurement at 6 rpm. 7

15 Dummy too circumference profie [µm] 9 6 1 6 4 Profie LDV 3 18 1 33 4 7 3 Figure 7. Poar pot of the roundness of the dummy too. Soid ine: Measured with a mechanica roundness tester. Dashed ine: Measured with LDV at a rotationa speed of 6 rpm. The distance between two circes in the pot is µm. 4 DISCUSSION AND CONCLUDING REMARKS Specke noise interference was avoided by poishing the surface of the dummy too (in effect a rotating shaft) unti an opticay smooth surface was achieved. The opticay smooth surface of the rotating too generated no repeated specke noise and hence no unwanted peaks at integer mutipes of the rotationa frequency. This aowed the axia misaignment and roundness of the dummy too to be measured at speeds of up to at east 7 rpm using an LDV. This impies that radia vibration measurements of the too can aso be conducted; for exampe, when investigating dynamics of the cutting process. The possibiity of extracting roundness and aignment information is based on the fact that at a rotationa speed of 6 rpm (1Hz) any misaignment woud be seen as a 1Hz signa. Since the too is not perfecty round, harmonics of 1Hz wi be present. The first component of the out of roundness is an eiptica form and resuts in a frequency peak at *1Hz ( Hz). The second roundness component, a tri-obed form, woud be seen as a peak at 3*1Hz (3Hz) and so on. In the experiments, no significant peaks at integer mutipes of the 8

rotationa speed coud be detected above the 6:th component at 7*1Hz. To eiminate the possibiity of structura vibrations being misinterpreted as misaignment or out of roundness, measurements were made at a number of rotationa speeds (7, 4, 6 and 7 rpm). At each of these speeds, the misaignment data woud be seen as a peak at a different frequency; namey 45Hz, 7Hz, 1Hz and 1Hz. Out of roundness data woud be at mutipes of the primary frequency. When measuring at speeds other than 6rpm, any structura vibration around 1Hz woud become cear, and the peak due to misaignment shifted. This makes it possibe to anayse the presence of structura vibrations overaying axia misaignment and out of roundness. No significant structura vibrations overaying the roundness and misaignment data were detected in the experimenta data. However, since some frequency components of structura vibrations are spinde speed dependant it is not possibe to draw firm concusions about misaignment and roundness based soey on the LDV measurements. This effect must be investigated further. Good correspondence was however seen between LDV measurements and direct mechanica measurements of misaignment and out of roundness made using a dia test indicator and roundness tester. This indicates that no significant structura vibrations are overaying the pitch and roundness data at the spinde speeds investigated. Severa probems must be overcome if LDV measurements are to be made of a rotating too when it is cutting. Firsty, the aser beam must have a cear ine of sight to the target surface on the too without interference from cooing fuid or meta chips generated by the cutting process. The target surface must aso be kept free of partices such as dust or process fuids. In miing machines where the too moves reative to the machine base it must be possibe to track the moving too either by physicay mounting the LDV on the moving axis of the machine or by some other tracking system. Finay, the aignment of the aser beam reative to the opticay smooth target surface on the too is aso an issue that has to be considered. Axia misaignment, roundness, in-pane vibrations or appied cutting forces can affect the direction in which the aser beam is refected from the target surface which coud ead to a poor signa eve or drop outs. In pane vibrations and defection due to cutting forces together with the geometry of the shaft can aso ead to misinterpretation of vibration data due to cross sensitivity. This has not been investigated in this work. Different spinde / cutting speeds, cutting forces and changing component geometry affect the dynamics of a machining system. Measurement techniques based on physicay mounting sensors, such as acceerometers, on the machine or workpiece can aso affect the system dynamics. The abiity to perform non-contact measurements of vibrations wi aow measurement of changes in machine dynamics to be made during the cutting process without affecting the process itsef. This is the subject of ongoing work. 5 ACKNOWLEDGEMENTS The financia and technica support for this work was provided by the Swedish Agency of Innovation Systems (Vinnova) and SKF Nova, respectivey. The LDV system and auxiiary equipment have been financed by the Kempe Foundations. 9

6 REFERENCES 1. Bäckström, M., On monitoring and contro of machining processes, in Department of Appied Physics and Mechanica Engineering. 1999, Lueå university of technoogy: Lueå. p. 148.. Huang, P.T. and J.C. Chen, Fuzzy ogic-base too breakage detecting system in end miing operations. Computers & Industria Engineering, 1998. 35(1-): p. 37-4. 3. Jeong, Y.-H. and D.-W. Cho, Estimating cutting force from rotating and stationary feed motor currents on a miing machine. Internationa Journa of Machine Toos and Manufacture,. 4(14): p. 1559-1566. 4. Li, X.L., A. Djordjevich, and P.K. Venuvinod, Current-sensor-based feed cutting force inteigent estimation and too wear condition monitoring. Ieee Transactions on Industria Eectronics,. 47(3): p. 697-7. 5. Byrne, G.D., D.;Inasaki, I.;Ketteer, G.;Konig, W.;Teti, R., Too condition monitoring (TCM) - the status of research and industria appication. CIRP Annas - Manufacturing Technoogy, 1995. 44(5): p. p 541-567. 6. Dornfed, D.A. Monitoring technoogies for inteigent machining. in CIRP/VDI Conference on Monitoring of Machining and Forming Processes. 1995. Dussedorf, Germany: VDI Berichte. 7. Prickett, P.W. and C. Johns, An overview of approaches to end miing too monitoring. Internationa Journa of Machine Toos and Manufacture, 1999. 39(1): p. 15-1. 8. Yan, D., T.I. E-Wardany, and M.A. Ebestawi, A muti-sensor strategy for too faiure detection in miing. Internationa Journa of Machine Toos and Manufacture, 1995. 35(3): p. 383-398. 9. DimaSnr., D.E., Sensor signas for too-wear monitoring in meta cutting operations-- a review of methods. Internationa Journa of Machine Toos and Manufacture,. 4(8): p. 173-198. 1. Agba, E.I., D. Ishee, and J.T. Berry, High speed machining of unsupported thin-waed structures. Technica Paper - Society of Manufacturing Engineers. MR. Proceedings of the 1999 3rd Internationa Machining and Grinding, Oct 4-Oct 7 1999, 1999(MR99-4): p. 99-4. 11. Atintas, Y. and E. Budak, Anaytica prediction of stabiity obes in miing. CIRP Annas - Manufacturing Technoogy, 1995. 44(1): p. 357-36. 1. Gren, P., Bending wave propagation in rotating objects measured by pused TV hoography. Appied Optics,. 41(34): p. 737-74. 13. Wiiams, D.C., Optica Methods in Engineering Metroogy. 1993, London: Chapman and Ha. 14. Gåsvik, K.J., Optica Metroogy. 3rd ed. : Wiey. 37. 15. Rothberg, S.J., J.R. Baker, and N.A. Haiwe, Laser vibrometry: Pseudo-vibrations. Journa of Sound and Vibration, 1989. 135(3): p. 516-5. 1

Paper III

Laser vibrometry measurements of an opticay smooth rotating spinde Kourosh Tatar*, Matti Rantatao** and Per Gren* Lueå University of Technoogy, SE-97187 Lueå, Sweden * Div. of Experimenta Mechanics ** Div. of Sound and Vibrations Abstract Laser Dopper Vibrometry (LDV) is a we-estabished non-contact method, commony used for vibration measurements on static objects. However, the method has imitations when appied to rotating objects. The LDV signa wi contain periodicay repeated specke noise and a mix of vibration veocity components. In this paper the crosstak between vibration veocity components in aser vibrometry measurements of a rotating dummy too in a miing machine spinde is studied. The spinde is excited by an active magnetic bearing (AMB) and the response is measured by LDV in one direction and inductive dispacement sensors in two orthogona directions simutaneousy. The work shows how the LDV crosstak probem can be avoided if the measurement surface is opticay smooth, hence the LDV technique can be used when measuring spinde dynamics. Keywords: Laser vibrometry; Crosstak; Specke noise; Spinde dynamics 1 INTRODUCTION To be abe to fuy investigate the behaviour of a rotating system, such as a miing machine spinde, it is necessary to make measurements directy on the spinde during rotation. This can be done either by eectronicay or opticay based non-contact measurement methods such as capacitive dispacement sensors [1], Laser distance sensors [] or Laser Dopper Vibrometry (LDV) [3]. The aser vibrometer is a powerfu too for measuring vibration veocities. The nature of the LDV system renders measurements without additiona mass oading and aows a wide range of distances between the sensor head and the object (from miimetres up to severa meters and scanning anges of about ± ). However, two major probems occur when performing 1

LDV measurements on rotating objects; the presence of specke noise and crosstak between vibration veocity components. In some cases a tracking system can be used, where the aser beam foows the rotating surface. Measurements on propeers and tiers have been demonstrated [4, 5]. Speckes are a random pattern of dark and bright spots formed in space when a diffusey refective surface is iuminated by coherent ight (aser ight). This is a resut of superimposing waveets of ight with different traveed path ength due to the surface structure. The specke noise from a rotating shaft is generated by the moving specke pattern on the LDV detector. This pattern is repeated for each revoution and wi create a repeated noise in the measurement signa, caed pseudo vibrations [6]. Larger surface structures than haf the aser waveength wi resut in fuy deveoped speckes. It has been shown by prior authors that the specke noise eve can be reduced or removed with different methods. By optimizing the target-detector separation within a aser vibrometer the noise eve can be reduced but not competey removed [7]. The specke noise in aser torsiona vibrometry measurements can be removed by randomising the path that the aser ight is undertaking during the revoutions, either by moving the aser aong the shaft [8] or simpy by adding a new surface structure. The atter can be achieved by continuousy appying e.g. oi or some other substances to the surface during the measurement [9]. In theory this technique shoud aso work in aser dopper measurements. The crosstak probem can be described as an error-term in the measurement caused by a veocity component due to the rotation [1-13]. The measured veocities of a rotating opticay rough shaft in the two orthogona directions, v x, v y, can be expressed as [13]: v y and v x = y& + Ω( x x ) (1a) = x& Ω( y y ), (1b) where x& and y& are vibration veocities, x and y are the vibration dispacements, x and y are the distances to the spin axis due to aignment errors and Ω is the tota anguar veocity incuding torsiona vibrations. x& and y& are the desired veocities to measure. The methods for specke noise reduction/remova described above cope with the specific specke noise probem but are not abe to neutraize the effect of crosstak in a singe beam LDV measurement. Consequenty; the signa obtained during measurements under these circumstances wi be a mix of the veocities in both directions. A method for resoving the true vibrations in the two x- and y-directions using a setup of two simutaneousy measuring asers in both directions and an accurate measurement of the rotationa anguar veocity has been deveoped by Hakon and Rothberg [13].

In [3] it is shown that the specke noise in aser vibrometry can be avoided by poishing the surface opticay smooth, i.e. the surface roughness is much smaer than the aser waveength. The out of roundness was measured and showed a good agreement with a mechanica roundness measurement. However, the crosstak was not investigated. In this work we investigate experimentay the crosstak between the two directions (x and y), in aser vibrometry measurements of a rotating spinde with a poished surface. The spinde is excited by an active magnetic bearing (AMB) manufactured by SKF Revove, and the response in the cross direction is measured by LDV and inductive dispacement sensors. EXPERIMENTAL SETUP AND PROCEDURE Fig. 1. (a) Photo of the setup. Active magnetic bearing (AMB), Coet hoder (CH), Dummy too (DT), LDV measurement point (LMP) and spinde (S). (b) Schematic representation of the AMB. Dispacement sensor (DS), Eectromagnet (EM), Ferromagnetic materia (FM). Fig. 1(a) is a photo showing the AMB mounted in the miing machine tabe and Fig. 1(b) is a sketch of the AMB. The measurements were made in a Dynamite; 3-axis vertica tabe-top miing machine. The spinde (S) is capabe of speeds of up to 7 rpm. The dummy too (DT) was mounted in a Coet hoder (CH) with a Morse taper which was in turn mounted in the machine. The surface of the LDV measurement position (LMP) was poished opticay smooth. The surface roughness was measured to Ra =.1 µm, using a Wyko NT11 optica profier (www.veeco.com). The surface coud be made temporary rough by spraying it with a deveoper for crack testing (paint). This paint coud easiy be removed without scratching the surface. The spinde was harmonicay excited by the AMB in the x- or y-direction with eectromagnets (EM) at a cyindrica segmented part of the dummy too consisting of a ferromagnetic materia (FM). The eectromagnets are arranged in two pairs opposite to each other. The vibrations of the spinde in the y-direction were measured by the vibrometer at LMP. Inductive dispacement sensors (DS) within the AMB measured the dispacements in the x- and y-directions simutaneousy. The inductive dispacement sensors are arranged in pairs, one pair measure the dispacement in the x-direction and the other pair in the y- 3

direction. The sensitivity of the inductive dispacement sensors are 11µmV -1. The measurement range is -5 khz and the imiting gap between the rotating dummy too and the dispacement sensors is 15 µm. A PSV 3 LDV system from Poytec GmbH incuding a dispacement decoder was used. The LDV scanning head was mounted on a sturdy tripod and paced 1 m from the poished dummy too. Specuar surfaces obey the aw of refection; ange of incidence = ange of refection. For successfu measurements on a specuar surface the aser vibrometer must be aigned propery with the target rotation axis. In practice the arrangement was aigned by ooking at the refected aser ight. A paper sheet with a hoe for passing the aser beam was mounted on the scanning head. The aser beam was focused on the poished surface behaving ike a cyindrica mirror. The refected ight sheet was adjusted so that the centra part passed the aperture of the scanning head. The signa quaity indicator of the LDV system showed a very high vaue at a spinde speeds so no dropout errors were present at the measurements. The LDV system was set up to perform samping with a frequency of 3 khz. The LDV sensitivity was set to 5mms -1 V -1 during the measurements on the smooth surface. The sensitivity was then decreased to 5mms -1 V -1 when measuring on the rough surface to avoid overoads due to the increased signa energy caused by the specke noise. The measurement ranges were 1V and 31.6V respectivey. Five different spinde speeds, 7, 14, 8, 56 and 7 rpm, were studied. The presence of specke noise was examined in the frequency domain of the LDV output during free run, which means that no forces where appied by the AMB. The eigenfrequencies of the dummy too/spinde were extracted from the frequency response functions shown in Fig., using the AMB controing system software. Three different excitation cases were examined; excitation cose to the first eigenfrequency of the too spinde system (4 Hz), excitation above the first eigenfrequency (7 Hz), and excitation at the rotation frequencies. 4

Fig. Frequency and phase response functions of the spinde in the x- and y-directions. The fact that the LDV and the dispacement sensors measure at different positions aong the z-axis (Fig. 1(b)) wi give different signa ampitudes. The dummy too was exited in the y-direction and the dispacement sensor output and the LDV output where compared. The veocity (differentiated dispacement) at the dispacement sensor is about four times greater than at the aser measurement point (LMP) at a measured spinde speeds during 4 Hz and 7 Hz excitation. When the excitation frequency coincides with the rotation frequency the recacuation factor is about seven. Henceforth, the data from the dispacement sensor is recacuated to the LMP. The crosstak in the LDV measurements of a rough and poished rotating shaft was studied. For each case the shaft was excited with different frequencies in the x-direction for the compete set of spinde speeds. 3 RESULTS Measurements were first performed at free run. Fig. 3 shows the dispacement ampitudes of the second to nineteenth rotationa harmonics of the poished dummy too. The ampitudes are independent of spinde speed and are rapidy decaying, which indicates that they are not caused by random specke noise. The profie of the poished measurement surface is not perfecty round. The actua deviation from a perfect circe wi be recorded by the LDV. These roundness components are seen in the FFT as rotation harmonics. The third harmonic has the highest vaue, which means that the trianguar component is the dominant one. By band-pass fitering the dispacement signa it is possibe to reconstruct the profie of the surface [3]. An Independent 5

measurement (at a ater time) of the surface using a mechanica roundness tester (C E Johansson) confirms these peaks and some scratches. The scratches are aso confirmed by the surface profie measurements performed by the Wyko equipment. Otherwise the surface was opticay smooth (Ra=1nm). Fig. 3. The dispacement ampitudes of the second to nineteenth rotationa order of the poished dummy too. To examine the crosstak, the dummy too was excited in the x-direction (cross direction to the LDV) at 4 Hz, 7 Hz and at the rotationa frequencies. In each case, two sets of measurements were made; firsty on the poished dummy too with an opticay smooth surface and secondy on the dummy too after being sprayed with paint to give an opticay rough surface. Fig. 4 iustrates the effect of crosstak in LDV measurements of a rotating rough surface for different spinde speeds. The vibration veocity measured on the dummy too after being sprayed with paint (triange up) shows a spinde speed dependent crosstak as expected from Eq. (1), whie the same measurements on the smooth surface (trianges down) does not. The outputs from the dispacement sensor (DS) in the y-direction for both sets of measurements (smooth and rough measurement surface) are aso presented in the graph (square and pentagram). The signa to noise ratio was checked to be 43dB in a typica case for the inductive dispacement sensors. The differentiation of the signa was performed in the frequency domain at the excitation frequency. Inserting the signas from the dispacement sensors y& and x into Eq. (1a) resuts in the expected veocity from the vibrometer when the surface is rough (circe). Numericay there is a good agreement between the cacuated and the measured veocity. In (a) the excitation frequency is at 4 Hz and the dispacement and veocity ampitudes in the x-direction are about 6 µm and 151 µms -1 at the LMP respectivey. 6

In (b) the excitation frequency is at 7 Hz and the dispacement and veocity ampitudes in the x-direction are about 4.5 µm and 198 µms -1 at the LMP respectivey. Despite the different excitation eve and frequency the resuts in (a) and (b) are consistent. There are some differences between the LDV output from the measurements on the opticay smooth surface and the veocities obtained by the dispacement sensor in the y-direction (triange down and square). These differences are due to a sma misaignment (few degrees) between the LDV and the dispacement sensor in the y-direction. The sensitivity to misaignment has been checked by cacuations. Since the excitation in the x-direction is comparativey arge, even sma misaignment anges do change the eve of the output consideraby. For exampe the difference in (a) can be compensated by an ange of about 4.5 degrees. In (c) the dummy too was excited at the rotation frequencies, 11.7, 3.4, 46.7, 93.3 and 116.7 Hz. The vibrometer output from the measurements on the smooth surface shows no crosstak and foows the differentiated dispacement sensor output. The crosstak effect owers the vibration ampitude eve in this case contrary to the previous two excitation frequencies. This shows that the crosstak can resut in either a higher vibration eve or a ower one than the correct one. 7

a b c Fig. 4. Crosstak in LDV measurements for different spinde speeds. Excitation at (a) 4 Hz, (b) 7 Hz and (c) at rotation frequencies. 8

4 DISCUSSION The aser beam aignment is one of the most important and critica step in rotating components measurement. In this study the aser vibrometer must aso be aigned with the inductive dispacement sensors for comparison. Further the ampitude of the orthogona excitation dispacement is even more important when studying the crosstak probem, which means that the inductive dispacement sensors measuring the dispacements in the cross direction must be perfecty orthogona to the aser vibrometer. This task showed to be difficut to overcome. The inevitabe instrument anguar misaignment was though minimized by try-and-error which was timeconsuming. The actua vaue of the anguar misaignment is unknown and difficut to contro now afterwards. Backward cacuations estimate the misaignment to be about 4.5 degrees in Fig. 4(a) which is not unreasonabe. However, the objective of this paper is to investigate the crosstak in aser vibrometry measurements on an opticay smooth rotating spinde. The crosstak is anguar veocity dependence and even sma dispacements in the orthogona direction resut in ampitude differences for different but sti high spinde speeds. Despite the systematic misaignment error the aser vibrometry measurements show no such spinde speed dependence. The differences between the eve of the aser vibrometry measurements and the eve of the dispacement sensor outputs associated with the instrument misaignment, noise and measurement error shoud not interfere with the crosstak investigation. It is shown that the crosstak can infuence the vibration measurement producing a vibration eve increase or decrease with respect to the correct eve. The atter trend can be expained by the foowing exampe. Suppose that the dispacements are x = Ax sin( ωt), () And y = Ay sin( ω t + ϕ), (3) where ω is the vibration frequency and A x and A y are the dispacements ampitudes. Differentiating (3) gives the transationa veocity in the y-direction y& = A ω cos( ωt + ϕ). (4) y Inserting expression () and (4) into (1a) and negecting the aignment error x gives the measured veocity v y = A ω cos( ωt + ϕ) + ΩA sin( ωt). (5) y x If the vibration frequency is equa to the rotation frequency, ω = Ω and if the phase difference between the dispacements in the x- and the y-direction is 9, the measured veocity (5) becomes 9

v y = Ω( A A )sin( Ωt), (6) x y and in the worst case; when the ampitudes A x and A y are equa the LDV output becomes zero. 5 CONCLUSIONS Laser vibrometry is normay used on stationary vibrating objects, athough synchronous tracking of the aser beam with the moving surface has been tried in number of cases by others. But to measure vibrations of a rotating spinde necessitates that the aser beam is stationary in space. If the rotating surface is opticay rough, a moving specke pattern wi occur on the detector, which gives a repeatabe specke noise in the measurement signa and aso crosstak from other veocity components. By using a poished surface, the crosstak in LDV measurements is avoided and the desired vibration can be measured. The fact that the cross vibrations appied by the AMB is removed from the LDV measurements automaticay impy the remova of torsiona vibration contributions. The frequency content in the signa due to roundness components did not indicate in any detectabe crosstak. The scanning LDV aso provides the possibiity to measure vibrations on different parts of a rotating machine, i.e. the miing machine spinde housing and other significant components during machining, which wi give an overa picture of the system. 6 ACKNOWLEDGEMENTS The work is supported by the Swedish Agency for Innovation System (Vinnova) and SKF Nova, respectivey. The purchase of the vibrometer is financed by the Kempefoundations. The authors wish to thanks Lars Frisk, for optimizing the poishing procedure of the dummy too, and aso Pär Markund, for surface roughness measurements. 7 REFERENCES [1] A. Abrecht, S.S. Park, Y. Atintas and G. Pritschow, High frequency bandwidth cutting force measurement in miing using capacitance dispacement sensors, Internationa Journa of Machine Toos and Manufacture 45 (5) 993-18. [] R.P.H. Faassen, N.van de Wouw, J.A.J. Oostering and H. Nijmeijer, Prediction of regenerative chatter by modeing and anaysis of high speed miing. Internationa Journa of Machine Toos & Manufacture 43 (3) 1437-1446. [3] M. Rantatao, K. Tatar, P. Norman, Laser dopper vibrometry measurements of a rotating miing machine spinde, in: Proceedings of the Eighth Internationa Conference on Vibrations in Rotating Machinery, University of Waes, Swansea, UK, 4, pp. 31-4. 1

[4] P. Casteini and C. Santoini, Vibration measurements on bades of a nava propeer rotating in water with tracking aser vibrometer, Measurement, 4 (1998) 43-54. [5] P. Casteini and R. Montanini, Automotive components vibration measurements by tracking aser Dopper Vibrometry: advances in signa processing, Measurement Science & Technoogy, 13 () 166-179. [6] S.J. Rothberg, J.R. Baker, and N.A. Haiwe, Laser vibrometry: Pseudovibration, Journa of Sound and Vibration, 135 (1989) 516-5. [7] M. Denman, N.A. Haiwe and S.J. Rothberg, Specke noise reduction in aser vibrometry: experimenta and numerica optimisation, in: Second Internationa Conference on Vibration Measurements by Laser Techniques: Advances and Appications, Washington, DC, USA 1996, pp 1-1. [8] S.J. Drew, and B.J. Stone, Remova of specke harmonics in aser torsiona Vibrometry, Mechanica systems and Signa Processing, 11 (1997) 773-776. [9] J. R. Be and S. J. Rothberg, Laser vibrometers and contacting transducers, target rotation and six degree-of-freedom vibration: what do we reay measure? Journa of Sound and Vibration, 37 () 45-61. [1] J. R. Be and S. J. Rothberg, Rotationa vibration measuring using aser Dopper Vibrometry: comprehensive theory and practica appication, Journa of Sound and vibration, 38 () 673-69. [11] S. J. Rothberg and J. R. Be, On the appication of aser vibrometry to transationa and rotationa vibration measurements on rotating shafts, Measurement, 35 (4) 1-1. [1] B. Hakon and S. J. Rothberg, Automatic post-processing of aser vibrometry data for rotor vibration measurements, in: Eighth Internationa Conference on Vibrations in Rotating Machinery, University of Waes, Swansea, UK, 4, pp. 15-9. 11

Paper IV

Miing machine spinde anaysis using FEM and noncontact spinde excitation and response measurement Matti Rantatao a, *, Jan-Oov Aidanpää b, Bo Göransson c, Peter Norman d a Lueå University of Technoogy, Division of sound and vibration, SE- 971 87 Lueå Sweden b Lueå University of Technoogy, Division of Computer Aided Design, SE- 971 87 Lueå Sweden c SKF Nova, HK1-6, 415 5 Göteborg, Sweden d Lueå University of Technoogy, Division of Manufacturing Systems Engineering, SE- 971 87 Lueå Sweden *Corresponding author. Te: +46 ()9-49 1 4, fax: +46 ()9-491 3, e-mai: matti.rantatao@tu.se ABSTRACT In this paper a method for anaysing atera vibrations in a miing machine spinde is presented incuding finite eement modeing (FEM), magnetic excitation and inductive dispacement measurements of the spinde response. The measurements can be conducted repeatedy without compromising safety procedures regarding human interaction with rotating high speed spindes. The measurements were anaysed and compared with the FEM simuations which incorporated a spinde speed sensitive bearing stiffness, a separate mass and stiffness radius and a stiffness radius sensitive shear deformation factor. The effect of the gyroscopic moment and the speed dependent bearing stiffness on the system dynamics were studied for different spinde speeds. Simuated mode shapes were experimentay verified by a scanning aser Dopper Vibrometer (LDV). With increased spinde speed, a substantia change of the eigenfrequencies of the bearing reated eigenmodes was detected both in the simuations and in the measurements. The centrifuga force that acted on the bearing bas resuted in a softening of the bearing stiffness. This softening was shown to be more infuentia on the system dynamics than the gyroscopic moment of the rotor. The study performed indicates that predictions of high speed miing stabiity based on rpm tap-test can be inadequate. Key words: Machine too spinde, centrifuga, gyroscopic, non-contact measurement, anguar contact ba bearings 1

1 INTRODUCTION Turning operations ike miing are common in the automotive and aerospace industry where arge meta work pieces are reduced to a fraction of its origina weight creating compex thin structures. It is important that unwanted behaviours ike too vibrations can be avoided during these operations. Especiay sef-excited machine too chatter caused by the waviness of the machined surface. This type of vibration wi cause poor surface finish and in some cases materia or machine damage. The phenomena is a significant issue and has been addressed and modeed by numerous authors during the past decades e.g. [1-4]. The deveoped modes predict a specific chatter free depth of cut which is governed by the transfer function of the too tip, assuming a rigid work piece. The chatter free depth of cut is cacuated for different spinde speeds which can be potted as a stabiity obe chart. The transfer function is normay measured manuay by tap-tests of a non-rotating spinde/hoder/too system where the too tip is excited by an impuse hammer and the response is measured by a vibration transducer. The assumption in this procedure is that the dynamics of the spinde/hoder/too system is independent of the spin speed. This is however not true for the case of high speed miing operations where the effect of gyroscopic moments and centrifuga forces must be taken into account [5-7]. To anayse the spinde speed dependency, the machine too must be anaysed in a rotating state that spans the whoe range of operating spinde speeds. Schmitz et a. [8] presented an experimenta method for the prediction of stabe cutting regions which refects the dynamic change that a rotating system undertakes. The method was based on impuse hammer excitation and capacitive probe response measurement of a too rotating during different spinde speeds. Stabiity obes for a discrete number of spinde speeds were cacuated and the imit of stabe cut corresponding to the actua spinde speed used was picked out to form a spinde speed dependent stabiity obe chart. Experimenta tests reveaed a changing stabe imit for stabe cut above 16 rpm due to changing spinde dynamics. An aternative method to anayse a spinde bearing system is by modeing. Wang and Chang [9] presents a spinde modeing method based on FEM. The mode did however not incude rotation and therefore no centrifuga forces and gyroscopic moments. In 1976 Neson and McVaugh [1] presented a FEM formuation of a rotor bearing system based on the Euer Bernoui beam theory where the effect of gyroscopic moments and centrifuga forces was incuded. Zorzi and Neson [11] ater on added interna damping and in 198 Neson [1] presented another formuation based on the Timoshenko beam theory which incuded the shear deformation effects. Xiong et a. [6] presented a way of combining this FEM representation and the miing cutting force mode formuated by Atintas []. The mode, which ony consisted of the rotor, predicted that the gyroscopic moment woud not affect the stabiity regions in miing but increases the rea part of the eigenvaues and so forth reducing the axia depth of cut. The mode aso predicted a change of the spinde resonance frequencies of about ± 1 Hz. Chi-Wei Lin et a. [13] integrated a thermo-mechanica-mode to the Timoshenko FEM description. Numerica and practica experiments verified an increase in bearing stiffness with increasing bearing preoad. The work aso predicted a softening of the spinde shaft with increasing spinde speed. It was shown that the softening of the bearing radia stiffness due to speed coud be compensated for by the thermay-induced preoad. Cao and Atintas [7] presented a genera method for the modeing of a spinde bearing system incuding the axia coordinate and a corresponding spinde speed and preoad dependent five degree of freedom bearing stiffness matrix. In the spinde mode a rotor reated centrifuga force was modeed by subtracting a term ( Ω mutipied by a radia version

of the transationa mass matrix) from the stiffness matrix. Simuations for different spinde speeds were performed but ony verified for a non rotating spinde. Simuations predicted that the centrifuga force of the rotor woud infuence the eigenfrequencies more than the gyroscopic moment of the rotor. This resut was not verified experimentay. This paper describes a method for anaysing atera machine too spinde vibrations based on a finite eement mode (FEM) and a contact-ess dynamic spinde testing equipment (CDST). The aim of this work is to study the effect of the gyroscopic moment and the speed dependent bearing stiffness on the system dynamics. The study were performed on a 5-axis Liechti Turbomi ST1 with an Fischer spinde (MFWS-35/4/8) with an integrated motor capabe of speeds up to 4 rpm and a pair of 5º anguar contact ba bearings. The spinde spring preoad was achieved experimentay and a speed dependent bearing stiffness was cacuated. The FEM eements were based on separate stiffness and mass radius. Simuations were performed with and without the speed sensitive bearing stiffness, together with a stiffness radius sensitive shear deformation factor. Tap tests with acceerometers and a scanning aser Dopper vibrometer were used to verify the CDST measurements and the FEM simuation. CDST MEASUREMENTS The CDST measured the frequency response function (FRF) of the too tip by exciting the rotor with eectromagnets whie inductive dispacements sensors registered the rotor position in the x and y direction. The use of eectromagnets and non contact dispacement sensors is common in the fied of active magnetic bearings (AMB). An AMB uses a contro system in order to keep the rotor in pace by adjusting the coi current in the magnets according to the measured rotor position and a desired ocation. In the segment of machine toos AMB s are mainy used as rotor support bearings but other appications have been deveoped during the past years. Auchet et a. [14] deveoped a method for indirect cutting force measurement by anaysing the command votage of magnetic bearing in a miing machine spinde supported by active magnetic bearings. Knospe [15] investigated the potentia of active chatter suppression by the use of AMB, and Chen and Knospe [16] presented an approach to estimate the cutting dynamics by both exciting the system and increasing the damping of the athe too using an AMB. 3

A B C -v1 +v1 +z -w3 +y +w3 DT +w1 -w1 Spinde & housing Too hoder Ax(t) Too hoder +v3 -v3 +x +z Dummy too Dummy too +y +x -w3 CDST Iy(t) ym(t) +w3 +dy +v3 +dx -v3 Ix(t) xm(t) CDST +y +x Fig 1. A: Photo of the setup with the dummy too in an eevated state. B: See-through sketch of the CDST with dummy too in pace. +dx and +dy denotes the dispacement sensors. C: Eectromagnet setup with dummy too (DT). Fig 1 shows a photo and a sketch of the experimenta setup used in the study. The spinde/hoder/too system dynamic was measured at the too tip in the x- and y-direction separatey by the use of a CDST. The excitation of the rotor was carried out by eectromagnets which were feed by frequency step vice sine sweep coi current, thereby introducing a magnetic force F m (t) which acted on the rotor. The rotor consisted of a speciay manufactured dummy too with a aminated rotor part in order to reduce the energy osses due to eddy current effects. In each direction two eectromagnets (e.g. x-direction v1 and v3) on opposite sides of the rotor were working out of phase with each other whie attracting the rotor to form the excitation. The force F m (t) appied to the rotor in the x direction (anaogous in the y direction) is expressed in terms of the stator coi current and the instantaneous air gap as [17]: I ( ) ( ) xt I xb F xm = Cxm, (1) d ( xm + xm) d + ( xm + xm) where C xm is a caibration factor, I xt is the measured current of top quadrant (v1), I xb is the measured current of bottom quadrant (v3). d is the effective gap of 15 µm between the magnets and the rotor, x m is the magnetic centre offset and x m is the instantaneous dispacement measured by the dispacement sensors. The coi current I xt and I xb is a superposition of the excitation current, a bias current and a compensation current for static oads, which are zero in this case. 4

-1 x -1 y Magnitude [og(m/n)] -13-14 -15-16 5 1 15-13 -14-15 -16 5 1 15 Mode 1 Phase [m/n] - Mode Mode 3 - Mode 4 5 1 15 Frequency [Hz] 5 1 15 Frequency [Hz] Fig. Tap test of an eevated dummy too compared to CDST measurements. Back: CDST, Gray: Tap test The instantaneous dispacement of the rotor in the x and y direction was measured by two dispacement sensors each which were couped together for each coordinate with opposite signs facing the rotor with 18 degrees apart. This arrangement enabes the canceation of any changes in dispacement due to therma expansion of the rotor diameter. The dispacement response and the exciting force for each frequency component in the interva 4- Hz were measured and a spectrum estimation (H1) of the transfer function was cacuated. This procedure was repeated for a speeds in the interva [::4] rpm and for each radia direction x and y. A reference acceerometer was mounted on the CDST housing to ensure that the assumption of a rigid CDST construction and a firm machine tabe mounting woud hod. The CDST were verified experimentay by tap tests of the mounted dummy too at rpm. Except for the third mode in the y direction the two different measurement methods resuted in simiar frequency response functions see Fig. The mode shapes for rpm were anaysed by a scanning LDV which performed a ine scan aong the z-axis of the visibe part of the dummy-too, hoder and rotor. 3 SPINDLE MODELLING A finite eement mode described by [1] is used to simuate the mode shapes and the eigenfrequencies of the rotor bearing system. The simuations are used to anayse and identify any detected gyroscopic or centrifugay induced speed dependency in the frequency response measured by the CDST. Fig 3 iustrates the finite eement mode and the eement division of the rotor bearing system potted aong the x-z pane. Each FEM eement of ength consists of two parts with an inner and outer radius see Fig 4. 5

Bearing position FEM node Stiffness & mass Mass.6.4. [m]..4.6.1..3.4.5.6 [m] Fig 3. Mode and FEM eement division of miing machine spinde The hoow shaft of the rotor with its inner stiffness radius r k and the outer stiffness radius R k governed the stiffness properties of the spinde. The physica outer mass radius R m which incuded the shaft, motor package, inner rings of the bearings and other additiona parts governed the mass together with an assumed physica inner mass radius r m. The assumed inner mass radius r m inside the spinde shaft modes the spring-package and the drawbar inside the spinde used for connecting the too hoder to the spinde. The connection between the spinde, hoder and the dummy too was not speciay modeed. Damping, gravity and the centrifuga effects of the rotor described by [7] was not incuded. The axia oad during free run was considered negectabe and therefore excuded in the study. The radia dispacement and rotation aong the radia coordinates of a singe eement is expressed in the generaised coordinates q = { xi, yi, φ x, i, φ y, i, xi+ 1, yi+ 1, φx, i+ 1, φ y, i+ 1 } where i is the node number. The area moment of inertia which is used in the forming of the stiffness matrix was 4 4 based on the stiffness radiuses I = ( R k r k ) π / 4. The poar moment of inertia J p = ( Rm rm ) / 4 and the diametra moment of inertia J d = ( Rm rm ) / were based on the mass radius. The shear deformation is expressed as Φ = (1EI) /( κg s ), where E is the moduus of easticity, G is the shear moduus and κ is the shear deformation factor. The shear deformation factor is normay determined experimentay and for a soid circuar shape a usua vaue is approximatey,9. However, the spinde studied in this paper consists of a hoow circuar shaft with a variabe stiffness radius. In 1 Hutchinson [18] proposed an anaytica expression for the shear deformation coefficients of a hoow circuar shaft expressed as ( R + r ) ( 1+ ν ) 6 k k κ = () 4 4 4 4 4 4 7r + 34r R + 7R + ν (1r + 48r R + 1R ) + ν (4r + 16r R + 4R ) k k k k k k k k k k k k which has been used in this work. 6

φ y,i φ y, i+1 φ x, i+1 φ x,i R m y x z r m R k r k Fig 4. FEM eement The homogenous equation of motion for the finite eement assemby used in this mode is expressed as: [ M ] q& + Ω[ G] q& + [ K ] q = {} & (3) K are the system matrixes of a shaft eement see Appendix. The assembed second order homogenous differentia equation was transformed into a first order differentia equation, using the state vector notation described in [19]. The equation of motion coud then be rewritten as: where [ M ], [ G ], and [ ] Ω[ G] [ M ] [ ] [ ] {} [ K ] [ ] {} = {} h + [] [ ] h M M &, (4) where {} h {} { &} = q q (5) is the state vector. Soving the obtained first order homogenous differentia equation gives compex eigenvectors with corresponding compex eigenvaues of the dispacement and the veocity of each node and its generaized coordinates. The front and the back bearing stiffness were added into the compete rotor stiffness matrixes at corresponding noda coordinates marked by trianges in Fig 3. 7

3.1 Preoad measurement To cacuate the bearing stiffness the preoad force of the bearings has to be known. However when deaing with a rea miing machine spinde the bearing preoad is normay not a known parameter and must therefore be measured. On ine measurements of the bearing oad woud be preferabe and a method for this has been presented by Chen and Chen []. For most spindes in operation, this faciity is however not incuded and the preoad must therefore be measured or provided by the designer of the spinde. The spinde used in this study was designed using spring oaded bearings as shown in Fig 5. The front and the back bearings are hybrid ange-contact bearings paced back to back. The preoad was measured by puing the spinde towards z whie the axia dispacement was measured using a dia indicator with a resoution of 1μm and a tota measuring range of +/- 5μm. The force used to pu the spinde was measured by a static force sensor with a range from to 5kN with a resoution of about N. The static force sensor was made out of strain gauges and connected to a strain gauge ampifier and an oscioscope. Data was coected manuay according to predefined steps. The estimated spring preoad was equa to the force needed to unoad the front bearing. The force needed to unoad the front bearing was estimated to 145N. Front bearing F Motor Preoad spring Housing Rotor Z+ Dia indicator Front bearing Back bearing Fig 5. Principa spinde drawing and preoad measurement setup. 3. Bearing stiffness cacuations Using the resut from the preoad measurement the bearing stiffness within the speed interva [::4] rpm were cacuated. The bearing stiffness cacuations were performed by SKF and an in house deveoped software Bearing Beacon based on the theory described in [1] and [] by de Mu et a. Inner ring Q i α i z F c Outer ring Fig 6. Contact anges together with the Hertzian and centrifuga forces which acts on one ba. α e Q e 8

4 RESULTS 4.1 Bearing stiffness With increasing speed the oad conditions between the ba and the rings in the bearing changes due to the centrifuga forces F c which act on the bas. See Fig 6. The centrifuga force induced by the rotating bas forces the rings to separate axiay. A new equiibrium state is reached where the contact forces Q i and Q e wi baance the new force condition which incudes both externa forces and the centrifuga force F c. When the ba reaches this new asymmetric position it wi act ike a spring when the externa force changes. This spring is in seria with the norma Hertz contact springs resuting in a decreased bearing stiffness for high spinde speeds. Tabe 1 shows the stiffness matrix of the back bearing and front bearing cacuated for rpm. Tabe 1 Stiffness matrix of the front and the back anguar contact bearing at rpm. Gray ces are potted in Fig 7 whie the other ces are eft out from the pot due to their simiar appearance. rpm x [1/m] y [1/m] yz [1/rad] zx [1/rad] Back Fx [N] 3.36e8.. -6.1e6 Fy [N]. 3.36e8 6.1e6. Myz [Nm]. 6.1e6 1.15e5 -. Mzx [Nm] -6.1e6.. 1.15e5 Front Fx [N] 3.38e8.. -6.76e6 Fy [N]. 3.38e8 6.76e6. Myz [Nm]. 6.76e6 1.39e5. Mzx [Nm] -6.76e6.. 1.39e5 The bearing stiffness for different spinde speeds reated to the stiffness at rpm are potted in Fig 7. The pot shows that the stiffness in the radia direction of the back bearing is decreasing to a eve of 6% of its origina ( rpm) vaue when the speed increases to 4 rpm. The corresponding vaue of the front bearing is 38%. The anguar stiffness shows approximatey the same amount of reduction for the same speed range. The reduction of the front bearing stiffness tends to diminish more than the back bearing above 18 rpm, giving it a characteristic shape (hereinafter denoted s-shaped ). 9

Percentage of stiffness at rpm [%] 11 1 9 8 7 6 5 4 Bearing stiffness -4 rpm (Preoad 145 N) Front bearing Back bearing Fx/x Mzx/zx Fy/yz Myz/y 3.5 1 1.5 Spinde speed [rpm] x 1 4 Fig 7. Changes in bearing stiffness with increasing spinde speed. 1% represents the bearing stiffness at rpm. The corresponding radia, anguar and couped stiffness in the other directions (se white ces in Tabe 1) shows an equa behaviour and are therefore not presented in the pot. 4. Gyroscopic and centrifuga effects In a rotor bearing system the modes often appears as a mix of rotor reated fexura modes and rigid body modes governed by the bearing properties [17]. The modes wi furthermore be effected by the gyroscopic moment of the rotor and spit up into two mode shapes [3]. The eigenfrequencies of these forward and a backward modes wi be infuenced by the driving frequency Ω. This can be seen in Fig 8 where the simuated eigenfrequencies with speed independent bearing stiffness are potted. The simuation shows an increasing difference with increasing speed between the two eigenfrequencies and their vaue at rpm. The deviation at 4 rpm compared to rpm is ±9 Hz, ±3 Hz, ±15 Hz and ± 3 Hz for modes 1-4 respectivey, see Tabe. The gyroscopic effect of the rotor can aso be seen in the eigenfrequencies of the simuation with speed dependent bearing stiffness, see Fig 9. In this simuation the infuence of the characteristic shape of the decreasing bearing stiffness can be seen added to the gyroscopic effect resuting in a deviation at 4 rpm of 4 and 19 Hz, and 19 Hz, 16 and 15 Hz and finay 1 and 55 Hz for the first four backward and forward modes. 1

18 16 Mode 4 Frequency [Hz] 14 1 1 Mode 3 8 6 Mode Mode 1 4 8 16 4 Spinde speed [rpm] Fig 8. Eigenfrequencies of the four first modes with the effect of the gyroscopic moment of the rotor. 18 16 Mode 4 Frequency [Hz] 14 1 1 Mode 3 8 6 Mode Mode 1 Intersection area 4 8 16 4 Spinde speed [rpm] Fig 9. Eigenfrequencies of the four first modes with the effect of the speed dependent bearing stiffness and the gyroscopic moment of the rotor. 11

4.3 Mode shape anaysis Anaysis of the simuated mode shapes, see Fig 1, reveas that the first mode shape starts at rpm as a mode governed by the back bearing with its node cose to the front bearing position. When the speed reaches 4 rpm the mode has transformed into a mode governed by the front bearing stiffness. According to the simuation the transformation is performed in the interva 1-16 rpm where the two first modes intersect. The first mode transforms its appearance by siding its front bearing node position towards the back bearing position. The second mode shape goes through a simiar transformation by starting as a front bearing mode at rpm, going through a cyindrica mode shape governed by both bearings around 14 rpm and finay end up as a mode shape governed by the back bearing at 4 rpm. The transformation area can be seen in Fig 9 as a disturbance of the shape of the changing eigenfrequency. Mode 3 and 4 are primariy fexura modes and are not changing its appearance in the same extent. The LDV measurement confirmed the simuated mode shapes at the visibe part of the spinde Fig 1. Mode LDV rpm Simuation rpm Simuation 14 rpm Simuation 4 rpm 4 3 1 LDV measured mode shapes Simuated mode shape Bearing position Fig 1. Mode shape anaysis of the four first eigenmodes. Where no compex conjugate (dashed shapes) is present in the simuation the mode shape is rea vaued. Drawings at top iustrate the positions of the mode shapes. LDV measured mode shapes are dispayed to the eft. 1

Hxx [og(m/n)] -1.5-13 16 Mode 4-13.5 Frequency [Hz] 1 Mode 3-14 -14.5-15 8 Mode -15.5-16 4 8 16 4 Spinde speed [rpm] -16.5 Fig 11. CDST measurement frequency response function Hxx. Hyy [og(m/n)] -1.5 16 Mode 4-13 -13.5 Frequency [Hz] 1 8 Mode 3-14 -14.5-15 Mode -15.5 4 8 16 4 Spinde speed [rpm] -16 Fig 1. CDST measurement frequency response function Hyy 13

4.4 CDST measurements The frequency response functions Hxx and Hyy of the spinde/hoder/dummy-too measured by the CDST are shown in Fig 11 and Fig 1. The two measurements made in the x and y-directions show a decreasing s-shaped pattern for the eigenfrequencies of the second mode (at 75 Hz) and the fourth mode (at app. 15 Hz). Ony the second front bearing reated rigid rotor mode can be seen in the measurement for a spinde speeds. The first back bearing reated mode is ony vaguey seen in the y-direction at 664 Hz. The same mode is detected in the tap test at 661 Hz see Fig. The third eigenmode at app. 9 Hz shows a reativey constant eigenfrequency even thought the spinde speed increases. A sma change of the eigenfrequency (,9-,%) of this mode (forward and backward) is detected in the simuation and in the CDST measurements. The simuated separation of the modes into a backward and a forward mode can not be detected in the measurements. A possibe intersection area where the two rigid rotor bearing modes intersects can be seen in the CDST measurement at 1 rpm and above as a change of magnitude. Furthermore the ampitude of this second mode increases with increasing spinde speed, especiay above 1 rpm. The CDST measurement aso shows a higher frequency for the fourth mode when excited in the y direction compared to the x direction. Tabe. Eigenfrequencies of CDST measurement and FEM simuation for and 4 rpm. FEM 1 is simuation with speed dependent bearing stiffness and FEM is without. (-) Not detected. Mode 1 [Hz] FEM 1 FEM CDST Backward Forward Backward Forward x y rpm 711 711 711 711-664 4 rpm 57 51 7 719 - - Frequency Change % -4, -36,5-1,3 +1,1 - - Mode [Hz] FEM 1 FEM CDST Backward Forward Backward Forward x y rpm 783 783 783 783 75 75 4 rpm 581 591 78 786 544 568 Frequency Change % -34,8-3,5 -,4 +,4-38, -3,4 Mode 3 [Hz] FEM 1 FEM CDST Backward Forward Backward Forward x y rpm 91 91 91 91 91 89 4 rpm 896 97 897 97 89 9 Frequency Change % -1,8 +1,6-1,7 +1,6 -, +,9 Mode 4 [Hz] FEM 1 FEM CDST Backward Forward Backward Forward x y rpm 1578 1578 1578 1578 1496 15 4 rpm 1478 153 1555 161 1384 144 Frequency Change % -6,8-3,6-1,5 +1,4-8,1-6,7 14

5 DISCUSSION AND CONCLUDING REMARKS In this paper, a method for anaysing atera vibrations in a miing machine spinde has been presented incuding a contact ess spinde dynamic measurement equipment substantiated with FEM simuation. The FEM formuation was based on reference [1] and was extended by incuding a separate mass and stiffness radius together with a stiffness radius dependent shear deformation factor. The frequency response functions in the radia directions for speeds in the interva [::4] rpm were measured by the CDST without compromising safety reguations regarding human interaction with high speed rotating spindes. The method was appied to a Liechti Turbomi ST1 equipped with a Fischer spinde capabe of 4 rpm. The machine too was designed with an integrated motorised spinde supported by a pair of 5º anguar contact ba bearings. In order to cacuate the bearing stiffness the spinde spring preoad was retrieved experimentay. 5.1 Gyroscopic and centrifuga effects The bearing stiffness was found to be sensitive to the centrifuga force acting on the bearing bas. This effect resuted in a substantia decrease in bearing stiffness (38-6%) and hence bearing reated eigenfrequencies with up to 4 %. This effect was confirmed by the CDST measurements. According to the simuations, the centrifuga effect in the bearings had a more significant effect on the eigenfrequencies of bearing reated modes than the gyroscopic moment of the rotor. E.g. mode shows a 33-35 % frequency change with centrifuga effect present compared to ony.4% when not. One coud expect that this softening coud be neutraised for high oads forcing the bas back to their nomina positions. Another possibiity of reducing the axia separation force of the rings woud be by using anguar contact bearings with a smaer ange (e.g. 15º). The softening of the rotor with increasing spinde speed and hence the reduction of the eigenfrequencies which was reported by previous authors coud not be detected. This is evident when anayzing the third simuated and measured mode, (fexura) at app. 9 Hz, where the eigenfrequency ony seems to be affected by the gyroscopic moment. 5. Mode anaysis The separation of the modes into a backward and a forward mode coud not be detected in the measurements most ikey due to damping which coud smears the signa energy from the two modes into a singe peak. The measured eigenmode at 75 Hz for rpm is assumed to be governed by the front bearing. This assumption is based on the mode shape anaysis where the front bearing mode normay woud resut in a arger dispacement at the too tip than the back bearing mode. Individua damping and phase conditions of these two modes coud reduce and smear the ampitude peak of the first mode. According to the simuation of the second mode, the shape of the frequency change woud begin by foowing the s-shape of the front bearing stiffness, and then jump over to foow the shape of the back bearing stiffness. The same but in the opposite order woud then aso appy to the first eigenmode. This tendency can be seen in the simuation see Fig 9. Due to this, a correct reading and identification of the eigenfrequencies of these two modes above the intersection area is difficut. Hence the CDST frequencies identified and isted in Tabe of the second mode at 4 rpm coud be infuenced by each mode individuay or by both modes together. The reativey constant eigenfrequency of the eigenmode at 91 Hz is consistent with the simuation which aso indicated that the third eigenmode was a fexura mode with itte infuence of the radia bearing stiffness. This was aso confirmed by the 15

mode shape simuation see Fig 1. The characteristic s-shaped pattern coud aso be seen in the CDST measurement of the fourth mode. This indicates that the mode in some way is reated to the front bearing. This indication was confirmed by the mode shape anaysis which showed that this mode was a mix of a front bearing mode and a fexura mode. 5.3 Accuracy and vaidation The CDST measurements were compared with traditiona tap-tests at rpm. The tap-tests showed e.g. sighty ower eigenfrequency for the third mode in the y-direction. This coud be due to the differences in dummy too positions for the two measurement cases. Despite this sma difference the tendency shown in the CDST measurements is not compromised and the change of spinde dynamic can ceary be seen. The different eigenfrequencies noted between the two radia directions indicates a non-symmetrica mounting of the spinde. This asymmetry was not considered in the mode. A deviation of approximatey -1% between the simuated and measured eigenfrequencies at rpm coud be seen in the resut. A possibe reason for the deviation coud be the stiff connection between dummy-too, hoder and spinde in the mode. The chosen vaue of the inner mass radius of the spinde and the absence of a modeed housing coud aso be a source of this deviation. Further studies wi incude the conversion of CDST measurements from a dummy too setup to a rea cutting too setup. The study performed indicates that predictions of high speed miing stabiity based on rpm tap-test can be difficut due to speed dependent system dynamic. 6 ACKNOWLEDGEMENT The study was financed by The Swedish Agency for Innovation Systems (Vinnova). SKF Nova is acknowedged for their technica support and for their provision of their deveoped CDST equipment. The Kempe Foundations is acknowedged for financing the LDV system. 16

7 NOMENCLATURE F m (t) Magnetic force x m, y m Instantaneous dispacement xm m A x y Magnetic centre offset Reference acceeration C xm Caibration factor CDST I xt I xb I yt Coi current quadrant v1 and w1 I yb Coi current quadrant v3 and w3 d Effective gap between rotor and magnets {} q Generaised coordinates R k r k R m r m Outer stiffness radius Inner stiffness radius Outer mass radius Inner mass radius [ M T ] Eement transationa mass matrix [ M R ] Eement rotationa mass matrix [ G ] Eement gyroscopic mass matrix [ K ] {} h μ I shaft J p J d Φ E G s κ ν F c Q e Q i α e α i Eement stiffness matrix Stat vector Eement mass per unit ength Area moment of inertia of a hoow Poar moment of inertia Diametra moment of inertia Shear deformation Moduus of easticity Shear moduus Shear deformation factor Eement ength Poisson s number Centrifuga force Contact force outer ring Contact force inner ring Contact ange outer ring Contact ange inner ring 17

8 APPENDIX 8.1 System matrixes Transationa mass matrix: [ ] = [ M ] + Φ[ M ] + Φ [ ] M T 1 M Rotationa mass matrix: [ ] = [ N ] + Φ[ N ] + Φ [ ] M R Mass matrix: [ M ] = [ ] + [ ] M T M R 1 N Gyroscopic matrix:[ G] = [ G ] + Φ[ G ] + Φ [ ] Stiffness matrix:[ K ] = [ ] + Φ[ ] where 1 G K K 1 [ M ] [ M ] 1 [ M ] μ = 4 1 μ = 4 1 ( + Φ) ( + Φ) μ = 4 1 ( + Φ) 156 54 13 38.5 16-31.5 94 14 17.5 7 17.5 156 54 13 94 38.5 16 31.5 14 17.5 7 17.5 4 13 3 7 7 4 13 3-31.5 3.5 17.5 3.5 7 7 156 31.5 3.5 17.5 3.5 Sym. 156 94 38.5 14 17.5 4 Sym. 94 38.5 Sym. 14 17.5 4 7 3.5 7 3.5 18

19 [ ] ( ) + Φ = 4 3 3 4 3 3 36 3 36 36 3 36 4 3. 4 3 36 36 1 3 Sym j N d [ ] ( ) + Φ = 1 5 15 5 15 5 15 5 15 15 15 5 15. 5 15 1 3 Sym j N d [ ] ( ) + Φ = 1 5 1 5 1. 1 1 3 Sym j N d [ ] ( ) + Φ = 4 3 3 3 3 36 3 36 3 36 4 3. 3 36 1 3 sym Skew J G p

[ ] ( ) + Φ = 5 15 5 15 15 5 15 15 15 5 15. 15 1 3 1 sym Skew J G p [ ] ( ) + Φ = 1 5 5 1. 1 3 sym Skew J G p [ ] = 3 4 6 6 4 6 6 1 6 1 1 6 1 4 6. 4 6 1 1 Sym EI K [ ] = 3 1. Sym EI K

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