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Condition Monitoring of Machine Tools and Machining Processes using Internal Sensor Signals JARI REPO Licentiate thesis Stockholm, Sweden, 2010

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Page 1: Condition Monitoring of Machine Tools and Machining ...319488/FULLTEXT02.pdf · machine tools are not used for their rigidity, but for their capacity of handling large and geometrically

Condition Monitoring of Machine Toolsand Machining Processes using Internal

Sensor Signals

JARI REPO

Licentiate thesisStockholm, Sweden, 2010

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TRITA IIP 10-01ISSN 1650-1888ISBN 978-91-7415-600-3

School of Industrial Engineering and ManagementSE-100 44 Stockholm

SWEDEN

Akademisk avhandling som med tillstånd av Kungliga Tekniska högskolanframlägges till offentlig granskning för avläggande av teknologie licentiatexa-men i produktionsteknik fredagen den 9 april 2010 klockan 10.00 i sal M312,Kungl Tekniska högskolan, Brinellvägen 68, Stockholm.

© Jari Repo, Mars 2010

Tryck: Universitetsservice US AB

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Abstract

Condition monitoring of critical machine tool components and machining pro-cesses is a key factor to increase the availability of the machine tool and achiev-ing a more robust machining process. Failures in the machining process andmachine tool components may also have negative effects on the final producedpart. Instabilities in machining processes also shortens the life time of thecutting edges and machine tool.

The condition monitoring system may utilise information from several sourcesto facilitate the detection of instabilities in the machining process. To avoidadditional complexity to the machining system the use of internal sensors isconsidered. The focus in this thesis has been to investigate if informationrelated to the machining process can be extracted directly from the internalsensors of the machine tool.

The main contibutions of this work is a further understanding of the directresponse from both linear and angular position encoders due the variations inthe machining process. The analysis of the response from unbalance testingof turn tables and two types of milling processes, i.e. disc-milling and slot-milling, is presented. It is shown that operational frequencies, such as cutterfrequency and tooth-passing frequency, can be extracted from both active andinactive machine axes, but the response from an active machine axis involvesa more complex analysis. Various methods for the analysis of the responsesin time domain, frequency domain and phase space are presented.

Keywords: Condition monitoring, machine tool, machining process, milling,position encoders, signal analysis

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Acknowledgement

The major part of the work presented in this thesis is carried out within theDLP-E project at Volvo Aero Corporation in Trollhättan, Sweden, duringthe years 2007-2009. The project was supported financially by VINNOVA1

through the MERA2 research programme, which is gratefully acknowledged.

First of all, I would like to thank my local supervisor Dr. Tomas Beno and co-supervisor Prof. Lars Pejryd at the University West for their excellent guidanceand encouragement during this work, even during the busiest periods. I alsothank Prof. Mihai Nicolescu for giving me the opportunity to become a PhD-student at the Royal Institute and for reviewing of this thesis. Project leaderAndreas Rudqvist at Volvo Aero Corporation also deserves special thanks forhis devoted participation in the project and for driving the project forward.

During this work I have had the opportunity to use modern equipment at theProduction Technology Centre in Trollhättan. Special thanks to Per Johans-son, Tomas Gustavsson, Jörgen Berg, and Ulf Hulling, for your assistanceduring the experimental work. I also appreciate the support from my col-leagues Mattias Ottosson and Hans Dahlin for solving some practical issues,and the support from Anna-Karin Christiansson. The discussions with NiklasEricsson at the University West and Arne Nordmark at the Royal Institute ofTechnology also gave valuable guidance in some of the theory, which is grate-fully appreciated.

Finally, I would like to thank my family for supporting me in this work. Veryspecial thanks to Linda and my children Robin and Ella for their patience andlove throughout this journey.

Jari RepoTrollhättan, Mars, 2010

1The Swedish Governmental Agency for Innovation Systems2Manufacturing Engineering Research Area

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Publications

The following papers are appended to the thesis.

Repo, J., Beno, T., Pejryd, L. (2009). New Aspects on Condition Monitoringof Machine Tools and Machining Processes. Proceedings of the 3’rd SwedishProduction Symposium (SPS’09), Göteborg, Sweden, 2-3 December 2009.

Repo, J., Beno, T., Pejryd, L. (2010). Machine Tool and Process ConditionMonitoring Using Poincaré Maps. The International Conference on Compet-itive Manufacturing (COMA’10), Stellenbosch, South Africa, 3-5 February2010.

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List of Symbols

ap Axial depth of cut [mm]αL Lissajous angle [rad]C(ε) Correlation sumd(ε) Correlation dimensionΔϕ(t) Modulation signal [rad]ftooth Tooth-passing frequency [Hz]fz Feed per tooth [mm/tooth]f0 Main frequency [Hz]Fs Sampling frequency [Hz]ϕ(t) Phase of an analytic signal [rad]ϕu(t) Unwrapped phase [rad]I(τ) Mutual information functionk Discrete time, signal segment indexλ1 Largest Lyapunov exponentm Embedding dimensionmu Unbalance mass [kg]n Spindle speed [rpm]ω Angular velocity [rad/s]N Number of samplesRL Lissajous radius [V]t Continuous time [s]Ts Sampling interval [s]τ Reconstruction delayU Arbitrary voltage signal [V]Ua, Ub, Ur Differentially measured voltage signals [V]vc Cutting speed [m/min]x = [x1, x2, . . . , xn] State vectorx, y, z, S1 Machine feed axis (x, y, z) and spindle axis (S1)z Number of cutting insertsz Complex/analytic signal

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Abbreviations

ACF Autocorrelation functionCBM Condition-based maintenanceCMS Condition monitoring systemCNC Computer numerical controlDAC Digital-to-analogue converterDAQ Data acquisitionDFT Discrete Fourier transformFAT Factory acceptance testFFT Fast Fourier transformFNN False nearest neighbourHHT Hilbert-Huang transformHT Hilbert transformIAT Installation acceptance testI/O Input/outputMI Mutual information functionRMS Root mean squareSNR Signal-to-noise ratioTCM Tool mondition monitoringTCMS Tool condition monitoring systemTFA Time-frequency analysis

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Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vAcknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiPublications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAbbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

I Introductory Chapters

1 Introduction 31.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Aim and scope . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Research approach . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Principles of condition monitoring 92.1 Acceptance testing of machine tool components . . . . . . . . 92.2 Role of condition monitoring systems . . . . . . . . . . . . . . 112.3 Sensorless condition monitoring . . . . . . . . . . . . . . . . . 13

2.3.1 Internal drive signals . . . . . . . . . . . . . . . . . . . 132.3.2 Encoder signals . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Position encoders in CNC machine tools . . . . . . . . . . . . 142.5 Principles for measuring of the position encoder output signals 16

2.5.1 Drive modules . . . . . . . . . . . . . . . . . . . . . . . 162.5.2 Counter card . . . . . . . . . . . . . . . . . . . . . . . 172.5.3 Data acquisition . . . . . . . . . . . . . . . . . . . . . . 18

3 Signal analysis methods 193.1 Characteristics of the output signals from position encoders for

CNC machine tools . . . . . . . . . . . . . . . . . . . . . . . . 193.1.1 General considerations regarding the analysis of position

encoder signals . . . . . . . . . . . . . . . . . . . . . . 233.1.2 Estimation of the SNR from measured signals . . . . . 24

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3.1.3 Filtering effects on the encoder signals . . . . . . . . . 253.2 Fourier analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Lissajous curves . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.1 Using Lissajous figures as vibration amplitude estimator 273.3.2 Formation of Lissajous figures from samples . . . . . . 31

3.4 Hilbert transform . . . . . . . . . . . . . . . . . . . . . . . . . 323.4.1 Separation of the modulation signal from the unwrapped

phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4.2 Scaling of the unwrapped phase . . . . . . . . . . . . . 36

3.5 Hilbert-Huang transform . . . . . . . . . . . . . . . . . . . . . 363.6 Nonlinear time series analysis . . . . . . . . . . . . . . . . . . 37

3.6.1 Mutual information . . . . . . . . . . . . . . . . . . . . 383.6.2 Embedding dimension . . . . . . . . . . . . . . . . . . 383.6.3 Chaotic invariants . . . . . . . . . . . . . . . . . . . . . 393.6.4 Poincaré sections . . . . . . . . . . . . . . . . . . . . . 41

3.7 Selection of signal analysis methods . . . . . . . . . . . . . . . 43

4 Exeperimental work 474.1 Linear encoder response to rotating unbalance . . . . . . . . . 47

4.1.1 Description . . . . . . . . . . . . . . . . . . . . . . . . 474.1.2 Signal analysis . . . . . . . . . . . . . . . . . . . . . . 494.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Machining of aerospace component - industrial trial . . . . . . 544.2.1 Description . . . . . . . . . . . . . . . . . . . . . . . . 544.2.2 Segmentation of the measured signals . . . . . . . . . . 554.2.3 Vibration amplitude estimation from the Lissajous figure 574.2.4 Analysis of the rotary encoder signals . . . . . . . . . . 584.2.5 Phase space reconstruction . . . . . . . . . . . . . . . . 604.2.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Slot-milling with various number of cutting inserts . . . . . . . 674.3.1 Description . . . . . . . . . . . . . . . . . . . . . . . . 674.3.2 Segmentation of the measured signals . . . . . . . . . . 704.3.3 Noise characterisation and SNR estimation . . . . . . . 714.3.4 Measuring of the Lissajous angle from the inactive feed

axis signals . . . . . . . . . . . . . . . . . . . . . . . . 724.3.5 Spectral analysis . . . . . . . . . . . . . . . . . . . . . 734.3.6 Nonlinear analysis of the active feed axis modulation

signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3.7 Phase plane analysis . . . . . . . . . . . . . . . . . . . 784.3.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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5 Conclusions and future work 835.1 Conclusions of experimental work . . . . . . . . . . . . . . . . 84

References 87

MATLAB script est_alpha 91

II Included Papers

New Aspects on Condition Monitoring of Machine Tools andMachining Processes 991 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002 Machine tool internal sensor signals . . . . . . . . . . . . . . . 101

2.1 Linear and rotary encoders for motion control . . . . . 1022.2 Encoder output signals . . . . . . . . . . . . . . . . . . 1022.3 Additional information from the encoder signals . . . . 103

3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 1033.1 Excitation of the machine tool structure . . . . . . . . 1033.2 Experiments with unbalance . . . . . . . . . . . . . . . 1043.3 Measurement setup . . . . . . . . . . . . . . . . . . . . 105

4 Time series analysis . . . . . . . . . . . . . . . . . . . . . . . . 1054.1 Measured time signals . . . . . . . . . . . . . . . . . . 1054.2 Fourier analysis applied to the time series . . . . . . . 1054.3 Poincaré analysis applied to measured time series . . . 106

5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 1096 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

References 113

Machine Tool and Process Condition Monitoring Using PoincaréMaps 1171 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . 119

2.1 Dynamical systems . . . . . . . . . . . . . . . . . . . . 1192.2 Phase space representation . . . . . . . . . . . . . . . . 1192.3 Phase space reconstruction . . . . . . . . . . . . . . . . 1202.4 Estimating the reconstruction delay . . . . . . . . . . . 1202.5 Estimating the embedding dimension . . . . . . . . . . 1212.6 Chaotic invariants . . . . . . . . . . . . . . . . . . . . . 1222.7 Poincaré sections . . . . . . . . . . . . . . . . . . . . . 1242.8 Visualising of Poincaré maps . . . . . . . . . . . . . . . 124

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3 Experimental studies . . . . . . . . . . . . . . . . . . . . . . . 1253.1 Preprocessing of the time series . . . . . . . . . . . . . 1253.2 Reconstruction of the phase space . . . . . . . . . . . . 127

4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 1285 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

References 131

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Introductory Chapters

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Chapter 1

Introduction

1.1 Background

Machine tools are composed of several subsystems, such as structures, elec-trical drive systems, controllers and actuators, which are all involved whenperforming the desired machining operations. The mechanical structure ofthe machine tool is often designed to be extremely rigid to withstand theforces created during the machining operation. Multitask machine tools aredesigned to perform several different machining operations such as turning,milling, drilling etc. in the same setup, which requires more degrees of freedomthan dedicated machine tools. The additional number of degrees of freedomhowever, comes with a price - some of the rigidity is sacrificed. Multitaskmachine tools are not used for their rigidity, but for their capacity of handlinglarge and geometrically advanced components and for their flexibility to al-low manufacturing in a single setup, i.e. without the need of refixturing thecomponent.

The availability and utilisation of machine tools are key factors which have adirect influence on the economy of the manufacturing company. Non-workingmachine tools due to scheduled and unscheduled maintenance, process or ma-chine tool component failure etc., have a negative effect on both availabilityand utilisation, which should be avoided as far as possible. The robustnessof the machining process is also a key factor in reaching an economicallyfavourable production simulation.

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Machine tool structural components, such as guideways, bearings and ballscrews, are subjected to gradual wear, which may be long-term. Testing ofmachine tool components on a regular basis is therefore important to reducethe risk of severe failures and breakdowns. Generally, the testing proceduresrequire that the machine tool must temporarily be taken out of service, thusreducing the availability of the machine tool. The testing is often carried out asa part of maintenance programs, but testing may also be needed when failures,such as unexpected collisions, have occured. Maintenance of machine tools isimportant to ensure high consistency between the produced parts, especiallywhen i) machine tools and spare parts are expensive ii) part consistency iscritical iii) downtime cost is extremely high.

Traditional methods to test different machine tool components include DoubleBall-Bar (DBB), Laser Doppler Vibrometry (LDV) and Laser Interferometry.These methods require mounting of additional equipment to perform the mea-surements, which is relatively time consuming. Appropriate maintenance ac-tivities, i.e. corrective actions, are then undertaken based on the results fromthe measurements. It can however be questioned when to motivate the use ofsuch detailed assessments of the machine tool because of the waste of valu-able production time. The preferred way is to use quick test of some criticalmachine tool component to indicate if more advanced test methods must beused. The main drawback with the traditional methods for testing of machinetools is that these tests are performed off-process and are not considering thespecific cutting parameters, and the spindle is not running.

The positional accuracy of machine tools is dependent on the function of crit-ical components, such as the guideways, ball screws, bearings and spindleshaft. Any deterioration, such as wear or misalignment, of these components,increases the risk of scrap production and later machine tool failures. Wearof spindle components has a strong influence on the performance of the spin-dle. Typical indicators of poor performance are i) increased temperature inthe spindle housing due to wear of spindle bearings and ii) increased powerconsumption and iii) rotational asymmetry (run out) caused by misalignmentof the spindle axis or iv) significantly increased vibration amplitudes.

To increase the availability and utilisation of machine tools, a maintenancefunction based on the actual condition (or health) of the machine tools istherefore desirable. A condition monitoring system, CMS, capable of in-process monitoring of the actual condition of the machine tool and machiningprocess, may not only provide early indications of problems, but may alsoactivate necessary control functions to perform the appropriate corrective ac-

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tion, e.g. temporarily halting the machining process, updating the machiningprocess parameters, or call for human assistance. This type of active con-trol mechanism has a substantial potential to increase the robustness of themachining process.

The aim with condition monitoring is early detection of disturbances in themachining process and wear of machine tool components. A machining systemis the interaction between the between the machining process and the elasticstructure of the machine tool. The cutting forces are created through theinteraction between the cutting tool and the workpiece.

Tool wear has been excessively researched in the past and have focused ontool wear detection, tool breakage detection, and the estimation of remainingtool life. Various techniques have been applied, with and without additionalsensors. Sensor based tool condition monitoring, TCM, are mainly basedon measuring of the cutting force components using a multi-channel table dy-namometer or rotating dynamometer, vibration amplitude using multi-channelaccelerometers, audible sound from the machining process, and high-frequencysound or acoustic emission, AE. Sensorless TCM are mainly based on mea-suring of internal drive signals, such as the feed motor current, spindle motorcurrent and spindle power. Combined measuring of multiple quantities is alsopossible.

The use of external sensors is however not always practical since it adds com-plexity to the overall machining arrangement - various number and types ofsensors must be mounted in the close vicinity of the machining process, makingthem subjected to the heat, chips and coolant, which may affect the lifetimeof the sensors and also quality of the measurements. The wirering of thesensors is another issue that must be considered especially in more advancedmachining operations. External sensors also require additional maintenanceand calibration in order to function properly.

A potentially attractive way to achieve a more robust solution to conditionmonitoring, compared with the traditional approach using external sensors,is to use the internal sensors and signals which are already available in themachine tool. Assuming that more information which is relevant to the mon-itoring task actually can be extracted from the signals, the complexity of themonitoring system may therefore be significantly reduced. Finding signaturesof specific phenomena, such as disturbances due to wear of critical machinetool components, and disturbance in the machining process due to tool wearor breakage, may also provide deeper insight into the health of machine tools

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and further understanding about the dynamics of machining processes due tothe choice of machining parameters. The information can then be used athigher level to support a condition based maintenance, CBM, function withinthe manufacturing company.

An issue often experienced in larger manufacturing companies with machineparks comprising multiple machine tools built on the same specification isthat even if these machine tools are expected to produce identical parts, theremay be some deviations in the dimensions of the produced parts, i.e. machinetools sometimes appear to behave more like individuals. This is detectedafter a dimensional check of the produced part and additional rework maybe needed to achieve the desired dimensions. Very often it is not alwayspossible with the currently used test methods to pin point the root cause ofthe problem. In general, this has a negative effect on the productivity in thatspecial versions of the numerical code and compensation schemes must bedeveloped and maintained for each of the supposedly identical machine tools.This additional complexity with variations among machine tools is howeverleft outside this thesis.

1.2 Aim and scope

The aim of this work is to investigate the possibilities to use internal machinetool signals for condition monitoring of machine tools and machining processes.This is important in order to achieve more robust machining processes withoutadding complexity to the overall machining system. In this work, a 5-axismultitask machine and various material removal processes, such as millingand drilling, are considered.

Condition monitoring involves measuring, processing and analysis of signals,the characterics of the measured signals must be known in order to selectappropriate methods for the processing and analysis of them. A major part inthis work is therefore to study the responses during various type of excitationsof the machine tool and present suitable strategies to extract the useful partfrom the signals.

The main research question is therefore whether it is possible to extract usefulinformation related to the health of the machine tool and stability of machiningprocesses from the internal sensors of the machine tool. A related question is

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also are what type of phenoma can actually be detected.

The quality of the information can be expressed in terms of its objectivity,repeatability, accuracy and errors, which must must also be considered inorder to evaluated the usefulness of the extracted information from the pointof view of machine tool availability.

The sensitivity of the method is related to the minimum detectable change ofthe wear and level of disturbance, which needs to be investigated.

The research questions for this thesis can be summarised as:

• Is it possible to detect deficient machine tool components using internalsensor signals?

• Is it possible to detect machining process instabilities using internal sen-sor signals?

• What signal analysis methods are suitable to extract the useful partfrom the internal sensors signals?

1.3 Research approach

The thesis has taken an experimental approach and is based on observationsobtained during machining in a modern 5-axis multitask machine tool. Variousexperiments have been performed which allow the systemtic study of certainphenomena. The main focus has been to study the time behaviour of theoutput signals due to vibration generated for various periodic excitation andvibration generated from rotating unbalance and vibrations generated fromimpacts.

The characteristic behaviour of the encoder output signals was initially un-known and needed a thorough investigation before any further analysis of themcould be undertaken. To get a fundamental understanding of the behaviourof the signals, initial experiments with minimal complexity have been carriedout, including both non-machining and various machining tests. Several pos-sibilities for the connection of the measurement equipment have been tried

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out until a final measurement chain were developed which allows high-qualityand reliable measurements without disturbing the overall machining system.

Various numerical methods have been applied to the encoder signals in order toextract the useful part from the signals. A subset of these methods have thenbeen selected when characterising various machining processes. The analysishas been carried out offline.

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Chapter 2

Principles of condition monitoring

Maintaining the health of macine tools and establishing stable machining pro-cesses is of major importance to reduce the risk of malfunctioning equipmentand ensure that high quality parts are produced. This can be achieved by testingof critical machine tool components and online measuring and analysis of oneor more quantities from the machining process in order to adjust the processtowards more stable machining regions. From the initial acceptance tests ofmachine tools this chapter reviews some principles of condition monitoring ofthe machining process using various methods found in the literature.

2.1 Acceptance testing of machine toolcomponents

Testing of machine tool components is important through its life cycle to avoidsevere breakdowns during operation. The testing procedure itself is carriedout both at the suppliers shop and after the installation. Generally, a fac-tory acceptance test, FAT, is carried out first at the suppliers shop beforethe delivery of the machine tool. After installation at the customers shop,an installation acceptance test, IAT, is performed as a final validation. Re-arrangement of machine parks at the customers shop, which may affect thealignment of structural components, is another reason when acceptance testsshould be performed. For a 5-axis multitask machine tool, the acceptance testprocedure may include measuring of the following properties [1]:

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• noise level during a well-defined and well-behaved machining operation,

• geometrical measurement,

• spindle speed,

• feed rate,

• idling power,

• radial and axial spindle vibrations (full spindle speed range),

• deflection of the machine tool structure,

• clamping force, i.e. the force that pulls the tool into the tool holder, and

• position accuracy of the linear and rotary axes

The duration of the FAT and IAT depends on the complexity of the machinetool and which properties are measured, but may take several days to com-plete. These tests are however performed only a few times during the life cycleof the machine tool.

The linear and angular position axes are tested for positional accuracy, re-peatability and backlash. Measuring of the alignment of the linear axes isnormally performed using LASER interferometry. The accuracy of the spindleshaft speed is measured using rotary encoder. The linear and circular interpo-lation capability of the machine tool is also measured to obtain the maximumdeviation from the programmed motion. The circularity test is normally per-formed using special measuring devices, such as the Renishaw™Double BallBar, DBB.

Machining of high quality parts is strongly dependent on that high relativeposition accuracy between the workpiece being machined and the cutting toolcan be achieved by the machine tool. The machine tool structure will howeverdeform over time due to thermal effects and wear of structural components,which makes the long-term behaviour of the machine tool difficult to predict.Deterioration of the machine tool condition may also affect its positional accu-racy. Thus, failing in maintaining the positional accuracy may result in thatthe dimensions of the produced part will fall outside the part specification.The manufacturing company may therefore not entirely rely on initial accep-tance tests, ATs, since the results from ATs will most probably only be validwithin a limited time window.

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2.2 Role of condition monitoring systems

The machining process is either continuous, such as turning or drilling, orintermittent, such as the milling operation. Continuous operations are per-formed with single cutting edge, removing material from one spindle revolutionto the next. Intermittent operations involve one or more cutting edges, remov-ing material from one tooth to the other. In both cases, material is removedfrom the workpiece under the generation of chips. The machining operationsare controlled by various parameters, such as the spindle speed, depth of cutand feed rate, etc.

If the correct machining parameters are set, the machining operation is ex-pected to perform well and the final produced part will meet the final require-ments given in the part specification. Depending on the machining processcharacteristics, the cutting tool may have a short or long life time. For costeffective production, the number of tool changes should be kept at minimumand the cutting tool inserts must be used close to the limiting tool life with-out violating the overall machining system. This requires in-depth knowledgeabout the tool wear rate and maximim tool life for the actual machining setupand machining conditions.

In well behaved machining processes, the tool life can be more or less accu-rately determined and the tool change interval may therefore be optimisedusing some tool wear criterion, such as the maximum flank wear. The simplecase will however require almost a gradual tool wear. For more complex ma-terials and demanding machining processes, tool wear may become excessiveand sudden events, such as tool chipping and breakage, will most likely occur.The monitoring of the tool condition may therefore be very difficult since suchunexpected events occur within a relatively short time interval.

The behaviour of the machining process is also dependent on the workpiecematerial, cutting tool material and geometry, actual machining process param-eters and the condition of the overall machining system. Hardness variationsin the workpiece material, which can be traced back to the manufacturing ofthe workpiece material itself, is another factor which may increase the unpre-dictability of the machining process, leading to drastically shorter tool life,tool chipping and tool breakage.

The cutting tool can be regarded as the limiting component of the machiningprocess and is the main reason why machining process condition monitoring

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Figure 2.1 – Role of condition monitoring systems.

is mostly concerned with the actual state of the cutting tool.

When the cutting tool is worn the cutting force and vibration amplitudes tendto increase and the machining process may become unstable. The main taskof the condition monitoring system, CMS, is to collect relevant data from themachining process, then process and analyse the data to detect symptoms oftroubles, but also to signal a control function to adjust the machining processparameters to a more stable machining region. Within the stable region,optimisation can be performed to meet some criterion, such as maximising thematerial removal rate, minimising the production cost, etc., see Figure 2.1.

Process instabilities are often recognised as increased vibration amplitudes,which may cause unexpected events such as tool failures, which can be harmfulto the workpiece and machine tool. To meet the increasing demands on higherproductivity and high quality of the produced parts, vast amount of researchhas been invested in the development of CMSs in order to prevent failures andcompensate for faults. A various number of sensors have been used for themonitoring of the state of the cutting tool. A review on the use of externalsensors for monitoring of the state of the cutting tool has been presented byByrne et al. [2].

The difficulties in designing reliable TCMs can be related to the complexityof the machining process itself, which may have one or more of the followingcharacteristics, apart from the changes of the machine tool itself [3]:

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1. complex to chaotic behaviour due to non-homogeneities in workpiecematerial,

2. sensitivity of the process parameters to cutting conditions, and

3. nonlinear relationship of the process parameters to tool wear.

2.3 Sensorless condition monitoring

Sensorless condition monitoring is about monitoring of the machining processby utilising the existing sensors and signals in the machine tool without usingexternal sensors, such as force sensors and accelerometers. The main advan-tage is that the additional complexity is kept to a minimum while reducingthe cost of the condition monitoing system. Two types of internal signals areconsidered, i.e. internal drive signals and encoder signals.

2.3.1 Internal drive signals

Internal drive signals, such as the spindle motor current and feed drive current,can be measured with a non-intrusive Hall effect sensor [4]. Measuring ofthe current signal from the servo drive motors and spindle motors has beenwidely used as a means of indirect measuring of the cutting force to avoidthe impracticability of force dynamometers. However, the observed currentsignal from the servo drive motor contains additional components related toacceleration and deceleration of the work table, friction force in the guideway, feed direction change, etc. Thus, to obtain a reliable estimation of thecutting force, the undesired components in the current signals must first beremoved, which can be accomplished using special pre-processing methods.The cutting force estimation method presented by Kim et al. [5] also makesuse of the internal feed rate signals to generalise the cutting force estimationto multi-axis machining. [6] utilised the spindle power consumption signal andfeed drive current signals to estimate tool wear in high-speed milling.

Internal signals may also be accessed directly from diagnostics sockets (DACoutputs) on the drive modules. However, these signals may have limited usedue to the relatively low internal sampling rate (a few milliseconds) and distor-tions due to internal digital-to-analogue conversion. The internal sensor may

13

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also be located far away from the component to be monitored. Internal drivesignals also represent a sum of signals originating from different sources [7].From a condition monitoring point of view, the use of the internal drive signalsmay be inappropriate due to the relatively low sampling rate and internal de-lays, which in turn may result in poor and incorrect results and assumptions.A TCM system that utilises the existing spindle speed and spindle load signalshave been reported by Amer et al. [8].

2.3.2 Encoder signals

The machine tool manufacturer have several options to achieve high positionaccuracy. The most popular way is to use position encoders due to their highreliability and sensitivity. Linear and angular position encoders are integralparts of the machine tool and used internally to close the position controlloops. These encoders measure the position along the feed axes rotating axes(spindle and turn table) respectively. Encoders have been used in the pastresearch. Kaye et al. [9] used the change in spindle speed (measured withan optical encoder) to detect tool wear in turning. Jang et al. [10] used thepulse signal from a rotary encoder to perform once-per-revolution sampling ofthe vibration signal. Plapper and Weck [7] found that backlash in the drivechain manifests itself as an increased difference between the position from themotor encoder and the encoder of the direct position when the axis movementis reversed. Verl et al. [11] used the output signals from the position encodersto quantify the wear of the ball screw drive. They concluded that the accuracyof positioning is a key factor to initiate maintenance. Klocke et al. [12] tooka step towards position-oriented process monitoring by utilising all positionencoder signals from a 5-axis milling machine for an in-depth analysis of a free-form milling operation. A complete measurement chain was also presented,allowing synchronisation of the position signals with other type of signals,such as cutting force and vibration signals.

2.4 Position encoders in CNC machine tools

Various types of position encoders are available, such as linear or rotary en-coders. The encoder may be either an absolute or an incremental encoders.Encoders are also based on different physical principles, such as light, mag-

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netism or induction. Rotary encoders are mounted directly on the motor shaftto measure the angular position and linear encoders are mounted on along thefeed axes, which gives a direct measuring of the position. Linear position canalso be measured indirectly using a rotary encoder on the driving shaft. In thelatter case the pitch of the lead screw must be known to get the position. Forhigh precision measuring of the position using encoders the direct measuringprinciple is prefered.

Figure 2.2 – Internal position encoders on a 5-axis multitask machine tool(DECKEL MAHO).

The 5-axis multitask machine tool1 used in the experimental work in thisthesis, is equipped with an rotary encoder2 to measure the angular position ofthe spindle shaft and turn tables, and linear encoders3 to measure the positionalong the feed axes. The physical structure of the 5-axis multitask machinetool and available position encoders is illustrated in Figure 2.2.

The position accuracy of machine tools is strongly dependent on the type ofposition encoders used. Modern machine tools are equipped with so calledSin/Cos-encoders, which give continuous sinusoidal outputs instead of squarewaves as is the case for incremental optical encoders. Position estimation fromthe signals is mainly based on interpolation. The main advantage with con-tinuous output signals is that the signals may be interpolated to an arbitraryresolution to achieve the desired accuracy.

1DMU/DMC 160 FD2WOELKE MINI-Sensor WG 053HEIDENHAIN LC 481

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2.5 Principles for measuring of the positionencoder output signals

2.5.1 Drive modules

The position encoders are initially connected to the drive modules of themachine tool as shown in Figure 2.3. The drive modules have special inter-polation circuits to calculate the position along the machine axis using theencoder output signals. For the connection of the encoder output signals toan external measurement system, direct measuring of the continuous outputsignals is considered in order to obtain as clean signals as possible.

Figure 2.3 – Drive modules in the DMU/DMC 160 FD machine tool.

Two options for the measuring of the encoder output signals are considered.One is the use of an external counter card, second is the use of traditionaldata acquisition, DAQ. A brief description of these measurement systems isgiven in the following subsections.

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2.5.2 Counter card

The first option is to connect the encoder signals to an external counter card,such as HEIDENHAIN IK 220 [13]. This card is used for special purposemeasuring of both angular and linear position and allows measuring of twomotion axes simultaneously. The card is also compatible with various signalinterfaces, such as 1Vpp signals. The input signals for each motion axis areencoder output signals ±A and ±B. To avoid damage to the original cablesand signals, breakout boards4 is used, from where the selected signals easilycan be identified and connected to the IK 220 board, see Figure 2.4.

Figure 2.4 – Measurement chain using the IK 220 counter card.

However, there is an issue with a disturbance on the control system when themeasurement computer is powered on when encoder signals are connected tothe IK 220 board. This is probably caused by an input impedance changedetected by the drive module. The machine tool will end up in a faultystate and will be unable to operate. A complete restart of the machine toolmay be required to resolve the problem. To avoid such problems the encodersignals may only be connected to the IK 220 board during measuring, whichof course is very inflexible. To overcome this issue, the drive modules mustbe isolated from the measurement system. The simplest solution is to add aswitch mechanism between the encoder signals and the IK 220 board. Theswitch can either be constructed using operational amplifiers which providenearly infinite input impedance, or using a manually controlled on-off buttonto control a set of relay coils to connect and disconnect the signals. A bettersolution to guarantee safe operation is to supplement the measurement chainwith opto-couplers in order to isolate the measuring system from the drivemodules.

4BRK15MF and BRK25MF from Winford Engineering

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The IK 220 interpolation card has been tested for calculation of the angu-lar position of the spindle shaft. The card was configured to sample every100 μs (10 kHz), which is the maximum sampling rate of the counter card.For monitoring of the encoder output signals at higher spindle speeds andfeed rates, the 10 kHz sampling rate may be insufficient. According to Eq. 3.1and Eq. 3.2, a Nyqvist frequency of 5 kHz (half of the maximum samplingfrequency) allows measuring of spindle speeds up to 1171 rpm and feed ratesup to 6000 mm/min. Furthermore, the size of the internal circular buffer inthe IK 220 card used to store the counter values is limited to only 8191 posi-tions, resulting in a buffer overflow if the measuring takes longer than 0.8191seconds. To make longer recordings possible, the data from the IK 220 buffermust be read into the PC RAM during measuring.

2.5.3 Data acquisition

The second option for measuring of the encoder output signals, is to diffe-rentially measure the encoder signals ±A,±B,±R using multi-channel digitialinput modules, see Figure 2.5. This is the preferred option since it allowssimultaneous measuring of all machine axes. This also offers higher flexibilitywhen setting the sampling rate. Selecting an appropriate sampling rate iscritical to obtain a good digital representation of the measured signals. Thedigital input module NI 9402 (4-ch/16-bit/100kHz) is used in the experimentalwork. The maximum Nyqvist frequency of the measured signals is 50 kHz,allowing measuring using a wider range of spindle speeds and feed rates.

Figure 2.5 – Measurement chain using a digital input module.

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Chapter 3

Signal analysis methods

This chapter presents an overview of signal analysis methods applicable tothe output signals from the position encoders described in Section 3.1. Bothtraditional linear methods and more advanced methods are presented. Thereare also some general considerations related to filtering of the encoder signals.Some of the methods will be used in the experimental work presented in thenext chapter in order to characterise and analyse the operational conditionsof machining processes where analysis in the time domain, frequency domainand phase space are considered.

3.1 Characteristics of the output signals fromposition encoders for CNC machine tools

The choice of appropriate methods for signal processing and signal analysis ismainly based on the characteristics of the measured signals and of course alsoon the nature of the phenomena being investigated. This section thereforepresents the most important characteristics of the output signals from bothrotary encoders and linear encoders.

Generally, the Sin/Cos-encoders for the linear and angular motion axes providethe differential continuous position signals ±A and ±B which are 90 degreeout of phase, also known as quadrature signals. The angular position encoderalso provides the reference mark signal ±R which provides a pulse to indicate

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a full revolution of the rotating axis being measured. The signal denoted witha minus (−) sign is basically the inverse of the signal denoted with a plus(+) sign and is a well known technique used especially in harsch industrialenvironments where the signals may be prone to various distrubances, suchas impulsive noise originating from electrical discharges or due to other typeof environmental noise. The basic idea is that the noise component will showup in both signals and can thus be easily removed by applying the differentialmeasurement principle on the signals [14]. In the continuation of this text, thedifferentially measured signals are denoted Ua, Ub and Ur. These signals areused by the drive modules to determine the position along the motion axis butalso to determine the direction of motion. In the positive motion direction, thesignal Ua is ahead of Ub and vice versa when moving in the negative direction.

The differentially measured output signals from the rotary encoder are thetwo continuous 1Vpp signals Ua and Ub. As already mentioned, there is also areference mark signal Ur available from the rotary encoder which gives a pulseafter each full spindle revolution, see Figure 3.1.

T ime

Am

plitu

de

Figure 3.1 – Typical rotary encoder signals Ua, Ub and Ur.

The rotary encoder signals are used internally by the machine tool drive mod-ules to determine both the angular position of the rotating axis and the di-rection of motion. The main frequency in the rotary encoder signals Ua andUb is related to the actual spindle speed. When the spindle speed is nearconstant or steady state, the output signals will be near sinusoidal with 256cycles per spindle revolution. The relation between the steady state spindlespeed n [rpm] and the main frequency f0 in the output signals is given as

f0 =n

60· 256 [Hz] (3.1)

From Eq. 3.1 it can be noted that the main frequency in the output signalsbecomes relatively high even for relatively low spindle speeds, see tabulated

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Table 3.1 – Main frequency f0 for some spindle speeds n.

n [rpm] f0 [Hz]100 427500 2133

1000 42675000 21333

10000 42667

values in Table 3.1. The output signals from the linear encoders are the two1Vpp signals Ua and Ub. The encoder signals are used internally by the drivemodules to determine the direction of the motion and the actual positionrelative to the machine coordinate system. The type of response from thelinear encoder depends on whether the signal is measured from an active orinactive machine axis. In this context, an active machine axis is a part ofeither a linear (feed axis) or angular (main spindle or turn table) motion.This motion itself can be either steady state motion, i.e. constant velocity,or time-varying. An inactive machine axis is regarded to be in a ”hold state”and not activly controlled to a specific location.

Figure 3.2 shows a typical response from an inactive feed axis during an in-termittent machining operation. The response is obviously nonlinear and pe-riodic.

T ime

Am

plitu

de

Figure 3.2 – Typical encoder output signals Ua, Ub from an inactive feed axisduring an intermittent machining operation, here plotted withthe spindle reference mark signal Ur.

Figure 3.3 shows a typical response from an active feed axis during an in-termittent machining operation. In this case the response is obviously near-sinusoidal.

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T ime

Am

plitu

de

Figure 3.3 – Typical encoder output signals Ua, Ub for an active feed axisduring an intermittent machining operation.

The main frequency f0 in the linear encoder output signals for an active feedaxis is directly related to the actual feed rate f and given as

f0 = f1.2

[Hz] (3.2)

where f denotes the steady state feed rate [mm/min]. Both Eq. 3.1 andEq. 3.2 have been found empirically by measuring the main frequency in theposition signals Ua and Ub from the main spindle and feed axes for steadystate spindle speeds and feed rates and performing the FFT on the measuredsignals. The proptionality constant 1

1.2 in Eq. 3.2 can be found by performing aleast squares linear fit of the feed rates (1000, 1500, . . . , 5000) mm/min and thecorresponding frequencies found by performing a spectral analysis. The valueof the proportionality constant may vary between different encoder types andtheir resolution since it includes the number of cycles per millimeter movement.The equivalence of Eq. 3.2 can be written as

f0 = f60· 50 [Hz] (3.3)

which shows the number of cycles of the sinusoid per mm movement, which is50 cycles/mm for the linear encoder in use. According to the linear encoderdata sheet1, one period of the output signals from the linear encoder corre-sponds to a relative linear movement of 20± 3μm, which is in agreement withEq. 3.3. Notice that the relations in Eq. 3.1 and Eq. 3.3 only apply for activemachine axes and give the main frequency for constant spindle speed and feedrate respectively.

1HEIDENHAIN, Position Encoders for Servo Drives, 12/2001

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For multi-tooth cutters with z cutting edges, such as milling tools, the feedrate f is given as

f = fz · z · n [mm/min] (3.4)

where n is the spindle speed [rpm] and fz denotes the feed per tooth [mm/tooth].

An important operational frequency is the tooth-passing frequency defined asthe inverse of the time between two subsequent tooth-passes. For a multi-tooth cutter with z teeth and uniform angular spacing between the individualcutting inserts, the tooth-passing frequency is defined as

ftooth = n60· z [Hz] (3.5)

3.1.1 General considerations regarding the analysis ofposition encoder signals

Some difficulties may be encountered in the analysis of position encoder sig-nals.

1. For active machine axes, the frequencies in the measured signals are notinitially related to any operational frequencies, why a transformation ofthe signals is needed in order to facilitate the extraction of the relevantinformation from the signals.

2. For inactive machine axes, the disturbances affect both the frequencyand amplitude of the signals. The operational frequencies may be there-fore found directly in the signals but the interpretation of the time-varying amplitude may be difficult due to the fact that encoder signalsin general are amplitudelimited, i.e. 1Vppsignals.

3. In the general machining case, it is most likely that the operationalfrequencies will be time-varying due to variations of the spindle speedand feed rate or various processrelated disturbances. Thus, in order touse the output signals from position encoders as the input for conditionmonitoring, constant speed and feed rates cannot be assumed.

4. For active machine axes, the main frequency in the signals acts as acarrier component, which will be modulated due various disturbances in

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the machining process. The useful information is then available in thefrequency of the signals as a modulation signal. In order to analyse theprocess dynamics from the signal, the modulation signal must be sepa-rated from the signal, i.e. signal must be demodulated. Once removed,the modulation signal can be analysed. This may however be difficultin the case when the carrier component is time-varying as in the generalmachining case.

5. Noise-free signals do not exist in the real world, especially not in harschindustrial environments, why a characterisation of the noise by means ofits probability distribution may be useful. A quantitive measure is thesignal-to-noise ratio (SNR) which is useful when evaluating the amountand quality of the information content in the signal.

Furthermore, in the general machining case, the number of simultaneous ac-tive feed axes will vary and the response from the individual feed axes mayalso be different due to varying rigidity of the machine axes. To reduce thecomplexity of the signal analysis due the aforementioned factors, the responsefrom single-axis2 machinining operations has therefore been considered in theexperimental work presented in Chapter 4.

3.1.2 Estimation of the SNR from measured signals

Any given measured signal is composed of two components, one is the usefuland meaningful signal s(t) and the other is the undesired noise signal v(t).The measured signal is then given as the sum s(t) + v(t). One measure usedto quantify how much the signal s(t) is correpted by the noise v(t) is thesignal-to-noise ratio, SNR, defined as the ratio between the signal power Psand the noise power Pv and is often expressed in dB. A definition of SNRwhich takes into account the presence of the noise, is given as [15]

SNRdB = 10 log10

(Ps + PvPv

)= 10 log10

(PsPv

+ 1)

(3.6)

which assumes that the noise can be measured by having the signal discon-nected. The other problem is that often the power in the signal and noiseis determined by measuring the voltage with and without the signal. If the

2a single active feed axis with the spindle running

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voltages are measured as true RMS-values, then the power values in Eq. 3.6can be replaced with the square of the RMS-values [15]

SNRdB = 10 log10

((srms + vrms)2

v2rms

)= 20 log10

(srms

vrms+ 1)

(3.7)

Notice that in Eq. 3.7 it has been assumed that the signal and noise are uncor-related, why (srms +vrms)2 = s2rms +v2rms since the double product, 2srmsvrms, iszero. The error in the estimation of SNR will be small if Ps/Pv is large. Thesize of the error also depends on whether the power or RMS values have beenmeasured. In practice, if the true RMS values are measured and the estimatedRMS values must be corrected [15].

Figure 3.4 shows 2000 voltage samples of a noise signal and a disturbed signal(with their mean value removed) recorded at 1024 Hz from the y-axis encoderof the machine tool.

−0.01

0

0.01

t

v(t)

[V]

−0.2

0

0.2

t

s(t)

+v(

t) [V

]

Figure 3.4 – Recorded noise signal (left) and noise-contamined signal (right)with their mean values removed.

In the case of N voltage samples {u1, u2, . . . , uN}, the RMS value is givenby urms =

√N−1(u2

1 + u22 + · · ·+ u2

N). The RMS values of the noise and dis-turbed signal are vrms = 0.002419 and srms = 0.1404 respectively. Eq. 3.7 thengives RMSdB = 35.42 dB. The histogram (or probability density function,PDF) of the noise indicates a nonuniform (Gaussian-like) noise distribution,see Figure 3.5.

3.1.3 Filtering effects on the encoder signals

Generellay, when measuring a physical quantity, it is advisable to prefilterthe input signal to suppress the noise to improve the signal-to-noise ratio,SNR, of the signal. There are several options where in the measurement

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−0.01

0

0.01

tv(

t) [V

]

−0.01

0

0.01

p(v)

Figure 3.5 – Noise signal (left) and its histogram (right) obtained using 50equidistant bins.

chain the filtering is performed but also regarding the type of filter to use.Analogue input signals may be filtered before the actual digitising by usingan anti-aliasing filter to avoid signal artefacts, such as aliasing. Typical anti-aliasing filters bandlimits the signal through lowpass filtering, which requiresan additional filter module as a part of the measurement chain. Anti-aliasingfilters are generally required if the bandwidth of the input modules of the dataacquisition system is low compared with the frequency of the signal.

The second option is to apply a digital filter after digitising of the signal.Using digital filters offers more flexibility in that both filter type, filter orderand filter cutoff frequencies can be specified and adjusted according to therequirements and therefore the preferred alternative due this flexibility.

However, prefiltering of the encoder output signals should be avoided as far aspossible since the signals may become severly disrupted and useless for furtheranalysis. Any filtering also has a price - loss of information.

3.2 Fourier analysis

sec:Fourier) One of the most important and widely used transforms in signalanalysis is the Fourier transform, FT3. The FT is a linear transformationwhich is capable of representing a time signal in the frequency domain. For aa finite discrete time series, the FT is defined as

X(k) = 1N

N∑k=1xke

2πjk/N (3.8)

3The Fast Fourier Transform, FFT, is a computationally efficient implementation of theFT, optimised for the case when N is a power of 2

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The discrete frequencies are fk = k/NΔt with k = −N/2, . . . , N/2 and Δtis the sampling interval. The FT gives the frequencies that contribute to thesignal and is useful for detecting signals buried in noise. One disadvantageis that FT is unable to tell when in time the frequencies occur since timeinformation is lost in the transformation. The FT may also show bad perfor-mance when applied to non-stationary signals and is therefore not well suitedto study transient effects due to the finite frequency resolution. One way toimprove the result and reduce the spectral leakage is to perform averagingover sliding windows in the time domain. The basic idea behind this is thatthe signal within a short time window may be regarded as stationary.

3.3 Lissajous curves

The Lissajous4 curve (or figure) is an old technique to study parametric equa-tions in the form

x = A sin(at+ ϕ), y = B sin(bt)

where A and B are the amplitudes, a and b are the driving frequencies and ϕis the phase shift. A Lissajous figure is produced by taking the two sine wavesand displaying them at right angles to each other, which can easily be doneusing an oscilloscope in XY mode. The shape in the Lissajous figure is stronglydependent on the frequency ratio and the phase difference, see Figure 3.6. Forthe special case when a/b = 1, the figure shows an ellipse. When x and y are90 degrees out of phase, i.e. when ϕ = π/2, a circle appears in the Lissajousfigure, which is a special case of the ellipse.

3.3.1 Using Lissajous figures as vibration amplitudeestimator

We know from Section 3.1 that the main frequency in the encoder outputsignals Ua and Ub, measured from an active machine axis, is related to themomentaneous spindle speed and feed rate. When the speed is constant,the main frequency in the signals can be calculated by using Eq. 3.1 andEq. 3.2. The frequency content in the signals must also be the same since

4The Lissajous figure was studied in 1857 by Jules Antoine Lissajous (1822-1880), aFrench physicist

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(a) (b) (c) (d)

Figure 3.6 – Lissajous figures for different values of the frequency ratio a/band phase difference ϕ. From left to right: a/b = 1, 1/2, 1/3, 1/4and ϕ = T/8, T/2, 3T/8, 0 where T is the driving period.

these signals co-exist as quadrature signals. The same reasoning applies forinactive machine axes with the exception that there is no carrier componentin the output signals.

A Lissajous figure of Ua and Ub from an active machine axis will always show acircle and will therefore not give any additional information. This applies forboth active feed axes and active spindle under both constant or time-varyingfeed rate or spindle speed. The response from an active machine axis containsa dominant frequency (a carrier) due to the active driving of the rotating orlinear axis.

The response from an inactive machine axis during excitation will show atime-varying amplitude caused mainly by forced periodic excitations from themachining process. The Lissajous radius, defined as RL =

√U2a + U2

b is nearconstant and thus invariant to the actual state of the machine axis, i.e. whetherthe axis is active or inactive. Thus, for an inactive machine during excitation,the Lissajous figure will have the shape of an arc.

In the following example from a rotating unbalance experiment, the encodersignals Ua and Ub were measured from the inactive feed axes x and y of a5-axis multitask machine tool during unbalance rotation of one of the turntables, see Figure 3.7.

As can be seen in Figure 3.7, both encoders sense the vibration due to inertialeffects but the energy in the responses are different due to varying rigiditybetween the x and y axes.

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−0.5

0

0.5

0 10 20 30 40−0.5

0

0.5

Time [s]

Am

plitu

de [V

olts

]

Figure 3.7 – Linear encoder signals measured during rotating unbalance ofa turn table. Shown are the x-axis signals (top) and the y-axissignals (bottom).

The least rigid machine axis in is the direction along the main guideway,which in this case corresponds to the y-axis. The output signals from thex-axis also contain the periodic behaviour due to the vibration but the energyis significantly lower in comparison with the responses from the y-axis.

A split-time view of the y-axis signals Ua and Ub and the corresponding Lis-sajous figure (arc shape) for increasing vibration amplitude (due to increasedrotational speed) is shown in Figure 3.15. It shows that the included angleof the arc increases with increasing vibration amplitude. Based on this ob-servation, the included angle of the arc is considered to reflect the vibrationamplitude sensed by the encoder.

There is obviously a relation between the shape in the Lissajous figure and thevibration amplitude, i.e. when the Lissajous figure of Ua and Ub shows an arc,the included angle αL of the arc is proportional to the maximum vibrationamplitude xmax [16, 17], i.e.

xmax ∝ αL (3.9)

For rotating unbalance, the vibration amplitude increases quadratically withthe rotational speed, which can be seen in Figure 3.8. For even larger vibrationamplitudes, the arc may even ”grow” into a circle. This situation will berefered to as the saturation level since it defines the limit for the maximumdetectable vibration amplitude by measuring of αL.

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0 100 200 300 4000

2

4

6

Rotational speed [rpm]

α L [rad

]

x−axisy−axis

Figure 3.8 – Experimentally measured Lissajous included angle as functionof rotational speed.

The close relation between the vibration amplitude and αL has been observedin practical machining tests, see Chapter 4.

Measuring of the Lissajous angle may provide additional information aboutthe stability of the machining process. However, the vibration amplitude isnot necessarily related to the stability of the machining process since i) astable machining process working close to resonance is still stable ii) a roughmachining system is still stable, but with higher dynamical forces causing thehigher vibration amplitudes. Furthermore, information about the vibrationamplitude may only be obtained up to the saturation level, i.e. the limit whenthe Lissajous figure shows an enclosed arc.

It can however be questioned whether Lissajous figures are suitable for condi-tion monitoring of machining processes due the the following factors:

• limited to inactive feed axes only,

• fails to produce a clear arc if the energy in the response from the encoderis low,

• ambiguous situation when saturation of αL occurs, which restricts themethod to smaller vibration amplitudes,

• unable to discriminate between forced vibration and more severe formsof vibrations

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Machining processes naturally generate forced vibrations which are generallynot severe. Instabilities on the other hand are often caused by self-excitedvibrations (chatter), which may lead to severe vibration amplitudes and con-sequently a machining process failure and poor surface quality and integrity ofthe machined part if not handled properly. The vibration amplitude may reacha level such that it cannot be detected by measuring of αL due to saturationwhen αL → 2π.

Even though the Lissajous method suffers from the aforementioned drawbacks,the implementation of the method is relatively straighforward since it onlytakes the encoder signals Ua and Ub as input, which are available in modernCNC machine tools.

3.3.2 Formation of Lissajous figures from samples

The formation of a Lissajous figure from scalar time series requires that asufficiently large number of samples are collected from the sample sequencies{Ua} and {Ub}.

Accumulation of all the data points from {Ua} and {Ub}, especially fromlong time series is not appropriate since the Lissajous diagram will soon beovercrowded and the Lissajous figure will most likely end up in an enclosedarc, thus giving no useful information at all. Instead, the input samples toform the Lissajous figure can be selected using a running window approach.Furthermore, the selected window length L must be large enough to containa sufficient number of pairs from the sequencies.

For practical machining cases, the spindle reference mark signal Ur can beused to identify each spindle revolutions. It is therefore convenient to have awindow length that corresponds to the period of one spindle revolution, e.g.a spindle speed of 280 rpm and sampling rate 20 kHz gives a window lengthof 4286 samples. Samples from the sequences {Ua} and {Ub} can then becollected between subsequent pulses to form the Lissajous figure.

When using the windowing appraoch, a function αL(k), i.e. the included anglein the Lissajous figure as function of the discrete time k, can be obtained,see Figure 3.9. Overlapping windows may also be considered to increase theresolution of αL(k). This technique is used when analysing the time series inChapter 4.

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10 20 30 40 50 60 70 80 90 1000

k

αL(k

)[ra

d]π

Figure 3.9 – Lissajous included angle as function of discrete time k measuredusing a running window over subsequent spindle revolutions.

It has also been observed in practical machining tests that the arc will notstay in a fixed location in the Lissajous diagram during the machining processfor subsequent spindle revolutions. Instead, the arc tend to circulate aboutthe periphery of the circle, which complicates the calculation of the includedangle from a set of points {Ua(n), Ub(n)} where n = 1, 2, . . . , L. A numericalmethod to carry out the computation has been developed and implementedby the author and included in the Appendix in this thesis.

As will be seen in Chapter 4, the resulting Lissajous angle, when using thewindowing technique, also shows a strong correlation with standard linearfirst-order statistical moments, such as the mean and variance, of the scalartime series. The Lissajous angle may therefore be used to measure the sta-tionarity in the data.

3.4 Hilbert transform

The Hilbert transform (HT) is a linear operator that can be used to extractthe instantaneous frequency of a signal. For an arbitrary signal x(t), its HTy(t) is defined as5

y(t) =1πP∫ ∞−∞x(τ)t− τ dτ (3.10)

5The P preceeding the integral in Eq. 3.10 denotes the Cauchy principal value

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from which it can be noticed that the HT is basically the convolution of thesignal x(t) with 1

πt, which makes it capable of identifying local properties of

x(t). The resulting analytical signal has a real and imaginary part, and givenas

z(t) = x(t) + jy(t) = a(t)ejϕ(t) (3.11)

The instantaneous properties of x(t) are defined as

a(t) =√x2(t) + y2(t) (3.12)

and

ϕ(t) = tan−1(y(t)x(t)

)(3.13)

where a(t) is the instantaneous amplitude of the signal x(t) and ϕ(t) is theinstantaneous phase of x(t). The instantaneous frequency ω(t) is defined asthe time derivative of the instantaneous phase ϕ(t) as follows

ω(t) =d

dtϕ(t) (3.14)

There are some restrictions regarding which type of signals that can be anal-ysed with the HT. Signals in general are classified as mono-component ormulti-component. The HT can only be applied to the first class of signals, i.e.mono-component signals, also known as ”Hilbert-friendy”. An example of thisclass of signals is the well known chirp-signal with a time-varying frequencyas shown in Figure 3.10, but also ordinary sinusoidal signals [18]. Chirp-likesignals generally arise from the acceleration and deceleration of moving orrotating masses.

0 2 4 6 8 10−1

0

1

t

y(t

)

Figure 3.10 – Chirp-signal with linearly increasing frequency,y(t) = sin(2πf(t) · t), f(t) = kt, k = 0.2.

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The response from position encoders for active machine axes are frequencymodulated sinusoids. Since signals with a frequency that varies nonlinearlywith time are mono-component, these signals may be analysed using the HT asthey represent a general form of a chirp-signal. The response from inactive feedaxes on the other hand are super-positioned sinusoids, i.e. multi-component,and cannot be analysed using the HT.

The process to obtain the modulation signal from a frequency-modulated sig-nal is illustraded in Figure 3.11. The phase of the mono-component signalin 3.11 (a) is shown in 3.11 (b). The unwrapped phase ϕu(t) in 3.11 (c) cor-responds to the position signal. The final modulation signal in 3.11 (d) isobtained by removing the linear component from the position signal.

U(t

)

t

(a)

ϕ(t

)

t

(b)

ϕu(t

)

t

(c)

Δϕ

(t)

t

(d)

Figure 3.11 – Demodulation process using the HT. (a) Original signal. (b)Phase of the analytical signal. (c) Unwrapped phase/positionsignal. (d) Modulation signal.

Since the resulting phase in Eq. 3.13 will be in the range [−π, π], it is necessaryto first ”unwrap” it to obtain the phase in the form

ϕu(t) = ω0t+ Δϕ(t) = 2πf0t+ Δϕ(t) (3.15)where ϕu(t) denotes the unwrapped phase in [rad], ω0 is the steady-stateangular frequency in [rad/s], Δϕ(t) is the modulation signal in [rad] .

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The HT is very useful when dealing with frequency modulated encoder signalsfrom various machining processes. During machining, the signals from activemachine axes will undergo frequency modulation. The modulation can beregarded as the sum of several effects in the machining process, such as vibra-tions, variations in the workpiece material and tool wear. The signals may alsocontain undesirable components, resulting from the acceleration/decelerationof moving masses, friction force in the guideway, direction change, etc.

3.4.1 Separation of the modulation signal from theunwrapped phase

When analysing the response from position encoders for an active machineaxis, the HT is can only be used to calculate the position signal (or unwrappedphase), see Figure 3.11 (c). Since the signature of the machining process isinitially ”hidden” in the position signal as the modulation signal, the mainproblem is to separate the modulation signal from the position signal. Thiswill be refered to as the demodulation process. Once the modulation signal hasbeen extracted, small variations, such as torsional vibrations and variationsin the feed directions, can be analysed.

The modulation signal is extracted by removing the linear term ω0t in Eq. 3.15through linear detrending, filtering or differentiation of the position signal.Filtering of the position signal requires that the carrier frequency is known,which can be determined by using Eq. 3.1 or Eq. 3.2. However, detrendingand filtering may fail or produce poor results in occasions when the spindlespeed or feed rate is non-constant. Differentiation of the position signal givesthe instantaneous velocity but is very noise-sensitive and a more advancedanalysis may be required in order to extract the useful information. Othermethods to separate the modulation signal Δϕ(t) from the position signalmay also exist.

If the demodulation process performs as expected, regardless which methodis used, the carrier component will be removed and the sideband frequen-cies will now appear in the lower frequency range. The modulation signal,which contains the process-related variations, may then be analysed withinan appropriate domain.

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3.4.2 Scaling of the unwrapped phase

As indicated by Eq. 3.1, there are 256 periods of the near-sinusoidal signalsduring one spindle revolution. The position signal, or unwrapped phase de-fined in Eq. 3.15, may be divided by this number to obtain correct scaling ofthe position signal. The same reasoning applies for the output signals fromthe linear encoders. Eq. 3.3 gives that the near-sinusoidal signals contain 50cycles per millimeter movement. To convert the modulation signal in lengthunits, the angular values must be divided by the number of periods per lengthunit to obtain correct scaling. However, correct scaling is not critical for theanalysis and can therefore be left out.

3.5 Hilbert-Huang transform

The Hilbert-Huang transform (HHT), was originally developed by Huang et al.[19]. The method is based on the HT (see Section 3.4) and results in a time-frequency representation of the data. It was originally developed to handleproblems due to insufficient number of samples, nonstationary data and non-linear data. The key strength of the method is the use of the instantaneousfrequency defined in the HT in Eq. 3.10.

As mentioned in Section 3.4, the drawback with the HT is the fact that it islimited to mono-component signals only. The HHT solves this issue by first de-composing the signal into so called Intristic Mode Functions (IMFs) througha process called the Empirical Mode Decomposition (EMD), also known asthe sifting process. The EMD is a pre-processing step which decomposes theoriginal signal into a series of separate oscillatory modes. In an ideal decom-position, all modes of oscillations, i.e. the IMFs, will be mono-component,to allow the use of the Hilbert Transform to determine all the instantaneousfrequencies in the signal.

A drawback with the HHT is that it suffers from not beeing able to tell inadvance the number of resulting IMFs generated in the EMD process. Further-more, some of the genaretd IMFs may not actually correspond to the physicalphenomena [20]. The results must therefore be carefully interpreted. Someimprovements to the original HHT have lately been reported, e.g. [18, 21],addressing the issue with the stop criterion in the sifting process.

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3.6 Nonlinear time series analysis

The HT and HHT methods presented in previous sections have shown to beuseful in the preprocessing step of the measured signals and used mainly totransform the original signals into some meaningful data, such as the mod-ulation part of the signals. TF analysis can provide some insight into theprocess dynamics, but must be supplemented with additional tools to get amore complete picture of the dynamics of the machining process. The set oftools used to analyse the measured signals must be expanded since the outputsignals from machining processes show nonlinear periodicites and linear toolsare not well suited for the task. Even the simplest machining process, suchas orthogonal cutting, may show chaotic behaviour where the level of chaos isdependent on the workpiece material [22].

A dynamic system, given as a differential equation, such as x = (x, t) is gen-erally analysed using its phase space description which gives time evolutionsof its state trajectory. The dimension of the phase space is related to thenumber of independent state variables. For systems with dimensions higherthan three it is difficult to look at their phase space. The study of the flowin phase space can reveal interesting details about the system, such as theexistence of limit cycles and stable, chaotic and strange attractors, but is alsouseful for the prediction of its future states.

For practical systems, mathematical models of the system are commonly un-available, making an analytical investigation very difficult. However, the non-linear theory allows the reconstruction of the system from measurements ofone or more representative and independent state variables. The most impor-tant reconstruction technique is the embedding of a single measured variableinto a delay coordinate vector

xn = [xn, xn−τ , . . . , xn−(m−2)τ , . . . , xn−(m−1)τ ] (3.16)

where m denotes the embedding dimension and τ is the reconstruction delay.The (m, τ) forms the embedding parameters. The delay coordinate vectorrepresents a time separation of the measured time series x. The most impor-tant property of this method is the preservation of the most important aspectsof the system dynamics, indicated by the fact that the reconstructed phasespace has similar geometric properties as the true phase space. The analysiscan then be performed using the pseudo phase space.

In the preceeding steps, the embedding parameters must be determined in the

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following order:

1. determine the optimal embedding delay

2. determine the minimum embedding dimension

3.6.1 Mutual information

The mutual information function presented by Fraser and Swinney [23], isunlike the autocorrelation function, a measure of the dependence betweensamples as it also takes into account nonlinear correlations. The computa-tion of the mutual information function for a scalar time series is based onpartitioning of the data into intervals (or bins), and defined as

I(τ) = −∑i,j

pi,j(τ) ln pi,j(τ)pipj

(3.17)

where pi is the probability of finding a sample in the ith interval, and pij(τ)is the joint probability of finding a sample in the ith interval which a time τlater is found in the jth interval. The optimal delay value is then be found byconsidering the first minimum of the mutual information function.

3.6.2 Embedding dimension

A method to determine the minimum required embedding dimension is themethod of false nearest neighbours (FNN) originally proposed by Kennel et al.[24]. The minimum dimension must be large enough to fully resolve the sys-tem’s phase space. Using a smaller value than the minimum required, pointson the phase space trajectory will be projected into the vicinity of other pointswith which they are not really neighbours, but false neighbours. The FNNmethod starts with a low dimension and then increases it step by step until asufficiently low number of false neighbours exist. If x(j) is the nearest pointto x(i) for an embedding dimension m, the distance between these points isgiven by

r2m(i, j) = (x(i)−x(j))2 + . . .+(x(i+(m−1)τ)−x(j+(m−1)τ))2 (3.18)

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Increasing the embedding dimension from m to m + 1 the distance will takethe form

r2m+1(i, j) = (x(i)− x(j))2 + (x(i+mτ)− x(j +mτ))2 (3.19)Then if

|x(i+mτ)− x(j +mτ)|rm

> RT (3.20)

the nearest neighbor at time i is considered as false. The threshold valueRT is set to 10 ≤ RT ≤ 50. The embedding dimension then converges to acharacteristic value as the percentage of FNNs drops to zero.

3.6.3 Chaotic invariants

Various methods for the quantification of dynamic systems are presented. Thecorrelation dimension is a measure of the complexity of the geometry andshape of the attractor, while Lyapunov characteristic exponents can be usedto distinguish between chaotic or nonchaotic behaviour.

A method to estimate the correlation dimension was originally presented byGrassberger and Procaccia [25]. A hyper sphere with a small radius of ε iscentered on the attractor. Let Nx(ε) denote the number of points on theattractor that are inside the sphere. When ε is increased, the number ofpoints inside the hyper sphere will increase exponentially. To account forthe variation of the pointwise dimension over the attractor, the average ofthe number of points is often computed, also known as the correlation sum,defined as

C(ε) =2

N(N − 1)

N∑i=1

N∑j=i+1

Θ(ε− ‖xi − xj‖) (3.21)

where Θ is the heaviside step function. The correlation sum is a measure ofthe number of pairs (xi − xj) whose distance is less than the radius ε. Foran infinite number of points, i.e. when N → ∞ and for small value of ε, thecorrelation sum is expected to scale according to the power law

C(ε) ∝ εd (3.22)The correlation dimension is then defined as the local slopes of the correlationsum, given as

d(N, ε) = logC(ε,N)log ε

(3.23)

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d = limε→0

limN→∞d(N, ε) (3.24)

In time series analysis, it is desirable to avoid temporal correlations, suchas those occurring when samples are close in time and correlation due to thegeometry of the attractor. A modified definiton of the correlation sum formulais used to exclude such temporal correlations.

C(ε) = 2(N − nmin)(N − (nmin − 1))

N∑i=1

N∑j=i+nmin

Θ(ε− ‖xi − xj‖) (3.25)

Here, the value of nmin can be chosen freely as long as

τ > nmin � N (3.26)

where τ is the lag that gives the first zero-crossing in the autocorrelationfunction, and N is the number of data items. To find an appropriate value ofnmin, the time-space separation plot can be used [26].

While correlation dimension is a measure of the complexity of the dynamicsystem, Lyapunov exponents measure the level of chaos in the system. Animportant property of chaotic systems is their sensitivity to small changes intheir initial conditions which have the effect that the state trajectory maydiverge after some time. The Lyapunov exponents measure the divergence ofnearby trajectories in the phase space. To measure the divergence, two pointsxn1 and xn2 in phase space, are considered. Their distance is ‖xn1 − xn2‖ =δ0 � 1. Following the trajectory a small time step Δn into the future fromthese two points, their distance is ‖xn1+Δn − xn2+Δn‖ = δΔn. The Lyapunovexponent is then defined as [26].

δΔn ∝ δ0eλΔn, δΔn � 1, Δn� 1 (3.27)

The number of Lyapunov exponents that can be expected for a dynamic sys-tem is determined by the dimension of the phase space. The Lyapunov spectrais therefore given as

λ1, λ2, . . . , λn, λ1 ≥ λ2 ≥ · · · ≥ λn (3.28)

It is however not always necessary to calculate the entire spectra of exponents.According to Rosenstein et al. [27] it is sufficient for most applications tocalculate the largest exponent λ1 to characterise the system. A positive valueof the largest exponent is an indicator of chaotic behavior. A large negativevalue reflects the existence of a stable fixed point, which is characteristic fordissipative systems.

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3.6.4 Poincaré sections

Poincaré sections are used extensively to transform complicated behaviour inthe phase space to discrete maps in a lower dimensional space, where thesystem dynamics can be analysed from a point-to-point basis, see Figure 3.12.

When the phase space is overcrowded, a Poincaré section can be used to revealthe underlying structure of the attractor. The Poincaré section is obtained byfollowing the trajectory and collecting successive intersection points with thesurface of section S. The state trajectory may cross the plane from two direc-tions - either from the negative or positive side of S. It is however sufficient torecord the crossings from one direction only, which produces the single-sidedPoincaré section.

−1−0.5

00.5

1

−1−0.5

00.5

1−1

−0.5

0

0.5

1

q1q2

q 3

Figure 3.12 – Making of Poincaré section with an arbitrary planar surfaceof section.

The mapping function P from one point on the surface S to the next is definedas

P : S → S (3.29)

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The discrete Poincaré map P is the mapping function between successivereturns on S and is defined as

rn+1 = P (rn) (3.30)

where P is the mapping function and rn is the nth return on S. The Poincarésectioning can be performed in various ways. For periodic non-autonomoussystems the system can be sampled using some natural periodicity of thesystem, which would not require any embedding of any state variables. Forautonomous systems, such as those represented by embeddings, the embeddedphase space is instead cut by a surface of section which is often fixed at one ofthe delay vectors, thus reducing the dimension by 1. Since time informationwill only be implicitly available in the return map, it can be supplemented withthe time of first return function τ(xn), sometimes referred to as the ceilingfunction, which gives the time between successive returns. For a trajectorystarting at point xn, the total time is given by the cumulative time, definedas

tn+1 = t0 + τ(xn), t0 = 0, xn ∈ S (3.31)

Other quantities, such as distance between successive returns, can be definedin a similar manner.

The Poincaré section is constructed using one of the following approaches [26]:

• delay embedding of time series and collecting of intersection with thesurface of section,

• stroboscopic map, collecting of one sample per driving period,

• collecting of the minima or maxima of the time series and plotting thevalues using a delay value of 1

For the time series from the slot-milling process, sampled at 20 kHz with a nearconstant driving period of 0.2143 seconds (60/280), there are approximately4286 possibilities to convert the data into map data by a stroboscopic view.

Visualisation of the Poincaré data becomes an issue for higher dimensionalembeddings, i.e. when the embedding dimension m > 4. In such cases, pro-jection of the data to lower dimensional subspaces is necessary to allow visualinspection of the phase space. Some important methods are the principal com-ponent analysis (PCA), singular value decomposition (SVD), and Wavelets.

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These methods allow the data to be represented in lower dimensional space,but may also involve some loss of information.

3.7 Selection of signal analysis methods

The signal analysis method presented in this chapter may be used to extractadditional information about the machining process condition from the re-sponse of the position encoders for multi-axis CNC machine tools. The typeof response to expect from the encoders is mainly dependent on whether theencoder is sitting on an active or inactive machine axis.

A position encoder mounted on an inactive feed axis senses the forced vibrationin the direction of the axis. Thus, the energy content in the response is thendependent on the size of the force component acting in the direction of the axisand therefore also dependent on the rigidity of the axis, which determines theability of the axis to withstand the vibration. Figure 3.13 gives a schematicoverview of the methods that may be applied when analysing the responsefrom an inactive feed axis.

Figure 3.13 – Schematic overview of signal analysis methods applicable tothe response from encoders for an inactive feed axis.

Another factor that influence the choice of analysis method is that the encoderoutput signals have a limited maximum amplitude. If the vibration amplitudeexceeds a critical level, a saturation of the signals will most likely occur. Thisis a source of uncertainty since the behaviour of the response due to saturationis not yet well understood.

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The Lissajous figure may provide a rough estimate of the vibration amplitudefor inactive feed axes. The formation of the arc requires accumulation ofsufficient number of samples, why it is unable to estimate the instantaneousvibration amplitude. The method also suffers from a saturation effect forlarger vibration amplitude. It also requires a clear response from the encoderwhich is not obtained the case in the more rigid feed axis directions.

Spectral analysis using the FT and HHT, may be applied to any of the sig-nals and is useful to detect the presence of periodicities in the signals. Foran active machine axis, the usefulness of spectral analysis is rather limited inthe sense that the signals contain a dominating frequency (a carrier), whichis directly related to the operational feed rate or spindle speed. The Lissajousmethod presented in Section 3.3 is not applicable to the response from anactive machine axis since the output produces an enclosed arc in the Lissajousfigure. For an active machine axis, the focused is instead on the analysis ofthe phase modulation of the carrier component. The HT combined with un-wrapping of the instantaneous phase provides an efficient means of evaluatingthe position signal. The main issue is however to separate the modulationsignal from the position signal. Analysis in the frequency domain and phasespace domain may then be carried out using the modulation signal as input.Figure 3.13 gives a schematic overview of the methods that may be appliedwhen analysing the response from an active machine axis, such as an activefeed axis or active spindle.

Figure 3.14 – Schematic overview of signal analysis methods applicable tothe response from encoders for an active machine axis.

The first criteion to extract relevant and reliable information about the ma-chining process is that the signal being analysed actually contains the process-characteristic frequencies. The response for an active machine axis do not

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initially contain the operational frequencies due to the existence of a carriercomponent, which in general may also be time varying. Additional prepro-cessing of the incoming signals is needed in this case to extract the useful partof the signal, which in this case is the modulation signal.

The HT is used to obtain the position signal for an active feed axis or activespindle. The disturbances that occurs in the machining process can be foundin the modulation signal which must be separated in order to be analysed. Theoperational frequencies can then be found by performing the FFT or HHT ofthe modulation signal. No general method for the separation of the modulationsignal has been presented. The simplest method is using linear detrending ofthe position signal but may lead to spurious results if applied to the whole timeseries. Best results are obtained when performing the detrending operationfor stationary parts of the measured time series.

Sicne the modulation signal is highly nonlinear, the analysis may be supple-mented by additional tools, such as nonlinear time series analysis in the phasespace obtained by reconstruction from the modulation signal. The Poincarésectioning technique may be used to produce a point-to-point representation ofthe dynamics, which also requires accumulation of a sufficiently large numberof returns on the surface of section.

Depending on the final objective with the condition monitoring, i.e. whetherthe objective is to characterise the machining process through ”fingerprinting”or detection of sudden failures, one of the following approaches may be used.

• monitoring from start to end of the machining process,

• monitoring per cutter revolution,

• monitoring using short time windows

Monitoring from the start to end of a machining operation allows to producea fingerprint of the machining operation which may later be compared withan ideal machining situation in order to reveal any deviations. Disturbanceswill most likely be visible in the fingerprint. Monitoring per cutter revolutionproduces a ”stroboscopic view” of the machining process based on once-per-revolution sampling. In order to detect events that occur within a short timeinterval, a short time window must be used.

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−0.5

0

0.5

Ua

50 r

pm

Ub

Ub

−0.5

0

0.5

100

rpm

Ub

−0.5

0

0.5

150

rpm

Ub

−0.5

0

0.5

200

rpm

Ub

−0.5

0

0.5

250

rpm

Ub

−0.5

0

0.5

300

rpm

Ub

−0.5

0

0.5

350

rpm

Ub

−0.5

0

0.5

400

rpm

t t Ua

Ub

Figure 3.15 – Output signals Ua and Ub from the y-axis encoder during un-balanced rotation at 50, 100, . . . , 400 rpm with the correspond-ing Lissajous figure shown in the third column.46

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Chapter 4

Exeperimental work

In this chapter, the experimental work conducted as a part of this thesis, ispresented. Much of the deltails regarding the experiments are left out but canbe found in the appended papers. Results from unpublished experimental workand later findings which have not been included in the individual papers, canbe found in the following subsections.

4.1 Linear encoder response to rotatingunbalance

4.1.1 Description

The focus in this experiment is primarly to study the response from linearposition encoders during periodic excitation. This is important since practicalmachining operations may invlove simultaneous use of active and inactivemotion axes. From Section 3.1 we know that the response from the encoderis different from active and inactive machine axes. To avoid the disturbancesor complexities from any machining process, this experiment is designed toallow the study of the response in the non-machining case with no activefeed axes or active spindle. During practical machining operations, especiallyintermittent machining operations, periodic excitations are created naturallyfrom the interaction between the tool and the workpiece. In this experiment

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however, no machining is taking place and the periodic excitation is thereforecreated by rotation of the turn table at various speeds and various amount ofunbalance, see Figure 4.1. Furthermore, the multitask machine tool used inthis experiment is equipped with two turn tables, which allows us to comparethe response between these two structural components.

Figure 4.1 – Turn table of the 5-axis multitask machine tool.

The centrifugal force Fc, resulting from inertial effects due to rotation, is afunction of the unbalance me and the angular velocity ω, and given as [28]

Fc = mueω2 (4.1)

wheremu denotes the unbalance mass and e is the eccentricity of the unbalancemass. This experiment allows us to play with three different parameters. Toget a significant increase of the centrifugal force, the rotational speed wasincreased in steps of 50 rpm starting at 50 rpm up to the maximum speed of400 rpm . The output signals Ua and Ub from the linear encoders for the xand y axes were sampled at 1024 Hz. To reduce the number of experiments,all rotational speeds were included in each measurement.

In the first experiment, the turn table was rotated according to the afore-mentioned scheme with no unbalance mass. In the following experiments, anunbalance mass mu (fixed mass) was placed at three different radial distancesfrom the center of rotation as illustrated in Figure 4.2. The excitations wererepeated three times per unbalance case and performed on both turn tables.

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ω

(a)

ω mu

(b)

ωmue

(c)

ω

mue

(d)

Figure 4.2 – Unbalance cases.

4.1.2 Signal analysis

The output signals Ua and Ub are measured from the linear encoders for bothfeed axes x and y. Since these axes are inactive, we know from Section 3.1that the encoder output signals are multi-component. The signals also includecomponents resulting from the acceleration and deceleration of the rotatingtable.

Only the y-axis signal is considered since the response from the x-axis encoderis relatively small due to the rigidity of the structure in this direction.

Figure 4.3 shows the power spectrum of the measured signal Ua for the max-imum unbalance case in Figure 4.2 (d). The power spectrum represents theenergy content in the signal and thus the amount of information contained ata given frequency. The energy in the signal should therefore increase whenincreasing the rotational speed of the turn table. From the power spectrumit can be noted that the lower frequencies are not represented, such as the0.83 Hz component corresponding to 50 rpm. One reason for this is that thesignal amplitude in time domain for this frequency is relatively low and willtherefore not show up as a distinct peak in the power spectrum. The stepwiseincrease of the rotational speed from one level to the next involves accelerationof the turn table and mass which will contribute to the energy at the higherfrequencies.

The power spectrum is unable to reveal when in time the higher frequenciesexist. A a time-frequency analysis, TFA, of the signal is therefore consideredin order to study the frequency variation in the signal over time. For thenonlinear signal in this case the STFT will produce a poor TF-representationdue to its poor frequency resolution. A decomposition of the signal into its

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Figure 4.3 – Power spectrum of the y-axis encoder signal Ua for the unbal-ance case (d) in Figure 4.2. The dotted vertical lines indi-cate the excitation frequencies corresponding to the turn tablespeeds 50, 100, . . . , 400 rpm.

intristic mode functions using the HHT1 described in Section 3.5 is thereforecarried out as shown in Figure 4.4.

0 10 20 30 400

10

20

30

40

t [s]

f[H

z]

Figure 4.4 – Instantaneous frequency for the y-axis position signal Ua for in-creasing vibration level, calculated with the HHT for unbalancecase (d).

Figure 4.4 shows a time-frequency representation on the signal obtained by theTFA. It shows that the higher frequency components (f > 6.67 Hz) originatefrom some ranges of the rotational speed, but also from the final decelerationof the rotating mass.

1MATLAB code for the HHT can be found at http://software.seg.org/2007/0003/

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To analyse if the measured signals carry any information about the vibrationlevel, the included angle in the Lissajous figure can be calculated. However,this must be performed within short duration time windows. Since the outputsignals from the rotary encoder of the turn table were not acquired in thisstudy, segmentation of the linear encoder signals based on the revolution ofthe turn tables is not possible. Instead, the measured time series were split intosegments of length 300 samples, giving a sufficient number of input samplesto the Lissajous figure. Then, by performing the calculation of the Lissajousincluded angle of each segment (see Section 3.3), will yield an approximationof the time evloution of the included angle from the start to the end of theexcitation, see Figure 4.5.

0 50 100 1500

αLi

ssaj

ous

π

k

Figure 4.5 – Calculated included angle of the Lissajous figures. The dottedcurve indicates a quadratic increase of the vibration amplitude.

The measured included angle in Figure 4.5 increases quadratically with in-creasing speed, which is in agreement with Eq. 4.1. There is obviously a strongcorrelation between the Lissajous included angle and the vibration level, atleast up to the saturation limit reached when the Lissajous figure shows anenclosed arc. Consequently, the measured time series contains informationabout the dynamics of the system of rotating unbalance.

Next, the scalar time series are analysed using nonlinear analysis presentedin Section 3.6. The mutual information and the FNN method are used toestimate the embedding parameters.

The first minimum M1 of the mutual information in Figure 4.6 varies for thedifferent unbalance cases (a)-(d) according to: (a) 37, (b) 49, (c) 40, (d) 49.The fraction of false neighbours also show small variations depending on theunbalance case, but tend to drop to zero when m > 3 with 1 − 2% false

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0 50 100 150 2000

0.25

0.5

τ

I(τ

)

M1

Figure 4.6 – Mutual information for the y-axis signal Ua for the unbalancecase (d) with first first minimum at τ = 49.

1 2 3 4 5 6 7 80

0.25

0.5

Fra

ctio

n of

f.n.

n.

m

Figure 4.7 – Determining the embedding dimension using the FNN method.

neighbours when m = 4. In this study, the time series is embedded in a 3-dimensional space, accepting 7 − 11% of false neighbours in the embeddingspace.

The Poincaré sectioning is then performed by following the trajectory in thereconstructed phase space and collecting the crossing points with the surfaceof section plane. The resulting Poincaré sections will be different dependingon the actual unbalance situation, as shown in Figure 4.8.

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(a) (b) (c) (d)

Figure 4.8 – Poincaré sections for unbalanced rotation of turn table I.

4.1.3 Results

The most important results from this experiment are that the vibration fromrotating unbalance is sensed by the position encoders, but it also shows thatthe output signals for an inactive machine axis is multi-component, i.e. theycan be represented by superpositioned sine waves, which is shown in the am-plitude spectrum of the signals.

From the analysis of the rotating unbalance experiment it has been observedthat

• the encoder response from the less rigid y-axis is stronger than the re-sponse from the x-axis,

• the amount of unbalance affects the strength of the response, whichresults in higher SNR values,

• the rotational frequencies appear as distinct peaks in the power spec-trum, and

• the Poincaré sections shows a unique pattern for each unbalance case

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4.2 Machining of aerospace component -industrial trial

4.2.1 Description

The work presented here is carried out during a pre-production by Volvo AeroCorporation in order to validate a machining process used to create specialfeatures on an aerospace component before real production. A disc-millingoperation is considered using a 9-flute milling cutter and spindle speed of 400revolutions per minute, see Figure 4.9.

Figure 4.9 – Disc milling cutter with 9 flutes.

During the testing, the response from the internal position encoders of themachine tool is measured in order to perform analysis of the signals and in-vestigate the usefulness of various analysis methods. The output signals froma linear position encoder and a rotary encoder are considered, see Table 4.1.The active machine axes in this machining process is the feed axis y and themain spindle axis S1. The second feed axis x is an inactive axis. The signalsare measured from the start to the end of the machining operation. No signalsare measured from the linear encoder for the active feed axis y during oper-ation. The actual sampling rate is set to 8192 Hz. According to Eq. 3.1 themain frequency in the S1-axis signals Ua and Ub is 1707 Hz which is below theNyqvist frequency 4096 Hz. The tooth-passing frequency is 60 Hz accordingto Eq. 3.5 which is expected in the response from the linear encoder for theinactive machine axis.

A typical response from the x-axis encoder is shown in Figure 4.10. A decrease

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Table 4.1 – Recorded signals during the disc-milling operation.

Machine axis Signalsx Ua, UbS1 Ua, Ub, Ur

of the amplitude can be noted at t ≈ 30 s. This effect is observed in all ninetests and may be due to variation of the actual cutting parameters for theactual machining operation.

It is desirable to remove all the non-machining segments from the measuredtime series so that only the most important aspects are included in the anal-ysis.

Figure 4.10 – Typical response from the x-axis position encoder during thedisc-milling operation. The initial end ending parts representnon-machining and the middle part represents machining.

The responses from the rotary encoder for the S1-axis is seen in Figure 4.11.

4.2.2 Segmentation of the measured signals

The segmentation of the measured signals is performed by considering thepulses found in the reference mark signal Ur. The pulses are well defined ascan be seen in Figure 4.11 and can be numerically detected by calculating thefirst-order difference of the sequence {Ur}. Spurious pulses may be removedusing thresholding and initial knowledge about the minimum time betweenthe pulses.

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Time (s)

Am

plitu

de (V

olts

) Reference mark signal

Figure 4.11 – Typical response from the S1-axis position encoder during thedisc-milling operation.

The spindle speed can then be estimated by evaluating the time between twosubsequent pulses. Table 4.2 shows the number of recorded spindle revolutionsafter removing the non-machining segments from the signals, minimum andmaximum spindle speed, the average and standard deviation of the spindlespeed for all nine machining cases. It can be noted that the spindle speed iskept almost constant at 400 rpm. Notice that the length of the recordingsvaries between the machining cases due to variations of some process parame-ters. Some operations are also interrupted leading to shorter recordings. Thisstep is always taken in the offline analysis of the signals to ensure that thesegmentation of the signals has performed well.

Table 4.2 – Spindle speed statistics calculated from the reference mark signalUr.

Case nrev nmin [rpm] nmax [rpm] n [rpm] σn [rpm]1 155 399.35 400.65 400.01 0.11992 127 397.74 401.96 400.01 0.28083 106 395.81 401.31 399.96 0.48764 81 399.03 400.65 400.01 0.16665 177 396.77 401.64 399.99 0.35346 179 399.03 400.65 400.01 0.13117 80 397.74 401.31 399.98 0.41828 47 399.03 400.98 400.02 0.24449 83 394.54 400.98 399.92 0.7179

The calculated spindle speeds confirm a stable spindle speed during the ma-chining operation but do generally not give any useful information about thedynamical aspects of the machining process.

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One way to obtain better insigt into the machining process is to study the ma-chining process per spindle revolution. The measured encoder signals Ua andUb from the x-axis and the position signal from the main spindle are thereforeconsidered. The signals are cut into shorter segments, where each segmentcorresponds to a cutter revolution. The length of the segments will vary dueto minor variations in the estimated spindle speed as given in Table 4.2.

4.2.3 Vibration amplitude estimation from the Lissajousfigure

Figure 4.12 shows the included angle in the Lissajous figure formed by thepairs {Ua, Ub} over subsequent revolutions of the milling cutter. A notablesaturation occurs within segments 80 < k < 100 which may be an indicationof relatively tough machining within the corresponding time interval.

20 40 60 80 100 120 1400

pi

2pi

α L(k)

(rad

)

k

Figure 4.12 – Included angle of the Lissajous figure obtained from the pairs{Ua, Ub} calculated over subsequent cutter revolutions.

In a similiar way, the sequences {Ua} and {Ub} can be evaluated by calculatingthe running variance and RMS-value over subsequent cutter revolutions, whichis shown in Figure 4.13 and Figure 4.14.

The running variance and RMS-value of the sequences Ua and Ub show strongcorrelation with the included angle of the Lissajous figure. Since the Lissajousincluded angle is proportional to the vibration amplitude (up to the satura-tion level 2π radians) as stated in Eq. 3.9 in Section 3.3, it is reasonable tobelieve that these quantities also carry some information about the vibrationamplitude in the machining process as well.

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20 40 60 80 100 120 1400

0.05

0.1

0.15

k

σ2

Figure 4.13 – Running variance of the x-axis signals Ua and Ub calculatedover subsequent cutter revolutions.

20 40 60 80 100 120 1400

0.1

0.2

0.3

0.4

0.5

k

URMS

Figure 4.14 – Running RMS-value of the time series Ua and Ub calculatedover subsequent cutter revolutions.

4.2.4 Analysis of the rotary encoder signals

The main spindle holding the milling cutter is responsible for the energy inputto the machining process. The spindle is designed to be very stiff in order toprovide enough torque to the milling cutter for the removal of material fromthe workpiece, but also to withstand torsional vibrations generated in theintermittent machining process.

Since the spindle is an active machine axis in this case, the output signals Uaand Ub from the rotary encoder will be near-sinusoids with a main frequencythat is proportional to the actual spindle speed according to Eq. 3.1. Usinga spindle speed of 400 rpm the main frequency in the signals will be 1707

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Hz. However, due to the intermittent nature of the process, it is likely thatadditional frequency components related to the machining process will appearin the output signals.

To carry out the analysis of the rotary encoder signals, the spindle position φis first calculated from either the Ua or Ub signals using the Hilbert Transform,HT, see Section 3.4. After unwrapping of the phase and subtracting the offset,the spindle position as function of time φ(t) where φ(0) = 0, is obtained.

Figure 4.15 shows a typical spindle position signal. The original signal fromthe HT process has been multiplied by the factor 1

256 to get correct scaling.To ensure the correctness of the result, one may also compare the maximumvalue φmax with 2πN whereN is the number of segments (or cutter revolutions)selected for the analysis. However, scaling is not necessary in order to carryout the analysis.

0 5 10 15 200

200

400

600

800

t

φ(t

)(r

adia

ns)

Figure 4.15 – Spindle position signal calculated from the Ua signal from therotary encoder.

The spindle position signal shown in Figure 4.15 resembles a linear function,starting at the origin where the slope of the line is proportional to the angularvelocity ω, i.e. φ(t) = ωt. Linear regression analysis of the spindle positionsamples also gives a strong linear correlation coefficient, which is very close to1, but gives also that there are some unexplained variation, i.e. deviation fromthe ideal straight line. In fact, these variations appear as small fluctuationson the straight line, originating from phase-modulation of the original spindlepositions signals Ua and Ub. Thus, in order to extract the process variationsusing the rotary encoder signals, the carrier component (or linear part of φ)related to the spindle speed, needs to be removed from the spindle positionsignal to produce the modulation part of the signal.

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10.35 10.4 10.45 10.5 10.55 10.6 10.65 10.7 10.75−2

−1

0

1

2

3x 10

−3

t

ϕ(t

)(r

adia

ns)

Figure 4.16 – Modulation signal extracted from the spindle position signalduring three cutter revolutions, obtained by detrending of theoriginal position signal.

Figure 4.16 shows a characteristic modulation signal during three cutter rev-olutions in the disc-milling process. The modulation signal shows a near-periodic behaviour between subsequent cutter revolutions, but also nine dis-tinct peaks within a cutter revolution. Hence, in this case a 9-flute millingcutter is used. Thus, the peaks may therefore originate from the intermittentnature of the milling process. As can be noted, the modulation signal alsocontains a noise component.

However, the process of extracting the modulation part from the spindle posi-tion signal by removing the linear component, i.e. through a detrending opera-tion, may produce some undesired effects on the final modulation signal, suchas linear trends and sudden drops of the amplitude as shown in Figure 4.17.

It is unrealistic to think that the effects shown in Figure 4.17 originate fromthe machining process. This observation leads to the conclusion that lineardetrending may not always be suitable in order to extract the modulation partfrom the position signal.

4.2.5 Phase space reconstruction

The spindle modulation signal in Figure 4.19 evolves nonlinearly with time.One way to analyse the signal is by using nonlinear time series analysis in thereconstructed phase space as desribed in Section 3.6.

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0 5 10 15−6

−4

−2

0

t

Δϕ

(t)

(rad

ians

)

Figure 4.17 – Artefacts in the modulation signal produced by detrending ofthe calculated position signal.

20 40 60 80 100 120 140

1

2

3

x 10−9

k

σ2

Figure 4.18 – Running variance of the de-noised S1-axis modulation signalcalculated over subsequent cutter revolutions.

The nonlinear analysis begins with an estimation of the optimal embeddingparameters, i.e. the reconstruction delay τ and the minimum embedding di-mension m. For a stationary part of the signal, the optimal embedding delayvalue gives a minimum in the mutual information function I(τ) defined inEq. 3.17.

However, applying the method directly to the modulation signal will lead topoor estimation of τ as shown in Figure 4.20. The MI function shows numerousspurious minima which, if used as a reconstruction delay, would lead to verypoor embeddings. The fluctuations in I(τ) originate from the noise contentin the original modulation signal. Thus, the noise must first be suppressed byfiltering of the modulation signal.

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0 500 1000 1500 2000 2500 3000 3500−1

−0.5

0

0.5

1

sample

ϕ[ra

d]

Figure 4.19 – Spindle modulation signal during three revolutions indicatedby the dotted vertical lines.

0 50 100 150 2000

1

2

τ

I(τ

) M1

Figure 4.20 – Mutual information of the modulation signal during threespindle revolutions (segments 220-222).

The mutual information of the filtered signal gives the first minima at τ = 34which is considered as the optimal delay when embedding the time series.

The use of mutual information to estimate the optimal reconstruction delay forthe final embedding requires careful judgement. It is essential to understandthat the first minimum will occur at different value of tau depending on whichpart of the time series is used as input to the calculation. A good initial guessis to set τ to approximately one quarter of the ”driving period”, as in thiscase the pariod of one tooth-pass (60/400/9/4 ≈ 34).

To estimate the embedding dimension, the merged segments 220− 222 of thefiltered modulation signal is used with the previously determined delay valueof 34. The embedding dimension is then estimated using the false nearest

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0 500 1000 1500 2000 2500 3000 3500−1

−0.5

0

0.5

1

sample

ϕ[ra

d]

Figure 4.21 – Lowpass filtered spindle modulation signal during three spin-dle revolutions (segments 220-222). The dotted vertical linesindicate a spindle revolution.

0 50 100 150 2000

1

2

τ

I(τ

)

M1

Figure 4.22 – Mutual information of the filtered modulation signal duringthree spindle revolutions. First minimum is found at τ = 34.

neighbours method.

As can be seen in Figure 4.23, the FNN method suggests an embedding intwo dimensions, for which the percentage of false nearest neighbours drops tozero. It is obvious that this is a too low value to be used for the embedding ofthe nonlinear modulation signal. Performing the calculation over subsequentsegments of the modulation signal show similiar results. The FNN method hasalso been found to produce varying results depending on whether an unfilteredor a filtered time series is used as input. The resulting dimension using theFNN method may also depend on the length of the input sequency and maynot always produce stable results [29].

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1 2 3 4 5 60

0.5

1

Fra

ctio

n of

f.n.

n.

m

Figure 4.23 – Minimum embedding dimension using the FNN method.

It becomes both necessary and interesting to evaluate the embedding dimen-sion using Cao’s method [29] instead. One of the reported strenghs with Cao’smethod is that it does not contain any subjective parameters and the result-ing embedding dimension is not strongly dependent on the length of the timeseries and produces more stable results. The OpenTSTOOL [30] implementa-tion of the Cao’s method is used for the calculation. The resulting dimensionestimating using Cao’s method is shown in Figure 4.24. As can be noted, thesuggested embedding dimension is somewhere between 4 − 5, indicated by abreakpoint in the graph. This is at least twice the value suggested by theFNN method.

1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

Dimension (d)

E1(

d)

Figure 4.24 – Minimum embedding dimension using Cao’s method.

Figure 4.25 shows the final reconstructed phase space in the three coordinatesq1, q2 and q3, corresponding to the time-delayed vectors x(k), x(k − τ) andx(k − 2τ) respectively. Here, shown for three spindle revolutions for the sakeof clarity.

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q1q2

q 3

Figure 4.25 – Reconstructed phase space from the spindle modulation sig-nal during three spindle revolutions including approx. 3700samples. Embedding parameters: m = 3, τ = 34.

4.2.6 Results

Practical machining of an aerospace component using a disc-milling operationhas been performed. The hardness of the workpiece material resulted in a verytough machining process creating large cutting forces and amplitudes of thevibrations. During machining, the output signals from the position encoderswas measured and analysed using both traditional and more advanced meth-ods. The most important findings from the analysis of the industrial trialscan be summarised as follows.

• The measured signals show a clear response for the disc-milling process.

• Linear detrending of the spindle position signal may produce poor rep-resentation of the modulation signal which is mainly due to noise

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• Filtering of the spindle modulation signal may be required in order toestimate the optimal reconstruction delay from its mutual informationfunction

• Spectrum analysis clearly shows the tooth-passing frequency. The exis-tence of both subharmonics and superharmonics also reveals the nonlin-ear effects in the machining process

• The calculated αL for the x-axis clearly reflects the tough machiningoperation.

• A strong correlation is found between αL and the running variance σ2

of the spindle modulation signal.

The phase space of the machining system can be reconstructed from the mod-ulation signal in order to analyse the dynamical aspects of machining system.Phase space analysis has not been carried out in this experiment.

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4.3 Slot-milling with various number of cuttinginserts

4.3.1 Description

The disc-milling tests presented in Section 4.2 clearly indicated that the im-patcs due to the intermittent interaction between the cutting tool and theworkpiece generate translational vibration of the work table and torsional vi-bration of the main spindle.

The experimental work in this section presents a more systematic study of theresponse from the position encoders for a simple slot-milling operation. Theslot-milling operation, also known as full-immersion milling, is an intermittentcutting process where the milling cutter is held in a rotating spindle. Theworkpiece, which is clamped on the table, is moved toward the cutter at aconstant feed rate, see Figure 4.26.

Figure 4.26 – Slot-milling process.

Furthermore, the milling cutter may have one or more teeth. The cuttingforces are produced when the cutting tool is in the cutting zone, i.e.Fx(φ), Fy(φ), Fz(φ) > 0 when φst ≤ φ ≤ φex, where φst and φex are the cutterentry and exit angles respectively [31]. For the slot-milling operation φst = 0and φex = π.

The size of the cutting forces will be time-varying and depends on the numberof teeth of the cutter that is cutting simultaneously, but also on the chip widthand the varying chip thickness. Thus, for a milling cutter with N teeth, the

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contribution from teeth j to the cutting forces can be formulated as [31]

Fx =N∑j=1Fx,j(φj), Fy =

N∑j=1Fy,j(φj), Fz =

N∑j=1Fz,j(φj)

where φj denotes the instantaneous angle of immersion for teeth j. Thus,when φst ≤ φj ≤ φex, the tooth j contributes to the cutting forces. When thetooth j is outside the immersion zone, the contribution is zero.

The general parameters of the slot-milling process used in this experiment aregiven in Table 4.3.

Table 4.3 – Slot-milling process parameters.

Milling tool 5-flute cutter, carbide insertsWorkpiece Inconel 718Depth of cut 2.0 mmSpindle speed 280 rpmFeed 0.08 mm/toothFeed rate 112 mm/minFeed direction yCutting length 50 mmCutting non-dry

All machining parameters are kept constant, except for the configuration ofthe milling cutter which is altered by reducing the number of teeth on themilling cutter. This will result in a notable increase of the cutting forcesand vibration amplitudes. Furthermore, all experimental trials are performedwith fresh inserts to minimise the effect from tool wear and to make theresults comparable between the different machining conditions or cutting toolconfigurations.

Table 4.4 – Milling cutter configurations.

Z1 1 0 0 0 0Z2 1 1 0 0 0Z3 1 1 1 0 0Z4 1 1 1 1 0Z5 1 1 1 1 1

The various configurations of the 5-flute cutter which correspond to the differ-ent experimental cases are listed in Table 4.4. The first column is the name of

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the cutter configuration and will be referred to in the text. The second columndenotes the cutter configuration, where all 0’s indicate the absence of an insertand all 1’s indicate the presence of an insert on the cutter with tooth number(1-5) from left to right. From this it can be noted that in configuration Z2 thesecond tooth takes three times larger chip thickness than in configuration Z5.

The main objective is to investigate the response of the position encoders whenperforming the machining operation using the aforementioned cutter configu-rations. The output signals from the rotary encoder on the main spindle (S1),linear encoders on the inactive feed axis x and active feed axis y are digitisedat 20 kHz. The measured signals are listed in Table 4.5.

Table 4.5 – Recorded signals during the slot-milling process.

Machine axis Signalsx Ua, Uby Ua, UbS1 Ua, Ub, Ur

At constant feed rate, the feed axis position encoder will give a near sinusoidaloutput signals, with a main frequency that is proportional to the actual feedrate. During machining, the encoder signals for active feed axes will be dis-torted by phase modulation, which originate from varying disturbances in themachining process.

The analysis of the measured signals is then carried out in the frequencydomain using traditional Fourier analysis and within the reconstructed phasespace of the system using nonlinear analysis methods. The modulation signalis extracted from the position signal of the active feed axis, which is usedas input for the nonlinear analysis. A Poincaré section is then obtained byusing the natural period of the machining process. A question here is howthe difference show up in the Poincaré sections due to the varying machiningconditions. The Lissajous angle is applied to the output signals from theinactive feed axis encoder in order to characterise the vibration amplitude forsubsequent revolutions of the milling cutter. The noise content in the outputsignals from the inactive feed axis encoder is also characterised. The SNR isthen estimated for subsequent cutter revolutions.

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4.3.2 Segmentation of the measured signals

As an initial step in the signal analysis, a segmentation of the measured signalsis performed by detecting the pulses in the spindle signal Ur, resulting insegment lengths corresponding to a spindle revolution. In the segmentationprocess, the starting and ending indices for each segment are recorded eachtime the spindle passes its reference position (zero angle). The spindle speedis also estimated by evaluating the time between two subsequent pulses, whichis an effective way to validate the segmentation process. The segmentation isalso useful for the exclusion of the initial and ending non-machining segmentsfrom the scalar time series.

Table 4.6 shows the number of recorded spindle revolutions nrev after removingthe non-machining segments from the signals, minimum and maximum spindlespeed nmin and nmax, the average and standard deviation of the spindle speedn and σn for all experimental trials. It can be noted that the spindle speedis kept almost constant at 280 rpm. Notice that the cutter configuration Z5only includes a single experimental trial. The small variations in nrev originatefrom the manual step when selecting the segments for analysis.

Table 4.6 – Spindle speed statistics calculated from the signal Ur.

Case Trial nrev nmin [rpm] nmax [rpm] n [rpm] σn [rpm]Z1 1 144 279.98 280.05 280.00 0.0312

2 147 279.92 280.11 280.00 0.03323 147 279.98 280.05 280.00 0.0313

Z2 1 145 279.92 280.11 280.00 0.03312 145 279.98 280.05 280.00 0.03133 146 279.98 280.05 280.00 0.0313

Z3 1 143 279.98 280.05 280.00 0.03142 144 279.92 280.05 280.00 0.03233 144 279.92 280.05 280.00 0.0323

Z4 1 143 279.92 280.05 280.00 0.03222 143 279.92 280.05 280.00 0.03223 144 279.92 280.05 280.00 0.0323

Z5 1 143 279.98 280.05 280.00 0.0313

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4.3.3 Noise characterisation and SNR estimation

Figure 4.27 shows a typical noise signal v(t) and its probability density func-tion (PDF) p(v) measured in the output signal Ua from the position encoderof the inactive feed axis x during this experiment.

−0.02

−0.01

0

0.01

0.02

Time

v(t)

Noise signal

0 0.1 0.2 0.3 0.4−0.02

−0.01

0

0.01

0.02

p(v)

Am

plitu

de

Noise histogram

−1000 −500 0 500 1000−0.5

0

0.5

1

τ

Rvv

(τ)

Autocorrelation of noise

0 50 100 150 2000

0.02

0.04

0.06

Frequency [Hz]

Pow

er

Power spectrum

Figure 4.27 – Typical time history, normalised probability density functionp(v), autocorrelation Rvv(τ) and power spectrum of the noisesignal v(t).

• The probability density function p(v) of the noise signal v(t) indicates anonuniform (Gaussian-like) noise distribution.

• The autocorrelation Rvv(τ) of the noise signal v(t), which is the cross-correlation of the noise with itself, clearly indicates the presence of aperiodic signal buried under the noise.

• The power spectrum of the noise signal v(t) obtained by using the FFTreveals that the signal is disturbed by the mains frequency (or utilityfrequency) 50 Hz. Its contribution to the signal is however consideredas relatively small.

Figure 4.28 shows a typical time evolution of the SNR value of the x-axissignals Ua and Ub over subsequent revolutions of the milling cutter during

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the slot-milling operation. The procedure to estimate the SNR value frommeasured signals is described in Section 3.1.2.

20 40 60 80 100 120 14020

25

30

35

40

45

k

SN

RdB

Figure 4.28 – Estimated SNR value of the signals Ua and Ub from the inactivefeed axis encoder over subsequent spindle revolutions.

The SNR values of the two channels Ua and Ub are strongly correlated andshow some variation during the machining process. SNR values up to 40 dBhave been observed. The lower SNR values found at the initial and final stageof the machining operation are a direct consequence of a weaker response fromthe encoder for an inactive machine axis.

4.3.4 Measuring of the Lissajous angle from the inactivefeed axis signals

The response from the inactive feed axis due to the various cutting conditionsis characterised by measuring of the Lissajous angle from the encoder outputsignals Ua and Ub. Figure 4.36 shows the typical time history of Ua and Ubduring three spindle revolutions and the time evolution of the corresponingLissajous angle αL(k) from the start to the end of the slot-milling process forall five cutter configurations Z1-Z5. A general trend is that the amplitude ofαL(k) decreases with increasing number of teeth on the cutter due to improvedstability in the machining process. One exception from the general trend canbe noted for the case Z1, which should generate the largest impacts, but isobviously not indicated by αL(k).

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4.3.5 Spectral analysis

The various cutter configurations give rise to different process dynamics due tothe nonuniform pitch angle of the cutter in the configurations Z2-Z4, causinguneven loads on the cutting inserts. The cutting insert that immediately fol-lows the gap of missing inserts, will remove more material from the workpiece,resulting in a higher impact on that particular insert.

Three of the measured signals has been considered for the analysis: the signalUa or Ub from the inactive feed axis x, the modulation signal from the activefeed axis y, and the the modulation signal from the main spindle S1, as listedin Table 4.7.

Table 4.7 – Naming of signals.

s1 x-axis signal Uas2 y-axis modulation signal in Uas3 S1-axis modulation signal in Ua

Figure 4.29 shows the amplitude spectra of the signals listed in Table 4.7. Adominant peak at 4.67 Hz which corresponds to the tooth-passing frequency(also the cutter frequency) can be noted for cutter configuration Z1. Thesignals also contain some higher frequency components with decreasing am-plitude.

Obviously, the frequency distribution is similiar in all these signals, but withvarying amount of energy at the specific frequencies. The cutter configurationZ5, which corresponds to the normal machining case, gives a dominant peakat the tooth-passing frequency 23.3 Hz, see Figure 4.30.

The amplitude spectra show similiar frequency distribution for the variouscutter configurations. The magnitudes at the distict frequencies of the s2signal are larger than the magnitudes of the corresponding frequencies in thes1 signal, which reflects the lower rigidity of the y-axis for the machine toolused. The spectrum of the s3 signal also indicates a higher damping of theS1-axis.

The impacts created in the intermittent milling process represent strong im-pulses representing a wide range of excitation frequencies. It is thereforepossible that some of these frequencies coincide with the natural frequency of

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−0.5

0

0.5

Time history

0

2500

5000

Pow

er

Power spectrum

−0.05

0

0.05

0

10

20

30

Pow

er

14.8 15 15.2 15.4 15.6 15.8−0.005

0

0.005

Time [s]0 5 10 15 20 25 30 35 40

0

0.25

0.5

Frequency [Hz]

Pow

er

s 1[V

olts

]s 2

[mm

]s 3

[rad]

Figure 4.29 – Time history of the signals s1, s2 and s3 during four spindlerevolutions and their power spectrum for the cutter configu-ration Z1. Modulation signals are scaled.

some of the components in the machining system. Figure 4.31 shows a closeview at the amplitude spectrum of the signal s3. The dominant peak foundat 1195 Hz may probably be related to the natural frequency of the spindle orthe tool holder system. However, this has not yet been verified by any means.This frequency is observed for all cutter configurations Z1-Z5, but the ampli-tude of the sideband frequencies tends to decrease with increasing number ofteeth on the cutter. No such high-frequency content have been observed inthe response from the feed axis encoders.

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−0.5

0

0.5

Time history

0

2500

5000

Pow

er

Power spectrum

−0.05

0

0.05

0

10

20

30

Pow

er

14.6 14.8 15 15.2 15.4 15.6−0.005

0

0.005

Time [s]0 5 10 15 20 25 30 35 40

0

0.25

0.5

Frequency [Hz]

Pow

er

s 1[V

olts

]s 2

[mm

]s 3

[rad]

Figure 4.30 – Time history of the signals s1, s2 and s3 during four spindlerevolutions and their power spectrum for the cutter configu-ration Z5. Modulation signals are scaled.

4.3.6 Nonlinear analysis of the active feed axis modulationsignal

To analyse the dynamical behaviour of the slot-milling process, nonlinear ana-lysis is used. In the current machining setup, the s2 signal shows a clearerresponse to process variations and is therefore used as input for the analysis.From the time signals it can be seen that the s2 signal also correlates relativelywell with the measured Lissajous angle of the x1 signal which itself is relatedto the vibration amplitude. The time behaviour of s2 also resembles the timeevolution of a vibration signal as shown in Figure 4.32. Notice that the signalis unscaled and the amplitude is presented in radians which originates fromthe HT demodulation process.

The optimum reconstruction delay used in the embedding of s2 is obtained byconsidering the first minimum M1 in the mutual information of s2.

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1180 1185 1190 1195 1200 1205 12100

0.005

0.01

f (Hz)

|S3(f

)|

Figure 4.31 – Amplitude spectrum of the spindle modulation signal s3. Thedistinct peak at 1195 Hz may correspond to the natural fre-quency of the spindle or tool holder system.

5 10 15 20 25−3

0

3

t [s]

s 1[ra

d]

Figure 4.32 – Time history of the y-axis modulation signal for the machiningcase Z1. The leading and ending transients are excluded.

The optimum reconstruction delay values for the different configurations ofthe milling cutter are tabulatad in Table 4.8. The experiment was repeatedthree times for all the cutter configuration except for the cutter configura-tion Z5. It can be noted that the value of τ varies slightly depending on thecutter configuration. Small variations also exists between the replicates. Theembedding dimension is then determined using the FNN method. The per-centage of false neighbours drops to zero when m = 5 which is the suggestedminimum embedding dimension. Figure 4.34 shows three different views ofthe phase portrait. Due to problems in visualising more than three dimen-sions, a 3-dimensional embedding is chosen. The final Poincaré section is thenobtained by cutting the phase space with a plane located at the average ofthe data. Figure 4.35 shows a Poincaré section of the y-axis modulation for

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0 100 200 300 400 5000

0.1

0.2

0.3

0.4

0.5

τ

I(τ

)

M1

Figure 4.33 – Mutual information of the y-axis modulation signal for themachining case Z1. First minimum occurs when τ = 212

Table 4.8 – Optimal reconstruction delays.

Cutter config. τ1 τ2 τ3Z1 212 207 202Z2 234 235 226Z3 222 227 234Z4 220 216 217Z5 196 - -

the machining case Z1. The section indicate the presence of a stable point atthe origin and shows also curve-like structures which indicate quasi-periodicmotion along the y-axis.

−1 0 1−2

0

2

−1 0 1−2

0

2

−1 0 1−2

0

2

x(k)x(k)

x(k−τ)

x(k − τ)

x(k−

2τ)

x(k−

2τ)

Figure 4.34 – Phase portrait of the y-axis modulation for the case Z1.

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−3 0 3−3

0

3

Figure 4.35 – Poincaré section of the y-axis modulation for the case Z1.

4.3.7 Phase plane analysis

An alternative appraoch to study the dynamics of the milling process is toobtain the phase plane (or phase portrait) in the form (x, x) where x is astate variable and x is the time derivative. The Poincaré section may then beconstructed by using the natural period of the system by sampling the statetrajectory once per cutter revolution, giving a stroboscopic view of the processdynamics. When using a spindle speed of 280 rpm and a sampling rate of 20kHz, there will be 4286 possibilities to create the Poincaré section. A naturalchoice in this case is to sample the state trajectory every time the cutterpasses the reference position (zero angle). The feed axis modulation signalis the state vector x. The phase portrait is then constructed by numericaldifferentiation of x. Since differentiation itself is very sensitive to noise, themodulation signal may first be prefiltered to minimise the effects due to noise.A 5th-order Butterworth lowpass-filter with cutoff frequency at 200 Hz maybe used to suppress the high-frequency content in the signal without loss ofthe major aspects of the dynamics.

Figure 4.37 shows the resulting phase plane and Poincaré section for eachconfiguration of the milling cutter. Notice that the variables are convertedin physical units by scaling of the modulation signal. A notable differencebetween the Poincaré sections can be observed for the different cutter confi-gurations.

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4.3.8 Results

In this experiment a single-axis slot-milling operation was considered in or-der to analyse the response from both linear and angular position encodersfor various machining conditions using a 5-axis multitask machine tool. Thestrength of the impacts between the cutting edges and the workpiece wasaltered by using various number of teeth on a 5-flute milling cutter. The mea-sured responses have been analysed in the frequency domain and also in thereconstructed 3-dimensional phase space of the machining system.

The results from this experiment can be summarised as follows.

• The clearest response is obtained from the feed axis encoder for the leastrigid machine axis direction, which in this case is the y-axis

• Active feed axis signals and spindle signals are phase-modulated andcontain information about the process characteristics

• Spectrum analysis using FFT indicate that the measured signals fromboth active and inactive machine axes contain operational frequencies,such as the cutter frequency and tooth-passing frequency, and their sub-harmonics

• The amplitude spectrum of the spindle modulation signal shows thatmost subharmonics, except the tooth-passing frequency, are highly damped

• The energy at the operational frequencies depends on the cutter config-uration

• The Lissajous angle of the inactive feed axis signals for subsequent cutterrevolutions decreases with increasing number of teeth on the cutter

• The milling process dynamics may be analysed using the reconstructedphase space of the machining system. A clear difference between thevarious cutter configurations is shown when creating Poincaré sectionsusing the natural period of the machining system, which is the periodof the cutter.

The sensitivity to detect phenomena, such as as tool wear, tool breakage,severe vibration has not been the focus in this experiment. This experimentmay however be refined to study the responses due to such phenomena.

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15.4 15.6 15.8−0.5

0

0.5

Time [s]

Ua [V

]

Time history

15.4 15.6 15.8−0.5

0

0.5

Time [s]

Ub [V

]

Time history

40 80 1200

Lissajous angle

k

α L(k)

[rad

]

π

(a)

15 15.2 15.4 15.6−0.5

0

0.5

Time [s]

Ua [V

]

Time history

15 15.2 15.4 15.6−0.5

0

0.5

Time [s]U

b [V]

Time history

40 80 1200

Lissajous angle

k

α L(k)

[rad

]

π

(b)

15 15.2 15.4 15.6−0.5

0

0.5

Time [s]

Ua [V

]

Time history

15 15.2 15.4 15.6−0.5

0

0.5

Time [s]

Ub [V

]

Time history

40 80 1200

Lissajous angle

k

α L(k)

[rad

]

π

(c)

14.8 15 15.2 15.4−0.5

0

0.5

Time [s]

Ua [V

]

Time history

14.8 15 15.2 15.4−0.5

0

0.5

Time [s]

Ub [V

]

Time history

40 80 1200

Lissajous angle

k

α L(k)

[rad

]

π

(d)

14.8 15 15.2 15.4−0.5

0

0.5

Time [s]

Ua [V

]

Time history

14.8 15 15.2 15.4−0.5

0

0.5

Time [s]

Ub [V

]

Time history

40 80 1200

Lissajous angle

k

α L(k)

[rad

]

π

(e)

Figure 4.36 – Lissajous angle αL from the signals Ua and Ub measured fromthe inactive feed axis x for the cutter configurations (a) Z1,(b) Z2, (c) Z3, (d) Z4 and (e) Z5.

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4.5 5 5.5 6−0.1

−0.05

0

0.05Time history

−0.1 −0.05 0 0.05

−10−5

05

10

Phase plane

−0.1 −0.05 0 0.05

−10−5

05

10

Poincaré section

t [s]

x [mm]

x [mm]

x[m

m]

x[m

m/s

]x

[mm

/s]

(a)

4.5 5 5.5 6−0.1

−0.05

0

0.05Time history

−0.1 −0.05 0 0.05

−10−5

05

10

Phase plane

−0.1 −0.05 0 0.05

−10−5

05

10

Poincaré section

t [s]

x [mm]

x [mm]x

[mm

]x

[mm

/s]

x[m

m/s

]

(b)

4.5 5 5.5 6−0.1

−0.05

0

0.05Time history

−0.1 −0.05 0 0.05

−10−5

05

10

Phase plane

−0.1 −0.05 0 0.05

−10−5

05

10

Poincaré section

t [s]

x [mm]

x [mm]

x[m

m]

x[m

m/s

]x

[mm

/s]

(c)

4.5 5 5.5 6−0.1

−0.05

0

0.05Time history

−0.1 −0.05 0 0.05

−10−5

05

10

Phase plane

−0.1 −0.05 0 0.05

−10−5

05

10

Poincaré section

t [s]

x [mm]

x [mm]

x[m

m]

x[m

m/s

]x

[mm

/s]

(d)

4.5 5 5.5 6−0.1

−0.05

0

0.05Time history

−0.1 −0.05 0 0.05

−10−5

05

10

Phase plane

−0.1 −0.05 0 0.05

−10−5

05

10

Poincaré section

t [s]

x [mm]

x [mm]

x[m

m]

x[m

m/s

]x

[mm

/s]

(e)

Figure 4.37 – Dynamic behaviour along the feed axis for various milling cut-ter configurations (a) Z1, (b) Z2, (c) Z3, (d) Z4, (e) Z5.

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Chapter 5

Conclusions and future work

The focus in this work has been on the analysis of the response from positionencoders of a 5-axis multitask machine tool for various machining processes.The encoder signals have been recorded during rotating unbalance of the turntables, but also during a disc-milling and a slot-milling process. The analysishas been carried out offline using both traditional and more advanced methodsfor the analysis of the measured signals.

It has not been possible to produce faults to individual machine tool compo-nents, why the condition monitoring of individual machine tool components,such as ballscrews, guideways and bearings has been left outside this thesis.

The effective signal analysis algorithm for the analysis of the internal sensorsignals uses information generated by several sources or sensors, which arelocated relatively far away from the machining process. The measurementapproach will therefore always include some unknown level of uncertainty.The type and amount of information that can be extracted from the sensors isalso limited due to the fact that the encoders already serves a specific purposein the drive system of the machine tool.

It has been shown that the sensitivity of each encoder to outer stimulus,such as impacts created in the machining process, depends on the locationof the encoder relative to the machining process, but also that the outputfrom the encoders depends on the state of the individual machine axes. Moreadvanced machining operations may involve all possible states of the machineaxes. Development of a general online CMS based on the position encoders,

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cannot be easily achieved due to the aforementioned factors. Due to the closeconnection between encoder response and the state of the machine axis, adistinction between active and inactive machine axes is therefore necessary tomake in order to select appropriate methods to analyses the responses.

The main finding in this work is that the encoder signals contain informationwhich can be related to the machining process, especially about the vibra-tion conditions of the machining process due to forced periodic excitations. Itmay however be difficult to detect the condition of specific machine tool com-ponents from the position encoder signals since the contribution from faultycomponents to the signals is not known. Unstable machining conditions, suchas chatter, may on the other hand be detected since this condition may gen-erates severe vibration levels.

The most reliable information may be found in the modulation signal fromactive machine axes. Both spectral analysis and phase space analysis may beapplied to the modulation signal in order to characterise the machining processcondition. The future work should focus on the sensitivity of these methodsto various phenomena, such as machine tool component wear/breakage andcutting tool wear/breakage.

Some of the nonlinear analysis methods are still computationally demanding,even for offline analysis, and cannot at this stage be used in in-process condi-tion monitoring without some adaptation of the numeric algorithms. Furtherinvestigation is also required regarding the sensitivity of the nonlinear signalanalysis methods due to varying machining parameters, such as spindle speed,feed rate, direction change, depth of cut, multi-axis feed, etc.

5.1 Conclusions of experimental work

The initial experiment with rotating unbalance using the turn tables, clearlyindicates that the linear position encoders pick up the vibration due to rotatingunabalance. The included angle in the Lissajous figure shows a strong corre-lation with the vibration amplitude. Different unbalance cases also produceda unique pattern in the final Poincaré sections.

The analysis carried out for two types of milling operations, i.e. disc-millingand slot-milling, shows that the response from the encoders contains process

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characteristic frequencies, such as the cutter frequency and tooth-passing fre-quency. The presence of subharmonics and superharmonics also reveals theexistence of nonlinearities in the measured signals.

• The modulation signal obtained from the S1-axis is more noisy than themodulation signal from the feed axes. In order to estimate the recon-struction delay using the mutual information function, MI, of the S1-axis modulation signal, prefiltering of the modulation signal is requiredto obtain a non-oscillating MI function.

• The value of the reconstruction delay, taken as the τ value that gives thefirst minimum in the MI function, depends on which part of the timeseries is used as input to the algorithm, but also on the length of theselected sequency of samples from the measured time series.

• In some cases, the response from machining operations requires a higherdimensional embedding space (up to 4 or 5) to be properly embedded, i.e.without self-intersection of the phase space trajectory. So far, embeddingis carried out using a 3-dimensional space, leading to the existence offalse neighbours (non-related points in the selected dimension). It isat this stage also unclear how to deal with higher-dimensional Poincarésections.

• The Lyapunov exponent shows a large negative value, meaning that theresponse from encoders during machining is non-chaotic and the machin-ing process is highly dissipative. Calculation of the Lyapunov exponentsand chaotic invariants have therefore have therefore been omitted in theanalysis.

• The Poincaré sectioning technique requires an accumulation of a rela-tively large number of intersection points with the surface of section toproduce a clear pattern.

The aforementioned issues are left outside this thesis, which only considerswell-controlled experiments and offline analysis of the signals. It is howevernecessary at this stage to establish a general method to analyse the encodersignals. Among the signal analysis methods presented in previous sections,the nonlinear methods are likely to be more general for practical signals.

In order to characterise the machining process, the operational frequenciesmust first be extracted from the signals using other methods, such as the

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Hilbert Transform (HT) which has shown to be powerful. The Wavelet anal-ysis method has for example not been covered in this work, but may also beuseful to identify events that occur in a short time interval [32]. For machiningprocess monitoring, this may be relevant for the detection of tool breakage.

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References

[1] Henry Brolin. Test instructions for machine tools. Volvo Aero Corpora-tion, Trollhättan, Sweden. IAT - Installation Acceptance Test.

[2] G. Byrne, D. Dornfeld, I. Inasaki, G. Ketteler, W. König, and R. Teti.Tool condition monitoring (TCM) – the status of research and industrialapplication. CIRP Annals - Manufacturing Technology, 44(2):541–567,1995.

[3] Dimla E. Dimla. Sensor signals for tool-wear monitoring in metal cuttingoperations–a review of methods. International Journal of Machine Toolsand Manufacture, 40(8):1073–1098, 2000.

[4] W. Li, D. Li, and J. Ni. Diagnosis of tapping process using spindle motorcurrent. International Journal of Machine Tools and Manufacture, 2003.

[5] Tae-Yong Kim, Joongwon Woo, Dongwon Shin, and Jongwon Kim. In-direct cutting force measurement in multi-axis simultaneous NC millingprocesses. International Journal of Machine Tools and Manufacture, 39(11):1717–1731, 1999.

[6] A. Rivero, L. N. López de Lacalle, and M. Luz Penalva. Tool wear detec-tion in dry high-speed milling based upon the analysis of machine internalsignals. Mechatronics, 18(10):627–633, 2008.

[7] Volker Plapper and Manfred Weck. Sensorless machine tool conditionmonitoring based on open NCs. In Proceedings of the 2001 IEEE Interna-tional Conference on Robotics & Automation, volume 3, pages 3104–3108,Seoul, Korea, May 21-26 2001.

[8] W. Amer, R. I. Grosvenor, and P. W. Prickett. Sweeping filters and toothrotation energy estimation (tree) techniques for machine tool condition

87

Page 104: Condition Monitoring of Machine Tools and Machining ...319488/FULLTEXT02.pdf · machine tools are not used for their rigidity, but for their capacity of handling large and geometrically

monitoring. International Journal of Machine Tools and Manufacture, 46(9):1045–1052, 2006.

[9] J. E. Kaye, D. H. Yan, N. Popplewell, and S. Balakrishnan. Predictingtool flank wear using spindle speed change. International Journal ofMachine Tools and Manufacture, 35(9):1309–1320, 1995.

[10] Dong Young Jang, Young-Gu Choi, Hong-Gil Kim, and Alex Hsiao. Studyof the correlation between surface roughness and cutting vibrations todevelop an on-line roughness measuring technique in hard turning. In-ternational Journal of Machine Tools and Manufacture, 36(4):453–464,1996.

[11] A. Verl, U. Heisel, M. Walther, and D. Maier. Sensorless automatedcondition monitoring for the control of the predictive maintenance ofmachine tools. CIRP Annals - Manufacturing Technology, 58(1):375–378,2009.

[12] Fritz Klocke, Stephan Kratz, and Drazen Veselovac. Position-orientedprocess monitoring in freeform milling. CIRP Journal of ManufacturingScience and Technology, 1(2):103–107, 2008.

[13] HEIDENHAIN. User’s Manual IK 220 PC Counter Card for HEIDEN-HAIN encoders. HEIDENHAIN, 1 edition, 7 2006. www.heidenhain.com.

[14] Lars Bengtsson. Elektriska mätsystem och mätmetoder. Studentlitteratur,Lund, 2., [rev. och utök.] uppl. edition, 2003.

[15] Anders Svardström. Modulation och teleteknik. Studentlitteratur, Lund,2. uppl. edition, 1996.

[16] I. Alejandre and M. Artés. Method for the evaluation of optical en-coders performance under vibration. Precision Engineering, 31(2):114–121, 2007.

[17] Zhijun Li, Shenglai Zhen, Bo Chen, Min Li, Renzhu Liu, and Benli Yu.Lissajous figures in the application of micro-vibration measurement. Op-tics Communications, 281(18):4744–4746, 2008.

[18] Z. K. Peng, Peter W. Tse, and F. L. Chu. An improved Hilbert-Huangtransform and its application in vibration signal analysis. Journal ofSound and Vibration, 286(1-2):187–205, 2005.

88

Page 105: Condition Monitoring of Machine Tools and Machining ...319488/FULLTEXT02.pdf · machine tools are not used for their rigidity, but for their capacity of handling large and geometrically

[19] N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Snin, Q. Zheng, N. C.Yen, C. C. Tung, and H. H. Liu. The empirical mode decompositionand the Hilbert spectrum for nonlinear and non-stationary time seriesanalysis. Proceedings of the Royal Society A: Mathematical, Physical andEngineering, 454(1971):903–995, 1998.

[20] Gary G. Leisk, Nelson N. Hsu, and Norden E. Huang. Application of theHilbert-Huang transform to machine tool condition/health monitoring.AIP Conference Proceedings, 615:1711, 2002.

[21] Li Lin and Ji Hongbing. Signal feature extraction based on an improvedEMD method. Measurement, 42(5):796–803, 2009.

[22] Dan B. Marghitu, Bogdan O. Ciocirlan, and Nicolae Craciunoiu. Nonlin-ear dynamics in orthogonal turning process. Chaos, Solitons and Fractals,12(12):2343–2352, 2001.

[23] Andrew M. Fraser and Harry L. Swinney. Independent coordinates forstrange attractors from mutual information. Physical Review A, 33:1134,1986.

[24] Matthew B. Kennel, Reggie Brown, and Henry D. I. Abarbanel. De-termining embedding dimension for phase-space reconstruction using ageometrical construction. Physical Review A, 45:3403, 1992.

[25] Peter Grassberger and Itamar Procaccia. Measuring the strangeness ofstrange attractors. Physica D: Nonlinear Phenomena, 9(1-2):189–208,1983.

[26] Holger Kantz and Thomas Schreiber. Nonlinear time series analysis.Cambridge Univ. Press, 2 edition, 2004.

[27] Michael T. Rosenstein, James J. Collins, and Carlo J. De Luca. A prac-tical method for calculating largest Lyapunov exponents from small datasets. Physica D: Nonlinear Phenomena, 65(1-2):117–134, 1993.

[28] Singiresu S. Rao. Mechanical vibrations. Pearson, Upper Saddle River,N.J., 4 edition, 2004.

[29] Liangyue Cao. Practical method for determining the minimum embed-ding dimension of a scalar time series. Physica D: Nonlinear Phenomena,110(1-2):43–50, 1997.

89

Page 106: Condition Monitoring of Machine Tools and Machining ...319488/FULLTEXT02.pdf · machine tools are not used for their rigidity, but for their capacity of handling large and geometrically

[30] C. Merkwirth, U. Parlitz, I. Wedekind, D. Engster, and W. Lauterborn.OpenTSTOOL User Manual. Drittes Physikalisches Institut, UniversitätGöttingen, http://www.physik3.gwdg.de/tstool/, 2 2009.

[31] Yusuf Altintas. Manufacturing automation : metal cutting mechanics,machine tool vibrations, and CNC design. Cambridge University Press,Cambridge, 2000.

[32] Michel Misiti. Wavelets and their applications. Digital signal and imageprocessing series. ISTE, London ; Newport Beach, CA, 2007.

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MATLAB script est_alpha

function [alpha,a1,a2,v] = est_alpha(A,B)% [ALPHA,A1,A2,V] = EST_ALPHA(A,B) estimates the included angle in% the Lissajous figure of A and B. If A and B are matrices the EST_ALPHA% operation is applied to each column.%% INPUTS% A - NxM matrixor containing N samples from the A-signal% B - NxM vector containing N samples from the B-signal%% OUTPUTS% ALPHA - The included angle of the arc/circle in the Lissajous figure% A1 - Stating angle [rad]% A2 - Ending angle [rad]% V - Vector containg individual angular values (sorted)

R2D = 180/pi;DENS = 1.0/R2D; % point density (angular gap between points)NMIN = 2; % minimum # of points

N = size(A,1); % # of samples in A (or B)d = size(A,2); % # of signals (columns in A or B)

alpha = zeros(1,d); % included anglea1 = zeros(d,1); % starting anglea2 = zeros(d,1); % ending anglev = zeros(N,d); % angular values (sorted in ascending order)n1 = zeros(d,1); % index for the starting anglen2 = zeros(d,1); % index for the ending angle

if ~(size(A)==size(B))disp(’(EST_ALPHA) A and B must be of the same size’)

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returnendif size(A,1)<NMIN

disp([’(EST_ALPHA) At least ’,num2str(NMIN),’ points are required’])return

end

% Calculate the angular values in the range [0,2pi] for each point (An,Bn)for m=1:d

x = A(:,m);y = B(:,m);for n=1:N

if x(n)==0if y(n)>0

a = pi/2;else

a = 3*pi/2;end

elsea = atan(y(n)/x(n));if x(n)<0

a = a + pi;else

if y(n)<0a = a + 2*pi;

endend

endv(n,m) = a;

endend

% Sort the vector with angular values in ascending order and find the% maximum values for the first-order difference of the sorted vector.[v,idx] = sort(v,’ascend’);[maxval,n1] = max(diff(v));

% Calculate the included angle.for m=1:d

% Look for a breaking point in the sorted vector.if abs(maxval(m))>DENS

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n2(m) = n1(m)+1;a = v(n1(m),m)-v(1,m) + v(N,m)-v(n2(m),m);

elsen1(m) = 1;n2(m) = N;a = v(n2(m),m)-v(n1(m),m);

enda1(m) = v(n1(m),m);a2(m) = v(n2(m),m);alpha(m) = a;

end

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