Active Vibration Control ofBoring Bar Vibrations

Linus AndrénLars Håkansson

Department of Signal ProcessingSchool of EngineeringBlekinge Institute of Technology

Blekinge Institute of TechnologyResearch Report No 2004:07

Active Vibration Control of BoringBar Vibrations

L. Andren and L. Hakansson

August, 2004

Research ReportDepartment of Signal Processing

Blekinge Institute of Technology, Sweden

Abstract

The boring operation is a cumbersome manufacturing process plagued by noise andvibration-related problems. A deep internal boring operation in a workpiece is a classicexample of chatter-prone machining. The manufacturing industry today is facing toughertolerances of product surfaces and a desire to process hard-to-cut materials; vibrationsmust thus be kept to a minimum. An increase in productivity is also interesting froma manufacturing point of view. Penetrating deep and narrow cavities require that thedimensions of the boring bar are long and slender. As a result, the boring bar is inclinedto vibrate due to the limited dynamic stiffness. Vibration affects the surface finish, leadsto severe noise in the workshop and may also reduce tool life.

This report presents an active control solution based on a standard boring bar withan embedded piezo ceramic actuator; this is placed in the area of the peak modal strainenergy of the boring bar bending mode to be controlled. An accelerometer is also includedin the design; this is mounted as close as possible to the cutting tool. Embedding theelectronic parts not only protects them from the harsh environment in a lathe but alsoenable the design to be used on a general lathe as long as the mounting arrangements arerelatively similar. Three different algorithms have been tested in the control system. Sincethe excitation source of the original vibrations, i.e. the chip formation process cannot beobserved directly, the algorithms must be constructed on the basis of a feedback approach.Experimental results from boring operations show that the vibration level can be reducedby 40 dB at the resonance frequency of a fundamental boring bar bending mode; severalof its harmonics can also be reduced significantly.

Contents

1 Introduction 2

2 Materials and Methods 52.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Measurement Equipment and Setup . . . . . . . . . . . . . . . . . . 62.1.2 Cutting Data and Machining Parameters . . . . . . . . . . . . . . . 7

2.2 Active Boring Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Control Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 PID Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 LMS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.3 Filtered-x LMS Algorithm . . . . . . . . . . . . . . . . . . . . . . . 152.3.4 IMC Controller based on the Filtered-x LMS Algorithm . . . . . . . 18

2.4 Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.1 Spectrum Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.2 Coherence Function Estimation . . . . . . . . . . . . . . . . . . . . 232.4.3 Frequency Response Function Estimation . . . . . . . . . . . . . . . 24

2.5 Nyquist Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Results 263.1 The Forward Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Active Vibration Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 Boring Bar Comparison . . . . . . . . . . . . . . . . . . . . . . . . 333.2.2 Algorithm Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.3 Stability and Robustness of Feedback Controllers . . . . . . . . . . 37

4 Discussion and Conclusions 51

1

Chapter 1

Introduction

The lathe is a very useful and versatile machine in the workshop, and it is able to performa wide range of machining operations. A boring operation is a metal cutting operationthat bores deep, precise holes in the workpiece. A boring bar is characterized by greatlength in comparison to its diameter. The boring bar is clamped at one end to a toolpost or a revolver and has a cutting tool attached at the free end. The cutting tool isused to perform metal cutting in a bore or cavity of the workpiece. Since a boring bar isusually long and slender, it is inclined to vibrate. Deep internal boring of a workpiece isa classic example of chatter-prone machining. Performing metal cutting under vibratingconditions will yield unsatisfactory results in terms of the surface finish of a workpiece, toollife and undesirable noise levels. The boring bar vibration was investigated in [1] and [2].The vibration may be characterized by a stochastic process with time varying statisticalproperties and with non-linear characteristics. [1]. In internal turning operations, theboring bar motion usually consists of components in both the cutting speed directionand the cutting depth direction [1, 2]. However, the motion of a boring bar during acontinuous machining operation is generally greatest in the cutting speed direction; andis related to one of the bar’s two fundamental bending modes [1, 2]. A frequent result ofthis resonant motion is extremely high boring-bar vibration levels [1]. A typical boringoperation is illustrated in Fig. 1.1.

A conventional countermeasure to the vibrations is to equip boring bars with a tunedvibration absorber. The absorber is tuned to the frequency range of the fundamentalbending mode of the boring bar by adjusting the weight of the reactive mass and thestiffness and damping properties of the elastic element. This will reduce the vibration levelduring the cutting operation. Active vibration absorbers based on inertial-mass actuatorshave also been investigated [3]. Active and passive vibration absorbers can provide somerelief and are most effective when placed near the tool-end of the bar [7]. Active vibrationcontrol of machine-tool vibration, however, comprises a number of different methods forthe introduction of a control force to the boring bar. In [8] the approach was to usean active clamping house, i.e. to let the clamping of the boring bar be the secondaryvibrating source; the results were good. A similar approach was proposed as early as 1975in [9]. The method uses a pivoting boring bar and an electro-hydraulic servo system as

2

The cutting speed direction

Workpiece

The feed direction

Boring bar

The cutting tool

The cutting depthdirection

Figure 1.1: A typical boring operation.

an actuator; the results are promising.This report presents a different method for the introduction of secondary vibration in

boring bars. Here the actuator is mounted in a milled space in a longitudinal directionbelow the centerline of the boring bar. When the actuator applies a load on the boringbar in its longitudinal direction due to the expansion of the actuator, the boring barwill bend and stretch. By introducing secondary anti-vibrations via the actuator appliedbending moment on the boring bar, the original vibrations from the cutting process canbe reduced [5, 6, 13].

A challenge is to incorporate electronic devices into the harsh environment of a lathe.An active vibration control application includes actuators and sensors in conjunction witha control system. The actuator and accelerometer must be protected from the metal chipsand cutting fluids. One of the goals was to make the active control system applicable toa general lathe. Embedding the active parts, i.e. the actuator and accelerometer, willnot only protect them from the surrounding environment but will also allow the design tobe used on a general lathe provided that the mounting arrangement is relatively similar.Due to the recent development of piezo ceramic actuators, the technique can be embeddedinto a boring bar. Milling a space in the boring bar reduces the bending stiffness; withpiezo ceramic actuator technology, however, the space can be kept small and the bendingstiffness reduction is moderate. The motion of the boring bar usually has componentsin both the cutting speed and the cutting depth directions. However, the boring bar

3

vibration is to a great extent dominated by the motion in the cutting speed direction [1, 2].In [5, 6] active vibration control using one actuator in the cutting speed direction indicatesthat the use of one actuator is a satisfactory solution.

When an active boring bar is used, a suitable control algorithm is needed. The firstrequirement with respect to the algorithm is that it should be able to handle the non-stationary environments which a boring process gives rise to. Since it is not possible to dis-tinguish between the the original boring bar vibrations and the secondary anti-vibrations,the algorithm must be based on a feedback approach. It may also be advantageous toconsider the forward path, i.e. the signal transfer from the adaptive filter to the errorsensor. A simple proportional or P controller, which is widely used in control theory,may help satisfy the first two requirements, i.e. handle non-stationary environments andbased on a feedback approach. Where variations in the forward path are considered, moreadvanced controllers may be preferred. An algorithm that meets all requirements is thefeedback filtered-x lms algorithm. This algorithm has proved successful in both the activecontrol of tool vibration in external longitudinal turning and the active control of boringbar vibration [5, 6, 13]. Both the feedback filtered-x LMS algorithm and the InternalModel Control (IMC) controller based on an adaptive control FIR filter and governed bythe Filtered-x LMS algorithm have been tested. Internal model control causes a feedbackcontroller to work as a feedforward equivalent, provided the estimate of the forward pathmatches the actual forward path.

This report deals with active vibration control in boring operations using three differ-ent control strategies and three different active boring bar designs.

4

Chapter 2

Materials and Methods

2.1 Experimental Setup

All the experiments have been carried out on a MAZAK 250 Quickturn lathe, see Fig. 2.1;this has 18.5 kW spindle power, a maximum machining diameter of 300 mm and 1007 mmbetween the centers. In order to save material, the cutting operations were performed asexternal turning operations, although a boring bar was used as a tool holder, see Fig. 2.2.

Figure 2.1: The lathe in which all the experiments have been carried out.

5

Figure 2.2: The turning operation used in the experiments.

2.1.1 Measurement Equipment and Setup

Three different control algorithms were used in the active control measurements: anordinary PID controller, a feedback filtered-x LMS algorithm, and an Internal ModelControl (IMC) controller based on an adaptive control FIR filter governed by the Filtered-x LMS algorithm. A block diagram of the experimental setup for the active vibrationcontrol in boring operations can be seen in Fig. 2.3. A signal conditioning unit is alsoincluded in the digital controller in order to be able to adjust the level of the inputand output of the DSP. When using the filtered-x LMS algorithms, the transfer functionof the signal chain D/A converter, amplifier, structural transfer path from the actuatorto the accelerometer and an A/D converter must be estimated in an initial phase priorthe active vibration control. The experimental setup for investigating the forward pathis illustrated in the block diagram in Fig. 2.4. The accelerometer in the cutting speeddirection was used when estimating the forward path. The following equipment was usedin the measurements.

• 2 Bruel & Kjær 4374 accelerometers.

• 1 Bruel & Kjær NEXUS 2 channel conditioning amplifier 2692.

• TEAC RD-200T DAT recorder.

• A custom-designed amplifier designed for capacitive loads.

6

Digital Controller

HP signalanalyzer

ch1 ch2

DAT recorder

ch1 ch3

AmplifierA = 10

Piezo actuator

Boring Bar

Accelerometerin the cutting

depth directionbk4374

NEXUSCharge Amplifier

Accelerometerin the cutting

speed directionbk4374

NEXUSCharge Amplifier

ch2

Figure 2.3: A block diagram describing the experimental setup for active vibration control.

• Texas Instruments DSP TMS320C32.

• Active boring bars with an embedded piezo ceramic actuator, see section 2.2.

2.1.2 Cutting Data and Machining Parameters

The workpiece material in the cutting experiments was chromium molybdenum nickelsteel. The diameter of the workpiece was large (< 200mm) to ensure that the workpiecevibrations were negligible. The workpiece material SS 2541-03, chromium molybdenumnickel steel, is a quenched and tempered steel. This material excites the machine-tool sys-tem with a narrow bandwidth in the cutting operation [27]. It facilitates the introductionof major narrow-banded tool vibration in a turning operation, resulting in a deteriorationin surface finish and severe acoustic noise levels [1, 27].

The cutting tools used were standard 55◦ diagonal inserts. These have a tool geometrydesignated by the ISO code DNMG 150618-SL and with chip breaker geometry for mediumroughing. The carbide grade was TN7015.

The cutting data was selected in order to produce significant tool vibrations. Theseresulted in an observable deterioration in the workpiece surface as well as severe acousticnoise. After a preliminary set of trials, a suitable combination of cutting data and toolgeometry was selected, see Table 2.1. Cutting data set No. 1 was selected for the pro-duction of significant tool vibrations for evaluating the three different control algorithmsused in active control of boring bar vibration. Cutting data set No. 1 was also selected tofacilitate investigation of the influence of the actuator location in the boring bar on thevibration control performance in the metal cutting process. To vary the tool vibration

7

Signal Source

HP signalanalyzer

ch1 ch2

DAT recorder

ch1 ch2

AmplifierA = 10

Piezo actuator(double P804.10)

Boring Bar

Accelerometerbk 4374

s/n 22439300.1418 pC/ms-2

NEXUSCharge Amplifier

1mV/ms-2

0.142 pC/ms-2

Figure 2.4: The block diagram describing the experimental setup for both offline andonline estimation of the forward path.

Cutting data Geometry Cutting speed depth of cut Feedset v (m/min) a (mm) s (mm/rev)

No. 1 DNMG 150608-SL 80 1.0 0.2-0.3No. 2 DNMG 150608-SL 100-150 0.5-1.5 0.2

Table 2.1: Cutting data and tool geometry

level in a controlled manner, a low cutting speed was selected, i.e. just beyond the limitof build up of edge effects. The initial feed rate was selected in accordance with thelower chip-breaking limit of the insert. In the cutting experiments the cutting depth wasincreased to the limit of the control of the active boring bar. The cutting depth was sub-sequently slightly reduced to the maximum machining depth where maintaining controlwas possible; the feed rate was then gradually increased to the limit of active control.

By using cutting data set No. 2, it was occasionally possible to perform the boringoperation without any large vibrations. Under such circumstances, it was possible torecord data that enabled online estimation of the forward path during continuous turning.

No cutting fluid was applied during machining.

8

2.2 Active Boring Bar

A boring bar is usually long and slender in order to facilitate metal cutting in the boreof a workpiece. The boring bar used in the experiments was based on standard WIDAXS40T PDUNR15 boring bars, see Fig. 2.5. The diameter of the boring bar was 40 mmand the length 300 mm; 100 mm is required for the clamping. The ovarhang part thusconstitutes 200 mm.

Figure 2.5: A CAD model of the standard boring bar WIDAX S40T PDUNR15.

To perform active vibration control, an actuator and accelerometer must be appliedto the vibrating object. The environment in which active control of boring bar vibrationis designed to operate is harsh. The actuator and accelerometer must be protected fromcutting fluids and metal chips resulting from the cutting operation. One possibility is toembed and seal the electronic parts into the boring bar. Accelerometers are usually sosmall that incorporating them into the design will have negligible effect on the bendingstiffness of the boring bar. The accelerometer was mounted 25 mm from the tool tip andsenses the vibrations in the cutting speed direction. The actuator, on the other hand,must be sufficiently large to produce adequate secondary vibrational forces to enable asufficient increase of the dynamic bending stiffness. There are several ways of mountingthe actuator in the boring bar. Three different mounting locations for the actuator havebeen tested in real-life cutting experiments. The difference between the active boring bars

Cutting depthdirection

Cutting speeddirection

40 m

m

15 mm

15 mm

18.5

mm

Actuator

α

Figure 2.6: Boring bar cross section with embedded actuator and α is the actuator offsetangle.

is the actuator offset angle, i.e. the angle between the cutting speed direction and the

9

boring bar cross section radius which intersects the actuator center towards the reversedcutting depth direction. The actuator offset angle is illustrated in Fig. 2.6 and is denotedα. In all three cases, the actuator was mounted in a longitudinal direction below thecenterline of the boring bar and adjacent to the clamping. When the actuator expandsin length, it applies a load to the boring bar in its longitudinal direction; as a result, theboring bar will bend and stretch. Secondary anti-vibrations may thus be introduced bythe actuator applied bending moment in order to reduce the original boring bar vibration,excited by the chip formation process during continuous machining. A schematic figureof the active boring bar control system is shown in Fig. 2.7

Boring Bar

Cutting speeddirection

Embeddedactuator

Workpiece

Primary excitationintroduced by the material deformation process

W

Feedbackcontroller

Secondary excitationvia active actuator

Figure 2.7: A schematic figure of the active boring bar control system.

The first active boring bar design was based on an embedded actuator with 0◦ actuatoroffset angle.

The second active boring bar design was based on an embedded actuator with 15◦

actuator offset angle.The third active boring bar design was based on an embedded actuator with 30◦

actuator offset angle.

2.3 Control Algorithms

Control is the process of causing a system variable to conform to some desired value knownas a reference value [17]. A special class of control theory is feedback control, which is

10

used in the active vibration control application discussed here. A block diagram of theelementary parts of feedback control is provided in Fig. 2.8. Feedback control can be

Referencesensor

Actuator Plant

Outputsensor

Disturbance

OutputControllerΣ

Figure 2.8: Block diagram of an elementary feedback control system.

found in a wide range of products ranging from simple heating systems to very complexprocesses. The central component is the plant, which must be controlled in some way.An output sensor senses a variable that is designed to imitate a reference signal. Thecontroller uses the information from the reference signal and the output of the plant toproduce a control signal to the actuator. The actuator is the physical part of the controlsystem that has direct control authority on the plant. A simple analogy is a heatingsystem. Here the reference is the desired temperature in the room, and a radiator is usedas the actuator. The plant would be the room and the output of the plant is the actualtemperature of the room. The temperature of the room is easily controlled by switchingthe radiator on and off.

Feedback control is not a new science. In 1788 James Watt invented the centrifugalgovernor; this was the first feedback device to attract the attention of the entire engi-neering community and be accepted internationally [23]. The earliest feedback deviceknown, however, can be found among the works of Ktesibios, Philon and Heron fromthe Hellenistic period ca 300 B.C. For further reading on the history of feedback control,see [23].

The choice of control algorithms must be based on the application. The boring opera-tion is a process which has non-stationary stochastic properties [1]; the algorithm must beable to handle variations in the plant being controlled. Another important factor whichhas strong effect the choice of control algorithm is that the excitation source, the chipformation process, cannot be observed directly. The accelerometer senses both the vibra-tions resulting from the cutting process and those induced by the actuator. The algorithmmust thus based on a feedback approach. The forward path, which is always present in anactive vibration control application [19] must also be considered. In the particular controlproblem under discussion, the forward path basically consists of an amplifier, actuatorand the structural transfer path in the boring bar, see Fig. 2.9. Strictly speakin, theforward path also includes D/A and A/D converters as well as an accelerometer. Threecontroller algorithms suitable for evaluating purposes with respect to the application un-

11

ActuatorAmplifierStructural

transfer path

Forward path

Figure 2.9: The physical components of the forward path in the boring bar vibrationcontrol system.

der discussion are the simple PID controller, the more advanced feedback filtered-x LMSalgorithm and the Internal Model Control, IMC, controller based on an adaptive controlFIR filter governed by the Filtered-x LMS algorithm. In the filtered-x LMS algorithmand the IMC-based controller the estimate of the forward path was a 40 coefficient FIRfilter. Both the adaptive controllers used an adaptiver FIR filter with 35 weights .

2.3.1 PID Controller

The proportional integral derivative PID controller is well-known and is, for instance,widely accepted in the processing industry [24]. Originally, it was implemented usinganalog technology [24]. However, today almost all controllers are implemented in com-puters [24]. The digital PID controller can be viewed as an approximation of its analogcounterpart.

In a boring operation, the plant or forward path may be observed by an accelerometermounted on the boring bar. Since the goal is to reduce vibration, acceleration shouldbe as small as possible. The acceleration signal is denoted e(t); the control signal to theactuator is denoted y(t). The PID controller in continuous time t can be written as [24]

y(t) = K(e(t) +

1

Ti

∫ t

0

e(τ)dτ + Tdde(t)

dt

)(2.1)

where K is the gain of the controller, Ti is the integration time and Td is the derivativetime. The proportional part of the controller sets the constant gain. The integral partin conjunction with a proportional part improves steady state properties; when combinedwith derivative control, it also improves the transient properties [17].

A discretized version of the analog PID controller can be approximated as [24]

y(n) = P (n) + I(n) + D(n) (2.2)

where

P (n) = Ke(n − 1) (2.3)

I(n) = I(n − 1) +K

FsTi

e(n − 1) (2.4)

D(n) =FsTd

FsTd + ND(n − 1) − FsKTdN

FsTd + N

(e(n − 1) − e(n − 2)

)(2.5)

12

where Fs is the sampling frequency and N is a high frequency gain limitation of thederivative part. e(n) is the resulting error; in active vibration control in a boring operationit is also the accelerometer signal. Fig. 2.10 shows a block diagram of a feedback control

Forwardpath C Σ

x(n) = e(n-1)

y(n)

d(n)

e(n)

Unit delayz-1

yC(n)K

Figure 2.10: Block diagram of a feedback control system based on a digital P controller.

system with a digital P controller. Here the box with the unit delay z−1 between e(n) ande(n− 1) at the input to the controller indicated that the subject is a digital controller ina feedback control system.

2.3.2 LMS Algorithm

The least mean square LMS algorithm was developed by Widrow and Hoff in 1960. Theleast square approach provides a powerful approach to digital filtering in situations wherea fixed, finite length filter is applicable. This approach has been widely used in manyareas [20]. It is an important member of the family of stochastic gradient algorithms [18].The LMS algorithm is very simple and has therefore been made the standard againstwhich other adaptive filtering algorithms are benchmarked [18].

In the active control of vibration application discussed here, an LMS algorithm wasused when estimating the forward path of the system. A forward path is always presentin active control applications [19]; if the filtered-x LMS algorithm is used as a controller,an estimate of the forward path is also needed. A block diagram of the forward pathestimation using an LMS algorithm is shown in Fig. 2.11. The task of the LMS algorithmis to imitate a desired signal d(n) by letting an input signal x(n) pass an adaptive filterwn(k) to produce an output y(n). The algorithm adjusts the weights wn(k) so that theerror signal e(n) is minimized in the mean square sense. The error signal can be writtenas

e(n) = d(n) − y(n) = d(n) −L−1∑l=0

wn(l)x(n − l), (2.6)

where the adaptive FIR filter has L coefficients. The weights are updated on averagein the negative direction of the gradient. The gradient estimate in the LMS algorithm

13

Noisegenerator

Adaptivefilter w(n)

LMS

Forwardpath

Σ−

+

x(n)

y(n)

d(n)

e(n)

Figure 2.11: Block diagram of the forward path estimation using an LMS algorithm.

consists of the derivatives of the square error signal with respect to each of the weights ofthe adaptive filter

∂e2(n)

∂wn(k)=

∂(d(n) −

L−1∑l=0

x(n − l)wn(l))2

∂wn(k)= −2x(n − k)e(n) (2.7)

for k = 0, 1, . . . , L − 1. The weights can now be updated in the negative direction of thegradient estimate as

wn+1(0)wn+1(1)

...wn+1(L − 1)

=

wn(0)wn(1)

...wn(L − 1)

+ µ

x(n)x(n − 1)

...x(n − L + 1)

e(n), (2.8)

To enable convergence in the mean square of the LMS algorithm the step size µ is usuallyselected according to the inequality [28]

0 < µ <2

LE[x2(n)](2.9)

where E[x2(n)] is the power of the input signal. In practice, the step size is selected lessthan 5 % of the upper bound in Eq. 2.9 [19].

The LMS algorithm can be summarized in vector notation as [21]

y(n) = wTnx(n) (2.10)

e(n) = d(n) − y(n) (2.11)

wn+1 = wn + µe(n)x(n) (2.12)

14

Insufficient spectral excitation of the LMS algorithm when implemented in limit nu-merical precision may result in a divergence of the adaptive weights [19], e.g. a noiselesssinusoid as the reference signal to an adaptive filter with more than two filter weightsmay have this effect. In that case, the unconstrained weights may grow out of bound [19].A solution to the problem is to incorporate a leakage factor γ into the algorithm. Theleaky LMS algorithm is obtained if the weight adjustment algorithm, Eq. 2.12, in theLMS algorithm is replaced by [19]

w(n + 1) = γw(n) + µe(n)x(n), (2.13)

where 0 < γ < 1; this usually selected close to unity [19].

2.3.3 Filtered-x LMS Algorithm

When the adaptive filter is followed by a forward path, the conventional LMS algorithmmust be modified in order to ensure convergence. In active control applications thereis always a forward path present [19]; the forward path must thus be compensated for.Morgan proposed two approaches in 1980 where one of the suggestions later resulted in thefiltered-x LMS algorithm [10]. The algorithm was independently derived by Widrow [11]for adaptive control and by Burgess [12] in an active noise control application, both in1981. The input signal to an adaptive algorithm is usually denoted x and when it isalso filtered, the name is straightforward. Another common name for the algorithm isfiltered-reference LMS since the input signal to the algorithm is commonly known as thereference signal.

The filtered-x LMS algorithm is illustrated by the block diagram in Fig. 2.12. Tocompensate for the forward path the input signal to the weight adjustment algorithm isfiltered by an estimate of the forward path.

Estimate offoward path C*

Adaptivefilter w(n)

LMS

Forwardpath C Σ

x(n) y(n)

d(n)

e(n)

yC(n)

xC*(n)

Figure 2.12: Block diagram of the filtered-x LMS algorithm.

As in the LMS, the filtered-x LMS is designed to minimize the mean square error. Theerror signal is produced by the summation of a desired signal d(n) and the output signalof the forward path C, i.e. the output signal of the adaptive filter filtered by the forwardpath C. The output of the adaptive filter is denoted y(n); after passing the forward path

15

it is denoted yC(n). By assuming that the adaptive filter weights are time invariant, theerror signal can be written approximately as

e(n) = d(n) + yC(n) ≈ d(n) +

LC−1∑lC=0

C∗(lC)y(n − lC) (2.14)

= d(n) +

LC−1∑lC=0

C∗(lC)Lw−1∑lw=0

wn(lw)x(n − lC − lw) (2.15)

= d(n) +Lw−1∑lw=0

wn(lw)

LC−1∑lC=0

C∗(lC)x(n − lC − lw) (2.16)

= d(n) +Lw−1∑lw=0

wn(lw)xC∗(n − lw) (2.17)

where LC is the length of the a FIR filter estimate of the forward path C*, xC∗(n) =∑LC−1lC=0 C∗(lC)x(n − lC) is the filtered reference signal and Lw is the length of the adap-

tive filter. Thus, the gradient estimate in the filtered-x LMS algorithm is based on thederivative of the squared error with respect to the adaptive filter weights

∂e2(n)

∂wn(k)=

∂(d(n) +

Lw−1∑lw=0

wn(lw)xC∗(n − lw))2

∂wn(k)= 2xC∗(n − k)e(n) (2.18)

for k = 0, 1, . . . , L − 1. The weights can now be updated as

wn+1(0)wn+1(1)

...wn+1(Lw − 1)

=

wn(0)wn(1)

...wn(Lw − 1)

− µ

xC∗(n)xC∗(n − 1)

...xC∗(n − Lw + 1)

e(n) (2.19)

where µ is the step size or convergence factor. The filtered-x LMS algorithm can besummarized in vector notations as

y(n) = wT (n)x(n), (2.20)

e(n) = d(n) + yC(n) (2.21)

w(n + 1) = w(n) − µxC∗(n)e(n) (2.22)

16

the filtered reference signal vector is produced as

xC∗(n) =

LC−1∑lC=0

C∗(lc)x(n − lC)

LC−1∑lC=0

C∗(lc)x(n − lC − 1)

...LC−1∑lC=0

C∗(lc)x(n − lC − Lw + 1)

(2.23)

In order to select a step size µ to enable the filtered-x LMS algorithm to converge, thefollowing inequality is commonly used [25]

0 < µ <2

E[x2C∗(n)](Lw + ∆)

(2.24)

where ∆ is the overall delay in the forward path in samples, Lw is the length of theadaptive FIR filter and E[x2

C∗(n)] is the mean square value of the filtered reference signalto the algorithm.

One way to justify this algorithm is to consider what happens when the adaptivefilter only changes slowly over time in comparison with the time duration of the forwardpath. Under these conditions, the estimate of the forward path and the adaptive filtercommute. Since the output of the adaptive filter is sensed after the forward path, anordinary LMS algorithm can be used provided that the reference signal to the weightadjustment algorithm is filtered by an estimate of the forward path. In practice, thefiltered-x LMS algorithm is stable even if the control filter changes within the time scaleassociated with the dynamic response of the forward path [26].

The feedback filtered-x LMS algorithm is obtained from the filtered-x LMS algorithmby using the error signal as reference signal. In Fig. 2.13 a block diagram of the feedbackfiltered-x LMS algorithm is shown and this algorithm is given by [25]

y(n) = wT (n)x(n), x(n) = e(n − 1) (2.25)

e(n) = d(n) + yC(n) (2.26)

w(n + 1) = w(n) − µxC∗(n)e(n) (2.27)

Here, the reference signal vector x(n) = [e(n−1), . . . , e(n−Lw)]T is the delayed estimationerror signal vector, where e(n) is the error or, in this case, the accelerometer signal. Forthe purpose of convergence of the feedback filtered-x LMS algorithm, the inequality inEq. 2.24 may be used as guidance when selecting the initial step size µ [25]. However,the estimate of the power in the filtered reference signal used in the upper bound of thisinequality, Eq. 2.24, should be estimated prior to control [25]. Where the error signal is

17

Estimate offoward path C*

Adaptivefilter w(n)

LMS

Forwardpath C Σ

x(n) = e(n-1)

y(n)

d(n)

e(n)

Unit delayz-1

yC(n)

xC*(n)

Figure 2.13: Block diagram of the feedback filtered-x LMS algorithm.

used as input to the control system, the algorithm acts as a feedback controller. Thiswill complicate the relation between the mean square error and the filter weights, i.e. themean square error will not be a quadratic function of the filter weights [25].

Incorporating a leakage factor stabilizes the algorithm [19] and improves the robustnessof the feedback control system [25]. The weight updating function for the leaky filtered-xLMS algorithm is defined as [19]

wn+1 = γwn − µe(n)xC∗(n), (2.28)

where 0 < γ < 1 and usually selected close to unity [19].

2.3.4 IMC Controller based on the Filtered-x LMS Algorithm

Internal Model Control, IMC, is a control structure that has been particularly popular inprocess control [24]. Extending a feedback control system with IMC theoretically enablesa feedback system to be transformed into a feedforward equivalent if the estimate of theforward path is identical to the actual forward path. The algorithm generates its referencesignal on the basis of the output of the adaptive filter and the error signal. Fig. 2.14 showsa block diagram of the IMC controller based on the filtered-x LMS algorithm.

The search for a minimum is made in the mean square sense. The error signal is thesummation of the desired signal d(n) and the output of the forward path yC(n), hence

e(n) = d(n) + yC(n). (2.29)

If one assumes that the estimate of the forward path is identical to the actual one, i.e.C∗ = C, the estimate of forward path output signal influencing the process yC∗(n) equalsthe true output signal of the forward path yC(n). The synthesized desired signal d∗(n) is

18

Estimate offoward path C*

Adaptivefilter w(n)

LMS

Forwardpath C Σ

x(n)

y(n)

d(n)

e(n)

Unit delayz-1

yc(n)

Estimate offoward path C*

Σd*(n) -

+

+

+

yc*(n)

xc*(n)

Figure 2.14: Block diagram of a IMC controller based on the filtered-x LMS algorithm.

then

d∗(n) = e(n) − yC∗(n) = d(n) + yC(n) − yC∗(n) = d(n). (2.30)

If C∗ = C, the feedback system in Fig. 2.14 can be transformed into a feedforwardequivalent as in Fig. 2.15 [19].

The difference between the above and not using IMC is that the reference signal x(n)is

x(n) = d∗(n − 1) = e(n − 1) − yC∗(n − 1)

= e(n − 1) −LC−1∑lC=0

C∗(lC)y(n − 1 − lC). (2.31)

The adaptive IMC controller algorithm based on the filtered-x LMS is given by

y(n) = wT (n)x(n), x(n) = d∗(n − 1) (2.32)

e(n) = d(n) + yC(n) (2.33)

w(n + 1) = w(n) − µxC∗(n)e(n) (2.34)

Here, the reference signal vector d∗(n − 1) = [d∗(n − 1), . . . , d∗(n − Lw)]T is the delayedsynthesized desired signal vector where d∗(n) is the synthesized desired signal. For con-vergence of the adaptive IMC controller algorithm, the inequality in Eq. 2.24 may be usedfor guidance when selecting the initial step size µ [25].

Unfortunately, perfect estimates of the forward path are seldom encountered [22].There are various reasons for this imperfection, e.g. the forward path response changes

19

Adaptivefilter w(n)

LMS

Forwardpath C

Σx(n) y(n)

d(n)

e(n)

Unit delayz-1

yc(n)

Forwardpath C

xc*(n)Estimate offoward path C*

Figure 2.15: Block diagram of feedforward equivalent of the feedback system using theIMC controller based on the filtered-x LMS algorithm.

during operation and with non-linearities in the forward path. This suggests that the feed-forward system in Fig. 2.15 is not a perfect match with the feedback system in Fig. 2.14.Imperfect plant response leads to an undesired feedback loop due to the difference betweenyC(n) − yC∗(n). Looking at Fig. 2.14 and assuming for a moment that the signals aredeterministic and that there is a time invariant adaptive filter (for conscience reasons),the synthesized desired signal can, from a frequency perspective, be written as [22]

D∗(f) = D(f) +(C(f) − C∗(f)

)Y (f) (2.35)

where

Y (f) = W (f)D∗(f)e−j2πf (2.36)

and the error signal can be written as

E(f) = D(f) + C(f)W (f)D∗(f)e−j2πf (2.37)

The closed loop frequency response function can then be established as

Hcl(f) =E(f)

D(f)= 1 +

C(f)W (f)e−j2πf

1 − (C(f) − C∗(f)

)W (f)e−j2πf

=1 + C∗(f)W (f)e−j2πf

1 − (C(f) − C∗(f)

)W (f)e−j2πf

(2.38)

The adaptation of the filter can be seen as a feedforward configuration with a residualfeedback loop around the adaptive filter as a result of the imperfect estimate of the forwardpath [22]. As the degree of mismatch between the forward path and the estimate of theforward path increases, the stable region of the adaptive control system decreases [22].One simple way of preventing the adaptive weights from becoming too large and improvingthe robustness of the feedback control system is to use a leakage term in the filtered-xLMS algorithm in accordance with Eq. 2.28 [22].

20

2.4 Spectral Properties

It is often considered desirable to be able to view the content of a signal or system in thefrequency domain. A spectrum shows how the power or energy of a signal is distributedversus frequency. The coherence function shows the correlation between two signals; thiscan also be interpreted as a measure of linearity [15]. The frequency response functionshows the amplitude and phase of a system versus frequency.

2.4.1 Spectrum Estimation

A common estimator of the spectral content of a signal is the Welch spectrum estimator.The data record is divided into segments; these are allowed to overlap and are windowedprior to computing the periodogram [14].

The Welch spectrum estimate is obtained by averaging a number of periodograms.Each periodogram is based on segments of a time sequence x(n), each segment consistingof N samples. The original time sequence of data must be divided into data segments asfollows

xl(n) = x(n + lD) where

{n = 0, 1, . . . , N − 1l = 0, 1, . . . , L − 1

where lD is the starting point for each periodogram and D is the overlapping increment. IfD = N there is no overlap and if D = N/2 there is a 50% overlap between the consecutivetime sequences or data segments xl(n) and xl+1(n).

Dividing the original time sequence into data segments is equivalent to multiplying thetime series by a window. The Welch method allows the use of arbitrary windows such askaiser, hanning, flattop, etc. Hence, each periodogram is based on a windowed sequencew(n)xl(n) where n = 0, 1, . . . N − 1.

The Welch power spectral density estimator P ∗(fk) is given by

P ∗xx(fk) =

1

FsLNU

L−1∑l=0

∣∣∣∣∣N−1∑n=0

xl(n)w(n)e−j2πnk/N

∣∣∣∣∣2

, fk =k

NFs (2.39)

where k = 0, . . . , N/2, L is the number of periodograms, N is the length of the peri-odogram, Fs the sampling frequency and

U =1

N

N−1∑n=0

(w(n)

)2

is the window-dependent normalization factor for power spectral density estimates.For lightly damped mechanical systems, the normalized bias error in a spectral density

estimate can be approximated, at the resonance frequencies fr using [15]

εb ≈ −1

3

(Be

Br

)2

, (2.40)

21

where Br is the half power bandwidth of the smallest resonance peak and Be is thewindow-dependent resolution bandwidth; the latter is defined as [1]

Be =

N−1∑n=0

w2(n)

(N−1∑n=0

w(n)

)2 Fs =

1

N

N−1∑k=0

|W (k)|2

W 2(0)Fs,

where W (k) is the Fourier transformed window function w(n).The normalized random error εr of the spectral estimator is dependent on the choice

of time window w(n) and the overlap between the periodograms D. εr can be expressedas [14]

εr

[P ∗

xx(fk)]

=

√√√√ 1

L

(1 + 2

L−1∑q=1

L − q

Lρ(q)

)(2.41)

where

ρ(q) =

( N−1∑n=0

w(n)w(n + qD)

)2

N−1∑n=0

w2(n)

(2.42)

is the correlation between the periodograms with overlap D and window function w(n).If the window is hanning and there is no overlap εr

[P ∗

xx(fk)]

= 1/√

L, and if the common

50% overlap is used εr

[P ∗

xx(fk)] ≈ 1/

√1.89L/2 [14]. The equivalent number of uncor-

related periodograms Le used in the average to produce the spectrum estimate is givenby

Le =L

1 + 2L−1∑q=1

L − q

Lρ(q)

(2.43)

where ρ(q) is defined by Eq. 2.42. Using the equivalent number of uncorrelated peri-odograms, the normalized random error can be expressed as

εr

[P ∗

xx(fk)]

= 1/√

Le (2.44)

for all choices of overlap.After preliminary trials, the following were selected: data length, data segment length

N , number of periodograms L, number of equivalent uncorrelated periodograms Le, digitalwindow and sampling rate Fs. The selection is shown in Table 2.2.

22

Parameter Value

Total record length, T 50 sData segment length, N 8192Number of periodograms, L 291Equivalent number of perodograms, Le 275Digital window w(τ) HanningOverlapping 50 %Sampling rate, Fs 24000 Hz

Table 2.2: Spectral density estimation parameters

2.4.2 Coherence Function Estimation

The coherence function estimate is calculated from spectral estimates of the input andoutput of the system and shows the degree of linear dependency between two signals. Thecoherence function estimate is defined as [15]

γ∗2xy(fk) =

| P ∗xy(fk) |2

P ∗xx(fk)P ∗

yy(fk)(2.45)

where P ∗xy(fk) is the cross power spectral density estimate between input x and output

y, P ∗xx(fk) and P ∗

yy(fk) are the power spectral density estimates of x and y respectively.

The coherence function satisfies the condition 0 ≤ γ∗2xy(fk) ≤ 1.

In practice, when the coherence function is greater than zero and less than unity, oneor more of the following four conditions exists [16]

• Extraneous noise is present in the measurements.

• Resolution bias errors are present in the spectral estimates.

• The system relating x(n) to y(n) is not linear.

• The output y(n) is due to other inputs than x(n).

Estimating the coherence between the potential reference signals and error signals ofa possible active control application enables a preliminary estimate to be made of amaximum theoretical performance of a potential feedforward active control system [22].Good coherence is thus important in active control applications.

The normalized random error of a coherence function estimate is [15]

εr

[γ∗2

xy(fk)] ≈

√2[1 − γ2

xy(fk)]

| γxy(fk) |√

Le

(2.46)

where Le is defined by Eq. 2.43.

23

2.4.3 Frequency Response Function Estimation

The dynamic characteristics of a constant parameter linear system that is physicallyrealizable and stable can be described by a frequency response function H(f) [15]. Thefrequency response function is usually complex valued and is commonly shown by itsamplitude- and phase function. A frequency response function may be estimated inaccordance with

H∗xy(fk) =

P ∗xy(fk)

P ∗xx(fk)

(2.47)

where P ∗xy(fk) is the cross spectral density of the input signal x and the output signal y,

and P ∗xx(fk) is the auto spectral density of the input signal. The amplitude function is

|H∗xy(fk)|; the phase function Θ∗

xy(fk) is defined as

Θ∗xy(fk) = arg

(H∗

xy(fk))

(2.48)

The random error in frequency response function estimates for the amplitude func-tion [15] are

εr(|H∗xy(fk)|) ≈

√1 − γ2

xy(fk)

γ2xy(fk)2Le

(2.49)

and for the phase function [15]

εr(|Θ∗xy(fk)|) ≈ arcsin

(εr(|H∗

xy(fk)|))

(2.50)

where Le is defined by Eq. 2.43.

2.5 Nyquist Diagram

The Nyquist stability criterion is a well-known stability test. It relates the open-loopfrequency response to the number of closed-loop poles of the system in the right half-plane in the laplacian domain [17]. For discrete time systems, the stability area in thez-plane is the unit disc instead of the left half plane in the laplacian domain. The Nyquistcriterion is especially useful for determining the stability of a closed-loop system whenthe open-loop system is given. Stability is determined using the frequency response of acomplex system, perhaps with one or more resonances, where the magnitude curve passesone several times and/or the phase crosses 180◦ several times [17]. The criterion is alsovery useful in dealing with open-loop systems, unstable systems, non-minimum phasesystems and systems with pure delays.

A Nyquist diagram may be created in accordance with the following three steps [29]

1. Estimate the frequency response functions for all parts of the system.

24

2. Determine the open-loop frequency response for the complete system.

3. Plot the real- versus the imaginary part of the open-loop frequency response for thecomplete system in the complex plane for f ∈ [−Fs/2, Fs/2].

W(f)Σ C(f)-

+Σ

ReferenceController Forward

path

Noise

Output

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Re{Hol(f)}Im

{Hol

(f)}

Critical point

a) b)

Figure 2.16: A simple feedback system in a) and an example of a nyquist plot in b).

A simple feedback system is given in Fig. 2.16a), where W (f) and C(f) are stabledynamic systems, e.g. controller and forward path. The open-loop Hol(f) and closed-loopHcl(f) transfer functions are

Hol(f) = W (f)C(f) (2.51)

Hcl(f) =W (f)C(f)

1 + W (f)C(f)(2.52)

By examining the closed-loop system Hcl(f) it becomes obvious that if W (f)C(f) = −1or Hol(f) = −1, at some frequency, then the response of the feedback control systemwould become unbounded at this frequency due to a division by zero. According to theNyquist stability criterion, the system is considered stable if the Nyquist plot does notencircle the critical point (−1, 0) [24]. An example of a Nyquist plot of a stable system isgiven in Fig. 2.16b).

When determining the open-loop system, several frequency response function esti-mates are normally included. When estimating each of the transfer functions it is impor-tant to produce estimates with a high level of accuracy in order to enable an accurateestimate of the open-loop response for the control system.

25

Chapter 3

Results

Three different feedback controllers have been tested in the active control of boring barvibration: two adaptive controllers based on the filtered-x LMS, algorithm and a timeinvariant digital P controller. The influence of the placement of the actuator in the activeboring bars when it comes to perfomance in active control of boring bar vibration hasbeen investigated. In addition, the properties of the forward path when the boring baris not in contact with the workpiece and during continuous cutting operations have alsobeen considered.

Time history records of typical boring bar vibration during continuous turning in boththe cutting speed and cutting depth directions are shown in Fig. 3.1. The correspondingspectra of the time history records are shown in Fig. 3.2.

3.1 The Forward Path

Three different active boring bars have been designed, all with an actuator located adja-cent to the clamping below the centerline and within the surface boundary of the boringbar. The difference between the designs is the actuator offset angle α, see Fig. 2.6 insection 2.2. The different designs introduce different forward paths in an active vibrationcontrol system. Fig. 3.3 shows offline frequency response function estimates of the forwardpath produced for the three different active boring bar designs when the boring bar is notin contact with the workpiece. Note that the first resonance peak of the forward pathwith 0◦ actuator offset angle of the active boring bar is lower both in frequency and am-plitude as compared with the other two. The coherence function estimates correspondingto the three different forward paths are presented in Figs. 3.4 a) - c). The coherencefunction estimates indicate that the forward path output signal may, to a great extent,be explained linearly from its input in all three cases.

In the adaptive controllers, the forward path is modeled as an FIR filter. The frequencyresponse function estimate and the corresponding Fourier transformed FIR filter estimateof the forward path are plotted in the same diagram for each of the active boring bardesigns, see Figs. 3.5 a) - c). In the subsequent results the active boring bar design 2has not been involved as it demonstrated a significant deterioration in dynamic stiffness

26

0 5 10 15 20−4000

−2000

0

2000

4000

Time [ms]A

ccel

erat

ion

[m/s2 ]

a)

0 5 10 15 20−500

0

500

Acc

eler

atio

n [m

/s2 ]

Time [ms]b)

Figure 3.1: Boring bar vibration as a function of time a) in the cutting speed direction andb) in the cutting depth direction. Workpiece material SS2541-03, feed rate s=0.2mm/rev,cutting depth a=1.0mm, cutting speed v=80m/min, cutting tool DNMG 150806-SL, gradeTN7015.

in the turning operation as compared to an unmodified standard boring bar. The 35coefficients of the FIR filter estimate of the forward path for boring bar designs 1 and 3,with actuator offset angles of 0◦ and 30◦ are shown in Fig. 3.6

Occasionally it is possible to perform a boring operation where the vibration level islow. Under such circumstances, it is possible to estimate the forward path during boringto see whether the forward path characteristics have changed or not as a result of thechanged boundary conditions at the tool tip. In Figs. 3.7 a) and b) both the offline andthe online frequency response function estimates as well as the corresponding Fouriertransformed FIR filter estimate of the forward path are plotted in the same diagramfor each respective active boring bar design. Note in particular that there is a slightchange in the first resonance peak for both versions of the boring bars when estimatingthe forward path during turning. When estimating the forward path during continuouscutting, the cutting operation will affect the estimation. The coherence functions of theonline estimates are given in Fig. 3.8. The errors associated with the online estimation ofcoherence functions and frequency response functions are shown in Fig. 3.9.

27

0 2000 4000 6000 8000−30

−20

−10

0

10

20

30

40

50

60

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

0 2000 4000 6000 8000−30

−20

−10

0

10

20

30

40

50

60

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

a) b)

Figure 3.2: Power spectral densities of boring bar vibration a) in the cutting speed di-rection and b) in the cutting depth direction. Workpiece material SS2541-03, feed rates=0.2mm/rev, cutting depth a=1.0mm, cutting speed v=80m/min, cutting tool DNMG150806-SL, grade TN7015.

0 500 1000 1500 2000 2500 3000 3500 4000−40

−20

0

20

40

H* xy

(f)

[m/s

2 /V]

Frequency [Hz]a)

α = 0o

α = 15o α = 30o

0 500 1000 1500 2000 2500 3000 3500 4000−π

0

π

Pha

se [r

ad]

Frequency [Hz]b)

α = 0o

α = 15o α = 30o

Figure 3.3: Frequency response function estimates of the forward path for the threedifferent active boring bar designs with a different actuator offset angle α, a) magnitudefunctions and b) phase functions.

28

0 1000 2000 3000 40000

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Coh

eren

ce γ*

2 xy(f

)

0 1000 2000 3000 40000

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Coh

eren

ce γ*

2 xy(f

)

a) b)

0 1000 2000 3000 40000

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Coh

eren

ce γ*

2 xy(f

)

c)

Figure 3.4: Coherence function estimates between the signal to the actuator and theacceleration signal in the cutting speed direction of the different active boring bar designsat a) 0◦, b) 15◦ and c) 30◦ actuator offset angle.

29

0 1000 2000 3000 4000

−60

−40

−20

|H* xy

(f)|

[dB

]

Frequency [Hz]

Direct FRFFIR based

0 1000 2000 3000 4000−π

0

π

Pha

se [r

ad]

Frequency [Hz]

Direct FRFFIR based

0 1000 2000 3000 4000

−60

−40

−20

|H* xy

(f)|

[dB

]

Frequency [Hz]

Direct FRFFIR based

0 1000 2000 3000 4000−π

0

π

Pha

se [r

ad]

Frequency [Hz]

Direct FRFFIR based

a) b)

0 1000 2000 3000 4000

−60

−40

−20

|H* xy

(f)|

[dB

]

Frequency [Hz]

Direct FRFFIR based

0 1000 2000 3000 4000−π

0

π

Pha

se [r

ad]

Frequency [Hz]

Direct FRFFIR based

c)

Figure 3.5: Frequency response function estimates of the forward path when the boringbar is not in contact with the workpiece, offline, and the Fourier transformed offline FIRfilter estimate of the forward path used in the controller at a) 0◦ actuator offset angle, b)15◦ actuator offset angle and c) 30◦ actuator offset angle.

30

0 10 20 30 40−0.01

−0.005

0

0.005

0.01

n

For

war

d pa

th c

*(n)

0 10 20 30 40−0.01

−0.005

0

0.005

0.01

n

For

war

d pa

th c

*(n)

a) b)

Figure 3.6: The 35 coefficient FIR filter estimate of the forward path of the boring bardesign 1 and 3 a) 0◦ actuator offset angle and in b) 30◦ actuator offset angle.

0 200 400 600 800 1000−40

−20

0

20

|H* xy

(f)|

[dB

rel

m/s2 /V

]

Frequency [Hz]

online FRF offline FRFFIR based

0 200 400 600 800 1000−π

0

π

Pha

se [r

ad]

Frequency [Hz]

online FRFoffline FRF FIR based

0 200 400 600 800 1000−40

−20

0

20

|H* xy

(f)|

[dB

rel

m/s2 /V

]

Frequency [Hz]

online FRF offline FRFFIR based

0 200 400 600 800 1000−π

0

π

Pha

se [r

ad]

Frequency [Hz]

online FRFoffline FRF FIR based

a) b)

Figure 3.7: Frequency response function estimates of the forward path during a continuouscutting operation (online) and when the boring bar is not in contact with the workpiece(offline), and the Fourier transformed offline FIR filter estimate of the forward path usedin the controller at a) 0◦ actuator offset angle and b) a 30◦ actuator offset angle. Theonline estimation of the forward path was produced using workpiece material SS2541-03,cutting tool DNMG 150806-SL, grade TN7015. In a) feed rate s=0.2mm/rev, cuttingdepth a=1.5mm, cutting speed v=100m/min and in b) feed rate s=0.2mm/rev, cuttingdepth a=0.5mm, cutting speed v=150m/min.

31

0 1000 2000 3000 40000

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Coh

eren

ce γ*

2 xy(f

)

0 1000 2000 3000 40000

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Coh

eren

ce γ*

2 xy(f

)

a) b)

Figure 3.8: Coherence function estimate of the online estimation of the boring bar. In a)active boring bar design 1 with 0◦ actuator offset angle and in b) active boring bar design3 with 30◦ actuator offset angle.

0 1000 2000 3000 40000

0.05

0.1

ε r[ γ*

2 xy(f

) ]

Frequency [Hz]a)

0 1000 2000 3000 40000

0.05

0.1

ε r[ H* xy

(f)

]

Frequency [Hz]b)

0 1000 2000 3000 40000

0.2

0.4

0.6

ε r[ γ*

2 xy(f

) ]

Frequency [Hz]c)

0 1000 2000 3000 40000

0.2

0.4

0.6

ε r[ H* xy

(f)

]

Frequency [Hz]d)

Figure 3.9: The random error associated with the coherence function estimate εr[γ∗2xy(f)]

and the frequency response function estimate εr[H∗xy(f)] of the the online estimation

of boring bar design 1 and 3 with 0◦ and 30◦ actuator offset angle respectively. In a)εr[γ

∗2xy(f)] active boring bar design 1 and in b) εr[H

∗xy(f)] of the same boring bar. In c)

εr[γ∗2xy(f)] active boring bar design 3 and in d) εr[H

∗xy(f)] of the same boring bar.

32

3.2 Active Vibration Control

The results of the active control of boring bar vibration are illustrated as power spectraldensities of boring bar vibration with and without active vibration control. In section3.2.1, a performance comparison of the different active boring bar designs is made, and insection 3.2.2 the reduction achieved using different algorithms is demonstrated. Finally,the stability and robustness of the feedback controllers are addressed in section 3.2.3.Here, the Nyquist diagrams are given for the feedback control system for each of the threedifferent feedback controllers. Also, the introduction of leakage in the adaptive algorithmsas a measure to improve robustness as well as the cost of this measure are addressed inNyquist diagrams and on power spectral densities of the boring bar vibration.

3.2.1 Boring Bar Comparison

Three different active boring bar designs were tested. The actuator was mounted withdifferent actuator offset angles, see Fig. 2.6 in section 2.2. The boring bar with a 15◦

actuator offset angle demonstrated a significant deterioration in dynamic stiffness in theturning operation as compared to an unmodified standard boring bar. This reductionin dynamic stiffness was not observed in the other two designs. The performance of theremaining two active boring bars with an actuator offset angle of 0◦ and 30◦ was evaluatedusing the feedback filtered-x LMS algorithm. Fig. 3.10 shows the power spectral densityof boring bar vibration with and without active vibration control using active boring bardesign 1 using the 0◦ actuator offset angle. The power spectral densities of boring barvibration with and without active vibration control using active boring bar design 3 using30◦ actuator offset angle, are shown in Fig. 3.11. Figs. 3.10 and 3.11 also demonstratethat the vibration level was not only suppressed in the cutting speed direction but alsothat significant vibration reduction was found in the cutting depth direction.

3.2.2 Algorithm Comparison

Three algorithms were tested in the active control of boring bar vibration: the feedbackfiltered-x LMS algorithm, an Internal Model Control (IMC) controller based on an adap-tive FIR filter governed by the filtered-x LMS algorithm, and a time invariant digital Pcontroller. The algorithms were compared using the active boring bar design which per-formed best, namely design 1 with a 0◦ actuator offset angle. Fig. 3.12 shows boring barvibration control results obtained using the feedback filtered-x LMS algorithm. Fig. 3.13shows the corresponding results obtained by using the IMC controller. Control resultsobtained using the P controller are illustrated in Fig. 3.14. Finally, a photograph of thesurface of a machined workpiece is shown in Fig. 3.15 with and without active vibrationcontrol.

33

400 600 800 1000 1200 1400 1600−20

−10

0

10

20

30

40

50

60

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

Active offActive on

400 600 800 1000 1200 1400 1600−20

−10

0

10

20

30

40

50

60

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

Active offActive on

a) b)

Figure 3.10: Power spectral densities of boring bar vibration with and without activevibration control using an active boring bar with 0◦ actuator offset angle and the feed-back filtered-x LMS algorithm, a) cutting speed direction and b) cutting depth direction.Workpiece material SS2541-03, cutting tool DNMG 150806-SL, grade TN7015, feed rates=0.3mm/rev, cutting depth a=1.0mm, cutting speed v=80m/min.

400 600 800 1000 1200 1400 1600−20

−10

0

10

20

30

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50

60

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

Active offActive on

400 600 800 1000 1200 1400 1600−20

−10

0

10

20

30

40

50

60

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

Active offActive on

a) b)

Figure 3.11: Power spectral densities of boring bar vibration with and without activevibration control using an active boring bar with 30◦ actuator offset angle and the feed-back filtered-x LMS algorithm, a) cutting speed direction and b) cutting depth direction.Workpiece material SS2541-03, cutting tool DNMG 150806-SL, grade TN7015, feed rates=0.2mm/rev, cutting depth a=1.0mm, cutting speed v=80m/min.

34

400 600 800 1000 1200 1400 1600

−20

−10

0

10

20

30

40

50

60

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

Active offActive on

400 600 800 1000 1200 1400 1600

−20

−10

0

10

20

30

40

50

60

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

Active offActive on

a) b)

Figure 3.12: Power spectral densities of boring bar vibration with and without activevibration control using an active boring bar with an actuator offset angle of 0◦ and thefeedback filtered-x LMS algorithm. a) cutting speed direction and b) cutting depth di-rection. Workpiece material SS2541-03, cutting tool DNMG 150806-SL, grade TN7015,feed rate s=0.3mm/rev, cutting depth a=1.0mm, cutting speed v=80m/min.

400 600 800 1000 1200 1400 1600

−20

−10

0

10

20

30

40

50

60

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

Active offActive on

400 600 800 1000 1200 1400 1600

−20

−10

0

10

20

30

40

50

60

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

Active offActive on

a) b)

Figure 3.13: Power spectral densities of boring bar vibration with and without activevibration control using an active boring bar with a 0◦ actuator offset angle and anIMC-based adaptive controller, a) cutting speed direction and b) cutting depth direc-tion. Workpiece material SS2541-03, cutting tool DNMG 150806-SL, grade TN7015, feedrate s=0.3mm/rev, cutting depth a=1.0mm, cutting speed v=80m/min.

35

400 600 800 1000 1200 1400 1600

−20

−10

0

10

20

30

40

50

60

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

Active offActive on

400 600 800 1000 1200 1400 1600

−20

−10

0

10

20

30

40

50

60

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

Active offActive on

a) b)

Figure 3.14: Power spectral densities of boring bar vibration with and without activevibration control using an active boring bar with a 0◦ actuator offset angle and a Pcontroller, a) cutting speed direction and b) cutting depth direction. Workpiece mate-rial SS2541-03, cutting tool DNMG 150806-SL, grade TN7015, feed rate s=0.2mm/rev,cutting depth a=1.0mm, cutting speed v=80m/min.

Figure 3.15: Photograph of a machined workpiece with and without active vibrationcontrol. The controller turned on to the right; there is no control to the left.

36

3.2.3 Stability and Robustness of Feedback Controllers

The stability of a feedback control system requires that its open loop frequency responseHol(f) does not violate the closed loop stability requirements, i.e. the Nyquist stabilitycriterion [24]. A closed loop system is said to be stable if the polar plot of the open loopfrequency response Hol(f) for the feedback control system does not enclose the (−1, 0)point in the Nyquist diagram. The greater the shortest distance between the polar plotand the (−1, 0) point, the more robust the feedback control system is with respect tovariation in forward path response and controller response. An estimate of the open loopresponse for a feedback control system may be produced based on the controller frequencyresponse function and the forward path frequency response function [17]. The open loopfrequency responses for the digital P controller were produced for the 6 different controllergains used for controlling of boring bar vibration. In the case of adaptive control, theadaptive FIR filter coefficients obtained after convergence were Fourier transformed toproduce the corresponding open loop frequency responses. The forward path frequencyresponse function was estimated both offline, when the boring bar was not in contactwith the workpiece, and online, during a continuous cutting operation with a low bor-ing bar vibration level. The online estimate of the forward path was conducted duringcontinuous turning in workpiece material SS2541-03; with cutting tool DNMG 150806-SLand grade TN7015, the cutting parameters were feed rate s=0.2mm/rev, cutting deptha=1.5mm and cutting speed v=100m/min. However, observe that during continuous cut-ting with severe boring bar vibration levels the dynamic response of the clamped boringbar generally has pronounced non-linear properties [1].

Open loop frequency responses for the boring bar vibration control system were pro-duced using 6 different P controller gains and an offline estimate of the forward path.The Nyquist diagram in Fig. 3.16 shows the polar plots of these P controller gains basedon open loop frequency responses. The corresponding magnitude and phase functionsare shown in 3.17. Based on the 6 different P controller gains and an online estimateof the forward path, open loop frequency responses for the boring bar vibration controlsystem were produced; these are shown in the Nyquist diagram in Fig. 3.18. The corre-sponding magnitude and phase functions are shown in 3.19. Fig. 3.20 illustrates powerspectral densities of boring bar vibration with and without P control for the 6 differentgain settings.

The adaptive control of boring bar vibration was carried out with and without aleakage factor in the adaptive weight update equation; the leakage factors γ = 0.9999and γ = 0.999 were used. The Nyquist diagram in fig. 3.21 shows the polar plots ofthe open loop frequency responses based on the feedback filtered-x LMS algorithm withand without leakage and with an offline estimate of the forward path. The magnitudeand phase functions corresponding to Fig. 3.21 are shown in Fig. 3.22. Using an onlineestimate of the forward path, the corresponding polar plots of the open loop frequencyresponses were produced; these are illustrated in the Nyquist diagram in Fig. 3.23. Thecorresponding magnitude and phase functions are shown in Fig. 3.24. Similarly, polarplots of open loop frequency responses based on the adaptive IMC controller with and

37

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Im{H

*ol

(f)}

Re{H*ol

(f)}

K=−2

K=−4

K=−6

K=−3 K=−5 K=−7

Figure 3.16: Nyquist diagram for a boring bar vibration control system based on a Pcontroller for 6 different gain factors K. An offline frequency response function estimate ofthe forward path was used based on an active boring bar with a 0◦ actuator offset angle.

without leakage and with an offline estimate of the forward path as well as an onlineestimate of the forward path were produced. The Nyquist diagram in Fig. 3.25 shows theopen loop frequency responses based on the IMC controller with and without leakage andwith an offline estimate of the forward path. The corresponding magnitude and phasefunctions are shown in Fig. 3.26. Finally, the open loop frequency responses produced forthe IMC controller with and without leakage and with an online estimate of the forwardpath are plotted in the Nyquist diagram in Fig. 3.27; the corresponding magnitude andphase functions are shown in Fig. 3.28.

One way of increasing the robustness of the adaptive control algorithms is thus toincorporate a leakage factor to the adaptive weight update equation. Figs. 3.29 a) and b)show power spectral densities of boring bar vibration with and without feedback filtered-x control using the leakage factors γ = 1, 0.9999, 0.999. The corresponding boringbar vibration spectra obtained with and without adaptive IMC control using the leakagefactors γ = 1, 0.9999, 0.999 are shown in Figs. 3.30 a) and b). It is clear that the increasein robustness is made at the expense of degraded performance.

38

0 500 1000 1500 2000−60

−40

−20

0

K=−2K=−3

K=−4K=−5K=−6K=−7

|H* ol

(f)|

[dB

]

Frequency [Hz]

0 500 1000 1500 2000−3π

−2π

−π

0

Pha

se [r

ad]

Frequency [Hz]

for all K

Figure 3.17: The estimated open loop frequency response for a boring bar vibrationcontrol system based on a P controller for 6 different gain factors K. An offline frequencyresponse function estimate of the forward path was used based on an active boring barwith a 0◦ actuator offset angle.

−0.2 0 0.2 0.4 0.6 0.8 1 1.2−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Im{H

*ol

(f)}

Re{H*ol

(f)}

K=−2

K=−4

K=−6

K=−3K=−5

K=−7

Figure 3.18: Nyquist diagram for a boring bar vibration control system based on a Pcontroller for 6 different gain factors K. An online frequency response function estimate ofthe forward path was used based on an active boring bar with a 0◦ actuator offset angle.

39

0 500 1000 1500 2000−60

−40

−20

0

K=−2K=−3

K=−4K=−5K=−6K=−7

|H* ol

(f)|

[dB

]

Frequency [Hz]

0 500 1000 1500 2000−3π

−2π

−π

0

Pha

se [r

ad]

Frequency [Hz]

for all K

Figure 3.19: The estimated open loop frequency response for a boring bar vibrationcontrol system based on a P controller for 6 different gain factors K. An online frequencyresponse function estimate of the forward path was used based on an active boring barwith a 0◦ actuator offset angle.

40

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Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

No ControlK=−2 K=−3 K=−4 K=−5 K=−6 K=−7

480 500 520 540 560 580 6000

10

20

30

40

50

Frequency [Hz]P

SD

[dB

rel

1 (

m/s2 )2 /H

z]

No ControlK=−2 K=−3 K=−4 K=−5 K=−6 K=−7

a) b)

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el 1

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/s2 )2 /Hz]

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480 500 520 540 560 580 600−10

0

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40

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

No ControlK=−2 K=−3 K=−4 K=−5 K=−6 K=−7

c) d)

Figure 3.20: Power spectral densities of boring bar vibration with and without active vi-bration control using a P controller for 6 different gain factors K, a) cutting speed directionand b) as a) but zoomed to the first resonance peak, c) cutting depth direction and d) asc) but zoomed to the first resonance peak. An active boring bar with a 0◦ actuator offsetangle was used. Workpiece material SS2541-03, cutting tool DNMG 150806-SL, gradeTN7015, feed rate s=0.2mm/rev, cutting depth a=1.0mm, cutting speed v=80m/min.

41

−2 −1 0 1 2 3 4 5 6−2

−1

0

1

2

3

4

5

6

Im{H

*ol

(f)}

Re{H*ol

(f)}

(−1,0)

γ = 1 γ = 0.9999γ = 0.999

Figure 3.21: Nyquist diagram for a boring bar vibration control system based on thefeedback filtered-x LMS algorithm for 3 different leakage factors γ. An offline frequencyresponse function estimate of the forward path based on the active boring bar with a0◦ actuator offset angle was used. Workpiece material SS2541-03, cutting tool DNMG150806-SL, grade TN7015, feed rate s=0.3mm/rev, cutting depth a=1.0mm, cutting speedv=80m/min.

42

0 500 1000 1500 2000−100

−50

0

|H* ol

(f)|

[dB

]

Frequency [Hz]

γ = 1 γ = 0.9999γ = 0.999

0 500 1000 1500 2000−6π

−4π

−2π

0

Pha

se [r

ad]

Frequency [Hz]

γ = 1γ = 0.9999 γ = 0.999

Figure 3.22: The estimated open loop frequency response for a boring bar vibration controlsystem based on the feedback filtered-x LMS algorithm for 3 different leakage factors γ.An offline frequency response function estimate of the forward path based on the activeboring bar with a 0◦ actuator offset angle was used. Workpiece material SS2541-03,cutting tool DNMG 150806-SL, grade TN7015, feed rate s=0.3mm/rev, cutting deptha=1.0mm, cutting speed v=80m/min.

43

−2 0 2 4 6 8 10 12−5

0

5

10

Im{H

*ol

(f)}

Re{H*ol

(f)}

(−1,0)

γ = 1 γ = 0.9999γ = 0.999

Figure 3.23: Nyquist diagram for a boring bar vibration control system based on thefeedback filtered-x LMS algorithm for 3 different leakage factors γ. An online frequencyresponse function estimate of the forward path based on the active boring bar with a0◦ actuator offset angle was used. Workpiece material SS2541-03, cutting tool DNMG150806-SL, grade TN7015, feed rate s=0.3mm/rev, cutting depth a=1.0mm, cutting speedv=80m/min.

44

0 500 1000 1500 2000−100

−50

0

|H* ol

(f)|

[dB

]

Frequency [Hz]

γ = 1 γ = 0.9999γ = 0.999

0 500 1000 1500 2000−6π

−4π

−2π

0

Pha

se [r

ad]

Frequency [Hz]

γ = 1γ = 0.9999 γ = 0.999

Figure 3.24: The estimated open loop frequency response for a boring bar vibration controlsystem based on the feedback filtered-x LMS algorithm for 3 different leakage factors γ.An online frequency response function estimate of the forward path based on the activeboring bar with a 0◦ actuator offset angle was used. Workpiece material SS2541-03,cutting tool DNMG 150806-SL, grade TN7015, feed rate s=0.3mm/rev, cutting deptha=1.0mm, cutting speed v=80m/min.

45

−1 −0.5 0 0.5 1 1.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Im{H

*ol

(f)}

Re{H*ol

(f)}

(−1,0)

γ = 1 γ = 0.9999γ = 0.999

Figure 3.25: Nyquist diagram for a boring bar vibration control system and an IMC-basedadaptive controller for 3 different leakage factors γ. An offline frequency response functionestimate of the forward path based on the active boring bar with a 0◦ actuator offset anglewas used. Workpiece material SS2541-03, cutting tool DNMG 150806-SL, grade TN7015,feed rate s=0.3mm/rev, cutting depth a=1.0mm, cutting speed v=80m/min.

46

0 500 1000 1500 2000−100

−50

0

|H* ol

(f)|

[dB

]

Frequency [Hz]

γ = 1 γ = 0.9999γ = 0.999

0 500 1000 1500 2000−4π

−2π

0

2π

Pha

se [r

ad]

Frequency [Hz]

γ = 1γ = 0.9999 γ = 0.999

Figure 3.26: The estimated open loop frequency response for a boring bar vibrationcontrol system and an IMC-based adaptive controller for 3 different leakage factors γ.An offline frequency response function estimate of the forward path based on the activeboring bar with a 0◦ actuator offset angle was used. Workpiece material SS2541-03,cutting tool DNMG 150806-SL, grade TN7015, feed rate s=0.3mm/rev, cutting deptha=1.0mm, cutting speed v=80m/min.

47

−1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

Im{H

*ol

(f)}

Re{H*ol

(f)}

(−1,0)

γ = 1 γ = 0.9999γ = 0.999

Figure 3.27: Nyquist diagram for a boring bar vibration control system and an IMC-basedadaptive controller for 3 different leakage factors γ. An online frequency response functionestimate of the forward path based on the active boring bar with a 0◦ actuator offset anglewas used. Workpiece material SS2541-03, cutting tool DNMG 150806-SL, grade TN7015,feed rate s=0.3mm/rev, cutting depth a=1.0mm, cutting speed v=80m/min.

48

0 500 1000 1500 2000−100

−50

0

|H* ol

(f)|

[dB

]

Frequency [Hz]

γ = 1 γ = 0.9999γ = 0.999

0 500 1000 1500 2000−4π

−2π

0

2π

Pha

se [r

ad]

Frequency [Hz]

γ = 1γ = 0.9999 γ = 0.999

Figure 3.28: The estimated open loop frequency response for a boring bar vibrationcontrol system and an IMC-based adaptive controller for 3 different leakage factors γ.An online frequency response function estimate of the forward path based on the activeboring bar with a 0◦ actuator offset angle was used. Workpiece material SS2541-03,cutting tool DNMG 150806-SL, grade TN7015, feed rate s=0.3mm/rev, cutting deptha=1.0mm, cutting speed v=80m/min.

400 600 800 1000 1200 1400 1600

−20

−10

0

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30

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60

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

No Control γ = 1 γ = 0.9999γ = 0.999

400 600 800 1000 1200 1400 1600

−20

−10

0

10

20

30

40

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60

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

No Control γ = 1 γ = 0.9999γ = 0.999

a) b)

Figure 3.29: Power spectral densities of boring bar vibration with and without activevibration control using the feedback filtered-x LMS algorithm for 3 different leakage factorsγ, a) cutting speed direction and b) cutting depth direction. The active boring bar witha 0◦ actuator offset angle was used, workpiece material SS2541-03, cutting tool DNMG150806-SL, grade TN7015, feed rate s=0.3mm/rev, cutting depth a=1.0mm, cutting speedv=80m/min.

49

400 600 800 1000 1200 1400 1600

−20

−10

0

10

20

30

40

50

60

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

No Control γ = 1 γ = 0.9999γ = 0.999

400 600 800 1000 1200 1400 1600

−20

−10

0

10

20

30

40

50

60

Frequency [Hz]

PS

D [d

B r

el 1

(m

/s2 )2 /Hz]

No Control γ = 1 γ = 0.9999γ = 0.999

a) b)

Figure 3.30: Power spectral densities of boring bar vibration with and without activevibration control using the IMC-based controller for 3 different leakage factors γ, a) cuttingspeed direction and b) cutting depth direction. The active boring bar with a 0◦ actuatoroffset angle was used, workpiece material SS2541-03, cutting tool DNMG 150806-SL, gradeTN7015, feed rate s=0.3mm/rev, cutting depth a=1.0mm, cutting speed v=80m/min.

50

Chapter 4

Discussion and Conclusions

Active vibration control in boring operations ia clearly one possible solution for reducingthe vibration present in the kind of machining discussed in the present report. By em-bedding the actuator and accelerometer into the boring bar the design can be applicableto a general lathe as long as the clamping arrangement is relatively similar. No expensivemodification is required to the lathe apart from wiring two cables from the control systemto the boring bar, one for the accelerometer and one for the actuator. Embedding theelectronic devices also protects them from the harsh environment in a lathe. The metalchips from the cutting process and the cutting fluid could otherwise constitute majorproblems for the actuator and accelerometer.

Three different active boring bar designs have been used in the experiments, all withan actuator embedded into the design and within the surface boundary of the boringbar. The actuator was mounted in a longitudinal direction of the boring bars. The dif-ference between the boring bar designs was the actuator offset angle α (see Fig. 2.6).Consequently, the direction of decreased bending stiffness will also differ between the ac-tive boring bars. Compared to the surrounding boring bar steel structure, the embeddedactuator has a lower modulus of elasticity and as a result, the bending stiffness in thedirection of the cross-section radius intersecting the actuator is reduced. The eigenfre-quency of the fundamental bending mode in the cutting speed direction decreases as theactuator offset angle α decreases, see Fig. 3.3. This was anticipated since the actuatorwas located adjacent to the clamping.

During continuous cutting operations, the workpiece apply boundary conditions onthe tool tip that are different to the free conditions obtained when the boring bar is notin contact with the workpiece [1, 2]. This will affect the fundamental bending modes ofthe boring bar, see Fig. 3.7. As a result, the forward path is usually different between con-tinuous cutting operations and when the boring bar is not in contact with the workpiece.A phase difference of approximately 90◦ generally occurs at the resonance frequency to becontrolled, see Fig. 3.7. To obtain a practical estimation of the forward path, the forwardpath must normally be estimated offline, i.e. the boring bar is not in contact with theworkpiece. With a phase error of approximately 90◦ in the forward path estimate thefeedback filtered-x LMS will not converge [19, 22], and adaptive control of boring bar

51

vibration is not possible. This problem may be reduced by using a forward path esti-mate with an increased 180◦ phase-change frequency interval at the offline forward pathresonance frequency. A short FIR filter offline estimate of the forward path is likely toproduce sufficient phase accuracy for the adaptive control of bar vibration, see Fig. 3.7.

Three active boring bar designs were developed for the experiments. They were allbased on a standard boring bar WIDAX S40T PDUNR15 with a diameter of 40 mmand an overall length of 300 mm; 200 mm constitutes the overhang part. The differencebetween the active boring bars is the actuator offset angle α. The boring bar with a 15◦

actuator offset angle demonstrated a significant deterioration in dynamic stiffness in theturning operation as compared to an unmodified standard boring bar. This reductionin dynamic stiffness was not observed with the other two designs. The active boring barwith a 0◦ actuator offset angle produced the highest material removal rate with maintainedvibration control. This might be seen by comparing the active control results in terms ofvibration spectra and cutting data given by Figs. 3.10 and 3.11 produced using the activeboring bars with 0◦ and 30◦ actuator offset angles.

By using the filtered-x LMS algorithm it is possible to attenuate the boring bar vi-bration level by up to 45 dB, see Fig. 3.12. The adaptive Internal Model Control (IMC)controller makes it possible to reduce the boring bar vibration level by up to 40 dB, seeFig. 3.13. A reduction of the vibration level by approximately 35 dB was attained usinga P controller, see Fig 3.14. The spectral pattern of severe boring bar vibration is usuallydominated by a resonance peak and several of its harmonics, see Fig. 3.2 [1]. In theactive control of boring bar vibration, the dominant first resonance peak was attenuatedconsiderably and its harmonics were almost completely removed, see Figs 3.12-3.14. Al-though the vibrations are controlled in the cutting speed direction, the vibration level inthe cutting depth direction is also reduced significantly, see Figs 3.12-3.14. In contrastto the adaptive feedback controllers, the P controller is time invariant with a manuallyadjustable gain factor K. Thus, for each P controller gain setting K, a fix loop gain isobtained. The boring bar vibration attenuation achieved is related to the P controllergain, see Fig. 3.20. By comparing the boring bar vibration control results obtained withthe adaptive controllers (see Figs 3.12 and 3.13) and the P controller (see Fig. 3.14) it isclear that there is an obvious difference in the attenuation of the vibration at the reso-nance frequency. Unlike the P controller, the adaptive controllers will adjust the loop gainmore selectively to de-correlate the residual boring bar vibration with the controller refer-ence signal, thus producing a higher attenuation of boring bar vibration at the resonancefrequency.

The robustness of the adaptive feedback controllers may be increased by incorporatinga leakage factor into the algorithm for the adaptive update of the filter weights. The useof leakage in the adaptive controllers reduces the loop gain of the adaptive control systemand increases the distance between the trajectory of the open loop frequency responseand the point (-1,0), see Figs. 3.21-3.28. Observe that the polar plot of the open loopfrequency response based on the feedback filtered-x LMS algorithm without leakage andan offline frequency function estimate of the forward path in Fig. 3.21 encircles the criticalpoint (-1,0). This suggests, however, that the active control of boring bar vibration using

52

the feedback filtered-x LMS algorithm without leakage was not stable. It follows fromthe offline and online estimates of the forward paths and the frequency response functionshown in Fig. 3.7 that the offline estimate of the forward path is likely to differ with theactual forward path during continuous turning. Thus, the polar plot of the open loopfrequency response based on the feedback filtered-x LMS algorithm without leakage andan online frequency function estimate of the forward path in Fig. 3.23 does not encirclethe critical point (-1, 0). This indicates stable active control of boring bar vibrationwhen using the feedback filtered-x LMS algorithm without leakage, which it actually wasaccording to experimental results.

The introduction of a leakage factor in the weight adjustment algorithm introducesbias in the coefficient vector (see Fig.3.6) and in this way causes a somewhat reducedattenuation of the tool-vibration. As expected, the introduction of leakage reduced theperformance of the adaptive control (see Figs. 3.29 and 3.30). Using a leakage of γ =0.9999 still results, however, in a good vibration reduction. At the first resonance peak, itis around 35dB both with and without IMC, and the harmonics are still almost completelyremoved, see Figs 3.29 and 3.30. The cost of adding more leakage to the algorithms interms of reduced attenuation of the vibration seems to be higher than the resulting increasein robustness (see Figs. 3.21, 3.23 and 3.29 and Figs. 3.25, 3.27 and 3.30).

The loop gains based on the online forward path estimate for the feedback controlsystem obtained for the three different algorithms is not, however, consistent with thecorresponding boring bar vibration attenuation. This might be seen by comparing theattenuation achieved with the three algorithms (see Figs. 3.12 - 3.14) with the magnitudeof the corresponding open loop frequency responses shown both as Nyquist and magnitudefunction plots in Figs. 3.18, 3.19, 3.23, 3.24, 3.27 and 3.28. The three control algorithmswould thus be expected to produce considerably larger loop gains.

The online frequency response function estimate of the forward path used to formthe open loop frequency responses was produced by turning characterized by a negligibleboring bar vibration level. The response of the boring bar has non-linear properties, andduring continuous cutting with severe boring bar vibration levels the non-linear propertiesare pronounced. Both the online and the offline frequency response function estimates ofthe forward path in Fig. 3.7 a) indicates a fundamental forward path resonance frequencybelow 500 Hz, while the power spectral densities for severe boring bar vibration shown ine.g. Figs. 3.12- 3.14 indicates that the fundamental resonance frequency of the forwardpath is above 500 Hz. Thus, it is likely that the dynamic properties of the forwardpath vary between low and high boring bar vibration levels. This might explain theinconsistency between the estimated loop gains of the feedback control system and thecorresponding boring bar vibration attenuation.

53

Bibliography

[1] L. Andren, L. Hakansson, A. Brandt, I. Claesson, ”Identification of Dynamic Prop-erties of Boring Bar Vibrations in a Continuous Boring Operation”, Journal of Me-chanical Systems & Signal Processing, vol. 18, no. 4, pp 869-901, 2004

[2] L. Andren, L. Hakansson, A. Brandt, I. Claesson, ”Identification of Motion of Cut-ting Tool Vibration in a Continuous Boring Operation - Correlation to StructuralProperties”, Journal of Mechanical Systems & Signal Processing, vol. 18, no. 4, pp903-927, 2004

[3] S. G. Tewani, T. C. Switzer, B. L. Walcott, K. E. Rouch, ”Active Control of MachineTool Chatter for Boring Bar: Experimental Results”, DE-Vol. 61, Vibration Controlof Mechanical Systems, ASME 1993

[4] Linus Pettersson, Lars Hakansson, Ingvar Claesson and Sven Olsson, ”Active Con-trol of Boring Bar Vibration in Cutting Operations”, Invited paper, InternationalConference on Sound and Vibration 9, Orlando, Florida, USA, 8-11 July, 2002

[5] Linus Pettersson, Lars Hakansson, Ingvar Claesson and Sven Olsson, ”Active Controlof Internal Turning Operations Using a Boring Bar”, Accepted paper, 9th Interna-tional Conference on Mechatronics and Machine Vision in Practive, Chiang Mai,Thailand, 10-12 September, 2002

[6] Linus Pettersson, Lars Hakansson, Ingvar Claesson and Sven Olsson, ”Active Con-trol of Boring Bar Vibration in Cutting Operations”, Invited paper, InternationalConference on Sound and Vibration 9, Orlando, Florida, USA, 8-11 July, 2002

[7] J. Redmond, P. Barney, D. Smith, ”Development of an active boring bar for increasedchatter immunity”, SPIE Vol. 3044

[8] D. R. Browning, I. Golioto, N. B. Thompson, ”Active chatter control system forlong-overhang boring bars”, SPIE Vol.3044

[9] Klein R G, Nachtigal C L, ”The Application of Active Control to Improve BoringBar Performance”, Journal of Dynamic Systems, Measurements and Control, Trans.ASME, Vol. 97, p.179, 1975

54

[10] D. R. Morgan, ”An analysis of multiple correlation cancellation loops with a filterin the auxiliary path”, IEEE Trans. Acoust., Speech, Signal Processing, ASSP-28,454-467, Aug. 1980

[11] B. Widrow, D. Shur and S. Shaffer, ”On adaptive inverse control”, in Proc. 15thAsilomar Conf. 1981, pp. 185-189

[12] J. C. Burgess, ”Active adaptive sound control in a duct: A computer simulation”, J.Acoust. Soc. Am., 70, 715-726, Sept 1981

[13] L. Andren, L. Hakansson, I. Claesson, ”Active Control of Machine Tool Vibrationsin External Turning Operations”, Journal of Engineering Manufacture, Vol. 217 NoB6, 2003

[14] P. D. Welch, ”The Use of Fast Fourier Transform for the Estimation of Power Spectra:A Method Based on Time Averaging Over Short, Modified Periodograms”, IEEETransactions on audio and electroaucustics, June 1967

[15] J. S. Bendat, A. G. Piersol, ”Random Data: Analysis and Measurement Procedures”,3rd ed, Wiley Interscience, 2000

[16] J. S. Bendat, A. G. Persol, ”Engineering Applications of Corelation and SpectralAnalysis”, 2nd ed, Wiley Interscience

[17] Gene F. Franklin, J. David Powell and Abbas Emami-Naeini, ”Feedback Control ofDynamic Systems”, Addison-Wesley, 1994

[18] Simon Haykin, ”Adaptive Filter Theory”, 4th ed, Prentice Hall, 2002

[19] Sen M. Kuo, Dennis R. Morgan, ”Active Noise Control Systems”, Wiley Interscience,1996

[20] Peter M. Clarkson, ”Optimal and Adaptive Signal Processing”, CRC Press, 1993

[21] Monson H. Hayes, Statistical Digital Signal ”Processing and Modelling”, John Wiley,1996

[22] Stephen Elliott, ”Signal Processing for Active Control”, Academic Press, 2001

[23] Otto Myer, ”The Origins of Feedback Control”, M.I.T. Press, 1970

[24] Karl J. Astrom, Bjorn Wittenmark, ”Computer Controlled Systems - Theory andDesign”, 3rd ed, Prentice Hall, 1997

[25] I. Claesson, L. Hakansson, ”Adaptive Active Control of Machine-Tool Vibration ina Lathe”, International Journal of Acoustic and Vibration, Vol. 3, No. 4, 1998, pp155-162

55

[26] S. J. Elliott, P. A. Nelson, ”Active Noise Control”, IEEE Signal Processing Magazine,pp 12-35, October 1993

[27] P-O. H. Sturesson, L. Hakansson, and I. Claesson, ”Identification of the statisticalproperties of the cutting tool vibration in a continuous turning operation - correlationto structural properties”, Journal of Mechanical Systems and Signal Processing, 1997,11(3), 459-489

[28] B. Widrow, S. D. Stearns, ”Adaptive Signal Processing”, Prentice Hall, 1985

[29] A. V. Oppenheim, A. S. Willsky, ”Signal & Systems”, 2nd ed, Prentice Hall, 1997

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Active Vibration Control of Boring Bar VibrationsLinus Andrén, Lars Håkansson

ISSN 1103-1581ISRN BTH-RES--07/04--SE

Copyright © 2004 by the individual authorsAll rights reserved

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