Vibrations in Metal Cutting Measurement, Analysis and Reduction

Linus Pettersson

Ronneby, March 2002 Department of Telecommunications and Signal Processing Blekinge Institute of Technology 372 25 Ronneby, Sweden c Linus Pettersson

Licentiate Dissertation Series No. 01/02 ISSN 1650-2140 ISBN 91-7295-008-0

Published 2002 Printed by Kaserntryckeriet AB Karlskrona 2002 Sweden v

Abstract

Vibration and noise in metal cutting are ubiquitous problems in the workshop. The turning operation is one kind of metal cutting that exhibits vibration related problems. Today the industry aims at smaller tolerances in surface finish. Harder regulations in terms of the noise levels in the operator environment are also central. One step towards a solution to the noise and vibration problems is to investigate what kind of vibrations that are present in a turning operation. The vibrations in a boring operation have been put under scrutiny in the first part of this thesis. Analytical models have been compared with experimental results and the vibration pattern has been determined. The second part of the thesis deals with active vibration control in external turning operations. By embedding a piezo-ceramic actuator and an accelerometer into a tool holder it was possible to obtain a solution that can be fitted in a standard lathe. The control system consists of the active tool holder, a control system based on the filtered-X LMS algorithm and an amplifier designed for capacitive loads. The vibration level using this technique can be reduced by as much as 40 dB during an external turning operation.

vii

Preface

The work presented in this licentiate thesis has been performed at the department of Telecommunications and Signal Processing at Blekinge Institute of Technology. This licentiate thesis summarizes my work within the ﬁeld of vibration measurement, analysis and control. It consists of two parts, which are based on one research report, two submitted articles and one accepted conference paper and the parts are Part I Vibration Analysis of a Boring Bar. Part II Active Control of Machine-Tool Vibration in a CNC Lathe Based on an Active Tool Holder Shank with Embedded Piezo Ceramic Actuators.

Acknowledgements

I am indebted to Lars H˚akansson, my nearest coworker, for the help and support in my research. Lars can be anything from a good friend to a nagging supervisor. I would also like to pay my gratitude to professor Ingvar Claesson for inspiration and guidance thruoghout my studies. I would like to thank all my colleagues at the Department of Telecommunica- tions and Signal Processing. They have helped me in several ways and without them my time at the department would have been more (of) boring. Many thanks to my family, for their support and encouragement. Finally I would like to express all my love to my ﬁanc´ee Cecilia, who helped me in many ways, especially in relaxing from work during the weekends.

Linus Pettersson Ronneby, March 2002

Contents

Publication list ...... 13

Introduction ...... 15

Part

I VibrationAnalysisofa BoringBar ...... 21-88

II Active Control of Machine-Tool Vibration in a CNC Lathe Based on an Active Tool Holder Shank with Embedded Piezo CeramicActuators ...... 89-101

Publication list

Part I is published as:

L. Pettersson, “Vibration Analysis of a Boring Bar” ResearchReport, ISSN 1103- 1581, Feb. 2002.

Parts of this research report has been revised and submitted for publication as:

L. Pettersson, L. H˚akansson A. Brandt and I. Claesson, “Identiﬁcation of Dy- namic Properties of Boring Bar Vibrations in a Continuous Boring Operation”, submitted to Mechanical Systems & Signal Processing, Dec. 2001.

L. Pettersson, L. H˚akansson A. Brandt and I. Claesson, “Analytical and Experi- mental Investigation of the Modal Properties of a Clamped Boring Bar”, submitted to Mechanical Systems & Signal Processing, March 2002.

Part II is published as:

Introduction

In turning operations the cutting tool is subjected to a dynamic excitation due to the deformation of work material during the cutting operation. The relative dynamic motion between the cutting tool and the workpiece will aﬀect the result of the machining, in particular the surface ﬁnish. Thus vibration related problems are of great interest in turning operations.

Introduction to the Lathe

The lathe is a very useful and versatile machine in the workshop, and is capable of performing a wide range of machining operations. The workpiece is held by a chuck in one end and when possible also by a tailstock at the opposite end. The chuck is mounted on a headstock, which incorporates the engine and gear mechanism. The chuck is holding the workpiece with three or four jaws and a spindle engine causes the chuck and workpiece to rotate. A tool-post is found between the headstock and tailstock, which holds the cutting tool. The tool-post stands on a cross-slide that enables it to move along the workpiece. An ordinary lathe can accommodate only one cutting tool at the time, but a turret lathe is capable of holding several cutting tools on a revolving turret. Two common types of turning operations are external longitudinal turning operations and boring operations. Bothturning operations are usually possible in general purpose lathes. External longitudinal turning operations are performed on the outside of a workpiece and the cutting tool is mounted on a tool holder shank. Boring operations are performed in pre-drilled holes in the workpiece, i.e. inside the workpiece, and the cutting tool is mounted on a boring bar.

PART I - Vibration Analysis of a Boring Bar

In internal turning or boring operations, vibration is a problem. The industries are having problems performing specific boring operations. The vibrations involved during the cutting operation influence the surface finish and the manufacturers are having problems with small tolerances in boring operations. When cutting in pre- drilled holes the cross sectional area of the boring bar is limited. Since a general boring bar is long and slender it is sensitive to external excitation and thereby inclined to vibrate. A thorough investigation of the vibrations involved in boring operations is therefore needed. Part I of this thesis scrutinizes the vibrations in boring operations. A solid foundation was achieved from both theoretical and experimental methods in order to analyze the vibrations involved. The theoretical methods derive from 16 knowledge of the dimensions of the system and its suspension or boundary conditions. The experimental methods are all derived from analysis of data acquired from accelerometers and force transducers mounted on the boring bar.

Evolution in Active Vibration Control in Turning Operations

A project in active vibration control in external turning operations was initiated in 1997. It concerned external longitudinal turning operations and was inspired by a project at Dept. of Mechanical Engineering, Lund Institute of Technology, LTH. A working solution was developed where the vibrations were reduced by approximately 40 dB and resulted in a PhD thesisfor Lars H˚ akansson. The magneto- strictive design was however not suitable for industrial purposes thus further improvements were needed. A schematic picture of the ﬁrst test design developed at LTH is presented in Fig. 1.

Figure 1: The ﬁrst working model developed at LTH.

The first magneto-strictive test design was not possible to incorporate into a standard lathe used in the industry without severe modifications. The design also needed improvements in its mean time between failure. A solution was found in 17 piezo-ceramic actuators. The first version of the new generation of active tool holders was based on piezo plates glued on the surface of the tool holder as in Fig 2. It had good potentials, but unfortunately the actuators were unsufficient for the forces in a cutting operation.

Figure 2: The ﬁrst attempt on an active tool holder solution using piezo ceramic technique.

PCB piezotronics who was also involved in developing the active tool holder with piezo plates was contacted to develop a more powerful active tool holder. The unsatisfactory result is presented in Fig. 3. This tool holder was to weak, both in structure and actuator and the mean time between failure was far from suﬃcient.

Figure 3: The PCB piezotronics active tool hoder solution using a piezo ceramic actuator.

The demand of having embedded actuators was abandoned at this stage. Fig. 4 shows the first external solution based on piezo ceramic actuators that was developed at BTH. This solution was possible to incorporate into the lathe used in the experiments without modifications. The experiments that were carried out showed that the vibrations were reduced by approximately 40 dB. The surface finish was also improved significantly. 18

Cutting tool

Tool holder shank Actuator

Figure 4: The ﬁrst solution developed at BTH using a piezo ceramic actuator mounted outside the tool holder.

Now, it was time to regain the demand of having the actuator embedded in the design. The used actuator in the previous design was small enough to ﬁt in a modiﬁed standard tool holder. By embedding the actuator into a standard tool holder enables the active vibration control technique to be used in a standard lathe in the industry. An accelerometer was also embedded in this design. A CAD model of the embedded design is presented in Fig. 5.

Accelerometer

Cutting tool

Tool holder shank

Embedded and sealed piezo ceramic actuator

Figure 5: A standard tool holder with an embedded piezo ceramic actuator.

The tool holder design has been improved by incorporating a socket into the design in cooperation withActive Control Sweden AB, a company trying to develop and market the active control technique in turning operations. The socket connects the power to the actuator and the acceleration signal from the accelerometer with 19 the ampliﬁer and control system. The result is called ActiCutTM and is presented in Fig. 6.

Figure 6: The active tool holder called ActiCutTM withan embedded piezo ceramic actuator and accelerometer.

Part II - Active Control of Machine-Tool Vibra- tion in a CNC Lathe Based on an Active Tool Holder Shank with Embedded Piezo Ceramic Ac- tuators

In the turning operation chatter or vibration is a common problem affecting the result of the machining, and, in particular, the surface finish. Tool life is also influenced by vibration. Severe acoustic noise in the working environment frequently occurs as a result of dynamic motion between the cutting tool and the workpiece. These problems can be reduced by active control of machine-tool vibration. How- ever, machine-tool vibration control systems are usually not applicable to a general lathe and turning operation. The physical features and properties of the mechanical constructions or solutions involved regarding the introduction of secondary vibration usually limit their applicability. An adaptive active control solution for a general lathe application has been developed. It is based on a standard industry tool holder shank with an embedded piezo ceramic actuator and an adaptive feedback controller. The adaptive controller is based on the well known filtered-x LMS-algorithm.

Part I

Vibration Analysis of a Boring Bar Part I is published as:

L. Pettersson, “Vibration Analysis of a Boring Bar” ResearchReport, ISSN 1103- 1581, Feb. 2002.

Parts of this research report has been revised and submitted for publication as:

Linus Pettersson

Department of Telecommunications and Signal Processing, Blekinge Institute of Technology 372 25 Ronneby Sweden

Abstract

In boring operations the vibrations are a cumbersome part of the manufacturing process. This paper puts the vibrations under scrutiny and the vibrations were measured both in the cutting speed direction and the cutting depth direction. The cutting process seems to be a time varying process and contains nonstationary as well as nonlinear parameters that are not under control. The experiments showed that the vibrations were usually dominated by the first resonance frequency in either of the two directions of the boring bar. Sample probability density estimates of the vibration records points to that the probability densities are varying from sinusiodal probability shape to gaussian shape. They also indicate that shape of the two directions could be of different probability density function from the same vibration record. Stationarity tests show, besides that the processes usually are nonstationary, that it is possible to extract short time stationary segments within a vibration record. The problem with force modulation in rotary machinery, which appears as side band terms is the spectrum, is also addressed. Furthermore, the resonance frequencies of the boring bar are correlated to an approximate theoretical beam model. The theoretical calculation of the boring bar deflection corresponds well with the experimental results from the modal analysis and the operating deflection shape analysis. 24 Part I

1 Introduction

A concern in the manufacturing industry today are the vibrations induced by metal cutting, such as turning, milling and boring operations. Turning operations, and especially boring operations, are facing tough vibration related problems. To reduce the problem of vibration extra care must be taken in the production planning and preparation regarding the machining of a workpiece in order to obtain a desired shape and tolerance. Thus, the vibration problem in metal cutting has a considerable influence on important factors such as productivity, production costs, etc. A thorough investigation of the vibrations involved is therefore an important step in challenging this problem. A schematic picture of a cutting operation using a boring bar can be seen in Fig. 1. The actual cutting is performed at the cutting tool mounted at the tip of the boring bar. During a cutting operation the boring bar is fed in the feed direction at a specific cutting depth and a specific rotational speed of the workpiece. The vibration of the boring bar is influenced by three parameters, feed rate, cutting depth and cutting speed. The vibration in the boring bar are in the cutting speed and the cutting depth direction.

The cutting speed direction

The cutting depth direction

Boring bar

The feed direction

Workpiece The cutting tool

Figure 1: A schematic view of a boring operation

During an internal turning operation the cutting tool and the boring bar are subjected to cutting forces due to the relative motion between the tool and workpiece in the cutting speed direction and in the feed direction. A desire of being able to perform a cutting operation into pre-drilled holes in a workpiece limits the Vibration Analysis of a Boring Bar 25 diameter or cross sectional size of the boring bar. Usually a boring bar is com- paratively long and slender, and is thereby more sensitive to excitation forces. In boring operations the cutting tool is placed on a boring bar. Since the boring bar is the weakest link in the boring bar - clamping system of the lathe, this is where the vibrations will of major concern. The boring bar motion may vary with time. The dynamic motion originates from the deformation process of the work material. The motion of the boring bar or vibrations will affect the result of the machining, and the surface finish in particular. The tool life is also likely to be influenced by the vibrations. Research in metal cutting is intensive. With new cutting technologies, especially high speed machining, where the cutting force dynamics can be significant and with ”hard-to-cut” materials such as aerospace alloys, there is a need for a comprehensive knowledge of the cutting process. The term ”chatter” is often used instead of vibration in the cutting process. 26 Part I

2 Materials and Methods

Internal cutting operations in a lathe is a complex manufacturing process. In order to achieve a comprehensive knowledge of the cutting process in internal turning many experiments have been carried out. The magnitudes of cutting process parameters are determined from many other parameters; some of them are observable or measurable in experiments. The inﬂuence of a process parameter can only be examined if the other parameters are ﬁxed. However, some parameters show linear dependence and it is therefore important to select the right parameters to study. Still, the number of parameters can be large and the number of experiments then has to be reduced by careful planning of the trials.

2.1 Experimental Setup The cutting trials have been carried out in a Mazak SUPERQUICK TURN- 250M CNC turning center, see Fig. 2, with 18.5 kW spindle power, maximal machining diameter 300 mm and 1007 mm between the centers. In order to save material the cutting operation was performed as external turning operation, although a boring bar, WIDAX S40T PDUNR15, was used.

Figure 2: Mazak SUPERQUICK TURN- 250M CNC lathe used in the experiments.

2.1.1 Measurement Equipment and Setup The measurements can be divided into two diﬀerent categories: vibration analysis of a boring bar and modal analysis of a boring bar. In the latter category an operating deﬂection shape, ODS, analysis is included. Vibration Analysis of a Boring Bar 27

In the vibration measurement the following measurement equipment have been used:

• 2 PCB U353B11 Accelerometers

• TEAC RD-200T DAT-recorder

The two accelerometers were mounted as close as possible to the cutting tool - one in the cutting speed direction and one in the cutting depth direction as in Fig. 3. The accelerometers were mounted on the boring bar using threaded studs.

Figure 3: Vibration measurement test setup.

The data collection for the modal analysis was performed using the following measurement equipment:

• 14 PCB 333A32 Accelerometers

• OSC Audio power ampliﬁer, USA 850

• Ling Dynamic Systems shaker v201

• Br¨uel & Kjær 8001 Impedance head

• Br¨uel & Kjær NEXUS conditioning ampliﬁer 2692

• HP VXI E1432 Front-end data acquisition unit

• PC with IDEAS Master Series version 6

In total, 7 accelerometers were glued equidistantly on the boring bar in the cutting speed direction and 7 in the cutting depth direction, see Fig. 4. In order to ex- cite the boring bar in both directions simultaneously the force was applied via an 28 Part I

Figure 4: Modal analysis test setup. impedance head at 45◦ angle from the cutting speed and the cutting depth directions, see Fig. 4. The covering of the cables was not critical in this measurement setup. The ODS data were collected using less sensitive accelerometers compared to the accelerometers used in the modal analysis measurement. The boring bar vibration was measured during continuous cutting operations and the data were collected using

• 14 PCB U353B11 Accelerometers

• HP VXI E1432 Front-end data acquisition unit

• PC with IDEAS Master Series version 6

As with the modal analysis the 14 accelerometers were glued on the boring bar, 7 in each direction with equidistant spaces, see Fig. 5. In both the modal analysis and the operating deﬂection shape measurement setups, the accelerometers were numbered starting at the position closest to the cutting tool as number 1 and ending at the position closest to the clamped end as number 7. The distance between adjacent accelerometers was 25 mm and the distance between the clamped end and the nearest accelerometer was also 25 mm, thus the accelerometers were mounted at 25 mm, 50 mm, ..., 175 mm distance from the clamped end. Since IDEAS is using X, Y and Z directions, the cutting speed direction and the cutting depth direction were deﬁned as Y and Z respectively leaving X as the feed direction, see Fig. 6. Vibration Analysis of a Boring Bar 29

Figure 5: Operating deﬂection shape test setup.

Z2 Z3 Z4 Z5 Z6 Z7

Y2 Y3 Y4 Y5 Y6 Y7

Figure 6: Accelerometer placement in the modal analysis and the operating deﬂec- tion shape measurements. 30 Part I

2.1.2 Work Materials In the cutting experiments, three different workpiece materials have been used, • SS 0727-02, nodular graphite cast iron • SS 2343-02, austenitic stainless steel • SS 2541-03, chromium molybdenum nickel steel The materials have different properties from a production point of view. The machinability, or more specifically the cuttability, of the materials differ and the chemical composition of the three materials, which is shown in Table 1, is also different. The diameter of the workpiece materials is chosen large (>200 mm). Thus, the workpiece vibrations may be neglected. Work materials are usually classified according to the three production engineering application classes: P (alloyed steels), M (stainless steels) and K (cast irons), all standardized by ISO. For the experiments, one material was selected from each class. Since the statistical properties of the cutting forces are to be investigated, the influence of the workpiece material are vital. The chemical composition and

Production Engineering Swedish American C Si Mn Cr Ni Mo Application Class Standard Standard (%) (%) (%) (%) (%) (%) K SS 0727-02 AISI 80-55-06 3.7 2.2 0.4 M SS 2343-02 AISI 316 0.05 18 12 2.7 P SS 2541-03 AISI 4340 0.36 0.27 0.62 1.53 1.41 0.17

Table 1: The composition of the workpiece materials. micro-structure of the materials as well as the strength and the thermo-dynamical properties determine the behavior of the cutting process. The optimal workpiece material should, from a production engineering point of view, induce small cutting forces, be capable of producing a proper surface ﬁnish, have a fair chip breaking property, and not deteriorate the cutting tool. The selected materials will have diﬀerent properties with respect to these four aspects [1].

2.1.3 Cutting Tool Materials In the cutting experiments, standard 55◦ diagonal inserts have been used. These have tool geometry with the ISO code DNMG 150608-SL with chip breaker geometry for medium roughing. Diﬀerent carbide grades have been used for diﬀerent materials but the geometries were the same. For cast iron and alloyed steel, carbide grade TN7015 was used and for stainless steel the carbide grade was TN8025. Vibration Analysis of a Boring Bar 31

2.1.4 Choice of Cutting Data The selection of the cutting data parameter space must be based on thorough knowledge of the cutting process itself. Excessive wear, catastrophic failure or plastic deformation may result if too high cutting speeds or feed rates are selected. This may lead to results, which are not characteristic to a turning operation during normal circumstances. After a preliminary set of trials the selection was made. The cutting data were chosen according to Table 2. No cutting ﬂuid was applied during machining. The aim was to ﬁnd a parameter space equal for all materials. However, it was found to be impossible to exceed 200 m/min cutting speed for stainless steel, SS 2343-02. Material Parameter Range SS 0727-02 Feed, s (mm/rev) 0.1-0.3 step 0.1 Depth of cut, a (mm) 2 Cutting speed, v (m/min) 50-300 step 25 SS 2343-02 Feed s, (mm/rev) 0.1-0.3 step 0.1 Depth of cut, a (mm) 2 Cutting speed, v (m/min) 50-200 step 25 SS 2541-03 Feed, s (mm/rev) 0.1-0.3 step 0.1 Depth of cut, a (mm) 2 Cutting speed, v (m/min) 50-300 step 25

Table 2: Cutting parameter space for the diﬀerent materials.

2.2 Force Modulation in Rotating Machinery Force modulation is inherent in certain types of mechanical systems and especially in rotating machinery [2, 3]. Rotating parts of mechanical systems introduce dynamic motion related to residual rotor mass imbalance [3, 4]. It is known that rotation speed may vary due to imperfections in gears [2, 3], e.g. variation in tooth spacing, roundness errors, etc. associated with gear wheels [2, 3, 5]. All gear- boxes give a varying output torque over one turn, given a uniform input torque. This causes a dynamic force (torque) by itself. These imperfections of rotating machines are known to introduce force modulation eﬀects into the vibration signature of rotating machinery, e.g. amplitude modulation and/or frequency or phase modulation properties in the vibration signature of the rotating machinery [2, 3]. Based on the fact that rotating parts of mechanical systems introduce dynamic motion related to residual rotor mass unbalance, it is reasonable to assume that 32 Part I dynamic motion related to residual rotor mass unbalance also exists in the spindle - chuck - workpiece system in a lathe. Hence, it is reasonable to believe that the motion of a boring bar is inﬂuenced by force modulation.

2.3 Structural Dynamic Properties of the Boring Bar In a boring operation the boring bar is subjected to dynamic excitation, due to the material deformation process during a cutting operation. This will introduce a time varying deflection of the boring bar. If the frequency of the excitation coincides with one of the natural frequencies of the boring bar, a condition of resonance is encountered. Under such circumstances the vibrations are at a maximum, thus the calculation of the natural frequencies is of major importance in the study of vibrations. There are two major types of vibrations in the boring bar caused by the forces from the cutting process, bending vibrations and torsional vibrations. The force is applied at the cutting tool and the force originates from the chip deformation process during a cutting operation. In order to model the structural dynamic properties of a boring bar, a simple model might be an Euler-Bernoulli beam. Most realistic structural systems are characterized by the ability to support transverse shear as well as having internal stiffness. The Euler-Bernoulli beam model assumes that the deflection of the cen- terline is small and only transverse. While this theory assumes the presence of a transverse shear force, it neglects any shear deformation due to it. Also the rotary inertia is neglected by this model [6, 7]. Thus the boring bar can be modeled as a clamped free slender beam with rigid body as in Fig. 7. By knowing the material and size of the boring bar the Euler-Bernoulli calculations can be used to obtain information regarding the modal properties of the boring bar, i.e. estimates of the eigenfrequencies and corresponding mode shapes [6, 7, 8]. When the beam is vibrating transversely, the dynamic equilibrium condition for forces in the y direction acting on an infinite small part of the beam, dx,isas illustrated in Fig. 7, ∂V (x, t) ∂2u(x, t) V (x, t)− V (x, t)+ dx −ρAdx = f(x, t)dx (1) ∂x ∂t2 where V (x, t) is the shear force, u(x, t) is the beam deflection in the y direction, ρ is the density of the material of the boring bar, A is the cross-sectional area of the beam in the y-z plane and f(x, t) is the total external force applied to the beam per unit length. The bending moment, M(x, t), acting on an element dx at a distance x from Vibration Analysis of a Boring Bar 33

l x dx

∂2 ρAdx u(x,t) ∂M(x,t) ∂ 2 M(x,t) + dx M(x,t) t ∂x y

x ∂V(x,t) V(x,t)+ dx z V(x,t) ∂x

Figure 7: Euler-Bernoulli beam model the clamped end around the z-axis is ∂M(x, t) ∂V (x, t) dx M(x, t)+ dx −M(x, t)− V (x, t)+ dx dx−f(x, t)dx =0, ∂x ∂x 2 (2) since dx is small (dx)2 can be assumed to be zero which simpliﬁes the equation 2 to ∂M(x, t) = V (x, t). (3) ∂x From the theory of mechanics of material, the beam is subjected to a bending moment which is related to the beam deﬂection by ∂2u(x, t) M(x, t)=EI(x) (4) ∂x2 where E is young’s elastic modulus and I(x) is the cross sectional area moment of inertia around the z-axis. I(x) is assumed to be constant along the boring bar. If no external force is applied i.e. f(x, t) = 0 and by using the relations in equations 3 and 4 in equation 1, the free vibration is governed by ∂2u(x, t) EI ∂4u(x, t) + =0. (5) ∂t2 ρA ∂x4 34 Part I

To be able to solve the spatial part of equation 5 four equations are needed and those are found in the boundary conditions of a clamped free beam. At the clamped end, x =0, u(x, t)|x=0, the deﬂection and slope of the boring bar are zero and consequently the boundary conditions at the clamped end are given by | u(x, t)x=0 =0, (6) ∂u(x, t) =0 (7) ∂x x=0 and at the free end, x = l, u(x, t)|x=l, there is no bending moment nor shear force which results in the following boundary conditions 2 ∂ u(x, t) EI(x) 2 =0, (8) ∂x x=l 2 ∂ ∂ u(x, t) EI(x) 2 =0. (9) ∂x ∂x x=l

Assuming that the time and space dependent deﬂection u(x, t) can be split up as u(x, t)=u(x)u(t) and using this substitution in equation 5 will yield

∂2u(x)u(t) EI ∂4u(x)u(t) ∂2u(t) 1 EI ∂4u(x) 1 + =0⇒ + =0. (10) ∂t2 ρA ∂x4 ∂t2 u(t) ρA ∂x4 u(x)

The time dependent deﬂection, u(t), which is dependent of the natural or resonance frequencies of a beam is given by [7]

2 ∂ u(t) 1 2 u(t)=A sin(2πfrt)+B cos(2πfrt) ⇒ = −(2πfr) . (11) ∂t2 u(t) where fr is the resonance frequency for mode r. By using the above result, equation 10 can be rewritten as ∂4u(x) − β4u(x) = 0 (12) ∂x4 where β4 is

2 ρA(2πfr) β4 = . (13) EI A general solution to equation 12 is [7]

u(x)=a1 sin(βx)+a2 cos(βx)+a3 sinh(βx)+a4 cosh(βx). (14) Vibration Analysis of a Boring Bar 35

Applying the boundary conditions to equation 14 will form the following four equations | u(x)x=0 =0=a2 + a4 (15) ∂u(x) =0=β(a1 + a3) (16) ∂x x=0 2 ∂ u(x) 2 − − 2 =0=β a1 sin(βl) a2 cos(βl)+a3 sinh(βl)+a4 cosh(βl) (17) ∂x x=l 3 ∂ u(x) 3 − 3 =0=β a1 cos(βl)+a2 sin(βl)+a3 cosh(βl)+a4 sinh(βl) (18) ∂x x=l Setting up the boundary conditions to equation 14 in matrix form will yield      0101a1 0       β 0 β 0  a2   0   2 2 2 2   =   (19) −β sin(βl) −β cos(βl) β sinh(βl) β cosh(βl) a3 0 3 3 3 3 −β cos(βl) β sin(βl) β cosh(βl) β sinh(βl) a4 0

T The vector [a1,a2,a3,a4] has a nonzero solution if the determinant of the matrix is zero. Solving the determinant equal to zero yields the characteristic equation cos(βl)cosh(βl)=−1 [7]. The solution for the characteristic equation yields a βr value for each mode, r, and the ﬁrst three values are

β = {1.8752/l, 4.6941/l, 7.8548/l}. (20)

Based on this information, the mode shape function φr(x) for the rth resonance frequency fr becomes cosh(βrl)− cos(βrl) φr(x)= sinh(βrx)−sin(βrx) −cosh(βrx)+cos(βrx), sinh(βrl)− sin(βrl) (21) To be able to calculate the resonance frequencies and corresponding mode shapes the cross sectional dimensions of the boring bar are needed in order to obtain the cross-sectional area moment of inertia. The cross section of the boring bar is given in Fig. 8 The values used in the Euler-Bernoulli calculation can be found in Table 3.

2.4 Experimental Modal Analysis The study of vibrations mainly consists of two approaches; theoretical and experimental approach. The theoretical approach i.e. analytical models and numerical ﬁ- nite element models are starting from the knowledge of the geometry of a structure, 36 Part I

15 mm

Cutting speed direction

40 mm

Cutting depth 18.5 mm direction

15 mm

Figure 8: The cross section of the boring bar used in the experiments. The cutting speed direction corresponds to the y direction and the cutting depth direction corresponds to the z direction in Fig. 7.

Variable Value Unit Cutting speed Cutting depth direction direction E 206 · 109 N/m2 l 0.2 m I 1.0958 · 10−7 1.1958 · 10−7 m4 ρ 7800 kg/m3 A 1.2386 · 10−3 m2

Table 3: Values used in the theoretical Euler-Bernoulli calculations. material properties such as mass and damping distribution and elastic modulus. Solving a theoretical model, the result is based on mathematical models deriving from perfect structures and conditions. The experimental approach starts from measurements of dynamic inputs forces and output responses. Measuring forces and responses usually includes extra masses applied to the structure such as accelerometers and force transducers and other reality challenges. Experimental modal analysis is a powerful tool for the derivation of reliable models representing dynamic properties of structures. It is the process of determining the modal parameters, i.e. the natural frequencies, the damping ratios, the mode shapes, and the modal scaling of linear time invariant systems based on estimates obtained from experimental measurements [9, 10, 11]. Every deformable structure can be considered as a combination of an infinite Vibration Analysis of a Boring Bar 37 number of small rigid body masses, hence all structures have an infinite degrees of freedom. However, a structure can be approximated as a limited number of these masses or limited degrees of freedom. This number defines the dimensions of the analytical mass, damping and stiffness matrices and the number of theoretically present natural frequencies and mode shapes. An N degree-of-freedom system describing the motion of a boring bar can be written as [M]{u¨}+[C]{u˙ }+[K]{u} = {f(t)}, (22) where [M]istheN × N mass matrix, [C]istheN × N damping matrix, [K]isthe N × N elastic stiffness matrix, {u} the displacement vector, and {f(t)} the space and time dependent load vector. The experimental modal analysis of a system, described by an N degree-of- freedom system, usually relies on the simultaneous measurements of a sub set or sub sets of the forces and responses at the N degrees of freedom. Based on the time records either parts of the receptance matrix or the corresponding impulse response matrix are estimated [9, 10]. The receptance matrix for the N degree-of-freedom system is defined by −1 [H(ω)] = [M](jω)2 +[C](jω)+[K] , (23) the elements Hpq(ω) of the receptance matrix [H(ω)] are the frequency response function between the input force q and the response p according to   H11(ω) H12(ω) ... H1N (ω)    H21(ω) H22(ω) ... H2N (ω)  Xp(ω) [H(ω)] =  . . . .  where Hpq(ω)=  . . .. .  Fq(ω) HN1(ω) HN2(ω) ... HNN(ω) (24) and where N is the number of measurements points or degrees of freedom i.e. the maximum of the number of inputs and the number of outputs. The frequency response function Hpq(ω) is related to force applied at q, Fq, and measured response at p, Xp. The frequency response function of a SDOF system can be written as 1/m H(ω)= 2 2 , (25) (jω) +2jζω0ω + ω0 where m is the mass, ω0 =2πf0 and f0 denotes the undamped system natural frequency and ζ denotes the relative damping. ω0 and ζ are defined as k c ω0 =2πf0 = ,ζ= √ . m 2 mk 38 Part I where k is the stiffness and c is the damping. The roots of the denominator of equation 25 are λ, λ∗, which implies that H(ω) can be factorized into 2 terms using partial fraction expansion as 1/m A A∗ H(ω)= = + . (26) (jω − λ)(jω − λ∗) jω − λ jω − λ∗ As in the SDOF case all elements of [H(ω)] can be decomposed using partial fraction expansion. The receptance matrix for a more complex case is illustrated as a 2DOF system as   2 ∗ 2 ∗ A11r A A12r A  11r 12r  + ∗ + ∗  jω − λr jω − λr jω − λr jω − λr   r=1 r=1  H(ω)= 2 ∗ 2 ∗  (27)  A21r A A22r A  + 21r + 22r − − ∗ − − ∗ r=1 jω λr jω λr r=1 jω λr jω λr where A1kr,A2kr,...,ANkr are the residue for mode r in column k of the receptance matrix, r, k ∈{1, 2,...,N}, in the 2DOF case we have the residue A1kr,A2kr. It is not necessary to measure the complete receptance matrix [H(ω)]. From one roworcolumnin[H(ω)] it is possible to synthesize the unknown rows or columns if the following requirements are met. • A complete row or column must be measured in [H(ω)].

• All frequency responses within a row or column must contain non-zero residue information in the frequency range of interest. Assuming that the above conditions are satisﬁed and that the k:thcolumninthe receptance matrix has been measured it can be represented as          H1k(ω)   A1k    N ∗   H2k(ω) { } { } A2k Ak r Ak r { } . = + ∗ where Ak r = . .  .  (jω − λr) (jω − λr)  .    r=1   HNk(ω) ANk r (28) {Ak}r is the residue for mode r originating from column k in the receptance matrix ∗ and λr,λr are the poles of the receptance matrix. The residue matrix is directly related to the modal vector {ψ}r as { } { } { } { } { }T A r = A1 r A2 r ... AN r = Qr ψ r ψ r , (29) where Qr is a scaling constant. Vibration Analysis of a Boring Bar 39

Measuring the k:th columns of [H(ω)] enables the determination of the k:th column of each mode of the residue matrix. By rewriting equation 29 only for the k:thcolumnofther:th mode yields        A1k   ψ1ψk   ψ1        A2k ψ2ψk ψ2 . = Qr . = Qrψkr . (30)  .   .   .        ANk r ψN ψk r ψN r Since there are one more variable than equations, an additional equation is needed. This equation is found in the modal vector criteria. Three common methods exists and are referred to as unity modal mass, unity modal coeﬃcient and unity modal vector length. Once the modal scaling method have been selected, the scaling factor, Qr,can be determined and all residues can be calculated as

Apqr = Qrψprψqr (31) and the complete receptance matrix can be calculated. To summarize the frequency response function matrix can be written as N T {ψ}r{ψ} H ω r . [ ( )] = 2 − 2 (32) r=1 ωr ω + j2ξωrω where ωr is the undamped natural frequency for mode r. The numerator of equation 32 is related to the spatial solution u(x) and the denominator is related to the temporal solution u(t) in the Euler-Bernoulli calculations. There are several methods for extracting the modal parameters, both in the time and in the frequency domain. The method used to determine the frequency and damping was a method called polyreference and is described in [10]. This method has a good ability to separate two modes that are close together in the frequency plane based on the fact that more than one excitation degree is used [10]. In order to obtain a quality measure on the mode shapes received from an experimental modal analysis the Modal Assurance Criteria, MAC, matrix is usually used [9, 10] This is a way to evaluate the orthogonality between modes in modal analysis and the elements of the MAC matrix are deﬁned as { }T { } 2 ψ iψ j MACij = T T . (33) {ψ}i {ψ}i {ψ}j {ψ}j

If {ψ}i and {ψ}j are estimates of the same mode shape the MACij should equal unity, i.e i = j. All oﬀ-diagonal elements in MACij should approach zero since {ψ}i and {ψ}j are estimates of diﬀerent mode vectors for i = j. 40 Part I

The analysis of the dynamic properties of the boring bar was carried out in a experiment analysis program called I-DEAS test. The parameters used in the modal analysis can be found in Table 4, and were ﬁrst chosen based on experience and later tuned on a trial and error basis. Parameter Value Excitation signal Burst Random (80/20) Frequency range 0 - 1000 Hz Number of spectral lines 1601 Number of averages 100 Window None Modal parameter estimation method polyreference Modal scaling method Unity modal mass Frequency range curve ﬁtting 500 - 700 Hz

Table 4: Parameters used in the modal analysis.

2.5 Evaluation of the Dynamic Motion of the Boring Bar The properties of the boring bar vibration may be investigated by applying the theory of time series analysis. It is obvious that there exists a time dependency, or correlation, in the cutting process. This time dependency is influenced by many parameters. In a continuous boring operation, the influence of different work materials and cutting data on the boring bar vibrations have been studied. When analyzing random data, and interpreting the results, it is very important to use the correct procedures. The procedures are strongly determined by some basic characteristics. Some important characteristics of the measured data are stationarity of the data and presence of periodicities in the data. Stationarity simplifies the analysis procedure since the analysis of nonstationary data generally are more complicated. To avoid erroneous interpretations in the analysis of the sampled data, periodicities must be identified.

2.5.1 Stationarity Test A stationary process is a process having statistical properties that are time invariant. Many data analyzing tools, such as spectrum estimation and sample probability density estimation or histogram, are all working under the assumption that the data are stationary. It is apparent that a cutting process is time dependent but usually a nonstationary process can be divided into segments that can be considered as short time stationary. The tests for stationarity used on the boring bar Vibration Analysis of a Boring Bar 41 vibration records are nonparametric and no assumptions need to be made concerning the probability distribution of the data. The stationarity can be examined by an investigation of a single time record of the boring bar vibration, x(n), as follows [12]; 1. Divide the sample record into N intervals x1, x2,...,xN where the length of each interval is L as xn = x (n−1)L+1 ,x (n−1)L+2 ,...,x (n−1)L+L . (34)

2. Calculate the mean square value for each interval and put these values into a vector L ¯2 ¯2 ¯2 ¯2 ¯2 2 x = {x1, x2,...,xN} where xn = x (n−1)L+l . (35) l=1

3. Test the sequence x¯2 for trends and fluctuations. By testing the mean square value of a time record for stationarity incorporates stationarity tests on both the mean and the variance of the time record. Fur- thermore, it is straightforward to test the stationarity for higher order statistical quantities of a time record using the methods described below [12]. Two common methods have been used on the boring bar vibration records to test the stationarity. The methods are called the run test and the reverse arrangements test. The run test is powerful in detecting fluctuations in the data. The detection is achieved by counting the number of consecutive sequences of identical observations that are followed by a different observation or no observation. An observation is related to the mean square value of an interval and is categorized into two groups, less than the median value of the total amount of N observations or larger than the median value. These observations are called minus (-) and plus (+) respectively. The median value is preferable as it is not sensitive to outliers. To illustrate the definitions of observations and runs consider the following 10 observations categorized as

++ − ++ −−− ++ 1 2 3 4 5 This sequence yields that there are 5 runs in this sequence of 10 observations. The run test produces the number of runs r in the sequence of observations. If the observations are from a stationary random process, the number of runs r should have a distribution with a mean [12], N E(r)= + 1 (36) 2 42 Part I and variance [12],

N(N − 2) V (r)= . (37) 4(N − 1)

The sequence is accepted as stationary according to the run test at a signiﬁcance level α if the total number of runs is in the interval

rn;α/2 x¯m hnm = (41) 0otherwise

A stationary sequence has a mean and variance of the total number of reverse arrangements as [12]

N(N − 1) E(A)= (42) 4 and N(2N +5)(N − 1) V (A)= . (43) 72 Vibration Analysis of a Boring Bar 43

The sequence of observations is accepted as stationary according to the reverse arrangements test at signiﬁcance level α if the total amount of reverse arrangements is in the interval of

AN;α/2

AN;α/2 and AN;1−α/2 are the upper and lower bound for the reverse arrangement test to be accepted as stationary at significance level α. These values can be found in various literature covering statistics e.g. [12]. The reverse arrangements test is more powerful for detecting monotonic trends than the run test. On the other hand the run test is more powerful in detecting fluctuations than the reverse arrangements test. If both stationarity tests indicates that the sequence of N observations is stationary, the original process can be considered stationary. The significance level in the stationarity tests was α =0.05.

2.5.2 Mean Square Value Properties Analysis of individual nonstationary time history records usually include a time averaging operation. The mean square value of a nonstationary time record may be estimated by using a moving average procedure. In e.g. stationarity tests it is not uncommon that the same time record can be classiﬁed as both stationary and nonstationary depending on the length of the averaging time. By using diﬀerent lengths of the moving average it is possible to gain information about the time varying properties of the mean square value or power of the time record. Usually it is desirable to follow an underlying curve and smooth out the variations in the mean square value. One approach in selecting an average time is a trial and error based method [12, 13], as follows 1. Compute a moving average for the mean square value of the nonstationary data with an averaging time, Ta, that is too short to smooth out the variations in the mean square value versus time. 2. Continuously recompute the moving average with an increasing averaging time until it is clear that the averaging time is smoothing out variations in the mean square value versus time.

2.5.3 Sample Probability Density Estimate The sample probability density estimate shows the distribution of a sampled time record in terms of amplitude values. A histogram, which is of frequent occurance, is an unscaled version of the sample probability density estimate. Both quantities reveals a lot about the process. By analyzing the shape of either a histogram or a sampled probability density estimate the kind of distribution can be established. 44 Part I

Other statistical measures, such as skewness and kurtosis, are also exposed. The probability distribution usually depends on process parameters, such as, in the case of metal cutting, feed rate and cutting speed. To obtain a sample probability density estimatep ˆ the amplitude of the time record, x(n) is divided into K + 2 intervals. The number of occurencies, the amplitude of the time record falls in each interval, is denoted Nk and each interval have the probability of

Nk pˆ(k)= (45) NW where W is the width of the intervals and N is the total amount of samples in the time record. If the division by N in equation 45 is excluded the result would be a histogram. The width of each interval is dependent on both the number of intervals and the amplitude range of the time record. Let [a, b] be the amplitude range of the time record then the width of each interval is given by b − a W = . (46) K

The interval ranges Ik can then be deﬁned as  ] −∞,a],k=0  Ik = ]a +(k − 1)W, a + kW],k∈{1, 2,...,K} (47)   ]b, ∞[,k= K +1

2.5.4 Spectral Properties The spectral properties of a signal express how the power or energy of the signal is distributed versus frequency. Typically, periodic and random signals have a power distribution and transient signals have an energy distribution. If periodic signals are discussed, the power of the signal is distributed on the fundamental frequency of the signal and its harmonics, i.e. the power of a periodic signal will be distributed at discrete frequencies. In order to estimate the spectral properties of a signal it is important to select an appropriate spectrum estimator. The selection of spectrum estimator is related to signal properties such as the signal being periodic, random and transient. If the signal is periodic it is usually preferable to estimate the spectrum with a power spectrum estimator PPS(f). On the other hand if the signal has a random nature the spectral properties is usually estimated by a power spectral density estimator PPSD(f) and in the case of transient signals it is preferable to use an energy spectral density estimator PESD(f)[14]. Vibration Analysis of a Boring Bar 45

In order to estimate the spectral properties of a analog signal xc(t)itiscon- venient to sample the signal and estimate the spectral properties of the A/D- converted signal x(n). If it is assumed that the quantization error is negligible the time discrete signal x(n) is related to the the analog signal as

x(n)=xc(nTs),n∈{0, 1,...}, (48) where Ts is the sampling interval. The fourier transform of a signal is usually obtained through the use of a FFT, fast fourier transform. The FFT is an eﬃcient way to calculate the discrete fourier transform DFT. The DFT is deﬁned by [15] N−1 X(k)= x(n)e−j2πnk/N ,k ∈{0, 1,...,N−1} (49) n=0 and is related to the fourier transform of the analog signal Xc(f)by   TsX(k), |fk|≤Fs/2 Xc(fk)= (50) 0, otherwise

k where fk = N Fs and Fs is the sampling frequency. In practice, however, only a finite duration sequence x(n), 0 ≤ n ≤ N − 1 is available for computing the spectrum of an analog signal. This limitation in time is equivalent to multiplying thetimesequencex(n) with a window w(n), thus the spectrum is based on a windowed time sequence x(n)w(n). Nonparametrical spectral density estimation can be done with e.g. the Bartlett method, Welch method or Blackman-Tukey method. The Welch method is a further development of the Bartlett method, where the data segments are allowed to overlap and the data are windowed prior to computing the periodogram [16], which usually results in a better estimate. 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 con- sisting of N samples. Thus the original time sequence of data must be divided into data segments according to n =0, 1,...,N − 1 xl(n)=x(n+lD)where l =0, 1,...,L− 1 where lD is the starting point for each periodogram and D is the overlapping increment. If D = N there is no overlap and if D = N/2 there is a 50% overlap between the consecutive data segments xl(n)andxl+1(n). Dividing the original time sequence into data segments is equivalent to multiplying the time series by a window. Actually even the Bartlett method can be 46 Part I considered to be windowed with a rectangular window when the data sequence is limited to N samples. Welch method usually uses other kinds of windows such as Kaiser, Hanning, Flattop, etc, which reduces the leakage in the frequency domain. Hence each periodogram is based on a windowed sequence w(n)xl(n)where n =0, 1,...N − 1. The Welch power spectral density estimator PPSD(f)isgiven by L−1 N−1 2 PSD Ts −j2πnk/N k Pxx (fk)= xl(n)w(n)e ,fk = Fs, (51) PSD LNU l=0 n=0 N where k =0,...,N/2, L the number of periodograms, N the length of the periodogram and N−1 1 2 UPSD = w(n) N n=0 is the normalization factor for the power in the window function used in the power spectral density estimate. For transient signals where the energy is finite the energy spectral density PESD(f) is estimated instead of the power spectral density. The energy spectral density estimator is easily obtained from the power spectral density estimator PSD P (f) by multiplying it with the periodogram time duration TsN and not use overlapping as in Welch method. The energy is distributed on discrete frequencies and if the signal has petered out within the sampling time no window is necessary, in the case of lightly damped systems an exponential window is appropriate. The energy spectral density function estimate is defined as 2 2 L N−1 ESD NTs −j2πnk/N k Pxx (fk)= xl(n)w(n)e ,fk = Fs (52) ESD LNU l=1 n=0 N where k =0,...,N/2, L the number of periodograms, N the length of the periodogram, xl(n)isthelth record of the transient signal i.e. there is no overlap in energy sprectral density functions and N−1 1 2 UESD = w(n) . N n=0

In the case where the signal consists of pure sinusoids or tonal components it is necessary to use power spectrum PPS(f) in order to get an unbiased estimate of Vibration Analysis of a Boring Bar 47 the tonal components. The power spectrum is deﬁned as L−1 N−1 2 PS 1 −j2πnk/N k Pxx (fk)= xl(n)w(n)e ,fk = (53) PS LNU l=0 n=0 N where w(n) is the window and

N−1 1 2 UPS = w(n) N n=0 is the normalization factor for the power in the window used in the power spectrum estimate. PPSD(f), PESD(f)andPPS(f) are all estimates and will introduce errors. There are two major errors involved in the estimation process, bias errors and random errors. Bias errors can be deduced from the window w(n) applied to the time series in order to obtain a certain number of data segments. The window is assumed to have a constant bandwidth Be centered at a frequency f which may be varied over the frequency range of interest. The ﬁnal estimate describes the average power in terms of frequency components lying inside the frequency band f − Be/2tof + Be/2. The random error can be derived from the fact that for a random stationary process the time record is ﬁnite and therefore the uncertainty will increase with decreasing time record. For lightly damped mechanical systems the normalized bias error εb in a spectral density estimate can, at the resonance frequencies fr be approximated using [12, 14] 2 b[P (fr)] 1 Be εb = ≈− , (54) P (fr) 3 Br where Br is the half power bandwidth of the sharpest resonance peak, the bias b[P (fr)] = E[P (fr)]−P (fr)andBe is the equivalent noise bandwidth of the spectral estimate [12]

N−1 N−1 1 w2(n) |W (k)|2 n=0 N k=0 Be = Fs = Fs, N−1 2 W 2(0) w(n) n=0 where W (k) is the Fourier transformed window function w(n). In ﬁgure 9 the resolution bandwidth Be is illustrated with logarithmic amplitude and frequency on the x-axis. The shaded areas to the left represents the approximate analysis 48 Part I

Log Mag Log Mag

W2(0) W2(0)

f - Be f f + Be f - Be f f + Be

Figure 9: Illustrating the resolution bandwidth Be bandwidth of the window and to the right the weighting function of the window is shown. 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 and is given by [12, 14] V [P (f)] ≈ √ 1 εr = 2 , (55) P (f) BeTe where V [P (f)] is the variance of the spectral estimate and Te is the eﬀective record length. The selection of data length, data segment length as well as the overlapping and averaging has to be done carefully. There is usually a compromise between the variance and the bias. After preliminary trials, a selection of data length, data segment length N, number of periodograms L, digital window, overlapping and sampling rate Fs was made. The selection is shown in Table 5 and 6.

2.6 Operating Deflection Shape A powerful tool in analyzing the motion of a structure during operation is Operat- ing Deflection Shapes, ODS, analysis. An ODS yields an estimate of the deflection shape of a structure during operation at frequencies of interest. By mapping a structure with N transducers and measure the vibration at these predefined locations it is possible to estimate the operating deflection shape. The operating deflection shape ODS(f)atfrequencyf at the measurement points k =2, 3,...,N on the boring bar and using the first measurement point r = 1 as reference may be written Vibration Analysis of a Boring Bar 49

Parameter Value Data length, T 60 s Data segment length, N 16384 Number of periodograms, L 350 Digital window w(τ) Hanning Overlapping 50 % Eﬀective data length, Te T/1.08 Sampling rate, Fs 48000 Hz

Table 5: Power spectral density estimation parameters.

Parameter Value Data length, T 60 s Data segment length, N 65536 Number of periodograms, L 87 Digital window w(τ) Hanning Overlapping 50 % Eﬀective data length, Te T/1.08 Sampling rate, Fs 48000 Hz

Table 6: Power spectral density estimation parameters for the spectra in high resolution. 50 Part I as      0   0       ODS2(f)   P22(f) · ∠P2r(f)  1 ODS3(f) = P33(f) · ∠P3r(f) (56)  .  (j2πf)2  .   .   .     .  ODSN (f) PNN(f) · ∠PNr(f) where the ﬁrst fraction on the right hand side is the conversion from acceleration to deﬂection. Pkk(f) is the autospectrum at the measurement point and ∠Pkr(f)is the phase of the cross spectrum between the reference point and the measurement point k. An ODS can also be obtained by using the frequency response functions Hkr(f)as      0   0       ODS2(f)   H21(f)  ODS3(f) = H31(f) (57)  .   .   .   .      ODSN (f) HN1(f)

The workpiece material and the cutting data used are summarized in Table 7 and the parameters used when obtaining the data to the ODS analysis can be found in Table 8.

Workpiece Cutting speed Cutting depth Feed rate material v, (m/min) a, (mm) s, (mm/rev) SS 0727-02 200 2 0.1 SS 2541-03 75 2 0.1

Table 7: Cutting data used in the operating deﬂection shape measurements. Vibration Analysis of a Boring Bar 51

Parameter Value Type of Spectrum Power Spectral Density Frequency range 0 - 1000 Hz Number of spectral lines 3201 Number of averages 50 Window Hanning

Table 8: Operating deﬂection shape analysis parameters. 52 Part I

3 Results

The results are discussed in the following order: the Euler-Bernoulli calculations, the experimental modal analysis, the outcome of the stationarity tests, mean square value properties, segmentation of nonstationary data records into short time stationary data segments based on stationarity tests, sample probability density plots, spectral properties, and operating deﬂection shapes analysis.

2500

2000

1500

] 1000 2 500

-500

-1000 Acceleration [m/s Acceleration -1500

-2000

-2500 0 2 4 6 8 10 12 14 16 18 20 Time [ms]

Figure 10: Typical boring bar vibration in the cutting speed direction as a function of time, material SS 0727-02, s = 0.2 mm/rev, a =2mmandv = 175 m/min.

In Fig. 10 a typical plot of the boring bar vibration in the cutting speed direction is shown as a function of time.

3.1 A Simple Model of a Boring Bar In order to obtain information on the dynamic properties of the vibration of the boring bar, the first approach was an Euler-Bernoulli beam model. This theoretical approach to the vibration analysis of a boring bar furnished the mode shapes and resonance frequencies for the first three modes. Fig. 11 shows the beam deflection shape for the first three modes. Using the values tabulated in Table 3 the frequencies for the first three modes were calculated and the result is presented in Table 9. The calculation of the resonance frequencies of the boring bar made it plain that the theoretical approach yielded too high estimates. The reason could be due to the clamping of the boring bar. Clamping using clamping bolts does not correspond completely to clamped boundary conditions. Furthermore, the clamping was attached to a revolver of Vibration Analysis of a Boring Bar 53

0.25

0.2

0.15

0.1

0.05

0.1 Deflection envelope 0.15

0.2

0.25 0 0.05 3 Feed direction0.1 [m] 0.15 2

0.2 1 Mode number

Figure 11: Deﬂection shapes of the ﬁrst three modes of a clamped free beam according to the Euler-Bernoulli calculations.

Direction of f1 f2 f3 mode deﬂection [Hz] [Hz] [Hz] Cutting speed direction 676 4238 11866 Cutting depth direction 708 4427 12369

Table 9: Euler-Bernoulli calculations of the ﬁrst three resonance frequencies in each direction using the values given in Table 3. 54 Part I

finite mass. It was also noted during the cutting experiments that even the whole revolver vibrated itself. The estimated deflection of the boring bar turned out to correspond well with the results from the modal analysis and the operating deflection shape analysis, see section 3.9.

3.2 Experimental Modal Analysis In the preliminary experimental modal analysis frequency shifts in the resonance frequencies of the boring bar were observed. In Fig. 12 the magnitude of the imaginary part of diﬀerent frequency response functions are presented to illustrate the encountered frequency shifts.

10*log10(|Imaginary part of FRF|) 40 imp head->csd imp head->csd 20 imp head->imp head a) 0 -20 Modal analysis

540 560 580 600 620 640 660 Frequency [Hz] 30 cdd->csd 20 cdd->cdd csd->csd 10 b) csd->cdd 0

Linearized clamping -10 540 560 580 600 620 640 660 Frequency [Hz] cdd->csd 5 cdd->cdd csd->csd c) 0 csd->cdd

-5 Unmodified clamping 540 560 580 600 620 640 660 Frequency [Hz]

Figure 12: The magnitude of the imaginary part of the frequency response function of the response of the boring bar between diﬀerent directions. The curve csd denotes the cutting speed direction, cdd denotes the cutting depth direction and imp head denotes the impeadance head used in the modal analsysis which was applied at a 45◦ angle from each of the csd and cdd. Observe the resonance frequency shifts between the three measurements.

The boring bar was originally clamped with six clamp bolts, three on the tool side and three on the opposite side of the tool. The axial direction of the six clamp bolts is parallel with the cutting speed direction. The frequency response function for the unmodiﬁed boring bar clamping using hammer excitation is presented in Vibration Analysis of a Boring Bar 55

Fig. 12 c). In order to linearize the response of the boring bar, the clamping was modified. The clamp bolts were principally replaced by steel wedges machined to match the space between the clamping house and the boring bar. Thus, the boring bar was clamped by glue joining the steel wedges in between the clamping house and the boring bar. A new set of preliminary frequency response functions were estimated for the boring bar with linearized clamping using hammer excitation in Fig. 12 b). The linearization of the boring bar did not completely removed the nonlinearities, thus the boring bar needed to be excited in both directions at the same time. The force was applied via a shaker at 45◦ angle from the cutting speed and the cutting depth direction. If a shaker was exciting both directions separately, frequency shifts were still observed. We are not certain however what the result would have been if two shakers were simultaneously exciting the boring bar. Probably this action would have provided a result as good as one shaker exciting both directions at the same time. Applying a shaker and an impedance head to the boring bar altered the resonance frequencies again as seen by comparing Fig. 12 a) and b). Furthermore, the linearization resulted in an increased damping of the first eigenmode. The orientation of the modes was also affected, in the unmodified clamping case it appears like both ”modes” contributes significantly in both the cutting speed and the cutting depth directions, see Fig 12 b). While in the case with linearized clamping the first mode vector was to a large extent in the cutting depth - feed direction plane and the second mode vector was mainly in the cutting speed - feed direction plane, see Figs 12 a), 16 and 17. The experimental modal analysis was carried out for the boring bar with linearized clamping. The obtained estimates of the eigenfrequencies and corresponding dampings are given in Table 10.

Mode Frequency Damping ratio [Hz] [%] 1 570 1.85 2 595 0.84

Table 10: Frequency and damping estimates obtained in the modal analysis.

The estimates of modal data are based on linear estimates of the transfer paths. By computing coherence function estimates of the involved transfer paths, a measure of the linear relationship between the input and output signals for these transfer paths is obtained. In Table 11 the coherence values for the involved transfer paths at the respective eigenfrequency are given. Furthermore, typical examples of coherence function estimates between input force and measured acceleration re- 56 Part I

570 Hz 595 Hz

γsz1 0.999 0.997 Driving point 0.999 0.998

γsz3 0.999 0.997

γsz4 0.999 0.997

γsz5 0.999 0.997

γsz6 0.999 0.997

γsz7 0.999 0.997

γsy1 0.999 0.998

γsy2 0.999 0.998

γsy3 0.999 0.998

γsy4 0.999 0.998

γsy5 0.999 0.998

γsy6 0.999 0.998

γsy7 0.999 0.998

Table 11: Coherence values at the eigenfrequencies in the modal analysis. γsz1 denotes the coherence between the force transducer signal and the accelerometer placed at location Z1, etc. Sensor labelling and location are given in section 2.1.1. sponse are given in Fig. 13. In order to emphasize the relevance of the extracted modal data, the underlying curve ﬁt using a method called polyreference is given in Fig. 14. Another quality measure of the estimated mode shapes is the Modal Assurance Criterion (MAC) matrix, which is shown in Fig. 15. In order to simplfy the interpretation of the two mode shape estimates, each mode shape is presented from four diﬀerent perspectives. In Fig. 16 the estimated mode shape of mode 1 at 570 Hz is shown and the estimated mode shape of mode 2 at 595 Hz can be seen in Fig. 17. Vibration Analysis of a Boring Bar 57

Cutting Speed Direction Cutting Depth Direction

1 1

0.8 0.8

0.6 0.6

Coherence 0.4 Coherence 0.4

0.2 0.2

0 0 400 500 600 700 800 400 500 600 700 800 Frequency [Hz] Frequency [Hz] a) b) Driving Point

0.8

0.6

Coherence 0.4

0.2

0 400 500 600 700 800 Frequency [Hz] c)

Figure 13: Coherence function estimates between force transducer signal and in a) accelerometer located at Y1, i.e. the accelerometer that measures the acceleration in the cutting speed direction closest to the cutting tool, in b) accelerometer located at Z1, i.e. the accelerometer that measures the acceleration in the cutting depth direction closest to the cutting tool, and in c) the driving point. Sensor labelling and location are given in section 2.1.1. Observe the good coherence around the resonance peaks. The coherence dip at 445 Hz is probably due to external disturbance. 58 Part I

Cutting Depth Direction Cutting Speed Direction

/N] 40 /N] 40 2 Experimental 2 Experimental Synthesized Synthesized 30 30

20 20

10 10

0 0

−10 −10

−20 −20

Frequency response function [dB rel m/s 400 500 600 700 800 Frequency response function [dB rel m/s 400 500 600 700 800 Frequency [Hz] Frequency [Hz] a) b) Figure 14: Synthesized versus experimental amplitude function estimates for the two modes in the modal analysis. In a) the ﬁrst mode at 570 Hz in the cutting depth direction, and in b) the second mode at 595 Hz in the cutting speed direction.

MAC matrix

0.8

0.6

0.4

0.2

570 595

595 570

Figure 15: The MAC matrix for the extracted mode shapes corresponding to the eigenfrequencies at 570 Hz and 595 Hz in the modal analysis. Vibration Analysis of a Boring Bar 59

Mode 1 Mode 1

20 20

15 15

10 10

5 5

0 0

-5 -5

-10 -10

-15 -15

-20 -20

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 in the cutting depth direction Deflection envelope 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Deflection envelope in the cutting speed direction Deflection envelope Feed direction [meter] Feed direction [meter] a) b) Mode 1 Mode 1

20 20 15 10 10 0 5 -10 0

Deflection envelope -20 -5

Deflection20 envelope Cutting depth direction -10 10 Feed direction -15 Cutting speed direction 0 -20 -10 0.15 0.1 -20 -15 -10 -5 0 5 10 15 20 -20 0.05 Deflection envelope in the cutting speed direction Deflection envelope Deflection envelope in the cutting depth direction 0 meter c) d)

Figure 16: Mode shape estimate of mode 1 at 570 Hz in the modal analysis. In a) mode shape viewed in the feed direction and cutting speed direction plane, in b) mode shape viewed in the feed direction and cutting depth direction plane, in c) mode shape viewed in the cutting depth direction and cutting speed direction, and in d) a 3D view of the mode shape. Observe that the ﬁrst mode is in the cutting depth direction. 60 Part I

Mode 2 Mode 2 50 50

40 40

30 30

20 20

10 10

0 0

-10 -10

-20 -20

-30 -30

-40 -40

-50 -50 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Deflection envelope in the cutting speed direction Deflection envelope Feed direction [meter] in the cutting depth direction Deflection envelope Feed direction [meter] a) b) Mode 2 Mode 2 50 50 40

20 0 10

0 Deflection envelope

-10 -50 50 Cutting depth direction -20 Deflection envelope Feed direction -30 Cutting speed direction 0 -40 0.15 -50 0.1 -50 0.05 -50 0 50 0 meter Deflection envelope in the cutting speed direction Deflection envelope Deflection envelope in the cutting depth direction c) d) Figure 17: Mode shape estimate of mode 2 at 595 Hz in the modal analysis. In a) mode shape viewed in the feed direction and cutting speed direction plane, in b) mode shape viewed in the feed direction and cutting depth direction plane, in c) mode shape viewed in the cutting depth direction and cutting speed direction, and in d) a 3D view of the mode shape. Observe that the second mode is in the cutting speed direction. Vibration Analysis of a Boring Bar 61

3.3 Stationarity The stationarity tests were first designed to filter out the influence of the lowest resonance frequency of the clamped boring bar. In order to accomplish that the used size of the interval, where the sum of the mean square values were calculated, was 2400 samples or 0.05 s of continuous cutting operation. The stationarity tests have been carried out for all time records from the continuous cutting operations for all of the three materials SS 0727-02, SS 2343-02 and SS 2541-03, and the cutting data used are given in Table 2. Figs. 18, 19 and 20 show the outputs from run test and reverse arrangements test. If the observed number of runs is in the interval between rn;1−α/2 and rn;α/2 the data was considered as stationary according to the run test. If the observed number of reverse arrangements is in an interval between AN;1−α/2 and AN;α/2 the data was considered stationary according to the reverse arrangements test. Based on 1200 observations of mean square values, each calculated using 2400 samples, the number of boring bar vibration records considered as stationary, was very poor. The number of vibration records considered as stationary was 2.3% in the cutting speed direction and 1.1% in the cutting depth direction of the total amount of the vibration records. The significance level was α =0.05.

5 x 10 Reversed arrangements test in the cutting speed direction 6

4 A N;α/2 A N;1-α/2 3

Reversed arrangements Reversed 2 50 100 150 200 250 300 Cutting speed [m/rev] Run test in the cutting speed direction 700 r n;α/2 600 r n;1-α/2 500

400

Number of runs 300

200 50 100 150 200 250 300 Cutting speed [m/rev]

Figure 18: Stationarity tests in SS 0727-02, s = 0.1 mm/rev, a = 2 mm, v =50 m/min - 300 m/min, cutting tool DNMG 150608-SL, grade 7015. The number of mean square observations is 1200 and each mean square value has been extracted from 2400 samples. 62 Part I

5 x 10 Reverse arrangements test in the cutting speed direction 6

4 A N;α/2 A N;1-α/2 3

Reverse arrangements Reverse 2 50 100 150 200 Cutting speed [m/rev] Run test in the cutting speed direction 700 r n;α/2 600 r n;1-α/2 500

400

Number of runs 300

200 50 100 150 200 Cutting speed [m/rev]

Figure 19: Stationarity tests in SS 2343-02, s = 0.2 mm/rev, a = 2 mm, v =50 m/min - 200 m/min, cutting tool DNMG 150608-SL, grade 8025. The number of mean square observations is 1200 and each mean square value has been extracted from 2400 samples.

5 x 10 Reverse arrangements test in the cutting speed direction 5

3 A N;α/2 A N;1-α/2 2

Reverse arrangements Reverse 1 50 100 150 200 250 300 Cutting speed [m/rev] Run test in the cutting speed direction 1000

800

600 r n;α/2 r n;1-α/2 400

Number of runs 200

0 50 100 150 200 250 300 Cutting speed [m/rev]

Figure 20: Stationarity tests in SS 2541-03, s = 0.3 mm/rev, a = 2 mm, v =50 m/min - 300 m/min, cutting tool DNMG 150608-SL, grade 7015. The number of mean square observations is 1200 and each mean square value has been extracted from 2400 samples. Vibration Analysis of a Boring Bar 63

It was suspected that revolution dependent spindle vibrations affected the result of the stationarity tests. These kinds of vibrations are always present in rotating machinery. From the spindle engine vibration power spectral density estimates shown in Fig. 42 it is obvious that these kind of vibrations are present in the CNC turning center used in the experiments. Low frequency periodicities in the boring bar vibration records deriving from the spindle engine vibration may be filtered out by increasing the averaging time used to compute the mean values for the stationarity tests. The number of samples used to compute the mean square values for the stationarity tests was increased to 96000 samples or 2s of continuous cutting operation. The number of mean square value observations used in the stationarity tests was 30. In Figs. 21, 22 and 23 the outputs from run tests and reverse arrangements tests, using the increased number of samples to compute the mean square values, are shown. There was an increase in the number of vibration records that were considered as stationary, in the cutting speed direction from 2.3% to 16.1% and in the cutting depth direction from 1.1% to 12.6% of the total amount of vibration records. The significance level was α =0.05.

Reverse arrangements test in the cutting speed direction 400

300 A N;α/2 200 A N;1-α/2 100

Reverse arrangements Reverse 0 50 100 150 200 250 300 Cutting speed [m/rev] Run test in the cutting speed direction 25 r 20 n;α/2

15 r 10 n;1-α/2

Number of runs 5 0 50 100 150 200 250 300 Cutting speed [m/rev]

Figure 21: Stationarity tests in SS 0727-02, s = 0.1 mm/rev, a = 2 mm, v =50 m/min - 300 m/min, cutting tool DNMG 150608-SL, grade 7015. The number of mean square observations is 30 and each mean square value has been extracted from 96000 samples. 64 Part I

Reverse arrangements test in the cutting speed direction 500

400

300 A N;α/2 200 A N;1-α/2

Reverse arrangements Reverse 100 50 100 150 200 Cutting speed [m/rev] Run test in the cutting speed direction 25

r n;α/2 20

15 r 10 n;1-α/2

5 Number of runs

0 50 100 150 200 Cutting speed [m/rev]

Figure 22: Stationarity tests in SS 2343-02, s = 0.2 mm/rev, a = 2 mm, v =50 m/min - 200 m/min, cutting tool DNMG 150608-SL, grade 8025. The number of mean square observations is 30 and each mean square value has been extracted from 96000 samples.

Reverse arrangements test in the cutting speed direction 500

400

300

200 A N;α/2 A α 100 N;1- /2

Reverse arrangements Reverse 0 50 100 150 200 250 300 Cutting speed [m/rev] Run test in the cutting speed direction 20 r n;α/2

10 r n;1-α/2

5 Number of runs

0 50 100 150 200 250 300 Cutting speed [m/rev]

Figure 23: Stationarity tests in SS 2541-03, s = 0.3 mm/rev, a = 2 mm, v =50 m/min - 300 m/min, cutting tool DNMG 150608-SL, grade 7015. The number of mean square observations is 30 and each mean square value has been extracted from 96000 samples. Vibration Analysis of a Boring Bar 65

3.4 Mean Square Value Property In order to facilitate the nonstationary mean square value or power of the boring bar vibration records the mean square value of the boring bar as a function of time was estimated by using moving average. The mean square value was estimated by using three diﬀerent averaging lengths. The reason for choosing 96000 samples or 2 s of continuous boring operation to compute the mean square values in the stationarity tests was that this size smoothes out the low frequency variations in the data. Figs. 24 and 25 shows the mean square acceleration for three diﬀerent averaging times, Ta.

6 x 10

] Ta = 0.05 s 2 ) T = 0.5 s 2 7 a Ta = 2 s 6

4 Mean square [(m/s Acceleration 0 0.5 1 1.5 2 2.5 3 3.5 4 Time [s]

5000 ] 2

0 Acceleration [m/s Acceleration

-5000 0 0.5 1 1.5 2 2.5 3 3.5 4 Time [s]

Figure 24: Mean square values of the acceleration for diﬀerent averaging times Ta in order to illustrate the smoothing eﬀect of the variations in the cutting speed direction in SS 2541-03, s = 0.2 mm/rev, v = 125 m/min and a = 2 mm.

The periodic dynamic motion related to residual rotor mass imbalance in the spindle - chuck - workpiece system is obvious in Fig. 24, showing the mean square value versus time of the boring bar vibration. When the length of the moving average was increased to 96000 samples or 2 s, the periodic dynamic motion related to residual rotor mass imbalance in the spindle - chuck - workpiece system was ﬁltered out. However, observe that the mean square value of the boring bar vibration produced by the sliding average 96000 samples varies slowly with time, as in Fig. 25. This probably indicates nonstationarities in the boring bar vibration related to the chip formation process. 66 Part I

5 x 10

] 16

2 Ta = 0.05 s )

2 14 Ta = 0.5 s Ta = 2 s 12 10 8 6 4

Mean square [(m/s Acceleration 2 0 0.5 1 1.5 2 2.5 3 3.5 4 Time [s]

5000 ] 2

0 Acceleration [m/s Acceleration -5000 0 0.5 1 1.5 2 2.5 3 3.5 4 Time [s]

Figure 25: Mean square values of the acceleration for diﬀerent averaging times Ta in order to illustrate the ability to follow an underlying trend in the cutting speed direction in SS 0727-02, s = 0.3 mm/rev, v = 250 m/min and a = 2 mm.

3.5 Segmentation of the Boring Bar Vibration Records The boring bar vibration records were divided into segments considered as stationary according to the run test and the reverse arrangements test. A stationary segment is, from an arbitrary starting point, the largest number of consecutive observations considered as stationary. This results in short-time stationary segments of the boring bar vibration records. To ﬁlter out the inﬂuence of low frequency periodicities deriving from the spindle engine, each mean square value observation was extracted from 96000 samples or 2 s of continuous cutting operation. Figs. 26, 27 and 28 show the power or mean square value of boring bar vibration records versus time and corresponding data segments considered as stationary for all three materials. Vibration Analysis of a Boring Bar 67

7 4 x 10 Cutting speed direction x 10 Cutting depth direction 1.156 6.1

1.154 6 5.9 1.152 5.8 Power

1.15 Power 5.7

1.148 5.6

1.146 5.5 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Time [s] Time [s]

1 1 0.5 0.5 stationary

stationary 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Time [s] Time [s] a) b)

Figure 26: Mean square value observations or power of boring bar vibration, where each mean square value is based on 96000 sampel and the corresponding time intervals of the boring bar vibration considered as stationary are illustrated as shaded areas. In a) the cutting speed direction and in b) the cutting depth direction, in SS 0727-02, s = 0.2 mm/rev, a = 2 mm, v = 100 m/min, cutting tool DNMG 150608-SL, grade 7015.

6 x 10 Cutting speed direction 4 Cutting depth direction 2.36 1.22 x 10

2.34 1.2 2.32 1.18 2.3 Power Power1.16 2.28 1.14 2.26

2.24 1.12 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Time [s] Time [s]

1 1 0.5 0.5

stationary 0

stationary 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Time [s] Time [s] a) b)

Figure 27: Mean square value observations or power of boring bar vibration, where each mean square value is based on 96000 sampel and the corresponding time intervals of the boring bar vibration considered as stationary are illustrated as shaded areas. In a) the cutting speed direction and in b) the cutting depth direction, in SS 2343-02, s = 0.1 mm/rev, a = 2 mm, v = 50 m/min, cutting tool DNMG 150608-SL, grade 8025. 68 Part I

7 x 10 Cutting speed direction 4 Cutting depth direction 1.2 8.6 x 10 1.19 8.5 1.18 8.4 8.3 1.17 8.2 Power Power1.16 8.1 1.15 8 1.14 7.9 1.13 7.8 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Time [s] Time [s]

1 1 0.5 0.5 stationary

stationary 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Time [s] Time [s] a) b)

Figure 28: Mean square value observations or power of boring bar vibration, where each mean square value is based on 96000 sampel and the corresponding time intervals of the boring bar vibration considered as stationary are illustrated as shaded areas. In a) the cutting speed direction and in b) the cutting depth direction, in SS 2541-03, s = 0.3 mm/rev, a = 2 mm, v = 100 m/min, cutting tool DNMG 150608-SL, grade 7015. Vibration Analysis of a Boring Bar 69

3.6 Sample Probability Density Estimate Based on the segmentation of the boring bar vibration records, sample probability density estimates were produced for each of the data segments considered as short-time stationary. The sample probability density estimates of the boring bar vibration showed different shapes for the different segments that were considered as stationary. The characteristic shapes, see Fig. 29, was the sine wave probability density shape in a), the gaussian shape in b), and a combination of these in c). Examples of sample probability density estimates for the three materials are presented in Figs. 30, 31 and 32. An essential feature of the vibration data was that the shape of the sample probability density estimates could be totally different in the cutting speed and the cutting depth direction during a continuous cutting operation with constant cutting data as in Fig. 31 and 32. A reason for differences between the shapes in the cutting speed and in the cutting depth direction might be that in one of the directions the vibration is dominated by an eigenmode but in the other direction there is none or little influence of this eigenmode or other egenmodes. However, similar shapes of the sampled probability density in the cutting speed and the cutting depth direction were also observed, see Fig. 30. From the presented sample probability density estimates, it it also obvious that the boring bar vibration is time dependent. This may be seen as an indication on the nonstationarities of the boring bar vibration. Furthermore, the sample probability density estimate presented as a waterfall in Fig. 33 a) clearly shows the nonstationary behavior of the boring bar vibration. Another interesting feature was found in the response of the boring bar during machining in workpiece material SS 0727-02. At high cutting speeds, typically higher than 150 m/min, the sample probability density estimate tends to have a gaussian shape. For the other two materials the shape at higher cutting speeds were more undetermined. At low cutting speeds the sample probability density estimate in all workpiece materials was usually of sine wave probability density shape. 70 Part I

Cutting speed direction

0.03

0.025

0.02

0.015

0.01 60

0.005 40

0 20 Sample probability density estimate -3000 -2000 -1000 0 0 Time [s] 1000 2000 3000 Acceleration [m/s2 ] a)

Cutting speed direction Cutting speed direction

0.04 0.025

0.02 0.03

0.015 0.02 0.01 60 60 0.01 0.005 40 40

20 0 20 Sample probability density estimate 0 Sample probability density estimate -5000 -3000 -2000 -1000 0 0 Time [s] 0 0 Time [s] 1000 2000 3000 Acceleration [m/s2 ] 5000 Acceleration [m/s2 ] b) c)

Figure 29: Examples of the three characteristic shapes in the sample probability density estimates versus time for the data segments considered as short time stationary of the the boring bar vibration during a 60 second continuous cutting operation in: a) SS 2343-02, s = 0.1 mm/rev, a = 2 mm, v =50m/min,tool DNMG 150608-SL, grade 8025. b) SS 0727-02, s = 0.3 mm/rev, a = 2 mm, v = 300 m/min, tool DNMG 150608-SL, grade 7015. c) SS 0727-02, s = 0.2 mm/rev, a = 2 mm, v = 225 m/min, tool DNMG 150608-SL, grade 7015. Vibration Analysis of a Boring Bar 71

Cutting speed direction Cutting depth direction

0.05 0.05

0.04 0.04

0.03 0.03

0.02 0.02 60 60 0.01 40 0.01 40

0 20 0 20 Sample probability density estimate Sample probability density estimate -2000 -1500 -1000 -1000 -500 0 500 0 Time [s] -500 0 Time [s] 1000 1500 2000 500 1000 0 Acceleration [m/s2 ] Acceleration [m/s2 ] 1500 a) b)

Figure 30: Sample probability density estimates versus time for the data segments considered as short time stationary of the dynamic response of the boring bar during a 60 second continuous cutting operation in SS 0727-02, s = 0.2 mm/rev, a = 2 mm, v = 250 m/min, tool DNMG 150608-SL, grade 7015. Observe that the stationary segments diﬀer between the cutting speed in a) and the cutting depth in b) directions.

Cutting speed direction Cutting depth direction

0.025 0.07 0.06 0.02 0.05

0.015 0.04

0.01 0.03 60 60 0.02 0.005 40 0.01 40

20 0 20 Sample probability density estimate 0 Sample probability density estimate -3000 -2000 -1500 -1000 Time [s] -1000 0 1000 0 Time [s] -500 0 0 2000 3000 4000 500 1000 Acceleration [m/s2 ] Acceleration [m/s2 ] a) b)

Figure 31: Sample probability density estimates versus time for the data segments considered as short time stationary of the dynamic response of the boring bar during a 60 second continuous cutting operation in SS 2343-02, s = 0.2 mm/rev, a = 2 mm, v = 50 m/min, tool DNMG 150608-SL, grade 8025. Observe that the stationary segments and the shape diﬀer between the cutting speed in a) and the cutting depth in b) directions. 72 Part I

Cutting speed direction Cutting depth direction

0.03 0.03

0.025 0.025

0.02 0.02

0.015 0.015

0.01 60 0.01 60

0.005 40 0.005 40

20 20 Sample probability density estimate 0

Sample probability density estimate 0 -5000 -600 -400 0 Time [s] -200 0 200 0 Time [s] 0 400 600 800 Acceleration [m/s2 ] 5000 Acceleration [m/s2 ] a) b)

Figure 32: Sample probability density estimates versus time for the data segments considered as short time stationary of the dynamic response of the boring bar during a 60 second continuous cutting operation in SS 2541-03, s = 0.1 mm/rev, a = 2 mm, v = 175 m/min, tool DNMG 150608-SL, grade 7015. Observe that the shape is diﬀerent between the cutting speed in a) and the cutting depth in b) directions.

Cutting speed direction Cutting speed direction

0.02 60 /Hz] 2 )

2 40 0.015 20

0.01 0

60 -20 60 0.005 40 40 -40

0 20 PSD [1 dB rel 1 (m/s 20 Sample probability density estimate 0 5 -10000 10 Time [s] -5000 Time [s] 15 0 0 5000 0 20 Acceleration [m/s2 ] 10000 Frequency [kHz] a) b)

Figure 33: Sample probability density estimate versus time in a) and the corresponding power spectral density versus time in b) for the data segments considered as short time stationary of the dynamic response of the boring bar during a 60 second continuous cutting operation in SS 2541-03, s = 0.2 mm/rev, a = 2 mm, v = 275 m/min, tool DNMG 150608-SL, grade 7015. Vibration Analysis of a Boring Bar 73

3.7 Spectral Properties The spectral density estimates are based on the segmentation, see section 3.5, of the boring bar vibration records. The normalized bias error, εb, of the spectral density estimates was calculated using equation 54 where Br was obtained in frequency response functions from hammer excitation of the boring bar, see Fig. 34. The normalized random error, εr, was calculated using equation 55 and was based on the length of the stationary segments in each vibration record. The calculated errors in the spectral density estimates are shown in Table 12 and under the random error εr assumption the longest stationary segment of each vibration record have been used.

Hammer excitation 16 20 15 /N] 2 14 10 13 3 dB 12 Br = 9.8 Hz 0 11

10 560 570 580 590 -10

-20

-30 Frequency response function [dB rel m/s Frequency -40 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency [Hz]

Figure 34: The response of the boring bar using hammer excitation in the cutting speed direction, showing the half power bandwidth of the sharpest resonance peak.

εb εr 0.07 0.06 - 0.16

Table 12: Normalized bias error, εb, and normalized random error, εr, in the power spectral density estimates.

The power spectral densities presented here are based on the longest stationary segment of each of the vibration records. The reason for not presenting the spectra 74 Part I as in the sample probability density estimate case, i.e. spectrum versus time, is that the nonstationarities are not illustrated well in such spectral density estimates. Fig. 33 a) and b), show sample probability density estimate and power spectral density of a typically nonstationary data record. Fig. 35 shows the power spectral densities as a function of frequency for in a) versus cutting speeds (50 - 300 m/min), and in b) versus feed rates (0.1 - 0.3 mm/rev). In order to facilitate observations of the inﬂuence of the parameter space, the spectral density estimates are presented as waterfall diagrams. Figs. 36, 37 and 38 show typical waterfall diagrams for the three materials used in the experiments. According to the spectrum analysis the vibration energy was usually dominated by the resonance peak at approximately 600 Hz and forces in the cutting speed direction were usually larger than the forces in the cutting depth direction. For increasing cutting speeds the vibration energy was increased for frequencies above 600 Hz. Also, the high frequency content in the boring bar vibration excited by the diﬀerent workpiece materials varied. The workpiece materials SS 0727-02 and SS 2541-03 resulted in similar vibration spectrum pattern, see Figs. 36 and 38, while SS 2343-02 has a more smooth spectra at higher frequencies, see Fig. 37. Another important matter is the harmonics, these are probably due to that the motion of the boring bar was limited by the workpiece in both cutting speed and cutting depth direction. All materials showed harmonics during cutting operations but most harmonics was discovered in SS 2541-03 as in Fig. 38.

Cutting speed direction Cutting speed direction 60 60

50 50

40 40 /Hz] /Hz] 2 2 ) 30 )

2 30 2 20 20

10 10

0 0

-10 -10

-20 -20 PSD [1dB rel 1 (m/s PSD [1 dB rel 1 (m/2 -30 -30

-40 -40

-50 -50 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 Frequency [kHz] Frequency [kHz] a) b)

Figure 35: Estimated power spectral densities of the dynamic response of the boring bar during a cutting operation in SS 0727-02, a) s = 0.3 mm/rev, a = 2 mm, v = 50 m/min to 300 m/min, b) s = 0.1 mm/rev to 0.3 mm/rev, a = 2 mm, v =50 m/min, tool DNMG 150608-SL, grade 7015. Vibration Analysis of a Boring Bar 75

Cutting speed direction

60 /Hz]

2 40 ) 2

0 300 -20 250 200

PSD [1 dB rel 1 (m/2 -40 150 100

0 2 4 6 50 8 10 12 14 16 18 20 Frequency [kHz] Cutting speed [m/min]

Figure 36: Power spectral density versus frequency and cutting speed of the dynamic response of the boring bar during a cutting operation in SS 0727-02, s = 0.3 mm/rev, a = 2 mm, v = 50 - 300 m/min, tool DNMG 150608-SL, grade 7015. Observe the growth of the high frequency contents at increasing cutting speeds.

Cutting speed direction

100 /Hz] 2 ) 2 50

200 0

150

PSD [1 dB rel 1 (m/2 100 -50 0 2 4 6 50 8 10 12 14 16 18 20 Frequency [kHz] Cutting speed [m/min]

Figure 37: Power spectral density versus frequency and cutting speed of the dynamic response of the boring bar during a cutting operation in SS 2343-02, s =0.1 mm/rev, a = 2 mm, v = 50 - 200 m/min, tool DNMG 150608-SL, grade 8025. 76 Part I

Cutting speed direction

100 /Hz] 2 ) 2 50

300 0 250 200

PSD [1 dB rel 1 (m/2 150

-50 100 0 2 4 6 50 8 10 12 14 16 18 20

Frequency [kHz] Cutting speed [m/min]

Figure 38: Power spectral density versus frequency and cutting speed of the dynamic response of the boring bar during a cutting operation in SS 2541-03, s = 0.1 mm/rev, a = 2 mm, v = 50 - 300 m/min, tool DNMG 150608-SL, grade 7015. Observe the harmonics of the ﬁrst resonance frequency

Figs. 39 and 41 are with high frequency resolution and zoomed in frequency to the ﬁrst peaks in the spectra or the ﬁrst resonance frequency of the boring bar. The size of each periodogram was 65536 samples. The normalized bias error and random error in these high resolution spectra are given in Table 13. In Fig. 35 the

εb εr 0.004 0.13 - 0.31

Table 13: Normalized bias error, εb, and normalized random error, εr, in the high resolution power spectral density estimates. resonance peaks appear to be independent of both feed rate and cutting speed. This is not the case, however, which is shown in the zoomed high resolution spectra in Figs. 36, 37 and 38. In high frequency resolution also the side band terms become visible. Fig 42 shows the low frequency spindle engine vibration measured by an accelerometer mounted on the spindle engine, when no cutting operation was performed. During a continuous cutting operation these low frequency vibrations usually appear as sidebands of the resonance frequency as shown in Figs 39 - 43. This phenomena may be explained by modulation of the spindle vibrations through the coupling between cutting tool and workpiece. Zooming the first resonance peak Vibration Analysis of a Boring Bar 77 in the power spectral density estimates showed that the first peak was either of two peaks, which could be suspected from the results from the Euler-Bernoulli calculations used to model the boring bar. At high cutting speeds the second resonance peak was active and at low cutting speeds the first resonance peak was active. Observe however that the resonance frequencies of the boring bar varies both at different feed rates and at different cutting speeds. It is evident that the coupling between cutting edge and workpiece alters at different feed rates, i.e. the boundary conditions changes. The two first resonance peaks were at approximately the same frequency for SS 0727-02 and SS 2541-03 but for material SS 2343-02 the frequencies for the first two resonance peaks were slightly higher, about 20 Hz. The motion of the boring bar was mainly characterized by either the first or the second resonance frequency as in Figs. 39 - 41. From Figs. 35 - 38 it is clear that either of the first or the second resonance frequency were dominating the spectra.

Cutting speed direction Cutting speed direction

80 70

/Hz] 70 /Hz] 60 2 2 ) ) 2 60 2 50 50 40 40 30 30 300 20 0.3 20 250 10 10 200 0.2 0 150 0 PSD [1dB rel 1 (m/s PSD [1dB rel 1 (m/s -10 100 -10 460 480 50 460 480 0.1 500 520 540 560 580 600 620 640 500 520 540 560 580 600 620 640 Feedrate [mm/rev] Frequency [Hz] Cutting speed Frequency [Hz] [m/min] a) b)

Figure 39: Power spectral densities zoomed in frequency to the ﬁrst resonance peak. In SS 0727-02, tool DNMG 150608-SL, grade 7015, a) s = 0.1 mm/rev, a = 2 mm, v = 50 m/min to 300 m/min, b) s = 0.1 mm/rev to 0.3 mm/rev, a =2 mm, v = 100 m/min. Observe the sideband terms. 78 Part I

Cutting speed direction Cutting speed direction

60 70

/Hz] 50 /Hz] 60 2 2 ) ) 2 2 50 40 40 30 30 20 200 20 0.3 10 150 10 0 0 0.2

PSD [1dB rel 1 (m/s 100 PSD [1dB rel 1 (m/s -10 -10 50 0.1 500 520 540 560 580 600 620 460 480 500 520 540 560 580 640 660 680 Cutting speed 600 620 640 [m/min] Frequency [Hz] Frequency [Hz] Feedrate [mm/rev] a) b)

Figure 40: Power spectral densities in SS 2343-02 in cutting speed direction, a) s = 0.1 mm/rev, a = 2mm, v = 50 m/min to 200 m/min, b) s = 0.1 mm/rev to 0.3 mm/rev, a =2mmv = 50 m/min, tool DNMG 150608-SL, grade 8025.

Cutting speed direction Cutting speed direction

80 70

/Hz] 70 /Hz] 60 2 2 ) ) 2 60 2 50 50 40 40 30 30 300 20 0.3 20 250 10 10 200 0.2 0 150 0 PSD [1dB rel 1 (m/s PSD [1dB rel 1 (m/s -10 100 -10 50 0.1 460 480 500 520 540 560 460 480 500 520 540 560 580 600 620 640 Cutting speed 580 600 620 640 Frequency [Hz] [m/min] Frequency [Hz] Feedrate [mm/rev] a) b)

Figure 41: Power spectral densities zoomed in frequency to the ﬁrst resonance peak. In SS 2541-03, tool DNMG 150608-SL, grade 7015. a) s = 0.3 mm/rev, a = 2 mm, v = 50 m/min to 300 m/min, b) s = 0.1 mm/rev to 0.3 mm/rev, a =2 mm, v = 125 m/min. Observe the resonance frequency shift and sideband terms. Vibration Analysis of a Boring Bar 79

Low frequency spindle vibrations at 3.95 rev/s Low frequency spindle vibrations at 2.6667 rev/s 20 20 3.95 Hz 15 15

10 10 2.67 Hz 2. 3.95 Hz

/Hz 5 /Hz 2 2

) 5 ) 2 2 0 2. 2.67 Hz 0 5 3. 3.95 Hz 5 10 10 3 .2.67 Hz 15 PSD (dB rel (m/s PSD (dB rel (m/s

20 15

25 20

30 25 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Frequency [Hz] Frequency [Hz] a) b)

Figure 42: Power spectral densities showing the low frequency spindle vibrations and the harmonics of the spindle revolution frequency when no cutting operation was performed. The accelerometer measuring the spindle vibrations was mounted vertically on the spindle engine. In a) spindle speed = 3.95 rev/s and in b) spindle speed = 2.667 rev/s.

Cutting speed direction 80

70 609.4 Hz 60 616.0 Hz 602.8 Hz 50 596.8 Hz /Hz] 2 ) 2 590.4 Hz 621.8 Hz 40 628.4 Hz 30

20 PSD [1dB rel 1 (m/s

0 6.4 Hz 6.0 Hz 6.6 Hz 6.6 Hz 5.8 Hz 6.6 Hz

10 560 570 580 590 600 610 620 630 640 650 Frequency [Hz]

Figure 43: Power spectral density showing the force modulated resonance peak, where the sidebands are equally spaced with the speed of the spindle which in this case was 6.2 rev/s during a cutting operation in SS 2343-02 at a = 2 mm, s = 0.3 mm/rev, v = 200 m/min, tool DNMG 150608-SL, grade 8025. 80 Part I

3.8 Operating Deflection Shape Analysis In order to get further insight into the motion of the boring bar during a continuous boring operation, an operating deflection shape analysis was carried out. In Fig. 44 estimates of the spectral densities of the boring bar vibration in the cutting speed and cutting depth direction during a boring operation in workpiece material SS 2541-03, v = 75 m/min, are shown. An estimate of the operating deflection shape at 536 Hz for the deflection of the boring bar corresponding to the spectra presented in Fig. 44 is given in Fig. 45, where it is presented from four different perspectives.

Cutting Speed Direction Cutting Depth Direction 80 80 /Hz] /Hz] 2 2 60 60

40 40

20 20

0 0

Power Spectral Density [dB rel g −20 Power Spectral Density [dB rel g −20 400 500 600 700 800 400 500 600 700 800 Frequency [Hz] Frequency [Hz] a) b)

Figure 44: Power spectral density estimates of the dynamic response of the boring bar, a) in the cutting speed direction, b) in the cutting depth direction, during a continuous cutting operation in SS 2541-03, s = 0.1 mm/rev, a = 2 mm, v =75 m/min, tool DNMG 150608-SL, grade 7015.

If the cutting speed is increased to above approximately 150 m/min, the dominating spectral line of the boring bar vibration spectral density is shifted upwards in frequency. Fig. 46 a) and b) show power spectral density estimates when machining in SS 0727-02 using a cutting speed of 200 m/min. The dominating spectral line was at 588 Hz and the estimated operating deflection shape at that frequency is presented in Fig. 47. The operating deflection shape is presented in four different perspectives. The deformation pattern of the boring bar was, according to the ODS analysis, at both resonance peaks, mainly in the cutting speed direction, see Figs 45 and 47. The reason is probably that the largest part of the excitation of the boring bar was in the cutting speed direction. In comparison to the modal analysis the resonance frequencies were different, which is probably partly due to the fact that the ODS Vibration Analysis of a Boring Bar 81

4 4 x 10 ODS for mode 1 x 10 ODS for mode 1 2 2

1.5 1.5

1 1

0.5 0.5

0 0

-0.5 -0.5

-1 -1

-1.5 -1.5

-2 -2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Deflection envelope in the cutting speed direction Deflection envelope

Deflection envelope in the cutting depth direction Deflection envelope 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Feed direction [meter] Feed direction [meter] a) b) 4 x 10 ODS for mode 1 ODS for mode 1 4 2 x 10 2 1.5 1 1 0 0.5 -1 0

-0.5 Deflection envelope -2 2 Cutting depth direction -1 Deflection envelope 1 Feed direction Cutting speed direction -1.5 0 4 x 10 -1 0.15 -2 0.1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 0.05 Deflection envelope in the cutting speed direction Deflection envelope 4 0 meter Deflection envelope in the cutting depth direction x 10 c) d) Figure 45: Estimate of the operation deﬂection shape for the motion of the boring bar at 536 Hz, during a continuous cutting operation in SS 2541-03, s = 0.1 mm/rev, a = 2 mm, v = 75 m/min, tool DNMG 150608-SL, grade 7015. Observe that the ODS at 536 Hz was in the cutting speed direction. 82 Part I

Cutting Speed Direction Cutting Depth Direction 80 80 /Hz] /Hz] 2 2 60 60

40 40

20 20

0 0

Power Spectral Density [dB rel g −20 Power Spectral Density [dB rel g −20 400 500 600 700 800 400 500 600 700 800 Frequency [Hz] Frequency [Hz] a) b)

Figure 46: Power spectral density estimates of the dynamic response of the boring bar, a) in the cutting speed direction, b) in the cutting depth direction, during a continuous cutting operation in SS 0727-02, s = 0.1 mm/rev, a = 2 mm, v = 200 m/min, tool DNMG 150608-SL, grade 7015. is carried out during machining a workpiece. Machining a workpiece introduces diﬀerent boundary conditions, i.e. the workpiece apply boundary conditions on the cutting tool which are not present in the modal analysis. Vibration Analysis of a Boring Bar 83

ODS for mode 2 ODS for mode 2

1500 1500

1000 1000

500 500

0 0

-500 -500

-1000 -1000

-1500 -1500

1500 1500 1000 1000 500 500 0 -500 0 -1000

Deflection envelope -1500 -500 Cutting depth direction Deflection envelope Feed direction -1000 1000 Cutting speed direction 0 -1500 -1000 0.15 0.1 -1500 -1000 -500 0 500 1000 1500 0.05 Deflection envelope in the cutting speed direction Deflection envelope 0 meter Deflection envelope in the cutting depth direction c) d) Figure 47: Estimate of the operation deﬂection shape for the motion of the boring bar at 588 Hz during a continuous cuttng operation in SS 0727-02, s = 0.1 mm/rev, a = 2 mm, v = 200 m/min, tool DNMG 150608-SL, grade 7015. Observe that the ODS at 588 Hz was in both the cutting speed and cutting depth direction. 84 Part I

3.9 Beam Deflection Comparison To see how well the Euler-Bernoulli calculations were and to what extent the modal analysis corresponded to the true deflection during machining, the deflection shape of the boring bar is shown for all three cases in Fig. 48. Since the envelope of the deflection shape varied between the three cases, the deflection envelope in Fig. 48 was scaled to unity, 0.15 m from the clamped end of the boring bar. In order to fit the Eular-Bernoulli result the clamping had to be moved or biased, i.e. the boring bar was made longer. The bias was in the cutting speed direction 0.025 m into the clamping house and in the cutting depth direction the bias was 0.03 m. This could indicate that the true clamping is a bit inside the clamping house.

Cutting Speed Direction Cutting Depth Direction

1.2 EulerBernoulli 1.2 EulerBernoulli Modal Analysis Modal Analysis ODS ODS 1 1

0.8 0.8

0.6 0.6

0.4 0.4

Deflection envelope Bias Deflection envelope Bias 0.2 0.2

0 0

0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 Feed direction [m] Feed direction [m] a) b) Figure 48: The deﬂection of a boring bar according to Euler-Bernoulli calculations, modal analysis results and operating deﬂection shape results. In a) the cutting speed direction and in b) the cutting depth direction. The dot marks the bias in the Euler-Bernoulli calculations.

The resulting resonance frequencies from the three different approaches varies more than the results of the deflection shapes. The first two resonance frequencies from all three investigations made is summarized in Table 14. An additional result from the Euler-Bernoulli calculations are included in this table, due to that in Fig. 48 the clamping was moved or biased i.e. the boring bar was made longer which of course altered the resonance frequencies. The bias was in the cutting speed direction 0.025 m and 0.03 m in the cutting depth direction. Vibration Analysis of a Boring Bar 85

Resonance frequency Result deriving Cutting speed direction Cutting depth direction from [Hz] [Hz] Euler-Bernoulli 676 708 Biased Euler-Bernoulli 530 508 Modal Analysis 595 570 ODS 536 588

Table 14: Resonance frequencies of the boring bar obtained in the three diﬀerent investigations and in the biased Euler-Bernoulli model. 86 Part I

4 Discussions and Conclusions

From the mixed result it is rather evident that the vibration of a boring bar is a complex process. The coupling between the cutting tool and the workpiece is time varying during a cutting operation. The boring bar clamping does not match theoretical clamped conditions. Low frequency periodicities related to the speed of the spindle also affect the boring bar vibrations. The low frequency vibrations are traceable to the residual rotor mass imbalance in the spindle - chuck - workpiece system. Other factors such as feed rate, cutting depth and cutting speed also affects the vibrations as well as the material of the workpiece. The theoretical approach gave resonance frequencies and deflection shapes of the boring bar. The resonance frequencies yielded too high estimates, which is probably due to clamped boundary conditions. It was discovered during the cutting trials that the whole clamping system also vibrated. The deflection calculations on the other hand exhibits a decent match to the experimental results. The experimental modal analysis yielded that the first mode was in the cutting depth direction at 570 Hz. The second mode was found in the cutting speed direction at 595 Hz. The two modes were well separated according to the MAC matrix and the synthesized frequency response functions corresponded well with the experimental results. The mode shapes were distinct in each direction. From the stationarity tests it is obvious that boring bar vibration cannot be considered to be stationary. The first approach in the stationarity tests was to filter out the dominating resonance frequency of the boring bar at approximately 600 Hz. The result was not satisfactory. It was suspected that the low frequency spindle vibrations might also affect the result. By increasing the averaging time in the stationarity tests to also filter out these vibrations, an increase in the number of vibration records considered as stationary was observed. There are a large variation in the shapes of the sample probability density estimates for the short-time stationary segments of the boring bar vibration. The characteristic shapes were the sine-wave probability shape, the gaussian shape and a combination of these. The sample probability density could be of different shapes in the cutting speed direction and the cutting depth direction. The reason might be that one of the boring bar directions was dominated by one of two resonance frequencies but in the other direction there was none or little influence from this resonance frequencies. Similar shapes in the two directions were also encountered. At low cutting speeds, typically lower than 150 m/min, the sample probability density tends to have a sine wave probability shape. At high cutting speeds the shape are more undetermined in workpiece materials SS 2343-02 and SS 2541-03, but in SS 0727-02 the shape of the sample probability density tends to have a gaussian shape. According to the spectrum analysis, the major part of the vibration energy Vibration Analysis of a Boring Bar 87 was in the cutting speed direction and was dominated by a resonance peak at approximately 600 Hz. Harmonics of the resonance peak was encountered in all three materials but was most apparent in workpiece material SS 2541-03. A reason to harmonic behavior might be that the motion of the cutting tool is limited by the workpiece in both the cutting speed and the cutting depth directions. For increasing cutting speeds the high frequency content of the vibration energy was increased. Zooming in around the first resonance peak and increasing the frequency resolution reveals that the resonance frequencies are neither independent of the cutting speed nor the feed rate. Presumably different cutting speeds and feed rates apply different boundary conditions to the boring bar and thus alter the resonance frequencies. High frequency resolution also reveals the side band terms, which may be explained by imbalances in the spindle engine. During a continuous boring operation such low frequency spindle vibrations appear as sideband terms spaced with the speed of the spindle. From the operating deflection shape, ODS, analysis, it follows that the motion of the boring bar was mainly in the cutting speed direction during a cutting operation at both resonance peaks. The reason is probably that the largest part of the excitation of the boring bar is in the cutting speed direction. In comparison to the modal analysis the resonance frequencies were differing. This is probably partly due to that the ODS analysis was carried out during machining which introduce different boundary conditions, i.e. the workpiece apply boundary conditions to the cutting tool which are not present in the modal analysis. The Euler-Bernoulli calculations, the modal analysis and the operating deflection shape analysis all gave an estimate of the beam deflection. The results of these three analyses were rather similar. Thus the deflection shape of the boring bar do not seem to be influenced to a large extent by the different boundary conditions of these three analyses. 88 Part I

Acknowledgment

The project is sponsored by PCB Piezotronics and the Foundation for Knowledge and Competence Development.

References

[1] P-O. H. Sturesson, L. H˚akansson, and I. Claesson. Identiﬁcation of the statistical properties of the cutting tool vibration in a continuous turning operation - correlation to structural properties. Journal of Mechanical Systems and Signal Processing, Academic Press, 11(3), July 1997.

[2] G.W. Blankenship and R. Singh. Analytical solution for modulation sidebands associated with a class of mechanical oscillators. Journal of Sound and Vibrationes, 179(1):13–36, 1995.

[3] S. Goldman. Vibration Spectrum Analysis. Industrial Press Inc., 2nd edition, 1999.

[4] JR. M.L Adams. Rotating Machinery Vibration From Analysis to Trobleshoot- ing. Marcel Dekker, 2001.

[5] R.B. Randall. A new method of modeling gear faults. Journal of Mechanical Design, Transactions of the ASME, 104:259–267, April 1982.

[6] J.F. Doyle. Wave Propagation in Structures - Spectral Analysis Using Fast Discret Fourier Transforms. Springer, second edition, 1997.

[7] D.J. Inman. Engineering Vibration. Prentice-Hall, second edition, 2001.

[8] R.W. Clough and J. Penzien. Dynamics of Structures. McGraw-Hill, second edition, 1993.

[9] N.M.M. Maia and J.M.M. Silva. Theoretical and Experimental Modal Analysis. Research Studies Press Ltd., 1997.

[10] S. Lammens W. Heylen and Paul Sas. Modal Analysis Theory and Testing. Katholieke Universiteit Leuven, Division of Production Engineering, Machine Design &Automation, 2nd edition, 1997.

[11] R. J. Allemang. Vibrations: Analytical and Experimental Modal Analysis, 1994. Vibration Analysis of a Boring Bar 89

[12] J.S. Bendat and A.G. Piersol. Random Data Analysis And Measurement Pro- cedures. John Wiley & Sons, 2000.

[13] J.S. Bendat and A.G. Piersol. Engineering Application of Correlation and Spectral Analysis. John Wiley & Sons, second edition, 1993.

[14] A. Brandt. Ljud- och Vibrationsanalys I. Course material, SAVEN EduTech AB, Sweden, 1999.

[15] D. G. Manolakis J. G. Proakis. Digital Signal Processing - Principles, Algo- rithms and Applications. Prentice Hall, third edition, 1996.

[16] 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. IEEE Transactions on Audio and Electroacoustics, pages 70–73, June 1967.

Part II

Active Control of Machine-Tool Vibration in a CNC Lathe Based on an Active Tool Holder Shank with Embedded Piezo Ceramic Actuators Part II is published as:

L. Pettersson, L. H˚akansson, I. Claesson ans Sven Olsson, “Active Control of Machine-Tool Vibration in a CNC Lathe Based on an Active Tool Holder Shank withEmbedded Piezo Ceramic Actuators”, The8thInternational Congress on Sound and Vibration, Hong Kong SAR, China, 2-6 July 2001. Active Control of Machine-Tool Vibration in a CNC Lathe Based on an Active Tool Holder Shank with Embedded Piezo Ceramic Actuators

L. Pettersson†,L.H˚akansson†, I. Claesson† and Sven Olsson‡

†Department of Telecommunications and Signal Processing Blekinge Institute of Technology 372 25 Ronneby Sweden

‡Active Control Sweden AB, IDEON 223 70 Lund Sweden

Abstract

In the turning operation chatter or vibration is a frequent problem affecting the result of the machining, and, in particular, the surface finish. Tool life is also influenced by vibration. Severe acoustic noise in the working environment frequently occurs as a result of dynamic motion between the cutting tool and the workpiece. These problems can be reduced by active control of machine-tool vibration. However, machine-tool vibration control systems are usually not applicable to a general lathe and turning operation. The physical features and properties of the mechanical constructions or solutions involved regarding the introduction of secondary vibration usually limit their applicability. An active control solution for a general lathe application has been developed. It is based on a standard Industry tool holder with an embedded piezo ceramic actuator and an adaptive feedback controller. The adaptive controller is based on the well known filtered-x LMS-algorithm. It enables substantial reduction of the vibration level by up to 40 dB at 3.4 kHz. 94 Part II

1 Introduction

In turning operations the tool and tool holder shank are subjected to dynamic excitation due to the deformation of work material during the cutting operation. The stochastic chip formation process usually induces vibrations in the machine- tool system. Energy from the chip formation process excites the mechanical modes of the machine-tool system. Modes of the workpiece may also influence the tool vibration. The relative dynamic motion between cutting tool and workpiece will affect the result of the machining, in particular the surface finish. Severe acoustic noise is also introduced, the noise level is sometimes almost unbearable to the machine operator. The tool life is also likely to be correlated with the amount of vibrations. It is well known that vibration problems are closely related to the dynamic stiffness of the structure of the machinery and workpiece material. The vibration problem may be solved in part by proper machine design which stiffens the machine structure. In order to achieve further improvements the dynamic stiffness of the tool holder shank can be increased more selectively. Active control of machine-tool vibration is a solution to these problems. Gen- erally, machine tool systems are classified as narrow band systems and as a conse- quence tool shank vibrations can usually be described as a superposition of narrow band random processes at each modal frequency [1]. These when added together form a more wide band random process. The tool vibrations in a turning operation mainly comprise vibrations in two directions; the cutting speed direction and the feed direction [1]. Usually, the vibrations in the cutting speed direction and the feed direction are linearly independent, except at some of the eigenfrequencies [1]. Consequently, the control problem involves the introduction of secondary sources driven in such way the anti-vibrations generated by means of these sources will interfere destructively withthetool vibration. However, in external longitudinal turning, most of the vibrations are induced in the cutting speed direction. Thus, the control of the vibrations in the cutting speed direction is an adequate solution to the vibration problem [2]. In order to control the vibrating modes of a tool holder it is essential to select a location for the actuators that enables the introduction of secondary vibration in to these modes. However, the location for the mounting of the actuators must be selected carefully to avoid unnecessary reconstructions and/or performance reductions of the the lathe. By embedding piezo ceramic actuators in a standard industry tool holder the active control of tool vibration is enabled to a general lathe. A complication in the turning operation is that the original excitation of the tool vibration cannot be observed directly and can therefore not be used as a feedforward control signal. A solution to the controller problem is to control the adaptive FIR filter with the leaky version of the well-known filtered-x LMS-algorithm [2] This paper discusses the single-channel feedback control of tool vibration in the Active Control of Machine-Tool Vibration ... 95 cutting speed direction. The single channel control system is illustrated in Fig. 1 below. The tool holder used in this application has an embedded piezo ceramic

Signal from sensor detecting the tool holder response

Primary excitation Feedback W controller Tool holder shank

Secondary excitation via active actuator

Figure 1: The machine-tool feedback control system[2]. actuator, i.e. secondary source, which have been developed at the department of telecommunications and signal processing. The construction of the tool holder is showninFig.2.

Tool

a s Tool holder shank v Embedded and sealed piezo ceramic actuator

Figure 2: The Tool holder with embedded actuator for the control of tool vibration in the metal cutting process. 96 Part II

2 Materials and Methods

2.1 Experimental Set-Up The cutting trials have been carried out in a Mazak SUPER QUICK TURN - 250M CNC turning centre with18.5 kW spindle power, maximal machiningdiameter 300mm, 1007 mm between the centres. The tool holder construction is based on an embedded design withan piezo ceramic stack actuator and an accelerometer mounted on the cutting tool to make it possible to measure the vibrations in the cutting speed direction. In order to operate the piezoelectric stack actuator a custom designed ampliﬁer was used. A digital signal processor controller was used and the measurements were carried out on a two-channel signal analyzer. Furthermore, a two channel low-pass ﬁlter was used to adjust the input level to the A/D converter and the output level from the D/A converter.

2.1.1 Work Material - Cutting Data - Tool Geometry The workpiece material SS 2541-03, chromium molybdenum nickel steel [1], was used in the experiments. This work material excites the machine-tool-system with a narrow bandwidthin thecutting operation. After a preliminary set of trials a suitable combination of cutting data and tool geometry was selected, see table 1.

Geometry Cutting Depth of Feed speed, v cut, a s (m/min) (mm) (mm/rev) DNMG 150508-SL 7015 80 0.9 0.25

Table 1: Cutting data and tool geometry.

The combination was selected to cause signiﬁcant tool vibrations which resulted in an observable deterioration of the workpiece surface and severe acoustic noise. The diameter of the workpiece was deliberately chosen large (over 100 mm), in order to render the workpiece vibrations negligible.

2.2 Active Tool Holder In order to control the vibrating modes of a tool holder it is essential to select a location for the actuator that enables the introduction of secondary vibration into these modes. However, the location for the mounting of the actuator must be selected carefully to avoid unnecessary reconstructions and/or performance reductions of the the lathe. The tool vibration or bending motion in the tool holder, Active Control of Machine-Tool Vibration ... 97 introduced by the stochastic chip formation process may be attenuate by introducing a opposite bending moment in the tool holder. By mounting the actuator in the area of peak modal strain and optimizing the actuator oﬀset distance to the centre axis of the tool holder, a suitable control force my be introduced by a voltage induced actuator strain. Hence, the bending deformation of the tool holder introduce a axial deformation of the actuator and by producing an equal and opposite control force, the tool vibration is reduced. The principle of the active tool holder with embedded actuator is illustrated in Fig. 2.

2.3 Adaptive Control of Tool Vibration The original excitation of the tool vibrations, originating from the material deformation process, cannot be directly observed. Consequently, the controller for the control of machine-tool vibration is based on a feedback approach. The response of the tool holder can be measured with a sensor mounted on the machine-tool. By introduction of secondary anti-vibrations witha secondary source, actuator, the response of the tool holder can be modified [2]. The actuator is steered by a controller which is fed with the accelerometer signal sensing the vibrations of the tool holder. A block diagram of the feedback control system is shown in Fig. 3. The objective of the control is to minimize the mean square error. The use of the error signal as input signal to the adaptive FIR filter controlling the plant, will cause the adaptive FIR filter to act as a feedback controller. This will complicate the relation between the mean square error and the filter coefficients, i.e. the mean square error will not be a quadratic function of the filter coefficients. In fact the mean square error function may be multimodal in the filter coefficients [3]. The search for a minimum on the mean square error surface can be performed by the well-known filtered-x LMS algorithm defined by [2]:

y(n)=wT (n)x(n)(1)

e(n)=d(n) − yC (n)(2) ∗T xC∗ (n)=c x(n)(3)

w(n+1)= w(n)+µxC∗ (n)e(n)(4) where xC∗ (n) is the filtered reference signal vector. A block diagram of the feedback control system with the filtered-x LMS algorithm is shown in Fig. 3. In Fig. 3 the box with the unit delay operator q−1 at the input to the controller handle the fact that we are dealing with an adaptive digital filter in a feedback control system. Observe the feedback relation from x(n)=e(n−1). Furthermore, C represents the dynamic secondary system (forward path) under control, i.e. the electro-mechanic 98 Part II

d(n) Forward path y(n) y (n) FIR filter c Σ w(n) C

x(n) = e(n-1)

x C * (n) e(n) C* Adaptive algorithm Unit delay Model of the forward path q-1

Figure 3: Equivalent block diagram of the feedback control situation with the filtered-x LMS algorithm[2]. response. The estimate of this path is denoted C∗. It is in practice customary to use an estimate of the impulse response for the forward path. As a result, the reference signal xC∗ (n) will be an approximation, and differences between the estimate of theforward pathand thetrue forward pathinfluence boththestability properties and the convergence rate of the algorithm [4, 5, 6, 7]. The estimation error can be expressed as follows [2]; 1 e(n)= d(n)(5) 1+C(q)W (n, q)q−1 where delay-operator notation is used. From this expression it is obvious that the poles of filter, i.e. the poles of the transfer function between the desired signal d(n) and the estimation error e(n), are affected by the controller response. The stability of the feedback control systems thus depends on the ability of the filtered-x LMS-algorithm to control the adaptive FIR filter, the time varying controller response, without violating the closed loop stability requirements, i.e. the Nyquist stability criterion [8]. In feedback control, limiting the energy in the control signal to the plant yields a more robust behaviour. By introducing leakage in the filtered-x LMS-algorithm the “memory” of the adaptive algorithm is reduced thereby reducing the energy in the response of the adaptive FIR filter and also the energy in the control signal to the plant. The leaky version of the filtered-x LMS-algorithm is obtained through a modi- fication of the algorithm for the coefficient vector adaption of the filtered-x LMS- algorithm with a leakage factor γ. As a result, the algorithm for the coefficient Active Control of Machine-Tool Vibration ... 99 vector adaption of the leaky version of the filtered-x LMS-algorithm is given by [7]:

w(n+1) = γw(n)+µxC∗ (n)e(n)(6)

The leakage factor γ is a real positive parameter which satisﬁes the condition:

0 <γ<1(7)

The secondary path was estimated in an initial phase and was carried out by a second adaptive FIR filter steered by the LMS algorithm. In the estimation, a broadband PN-signal was used. The fixed FIR filter estimate of the forward path was subsequently used to prefilter the input signal to the algorithm for the adaptation of the coefficient vector in the filtered-x LMS algorithm. For the control of tool vibration a 20-tap adaptive FIR filter was used together with a 35-tap FIR filter estimate of the secondary path [2]. These filter lengths were at the limit for the processing capacity of the signal processor used. A 16 kHz sampling rate was chosen for the digital filter. In order to minimize delay in the loop, no anti-aliasing or reconstruction filters were used. Obviously, this necessitates extra care being taken in order to avoid aliasing.

3 Results

The tool shank vibrations considered in this paper originate from the cutting speed direction of the tool holder shank. To illustrate the effect of feedback control of tool vibration in the cutting speed direction, the spectral densities of the tool vibrations with and without feedback control are shown in the same diagram. Figure 3 shows a typical result obtained withadaptive feedback control of tool-vibration. It performs a broad-band attenuation of the tool-vibration and manage to reduce the vibration level withup to approximately 40 dB at 3.4 kHz. If a leakage factor is introduced in to the filtered-x LMS-algorithm the performance of the adaptive control system will be reduced. This is illustrated by the spectral densities given in Fig. 3. In order to illustrate the influence of leakage on the stability of the feedback control system, the significant part of the Nyquist plot, i.e. the part of the Nyquist plot closest to the point (−1, 0), is given for estimates of the open loop frequency response with and without leakage in the filtered-x LMS algorithm. Fig. 6a shows the Nyquist plot for the case of no leakage; and Fig. 6b shows the Nyquist plot for the case of leakage in the filtered-x LMS algorithm. In the experiments, it was observed that the adaptive feedback control of tool vibration resulted in a significant improvement of the workpiece surface. In Fig. 7 a photo of the workpiece used in the experiments is shown. 100 Part II

60 active off active on 50

40 /Hz] 2 )

2 30

10 PSD [dB rel 1 (m/s 0

20 2500 3000 3500 4000 4500 5000 5500 Frequency [Hz]

Figure 4: The spectral density of tool vibration with 20 tap FIR ﬁlter feedback control (solid) and without (dashed). Step length µ = -1, cutting speed v =80 m/min, cut depth a = 0.9 mm, feed rate s = 0.25 mm/rev, tool DNMG 150508-SL , grade 7015. Withleakage factor γ =1.

60 active off active on 50

40 /Hz] 2 )

2 30

10 PSD [dB rel 1 (m/s 0

20 2500 3000 3500 4000 4500 5000 5500 Frequency [Hz]

Figure 5: The spectral density of tool vibration with 20 tap FIR ﬁlter feedback control (solid) and without (dashed). Step length µ = -1, cutting speed v =80 m/min, cut depth a = 0.9 mm, feed rate s = 0.25 mm/rev, tool DNMG 150508-SL , grade 7015. Withleakage factor γ =0.9999. Active Control of Machine-Tool Vibration ... 101

Nyquist diagram, γ = 1 Nyquist diagram, γ = 0.9999 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

Im[W(f)C(f)] −0.2 Im[W(f)C(f)] −0.2

−0.4 −0.4

−0.6 −0.6

−0.8 −0.8

−1 −1 −1.5 −1 −0.5 0 0.5 −1.5 −1 −0.5 0 0.5 Re[W(f)C(f)] Re[W(f)C(f)] a) b) Figure 6: Nyquist diagram for the feedback control system with and without leakage, step length µ = -1. In a) γ =1andinb)γ =0.9999

Figure 7: The workpiece surface with and without adaptive feedback control 102 Part II

4 Conclusions and Future Work

It is clear from the results presented that tool vibrations in a lathe during metal cutting can be controlled by using a tool hoder with an embedded piezoelectric actuator and a adaptive FIR filter feedback controller. Furthermore, by embedding the actuator in the tool holder, the active control of tool vibration is likely be applicable in a arbitrary lathe. The adaptive feedback control performs a broad-band attenuation of the tool- vibration, and is able to reduce the vibration level by almost 40 dB at 3.4 kHz (see Fig. 3). It is thus essential that the adaptive feedback control of machine-tool vibration handles the time varying environment. And indeed, the leaky filtered-x LMS algorithm appears to have great potential with respect to the feedback control of tool vibrations in the turning operation. It has been reported that the leaky version of the feedback filtered-x LMS-algorithm is robust to large variations in the spectral properties of tool vibration [2]. The leakage factor reduce the magnitude of the frequency response of the adaptive FIR filter, i.e. causes the loop gain of the control system to be reduced and is so doing increases the distance between the trajectory of the open loop frequency response and the point (−1, 0). This can be observed by comparing the Nyquist diagram for the open loop response for the control system when the filtered-x LMS algorithm is used to control the response of the adaptive FIR filter shown in Fig. 6. The control system is thus likely to be more robust in the Nyquist sense when the leaky filtered-x LMS algorithm controls the response of the adaptive FIR filter. From a manufacturing point of view the improvements of the surface is of great importance. The surface is a result of a more stable cutting operation due to adaptive control system. The reduced noise level around the lathe is also important. Today, the industry are facing more and more regulations concerning the noise level in the working area of the employees . It is also interesting to note that the adaptive technique does not affect the cutting data, it may even allow an increase of the material removal rate. It is also likely that the tool life, which is an important cost to the manufactures, will increase. Future work in this project is for example to transfer the active vibration control technique to boring operations. Further study of different control algorithms is also urgent.

Acknowledgment

The project is sponsored by The Foundation for Knowledge and Competence De- velopment. Active Control of Machine-Tool Vibration ... 103

References

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[3] S.J. Elliott. Active control using feedback. Technical Report 732, Institute of Sound & Vibration Research, University of Southampton, January 1994.

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