Smirnova_diss Machini Dynamics Boring Bar

8/11/2019 Smirnova_diss Machini Dynamics Boring Bar

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Dynamic Analysis and Modeling

of Machine Tool Parts

Tatiana Smirnova


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2008 Tatiana SmirnovaDepartment of Signal ProcessingSchool of EngineeringPublisher: Blekinge Institute of TechnologyPrinted by Printfabriken, Karlskrona, Sweden 2008ISBN 978-91-7295-128-0


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Abstract

Boring bar vibration during internal turning operations in machinetools is a pronounced problem in the manufacturing industry. Vibra-tion may easily be induced by the workpieces material deformation pro-cess, due to the bars normally slender geometry. In order to overcomethe vibration problem in internal turning active or/and passive controlmethods may be utilized. The level of success achieved by implement-ing such methods is directly dependent on the engineers knowledge of the dynamic properties of the system to be controlled.

This thesis focuses predominantly on three steps in the development

of an accurate model of an active boring bar. The rst part considersthe problem of building an accurate 3-D FE model of a standardboring bar used in industry. The inuence of the FE models meshdensity on the accuracy of the estimated spatial dynamic properties isaddressed. With respect to the boring bars natural frequencies, the FEmodeling also considers mass loading effects introduced by accelerom-eters attached to the boring bar. Experimental modal analysis resultsfrom the actual boring bar are used as a reference.

The second part discusses analytical and experimental methods formodeling the dynamic properties of a boring bar clamped in a machinetool. For this purpose, Euler-Bernoulli and Timoshenko distributed-

parameter system models are used to describe the dynamics of the bor-ing bar. Also, 1-D FE models with Euler-Bernoulli and Timoshenkobeam elements have been developed in accordance with distributed-parameter system models. A more complete 3-D FE model of thesystem boring bar - clamping house has also been developed. Spatialdynamic properties of these models are discussed and compared withadequate experimental modal analysis results from the actual boringbar clamped in the machine tool. This section also investigates sensi-tivity of the spatial dynamic properties of the derived boring bar modelsto variation in the structural parameters values.

The nal part focuses on the development of a 3-D FE model of the system boring bar - actuator - clamping house, with the pur-pose of simplifying the design procedure of an active boring bar. Alinear model is addressed along with a model enabling variable con-tact between the clamping house and the boring bar with and withoutCoulomb friction in the contact surfaces. Based on these FE modelsfundamental bending modes, eigenfrequencies and mode shapes, controlpath frequency response functions are discussed in conjunction with thecorresponding quantities estimated for the actual active boring bar.


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Preface

This licentiate thesis summarizes my work at the Department of SignalProcessing at Blekinge Institute of Technology. The thesis is comprised of three parts:

Part

I On accurate FE-modeling of a Boring Bar with Free-Free BoundaryConditions.

II Dynamic Modeling of a Boring Bar Using Theoretical and ExperimentalEngineering Methods.

III Modeling of an Active Boring Bar.


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Acknowledgments

I would like to express my sincere gratitude to all people who inuencedmy studies and helped me during the years towards Licentiate Candidacy.

First of all I would like to express my deep gratitude to Professor IngvarClaesson for giving me the opportunity to conduct research in the form of aPh. D. position at the Blekinge Institute of Technology and for his supervi-sion throughout my work. I would also like to thank my research supervisorand friend Associate Professor Lars Hakansson for his constant guidance, hisprofound knowledge and experience in the elds of applied signal processingand mechanical engineering, his support in writing this thesis, and also for hiscontinuous care. I am most grateful to my dear friends Associate ProfessorNedelko Grbic and his wife Marina for their warmth, inspiration, encourage-ment and all their help during my stay in Sweden. Many thanks goes to allmy present and former colleagues at the department of Signal Processing forbeing so helpful, friendly and cheerful, creating a great working environment.Especially, I am indebted to my colleague and friend Henrik Akesson for allhis help, patience and many fruitful discussions.

I am grateful to my parents Nadezhda and Alexandr and my sister Anas-tasia for their endless love, support and care. Finally, I would like to thankmy husband Sergey for acceptance of my choices, patience, understanding andall his love.

Tatiana Smirnova Karlskrona, December 2007


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Contents

Publication list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Part

I On accurate FE-modeling of a Boring Bar with Free-Free BoundaryC o n d i t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9

II Dynamic Modeling of a Boring Bar Using Theoretical and ExperimentalEngineering Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

III Modeling of an Active Boring Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97


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Other Publications

T. Smirnova, H. Akesson, L. Hakansson, I. Claesson and T. Lag o, Investi-gation of Boring Bar Mode Shape Rotation by Experimental Modal Analysis in Correlation with Finite Element Modeling , Noise and Vibration Engineer-ing Conference, ISMA2006, Leuven, Belgium, 18 - 20 September 2006.

H. Akesson, T. Smirnova, L. Hakansson, T. Lag o, I. Claesson, Analog and Dig-ital Approaches of Attenuation Boring Bar Vibrations During Metal Cutting Operations , published in proceedings of The Twelfth International Congresson Sound and Vibration, ICSV12, Lisbon, Portugal, July 11 - 14, 2005.

H. Akesson, T. Smirnova, L. Hakansson, I. Claesson and T. Lag o, Analog versus Digital Control of Boring Bar Vibration , Accepted for publication inproceedings of the SAE World Aerospace Congress, WAC, Dallas, Texas, USA,October 3-6, 2005.

H. Akesson, T. Smirnova, L. Hakansson, I. Cleasson, Andreas Sigfridsson,

Tobias Svensson and Thomas Lag o, Active Boring Bar Prototype Tested in Industry , Adaptronic Congress 2006, 03-04 May, Gottingen, Germany.

H. Akesson, T. Smirnova, L. Hakansson, I. Claesson, A. Sigfridsson, T. Svens-son and T. Lago, A First Prototype of an Active Boring Bar Tested in In-dustry , In Proceedings of the Twelfth International Congress on Sound andVibration, ICSV12, Lisbon, Portugal, 11 - 14 July, 2006.

H. Akesson, T. Smirnova, L. Hakansson, I. Claesson and T. Lag o, Comparison of different controllers in the active control of tool vibration; including abrupt changes in the engagement of metal cutting , Sixth International Symposiumon Active Noise and Vibration Control, ACTIVE, Adelaide, Australia, 18-20September, 2006.

H. Akesson, T. Smirnova, L. Hakansson, I. Claesson and T. Lag o, Vibration in Turning and the Active Control of Tool Vibration published in proceedingsof WCEAM-CM2007, Harrogate, 2007.


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H. Akesson, T. Smirnova, L. Hakansson, I. Claesson and T. Lag o, Inves-tigation of the Dynamic Properties of a Boring Bar Concerning Different Boundary Conditions published in proceedings of The Fourteenth Interna-tional Congress on Sound and Vibration, ICSV14, Cairns, Australia, 2007.

H. Akesson, T. Smirnova, L. Hakansson, and T. Lag o, Analysis of Dynamic Properties of Boring Bars Concerning Different Clamping Conditions Re-search Report 2007:6, ISSN: 1103-1581.

H. Akesson, T. Smirnova, I. Claesson and L. Hakansson, On the Develop-ment of a Simple and Robust Active Control System for Boring Bar Vibrationin Industry, IJAV-International Journal of Acoustics and Vibration , 12(4),pp. 139-152, 2007.


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Introduction

Analyzing the dynamic properties of mechanical systems involves an exam-ination of the time-varying response of a system under applied time-varyingexcitation force. In this case, the response of the system represents a repeti-tive motion in time about the systems position of equilibrium, and is referredto as a vibration or an oscillation [1,2]. Dynamic analysis serves to e.g. pre-dict the dynamic response of the structure by calculating its spatial dynamicproperties; i.e. its natural frequencies and mode shapes, etc.

Vibration is a common problem in the manufacturing industry, in partic-ular, with respect to metal cutting (e.g., during turning, milling and grindingoperations). Internal turning involves machining cavities inside workpiece ma-terials to pre-dened geometries by means of a tool holder usually referred toas a boring bar, see Fig 1. Traditionally, the interface between cutting insertand machine tool, i.e., the tooling structure, is considered to be the weakestlink in the machining system [3,4]. With respect to internal turning, the crit-ical component of the tooling structure is usually the boring bar, which maybe clamped inside the clamping house by means of screws, hydraulic pres-sure, spring-clamping, etc. The tooling structure may exhibit vibrations of different kinds: free and forced vibrations, as well as self-excited chatter [36].Vibrations such as these may result in the following: failure to maintain ma-chining tolerances, unsatisfactory surface nish, excessive tool wear (and thusdecreased productivity of the lathe). Different methods may be suggestedto reduce degrading vibration problems in internal turning and improve pro-ductivity and working environment. For instance, active and passive controlmethods may be utilized [3,79]. The level of success of utilizing active and/orpassive methods for the reduction of tool vibration is closely related to theengineers knowledge of the dynamic properties of the tooling structure [3,7].

The dynamic properties of boring bars may be estimated using a numberof different approaches. Simple models of the boring bar can be created us-ing Euler-Bernoulli or Timoshenko beam theory. However, these distributed-parameter system models are not capable of accurately describing the actualboring bars geometry and boundary conditions (such as interfaces and jointsbetween machine tool parts, e.g., the screw clamping of the boring bar in-side the clamping house). Assuming rigid boundary conditions while utilizingsuch distributed-parameter system models leads to oversimplication of thereal structure, and rough estimates of the dynamic properties. Thus, the


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Figure 1: The part of the lathe Mazak SUPER QUICK TURN - 250M CNCwhere machining is carried out.

inuence of joints should not be underestimated, because they not only in-troduce damping in the structure, but may also contribute to nonlinearityin the structures response [10, 11]. More accurate models of machine toolparts together with joints and contacting interfaces can be developed usingnumerical methods, e.g., nite element analysis [11]. In order to verify andupdate models of dynamic systems, modal testing techniques or experimentalmodal analysis is generally used to provide information regarding the actualdynamic behavior of a system [10].

Experimental Modal Analysis

Experimental Modal Analysis (EMA) is usually referred to as the processof identifying a systems dynamic properties (such as natural frequencies,relative damping ratios and mode shapes) based on the experimental vibrationmeasurements of the time-varying excitation and systems response signals[12].


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Experimental modal analysis is built on the following assumptions: thatthe system is linear and time invariant, the system is observable, and Maxwellsreciprocity principle holds [13]. The experimental modal analysis procedurebasically consists of three stages: measurement planning, frequency responsemeasurements, modal data extraction [14]. The concept of experimentalmodal analysis will be described in the subsequent text based on an examplestructure, a simple clamped-free beam or cantilever beam, see Fig. 2. Therst three bending modes of the cantilever beam provided as an example willbe considered in the experimental modal analysis description.

Measurement planning In order to extract estimates for the rst threenatural frequencies, damping ratios and mode shapes of the cantilever beam,it is generally sufficient to measure the transverse response of the beam in atleast three spatial locations [10]. The most widely used transducers in vibra-tion measurements are as follows: the accelerometer and the force transducer.Usually, force transducers are used to measure the force produced by excita-tion sources such as impulse hammers and electrodynamic shakers. Charac-teristics such as the accelerometers weight, axial sensitivity and transversesensitivity, frequency range, etc., should be considered during measurementplanning. Other considerations include the selection of suitable transducerlocations, and the method of attaching the transducers to the structure (can-tilever beam). It is generally preferable to avoid attaching accelerometers andforce transducers to the structure in the vicinity of nodes of eigenmodes thatare important to identify by the experimental modal analysis. Methods suchas distributed-parameter system modeling and nite element modeling of thestructure may be utilized prior to measurement in order to predict the posi-tions of relevant structural nodes. Now, let us assume that we have selectedthree adequate response and excitation positions for measurement and excita-tion in the transverse direction of the example cantilever beam, as illustrated

in Fig. 2. In the setup of an electrodynamic shaker it is usually important tosuspend the shaker by exible strings in order to isolate eventual excitationof the structure to be analyzed via the shaker suspension. The shaker shouldideally only apply force strictly in the desired excitation direction to the struc-ture via a force transducer, and, if possible, the shaker should not impose anymass loading on a structure. The spatial dynamic properties of a structuremodeled as an N degree-of-freedom system may usually be described by sym-metric stiffness [ K ], mass [M ] and damping [C ] matrices [10, 15]. Thus, inthe frequency domain, the relation between the input forces and the output


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accelerations of the system approximated by the N degrees-of-freedom may,principally, be described by a N N symmetric accellerance matrix or fre-quency response matrix [ H (f )] [12]. Hence, it may be sufficient to measureone row or one column in [H (f )] in order to reconstruct the complete acceller-ance matrix [10]. In order to estimate the required accelerance functions of one row or one column in [H (f )], a suitable set of acceleration responses andexcitation forces have to be measured and recorded. There are two commonways to perform the measurements. In the rst case, the excitation force isapplied and measured in the transverse direction in one of the cantilever beammeasurement positions simultaneous with its response at the three measure-ment locations using three vibration sensors, illustrated in Fig. 2. In thesecond case, only one vibration transducer is used to measure the cantileverbeam response in one of the three measurement positions. The excitation forceis applied and measured in the transverse direction in one of the three po-sitions simultaneous with its response at the selected response measurementlocation. Subsequently, excitation force is moved to the next measurementposition and the force and response are again measured simultaneously. Fi-nally, this is repeated, with the excitation force moved and applied to theremaining measurement position. Selecting a method for carrying out the

measurements depends upon the number of available vibration transducersand the type of excitation signal that is required to extract a sufficient modalmodel [10,12] for the system, etc. Basically, a modal model is an approach todescribe frequency response function matrix of a system in terms of partialfraction expansion where residues are dependent on mode shapes and polesare damped natural frequencies of the system [10,12].

Figure 2: Experimental setup for experimental modal analysis of examplecantilever beam.


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Frequency Response Function Measurement The excitation forces andstructures response signals from the transducers are recorded by means of asignal analyzer. Modern signal analyzers are usually based on a PC with (forexample) experimental modal analysis software, connected to a data acquisi-tion system with a suitable number of input and output channels. The sensoroutput signals are basically low-pass ltered, analog-to-digital converted bya data acquisition system. Generally, parallel with the measurements, theacquired data are transferred to the PC. Fast Fourier Transform (in combi-nation with windowing and averaging) is utilized to estimate power spectraldensity for the force signals or excitation signals, and cross power spectraldensity between the response signals and excitation signals. These quanti-ties are used to produce an estimate of the accelerance matrix or frequencyresponse function matrix. Coherence function and random error are usuallycalculated to ensure the quality of the estimates produced [10,16].

Figure 3: Measured force and responses for the cantilever beam example,and corresponding frequency response functions estimates or magnitude andphase functions of accelerance functions.


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Modal Data Extraction At this phase of experimental modal analysis(EMA) the dynamic properties of the structure are estimated. This process isalso often referred to as a curve tting procedure. The curve tting procedurecan be carried out using various techniques both in the time and frequencydomain [10, 13]. However, simple single mode or SDOF methods are suffi-cient to explain the concept of the modal data extraction. In the case of EMA of the cantilever beam example (with well separated modes) the natu-ral frequencies and relative damping ratios may be estimated from the drivingpoint frequency response function (the FRF estimate between the excitationsignal and the response signal, measured along the same direction and at thesame position on the structure). The three damped natural frequencies canbe estimated simply by choosing the frequencies corresponding to the threemaximum values of the magnitude function |H (f )|, illustrated for one peak inFig. 4. The relative damping can be estimated using the Half-power band-width method (see Fig. 4). Subsequently, with the aid of this information andthe damped eigenfrequencies, estimates of the undamped natural frequenciescan be produced [12,17].

Figure 4: Magnitude function of the driving point accelerance function for thecantilever beam example; identication of a damped natural frequency andcorresponding relative damping using the Half-power bandwidth method.

The next step involves extracting the mode shapes from the measureddata. An illustrative but rough approach is to construct a mode shape asfollows: rst, one of the damped eigenfrequencies, f di where i {1, 2, 3}, is


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selected. Subsequently, the peak values of the imaginary part of the acceler-ance functions Im{H s 1(f di )} where s {1, 2, 3} (assuming that the cantileverbeam is excited at position 1) corresponding to the selected eigenfrequencyare extracted. The mode shapes are nally constructed as { Im{H 11 (f di )},Im {H 21 (f di )}, Im{H 31 (f di )}}T where i {1, 2, 3}. The three extracted modeshapes are illustrated in Fig. 5. The validity of this statement follows fromthe modal model [12].

0200

400600

8001000

0.4

0.3

0.2

0.10

0

1000

2000

F r e q u e n c y, [ H

z ]

L e n g t h o f t h e b e a m , [ m ]

I m a g i n a r y p a r

t o f a c c e

l e r a n c e , [

m / s 2 / N ]

Mode 1Mode 2

Mode 3

Figure 5: Illustration of the mode shape extraction.

Distributed-Parameter System ModelIn practice, all engineering structures can be considered as systems with dis-tributed parameters. This implies that a structure consists of an innitenumber of continuously distributed innitesimal mass particles connected toeach other with some elasticity and energy dissipation mechanism. Thus,structures inertial, elastic and damping properties are distributed in spaceand often referred to as distributions [2].

The distributed-parameter system model is considered to be a model withinnite number of degrees-of-freedom, this relates to another distinctive fea-


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ture of these models - they are characterized by an innite number of eigen-modes [12]. The displacement of the structure is described by a continuousfunction dependent on time and spatial variables. In the simplest case (if thetransverse vibration of a 1-D beam is considered, see Fig. 6) the transversaldisplacement is w(z, t ).

Figure 6: Distributed-parameter model of transverse vibration of a cantileverbeam.

The equation of motion for a system with distributed parameters such

as transverse vibration of a simple beam may conveniently be derived basedon Newtons second law, and is a partial differential equation [12]. For in-stance, the Euler-Bernoulli model of transverse vibration of the beam is de-rived considering an innitesimal element of the beam with length dz, seeFig. 6. Basically, two equilibrium equations are formed: all forces acting onthe element in vertical direction (inertial force A

2 w(z,t )t 2 dz and shear forces

Q(z, t ) respective Q(z, t ) + Q (z,t )z dz) are summed, and the moments M (z, t ),M (z, t ) + M (z,t )z dz and (Q(z, t ) +

Q (z,t )z dz)dz acting on the element about

the x-axis through point PP are summed. The summation of moments andforces are carried out based on right-hand rule and general positive rotation

convention [7]. Shear deformation is neglected by this model, yielding a shearforce Q(z, t ); that is proportional to the spatial change in the bending momentQ(z, t ) = M (z,t )z . The inuence of rotary inertia is also neglected [12]. Themodel assumes the bending moment is inversely proportional to the radiusof curvature of the bent element M (z, t ) = EI xR = EI x

2 w(z,t )z 2 [7]. The

equation of motion for the free transverse vibration of the distributed param-eter cantilever beam is given by the following fourth order partial differentialequation:


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A(z) 2w(z, t )

t 2 + EI x

4w(z, t )z 4

= 0 (1)

In order to nd a closed-form solution to this equation, four boundaryconditions and two initial conditions are required. The combination of thepartial differential equation (describing the dynamic motion of the structure)and the boundary conditions (which are imposed upon the structure) is oftenreferred to as a boundary-value problem [2].

A closed-form solution to the boundary-value problem may only be foundif the structures material is homogeneous, elastic and isotropic [18]. Dy-namic response can, in this case, be produced as a sum of the normal modecontributions [12].

w(z, t ) =

n =1T n (t)Z n (z) (2)

where T n (t) is nth temporal solution and Z n (z) is nth normal mode. Theclosed-form solution contains an innite number of mode shapes. However, inmost cases it is sufficient to consider only a few of them, i.e., those contributing

the most to the structures dynamic response [18].A closed-form solution is often impossible to obtain for a general typeof structure, e.g., a structure combining various boundary conditions [18].In the case of modeling nonlinear systems, discrete-parameter systems withapproximate solutions are suggested.

Finite Element Model

The nite element method (FEM) was developed for the modeling and analysisof complicated structures when closed form solutions are difficult to obtain.This method is based on the approximation of a continuum structure by the

assembly of a nite number of parts (elements), and is based on the variationaland interpolation methods for modeling and solving boundary value problems[12,15].

In the modeling of a structure with the nite element method, a spatialmodel, assembled with discrete nite elements connected via the endpointscalled nodes (should not be mistaken for the nodes of vibrating modes of astructure) that approximate the actual structures spatial geometry, is pro-duced. The force-displacement relationships are established for each niteelement based on the principal of virtual work [2, 15]. A spatial solution is


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assumed for each nite element and approximated by a low-order polynomialknown as a shape function . At this stage local stiffness and mass matricescan be derived based on relations for the kinetic and strain energy, and shapefunctions. The nite elements are assembled into a nite element model of the structure. Global stiffness and mass matrices are constructed based onthe local ones. The model of the structures dynamic response, unlike in thecase of the distributed-parameter system, is governed by a system of ordinarydifferential equations. During the solution process, the equilibrium of forcesat the joints and compatibility of displacements between the elements aresatised, so the assembled nite elements are made to behave as a completestructure. The time response can be found using well-developed numericalintegration techniques [1, 14].

The concept of the nite element method can be described using the ex-ample of the transverse vibration of the cantilever beam. The nite elementmodel of the clamped-free beam consists of four nodes and three nite ele-ments, see Fig. 7.

Figure 7: Finite element model of a cantilever beam.

In order to describe transverse vibrations of the beam, each node has onetranslational and one rotational degree-of-freedom. Thus, the simplest beamelement has two nodes, with four degrees-of-freedom in total.

The displacement of any point within the nite element can be describedby the function [2]

w(z, t ) =4

i=1

q i (t)n i (z) (3)

where q i (t) are generalized coordinates , or degrees-of-freedom and ni (z)


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are shape functions [2]. In the case of the Euler-Bernoulli beam element, thegeneralized coordinates are translational displacements w1(t) and w2(t) androtational displacements 1(t) and 2(t) at the rst and the second node of the element, respectively; see Fig. 8. The shape functions are determinedover a nite element. They have a maximum amplitude equal to unity, andare equal to zero outside the nite element. However, the shape functions arethe same for all elements of a certain type.

In this case, the shape functions can be derived from the fact that thetransverse displacement must satisfy

2

z2 EI

x

2 w(z,t )

z2 = 0 and boundary

conditions at the ends of the element with length l, see Fig. 8.

Figure 8: Shape function of the Euler-Bernoulli beam element.

The element stiffness [ K ]e

and mass matrices [ M ]e

can be derived basedon the expressions for the strain and kinetic energy of the Euler-Bernoullibeam element [12].

V = 12

l

0EI x

2w(z, t )z 2

2

dz (4)

T = 12

l

0A

w(z, t )t

2

dz (5)


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The dimension of the element matrices [ K ]e and [M ]e is 4 4 in correspon-dence with the amount of degrees-of-freedom assigned to an Euler-Bernoullibeam nite element. The system of differential equations describing the freevibration of the single element can be written as follows:

[M ]e{w (t)}e + [K ]e{w (t)}e = {0} (6)where {w (t)}e = [w1(t), 1(t), w2(t), 2(t)]T is the vector of the Euler-Bernoullibeam element displacements.

The global stiffness and mass matrices are assembled from individual ele-ment matrices by summarizing their elements at common degrees of freedom,see Fig. 9.

[K ] =N e

e=1

[K ]e , (7)

for common degrees of freedom.

[M ] =N e

e=1[M ]e , (8)

for common degrees of freedom, where N e is the number of nite elements inthe model.

Boundary conditions, for instance for a xed support, may be applied inthe following manner: rows and columns corresponding to restricted degrees-of-freedom are removed from the global stiffness and mass matrices, see Fig.9.

The equations of motion for the free vibration of an undamped mechanicalsystem can now be described by a system of linear differential equations:

[M ]{w (t)} + [K ]{w (t)} = {0} (9)where {w (t)} is the vector containing unknown displacements of all degrees-of-freedom in the nite element model.The natural frequencies and mode shapes can be calculated based on Eq.

9, assuming that that the temporal solution is harmonic, yielding [15]:

((2f )2[M ] + [K ]){} = {0} (10)Where is a normal mode of the system [15]. This general linear eigen-

value or characteristic value problem can be solved using standard modal


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Figure 9: Illustration of the process of assembling global stiffness and massmatrices including application of the boundary conditions.

analysis procedure [12]. In nite element analysis software methods, such asInverse Power Sweep or Lanczos are implemented for this purpose [1].

PART I - On accurate FE-modeling of a Boring

Bar with Free-Free Boundary ConditionsVibration problems encountered during internal turning operation in manu-facturing industry require adequate passive and/or active control techniquesto increase the productivity of machine tools. Passive control is frequentlytuned to increase the dynamic stiffness of a particular boring bar at a cer-tain eigenfrequency. This may result in a redesigning of the system, whichis a costly and inexible solution. Active control based on, for example, anadaptive feedback controller and a boring bar with integrated piezoceramicactuator and vibration sensor can easily be adapted to different conditions.

This solution is more exible and may, therefore, prove to be preferable. Inorder to simplify the process of designing of an active boring bar, an accuratemathematical model of the active boring bar is required. This thesis addressesthe procedure involved in developing such a model.

The rst part of this thesis focuses on the development of an accuratemodel of a standard boring bar used in industry. The nite element method,utilizing 3-D nite elements is suggested, in order to obtain a precise de-scription of the boring bars geometry. The natural frequencies, mode shapesand rotational angles of the mode shapes of the boring bar were estimated


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distributed-parameter system model, the exibility of the section of the boringbar clamped inside the clamping house was described by means of pinned-pinned-free boundary conditions. The boring bars mode shapes and naturalfrequencies were estimated using 1-D FE models with Euler-Bernoulli andTimoshenko beam elements, with clamped-free and pinned-pinned-freeboundary conditions. In order to further improve the spatial dynamic proper-ties estimates, a 3-D FE model of the boring bar with boundary conditionsimposed by the rigid clamping house, and a 3-D FE model of the systemboring bar - deformable clamping house were utilized (see Fig. 11). Esti-mates of the natural frequencies and mode shapes obtain by means of variousmodels were compared to the estimates produced by experimental modal anal-ysis. Finally, the sensitivity of the spatial dynamic properties was investigatedwith respect to the variation in the structural parameters values.

Figure 11: 3-D nite element models of the system boring bar - clamping

house with deformable clamping house.

PART III - Modeling of an Active Boring Bar

Part III summarizes the development of a 3-D FE model of the active boringbar. A mathematical model, such as this, is required in order to simplify thedesign procedure of an active boring bar: i.e., the choice of the actuators


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characteristics, the actuator size, the position of the actuator in the boring bar,etc. The 3-D FE model contains the sub-models of the boring bar, actuatorand clamping house, and incorporates the piezo-electric effect. The spatialdynamic properties are predicted using the 3-D FE model and comparedto estimates produced by means of experimental modal analysis of the actualactive boring bar clamped in the lathe. Control path transfer functions (i.e.,frequency response functions between the actuator voltage and accelerationsat the error sensors positions) are calculated based on the 3-D FE model bymeans of harmonic response and transient response simulations, and comparedto those estimated experimentally. The inuence of the Coulomb friction forceon the active boring bars dynamics was investigated by means of arctangentand bilinear models: rstly, with respect to the example of the SDOF model,and subsequently on the 3-D FE model of the active boring bar. Finally,receptance functions for the boring bar - actuator interfaces were estimatedusing the 3-D FE model.

References

[1] J.W. Tedesco, W.G. McDougal, and C.A. Ross. Structural Dynamics:Theory and Applications . Addison Wesley Longman, Inc., 1999.

[2] L. Mierovitch. Fundamentals of Vibrations . McGraw-Hill Companies,Inc., 2001.

[3] E. I. Rivin. Tooling structure: Interface between cutting edge and ma-chine tool. Annals of the CIRP , 49(2):591634, 2000.

[4] L. Hakansson, S. Johansson, and I. Claesson. Chapter 78 - Machine Tool Noise, Vibration and Chatter Prediction and Control to be published in John Wiley & Sons Handbook of Noise and Vibration Control, Malcolm J. Crocker (ed.) . John Wiley & Son, 2007.

[5] S.A. Tobias. Machine-Tool Vibration . Blackie & Son, 1965.

[6] H. E. Merritt. Theory of self-excited machine-tool chatter. Journal of Engineering for Industry , pages 447454, 1965.

[7] C.H. Hansen and S.D. Snyder. Active Control of Noise and Vibration .E& FN Spon, 1997.


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Part I is published as:

T. Smirnova, H. Akesson, L. Hakansson, I. Claesson, and T. Lag o, Accurate FE-modeling of a Boring Bar Correlated with Experimental Modal Analysis ,In proceedings of the IMAC-XXV A Conference and Exposition on StructuralDynamics, February 19-22, 2007, Orlando, Florida USA.


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On accurate FE-modeling of a BoringBar with Free-Free Boundary

ConditionsT. Smirnova, H. Akesson, L. Hakansson, I. Claesson, and T. Lag o*

Department of Signal Processing,Blekinge Institute of Technology

372 25 RonnebySweden

*Acticut International ABGjuterivagen 7

311 32 Falkenberg, Sweden

Abstract

In metal cutting, the problem of boring bars vibration leads to sig-nicant degrading of productivity. A boring bar is very exible andeasily subject to vibrations, due to the large length to diameter ratio,required to perform internal turning. Boring bar vibrations appear atthe bars rst eigenfrequncies, which correspond to the boring barsrst bending modes affected by boundary conditions applied by theclamping and workpiece in the lathe. Therefore, investigation of thespatial dynamic properties of boring bars is of great importance for theunderstanding of the mechanism and nature of boring bar vibrations.This paper addresses the problem of building an accurate 3-D niteelement model of a boring bar with free-free boundary conditions.Considerations related to appropriate meshing and its inuence on theboring bar FE models spatial dynamic properties, as well as modelingthe effect of mass loading are discussed. Results from simulations of the 3-D nite element model of the boring bar (i.e., its rst eigen-modes and eigenfrequencies) are correlated with results obtained both


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22WhitePart I

from experimental modal analysis and analytical calculations using anEuler-Bernoulli model.

1 Introduction

A boring bar is the tool holder used when performing metal cutting in inter-nal turning operations. It is clamped inside the clamping house with boltsat one end and has a cutting tool attached to the other end, and is used formachining deep precise geometries inside the workpiece material. High levels

of boring bar vibration frequently occur under the load applied by the ma-terial deformation process. Boring bar vibrations are easily excited due tothe bars large length-to-diameter ratio (usually required to perform internalturning operations) and also because of exibility in the clamping system,i.e., clamping house and clamping screws, etc. High levels of vibrations resultin a poor surface nish, reduced tool life, severe acoustic noise in the work-ing environment, and occur at frequencies related to the boring bars naturalfrequencies, which correspond to its low-order bending modes [5,7].

Conventional techniques of vibrational suppression which could be appliedin this application, i.e., incorporation of a passive vibrational absorber into theboring bar [1,8] or use of an active boring bar [1,6], require detailed knowledgeof the spatial dynamic properties of the system boring bar - clamping house.

The natural frequencies and mode shapes of this system can be estimatedusing different approaches, such as experimental modal analysis, distributed-parameter system modeling (e.g., an Euler-Bernoulli model) and numericalmodeling (for instance, using nite element analysis). Finite element analysisoffers the possibility to develop an accurate model of the desired system inorder to obtain its spatial dynamic properties, and to use this model later forthe design of active tool holders.

The paper is focused on the development of a 3-D nite element modelof the boring bar with free-free boundary conditions as a rst step towardsthe construction of a 3-D nite element model of the system boring bar -clamping house. The accuracy of the model is veried based on results ob-tained using experimental modal analysis and a distributed-parameter systemEuler-Bernoulli model. Modication of the nite element model (incorporat-ing the effect of mass loading of the structure by 14 accelerometers) is per-formed in order to obtain a higher correlation to the results of experimentalmodal analysis.


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On accurate FE-modeling of a Boring Barwith Free-Free Boundary Conditions 23

2 Materials and Methods

This section describes the following: experimental setup used in experimentalmodal analysis of the boring bar with free-free boundary conditions, phys-ical properties of the boring bars material, and methods of identifying theboring bars spatial dynamic properties.

2.1 Physical Properties of Boring Bar Material

The boring bar used in experiments and modeling is a standard boring barS40T PDUNR15 F3 WIDAX, composed of 30CrNiMo8 material with the fol-lowing physical properties: Youngs elastic modulus E = 205 GP a , density = 7850 kg/m 3 , Poissons coefficient = 0 .3.

2.2 Measurement Equipment and Experimental Setup

Experimental modal analysis was carried out on a boring bar suspended bywire bands attached to the ceiling of the laboratory; see experimental setupin Fig. 1. The following equipment was used to conduct the experimentalmodal analysis.

14 PCB 333A32 accelerometers; 1 Kistler 9722A500 Impulse Force Hammer; HP VXI E1432 front-end data acquisition unit; PC with IDEAS Master Series version 6.

The spatial motion of the boring bar was measured by 14 accelerometers.The accelerometers were glued to the boring bar with the distance of 0.045m: 7 in the cutting depth direction and 7 in the cutting speed direction. Theboring bar was excited using an impulse hammer. The excitation force andacceleration signals were collected simultaneously.

2.3 Euler-Bernoulli Model

Since the boring bar is long and slender (i.e., its length-to-diameter ration is7.5) and only the rst bending modes are of interest, an Euler-Bernoulli modelcan be used to obtain a sufficiently accurate estimate of its low-order naturalfrequencies. However, if a shorter beam is under consideration, or if higher


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24WhitePart I

Figure 1: Setup for the experimental modal analysis.

eigenfrequencies are of interest, the Timoshenko beam model (which describesbending deformation, shear deformation and rotatory inertia) is preferablein order to obtain accurate estimates. According to Euler-Bernoulli beamtheory, the free vibration of the boring bar in the cutting speed direction canbe described by the following equation (bending motion in the cutting depthdirection is described by the same equation, where I x is replaced by I y ) [2]:

A 2w(z, t )

t 2 +

2

z 2[EI x

2w(z, t )z 2

] = 0, (1)

where w(z, t ) is the bending deformation; A is the area of the boring barscross section; I x is the cross-sectional area moment of inertia about x-axis;I y is the cross-sectional area moment of inertia about y-axis; is the densityof the material; E is the Youngs modulus of elasticity.

The area and cross-sectional moments of inertia were calculated based ongeometric dimensions of the boring bars cross section (see Fig. 2). The

following properties were used in Euler-Bernoulli model calculations:

Property Cutting speed direction Cutting depth direction Unitsl 0.3 mA 1.1933 10 3 m2I 1.1386 10 7 1.1379 10 7 m4

Table 1: Properties used in Euler-Bernoulli model calculations.


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26WhitePart I

where N is the total number of degrees-of-freedom; {}n is a mode shape vec-tor; n is a modal damping ratio; f n is an undamped systems eigenfrequency;Qn is a modal scaling factor.

An estimate of the accelerance matrix [ H (f )] is obtained experimentallybased on power spectral density and cross-power spectral density estimatesobtained from excitation force signal applied to the boring bar by impulsehammer, and 14 accelerometer response signals recorded simultaneously. Thespatial dynamic properties of the boring bar were identied using the time-domain polyreference least squares complex exponential method.

The orthogonality of the extracted mode shapes {EMA }k and {EM A }lwas checked using Modal Assurance Criterion [3]:

MAC kl = |{EMA }T k {EM A }l |2

({EM A }T k {EM A }k )({EM A }T l {EM A }l ) (4)

The Modal Assurance Criterion can also be used to provide a measure of correlation between the experimentally-measured mode shape {EM A }k andthe numerically-calculated mode shape {F EM }l

MAC kl = |{EM A

}T k

{F EM

}l

|2

({EM A }T k {EM A }k )({F EM }T l {F EM }l ) (5)2.5 Finite Element Analysis

The nite element method was used to develop a numerical model of the bor-ing bar in order to predict the systems dynamic behavior. The boring barsnite element model with free-free boundary conditions was constructedand veried for later use in the nite element model of the complete systemof interest boring bar-actuator-clamping house. The 3-D nite elementmodel was developed in order to describe the actual geometry of the boringbar. The nite element method is advantageous in that it allows the approxi-mation of a system with distributed parameters (i.e., with innite number of degrees-of-freedom) with a discrete system of elements with large but nitenumber of degrees-of-freedom. Thus, mode shapes can be estimated with con-siderably higher resolution (which depends only on element size) than modeshapes extracted by experimental modal analysis, where the resolution is lim-ited by the amount and physical dimensions of sensors used.

The rst two natural frequencies and mode shapes of the boring bar werecalculated based on the general matrix equation of free vibrations for an un-damped system


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[M ]{w (t)} + [K ]{w (t)} = {0}, (6)where [M ] is the global mass matrix of the nite element model of the system;[K ] is the global stiffness matrix of the nite element model of the system;

{w (t)} is the space- and time-dependent displacement vector.The tetrahedron with 10 nodes and quadratic shape functions was used asa basic nite element to develop the model of the boring bar. To simplify themeshing process, the nite element model of the boring bar consisted of two

sub-models: the sub-model of the boring bar with the constant cross-section- body, and the sub-model of the head. These sub-models were gluedtogether; i.e., contacting nodes from the sub-models were tied to each otherto avoid relative normal or tangential motion between the sub-models in thesenodes. The nite element model of the boring bar is shown in Fig. 3 a).

It is well known that the procedure of experimental modal analysis can af-fect the dynamic properties of the boring bar. For instance, the 14 accelerom-eters attached to the boring bar will result in an unwanted effect known asmass-loading of the structure, in which the boring bars natural frequenciesare altered by the attached accelerometer masses. In order to correlate theresults obtained from nite element analysis and experimental modal analy-sis, the nite element model was modied. 14 acclerometers were modeled ashomogeneous cubes with a certain material density to equate the mass of theaccelerometer to 5 g. The modied nite element model of the boring bar isshown in Fig. 3 b).

Natural frequencies and mode shapes were extracted using Lanczos itera-tive method with the use of MSC.MARC software [4,9].

3 Results

3.1 Mesh DevelopmentThis section presents results concerning the inuence of different boring barFE model mesh densities on estimated natural frequencies. The 3-D niteelement model of the boring bar consisted of the two sub-models the bodyand the head.

These two sub-models were meshed separately with different element sizesvarying from 0 .01 m to 0.003 m. In total, four models of the boring barwere created. The estimated fundamental boring bar natural frequencies arepresented in Table 2.


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28WhitePart I

a) b)

Figure 3: The nite element model of the boring bar with free-free boundaryconditions a) without, and b) with nite element models of the accelerometers.

Sub-model body, Sub-model head Total Mode 1 Mode 2element edge element edge number of Freq., [ Hz ] Freq., [ Hz ]length, [ m ] length, [ m ] elements

0.01 0.005 7366 2006.68 2009.550.005 0.005 19248 2007.51 2010.430.005 0.003 27481 2007.06 2009.50.003 0.003 53009 2007.36 2009.83

Table 2: The estimates of the boring bars st two natural frequencies usingfour different FE model meshes.

3.2 Spatial Dynamic Properties Estimates

Table 3 presents the spatial dynamic properties (i.e., natural frequencies,mode shapes, angles of mode shape rotation relative the chosen coordinatesystem) of the boring bar with free-free boundary conditions estimatedusing experimental modal analysis, the distributed-parameter system Euler-Bernoulli model, the nite element model of the boring bar, and the modiednite element model of the boring bar with incorporated effect of mass loadingby 14 accelerometers.

The rst two mode shapes are shown in Fig. 4. Since nite elementanalysis allows the construction of a model of the system with a large butnite number of degrees of freedom, mode shapes were obtained with signi-


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Model Mode 1 Mode 2Freq., Angle Relative Freq., Angle Relative[Hz ] between natural [ Hz ] between natural

mode freq. mode freq.shape error, shape error,

and CDD, [%] and CDD, [%][ ] [ ]

EMA 1969.60 -10 - 1970.43 80 -Euler-Bernoulli 1974.36 0 0.24 1974.97 0 0.23FE 2006.68 -23.9 2.05 2009.55 67.5 1.76Modied FE 1976.19 -22 0.49 1979.5 67 0.24

Table 3: Calculated eigenfrequencies, estimated angles of mode shape rota-tion, relative error between natural frequencies estimated by experimentalmodal analysis and calculated based on basic and modied nite elementmodel as well as Euler-Bernoulli model.

cantly higher resolution than those obtained by experimental modal analysis.However, in order to compare numerically calculated mode shapes with thoseobtained experimentally, deection was considered only in the nodes of the

nite element model corresponding to the positions of the accelerometersattachment.

The MAC-matrix was used as a quality measure for mode shapes estimatedby experimental modal analysis.

[MAC ]1 = MAC EMA 1 ,EMA 1 MAC EM A 1 ,EMA 2MAC EMA 2 ,EMA 1 MAC EM A 2 ,EMA 2

= (7)

= 1.000 0.0010.001 1.000

Corresponding cross-MAC matrices were calculated as a measure of corre-lation between the two rst mode shapes calculated based on the basic niteelement model ( F EM 1 , F EM 2), modied nite element model ( FEMM 1 ,FEMM 2), Euler-Bernoulli model ( EB 1 , EB 2) and the two rst mode shapesestimated from the experimental modal analysis ( EM A 1 , EM A 2).


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30WhitePart I

0 0.05 0.1 0.15 0.2 0.25 0.31

0.5

0

0.5

1

Distance from the end of the boring bar, [m] M o d e s h a p e

1 i n t h e c u

t t i n g

d e p t

h d i r e c t

i o n

X

EMAEuler BernoulliFEModified FE

0 0.05 0.1 0.15 0.2 0.25 0.31

0.5

0

0.5

1


1 i n t h e c u

t t i n g s p e e

d d i r e c t

i o n

Y


a) b)

0 0.05 0.1 0.15 0.2 0.25 0.31

0.5

0

0.5

1


2 i n t h e c u

t t i n g

d e p t

h d i r e c t

i o n

X


0 0.05 0.1 0.15 0.2 0.25 0.31

0.5

0

0.5

1


2 i n t h e c u

t t i n g s p e e

d d i r e c t

i o n

Y


c) d)Figure 4: First two mode shapes of the boring bar with free-free boundaryconditions a) component of mode shape 1 in the cutting depth direction b)component of mode shape 1 in the cutting speed direction, c) component

of mode shape 2 in the cutting depth direction, and d) component of modeshape 2 in the cutting speed direction estimated based on Euler-Bernoullimodel, experimental modal analysis and nite element models.

[MAC ]2 = MAC EMA 1 ,FEM 1 MAC EM A 1 ,FEM 2MAC EMA 2 ,FEM 1 MAC EM A 2 ,FEM 2

= (8)

= 0.787 0.2250.235 0.745


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32WhitePart I

the boring bar allows reduction of the relative error of natural frequencies esti-mates from 2.05 and 1.76, to 0.49 and 0.24 % in the cutting depth and cuttingspeed directions, correspondingly. The discrepancy between results obtainedfrom the modied nite element model and experimental modal analysis (seeFig. 4, Table 3) can be explained by following: imperfection of the geometri-cal model of the boring bar used in the nite element analysis, differences inmaterial properties and uncertainty in measurements.

The angles of rotations of the experimentally estimated mode shapes withrespect to the chosen coordinate system can be explained partly by the trans-verse sensitivity of the accelerometers, and partly by uncertainty in the mea-surements (suspension of the structure by cables); see Fig. 4. The rotationangles of the mode shapes of the nite element model differ from the rotationangles of the mode shapes obtained from the experimental modal analysis,thus resulting in signicant off-diagonal element values of the cross-MAC ma-trix in Eq. 8. However, this may be explained by the fact that the mass dis-tribution of the nite element model, related to the mesh used in the model, isnot identical with the actual mass distribution of the modeled boring bar. Itis possible to reduce errors in the rotation angles of the nite element modelmode shapes by, for instance, improving the model using a mesh symmetric

about x-z plane in the section of the boring bar with a constant cross-section, i.e. the body. However, utilizing the symmetric mesh does notimprove accuracy of the natural frequency estimates, and leads to a tremen-dous increase in model size. This is undesirable and problematic with respectto, for instance, calculating the boring bars transient response.

From the results presented it may be concluded that it is possible to es-timate the natural frequencies of the rst two bending modes of the boringbar from the 3-D nite element model with sufficient accuracy. It is alsopossible to predict the correct direction of the extracted mode shapes.

AcknowledgmentsThe present project is sponsored by the Foundation for Knowledge and Com-petence Development and the company Acticut International AB.


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Part II

Dynamic Modeling of a BoringBar Using Theoretical and

Experimental EngineeringMethods


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Part II is submitted as:

T. Smirnova, H. Akesson and L. Hakansson Dynamic Modeling of a Bor-ing Bar Using Theoretical and Experimental Engineering Methods , submittedto Journal of Sound and Vibration, January 2008.

Parts of this article have been published as:

T. Smirnova, H. Akesson, L. Hakansson, I. Claesson and T. Lag o, Identi-cation of Spatial Dynamic Properties of the Boring Bar by means of Finite Element Model: Comparison with Experimental Modal Analysis and Euler-Bernoulli Model , In Proceedings of the Thirteenth International Congress onSound and Vibration, ICSV13, Vienna, Austria, 2 - 6 July, 2006.


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Dynamic Modeling of a Boring BarUsing Theoretical and Experimental

Engineering Methods

T. Smirnova, H. Akesson and L. HakanssonDepartment of Signal Processing,Blekinge Institute of Technology

372 25 RonnebySweden

Abstract

Boring bar vibrations is a common problem experienced during in-ternal turning operation. Also referred to as self-excited chatter, thisis a major problem for the manufacturing industry. High levels of bor-ing bar vibration generally occur at frequencies related to the rst twonatural frequencies of a boring bar. This article addresses differentmethods for the dynamic modeling of a clamped standard boring bar,including: the Euler-Bernoulli and Timoshenko models with clamped-free and pinned-pinned-free boundary conditions; 1-D nite ele-ment models with Euler-Bernoulli and Timoshenko beam elements withclamped-free and pinned-pinned-free boundary conditions; and the3-D nite element model of the boring bar-clamping house system.The sensitivity of these models (with respect to variations in materialdensity and the Youngs elastic modulus) has also been addressed. Thederived boring bar models have been compared to results obtained bymeans of experimental modal analysis, conducted on the actual boringbar clamped in a lathe. The results indicate a correlation between themode shapes produced by the different models. However, the orien-tation of the mode shapes and the resonance frequencies demonstratedifferences between the models and the experimental results. Further,the the accuracy of the model occurs to be partly determined by themodeling of boundary conditions.


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38WhitePart II

1 Introduction

The internal turning operation is known to be one of the most troublesomewith regard to vibrations in metal cutting. During such an operation a bor-ing bar tool cuts deep precise cavities into a workpiece material. However,due to geometric dimensions that a boring bar is required to have in orderto perform the boring operation (i.e. large length to diameter ratio), the baris easily subjected to vibrations. Classes of boring bar vibrations include:free or transient vibrations induced by shock loads, e.g., from engagementof the cutting tool and workpiece; forced vibrations which occur due to theperiodic excitation resulting from unbalanced rotating parts of the lathe; andself-excited vibrations known as chatter [13]. The latter class of vibrations -chatter - can be of primary or regenerative types [24]. Primary chatter oc-curs, for example, under random excitation applied by the workpiece materialdeformation process, whereas regenerative chatter is the result of undulationof the workpieces surface produced during a previous cut [3,5]. Boring barvibrations commonly lead to poor workpiece surface nish, reduced tool life,and severe acoustic noise levels, and have a negative impact on factors suchas productivity and production costs.

Extensive research has been conducted concerning development of meth-

ods and strategies for reducing the problem of self-excited chatter. Theseinclude, for instance:

prediction of limits for stable cutting with respect to cutting data, op-timal cutting tool insert angles, etc. [6,7]; passive control; i.e., utilizing composite boring bars and/or incorporat-ing a tuned vibrational absorber, etc. [8,9]; active control; i.e., selective increase of the dynamic stiffness of theboring bar [2,10].These strategies to avoid or suppress vibrations rely on mathematical mod-

els of the machine tools dynamics. Usually, the dynamics of the machine toolduring cutting is described by means of a closed-loop system containing thesub-model of the boring bar and the sub-model of the cutting process [6,7,11].

In internal turning, the boring bar - clamping system is usually the mostexible link in the machine-tool [4,5,7,9]. As a consequence, boring bar vibra-tions are generally dominated by the rst fundamental bending modes of thebar in the cutting speed direction [3,7]. A fundamental factor in the success


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Dynamic Modeling of a Boring Bar Using Theoreticaland Experimental Engineering Methods 39

of boring bar vibration reduction is, thus, the capacity of accurate dynamicmodeling of boring bars clamped in a lathe.

Literature overview

The following literature overview concerns models used to describe the dy-namics of the boring bar, as well as methods of experimental modal anal-ysis used to identify modal parameters (e.g. natural frequencies and modeshapes) in order to match analytical models with real boring bars. The liter-ature overview covers three groups of boring bar models: lumped-parametermodels, distributed-parameters models and numerical nite element models.

The overview begins with research works involving lumped-parameter mod-els.

Tobias [3] claimed that chatter can develop at the natural frequencies of one of the following systems: spindle-workpiece, workpiece, tool. He, there-fore, attempted to identify modal parameters under test conditions. Inter-rupted cutting (previously introduced by Salie [12]) was used in order to ex-cite the system, implying that broadband excitation was achieved by cuttinginto the pre-milled workpiece.

Parker [13] developed an analytical model for cutting dynamics in thecase of regenerative chatter induced by mode coupling. The model describedboring bar dynamics as a two-degree-of-freedom mass-spring-damper system,the workpiece-spindle-machine structure was considered rigid. This modelallowed the prediction of a favorable setting angle for the boring bar headwith respect to the two planes of vibrations, in order to achieve maximumcutting depth while maintaining stable cutting for the given range of cuttingtool setting angles and cutting speeds. The modal parameters used in themodel were estimated based on measured point receptances. The analyticalmodel was claimed to give fairly satisfactory prediction for stable behavior incomparison with experimental results.

Zhang [7] modeled a boring bar as a system with two degrees-of-freedom.A linear state-variable system model was used to describe the response of theboring machining system. As state variables displacements, and velocities of the two rst principal modes of the boring bar were used. Modal parameterswere obtained by experimental modal analysis conducted using the impulseresponse method. Cutting force was described by means of a model consistingof four components, one of which was proportional to vibrations in the cuttingspeed direction. A procedure was suggested for identifying the critical gainfactor of the cutting force component, proportional to vibrations in the cutting


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speed direction. This procedure was based on the developed state variablesystem model, the Lyapunov energy method, and the Nyquist criterion. Thestate variable system model was also used to predict critical cutting stiffness,i.e., to identify the limit width of cut leading to instability occurrence. Zhangclaimed to achieve fairly good agreement between predicted and estimatedresults.

Minis [11] modeled the response of a boring bar under applied cutting forceby a three-degree-of-freedom model, identifying modal parameters experimen-tally by means of two techniques. Firstly, an approach similar to Tobiass wasused to apply pseudo-random broadband excitation to the boring bar duringorthogonal turning. The boring bars natural frequencies and damping ra-tios were estimated based on the measured averaged cross- and auto-powerspectrums of excitation force, and the boring bars response signals. Minisclaimed that the modal parameters of the boring bar can be identied duringmachining, that the effect of coupling of the structure with the cutting pro-cess can be neglected due to interrupted cutting. Miniss second techniqueinvolved identifying modal parameters from the impact excitation applied tothe boring bar using a curve-tting technique. There was no coupling betweenthe boring bar and workpiece during impact test. Minis achieved good agree-

ment between estimates of natural frequency, however the estimates of modaldamping varied greatly between the two methods. This fact was explainedby the nonlinearity of the machining system introduced through its couplingwith the cutting process. Minis generalized the linear stability theory in orderto describe the orientation of the tool with respect to the structure and usedit to accurately predict critical depth of cut for both left- and right-handedorthogonal turning.

Pratt [14] developed a two-degrees-of-freedom analytical model of boringbar dynamics with the purposes of chatter stability analysis, simulation of boring bar response, and design of biaxial feedback controller. Modal param-eters were estimated experimentally using impact tests and circle ts of themeasured frequency response function. Pratt noticed that frequency of thechatter differs between cases of heavy cutting (corresponds to the natural fre-quency in the cutting speed direction) and light cutting (corresponds to thenatural frequency in the cutting depth direction), a fact which may be theresult of the mode coupling effect. He also noticed that modal characteristicsof the boring bar vary greatly depending on clamping conditions.

Parallel to the development of lumped-parameter models, distributed-parameter models of boring bars were developed.

Rao et al. [15] approximated a boring bar as a continuous system cantilever


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42WhitePart II

mode.Nagano et al. [20] utilized pitched-based carbon ber reinforced plastic

(CFRP) material to develop a boring bar with large overhang resistant tochatter. He attempted to create a 3-D nite element model to predict nat-ural frequencies and improve dynamic characteristics of the boring bar bymodeling embedded steel cores of various shapes. The cutting performanceand stability of designed boring bars against chatter were investigated exper-imentally. He claims that utilization of the CFRP material in combinationwith the cross-shaped steel core yields the successful chatter suppression forboring bars with a length-to-diameter ratio greater then seven. Nagano men-tioned the necessity of developing improved models for clamping of the boringbar.

Later, Sturesson et al. [21] developed a 3-D nite element model of atool holder shank, utilizing normal mode analysis to evaluate natural frequen-cies, modal masses and mode participation factors of the tool holder shank.Modal damping was estimated by means of a free vibration decay method.Spectral densities estimates were also utilized to obtain natural frequencies.The results of normal mode analysis and spectral density estimates were well-correlated.

Openings

As mentioned above, the identication of modal parameters is an importantstep in building accurate mathematical models of a boring bar intended to pre-dict stability limits (i.e., optimal removal rate, geometric conguration of thecutting tool, cutting data), design of controllers for the vibration suppression,and simulation of the machine-tool system response. Since the utilization of a lumped-parameter system has several disadvantages, the development of aproper dynamic model of the system with distributed parameters is required.

Up to date it seems that Euler-Bernoulli beam theory has been the onlytheory used to describe dynamics of the boring bar as a continuous system.However, Euler-Bernoulli beam models ignore the effects of shear deformationand rotary inertia and, as a consequence, they tend to slightly overestimatethe eigenfrequencies; this problem increases for the eigenfrequencies of highermodes [22]. The Euler-Bernoulli model is adequate for beams that are consid-ered to be long and slender (with a length-to-diameter ratio grater than 10)in the frequency range of the lower modes, where the inuence of shear defor-mation and rotary inertia are negligible. However, in the case of distributedparameters modeling of beams with a length-to-diameter ratio below 10, the


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effects of shear deformations and rotary inertia are signicant and should beconsidered by the model [22]. This suggests that Timoshenko beam theoryshould be utilized for the modeling of boring bars with overhangs below 10.

As evidenced by the literature overview, in the case of rigid clamping, as-sumed boundary conditions do not necessary correspond to the actual clamp-ing conditions of the boring bar. It, therefore, seems important to investigatethe possibility of developing models that describe the actual boundary condi-tions of a boring bar more accurately, i.e., to incorporate boundary conditionsapproximating the exibility of the actual clamping of the boring bar end in-side the clamping house. This may, for example, involve the developmentof Euler-Bernoulli and Timoshenko multi-span beams with various boundaryconditions. However, the derivation of closed-form solutions for the Euler-Bernoulli and, in particular, the Timoshenko multi-span beam model is time-consuming, and usually performed by symbolic arithmetics calculation on acomputer [23]. As an alternative to distributed-parameter system modeling of the boring bar, the utilization of the nite element method may be suggested.The nite element method can be used for developing 1-D models (e.g., forcalculating multi-span beam), as well as 3-D models [21]. The latter case isof great interest since it enables the spatial dynamic modeling of not only the

boring bar with its actual dimensions, but also the combined system boringbar - clamping house. Further, the 3-D nite element model is likely tofacilitate the modeling of the actual boundary conditions of a clamped boringbar.

This paper discusses different approaches to the dynamic modeling of aboring bar, utilizing several different methods. Firstly, the Euler-Bernoulliand Timoshenko beam theories for the modeling of a clamped boring barusing clamped-free boundary conditions are considered. In order to in-corporate clamping exibility in the distributed-parameter models, two-spanEuler-Bernoulli and Timoshenko boring bar models with pinned-pinned-freeboundary conditions have been derived. The nite element approach is alsoused to model a clamped boring bar. Several 1-D nite element models weredeveloped and compared with corresponding distributed-parameter models.The 1-D nite elements models are based on Euler-Bernoulli and Timo-shenko beam nite elements [22, 24]. Finally, a 3-D nite element modelof the clamped boring bar is presented. Two different 3-D nite elementboring bar models are investigated; the rst utilizes a rigid clamping housemodel, and the second a deformable clamping house nite element model. Thelatter 3-D nite element boring bar model illustrates the importance of theexibility assumption in the clamping system. A sensitivity analysis was car-


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44WhitePart II

ried out on the distributed-parameter system and numerical nite elementboring bar models with respect to variation in material properties. Exper-imental modal analysis of a clamped boring bar is conducted. The modalparameters provided by the experimental modal analysis and the developeddistributed-parameter and nite element models of the clamped boring barare compared.

2 Materials and Methods

This section describes the experimental setup used in the experimental modalanalysis, physical properties of the boring bar material and methods of mod-eling and identication of the boring bar modal parameters.

2.1 Measurement Equipment and Experimental Setup

The experimental modal analysis was carried out in a MAZAK 250 Quickturnlathe with 18.5 kW spindle power, a maximum machining diameter of 300mm and 1007 mm between the centers. A standard boring bar, WIDAXS40T PDUNR15 F3, was used. The boring bar was clamped with a standardclamping housing, and a Mazak 8437-0 40 mm holder attached to the turretin the lathe. The following equipment was used to carry out experimentalmodal analysis:

14 PCB 333A32 accelerometers; 2 Ling Dynamic Systems shakers v201; 2 Bruel & Kjr 8001 impedance heads; HP VXI E1432 front-end data acquisition unit; PC with IDEAS Master Series version 6.Experimental modal analysis was performed on the boring bar clamped

in the clamping house with four bolts in the direction corresponding to thecutting speed direction. The boring bar was simultaneously excited in thecutting speed and cutting depth directions by two shakers (see Fig. 1) via twoimpedance heads situated at the distance l1 = 100 mm from the clamped endof the boring bar (see Fig. 2). Spatial motion of the boring bar was measuredby accelerometers glued in the following order: two accelerometers were placed


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in the cutting speed and cutting depth direction corresponding to l2 = 10 mmfrom the clamped end of the boring bar, the other two accelerometers wereplaced at a distance of l3 = 40 mm from the rst two and the rest of theaccelerometers were equidistantly placed at l4 = 25 mm from each other (seeFig. 2). In total 14 accelerometers were used; 7 in the cutting speed directionand 7 in the cutting depth direction.

Figure 1: Setup for the experimental modal analysis.

The following notation for the coordinate system is utilized in this paper(including Fig. 2): x - cutting depth direction (positive direction pointingoutside the gures plane towards the reader), y - cutting speed direction, z -feed direction.

2.2 Physical Properties of the Boring Bar and ClampingHouse Material

The boring bar used in experiments and modeling is a standard boring barWIDAX S40T PDUNR15 F3. It is composed of 30CrNiMo8 material withthe following physical properties: Youngs elastic modulus E = 205 GP a ,density = 7850 kg/m 3 , Poissons coefficient = 0 .3. The clamping house isa standard clamping house 8437-0 RAWEMA. For simplicity, the propertiesof the clamping house material were assumed to be identical to those of the


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46WhitePart II

Figure 2: Drawing of the clamped boring bar with accelerometers and cementstuds for attachment of impedance heads.

boring bar.

2.3 Euler-Bernoulli Boring Bar Models

An analytical Euler-Bernoulli model may be utilized in order to provide es-timates of the lower order natural frequencies and mode shapes of a boringbar [18,25]. In the Euler-Bernoulli model the boring bar is considered to bea system with distributed mass and an innite number of degrees-of-freedom.This classical beam model considers only transverse beam vibrations, ignoringthe shear deformation and rotary inertia [22]. Thus, the Euler-Bernoulli model

tends to slightly overestimate the eigenfrequencies; this problem increases forthe eigenfrequencies of the higher order modes [18, 22]. An Euler-Bernoullimodel of the boring bar with clamped-free boundary conditions is shown inFig. 3.

The boring bars transversal motion in the cutting speed direction can bedescribed by the following equation (transversal motion in the cutting depthdirection is described by the same equation, where I x is replaced by I y ) [22]:

A 2w(z, t )

t 2 +

2

z 2EI x

2w(z, t )z 2

= f (z, t ) (1)


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Figure 3: A clamped-free model of a boring bar, and the cross-section shapeof the boring bar used in the experiments.

where - density of boring bar material, A - boring bars cross sectionarea, l - overhang of the boring bar, lc - clamped length of the boring bar,I x and I y - cross-sectional area moments of inertia about x axis and abouty axis, E - Youngs elastic modulus, w(z, t ) - deection in the y-direction,f (z, t ) - external excitation force. The area and cross-sectional moments of inertia were calculated based on geometric dimensions of the boring bars crosssection (see Fig. 3), and are summarized in Table 1.

Quantity Cutting speed direction Cutting depth direction Unitsl 0.2 mlc 0.1 mA 1.1933 10 3 m2I 1.1386 10 7 1.1379 10 7 m4

Table 1: Quantities related to the boring bar geometry.The boring bars eigenfrequencies and eigenmodes are calculated from Eq.

1 by setting f (z, t ) = 0, which corresponds to the case of free vibrations. Thesolution of Eq. 1 is found using separation-of-variables procedure in the formof w(z, t ) = T (t)Z (z) [22]. General forms of the temporal and spatial solutionare given by

T (t) = B1 sin(2f t ) + B2 cos(2f t ) (2)


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respective

Z (z) = C 1 sin(z ) + C 2 cos(z ) + C 3 sinh(z ) + C 4 cosh(z ) (3)

where the eigenvalue 4 or wave number is related to the frequency f bythe dispersion relationship:

4 = (2f )2A

EI . (4)

The natural frequencies and mode shapes can be determined from Eq. 3by applying the particular boundary conditions suitable for the purpose of modeling. Two Euler-Bernoulli boring bar models will be considered: oneclamped-free model and one pinned-pinned-free model.

2.3.1 Euler-Bernoulli Model: Clamped-Free Boundary Condi-tions

Utilization of the clamped-free boundary conditions in Euler-Bernoulli mod-eling of a boring bar results in a deection and slope restricted to zero at theclamped end of the boring bar model, and zero bending moment and shearforce at the free end of the boring bar model:

w(z, t )|z =0 = 0 (5)w(z, t )

z |z =0 = 0 (6)EI x

2w(z, t )z 2 |z = l = 0 (7)

z

EI x 2w(z, t )

z 2 |z = l = 0 (8)The expressions for the eigenfunctions, and the frequency equation for the

clamped-free Euler-Bernoulli model of the boring bar are derived and givenin Appendix A.


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2.3.2 Euler-Bernoulli Model: Pinned-Pinned-Free Boundary Con-ditions

A two-span model of the boring bar with pinned-pinned-free boundary con-ditions is shown in Fig. 4.

Figure 4: A pinned-pinned-free model of a boring bar.

where lc = 0 .1 m and corresponds to the length of the part of the boringbar clamped inside the clamping house, and l = 0 .2 m corresponds to theoverhang of the boring bar.

The general spatial solution of Eq. 3 can now be written for two spans asfollows:

Z 1(z1) = C 1 sin(z 1) + C 2 cos(z 1) + C 3 sinh(z 1) + C 4 cosh(z 1) (9)

Z 2(z2) = C 1 sin(z 2) + C 2 cos(z 2) + C 3 sinh(z 2) + C 4 cosh(z 2) (10)

where 0 z1 lc and 0 z2 l. The boundary conditions at the supportedend z1 = 0 are as follows:

w(z1 , t )|z1 =0 = 0 (11)EI x

2w(z1 , t )z 21 |z1 =0 = 0 (12)


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At the free end z2 = 0 of the beam model, the bending moment and shearforce are zero, i.e.,

EI x 2w(z2 , t )

z 22 |z2 =0 = 0 (13)

z2EI x

2w(z2 , t )z 22 |z2 =0 = 0 (14)

The intermediate support at z1 = lc and z2 = l is modeled by applyinga boundary condition that restricts the deection to zero, and continuityconditions governing slope and the bending moment at the position of theintermediate support, according to:

w(z1 , t )|z1 = lc = 0 (15)w(z2 , t )|z2 = l = 0 (16)

w(z1 , t )z 1 |z1 = lc =

w(z2 , t )z2 |z2 = l (17)

EI x 2w(z1 , t )z 21 |z1 = lc = EI x 2w(z2 , t )

z 22 |z2 = l (18)The expressions for the eigenfunctions and the frequency equation for the

pinned-pinned-free Euler-Bernoulli model of the boring bar are derived andgiven in Appendix A.

2.4 Timoshenko Boring Bar Models

In the case of short beams or non-slender beams, i.e., beams with a length-to-diameter ratio less than ten, or when higher order bending modes are con-sidered for slender beams, the effects of shear deformation and rotary inertia(ignored by Euler-Bernoulli beam theory) are not negligible [22]. The Tim-oshenko beam model describes these effects. A clamped-free beam modelthat displays bending and incorporates shear deformation is illustrated in Fig.5.

Here, w(z, t ) is transverse displacement of the cross-section located at thedistance z from the clamped end of the boring bar; (z, t ) is the angle dueto pure bending deformation of the beam, dw (z,t )dz is the total slope of thecenterline of the beam and the difference dw (z,t )dz (z, t ) provides the shear


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Figure 5: A clamped-free model of a boring bar displaying shear and bend-ing deformation.

angle. The transverse vibration of the beam under a applied force f (z, t ) (N per unit length) can be described by a system of coupled partial differentialequations [26].

EI 2(z, t )

z 2 + kAG(

w(z, t )z (z, t )) = I

2(z, t )t 2

(19)

kAG( 2w(z, t )

z 2 (z, t )

z ) A

2w(z, t )t 2

= f (z, t ) (20)

where k is the shear coefficient which describes the distribution of the shearstress in the cross-section depending on the cross-section shape [27]. Sincethe cross-section shape of the boring bar is almost circular, the value of k wasselected for a circular cross-section shape, i.e., k = 6(1+ )7+6 . Here, is Poissonscoefficient, and G = E 2(1+ ) is the modulus of elasticity in shear. In order toobtain the natural frequencies and mode shapes, the homogeneous problemis considered by setting f (z, t ) = 0 and utilizing the method of separation of

variables. By separating both w(z, t ) and (z, t ) into two functions (such thatw(z, t ) = Z (z)T (t) and (z, t ) = ( z)T (t), and assuming that w(z, t ) and(z, t ) are synchronized in time, the equations of motion (Eqs. 19 and 20)can be separated into three ordinary differential equations [26]. This yieldsone temporal equation identical to Eq. 2, and two spatial equations that maybe decoupled and written in a non-dimensional form as follows:

Z ( ) + b2(r 2 + s2)Z ( ) b2(1 b2r 2s2)Z ( ) = 0 (21) ( ) + b2(r 2 + s2) ( ) b2(1 b2r 2s2)( ) = 0 (22)


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52WhitePart II

where = z/l is a non-dimensional variable,

b2 = Al4 (2 f )2

EI , r2 = I Al 2 and s

2 = EI kAGl 2 (23)

The decoupled equations Eqs. 21 and 22 have identical roots. Below theso called cut-off frequency f c = 12 kAGI dened by beam material and geom-etry, there exist two real and two imaginary roots, however, above the cut-off frequency the four roots are all imaginary [26]. Since only the low-order bend-ing modes of the boring bar are of interest, only lower frequency vibrations areconsidered, i.e., for frequencies below f c when (r 2 s2)2 + 4 /b 2 > (r 2 + s2).Thus, the characteristic equation for Eq. 21 has four roots given by:

1, 2 = bg (24)respective

3, 4 = jbo (25)where

g = 1 2 (r 2 + s2) + (r 2 s2)2 + 4 /b 2 (26)

ando = 1 2 (r 2 + s2) + (r 2 s2)2 + 4 /b 2 . (27)

Thus, the the spatial solutions to equations Eq. 21 and Eq. 22 can bewritten in a general form as follows:

Z ( )( ) =

C 1D1

sin(bo ) + C 2D2cos(bo ) + (28)

+ C 3D3sinh( bg ) + C 4D4

cosh(bg )

The general spatial solutions have eight unknown constant coefficients,C 1 ,..., C 4 , D1 , ..., D4 . However, only 4 constants are independent, and de-pendency between constants can be found by means of Eq. 19 and Eq. 20, inwhich the general solutions to Z ( ) and ( ) are inserted into Eq. 28. Equa-tions expressing the dependency of the constants C 1 ,..., C 4 on the constantsD1 , ..., D4 Eq. 88 - Eq. 91 are shown in Appendix A.

To obtain the frequency equations, eigenfunctions and eigenfrequencieswhich correspond to the suggested Timoshenko models of the boring bar, theadequate boundary conditions are applied to the spatial solution in Eq. 28.


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interest, the spatial solution provided by Eq. 28 will be used. In the case of the two-span beam, there are two spatial solutions (one for each span) which

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