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International Journal of Machine Tools & Manufacture 48 2008 832–840Dynamic model of a centerless grinding machine based on an updated FE model I.. Using as reference results obtained fr

International Journal of Machine Tools & Manufacture 48 (2008) 832–840

Dynamic model of a centerless grinding machine based on

an updated FE model

I Garitaonandiaa, , M.H Fernandesa, J Albizurib

a Department of Mechanical Engineering, Faculty of Mining Engineering, University of the Basque Country, Colina de Beurko s/n 48902 Barakaldo, Spain

b Department of Mechanical Engineering, Faculty of Engineering, University of the Basque Country, Alameda de Urquijo s/n 48013 Bilbao, Spain

Received 20 August 2007; received in revised form 29 November 2007; accepted 4 December 2007

Available online 15 December 2007

Abstract

Centerless grinding operations present some characteristic features that make the process especially susceptible to regenerative chatter instabilities Theoretical study of these vibrations present some difficulties due to the large amount of parameters involved in the process and, in addition, such a study requires a precise determination of dynamic properties of the particular machine under study Taking into account the important role of the dynamic characteristics in the process, in this paper both analytic and experimental approaches are used with the aim of studying precisely the vibration modes participating in the response Using as reference results obtained from an experimental modal analysis (EMA) performed in the machine, the finite element (FE) model was validated and improved using correlation and updating techniques This updated model was used to obtain a state space reduced order model with which several simulations were carried out The simulations were compared with results obtained in machining tests and it was demonstrated that the model predicts accurately the dynamic behavior of the centerless grinding machine, especially concerning on chatter

r2007 Elsevier Ltd All rights reserved

Keywords: Centerless grinding; Finite element; Experimental modal analysis; Model reduction

1 Introduction

Centerless grinding is a chip removal process in which

the workpiece is not clamped, but it is just supported by the

regulating wheel, the grinding wheel and the workblade

(Fig 1)

This configuration allows a simple and easy way to load/

unload workpieces with minimal interruption of the

process, providing high flexibility in the sense that high

productivities together with precise dimensional tolerances

of the parts can be obtained

On the other hand, problems associated with roundness

errors are very common in these machines because of the

floating center of the workpiece As a consequence, surface

errors of the workpiece, after contacting the workblade and

the regulating wheel, produce a displacement of its center

that can lead to an error regeneration mechanism

Several researchers have studied the origin and evolution

of roundness errors [1–4] These researches have shown that instabilities are produced due to geometric and dynamic causes Geometric instabilities are specific of centerless grinding and they are produced as a consequence

of the geometric configuration of the machine, independent

of the structural characteristics and the workpiece angular velocity Dynamic instabilities have their origin in the interaction between the regenerative process and the dynamic of the structure In this case, self-excited vibra-tions appear limiting the surface quality of the workpieces The study of this last problem requires an adequate knowledge of the dynamic properties of the machine, so it

is of great assistance to have numerical models including these properties in order to predict the dynamic response of the machine-process system both in its original design and

in a design with modifications For the purpose of obtaining an adequate model, in this paper finite element (FE) models correlation-updating techniques are used by means of experimental data

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0890-6955/$ - see front matter r 2007 Elsevier Ltd All rights reserved.

doi: 10.1016/j.ijmachtools.2007.12.001

Corresponding author Tel.: +34 946 014 967; fax: +34 946 017 800.

E-mail address: iker.garitaonandia@ehu.es (I Garitaonandia).

2 System modeling

Previously to the development of this work, Albizuri

et al [5] studied the vibratory response of the machine

under study using a lumped mass model, which

character-ized the moving components guided by the two ball screws

in the feed direction (seeFig 1) These are the components

with larger vibration amplitudes in chatter conditions

This model, which has the advantage of its simplicity, is

somewhat limited because it supposes several infinitely

rigid components of the machine as the bed, the grinding

wheel and the grinding wheel head, so it does not model

adequately the force transmission path from the cutting

point between the workpiece and the grinding wheel to the

contact point between the workpiece and the regulating

wheel

Due to the mentioned limitation, in this work special attention has been paid to the development of an updated

FE model that will predict the dynamic response of the machine under operation conditions, giving an insight into the real behavior of different components

2.1 FE model Dynamic characteristics of centerless grinding machine were studied by means of a FE model using ANSYS software This model, which consists of 53,200 nodes and 37,807 elements, is depicted inFig 2

This figure shows the global coordinate system used in the model, where z-axis was defined as the longitudinal axis

of the machine, x-axis as the transversal one and y-axis as the vertical one

y state space input vector

x state space state vector

j1, j2, g, y, h centerless grinding geometric parameters

(seeFig 8)

k0

cr contact stiffness per unit width between grind-ing wheel and workpiece (N/mm/mm)

k0

cs contact stiffness per unit width between reg-ulating wheel and workpiece (N/mm/mm) G(s) machine transfer function

Fig 1 Scheme of the centerless grinding machine.

A total of 15 mode shapes and natural frequencies were

extracted in the 0–160 Hz frequency range This range of

interest was defined taking into account that for the

grinding machine under study, in chatter conditions, the

most severe vibrations were observed in the neighborhood

of 60 Hz, while other less severe vibrations appeared at

about 130 Hz

This model predicts adequately the elastic-inertial

properties of different components of the machine A

major problem arises when introducing the stiffness and

damping properties of joint elements into the model

because when studying joints, there are a lot of

uncertain-ties that make not possible to model them precisely[6] In

order to overcome this obstacle, the development of a good

FE model requires the use of experimental data

2.2 Experimental analysis

An impact testing experimental modal analysis (EMA)

was performed in the machine Responses were measured

in 69 points using triaxial accelerometers, so aceleration/

force frequency response functions (FRF) corresponding

to 207 degrees of freedom were obtained.Fig 3illustrates

the geometry used in the analysis, in which the arrow shows

the excitation point and direction This excitation direction

was selected in order to excite modes with high modal

displacement components in z direction

The FRFs obtained in the analysis were studied in LMS

Cada-X software, obtaining 10 natural frequencies, mode

shapes and damping factors in the frequency range of

interest

2.3 Theoretical–experimental correlation

Numerically obtained mode shapes were correlated with

the experimental ones in FEMtools software using the

modal assurance criterion (MAC)[7]:

MAC FEM; EMAð Þ ¼ ðfTFEMfEMAÞ2

ðfTFEMfFEMÞðfTEMAfEMAÞ (1) MAC values obtained between the first 15 numerical mode shapes and the first 10 experimental ones are shown

inTable 1 In this table, four values of MAC corresponding

to as many paired mode shapes have been highlighted due

to their importance in the development of this work The first three pairs show MAC values above 85%, while the last pair shows a lower value Although these MAC values point out that the correlation between the corresponding numerical and experimental mode shapes is adequate, it can be seen that there are significant differences in the natural frequencies of these mode shapes, so it was necessary to improve the FE model by means of an updating process

In the updating process, the three numerical natural frequencies with higher MAC values were selected as responses to be improved To select adequate parameters

to be updated, a sensitivity analysis was performed and it was concluded that the parameters most influencing the estimation of the mentioned natural frequencies were the stiffness values of the joint elements connecting the bed to the foundation and the axial stiffness of the lower slide ball screw

These stiffness values were improved iteratively in order

to match the numerical natural frequencies to the experi-mental ones A Bayesian parameter estimation technique was used[7]for this purpose.Fig 4shows the MAC matrix obtained after the updating process

In this figure, it can be seen that an adequate corre-lation remains between the previously paired mode shapes Moreover, Table 2shows that the difference between the updated natural frequencies and the experimental ones are

Fig 2 FE model of the machine.

Z Y X

Fig 3 Geometry of the EMA.

small Numerical frequency 13 also was improved in the

updating process

3 Updated FE model characteristics

Once an updated FE model has been obtained, in the

development of a dynamic analysis it is important to study

the model in order to identify the modes with higher

contribution to the response in operation conditions In

this analysis, it is necessary to take in mind that in

centerless grinding operations the normal force generated

in the cutting point between the workpiece and the grinding wheel is produced mainly in z direction, so the relative excitability of the different modes in that direction was evaluated calculating their modal participation factors (MPF) The result is shown in Fig 5, where the MPF have been normalized so that the largest value has unit magnitude

From this figure it can be concluded that there are three modes with highest contribution to the dynamic response

In the mode shape at 33.48 Hz all the components of the machine move in phase in the longitudinal direction in a suspension movement with respect to the supports of the bed The mode shape corresponding to natural frequency

of 58.59 Hz is the one which is excited normally when chatter vibrations appear in the centerless grinding machine, so it is called the main chatter mode Fig 6

shows the animation for this mode shape It is seen that this mode corresponds to an out of phase movement between the heads of the two wheels

The mode shape at 127.41 Hz corresponds to a mode, which has been excited only in some tests performed in the machine, always leading less vibration amplitudes than the previous one

Bold numbers correspond to important mode shape pairs.

Fig 4 MAC matrix after the updating process, in %.

Table 2 Comparison between updated numerical values and experimental results Pair FE

mode

mode

Hz Diff.

(%)

MAC (%)

4 FE model order reduction

The large number of degrees of freedom of the updated

FE model implies computationally expensive calculations

in order to simulate the vibration behavior of the machine

This restriction makes necessary the reduction of the size in

such a way that the reduced model and the original model

will have the same frequency response characteristics in the

frequency range of interest

The big size of the FE model was reduced using the

modal coordinate reduction [8], which is based on the

fact that the dynamic response of the centerless grinding

machine in the frequency range of interest is dominated by

the first 15 structural modes, so it is possible to simulate

its dynamic behavior using these modes and neglecting

the rest

Several representative degrees of freedom were selected

defining the points in which application of forces or

acquisition of responses was required Using as reference

the different lists obtained from the updated FE model,

truncated U matrix was created retaining the modal contributions of the mentioned degrees of freedom for the first 15 mode shapes The first 15 updated natural frequencies were used to create X matrix and the damping properties obtained experimentally were used to construct

n matrix

These matrices were used in MATLAB environment

to obtain a modal model of the structure in state space defined by

_

x ¼ Ax þ Bu;

The state vector was selected as follows[9]:

x ¼ Xq _q

" #

In Eq (3), the size of the state vector (and thus the order

of the modal model) is twice the modes included in the model, being this size much less than the order of the

Fig 5 Modal participation factors.

Fig 6 Main chatter mode animation.

updated FE model used as reference The resulting state

space model A and B matrices are

A ¼ 0 X

X 2nX

" #

; B ¼ 0

UTLu

" #

C and D matrices of system (2) depend on the required

outputs, so the described model can be used to simulate

displacements, velocities and accelerations of selected

degrees of freedom

FRFs acceleration/force obtained both experimentally

and using the state space modal model between input degree

of freedom j and output degree of freedom k (see Fig 2)

were compared The result is displayed inFig 7

This figure shows that the state space modal model

reflects adequately the system dynamic below

approxi-mately 70 Hz, while above this frequency the discrepancies

with the experimental results are higher Likewise, it can be

seen that both the FRFs show three important resonance

peaks corresponding to the mode shapes with higher MPF

obtained inFig 5

5 Experimental verification

In order to evaluate the effectiveness of the state space

reduced model obtained in previous section, it was used to

perform a theoretical study of chatter instabilities in the

centerless grinding machine under study Fig 8 shows a

detail of the configuration of the machine andFig 9shows

the block diagram used to study its stability, which is based

on similar diagrams presented in previous works[3,4]

The term G(s) takes into account the dynamic flexibility

of the machine and it was obtained considering the three

modes with major dynamic contribution in feed direction

(seeFig 7) This idea was carried out using controllability

and observability criteria[10] The state space modal model

defined by Eq (2) was transformed in the balanced

realization, in which the controllability and observability

matrices are equal and strictly diagonal, being the terms of

the diagonal a quantitative measure of the relative

importance of the different states in the input–output

behavior of the system This realization was divided into a dominant subsystem formed by the six more controllable and observable states, and a weak one, formed by the rest

of the states This last subsystem was eliminated residualiz-ing the least controllable and observable states in order to include their static contribution in the response[11]

5.1 Analysis of chatter frequencies The characteristic equation of the block diagram shown

in Fig 9is

1  e2pðs=op Þ

K ð1  Þ

bk0 cr

þ 1

bk0 cs

þGðsÞ

þ1  0ej1 ðs=o p Þ

þ ð1  Þej2 ðs=o p Þ¼0 ð5Þ

To guarantee stable cutting conditions, all the roots of this equation must be in the left side of the complex plane

If one of the roots is located in the right side of the complex plane, the system is unstable and during the grinding process the response grows in time causing the regeneration

of a roundness error in the workpiece

The complete resolution of the roots of the characteristic equation is not an easy task due to the transcendental nature of the equation to be solved, formed by three time delays, so there are infinitely many solutions satisfying it

In this application, these roots were solved using the root locus technique [12], obtaining the solutions of the characteristic equation (Eq (5)) for increasing values of cutting stiffness in the 0-N range This technique plots the evolution of the different roots, so it can be determined which one becomes unstable

The application of this method requires the resolution of the characteristic equation for a zero cutting stiffness value These solutions are:

 the poles of the transfer function G(s),

 an infinite number of poles of ð1  e2pðs=o p ÞÞlocated at minus infinity,

 the roots of the geometric characteristic equation

1  0ej1 ðs=o p Þþ ð1  Þej2 ðs=o p Þ¼0

The initial esti-mations of these roots were obtained using an iter-ative graphical procedure, which consisted in modifying

Fig 7 FRFs between j–k degrees of freedom.

sequentially the values of the possible roots until the real

and imaginary parts of the geometric characteristic

equations were annulated These initial estimations

were used to obtain the final solutions using Newton–

Raphson method [13]

The first set of roots obtained for K ¼ 0 were used as initial estimations for the next increment of the cutting stiffness using Newton–Raphson method, and so on until reaching the cutting stiffness value obtained experimentally for the particular geometric configuration of the machine under study

In order to compare the results obtained theoretically with those obtained experimentally, several simulations were performed programming geometric conditions with which previously cutting tests had been done in the machine These conditions are shown inTable 3

Contact stiffness values assumed in the simulations corresponded to typical values for centerless grinding of steel components using a vitrified grinding wheel together with a rubber-bonded regulating wheel[13]

For illustration purposes,Fig 10shows the evolution of the root loci for a workpiece angular velocity of 6.2 Hz In this figure, structural pole on 58.59 Hz migrates towards the imaginary axis for increasing values of cutting stiffness

Fig 9 Block diagram of centerless grinding.

Table 3

Cutting conditions

Regulating wheel diameter 340 mm

Fig 10 Root locus for increasing cutting stiffness o p ¼ 6.2 Hz.

Fig 11 Comparison between theoretical and experimental chatter frequencies.

until the pole crosses it and thus, it is instabilized Chatter

frequency is determined by the imaginary part of the

unstable root after concluding the loci

This procedure was repeated for different values of

workpiece angular velocity in the 0–20 Hz range Chatter

frequencies obtained theoretically were compared with the

experimentally measured ones, as it is shown inFig 11

This figure shows that theoretical predictions are in

agreement with experimental results

5.2 Time domain simulations

The reduced order model validation was completed with

various time domain simulations in order to quantify the

number and the amplitudes of the undulations produced in

the workpiece in chatter conditions The workpiece was

discretized in 360 equal segments and its rotation was

simulated segment by segment Taking as reference the

process block diagram shown inFig 9, time evolution of

the roundness errors of the workpiece was obtained and in

each segment rotation, integrating numerically this

evolu-tion using Runge–Kutta algorithm [14], the errors were

calculated as a function of the dynamic response of the

machine and the errors in the previous pass and in the

contact points with the workblade and the regulating

wheel Nonlinear effects as contact loss between the

workpiece and the grinding wheel and spark-out process

were taken into account [15] The simulations were done

programming a regulating wheel infeed rate of 1 mm/min, a

total feed of 0.2 mm and a spark-out time of 2 s.Fig 12a

shows the final theoretical profile obtained for a workpiece

angular velocity of 6.2 Hz, while Fig 12b shows the real

profile obtained under the same conditions programmed in

the simulations

It is shown that workpiece profiles obtained both

theore-tically and experimentally are quite similar Moreover,

theoretically predicted roundness error is within the same order of magnitude of the experimentally measured one

6 Conclusions

In this work, a dynamic model of a centerless grinding machine has been developed performing a detailed study of mode shapes that are excited in chatter conditions The combined use of numerical FE model updating techniques via experimental modal data and model reduc-tion techniques resulted in a state space model representing adequately the modes with major modal contribution in machine vibrations The presented methodology advances the state-of-the-art in modeling procedures of centerless grinding machines

Simulations demonstrated that the model predicts accurately both the appearance of chatter vibrations for different machine configurations and the time evolution of workpiece roundness errors under unstable operation conditions Thus, this model represents a powerful tool

to define optimal set up conditions in order to increase the productivity in centerless grinding practice

Acknowledgments The authors are grateful to IDEKO Technological Center for the provision of numerical and experimental facilities to conduct this work

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