International Journal of Machine Tools & Manufacture pot

Simulation of an active vibration control system in a centerless grindingmachine using a reduced updated FE model M.H.. Based on the updated finite element FE model of the machine, the st

Simulation of an active vibration control system in a centerless grinding

machine using a reduced updated FE model

M.H Fernandesa, I Garitaonandiaa, , J Albizurib, J.M Herna´ndeza, D Barrenetxeac

a

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

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

c

IDEKO Pol Industrial de Arriaga, 2 20870 Elgoibar, Spain

a r t i c l e i n f o

Article history:

Received 22 August 2008

Received in revised form

28 October 2008

Accepted 13 November 2008

Available online 27 November 2008

Keywords:

Centerless grinding

Active control

Modelling

Vibration simulation

a b s t r a c t

In this paper, a novel and complete process to simulate an active vibration control system in a centerless grinding machine is presented Based on the updated finite element (FE) model of the machine, the structural modifications performed to incorporate active elements are detailed, as well as the subsequent reduction procedure to obtain a low-order state space model This reduced structural model was integrated in the cutting process model giving a tool adapted for the purpose of simulating different control laws Using the developed model, a control algorithm, which previously had been implemented

in the centerless grinding machine under study, was checked The simulation results were in agreement with the experimentally obtained ones, showing that the designed model is able to reproduce machine behaviour with the control activated This model constitutes a powerful tool to evaluate the effectiveness of different approaches to that of the described one, making it possible to tackle an optimisation process of the control system by means of simulations and, thus, avoiding the costs that would involve the practical implementation of each one

&2008 Elsevier Ltd All rights reserved

1 Introduction

One of the most important problems limiting the circularity of

round parts in centerless grinding is the occurrence of self-excited

vibrations caused by regenerative chatter Due to the fact that this

process is usually used to machine cylindrical elements with high

surface finish requirements, the appearance of roundness errors is

undesirable, and the study of procedures to minimize this

negative effect has aroused the interest of several researchers

Traditionally, the developed solutions consist of avoiding the

unstable operating regions through an adequate selection of

set-up conditions, combining properly the geometric configuration of

the machine with workpiece rotational speed [1–5] However,

from a productivity point of view, these solutions are not

necessarily the recommended ones because the optimum cutting

conditions can differ from the chatter-free configurations

Other procedures have been based on structural modifications

to stiffen the most flexible components in the force transmission

loop [6,7] This stiffening produces an increase in the first

resonant frequency, widening the low workpiece rotational speed

stability zone Nevertheless, such an alternative involves a

redesign of the machine structure, giving rise to solutions that can be economically unfeasible

Taking into account the above-mentioned limitations, Albizuri

et al.[8]proposed a novel approach based on the application of active vibration control Using commercial piezoelectric actuators (A) in a feedback loop, they reduced the structural vibration level and, consequently, the roundness errors of the workpieces Although the results they obtained were promising, they were presented without a previous mathematical development, which would allow them to optimise the active vibration system by means of simulations, concerning both the programmed con-trollers and the used sensors and actuators

Later, Garitaonandia et al [9] characterized the dynamic properties of the machine combining finite element (FE) model updating and model order reduction techniques The proposed approach was effective for the estimation of machine dynamic behaviour under usual cutting conditions, but the study was not extended to the simulation of any active vibration control scheme The availability of a theoretical model describing both the structural and the controller characteristics is of major interest, as

it permits one to predict the effectiveness of different control alternatives, giving insight into their behaviour before their practical implementation and providing valuable information to select the most suitable one Therefore, the high costs (both economic and time costs), involving the experimental evaluation

of each possible solution to make this selection, can be avoided

Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/ijmactool

International Journal of Machine Tools & Manufacture

0890-6955/$ - see front matter & 2008 Elsevier Ltd All rights reserved.



Corresponding author Tel.: +34946017815; fax: +34946017800.

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

Motivated by the benefits that can be obtained with a model

having the described characteristics, in this paper a theoretical

model derived from the FE formulation is developed to simulate

the active vibration control system that had been previously

implemented in [8] Practical results obtained in the same

reference will be used to evaluate the accuracy of the simulated

predictions, thus validating the model

2 FE model

The simulation procedure presented in this paper is based on

the updated FE model developed in[9].Fig 1shows this model,

where for greater clarity only solid models of the components

have been shown It consists of 53,200 nodes and 37,807

elements

As is reported in [9], this model predicted accurately the

chatter behaviour observed in different machining tests In these

tests, the most violent vibrations appeared when the main chatter

mode was excited at frequencies close to 55 Hz, where an

important displacement of the lower slide relative to the swivel

plate was presented in the longitudinal Z direction (see Fig 1)

Other less-severe vibrations were detected at about 130 Hz when

the secondary chatter mode was excited, dominated by a local

bending deformation of the workblade

3 Integration of the actuators in the FE model

To simulate the active control system, it is necessary to modify

the FE model in order to incorporate the piezoelectric actuators

With this purpose, the design presented in [8] was followed,

where two PI-247.30 piezoelectric actuators had been placed in

the upper spindle support (D) area to obtain a strong control

authority on the main chatter mode This design is depicted in

Fig 2, where it can be seen that the integration of the actuators

requires a design modification of the upper spindle support in

such a way that two holes are made in it

Each actuator, which is located collinear to a force transducer

(B), exerts the control force on its left side (according toFig 2)

over support D and on its right side over an auxiliary component

(E) that connects the two actuator-force transducer groups This

component transfers the two control forces to the lower spindle

(C) Therefore, both actuators are placed in a parallel configuration

in the loading path from the lower slide to the swivel plate, and they transmit the force to the lower spindle in a series configuration

Piezoelectric actuators like the ones used in this application consist of n-stacked ceramic layers of PZT material (lead–zirco-nate–titanate) that change in length when electrically charged An important aspect to consider is the modelling of the force they exert The relation among the externally applied voltage (Vpzt), the axial deformation of an actuator (DL) and the actuator-force (Fpzt)

is[10],

where Kpzt represents the actuator stiffness and d33 is the piezoelectric constant Defining z as the vector containing the displacements of the different nodes in the FE model, the axial deformation of an actuator can be expressed by the following relation:

where b is the influence vector of the axial end displacements of the actuator

Substituting Eq (2) into Eq (1) and considering the centerless grinding machine structure formed by its mass matrix M, its damping matrix C and its stiffness matrix K, the incorporation of the forces exerted by the actuators in the equation of motion governing the dynamic equilibrium leads to[11],

Fig 1 Updated FE model of the centerless grinding machine.

Fig 2 Detail of piezoelectric actuators location area.

M €z þ C _z þ ðK þ Kpztb1bTþKpztb2bTÞz ¼ LuFnþb1Kpzt

d33nVpzt_1þb2Kpztd33nVpzt_2, (3)

where Fnis the normal grinding force and Luis its influence vector

Subscripts ‘‘1’’ and ‘‘2’’ have been used in b and Vpztto make a

distinction between the two actuators

Eq (3) shows that the effect of each actuator in the structure

can be properly modelled through an axial stiffness Kpztbetween

its ends (modelling the passive behaviour) and a pair of opposite

forces of value Kpztd33nVpzt applied axially in the same ends

(modelling the active behaviour)

Following the design illustrated inFig 2, the lower spindle area

of the FE model (Fig 3a) was modified in order to incorporate the

actuators, as shown inFig 3b Each actuator was modelled with

an element having the same stiffness as the piezoelectric stack

and two lumped masses of half the total mass of the actuators

were placed at their ends The actuators were connected with

the upper spindle support and with two elements modelling

the auxiliary component These elements were idealized as

undeformable because it was considered that the displacement

of the lower slide relative to the swivel plate was much bigger

than the deformation of the auxiliary element Finally, the rigid

elements were connected to the lower spindle

It was verified that these structural modifications did not alter

appreciably the natural frequencies corresponding to the main

and the secondary chatter modes; hence, the passive behaviour of

the structure remained basically unchanged

4 Definition of the control strategy

The active damping control strategy, which had been

im-plemented by Albizuri et al [8], consisted of measuring the

machine vibrations with an accelerometer located in the upper

spindle support and passing this acceleration signal through a

controller based on a second-order filter (SOF) to generate a

voltage feedback to the actuators proportional to the output of the

filter[12].Fig 4shows the control loop sketch in the modified FE

model

The SOF controller is described by the general law given by

f

s2þ2 fofs þo2

f

The controller poles, which are defined by the filter properties

(of,xf), are located in the complex plane to produce an adequate

migration of closed loop poles as the feedback gain g is increased

The minus sign in Eq (4) is included to produce a negative

feedback of the acceleration

As is shown inFig 4, the accelerometer is located very close to the actuators The control configuration, where the measurement

is done in the same degree of freedom (dof) as the excitation, is called the collocated control, and it is very demanding in practical applications because it enjoys the very attractive property of unconditional stability [12,13] Physical constraints avoid exact collocation in this application

5 Reduction procedure Once the FE model has been modified incorporating the active elements and the control strategy has been described, it is necessary to apply a reduction procedure to complete the required simulations at a reasonable computer cost For this purpose, the modal truncation method was used [14], which is based on characterizing machine dynamics by the dominating vibration modes within the frequency range of interest To cover the main and the secondary chatter modes, this frequency range was established between 0 and 160 Hz, where 15 vibration modes were calculated For these modes, both natural frequencies and modal displacements corresponding to dof’s, where application of forces or acquisition of responses was required, were selected Following the procedure described in[9], the extracted modal parameters, together with modal damping factors obtained experimentally, were used to obtain a state space model of order

30 designed for the purpose of predicting displacements, velocities and accelerations of selected dof’s

The integration of the reduced order structural model in the cutting process model demanded the consideration as inputs of the normal grinding force and the voltages Vpzt_1 and Vpzt_2

supplied to the piezoelectric actuators The controller H(s)

Fig 4 Control strategy.

supplies the same voltage signals to both the actuators

(Vpzt_1¼Vpzt_2), so these two voltages were considered as one

unique input Vpzt

The variable describing machine deformation (ym) and the

acceleration of the dof corresponding to the accelerometer

location (a) were selected as model outputs

After defining the necessary inputs and outputs, the state space

model order was reduced, removing vibration modes with

unimportant contribution in the input–output behaviour using

the balanced truncation method[15] As a result, a reduced state

space representation was obtained, described by the three most controllable and observable modes (6 states), defined by _

x ¼Arx þBr

Fn

Kpztd33nVpzt

ym a

¼Crx þDr

Fn

Kpztd33nVpzt

where Ar, Br, Crand Drare the reduced matrices of the system This reduced order state space model was validated comparing several frequency response functions (FRFs) obtained both experimen-tally and using the model A good agreement was obtained between the responses, mainly at low frequencies and in the vicinity of the resonance peaks corresponding to the main and the secondary chatter modes

6 Control system evaluation Once a state space model describing the structural behaviour

of the modified centerless grinding machine has been developed, the following step in the simulation process consists of closing the loop between the acceleration and the voltages through the control law described in Eq (4), as illustrated inFig 5

The controller design requires adjusting the frequency of the filter poles to the natural frequency of the mode that is intended

to actively damp[12] This application is focused on the reduction

of vibrations related to the main chatter mode because this is the one responsible for the appearance and evolution of the most important roundness errors in workpieces, soofwas adjusted to

55 Hz Filter damping was fixed inxf¼0.5[12] 6.1 Evolution of structural roots

A very important feature to consider in the simulation process

is the choice of the feedback gain, as it modifies the structural dynamic behaviour, changing closed loop roots location The evolution of these roots for increasing values of feedback gain is shown inFig 6, where the trajectories of compensator poles and structural roots selected in the balanced truncation process can be distinguished The structural roots correspond to the following vibration modes:

a suspension mode at 209 rad/s (33.3 Hz),

the main chatter mode at 363 rad/s (57.8 Hz),

Fig 5 Feedback loop.

Fig 6 Root locus for increasing feedback gain.

 the secondary chatter mode at 797 rad/s (126.8 Hz).

None of the trajectories shown inFig 6crosses the imaginary

axis and thus all roots remain in the left part of the complex plane

It is interesting to go deep into the evolution of the structural

root associated with the main chatter mode, shown inFig 6in a

box and detailed inFig 7a It can be seen that the root reaches a

maximum damping of 22.5% for an optimum value of feedback

gain Damping of the root in open loop was 3.6%, and so the active

damping capability of the proposed control strategy is

demon-strated

representing the structural deformation and the normal grinding

force, both in open loop configuration and in closed loop with optimal gain Resonant peaks corresponding to the main chatter mode and the suspension mode are substantially reduced On the other hand, the peak corresponding to the secondary chatter mode remains practically unaltered, showing that actuators have

no control authority over this mode

6.2 Time domain simulations: experimental verification The procedure to obtain the maximum damping in the main chatter mode, described in the previous section, does not take into account the practical limitations that can be presented when the control system is implemented These limitations arise because force generation capability of the actuators is limited and, in practice, the required active forces to counteract self-excited vibrations can exceed the admissible limits before feedback gain reaches its optimum value

The above-mentioned restriction results in an upper limit of the voltage that can be applied to the actuators Taking into account that PI-247.30 piezoelectric actuators require an input voltage between 0 and 1000 V, it is very interesting to develop a methodology to predict the voltages demanded by the control system for different feedback gain values to assure that admissible limits are not exceeded For this purpose, time domain simulation

of the process was programmed, which is well suited to obtain such quantitative values This procedure implies integrating the control loop shown inFig 5in the chatter loop of the centerless grinding process, previously presented by several authors[6,9] Fig 8shows the integrated model, wheree0, (1e),j1andj2

are variables depending on the geometric configuration of the machine, s is the Laplace operator,opis the angular velocity of the workpiece, K is the cutting stiffness and keq is the equivalent contact stiffness The control system influence over the structure

Fig 8 Centerless grinding process chatter loop with control algorithm integrated.

Fig 9 Theoretical evolution of accelerations: (a) at the grinding wheel head, (b) at the regulating wheel head and (c) at the workblade.

can be seen as follows: when a feedback gain value is selected, the

relation between structural deformations and normal grinding

force is expressed by the damped FRF, thus originating a reduction

in vibration amplitudes

Several grinding cycles were simulated, increasing sequentially

the feedback gain and, consequently, the voltage applied to the

actuators For each gain value, the workpiece was divided into 360

equal radial segments and its rotation was simulated

segment-by-segment using the experimental cutting conditions programmed

previously in the frame of the work performed in[8] Roundness

errors of workpieces were calculated following the procedure

described in[9] and, additionally, the evolution of both normal

grinding forces and voltages applied to the actuators was

obtained

The last simulated grinding cycle corresponded to the one

where the voltage applied to the actuators reached its limit This

situation happened for a smaller feedback gain value than the one

with which maximum active damping had been obtained inFig 7

A subsequent increase in the feedback gain would saturate the

actuators, meaning that their physical limits are an important

constraint in this application

This last study was used to check the effectiveness of the

theoretical model comparing two important simulated results to

the corresponding experimental measurements The first test was

based on the comparison of vibration amplitudes in different

components of the centerless grinding machine, whereas the

second one was based on the comparison of final workpiece roundness errors The results are detailed below

6.3 Comparison of vibration amplitudes This study was undertaken using the normal grinding force evolution as input in the developed state space model to obtain some acceleration predictions as outputs Figs 9a–c show the theoretical evolution of accelerations in three dof’s located at the grinding wheel head, the regulating wheel head and the work-blade, respectively, both before and after the application of control law Experimental measurements of the same variables are shown

inFig 10 Theoretical results agree with experimental measurements quite well, as they predict adequately the quantitative values of accelerations in different components of the machine Further-more, the maximum reachable vibration reductions in these components are predicted correctly Additionally, it can be seen that no vibration reduction is obtained in the workblade, as it could be expected taking into account that vibration of this component is dominated by the secondary chatter mode, which cannot be actively damped (Fig 7b) In this case, simulation results show even an increase in acceleration amplitudes when control is applied

6.4 Comparison of final workpiece roundness errors The final profile of the workpiece gives a quantitative measurement of the maximum error reduction that can be achieved with the SOF controller.Fig 11a shows the theoretical shape simulated before the application of the control law whereas Fig 11b illustrates the profile after its application.Figs 12a and b show the profiles obtained experimentally

The theoretical model predicted a roundness error reduction of 41.7%, whereas the experimental error reduction had been of 32% This result shows that computer calculations were highly realistic, which is a statement that is reinforced comparing theoretical and experimental workpiece profiles

7 Conclusions The experimental testing of active vibration control

time-consuming task and, therefore, solutions based on virtual

Fig 11 Final theoretical workpiece profile: (a) without control and (b) with

control.

prototyping by numerical simulations are increasingly

demanded The work presented in this paper satisfies the demand

existing in the centerless grinding sector, providing an

inexpensive tool adapted for the purpose of evaluating the

performance of different control alternatives before their practical

implementation

The developed model integrates the description of both the

mechanical structure and the control system following a

mecha-tronic approach It was validated comparing the experimental

results obtained from a previously implemented second-order

filter controller with the ones estimated numerically Quantitative

predictions concerning vibration reduction capability of the

controller were in good agreement with experimentally obtained

results Additionally, the model has the ability to predict correctly

the roundness error improvement in workpieces once the

control has been activated Thus, it is proved that the simulation

procedure described in this paper gives a reliable model

capable of predicting the controlled dynamic behaviour of the

machine

Therefore, this work constitutes an important advance in the

field of design of controllers integrated in machine tool structures

The availability of the developed model is an essential

require-ment to tackle an optimisation process of the active vibration

control system in the centerless grinding machine, as it permits

one to evaluate in the design stage how different control

algorithms contribute to improve the dynamic behaviour of the

machine

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