Intelligent Centerless Grinding Global Solution for Process Instabilities and Optimal Cycle Design

7/30/2019 Intelligent Centerless Grinding Global Solution for Process Instabilities and Optimal Cycle Design

1/6Annals of the CIRP Vol. 56/1/2007 -347- doi:10.1016/j.cirp.2007.05.080

Intelligent Centerless Grinding: Global Solution for Process Instabilities and OptimalCycle Design

I. Gallego (3)

Manufacturing Department, Faculty of Engineering Mondragon University, Mondragon, SpainSubmitted by R. Bueno (1), San Sebastian, Spain

AbstractCenterless grinding productivity is largely limited by three types of instabilities: chatter, geometric lobing andworkpiece rotation problems. Regardless of its negative effect in manufacturing plants, no functional tool hasbeen developed to set up the process, because it involves the simultaneous resolution of several coupledproblems. In this paper, new simulation techniques are described to determine instability-free configurations,making it possible to guarantee that the final workpiece profile is round. With this information and taking intoaccount other process restrictions, like system static stiffness and workpiece tolerance, the optimal grindingcycle is designed. These results have been implemented into an intelligent tool to assist the application of thisresearch in industrial environments.

Keywords:Centerless Grinding, Productivity, Simulation

1 INTRODUCTION

In centerless grinding, the workpiece is not clamped, butsimply supported between the grinding wheel, the bladeand the regulating wheel (figure 1), reducing the machineidle time and avoiding the necessity of centring holes onthe workpiece. Due to the enormous manipulating timeand manufacturing cost saving that this implies,

centerless grinding is extensively used in the massproduction of components in automotive and bearingindustries for example.

Nevertheless, the process suffers from three kinds ofinstabilities that may limit its precision and productivity.

1) Chatter, whose growing is much more pronounced thanin conventional grinding.

2) Geometric lobing, which appears when the workpiecedoes not self-centre and begins to oscillate between thewheels.

3) Work-holding instability, which appears when theregulating wheel is not able to make the workpiece spin atits peripheral velocity.

In previous works, techniques to avoid geometric lobing ininfeed [1] and throughfeed [2] have been shown.Nevertheless, as stated by Hashimoto, all the problemsshould be solved at once, because stable conditions forone instability may not be so proper for another one [3].

Figure 1: Geometric configuration in infeed grinding.

In this work, a new technology has been developed togive a global solution to all instabilities at the same timeand design the optimal grinding cycle, so as to obtain therequired workpiece tolerance in the minimum time withoutthermal damage. The application of these results ingrinding industry has potentially a great impact, reducingprocess set-up time and decreasing production stops

motivated by out-of-roundness issues.

2 PROCESS SIMULATION

2.1 Process equations

In this section, the basic equations governing thecenterless grinding process are shown. A new and morecoherent notation has been employed to simplify some ofthe expressions.

Let rw(t) and Grw(t) be the radius and radius defect of theworkpiece in an infeed process (figure 1). The radiusdefect varies due to strictly geometric reasons, becausethe cutting forces cause changes in the deflection of themachine, workpiece and wheels, and because vibrations

may arise. This way, Grw(t) can be expressed as:

ttttr DKgw HHHG (1)

where Hg(t) is the geometric displacement of theworkpiece due to roundness errors passing through the

contact points with the blade and regulating wheel. HK(t)represents the time variation of the system static

deflection (along the cutting direction) and HD(t) is the termrepresenting the vibrations generated in the process.

The geometric term Hg(t) was derived by Dall [4] andrefined by Rowe et al. [5]. It can be expressed as:

rwrbwbg WGWGH trgtrgt (2)

being bw WG tr and rw WG tr the radius defect at thecontact points with the blade and regulating wheel. gb andgrare two geometrical parameters [1].

Geometric lobing stability has been studied in thefrequency domain [6,7]. Applying the Laplace transform toequation (2), it is obtained:

T

Jrh

M M

JJrJs

Grinding wheel Regulating wheelBlade

Workpiece


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rb rbwgWW

Hss

egegsRs (3)

In a grinding process, the cutting force Fc(t) and theinstant depth of cut, ae(t) are related by a parametercalled cutting stiffness (kw):

)(ewc taktF (4)

The cutting force produces a deflection of the machine,wheels and workpiece, which makes the radius reductionto accumulate a delay in relation to the programmed feed.From a stability point of view, it is not a matter of interestto know the delay, but the variations in deflection

generated by the radius defect evolution. Defining W asthe rotation period of the workpiece, it is obtained:

trtrktF wwwc GWGG (5)

The second term of equation (1) can be expressed as:

eq

cK

k

tFt

GH (6)

where keq is the equivalent stiffness of the system.

keq can be determined in two different ways:experimentally or theoretically with the help of a data-base. In the first case, several tests must be performedemploying different feed levels. Based on the differenceof diameters between tests with and without spark-out,the relation of the cutting force and the total deflection ofthe system can be obtained. In the second case, thevalue of keq can be predicted analytically using thisexpression:

1crr

1cs

1m

1eq

kgkkk (7)

This equation is derived from a spring-model, whichincludes the machine and blade stiffness (km) and thewheels/work contact stiffness (kcs, kcr). Determination ofequivalent stiffness and its dependencies on feed rate,wheel type, etc. is essential to predict instabilitiesaccurately [8,9].

Regarding cutting stiffness, its value may be estimatedwith analytical approximations, but a final experimentalcalibration is recommended to have chatter predictionmaps and the optimal grinding cycle very close to reality.

Introducing expression (5) in equation (6):

trtrKtrtrk

kt wwww

eq

wK GWGGWGH (8)

K parameter represents the flexibility of the system andrelates the amount of deformation of the system todifferent depths of cut.

Centerless grinding chatter was extensively studied byMiyashita, Hashimoto et al. [10,11] and Rowe et al.[12,13,14]. These authors developed successful modelsthat can be used to predict chatter-free configurations.Bueno et al. [8] and Nieto et al. [15] determined thethreshold stability value of the cutting stiffness as afunction of the workpiece rotation frequency andintroduced non linear effects in the model, such as thespark-out and lobe filtering. Hashimoto and Zhou [16]proved that filtering effects have a big effect on highwaviness stabilisation. According to these authors,contact stiffness is again an important factor on chattergrowing, as it has been stated for other types of grindingprocesses too [17].

To introduce vibrations in the model, the dynamicflexibility of the machine H(s) may be used. Centerlessgrinding process is usually well defined with a two-dimensional modal analysis at the three points of contactbetween the workpiece, blade and wheels. Nevertheless,ifH(s) is introduced in the model without any correction, itwill add an extra contribution to the static deflection of thesystem, already considered in equation (8). Subtracting

this contribution, the expression ofHD

in Laplace domainturns into:

m

12r

r

rr22

r

rwwD

21

N

r

s V

ss

VesRks

ZZ[ZH W

(9)

Nm: number of considered vibration modes.

mr, Zr,[r: modal mass, frequency and damping ofrmode.

Vrmay be expressed in this way:

^ ` ^ ` ` ^ `PXXCV TrrTr (10)

{Xr

}: vector containing the relative deformations at thecontact points ofrmode.

{C}: vector quantifying the real displacement at the cuttingpoint due to a displacement of the contact points.

{P}: vector relating the forces at the contact points withthe normal force at the cutting point.

The last term in equation (9) is the referred correction toH(s). We consider that this is an essential enhancementof preceding models, as it has a significant influence onthe dynamic stability maps shown in section 4. In addition,it should be pointed out that when vibrations areintroduced in the model, it is very important not to

disregard the static term HK(t), because it may be provedthat, in that hypothetical case, geometric lobing could

appear with lobe numbers very far from integer, which isnot physically possible.

Rearranging all the terms deduced in equation (1):

01

2

11

m

rb

12r

r

rr22

r

rw

rb

w

-

W

WWW

ZZ[Z

sN

r

sss

eV

ss

Vk

eKegeg

sR

(11)

As it is well known, the poles of'Rw(s) will define whetherthe process is stable or not. The poles can be expressed

in terms of defect regeneration frequency (Z) and

damping ([): s=-Z[+iZ(1-[2)1/2

. Regeneration will be

unstable for[


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but using a new type of functional blade to avoidgeometric lobing. Other authors, like Harrison and Pearce[20], have proposed changing machine configuration in-process to allow a faster correction of the initialroundness error of the workpiece.

Finally, it should be mentioned that the equations toestablish the limits for work rotation instabilities werededuced by Hashimoto et al. [21].

2.2 Instabilities determination

In contrast to milling process, where it is just necessary toknow the chatter limits, in centerless grinding it is alsonecessary to determine the absolute value of the stabilitydegree of the process. This is because at the optimalconfiguration, where all the lobes are stable with themaximum possible stability degree, initial roundness errorcorrection is faster. This way, the process is less sensitiveto changes in the roundness of entering workpieces and afinal round profile may be obtained.

Consequently, the development of a completely reliableand efficient method is the key to determine the bestworking configuration in a reasonable time. This is one ofthe major contributions of this work. In the past, several

types of techniques have been employed: graphicalmethods [11], numerical methods [22], Taylors seriesapproximation [9] or the Simplex method [23].

Established that f(s) is a complex function of complexvariable, those s values which verify f(s)=0 will also verify

|f(s)|2=0. Naming Dand E the real and imaginary parts of

s, the function is graphically shown in figure 2. This way,

pole-finding of 'Rw(s) has been transformed into rootfinding of a real function of two real variables. Being |f(s)|

2

positive, the problem is transformed into finding local

minima. |f(s)|2

has many minima close to D=0 axis. Those

configurations with positive real part roots (i.e. [


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0 5 10 15

Height (mm)

Figure 4. Chatter and geometric lobing stability map. Indark blue: stable areas. In red: unstable areas. Star sizeis proportional to the experimental vibration amplitude,

round points represent tests without chatter.

In figure 4 a stability map is shown as a function of h and

Zr for a blade angle of 30. Red areas representconfigurations susceptible to chatter, light blue zonescorrespond to geometric lobing, while dark blue areas are

stable for both chatter and geometric lobing. Black linesseparate stable and unstable areas.

Several tests have been performed in order to check thesimulations, using two grinders of different manufacturersin the next conditions:

1. Machine 1: small grinder, wheelhead power 8 KW,

wheelhead opening frequency: 90.8 Hz; T= 30; Ds == 325 mm, Dr = 220 mm, Dw = 24 mm; workpiece

length: Lw = 25 mm; Keq = 14.2 N/Pm; Q == 0.75 mm

2s

-1; feed: 1.2 mm min

-1.

2. Machine 2: large grinder, wheelhead power 60 kW,

wheelhead opening frequency: 58.3 Hz; T= 30; Ds == 628 mm, Dr= 340 mm; Dw = 47 mm; Lw = 368 mm;

Keq = 69.7 N/Pm; Q= 1.23 mm2

s-1

; feed: 1 mm min-1

.

In the theoretical maps obtained for these cases, chatterfree and geometric lobing free areas can be observedwhen using low heights and workpiece rotation speeds,as well as some transient areas at higher rotation speeds.There are also stable areas elsewhere, but they are toosmall from a practical point of view to be used.

For machine 1, the map is checked with experimentalresults in figure 4. The size of the stars is proportional tothe experimental vibration amplitude, while the round dotsmean that no dynamic instability is excited. A goodcorrelation is observed between simulation andexperimental results.

For machine 2, the simulation has been likewise reliable,in spite of the fact that the machine and process are very

different from the previous case.

A remarkable property observed in the theoretical maps isthat the size of unstable areas changes as a function of

the equivalent stiffness. For a given machine, changes incontact stiffness caused by workpiece length, feed rate orwheel type have an impact on stability maps and explaindynamic variations observed in production plants. Forexample, a more flexible regulating wheel reduces thesize of unstable areas significantly.

It should be noted the importance of the existence ofdynamically and geometrically stable zones on the maps

at high rotating speeds. It has been proved that in theseconfigurations, as the workpiece turns many revolutionsduring the process, better roundness and roughnessqualities are achieved. Moreover, a high rotating speedalso prevents the workpiece from thermal damage.

On the other hand, as we presented for geometric lobing|1,2], it is possible to solve the equations in time domain.This possibility opens new ways for chatter suppression,such as variable workpiece rotation velocity. This pointwill be discussed in a future work.

5 WORK ROTATION INSTABILITY AVOIDANCE

The regulating wheel is the element controlling therotational movement of the workpiece, exerting a brake

moment over it by friction. In certain conditions, forexample when high feed velocities or dull wheels areemployed, the workpiece dragging becomes unstable,generating shakes, irregular velocities, jumps andaccelerations, with risks for machine operators.

In the absence of other kind of instabilities, thesephenomena are the limiting factor to productivity, becausethe process feed defines the cutting force exerted by thegrinding wheel and the required brake moment to controlthe movement of the workpiece. On the other hand,another function of the regulating wheel is to rotate theworkpiece before beginning the grinding process. If theworkpiece does not rotate, a flat band may be generatedin the periphery of the workpiece.

As mentioned before, the model and equations describingthese phenomena were fully developed by Hashimoto etal. [21]. The model is conditioned by how precisely the

values of the friction coefficients in the contacts (Pb, Pr)are introduced. With this objective two methodologieshave been developed: one in laboratory and another insitu in the process. In laboratory, tribometermeasurements have been performed with disc-on-discgeometry, due to its similarity with the real process, usingpressures at the contact area identical to real processes.To obtain the values in situ, two force sensors in theslides and a Kistler plate under the workblade have beenemployed.

On tables 1 and 2 the results obtained for a specific wheel(rubber based A80 regulating wheel in contact with

hardened F-522 steel) are displayed. The frictioncoefficient depends strongly on the wheel topography(related to the dressing and the wheel wear) and thelubricant used, but not so strongly on the exertedpressure and the grain mesh. The working configurationemployed in table 2 belongs to a working regime in whichthe regulating wheel has no problem to hold back theworkpiece, as the relationship between tangential and

normal forces at the contact point ( nrt

r FF ) is smaller

than the limit friction coefficient.

Based on these friction coefficient values, it is easy to plot

stability maps as a function of feed, h, T, Zr or otherparameters, showing spinning free areas (see figure 5).

With the aid of these maps, it is possible to know theavailable margin to increase the feed without risks.

6

8

10

12

14

1618

2022 21

19

9

11

13

1517

2119

9

11

13

1517

7

Regulatingwheelspeed(min-)

-5


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Rough dressing Fine dressing

Pressure (MPa) R Pressure (MPa) R70 0,29 70 0,22

100 0,31 100 0,19

Table 1: Friction coefficients between regulating wheeland F-522 steel as obtained in tribometer for different

dressing conditions and pressures.

Rough dressing Fine dressing

Feedmm/min

PressureMPa n

r

tr

F

F Feedmm/min

PressureMPa n

r

tr

F

F

1,6 45 0,25 1,6 52 0,20

3,2 59 0,25 3,2 73 0,20

5,7 72 0,25 5,7 89 0,20

8 86 0,24 - - -

Table 2: Relationship between tangential and normalforces at regulating wheel-steel contact as obtained in asensorised grinder for different dressing conditions and

pressures.

In a future paper, a complete study of friction coefficientvalues for different wheels and conditions will be shown.

6 OPTIMAL CYCLE DESIGN

The main reference on optimal cycle calculation ingrinding processes is the work developed by Malkin [25].Based on this work, the author has successfullydeveloped the GrindSim software to set-up and optimisecylindrical grinding processes.

Two possible criteria can be used to design a centerlessgrinding cycle: 1) Minimise the process cycle time or 2)Adjust the cycle to a previously established process time,

minimising wheel wear. This last option is very common inproduction lines.

The process parameters to be optimised include the feedfor each stage of the infeed cycle, the stock removal ineach stage and the spark-out time. The restrictions toapply are given by the maximum power to employ in theroughing process free from burning problems [25] and therequired tolerance and roughness of the final workpiece.

To define the optimal cycle, the feeds to use are fixedfirst. The first feed will depend on the criteria chosen tooptimise the cycle. In the first case, it is deduced from themaximum power that is possible to use. If production timeis pre-established the strategy is similar, although it is

Figure 5: Work rotation stability map for 5 mm/min feedand a dull regulating wheel (friction coefficient: 0.20).

Figure 6. Grinding cycle evolution with 4 feeds and spark-out: (a) Deflection. (b) Real radius of workpiece (blue) and

programmed position for the wheel (black).

necessary to employ iterative methods. Once theroughing feed is known, the rest are obtained through aseries of given relations (depending on the wheel in use).Machining time for each feed can be calculated by

establishing a proportionality with the deflection (G)

accumulated in the previous stage (figure 6a), although itwould still be necessary to determine the time for the firststage and the spark-out. The referred proportionality canbe adjusted depending on how aggressively we want todesign the process.

The spark-out time is calculated establishing that theexponential reduction of the radius defect fits the required

radial tolerance (Gf


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Intelligent Centerless Grinding Global Solution for Process Instabilities and Optimal Cycle Design