Precision, Stability and Productivity Increase in Throughfeed Centerless Grinding docx

However, workpiece roundness is critically affected by geometric lobing and no practical tool has been developed to solve the problem in throughfeed working mode.. Keywords: Centerless

Precision, Stability and Productivity Increase in Throughfeed Centerless Grinding

I Gallego1 (3), R Lizarralde2, D Barrenetxea2, P J Arrazola1

1Manufacturing Department, Faculty of Engineering – Mondragon University, Mondragon, Spain

2Ideko Technological Center, Elgoibar, Spain Submitted by R Bueno (1), San Sebastian, Spain

Abstract

Centerless grinding is a high precision manufacturing process commonly applied to the mass production of

many industrial components However, workpiece roundness is critically affected by geometric lobing and no

practical tool has been developed to solve the problem in throughfeed working mode Based on simulation

methods previously applied to plunge grinding, a new software tool has been developed in this work The

software determines the optimal working configuration and can be used to reduce set-up time and improve

three important features: 1) Precision, as the roundness error is rapidly corrected at the optimal configuration

2) Productivity, since the workpiece stock can be significantly reduced 3) Stability, because the process is

less sensitive to the original roundness error of the workpiece

Keywords:

Centerless grinding, Productivity, Simulation

1 INTRODUCTION

Centerless grinding is a manufacturing process widely

used as a high-productivity finishing technology The main

advantage of this process comes from the fact that

clamping and centering operations are eliminated,

reducing operation times and allowing automation via

workpiece chargers and manipulators

However, workpiece roundness is critically influenced by

two types of instabilities: chatter and geometric lobing

Chatter appears as a result of the coupling between the

cutting process and the main frequencies of the machine

Simulation of dynamic instabilities and chatter avoidance

will be treated in an upcoming paper On the other hand,

geometric lobing may appear when the machine

configuration allows a number of lobes close to an integer

to be present in the workpiece surface (see Figure 1),

coinciding a lobe maximum in the regulating wheel with

two minima in the grinding wheel and in the blade

respectively Under these conditions, together with its

rotation, the workpiece oscillates between the grinding

and regulating wheels during the process, leading to lobe

regeneration On the contrary, under stable conditions,

the workpiece self-centers and high precision can be

rapidly achieved

Figure 1: Geometric lobing in centerless grinding (size of

lobes is exaggerated)

In a previous work [1], a new practical tool developed to facilitate the set up of plunge centerless grinding was presented Throughfeed grinding is much more complex

to simulate In this operation mode, the regulating wheel

is swivelled to give the workpiece an additional axial movement As a result, the grinder can be continuously fed with workpieces and productivity is significantly increased Nevertheless, the process may be more difficult to set up

This paper presents the basis of a new software tool

called SUA (Set-up Assistant), which is already

commercially available, developed to increase precision and productivity in centerless grinding

2 THROUGHFEED STABILITY SIMULATION

Several models have been successfully developed in the bibliography [2-11], specially concerning plunge grinding

In mid sixties, Rowe et al [2] derived an equation to obtain the roundness error as a function of the error at the contact points, which was applied to the grinder set up problem [3] In 1967, Reeka [4] presented geometric stability charts, which showed the number of lobes generated as a function of the blade angle and the angle between the two tangents of the workpiece with the wheels In the seventies and eighties, the existing models were refined by Miyashita, Hashimoto, Rowe and other authors [5-7], and dynamic effects were introduced in the

equations Both Bueno et al [8] and Zhou et al [9]

showed that the contact stiffness is an essential parameter to predict geometric lobing correctly More recently, Harrison and Pierce [10] applied the model of Zhou to plot stability maps in plunge grinding Finally, in

2004 Hashimoto et al [11] presented guidelines for determining proper set-up conditions to avoid spinners, chatter vibration and roundness problems

Following Furukawa et al notation [12], let rω(ϕ) be the radius defect in a point in the surface defined by the angle

ϕ In plunge operation (Figure 2), the equation that describes the process is:

( )ϕ ε ω(ϕ ϕ ) ( ε) (ωϕ ϕ ) ω( )ϕ

Support blade

Grinding wheel

0,0

0,4

0,8

1,2

Regulating wheel

Annals of the CIRP Vol 55/1/2006

Figure 2: Geometric configuration in plunge grinding

Figure 3: Geometric configuration in throughfeed grinding

(workpieces are machined in a continuous run)

This equation simply describes the displacement of the

contact point between the workpiece and the grinding

wheel due to the presence of defects on the blade and on

the regulating wheel ε′ and (1-ε ) are coefficients that

relate the movement of the workpiece with respect to the

grinding wheel originated by an unitary defect on the

blade and on the regulating wheel respectively Hω(ϕ)

represents the distortion in the cutting depth due to the

static deflection of the grinder, the deformation at the

contact point between the workpiece and the regulating

wheel and other minor effects

In throughfeed grinding (Figure 3), α angle remains

constant, but β varies as the workpiece goes through the

path between both wheels and the blade Consequently,

ϕ2 in equation (1) does not remain constant anymore and,

as a result, process simulation becomes more complex

The solution adopted in this work has been to transform

the plunge model, which is known to be very reliable for

this type of operation [1], into a more generalised

simulation scheme for throughfeed grinding The first step

is to calculate the actual value of ϕ2 as a function of the

workpiece position in its path:

λ λ

λ π

⋅

⋅ +

⋅

−

−

=

1 0 0

0

sin arctan

L

L

z h

λ0 and λ1 are the feed and penetration angles of the

regulating wheel after truing and positioning it (Figure 4)

The next stage is to analyse the process stability degree

all along the workpiece path in the grinder For that

purpose, the path is divided in a number of segments

(typically near one hundred) and equation (1) with the actual value of ϕ2 in each position is used

Every equation is then analysed as described in previous works [1, 13] Laplace transformation is applied to it with the aim of studying unstable frequency components of the defect function As it is well known, the poles of the function in the frequency domain will define whether the process is stable or unstable The function has a very complex analytical form and all the significant poles should be found by means of a highly efficient algorithm Levenberg-Marquardt is the most appropriate optimisation method, as it takes advantage from the safe local convergence given by first-order methods and the faster convergence of second-order methods This algorithm is much faster than Simplex method used by Harrison and Pearce [10], being more suitable for throughfeed grinding due to the number of calculations that must be done Using the output of these calculations, stability maps can

be plotted, as is described in next section

3 RESULTS 3.1 Stability diagram of a specific configuration

In general, ϕ2 angle changes considerably in throughfeed grinding Therefore, it is very difficult to prevent the workpiece from passing through one geometrically unstable segment at least Stability diagrams provide information about the less stable lobe component and its stability degree all along the workpiece path A positive value of stability degree (ωξ) means that roundness error

is corrected, while a negative value indicates that lobe regeneration is going to take place [1]

In Figure 5, three stability diagrams are plotted for different workpiece heights: 0.75 mm, 3.5 mm and 9.5

mm Other configuration parameters are: blade angle 30º, wheels and workpiece diameters 632 mm, 310 mm and

33 mm respectively, λ0 3º, ωr 49 min-1, wheel width 500

mm, feed 2500 mm/min The less stable component

values are shown at the bottom of the graphs

It is important to point out that the presence of a high order unstable component in a short section of the stability diagram (Figure 5c) is not critical for the process, because the amplitude of these type of components is rapidly corrected in the rest of the path The important configurations to avoid are those that have: 1) Low order components at the entry (typically 3 or 5), even in short segments, since lobe correction is much slower in this case 2) High order components in a relatively long segment (more than 20% of the path)

Figure 4: Some important parameters in throughfeed

grinding

Grinding wheel

Regulating wheel

Workpiece exit

Workpiece entry

Feed

α

α

λ1

λ0

λ

z h

hd

zL

x0

Diamond path Workpiece

axis

Regulating wheel axis

(x0,y0,z0)

Y

X

θ

γ=α+β

Workpiece

(a)

(b)

(c)

Figure 5: Stability diagrams, blade angle: 30º, workpiece

height: (a) h=0.75 mm, (b) h=3.5 mm and (c) h=9.5 mm

Due to the computational efficiency of

Levenberg-Marquardt algorithm, stability diagrams are obtained in a

few seconds with an average computer

In the absence of vibrations, these plots may facilitate to a

large extent process setting up As wheel diameter

changes after repeated dressing, stability diagrams may

also be used to correct workpiece height and remain at

stable configurations from the beginning to the end of

wheels life Furthermore, roundness error decreases very

fast at optimal configuration, so productivity of

manufacturing lines may be enhanced by reducing the

stock to grind

In addition, if high process capability (cpk) is required, the use of stability diagrams may be very helpful, because the final result is less sensitive to the initial profile of workpiece, which is usually a matter of concern for grinding companies

3.2 Stability maps

By repeating the previous calculation for different working configurations, stability maps can be plotted (Figure 6) Figure 6a shows the less stable lobe components and Figure 6b represents the global stability of the process The latter may be used at the first stages of the setting

up, as it clearly displays the most stable areas (in blue) and the working configurations to avoid (dark red)

Mathematically, global stability of each lobe component should be defined as the mean value of stability along the path However, to further ensure that unstable segments are avoided, a penalty weight is applied to negative stability values The weight value selected in our simulation is 2.5 for wheels of 500 mm or wider (as in Figure 6b), which decreases linearly to 1 for wheels of

100 mm or less The reason for this change is that, from a stability point of view, throughfeed grinding with narrow wheels is not very dissimilar to plunge grinding, as ϕ2 angle does not change very much

(a)

(b)

Figure 6: Stability maps, wheel with: 500 mm (a) Number

of lobes (b) Global stability degree (relative units) In blue: stable areas In dark red: unstable areas

30 35 40 45

Workpiece height (mm)

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

3 5

12 14 16 18

18 20

24

25

26

27 27 28

29

29 30

31 32

Workpiece height (mm)

30 35 40 45

0 50 100 150 200 250 300 350 400 450 500

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

STABLE

INSTABLE

Workpiece position (mm)

0 50 100 150 200 250 300 350 400 450 500

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

STABLE

INSTABLE

Workpiece position (mm)

0 50 100 150 200 250 300 350 400 450 500

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

STABLE

INSTABLE

Workpiece position (mm)

(a) (b)

0%

-75%

-30%

-32%

-90%

-80%

-70%

-60%

-50%

-40%

-30%

-20%

-10%

0%

n=3

n=5

n=27

Before After grinding

0%

-74%

-71%

-77%

n=3 n=5 n=27

Before After grinding

Figure 7: Experimental evolution of 3, 5 and 27 lobe

components, workpiece height: (a) 3.5 mm (b) 9.5 mm

Figure 8: Global stability degree, wheel width: 200 mm

Experimental tests were performed at the same working

conditions of Figures 5 and 6 (workpiece stock: 0.2 mm)

An amount of 36 workpieces with 3, 5 and 27 lobes were

prepared At low workpiece height, it is very well known

that 3 lobes are formed At h=3.5 mm (Figure 7a), the

process is stable, but the amplitude of 3 and 5

components decreases very slowly On the contrary,

closer to the optimal working configuration, all the lobes

decrease very rapidly (Figure 7b)

3.3 Influence of wheel width on stability

Preceding maps have been obtained for a wheel width of

500 mm For shorter wheels, there is a bigger risk of

finding a high order unstable component in a relatively

long segment

For the purposes of comparison, Figure 8 displays the

global stability degree when the wheel width is 200 mm

As the process is now more unstable, the information that

this graph provides is even more useful than for wider

wheels

4 CONCLUSIONS

A new software tool has been developed to predict

geometric lobing in throughfeed grinding Stability

determination of a specific working configuration requires

only a few seconds in an average computer

Stability maps can be plotted by repeating the same

calculation for different workpiece heights and blade

angles In the absence of vibrations and work rotation

instabilities, these maps are very helpful to select the

ideal working configuration and improve precision and productivity

Stability models provide the basis for a new simulation tool which has been practically implemented in industry by

a software program (SUA, Set-up Assistant), which can

be incorporated into the CNC control of centerless grinders

5 ACKNOWLEDGMENTS

This work has been carried out with the financial support

of the Basque Country Government (projects UE 2005-4 and PI 2004-9) and the Spanish Government (projects DPI2003-09676-C02-01 and FIT-020200-2003-72) The authors wish to thank J I Marquinez for his contribution

to this work

6 REFERENCES

[1] Lizarralde, R., Barrenetxea, D., Gallego, I., Marquinez, J.I., 2005, Practical Application of New Simulation Methods for the Elimination of Geometric Instabilities in Centerless Grinding, Annals of the

CIRP, 54/1:273-276

[2] Rowe W.B., Koeningsberger F., 1965, The Work Regenerative Effect in Centerless Grinding, Int J Mach Tool Des Res., 4:175-187

[3] Rowe W.B., Richards R.L., 1972, Geometric Stability Charts for the Centreless Grinding Process,

J Mech Eng Science, 14/2:155-158

[4] Reeka D., 1967, On the Relationship Between the Geometry of the Grinding Gap and the Roundness Error in Centerless Grinding , PhD Diss., Tech Hochschule, Aachen

[5] Miyashita, M., 1972, Unstable Vibration Analysis of Centerless Grinding System and Remedies for its Stabilisation, Annals of the CIRP, 21/1:103-104 [6] Miyashita M., Hashimoto F., and Kanai A., 1982, Diagram for Selecting Chatter Free Conditions of Centerless Grinding, Annals of the CIRP,

33/1:221-223

[7] Rowe, W B., Miyashita, M., Koenig, W., 1989, Centerless Grinding Research and Its Application, Annals of the CIRP, 38/2:617-624

[8] Bueno R., Zaratain M., Aguinagalde J M., 1990, Geometric and Dynamic Stability in Centerless Grinding, Annals of the CIRP, 39:395-398

[9] Zhou S.S., Gartner J.R., Howes T.D., 1996, On the Relatioship between Setup Parameters & Lobing behavior in Centerless Grinding, Annals CIRP, 45: 341-346

[10] Harrison A J L., Pearce T R A., 2002, Prediction

of lobe growth and decay in centreless grinding based on geometric considerations Proc Instn Mech Engrs., Part B: J Engineering Manufacture, 216:1201-1216

[11] Hashimoto F., Lahoti G.D., 2004, Optimization of Set-up Conditions for Stability of The Centerless Grinding Process Annals of the CIRP,

53/1:271-274

[12] Furukawa Y., Miyashita M., Shiozaki S., 1972, Chatter Vibration in Centerless Grinding, Bull of JSME, 15, 82:544-553

[13] Gallego, I., Barrenetxea, D., Rodríguez, A., Marquínez, J I., Unanue, A, Zarate, E., 2003, Geometric lobing suppression in centerless grinding

by new simulation techniques, The 36th CIRP-International Seminar on Manufacturing Systems, 163-170

30

35

40

45

Workpiece height (mm)

-0.1 0 0.1 0.2 0.3 0.4 0.5