Computer simulation of the deformation of slender, multidiameter rollers during grinding

Int. J. Much. Tools Manufact, Printed in Great Britain

Vol. 31. No. 1, pp.83-93. 1991.

11890--6955/9153.(10 + .00 Pergamon Press pie

C O M P U T E R SIMULATION OF THE D E F O R M A T I O N OF SLENDER, M U L T I D I A M E T E R ROLLERS D U R I N G G R I N D I N G Y. GAO* and K. FOSTER* (Received 20 February 1990; in final form 26 June 1990)

A~tr~et--The paper presents an easily implemented computer simulation method for grinding to high accuracy a class of long slender rollers with a wide variety of multidiameter structures used in the plastic film industry, the cost of which compares favourablywith normal trial-and-error in situ grinding. Mathematical models have been developed to describe the deformation of the roller when grinding conditions such as the rigidity of the supporting centres, the grinding force and also the oscillation of the workpiece in high speed rotation are taken into account. The proposed method has been verified by comparing it with the analytical results of a simple beam and also by verificationfrom an in situ experiment. Based on the results of simulation, a modified control method, which is used to compensate the deflections in order to achieve the required accuracy of parallelism, is also proposed in order to cope with different types of rollers and different grinding conditions.

INTRODUCTION IT IS WELL known that a plain long slender roller supported by two centres on the machine, which can be structurally represented as a simple beam, will have a barrel shape from a traverse grinding process as unequal deflections develop in the radial direction due to the grinding force. However, when the rollers are of the multidiameter structure as shown in Fig. 1, largely used in the film industry, the deflections of the roller along the longitudinal axis of the roller will be quite different and more complex from those of a plain roller. Therefore, deflections and the final shape of the roller are not easily predicted. In order to achieve a higher accuracy of parallelism, predictions of the shape are required to be made in advance, as the five correction steadies used to compensate the deflections of the roller in the opposite directions during the grinding process will be adjusted according to the predictions. It is obvious that the better the knowledge of the deflections, the easier and quicker it would be to achieve the required accuracy and therefore the less the cost. In the current method, the correction steadies are manually adjusted. The results of grinding from the manual operation are often unsatisfactory in both accuracy and efficiency. To achieve a constant productivity at a high grinding accuracy, a computer controlled form error correction system is to be used to replace the manual adjustment. Following an analysis and some experiments [1, 2], a trial was made of a fuzzy correction method, which aimed to copy and replace the experience of a skilled operator, but the grinding quality and accuracy achieved in practice proved not to be entirely satisfactory. This may be explained by errors between the correct compensation decision and the operator's decision, and also by errors between the operator's experience and the fuzzy decision. The unsatisfactory results may also be explained by lack of sufficient experiments to cover widely different types of rollers, which are costly and time consuming to carry out. Further, certain parameters affecting grinding such as the stiffness of the supporting centres, the grinding force and the workpiece oscillation when in rotation were not considered. In the present paper, an explicit analysis and a computer simulation method of the grinding process are presented. The paper also gives some guides for correction of the

* Mechanical Engineering Department, The University of Birmingham, PO Box U.K.

83

363,

Birmingham B15 2TT,

84

Y. GAO and K. FOSTER

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form error which will be more able to cope with the wide variety of rollers and grinding conditions. Finally the paper proposes a modified error correction system, in which an additional on-line shape prediction system and the previous fuzzy system are included. MATHEMATICAL MODEL

Mathematical model

When the roller is being ground, the grinding force developed between the roller and the wheel will deform the roller with an elastic deflection in the normal direction of the contact surface (Fig. 1). The roller, having a multidiameter structure, can be regarded as a nonprismatic beam [3] in mechanics, where the force applied on the roller can be supposed as a concentric load when the width of the wheel is small compared with the length of the roller. For small deflection, the curvature of the longitudinal axis of a beam may be assumed to be proportional to the bending moment and inversely proportional to the flexural rigidity. As the cross-section of the roller changes with x, the roller has to be mathematically divided into 10 sections so that within each section a constant moment of inertia is kept. Let the sequence number of the 10 sections be denoted by i. The joint points between sections can then be described by {X[i], i = 1, 2 . . . . . 10}. When the second order derivation is denoted by y", which is approximately equal to the curvature [3], the curvature equation can be represented separately as EJ[i]y"= - Pbx/L,

X[i-1] ~ x < X[i]

(1)

i = 1 , 2 ..... 5 and EJ[i]y"= - Pbx/L + P ( x - a ) ,

X [ i - iI -< x < X[i]

(2)

i = 6 , 7 ..... 10 where a is the wheel position, P is the load equal to the normal grinding force, and E J[i] is known as the flexural rigidity of the beam; E is the modulus of elasticity of

Computer Simulation of Rollers

85

the material of the roller; J[i] is the moment of inertia for the described section depending on the geometry of the cross-section of the roller. Therefore, as the workpiece is circular in cross-section, J[i] = ~rd4[i]/64 for i = 1, 2, 3, 4, 7, 8, 9, 10, and J[i] = "trda[i]t/8 for i = 5, 6, where d[i] is the diameter of the ith section. The data {X[i], i = 1, 2 . . . . , 10} and {d[i], i = 1, 2 . . . . . 10} are shown in Table 1. Further, L=a+b= 1622mm, t = 2 4 m m , E = 2 . 0 x 105N/mm 2 , a n d a = X [ 5 ] = 1005mm for the shown wheel position. The load P is firstly assumed to be an usual value 200 N. Integration twice according to (1) and (2), the following equations can be obtained as

EJIily = - Pbx3/(6L) + Cl[ilx + C2[il i = 1,2,...,5

(3)

EJ[i]y = - Pbx3/(6L) + P(x-a)3/6 + Cl[i]x + C2[i] i = 6 , 7 .... ,10

(4)

and

where {CI[i], i = 1, 2 . . . . . 10} and {C2[i], i = 1, 2, ..., 10} are constant arrays. The twenty constants of integration appearing in the preceding equations can be found from the following three categories of conditions. Firstly, at each joint point the slopes y'(x) of the two sections must be equal. Secondly, at the same point the deflections y(x) must be equal. The conditions can also be expressed as

Cl[i]/J[i I = K[i] + Cl[i+l]/J[i+l] i = 1 , 2 .... ,9 Cl[i]X[i]/J[i] + C2[i]/J[i] = KI[i] + CI[i+ 1]X[i]/J[i+ 1] + C2[i+ 1]/J[i+ 1] i = 1 , 2 ..... 9

(5) (6)

where

K[i] = -PbX2[il/(2LJ[i+ 1]) + PbX2[i]/(2LJ[i]) KI[i] = -PbX3[il/(6LJ[i+ 1]) + PbX3[i]/(6LJ[i]) i = 1,2,3,4,5 and

K[i] = {-PbXZ[iI/L + P(X[i]-a)2}/(Zl[i+ 1]) + {PbX2[i]/L - P(X[i] - a)2}/(2J[i]) KI[i] = {-ebX3[i]/L + P(X[i] - a)3}/(6J[i+ 1]) + { e b X 3 [ i l / L - P(X[i] - a)a}/(6J[i]) i= 6,7,8,9. Thirdly, at each end of the roller, where x = 0 mm and x = 1622 mm, the deflections must be equal to the deflections of the supporting centres. Thus the following expression can be found

TABLE 1. STRUCTURE DAtA OF THE ROLLER

i

X[i] d[i]

0 0

1 20 30

2 76 54

3 252 78

4 302 135

5 1005 111

Note. X[5] is specified for the wheel position shown in Fig. 1. d[5], d[6] are the central diameters of the ring structure.

6 1320 111

[mm] 7 1370 135

8 1546 78

9 1602 54

10 1622 30

86

Y. GAO and K. FOSTER

EJ[I] (Pb/L)/KH = C2[1] EJ[10] (Pa/L)/K-r = -PbL3/(6L) + P(L-a)3/6 + CI[10]L + C2[10]

(7) (8)

where KH and K-r are, respectively, the rigidity of the headstock and the tailstock of the grinding machine.

Algorithm The equations from (5) to (8) form a twenty simultaneous linear equation set. It is obviously a narrow band linear equation set with a band width of 2. Therefore a singular algorithm rather than the generally used Gaussian Elimination Method is needed to maintain a high accuracy of numerical calculation. An algorithm to obtain the integration constants, when KH and K-r are firstly assumed to be infinite, is proposed as (a) (b) (c) (d)

C2[1] = 0 C2[i] = J[il{C2[i-llIJ[i-1] - K I [ i - I ] + K[i-1]X[i-1]} i = 2, ..., 10 CI[10] = {PbL3/(6L) - P(L-a)3/6 - C2[10]}/L CI[i] -- J[i]{K[i] + Cl[i+l]/J[i+l]} i = 9 , 8 . . . . . 1.

Program verification Following the above algorithm, the computer programs for simulation have been developed in P A S C A L language. To verify the mathematical model and the programs, the calculations have been performed for a simple beam with a constant circular crosssection of 100 mm in diameter. The analytical solution for the simple beam, when a = b = L/2 = 811 mm, predicts a maximum deflection of pL3/(48EJ) = 18.111 p,m. To five significant figures this is identical to the result of y(812) = 18.111 ~,m obtained from the program R O L E R D E F . For a = 1335 mm and b = 287 mm, the maximum deflection P b ( L 2 - b 2 ) 3/2 /(9~'3LEJ) = 9.408 ~,m must be at x ((L2-b2)/3) I/2 921.7 mm, and again the result from the program y(922) = 9.408 Ixm compares well with the theoretical result. In the same case, at the middle point where x = 811 mm, the computer result of y(812) = 9.216 p,m shows only a small difference of 0.004 Ixm compared with the theoretical result of 9.212 I~m. Thus, the comparisons give confidence in the model and the software. =

=

COMPUTER SIMULATION OF THE GRINDING PROCESS

With two infinitely rigid supporting centres On the application of load, the program predicts the whole roller to deflect as shown in Fig. 2 for the three cases where the load is applied at a = 287 mm, a = 811 mm and also a = 1335 mm. However, the actual diameter of the roller will only depend on the deflection at point x = a where the grinding wheel is positioned. When the wheel moves o v e r t h e length of the sleeve, i.e. the working section which is from 252 mm to 1370 mm, there will be a variation in diameter over the length which then forms a saddle shape or a barrel shape. Such a process can be simulated on the digital computer [4]. From (3) and (4), the shape curve y(a) can be expressed as

y(a) = y(x,a)[{x=a},

a ~ [252,1370]

(9)

The shape curve for the given dimensions from program R O L E R S H P is shown in Fig. 3, where a barrel shape rather than a saddle shape is obtained. However, when the sleeve is very rigid, for example if one assumes a diameter of say 2000 mm, which is large (probably unrealistically large) compared with the remaining parts of the roller, the expected saddle shape [1] is realised, the results are shown in Figs 4 and 5.

Computer Simulation of Rollers !0 76

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8!1

87

1370

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1.00 2.00 3.00 4.00

,5.00 6.00 7.00 J, 8.00 ~,

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Kum) FIG. 2. Deflection curves with the load applied at different positions.

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811

1370

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1.00

2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.0

R(P)-R (~m)

with two finitely rigid ¢entre$ FtG. 3. Shape curves from traverse grinding process.

While the diameter of the sleeve is increased from 111 mm to 2000 mm, the shape curve changes from a barrel shape to a saddle shape. There must be a critical point in the diameter of the sleeve corresponding such a change. The shape of the sleeve may be described by the maximum difference of the shape curve, denoted by E(d). The result for E(d), which was obtained from the program D E L T A D I A using the data in Table 1, is shown in Fig. 6. For the particular roller structure chosen, the value of the diameter at the transition point is found to be approximately 210 mm.

With two finitely rigid supporting centres In practice, the deflections of the two supporting centres from the reaction of the load P should be considered. The rigidity KT and KH of the two centres was obtained from a shop floor experiment as KH = 35.6x 103 N/mm, KT = 47.9X 103 N/mm. It can be seen that the headstock is less rigid than the tailstock. The deflection curves y(x), where the load is acting at a = 287 mm, a = 811 mm and also a = 1335 mm, are shown in Fig. 7. The shape curve y(a) is also shown in Fig. 3 and a barrel shape is still found. Following the above idea a shape chart is also shown in Fig. 6 with a critical point of 146 mm in diameter.

88

Y. GAO and K. FOSTER

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3.00 4,00 5.00

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IR(P)-R (urn) FIG. 5. Shape curves from traverse grinding process when the diameter of the sleeve is assumed to be 2000 mm.

E(O)(um)

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400

200

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800

1000

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1800

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2000 D(mm)

E(d). la70

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FIG. 7. Deflection curves with the load applied at different positions when the rigidity of the supporting centres is considered.

Computer Simulation of Rollers

89

With workpiece being in rotation When the long slender roller is rotating, oscillation where the rotation centre does not coincide with the two supporting centres will occur. The effects of such an oscillation on the shape of the roller should be considered. When a driving moment at a rotation speed to = 2rm, in the unit of rad/s, is applied to the end of the workpiece, each single section with a length dx (Fig. 8) will have an initial velocity V = {-z(x)}00 and a radial acceleration. Thus a centrifugal force and an elastic force from the workpiece will be developed. If the workpiece were infinitely rigid, such a section will rotate to the x axis and keep a radius z(x). However, the workpiece can not be infinitely rigid. The revolution to the x axis will be ceased at some height in the z direction as the weight of the workpiece will pull itself down, the elastic force will pull the workpiece back to its equilibrium position. Therefore, oscillation happens. The longer and the less rigid the roller, the larger the deflection and thus the stronger the oscillation. When the mass density of the workpiece is represented as p, the total weight W of the workpiece can be obtained as I0

~rd2[i](X[i] - X[i-1]) W = pg ~ f ~rd2[i](X[il-X[i-ll) + i=7 l i=1 + ~r(d~o-~)(X[6]-X[4])}/4

(10)

where g = 9.81 m/s 2, do = 135 mm is the outer diameter of the sleeve, d~ = 87 mm is its inner diameter. When p is selected as 7850 kg/m 3 for steel, the weight is found as W = 917.787 N. For simplifying the calculation of the deflection due to the weight, a simple roller with a constant circular cross-section of De in diameter is assumed to have the same weight as the complex roller shown in Fig. 1, thus its weight is assumed to be uniformly distributed. The linear intensity q of a uniform load equal to the weight W can be obtained as q = W/L in the unit of N/mm, and q = 0.566 N/mm. Therefore, De can then be obtained as De = 2(W/(~rLpg)) ~/2= 96.724 mm. The inertia of the simple roller is also found as Je = ~rDe4/64 = 4296305.3 mm 4. The deflection curve - z ( x ) due to the weight can then be obtained as - z ( x ) = qx(L3-2Lx:+x3)/(24EJe), and shown in Fig. 8. The maximum deflection is found as {-z(L/2)}max = 5qL4/(384EJe) = 59.365 x 10 -6 m. In Fig. 9, a small part with a mass of m = qrDe2dxp/4 is considered. Here it has to be explained that the positional relationship between the grinding wheel and the

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x(rnm)

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-20.0 -.t0.0 -40.0 -50,0 -60.0

A~ F~G. 8. Deflection due to the weight of the roller and the oscillation from eccentric rotation.

90

Y. GAO and K. FOSTER

workpiece can be seen from Figs 1, 8 and 9. The elastic force F can be projected to the y axis and the z axis and represented as Fz and Fy, respectively. The oscillation of the workpiece when in rotation in y direction has to be prior considered as any displacement in y direction will be directly projected to the diameter of the workpiece. Thus the differential equation can be obtained as [5, 6] my"(t) + Ks(x)y(t) = 0, where Ks(x) is the stiffness of the workpiece at position x, and Ks(x) = q dx/{-z(x)}. Solving the above differential equation gives [7] y(t)= {-z(x)}to/(Ks(x)/m) w2 sin{(Ks(x)/m) w2t} for a zero initial position. It can be seen that a further radial reduction at position x due to the oscillation, represented by Eo(x), will be equal to the amplitude of the sine oscillation. Therefore, Eo(x), in the unit of mm, can be obtained as

Eo(x) = oJ{- z(x)}3/2 ('rrp) '/2 De/{2(lOOOq)'/2}.

(11)

The result of Eo(x) is shown in Fig. 10 for n = 212.8 rpm. However, if n is selected to be smaller, smaller Eo(x) can then be obtained. For example, when n is selected as 36 rpm, Eo(x) is then less than 1 txm and it may be ignored.

With increased grinding force Grinding force can be obtained from measuring the consumed power which is to drive the grinding wheel. However, the wheel power depends mainly on many conditions such as the cutting depth applied to the workpiece and the wheel speed [8]. Thus it may vary over a wide range. To see how the shape or the parallelism of the workpiece changes with the force, a set of grinding forces of Fn or P from 200 N to 1200 N [9] are selected to be applied in the program D E L T A Y P . The result is shown in Fig. 11.

Z (urn)

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001 =2.076

252

811

1370

______

2.00 3.00

y (urn) FIG. 10, The effect of workpiece oscillation on the shape.

1622

Computer Simulation of Rollers

91

Therefore, if the grinding force is measured, E(d) can then be easily obtained from the figure.

Another type of roller The shape curves of another type of roller, which has two small journal diameters of 30 mm at the two ends of the roller, are also obtained from the same program R O L E R S H P with a little modification in the data set procedures. The result is shown in Fig. 12. DISCUSSION AND RECOMMENDATION

From Fig. 3, it can be seen that the results of the maximum radial difference E(d) differ only a little, which are 3.672 ~m and 4.325 ~.m, respectively, for the case with two infinitely rigid centres and for the case with two finitely rigid centres. In Fig. 12, it also changes a little for the two cases, i.e. from -26.729 p.m to -28.430 ~m. Such results show that the two supporting centres are rigid enough or they have only a little effect on the parallelism. However, the deflections of the supporting centres will largely affect the position of the shape curves, thus the final diameter of the roller may largely be affected. Fortunately, the diameter of roller is not important in the present kind of practical use. Although the deflection in the vertical plane is not normal to the roller, the deflection due to the weight of the roller still affects the shape or parallelism of the roller (Fig. 10) when the workpiece is in high speed rotation. In particular it helps to produce a saddle shape. As the deflection depends on both the weight and the moment of inertia, the simple roller may be determined on either equal weight or equal moment of inertia. However, the latter case has not yet been investigated. Figures 5 and 12 show the results when the sleeve of the roller is much more rigid than the two journal shafts. In the former figure the diameter of the sleeve is far bigger

E(D) (um)

with two finitely rigid centre=

2o o

1

\ I

200

400

600

800

Ftc. 11. Grinding force vs ZO76

252

1000

1200

Force (N)

E(d).

811

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1370

1622

..............

10 20 30

40 50 60 with two infinitely rigid centre=

70 80 90 100

~(um)

Fro. 12. Shape curves of the roller with two small journal shafts.

xCmm)

92

Y. GAO and K. FOSTER

than that of the two journal shafts. The second case is that the diameters of the two journal shafts are very small. Both result in a saddle shape and show that the final shape is very sensitive to the roller structure. In particular, the effect is more pronounced for the case of Fig. 12. From Fig. 12, it can also be seen that the parallelism of the roller is 2×IE(lll)I = 56.86 ~m. The result coincides very well with the result obtained from the in situ tests [1]. The deflection due to the weight of the roller (Fig. 8) also has a good linear regression coefficient of 0.98 with the experienced operator's decision of adjusting the vertical pads of the five steadies, which were measured as 14.1, 31.7, 43.5, 29.4, 12.8 Ixm, respectively, although all the data are less than the corresponding point on the deflection curve. As the final shape is very sensitive to the structure, grinding tests must be done for every type of roller in order to copy the operator's experience when the fuzzy correction system is used. However, it would cost a great deal to grind such large and expensive rollers; it also requires operator's skill. To give better form error correction, a modified control system is proposed as shown in Fig. 13, in which there is an additional on-line shape prediction system using the proposed simulation method. The on-line shape prediction system aims to cope with the wide range of rollers. The previously proposed fuzzy system can be used to deal with the unmeasured and unpredictable factors. Both the theoretical decision and the fuzzy decision will be processed in the main controller by comparing and judging the decisions from both directions and then making the final correction decision. The system has been implemented, but not yet finally tested. CONCLUSION

The results from the computer simulation show that the final shape of the roller from the grinding process is very sensitive to the roller structure. It is not a saddle shape with the originally specified dimensions. However, when the sleeve is far more rigid than the remaining parts of the roller, a saddle shape is predicted. Therefore, for different types of roller, different control algorithms, which are used to compensate the deflection in the opposite direction to maintain a minimum parallelism, must be applied. The rigidity of the two supporting centres has only a small effect on the parallelism of the roller, but it will affect the final diameter of the roller. The weight of the roller itself can produce a large deflection in the vertical plane and such a deflection affects the parallelism of the roller when the roller is in high speed rotation. The closer to the ~

t

on-line shape Prediction

Roller structure

)

Grindingconditions ,) Machineperformance,)

_ -

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-

FtG. 13. The modified error correction system.

I

Computer Simulation of Rollers

93

centre of the roller, the stronger the effect. Therefore, it is suggested that a lower rotation speed of the workpiece is preferred when the roller is in free running with no supporting steadies. The deflection and the parallelism of the roller is proportional to the grinding force. Thus smaller cutting depth is also preferred. Acknowledgement--The authors would like to thank the support from the ORS Awards Scheme. REFERENCES [1] Y. W. ZnAo, In-process measurement and computer control of cylindrical grinding, PhD thesis, Birmingham University, U.K. (1989). [2] H. KALISZER,Y. W. ZHAOand J. A. WEBSTER,Computer aided correction of axial form error in traverse grinding, Third Int. Grinding Conf., MR88-613, SME, Wisconsin, U.S.A. (4-6 October 1988). [3] J. M. GERE and J. MONROE, Mechanics of Materials. Van Nostrand-Reinhold, New York (1987). [4] R. E. STEPHENSON,Computer Simulation for Engineers. Harcourt Brace Jovanovich, New York (1971). [5] W. T. T.OMSON, Mechanical Vibrations, 2nd edn, p. 23. Prentice Hall, Englewood Cliffs, New Jersey (1953). [6] R. BiSHOp, Vibration, 2nd edn, pp. 66-69. Cambridge University Press, Cambridge (1979). [7] W. T. THOMSON, Theory of Vibration with Applications, p. 15. Prentice-Hall, Englewood Cliffs, New Jersey (1981). [8] R. S. HAHN and R. P. LINDSAY, Principles of Grinding: Theory, Techniques and Troubleshooting, pp. 3-41. SME, Michigan (1982). [9] R. I. KISG and R. S. HAHN, Handbook of Modern Grinding Technology, pp. 368--371. Chapman and Hall, London (1986).