Edge effects with the Preston equation for a circular tool and workpiece

Transcription

1 Edge effects with the Preston equation for a circular tool and workpiece lberto Cordero-Dávila, Jorge González-García, María Pedrayes-López, Luis lberto guilar-chiu, Jorge Cuautle-Cortés, and Carlos Robledo-Sánchez In a polishing process the wear is greater at the edge when the tool extends beyond the border of the workpiece. To explain this effect, we propose a new model in which the pressure is higher at the edge. This model is applied to the case of a circular tool that polishes a circular workpiece. Our model correctly predicts that a greater amount of material is removed from the edge of the workpiece Optical Society of merica OCIS codes: , , Introduction The experimental results indicate that the amount of material removed is affected whenever a polishing tool extends beyond the edge of the workpiece. The first analysis of this effect was done by Preston in First, he calculated the wear on an imaginary glass assuming that the tool is always inside the workpiece. This means that he used a glass that is larger than the actual one. plot of the imaginary wear is shown in Fig. 1 a. Since the actual glass is narrower, the wear distribution takes the form shown in Fig. 1 b. Preston estimated the actual wear distribution from the following two requirements: a the total area of the curves shown in Figs. 1 a and 1 b remains unchanged and b the effect is present only within an annular glass region R whose width is equal to half of the tool diameter. Then Preston added the remaining area r to the annular region of the wear plot of the actual glass. In previous research this edge effect was explained as the result of excessive pressure applied by the tool when it is at maximum stroke, since the contact area. Cordero-Dávila acordero@fcfm.buap.mx, J. González-García jgonzale@fcfm.buap.mx, J. Cuautle-Cortés, and C. Robledo- Sánchez are with Facultad de Ciencias Físico-Matemáticas, enemérita Universidad utónoma de Puebla, partado Postal 1152, Puebla, Pue., México. M. Pedrayes-López and. guilar- Chiu are with the Instituto de stronomia, Universidad Nacional utónoma de México, Km 103 carretera Tijuana-Ensenada, aja California 22860, México. Received 9 May 2003; revised manuscript received 13 October 2003; accepted 14 October $ Optical Society of merica between the tool and the workpiece is reduced and therefore the wear is increased. 2,3 To obtain the proper profile at the edge of a workpiece while polishing, several methods exist that use a small tool size 3 8 that minimizes the variable contact area problem: the use of a stressed lap 9,10 that changes its shape actively as needed, or the use of an adjustable aluminum collar around the workpiece. 11 One of the most common ways is to embed the workpiece in a large domain and then discard that portion at which the edge effect occurs. 12 The edge effect is avoided by use of a vacuum-induced force at the tool interface 13,14 to cut the mirror along the straight edges and to sandblast outside the curved edges. 15 Wagner and Shannon 16 proposed that, when the tool is driven beyond the edge of the workpiece, the uniform pressure between the tool and the workpiece is altered, and the theoretical model must be changed accordingly. Using the force equation in conjunction with the torque equations for static equilibrium, one can determine the pressure distribution, assuming that it is linear. This model, however, presents an important problem whenever the tool center is near the workpiece border, since the minimum pressure can become negative, which means that this model is no longer valid. Cordero-Dávila et al. 17 presented a new pressure distribution model to predict the edge effect while avoiding the negative pressure problem. For this model, the pressure is significantly higher at the border points of the glass. The outer annular region of width s is the skin zone, where the pressure is increased. This model has two important problems: 1 The skin width s has to be drastically reduced to avoid negative pressures, and 2 it cannot be applied 1250 PPLIED OPTICS Vol. 43, No February 2004

2 Fig. 1. Preston calculated the wear distribution of an actual glass b beginning with that of an imaginary larger glass a, and adding the remaining area r to a border zone of the actual glass. to circular tools and workpieces. Here we report the results obtained by use of the skin model for a circular tool and circular workpiece. In Section 2 we present the equilibrium, force, and torque equations. In Section 3 we introduce a formula to calculate the skin width, and the formulas to calculate the pressure values are deduced. Finally, in Section 4 results of some simulations are shown. 2. Preston and Equilibrium Equations The theoretical basis for wear prediction h was provided by Preston 1 in The Preston equation can be written as h t1 t 2 Dpvdt, (1) where the value of D depends on technological factors, p is the pressure, and v is the relative velocity of the workpiece contact point with respect to the tool contact point. We suppose, as you can see in Fig. 2, that the circular workpiece has a radius a and rotates about its center of mass. The circular tool of radius b is a rigid body that rotates about its center of mass. On the workpiece an external force f 0 is applied, including the gravitational force. The reaction forces on the tool exerted by the workpiece are measured by the pressure distribution. The force and torque equilibrium equations are given by p x, y dxdy f 0, (2) xp x, y dxdy 0, (3) where the integrals are evaluated only in the contact area between the tool and the workpiece. We used a reference system whose origin is located at the center of mass of the tool. This is important since the equilibrium equations 2 and 3 are true even when the center of mass accelerates. 18 Therefore, Eqs. 2 and 3 become the force and torque equations for dynamic equilibrium. s was pointed out in Ref. 17, it is possible to find several solutions, such as a pressure that varies as a linear function with position. However, negative pressures appear since the wear is directly proportional to the pressure and the material deposition is then predicted. 3. Skin Model and Equation System for Pressure Values Negative pressures are not physical in this case, since the glass does not pull the tool. If the whole tool is inside the glass, then a constant pressure distribution is a solution for the equilibrium equations. However, when the tool border is beyond the glass Fig. 2. Parameters of the workpiece and tool when the tool border is beyond the workpiece border. The origin of the reference system is at the center of mass of the tool to establish the equilibrium equation. Fig. 3. Intersection region is divided into two regions, and C. In region the pressure is p 0, and in skin region C the pressure increases until p 0 p. The border points are located between R v s and R v, where R v is the radius of the workpiece and s is the thickness of the skin. Fig. 4. We show a plot of skin width s as a function of the separation between borders d. 20 February 2004 Vol. 43, No. 6 PPLIED OPTICS 1251

3 border, the torques can be equilibrated only if the pressure distribution is not homogeneous in the X direction defined along the line that connects the center of the tool and the workpiece ; see Fig. 3. Since the glass always pushes the tool, to accomplish the torque equilibrium the pressure has to be greater on the tool points nearest the glass border. We call this zone of increased pressure the skin zone; see C in Fig. 3. The skin zone is a circular annulus of width s whose center is located at the glass center and its outer border is the edge of the glass. The skin width s, the distance along the X direction between the glass and the tool borders d, and the semidiameter of the tool b have to satisfy the following conditions: a Skin width s is not larger than the distance between the tool center and the border of glass d. Therefore, when the tool center is at the glass border the skin width has to be zero s 0 when d b. b If the tool is completely inside the glass the pressure has a constant value. This is true even when the tool and workpiece borders coincide s 0 and d 0. ased on these two conditions, we propose that s is given by s 2 b d b d. (4) It is important to point out that the maximum value of s is b 2 when d b 2, according to the first condition. plot of s against d is shown in Fig. 4, where we used b 2.5 cm. In the rest of this section we deduce a linear equation system for p 0 and p, by using the skin model and equilibrium equations. The contact area between the tool and the workpiece is divided into two regions: the skin zone and the rest of the contact area; see Fig. 3. The former is called region C and is assumed to have a constant pressure equal to p 0 p, whereas the latter is region, which remains at pressure p 0. The left-hand term in Eq. 2 can be divided into the next two integrals: p x, y dxdy p 0 dxdy p 0 p dxdy. Since C, Eq. 5 is expressed by p x, y dxdy p 0 C (5) dxdy p dxdy. (6) whereas the force equilibrium equation can be finally written as p 0 dxdy p dxdy dxdy f 0. (7) Following a similar procedure, the integral for the torque equilibrium is expressed as p 0 xdxdy p xdxdy 0. (8) xdxdy Equations 7 and 8 are the system of equations for p 0 and p. y using the definitions I I I I I x I x Eqs. 7 and 8 can be rewritten as dxdy, (9) dxdy, (10) xdxdy, (11) xdxdy, (12) I I p 0 p I I I I f 0, (13) I x p 0 p I x I x 0, (14) In the ppendix we derive expressions for Eqs for the case of circular geometry. The system of Eqs. 13 and 14 can be solved for p 0 and p as p 0 f 0 I x I x I I I x I x I x I I I I, (15) f 0 I x p I I I x I x I x I I I I. (16) We developed a computer program that calculates the wear by use of the skin model. The total time t 2 t 1 of Eq. 1 is divided into N smaller time intervals. t the beginning of each, x tc is calculated to determine whether the whole tool is inside the workpiece. If the whole tool is inside the workpiece, then I I I I, (17) I x 0. (18) Substituting these values into Eq. 15, we obtain p 0 f 0 I I, (19) which means that the pressure is a constant value p 0 at the intersection points. If part of the tool is beyond the workpiece, then p 0 and p are evaluated from Eqs. 15 and 16, respectively. The wear is evaluated according to the pressure on each workpiece point that belongs to the intersection points PPLIED OPTICS Vol. 43, No February 2004

4 Fig. 5. Depiction of the tool travel on the workpiece. Fig. 6. Wear versus radial position by use of the skin model with a constant value of the skin width of 1.87 cm. The negative pressures are not avoided. Fig. 7. Wear versus radial position by use of the skin model with a constant value of the skin width of 0.5 cm. The negative pressures are avoided. 4. Simulations We calculate removed material h versus radial position R for three cases by using a circular workpiece and a tool with diameters of 30 and 5 cm. In all the cases, the center of the tool oscillates about a point located cm from the glass center with an oscillation amplitude of 2.5 cm; see Fig. 5. We show this example because it is the only experimental example reported in a paper 16 that is known to us. For the first case, we chose a fixed value for the skin width of s 1.87 cm. In Fig. 6 we show a plot of the wear versus R. s we can see, we still have negative pressures, since the value of the skin width is too large. 17 However, if we use a skin width of s 0.5 cm the negative pressures are avoided, as shown in Fig. 7. For the second case we assume that skin width s is expressed by Eq. 4. plot of the wear versus radial position on the glass is shown in Fig. 8. s can be seen, we obtained typical behavior of the experimental wear at the border 16 and the negative pressures were avoided. In the third case, see Fig. 9, the amplitude of the oscillation was increased to 2.65 cm. The wear on the border of the workpiece was increased because Fig. 9. Wear versus radial position by use of the variable skin width. In this case, the oscillation amplitude is increased to 2.65 cm. s can be observed, the wear increases significantly at the border points. the maximum separation between borders was also increased. 17 Finally, we point out that see Figs. 8 and 9 there is a decrease in the removal that occurs before the increase toward the edge. This effect can be explained with the aid of the force equilibrium equation. Equation 2 indicates that the integral on the left-hand side has to have a constant value f 0. Then, when the pressure in the skin zone is increased, the pressure outside this zone is decreased so that Eq. 2 is satisfied. 5. Conclusions We report the results that are obtained when the pressure distribution skin model is applied to a circular tool and a circular rotating workpiece. With this model we predict that the removed material is significantly increased and negative pressures are avoided. Fig. 8. Wear versus radial position by use of the variable skin width. The negative pressures are automatically avoided. We used an oscillation amplitude of 2.5 cm. ppendix The integrals in Eqs have the form I f x, y dxdy. (1) 20 February 2004 Vol. 43, No. 6 PPLIED OPTICS 1253

5 The integral in Eq. 1 can be written as I 2 xb,y 0 x x 0,yb 2 x 2 f x, y dxdy y y 0, xx tc a 2 y 2 2 f x, y dxdy, (2) y 0, x x0 where x tc is the x coordinate of the tool center, a is the radius of the workpiece, b is the radius of the tool, and x 0 and y 0 are intersection point coordinates. Note that, in Eq. 2, the order in which the variables are integrated must be interchanged for each integral. This is done to define univocally the upper border of the tool for the first integral and the right border of the glass for the second integral. x 0 and y 0 are given by x 0 a2 b 2 2 x tc, (3) 2x tc y 0 b 2 x (4) In the force equilibrium Eq. 2, f x, y 1. This integral can be written in the form I I 2 xb,y 0 x x 0,yb 2 x 2 dxdy y y 0, xx tc a 2 y 2 2 dxdy. (5) y 0, x x0 Calculating the integral we obtain the following equation: I I x 0 b 2 x b 2 arcsin x 0 x 0 y 0 a 2 y a 2 arcsin y 0 b b 2 2 2y 0 x tc. a (6) In the torque equilibrium Eq. 3, f x, y x, which results in the expression I x 2 xb,y 0 x x 0,yb 2 x 2 xdxdy y y 0, xx tc a 2 y 2 2 xdxdy. (7) y 0, x x0 fter the integration is done, we obtain I x 2 3 b2 x x 2 tc a 2 x 2 0 y 0 y x tc y 0 a 2 y a arcsin y 2 0 (8) a. To calculate the integrals in Eqs. 9 11, we substituted a R v into Eqs. 3 and 4 to calculate the intersection point coordinates x 0 and y 0. With these latter values, integrals I I and I x were calculated with the aid of Eqs. 6 and 8. Using a R v s and following an analogous way, we calculated the integrals I I and I x. We appreciate the recommendations and comments made by the reviewer. References 1. F. Preston, The theory and design of plate glass polishing machines, J. Soc. Glass Technol. 9, W. Rupp, Loose abrasive grinding of optical surfaces, ppl. Opt. 11, R. spden, R. McDonough, and F. R. Nitchie, Jr., Computer assisted optical surfacing, ppl. Opt. 11, R.. Jones, Optimization of computer controlled polishing, ppl. Opt. 16, N. J. rown, Computationally directed axisymmetric aspheric figuring, Opt. Eng. 17, S. Savel ev and. P. ogdanov, utomated polishing of large optical components with a small tool, Sov. J. Opt. Technol. 52, R.. Jones, Computer controlled optical surfacing with orbital tool motion, Opt. Eng. 25, D. W. Small and S. J. Hoskins, n automated asphere polishing machine, in Optical Manufacturing, Testing, and spheric Optics, G. M. Sanger, ed., Proc. SPIE 645, D. S. nderson, J. R. P. ngel, J. H. urge, W.. Davison, S. T. DeRigne,.. Hille, D.. Ketelsen, W. C. Kittrell, H. M. Martin, R. H. Nagel, T. J. Trebisky, S. C. West, and R. S. Young, Stressed-lap polishing of 3.5-m f 1.5 and 1.8-m f 1.0 mirrors, in dvanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE 1531, S. C. West, H. M. Martin, R. H. Nagel, R. S. Young, W. D. Davison, T. J. Trebisky, S. T. Derigne, and.. Hille, Practical design and performance of the stressed-lap polishing tool, ppl. Opt. 33, R.. Jones, Fabrication using the computer controlled polisher, ppl. Opt. 17, T.. Porsching, C.. Hall, T. L. ennett, and J. M. Ernsthausen, mathematical model of material removal with application to CNC finishing, Math. Comput. Modell. 18, R.. Jones and W. J. Rupp, Rapid optical fabrication with computer-controlled optical surfacing, Opt. Eng. 30, R.. Jones, Fabrication of a large, thin, off-axis aspheric mirror, Opt. Eng. 33, R.. Jones, Computer controlled polishing of telescope mirror segments, Opt. Eng. 22, R. E. Wagner and R. R. Shannon, Fabrication of aspherics using a mathematical model for material removal, ppl. Opt. 13, E. Luna-guilar,. Cordero-Dávila, J. Gonzalez-García, M. Núñez-lfonso, V. H. Cabrera-Pelaez, C. Robledo-Sánchez, J. Cuautle-Cortez, and M. H. Pedrayes-López, Edge effects with Preston equation, in Future Giant Telescopes, J. Roger, P. ngel, and R. Gilmozzi, eds., Proc. SPIE 4840, G. R. Fowles, nalytical Mechanics Wiley, Reading, Mass., PPLIED OPTICS Vol. 43, No February 2004