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J. Eur. Opt. Society-Rapid Publ. 21, 30( 2025)
The material removal z( x, y) can be calculated with the Preston equitation [ 5 ]:
zx ð; yÞ ¼
Z t
0 c p Pðx; y; tÞvðx; y; tÞdt ð1Þ
Figure
1. Schematic image of a lever arm machine. Description: A 1: work piece, A 2: area that point F can reach in the process; A and E: machine spindles; B – D: hinges, G: welded connection point, a – d: cranks.
work piece is smaller than the tool, it is called full-aperture polishing. Another possibility is that the work piece is rotating the tool is moved over the surface e. g. reconditioning the tool. In this setup, the polishing dish( polishing tool) shape is assumed to be constant, meaning the PV( Peak-to-valley) of the polishing tool is < ± 2 lm. The operator measures the tool shape between attempts and manually guides a grindstone over the surface until the required PV is again achieved. This process is called“ reconditioning.” The system’ s terminology is reversed: the polishing tool becomes the“ work piece,” and the grindstone becomes the tool. The setup is also reversed: the work piece is driven by the machine, and the grindstone is guided over the work piece.
Due to the tilting moment, the tool must be smaller than the part: this is then referred to as sub-aperture polishing such as in robot polishing or CCP polishing. Subsequently, the tool is always at the bottom and the work piece is guided above it. The bearing point F is usually a spherical cap so that the tool rests flat and the work piece can rotate freely. This leads to two challenges: The normal force is always a point load( a surface load would be better; it affects the constant polishing force) and no wired sensors can be attached to the work piece easily.
The lever arm machine can be used for lapping( manufacturing tool surface is as hard as the work piece) or polishing( polishing foil / pitch is soft: polishing grains can embed themselves into the surface). In this publication only the shape deviation and not the roughness is considered. For conventional optics, it is easier to adjust the roughness e. g. polishing agent concentration than the global shape. c p: Preston-coefficient; P: Pressure; v: relative velocity.
Looking at the Preston equation, it can be seen that material removal depends linearly on polishing pressure, tool speed and dwell time. In the following, only the relative speed and not the remaining factors, including the shape of the polishing dish, will be considered. Various notations and applications of the Preston equation( e. g. also for polishing wafers with already installed conductor tracks) are described by Luo and Dornfeld [ 6 ]. There are also other ways of writing the equation with weighting factors that better reflect the nonlinear friction state and the temperature dependence of the parameters. All unknown factors, like pH-value of the slurry or temperature behaviors are considered in the Preston-coefficient. However, the Preston factor is a process factor and not a machine factor or the like, i. e. the formula should be solved and applied differently for each work piece material, polishing or lapping materials and each machine.
The relative velocity in polishing was published in a different way in previous research [ 7, 8 ]. In 1980, Kaller wrote that the control of design is still subjective [ 9 ]. Not much has changed in literature since then, at least for lever arm machines.
Mathematically, a lever arm machine can be described as a closed-chain double crank gear or five-bar / five-crank mechanism [ 10 ]. Such a sketch of a lever arm machine with 2-DOF( degrees of freedom) is shown in Figure 2. Disadvantages of the method: The levers a, d and f( Fig. 2) cannotbe adjusted precisely( ~ ± 0.1 mm) due to the manufacturing tolerances, the statistically desirable irregularity and the rough adjustment of the lever arm machine’ sprojection. One possibility is to use linear scales, but this is currently not available on all machines. Furthermore, the work piece can rotate independently and relative speeds arise that cannot be calculated or detected using this method. These relative speeds arise from the tribology in the polishing agent gap. The advantage of the calculation method is that the lever lengths can be recalculated in real terms using three known positions.
Mathematically, the mechanism can be calculated as follows [ 11 ]. The distances between BC, CD are constant and the distance BD is known: with this information the angles qB and qE can be calculated and the algebraic system can be solved [ 12 ]. With the offset to Point F the reached position as well as the velocity of the lever arm machine can be calculated. The side lengths of the triangle BCD have the same length the whole time.
d cos ðq E Þþc cos ðq E þ q D Þ ¼ C x; ð2Þ
d sin ðq E Þþc sin ðq E þ q D Þ ¼ C y; ð3Þ a cos ðq A Þþb cos ðq A þ q B ÞþA x ¼ C x; ð4Þ a sin ðq A Þþb sin ðq A þ q B Þ ¼ C y: ð5Þ