(2) The last term is the systematic component of the uncertainty statement. This term is instances the interest is in the Y relative to some particular requirement. That is, if |Y is less than some limiting value, the block is used as if the length was N. Unfortunately, these two methods of interpretation are not well understood. In the first interpretation, the uncertainty of Y reflects all of the terms in the above relation. In the second interpretation, compliance with specification limits. is usually announced on the basis of a simple uncharacterized measurement procedure. There are several courses of action dependent upon the intended usage. When the uncertainty of Y is smaller than the tolerance limits, one can accept the Ŷ and its uncertainty in lieu of the specified limits. For example, a length 4.000 028.000 002 in as determined by measurement is a more precise basis for adjusting instruments, etc., than a statement that the length of the block does not deviate from a nominal 4 in excess of 0.000 005 in. Such action, however, carries the implication that all of the terms considered in establishing the uncertainty of Ÿ must also be considered in the local measurement process in which the block is to be used. In certain circumstances, one can use simplified procedures to establish tolerance compliance. If the measurement process used is free from significant systematic errors (the magnitude of known systematic effects is less than one s.d.) and if the process standard deviation is less than approximately one-tenth the tolerance limit, a simple sorting procedure should identify blocks which are significantly "out of tolerance." Reasonable tolerance limits should encompass the combined uncertainty of the production and inspection measurement process. Finally, one can evaluate the situation relative to a particular end use and accept those items which are adequate. Generally speaking, one cannot compare the results for the same measurement performed by two different processes unless both processes are well characterized. One cannot judge the difference between the results without a detailed knowledge of both processes and the methods of computation (round off rules, etc.). This is particularly true when one measurement is in essence a sorting operation according to a locally determined procedure. In many instances, the use of precise measurement processes to establish an "in tolerance by actuality" will not confirm an "in tolerance by local definition." Use Stating the defined length of a block or artifact in terms of (N + Y) suggests two different interpretations. Since N is exact (the nominal length), (N+Y) carries the uncertainty of Y, thus when disseminating a length unit, one is concerned with 11. Appendix 2. Gage Block Intercomparison Designs With the sequence of operations required to make a "single measurement" precisely defined, the schedule of "single measurements" to be made "unknowns" is called an intercomparison design. In general, the intercomparison design provides a means to obtain the most information from the fewest measurements. While many features can be incorporated, the formulation of efficient designs is not a trivial task. Discussion of design formulation. is beyond the scope of this paper. The sequence of operations required for a "single measurement" can be shown symbolically as: for a design with n observations on k objects. Since all of the comparisons required by a given design can usually be made on one instrument in a short time interval, the standard deviation computed from the residual from one sequence of comparisons is called the estimated within group precision, Sw, for a particular instrument. This standard deviation applies to the defined "single measurement.” For a given instrument, each defined "single measurement" procedure will have a distinctive standard deviation. Collections of sw can be combined to obtain a long term or accepted within group standard deviation, σw, one of the important process performance parameters. The flexibility of intercomparison designs provides a means for the metrologist to obtain long sequences of repeated measurements on the same objects with little additional measurement effort. If one is to believe that the values assigned to the “unknowns" are valid over time, the fact must be demonstrated. The idea of a "check standard" refers to a difference between two objects, or the similar in all respects to the "unknown" and always used in a particular measurement. For example, in the design shown, an object with known value could be designated 1. This object would be called the "starting standard" since its assigned value, 1, would be the restraint, 2 could be the "check standard," assumed unknown and always used with 1. The sequence of measurements called for by the design would assign values to X2, 3, and 4 relative to 1. While the objects 3 and 4 together with their assigned values are passed on to others, X2 2 remains with the process. The collection of values for 2 reflects not only the variation of both 1 and 2 but also the variability of the process over time. The standard deviation of this collection of values is called the "total standard deviation of the process," σT. The appropriate choice of location within the design for the "starting standard(s)" and the "check standard" is part of the design formulation. Where possible, for the type of design shown, both and 2 are used for "starting standards. The restraint is taken as the sum of the assigned values, (+2) and the difference between 1 1 and 2 as determined from the measurements serves as a "check standard." This procedure can sometimes reduce the systematic component of the uncertainty of the values assigned to the unknowns. (The systematic error of the restraint is prorated between the unknowns in proportion to the ratio of the value of the unknown to the value of the restraint. This will be discussed in detail elsewhere.) The design shown is usually called a "four one's" design, four being the number of objects involved and the one being associated with the limitations of the "on scale" range of the various measurement instruments. For the most part, available precise instruments have a limited on scale range so that the objects being compared must be nominally equal. It should be noted that this is not a limitation imposed by the statistical design. This would be interpreted as meaning the difference, as measured by the prescribed procedure, between object 1-1 and object 1-2 is called A(1), and so on. The restraint, R, is the sum of the values currently assigned to 1-1 and 1-2. The check standard, C, is the difference between the values determined in the process for 1-1 and 1-2. The position of the restraint would be shown in vector form, R(1,1,0,0), and the check standard location would be shown as C(1,-1,0,0). If only the first object, 1-1 had an assigned value, the restraint vector would be R(1,0,0,0), and the check standard would most likely be the second object, 1-2, designated by the vector C(0,1,0,0). For a fuller treatment of this subject see reference [6]. 12. Appendix 3. Gage Block A typical gage block interferometer is shown schematically in figure 1. A beam of collimated monochromatic light impinges on a beam splitter part of which passes through to reference mirror M1, and part of which is reflected to the platen or reference mirror Pl. The reflected beam from Pl passes through the beam splitter into the viewing system. For the purpose of this discussion, the reflected light from M1 can be thought of as coming from the Virtual Image M1, hence also passing through the beam splitter into the viewing system. The beams are recombined in the viewing system to produce the observed interference fringe patterns. It should be noted that in such a schematic diagram, all angles must be shown very large. In the real instrument, all angles are very small so that cosine length errors are practically negligible. In detail, an arbitrary ray (1) divides at the beam splitter, one part being reflected from Pl along path (P1) to the focal plane, and one part being reflected from Virtual M1 along path (M1). The difference in path length, starting at the beam splitter and ending at the focal plane is of interest. For some position across the face of the platen, this difference in path length will be an odd multiple of the half wavelength of the light. Under this condition, the two ray components will interfere destructively at the focal plane, and at that point in the observed field would be the dark center of an interference fringe. Because of the included angle (a + ß), between Pl and Virtual M1, the difference in path length for the two ray incident components continually changes as one moves across the viewing field. For ray (2) the difference will again be an odd multiple of the half wavelength, indicating the center of the second fringe. Midway between ray (1) and ray (2), the difference in path length is an even multiple of the half wavelength, therefore in this region there is no destructive interference, thus the color of the light is seen. In the field of view the resulting fringe pattern appears as alternate rows of dark and colored bands. The fringe pattern can be interpreted as shown in figure 2. Starting with Virtual M1, one can construct a series of parallel planes representing the difference in path length in odd multiples of the half wavelength. Except for the first plane, these planes represent incremental changes in elevation above Virtual M1 of one wavelength. The intersection of surface Pl with these elevation planes, at points a spread out, as shown in the top part of the figure 2. If the operation is done slowly, one can observe fringe P+1 move to a new location. Fringe P+4 would move completely out of view. With a gage block on the platen, as shown in figure 3, the top surface of the block intersects another set of parallel elevation planes in a similar manner. If the block length L, was shortened by the amount dL, as shown, fringe B + 1 would be coincident with fringe P+ 2. In like manner, if L was increased an appropriate amount, fringe B + 1 would become coincident with fringe P+3. From this, it follows that the difference in optical path length associated with fringes B+ 1 and P + 2 is: ((B + 1) − (P + 2) + (a/b))λ where (B+ 1)(P+ 2) is a large integer (Int. F), which must be determined by other means and (a/b) is the observed fractional fringe. This path length difference is equivalent to 2L, so that: L= (Int. F+ (a/b)) (λ/2). and the platen have the effect of changing the path length differences in a manner not related to L. For long blocks, small platens are used which are of similar material and surface finish as the blocks in order to obtain nearly the same optical properties on both surfaces. For a given setup, one must determine experimentally the direction of increasing fringe order. For a particular measurement, a "tentative" assigned length, L(t, T) is expressed in "fringe" by: F=2(L(t, T))/AT.p=(Int. F+0) where Arp is the wavelength of the laser radiation at the time of the measurement; T, p and f being the air temperature, pressure and relative humidity at the time the fringe photograph is taken, and is the computed fraction. In practice for well known reference blocks such as the NBS (.) blocks considered in section 5.0, the accepted value, Li(t, 20), is normalized to temperature T for L(t, T). For other blocks, such as the NBS(. .) group considered in section 6.2, L(t, 20) is determined by mechanical comparison with suitable reference blocks. |