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The spatial separation between the measured point and reference line is known as the Abbeoffset. Figure 3.3 shows the effect of Abbe error on an interferometricmeasurement of length. To ensure zero Abbe error, the reflector axis ofmovement should be co-linear with the axis of measurement. To account forthe Abbe error in an uncertainty analysis relies on knowing the magnitude ofthe Abbe offset and the magnitude of the errors in motion of the positioningsystem (for example, straightness).FIGURE 3.3 Effects of Abbe error on an optical length measurement.Elastic compressionThe Abbe Principle is, perhaps, the most important principle in precisioninstrument design and is also one that is commonly misunderstood – Bryan[14] described it as ‘the first principle of machine design and dimensionalmetrology’.
Abbe’s original paper concentrated on one-dimensional measuringinstruments. Bryan re-stated the Abbe Principle for multi-dimensionalsystems as:The displacement measuring system should be in line with thefunctional point whose displacement is to be measured. If this is notpossible, either the slideways that transfer the displacement must befree of angular motion or angular motion data must be used tocalculate the consequences of the offset.Many three-axis instruments, especially coordinate measuring machines(CMMs), attempt to minimize the Abbe error through good design principles(see chapter 8).
Two good examples of this are the Zeiss F25 CMM [16] andan elastically guided CMM developed at the Eindhoven University of Technology [17].3.5 Elastic compressionWhen any instrument uses mechanical contact, or when different parts of aninstrument are in mechanical contact, there will be some form of compression due to any applied forces. With good design such compression will beminimal and can be considered negligible, but when micrometre or nanometre tolerances or measurement uncertainties are required, elasticcompression must be accounted for, either by making appropriate correctionsor taking account of the compression in an uncertainty analysis.
In somecases where the applied load is relatively high, irreversible, or plastic,deformation may occur. This is especially probable when using either highforces or small contact areas, for example when using stylus instruments (seesection 6.6.1) or atomic force microscopes (see section 7.3). The theorybehind elastic and plastic deformation can be found in detail elsewhere [18].The amount that a body compresses under applied load depends on:-the measurement force or applied load;-the geometry of the bodies in contact;-the material characteristics of the bodies in contact;-the type of contact (point, line, etc.);-the length of contact.4142C H A P T ER 3 : Precision measurement instrumentation – some design principlesThe formulae for calculating the amount of compression for mostsituations can be found in [18] and there are a number of calculators availableon the Internet (see for example emtoolbox.nist.gov/Main/Main.asp).
Themost common cases will be included here. More examples of simplecompression calculations are given elsewhere [2]. For a sphere in contactwith a single plane (see Figure 3.4), the mutual compression (i.e. thecombined compression of the sphere and the plane) is given bya ¼ 1=3ð3pÞ2=3 2=31P ðV1 þ V2 Þ2=32D(3.1)where D is the diameter of the sphere, P is the total applied force and V isdefined asV ¼ð1 s2 ÞpE(3.2)where E is the Young’s modulus of the material and s is Poisson’s ratio. Notethat the assignment of the subscript for the two materials is arbitrary due tothe symmetry of the interaction. For a sphere between two parallel planes ofsimilar material, equation (3.1) is modified by removing the factor of two inthe denominator.For a cylinder in contact with a plane, the compression is given by8a2(3.3)a ¼ PðV1 þ V2 Þ 1 þ lnðV1 þ V2 ÞPDFIGURE 3.4 Mutual compression of a sphere on a plane.Force loopswhere 2a is the length of the cylinder and the force per unit length is given byP ¼P:2a(3.4)Plastic compression is much more complicated than elastic compressionand will be highly dependent upon the types of materials and surfacesconsidered.
Many examples of both elastic and plastic compression areconsidered in [19].3.6 Force loopsThere are three types of loop structures found on precision measuringinstruments: structural loops, thermal loops and metrology loops. Thesethree structures are often interrelated and can sometimes be totally indistinguishable from each other.3.6.1 The structural loopA structural loop is an assembly of mechanical components that maintainrelative position between specified objects.
Using a stylus surface texturemeasuring instrument as an example (see section 6.6.1) we see the structuralloop runs along the base-plate and up the bridge, through the probe, throughthe object being measured, down through the x slideway and back into thebase-plate to close the loop.
It is important that the separate components inthe structural loop have high stiffness to avoid deformations under loadingconditions – deformation in one component will lead to uncompensateddimensional change at the functional or measurement point.3.6.2 The thermal loopThe thermal loop is described as: ‘a path across an assembly of mechanicalcomponents, which determines the relative position between specifiedobjects under changing temperatures’ [5]. Much akin to mechanical deformations in the structural loop, temperature gradients across an instrumentcan cause thermal expansion and resulting dimensional changes. It ispossible to compensate for thermal expansion by choosing appropriatecomponent lengths and materials. If well designed, and if there are notemperature gradients present, it may just be necessary to make the separatecomponents of an instrument from the same material.
Thermal expansioncan also be compensated by measuring thermal expansion coefficients andtemperatures, and applying appropriate corrections to measured lengths.4344C H A P T ER 3 : Precision measurement instrumentation – some design principlesThis practice is common in gauge block metrology where the geometry of theblocks being measured is well known [20]. Obviously, the effect of a thermalloop can be minimized by controlling the temperature stability of the room inwhich the instrument is housed.3.6.3 The metrology loopA metrology loop is a reference frame for displacement measurements,independent of the instrument base.
In the case of many surface texturemeasuring instruments or CMMs, it is very similar to the structural loop.The metrology loop should be made as small as possible to avoid environmental effects. In the case of an optical instrument, relying on the wavelength of its source for length traceability, much of the metrology loop may bethe air paths through which the beam travels. Fluctuations in the airtemperature, barometric pressure, humidity and chemical composition ofthese air paths cause changes in the refractive index and correspondingchanges to the wavelength of the light [21,22]. This can cause substantialdimensional errors.
The last example demonstrates that the metrology andstructural loops can be quite different.3.7 MaterialsNearly all precision measuring instrument designs involve minimizing theinfluence of mechanical and thermal inputs which vary with time and whichcause distortion of the metrology frame. Exceptions to this statement are, ofcourse, sensors and transducers designed to measure mechanical or thermalproperties. There are three ways (or combinations of these ways) to minimizethe effects of disturbing inputs:-isolate the instrument from the input, for example using thermalenclosures and anti-vibration tables;-use design principles and choose materials that minimize the effect ofdisturbing inputs, for example, thermal compensation designmethods, materials with low coefficients of expansion and stiffstructures with high natural frequencies;-measure the effect of the disturbing influences and correct for them.The choice of materials for precision measuring instruments is closelylinked to the design of the force loops that make up the metrology frame.Materials3.7.1 Minimizing thermal inputsThermal distortions will usually be a source of inaccuracy.
To find a performance index for thermal distortion consider a horizontal beam supported atboth ends of length L and thickness h [23]. One face of the beam is exposed toa heat flux of intensity Q in the y direction that sets up a temperature, T,gradient, dT/dy, across the beam. Assuming the period of the heat flux isgreater than the thermal response time of the beam, then a steady state isreached with a temperature gradient given byQ ¼ ldTdy(3.5)where l is the thermal conductivity of the beam. The thermal strain is given by3 ¼ aðT0 TÞ(3.6)where a is the thermal expansion coefficient and T0 is the ambienttemperature. If the beam is unconstrained, any temperature gradient willcreate a strain gradient, d3/dy in the beam causing it to take up a constantcurvature given byd3dTa¼ a¼ Q:(3.7)K ¼dydylIntegrating along the beam gives the central deflection ofad ¼ C1 L2 Ql(3.8)where C1 is a constant that depends on the thermal loads and the boundaryconditions.
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