Metrology pocket reference guide Date: December 2014 Issue: 1:0
Contents Co-ordinate systems Rotation convention Six Degrees of Freedom Alignment principles Alignment through circle centres Alignment principles Geometric tolerancing Trigonometry Stylus considerations Conversion factors Thermal coefficients Notes........................... 1 2 3 4 5 6 7 8 9... 11... 12... 15
Co-ordinate systems 3D cartesian co-ordinates Point A can be represented as: X 10, Y 10 or θ 45 r 14.14 (polar co-ordinates) +Y 10 A 14.14 10 X 45 +X 1 Y
Rotation convention Right-hand grip rule - for a rotating object, the fingers follow the curve of a point on the object and the thumb points along the axis of rotation. Primary axis in XY plane is X Rotation around Z axis Primary axis in YZ plane is Y Rotation around X axis Primary axis in ZX plane is Z Rotation around Y axis Angle of rotation is from the primary axis and the secondary axis. Primary axis is 0 Secondary axis is 90 2
Six degrees of freedom +Z Roll +Y +X Yaw X Y Pitch Z Prior to alignment a component is free to rotate and translate in each of the 3 axes, the Six Degrees of Freedom (6 DoF). 3
Alignment principles To accurately define any alignment, all 6 DoF must be constrained. Plane, Line, Point (PLP) Alignment Z + X + Plane = Z position + X/Y rotation (3 DoF) = Z up/down, X roll, Y pitch = constrained Line = Z rotation, Y postion (2 DoF) = Z yaw, Y forward/back = constrained Point = X position (1 DoF) = X left/right = 6 DoF 4
Alignment through circle centres ARCTAN (4.2 / 5.5) = 37.367 Note: There is always a level of uncertainty with every measurement. Ideally you would take multiple points for higher accuracy. If not, spacing points as far apart as possible minimises this uncertainty, and therefore the alignment, as the inherent uncertainty of the measuring system, and any misalignment of the circle centres will have a less pronounced effect on the angle. 5
Alignment principles Z Y + Y + X + Plane = Z position + X/Y rotation (3 DoF) = Z up/down, X roll, Y pitch = constrained Circle 1 + Circle 2 = construct Line. Construct Line = Z rotation (1 DoF) = Z yaw = 1 constrained Circles and axis not aligned - need to rotate line 45 Circle 1 = X/Y Position (2 DoF) = X left/right, Y forward/back = 6 DoF 6
Geometric tolerancing Straightness Flatness Circularity Cylindricity Profile of a line Profile of a surface Angularity Perpendicularity Parallelism Position Concentricity Symmetry Runout Total runout Form Form Form Form Form Form Orientation Orientation Orientation Location Location Location Runout Runout 7
Trigonometry Known data Sides Formula to find Angles a & b c = (a² - b²) B = Sin ¹(b/a) C = 90 - B a & c b = (a² - c²) B = 90 - C C = Sin ¹(c/a) b & c a = (b² + c²) B = Tan ¹(b/c) C = Tan ¹(c/b) a & B b = a * Sin (B) c = a * Cos (B) C = 90 - B a & C b = a * Cos (C) c = a * Sin (C) B = 90 - C b & B a = b/ Sin (B) c = b * Cot (B) C = 90 - B b & C a = b / Cos (C) c = b * Tan (C) B = 90 - C c & B a = b/ Cos (B) b = c * Tan (B) C = 90 - B c & C a = b / Sin (C) b = c * Cot (C) B = 90 - C 8
Stylus considerations 5 ways to maximise the accuracy of your CMM by your choice of styli 1. Shortest possible stylus for application 2. Largest possible ball for application 3. Minimise joints 4. Use star styli only where necessary 5. Use high quality balls, grade 5 or better Did you know? Most CMMs are supplied with grade 5 or grade 3 stylus balls to achieve usable accuracy. Replacing these styli with a lower grade can reduce the operating accuracy of the CMM by up to 15%. 9
Stylus considerations Styli ball considerations Ruby Silicon Nitride Zirconia Suitable for most applications Resists general wear Suitable for scanning aluminium Resists Adhesive wear Suitable for scanning rough surfaces Resists Abrasive wear Styli stem considerations Low weight + Low stiffness = Large balls and long length High Stiffness + High Weight = Small stem therefore small ball and short length Carbon fibre - used on large styli up to 800 mm Ceramic - used with 3 mm balls on styli exceeding 30 mm in length Stiffness Carbon Fibre Ceramic Steel Weight Tungsten Carbide Steel - standard for styli up to 30 mm long and 2 mm balls Tungsten carbide - used for balls of <1 mm and stems of up to 50 mm long 10
Conversion factors Metric to Imperial 1 millimetre = 0.039 370 in. 1 centimetre = 0.393 701 in. 1 metre = 3.280 84 ft 1 centimetre 2 = 0.15500 sq. in 1 decimetre 2 = 0.107639 sq. ft 1 metre 2 = 10.7639 sq. ft * Exactly Length Area Imperial to Metric 1 inch = 2.54 cm* 1 foot = 30.48 cm* 1 yard = 0.9144 m* 1 sq. inch = 6.4516 cm 2 * 1 sq. foot = 929.0304 cm 2 * 1 sq. yard = 0.836127 m 2 NB: Conversion factors provided to 6 significant figures to provide accuracy where necessary. 11
Thermal coefficients Material CTE (ppm/k) Alumina ceramics 6-7 Aluminium and its alloys 21-25 Beryllium 11 Beryllium copper 17 Brass 18-21 Bronze, alumium (cast) 16-17 Bronze, phosphor-silicon 17-18 Bronze, tin (cast) 18 Cast iron, nodular or ductile 10-19 Cermet, alumina 8-9 Cermet, chromium carbide 10-11 Cermet, titanium carbide 8-13 Cermet, tungsten carbide 4-7 Copper 17 Cupro-nickel and nickel silver 16-17 Diamond 1 Gauge block, CERA block* 9.3 ± 1 Gauge block, steel 10.9 ± 1 Gauge block, tungsten carbide 4.2-4.3 Glass, fused quartz 0.55-0.59 Glass, pyrex 3.3 Iron, grey cast 11 Magnesium alloys 25-28 * Zirconia ceramic.
Thermal coefficients Material CTE (ppm/k) Molybdenum and its alloys 5-6 Nickel and its alloys 12-17 Nickel alloy, low expansion 10 Nitriding steel 12 Platinum 9 Stainless steel, age hardenable 10-15 Stainless steel, austenitic 14-18 Stainless steel, cast 11-19 Stainless steel, ferritic 10-11 Stainless steel, martensitic 10-12 Steel, alloy 11-15 Steel, alloy, cast 14-15 Steel, carbon, free cutting 15 Steel, high temperature 11-14 Steel, ultra-high strength 10-14 Superalloys, cobalt based 12-17 Superalloys, Cr-Ni-Co-Fe 17-19 Superalloys, Cr-Ni-Fe 14-16 Superalloys, nickel based 14-18 Tantalum carbide 8 Tin and its alloys 23 Titanium and its alloys 9-13 Titanium carbide 7 Tungsten 4
Thermal coefficients Material CTE (ppm/k) Zerodur, glass ceramic** 0.05 ± 0.10 Zinc and its alloys 5.5-6 Zirconium and its alloys 5.5-6 ** Used for the XYZ scales of ultra-high accuracy CMMs, such as Mitutoyo s LEGEX series. 14
Notes 15
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