B-67BMitutoyo reserves the right to change any or all aspects of any product specication, including prices, designs and service content, without notice.B-67BAbbe’s PrincipleAbbe’s principle states that “maximum accuracy is obtained when the scale and the measurement axes are common”.This is because any variation in the relative angle () of the moving measuring jaw on an instrument, such as a caliper jaw micrometer, causes displacement that is not measured on the instrument’s scale and this is an Abbe error ( = − L in the diagram). Spindle straightness error, play in the spindle guide or variation of measuring force can all cause () to vary, and the error increases with R.The above graph shows micrometer frame expansion due to heat transfer from hand to frame when the frame is held in the bare hand which, as can be seen, may result in a signicant measurement error due to temperature-induced expansion. If the micrometer must be held by hand during measurement then try to minimize contact time. A heat insulator will reduce this effect considerably if tted, or gloves may be worn. (Note that the above graph shows typical effects and is not guaranteed.)LεRθℓHooke's LawHooke’s law states that strain in an elastic material is proportional to the stress causing that strain, providing the strain remains within the elastic limit for that material.Hertz's FormulaeHertz’s formulae give the apparent reduction in diameter of spheres and cylinders due to elastic compression when measured between plane surfaces. These formulae are useful for determining the deformation of a workpiece caused by the measuring force in point and line contact situations.PSøD22δP22δδδσδ2LøD(b)Cylinder betweentwo planes(a)Sphere betweentwo planesEffect of Changing Support Method and Orientation (Unit: µm)Difference in Thermal Expansion between Micrometer and Length Standard125-3-2-10+1+2+3225Difference in expansion (µm)Nominal length (mm)3250 °C10 °C20 °C425525In the above experiment, after the micrometer and its standard were left at a room temperature of 20 ºC for about 24 hours for temperature stabilization, the start point was adjusted using the micrometer standard. Then, the micrometer with its standard were left at the temperatures of 0 ºC and 10 ºC for about the same period of time, and the start point was tested for shift. The above graph shows the results for each of the sizes from 125 through 525 mm at each temperature. This graph shows that both the micrometer and its standard must be left at the same location for at least several hours before adjusting the start point. (Note that the graph values are not guaranteed values but experimental values.)Length Standard Expansion with Change of Temperature (for 200 mm bar initially at 20 ˚C)The above experimental graph shows how a particular micrometer standard expanded with time as people whose hand temperatures were different (as shown) held the end of it at a room temperature of 20 °C. This graph shows that it is important not to set a micrometer while directly holding the micrometer standard but to make adjustments only while wearing gloves or lightly supporting the length standard by its heat insulators.When performing a measurement, note also that it takes time until the expanded micrometer standard returns to the original length.(Note that the graph values are not guaranteed values but experimental values.)Micrometer Expansion due to Holding Frame with the Bare Hand141312111098765432102468101520Time (minutes)Expansion (µm)3010020030050Supporting methodSupported at the bottom and centerSupported only at the centerAttitudeMaximummeasuringlength (mm)3250−5.54250−2.55250−5.56250−11.07250−9.58250−18.09250−22.510250−26.0Supporting methodSupported at the center in a lateral orientation.Supported by hand downward.AttitudeMaximummeasuringlength (mm)325+1.5−4.5425+2.0−10.5525−4.5−10.06250−5.5725−9.5−19.0825−5.0−35.0925−14.0−27.01025−5.0−40.0Assuming that the material is steel and units are as follows:Modulus of elasticity: E =205 GPaAmount of deformation: (µm)Diameter of sphere or cylinder: D (mm)Length of cylinder: L (mm)Measuring force: P (N)a)Apparent reduction in diameter of sphere2 = 0.82 3√P2/Db)Apparent reduction in diameter of cylinder1 = 0.094·P/L 3√1/D Changing the support method and/or orientation of a micrometer after zero setting affects subsequent measuring results. The tables below highlight the measurement errors to be expected in three other cases after micrometers are zero-set in the ‘Supported at the bottom and center’ case. These actual results show that it is best to set and measure using the same orientation and support method.31 °C27 °C21 °CLapse of time (minutes)Thermal expansion (µm)10987654321020151050

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