Machinery's Handbook, 31st Edition
DIMENSIONAL–TEMPERATURE CHANGE 375 Adjusting Lengths for Reference Temperature.— The standard reference temperature for industrial length measurements is 20 degrees Celsius (68 degrees Fahrenheit). For other temperatures, corrections should be made in accordance with the difference in ther mal expansion for the two parts, especially when the gage is made of a different material than the part to be inspected. Example: An aluminum part is to be measured with a steel gage when the room tempera ture is 30 ° C. The aluminum part has a coefficient of linear thermal expansion, a Part = 24.7 3 10 −6 mm/mm- ° C, and for the steel gage, a Gage = 10.8 3 10 −6 mm/mm- ° C. At the reference temperature, the specified length of the aluminum part is 20.021 mm. What is the length of the part at the measuring (room) temperature? D L , the change in the measured length due to temperature, is given by:
L L T T ^
h
h
^
D = −
α α −
0
R
Part
Gage
. 20021 30 20 247 108 10 2782 919 10 0 003 . . . . mm 6 # # . − − − ^ ^ h h
mm
6
= =
−
where L = length of part at reference temperature; T R = room temperature (temperature of part and gage) and T 0 = reference temperature. Thus, the temperature-corrected length at 30 ° C is L + D L = 20.021 + 0.003 = 20.024 mm. Length Change Due to Temperature.— Table 14 gives changes in length for variations from the standard reference temperature of 68 ° F (20 ° C) for materials of known coeffi cients of expansion, a . Coefficients of expansion are given in tables on pages 372, 373, 374, 386, 387, and elsewhere. Example: In Table 14 , for coefficients between those listed, add appropriate listed val ues. For example, a length change for a coefficient of 7 is the sum of values in the 5 and 2 columns. Fractional interpolation also is possible. Thus, in a steel bar with a coefficient of thermal expansion of 6.3 3 10 − 6 = 0.0000063 in/in = 6.3 m in/in of length/ ° F, the increase in length at 73 ° F is 25 + 5 + 1.5 = 31.5 m in/in of length. For a steel with the same coefficient of expansion, the change in length, measured in degrees C, is expressed in microns (micrometers)/meter ( m m/m) of length. Alternatively, and for temperatures beyond the scope of the table, the length difference due to a temperature change is equal to the coefficient of expansion multiplied by the change in temperature, i.e., D L = aD T . Thus, for the previous example, D L = 6.3 3 (73 − 68) = 6.3 3 5 = 31.5 m in/in. Change in Radius of Thin Circular Ring with Temperature.— Consider a circular ring of initial radius r , that undergoes a temperature change D T . Initially, the circum- ference of the ring is c = 2 p r . If the coefficient of expansion of the ring material is a , the change in circumference due to the temperature change is D c = 2 p r aD T. The new circumference of the ring will be: c n = c + D c = 2 p r + 2 p r aD T = 2 p r (1 + aD T ). Note: An increase in temperature causes D c to be positive, and a decrease in temperature causes D c to be negative. As the circumference increases, the radius of the circle also increases. If the new radius is R , the new circumference is 2 p R . For a given change in temperature, D T , the change in radius of the ring is found as follows: c n = 2 π R = 2 π r (1 + α∆ T ) R = r + r α∆ T ∆ r = R r = r α∆ T
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