Linear Regression-More Examples: Mechanical Engineering
Chapter 06.03
Linear Regression-More Examples
Mechanical Engineering
Example 1
The coefficient of thermal expansion,[pic],of steel is given at discrete values of temperature in Table 1.
|Table 1 Coefficient of thermal expansion versus temperature for steel. |
|Temperature, [pic] |
|Coefficient of |
|thermal expansion, [pic] |
| |
|[pic] |
|[pic] |
| |
|80 |
|[pic] |
| |
|60 |
|[pic] |
| |
|40 |
|[pic] |
| |
|20 |
|[pic] |
| |
|0 |
|[pic] |
| |
|−20 |
|[pic] |
| |
|−40 |
|[pic] |
| |
|−60 |
|[pic] |
| |
|−80 |
|[pic] |
| |
|−100 |
|[pic] |
| |
|−120 |
|[pic] |
| |
|−140 |
|[pic] |
| |
|−160 |
|[pic] |
| |
|−180 |
|[pic] |
| |
|−200 |
|[pic] |
| |
|−220 |
|[pic] |
| |
|−240 |
|[pic] |
| |
|−260 |
|[pic] |
| |
|−280 |
|[pic] |
| |
|−300 |
|[pic] |
| |
|−320 |
|[pic] |
| |
|−340 |
|[pic] |
| |
The data is regressed to a first order polynomial.
[pic]
Find the constants [pic] and [pic] of the regression model.
Solution
Table 2 shows the summations needed for the calculation of the constants of the regression model.
|Table 2 Tabulation of data for calculation of needed summations. |
|I |
|[pic] |
|[pic] |
|[pic] |
|[pic] |
| |
|− |
|[pic] |
|[pic] |
|[pic] |
|[pic] |
| |
|1 |
|80 |
|[pic] |
|[pic] |
|6400 |
| |
|2 |
|60 |
|[pic] |
|[pic] |
|3600 |
| |
|3 |
|40 |
|[pic] |
|[pic] |
|1600 |
| |
|4 |
|20 |
|[pic] |
|[pic] |
|400 |
| |
|5 |
|0 |
|[pic] |
|[pic] |
|0 |
| |
|6 |
|-20 |
|[pic] |
|[pic] |
|400 |
| |
|7 |
|-40 |
|[pic] |
|[pic] |
|1600 |
| |
|8 |
|-60 |
|[pic] |
|[pic] |
|3600 |
| |
|9 |
|-80 |
|[pic] |
|[pic] |
|6400 |
| |
|10 |
|-100 |
|[pic] |
|[pic] |
|10000 |
| |
|11 |
|-120 |
|[pic] |
|[pic] |
|14400 |
| |
|12 |
|-140 |
|[pic] |
|[pic] |
|19600 |
| |
|13 |
|-160 |
|[pic] |
|[pic] |
|25600 |
| |
|14 |
|-180 |
|[pic] |
|[pic] |
|32400 |
| |
|15 |
|-200 |
|[pic] |
|[pic] |
|40000 |
| |
|16 |
|-220 |
|[pic] |
|[pic] |
|48400 |
| |
|17 |
|-240 |
|[pic] |
|[pic] |
|57600 |
| |
|18 |
|-260 |
|[pic] |
|[pic] |
|67600 |
| |
|19 |
|-280 |
|[pic] |
|[pic] |
|78400 |
| |
|20 |
|-300 |
|[pic] |
|[pic] |
|90000 |
| |
|21 |
|-320 |
|[pic] |
|[pic] |
|102400 |
| |
|22 |
|-340 |
|[pic] |
|[pic] |
|115600 |
| |
|[pic] |
|-2860 |
|[pic] |
|[pic] |
|726000 |
| |
[pic]
[pic]
[pic]
[pic] [pic]
[pic]
[pic][pic]
[pic]
[pic]
[pic][pic]
[pic]
[pic]
[pic][pic]
| |
|[pic] |
|Figure 4 Linear regression of coefficient of thermal expansion vs. temperature |
|data. |
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