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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