Lubricating characteristics of oils deteriorate at elevated temperatures, and the amount of bearing wear, y, is normally a linear function of the operating temperature, t. That is, y = mt + b. By weighing bearings before and after operation at various temperatures, the following table was

Question

Lubricating characteristics of oils deteriorate at elevated temperatures, and the amount of bearing wear, y, is normally a linear function of the operating temperature, t. That is, y = mt + b. By weighing bearings before and after operation at various temperatures, the following table was constructed:

Operating temperature, [latex]{ }^{\circ} C[/latex]	120	148	175	204	232	260	288	316	343	371
Amount of wear, gm/10,000 hr	3	4	5	5.5	6	7.5	8.8	10	11.1	12

Determine the least-squares linear fit from these readings and use it to determine an operating temperature that should limit bearing wear to 7 gm/10,000 hr of operation.

Accepted Answer

For the system Ax = b, we see that A and b are given by

[latex]A=\left[\begin{array}{rrrrrrrrrr}120 & 148 & 175 & 204 & 232 & 260 & 288 & 316 & 343 & 371 \1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\end{array} ight]^{T}[/latex]
[latex]b =[3, \quad 4, \quad 5, \quad 5.5, \quad 6, \quad 7.5, \quad 8.8, \quad 10, \quad 11.1, \quad 12]^{T}[/latex]

The least-squares solution to Ax = b is found from the MATLAB commands in Fig. 3.24.

From Fig. 3.24 we see that the least-squares linear fit is
y = (0.0362)t − 1.6151.
Setting y = 7 yields t = 237.986. Hence an operating temperature of about 238 [latex]{ }^{\circ} C[/latex] should limit bearing wear to 7 gm/10,000 hr.

Question 3.8.3: Lubricating characteristics of oils deteriorate at elevated ...

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