# Effective Matching of Surface Roughness

Concerning the effective matching of surface roughness using a lathe, I have been asked to quantify Ra = C1* s^C2 v^C3 Re^C4, where Ra is surface roughness, s is feedrate, v is cutting velocity and Re is the nose radius of the tool.

I had performed an experiment to get the values of Ra for combinations of different v, s and Re. My problem is that I don't know how to use this data to define the constants in the relationship above. Can you tell me how to do this?

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#### Solution Preview

Please see the attachment for the least squares methods in computing the approximate estimations of the lathe tool's nose radius, cutting velocity, and feedrate.

Solution:

You have to determine a mathematical expression with parameters (C1, C2, C3, C4), based on a set of experimental data.

If you should have 4 sets of experimental data with various values of (s, v, Re), you should have to solve an algebraic system with unknown (C1, C2, C3, C4).

The system is not linear, but could be reduced to a linear one by applying the logarithm function:

(1)

(2)

By writing the above relation for every set of data, we will get a linear algebraic system from where we can determine the unknown (Ci), i = 1,...4.

Note that we can denote

C1' = lnC1 ...

#### Solution Summary

The answers are found by using the least squares method, to compute approximate estimations of feedrate, velocity, and nose radius of the lathe tool.