Surface roughness parameters Ra, Rq

Surface texture has roughness (fine irregularities from the process), waviness (longer wavelength, from vibration or deflection) and lay (direction of the pattern).

Definitions

$R_a$ is the arithmetic mean of the absolute deviations of the profile from the mean line:

$$R_a = \frac{1}{n}\sum_{i=1}^{n}|y_i|$$

$R_q$ (also RMS) is the square root of the mean of the squares of the deviations:

$$R_q = \sqrt{\frac{1}{n}\sum_{i=1}^{n} y_i^2}$$

Check of the mean line

Sum of ordinates: $2.0-1.5+3.5-2.5+1.0-3.0+2.5-1.0+3.0-4.0 = 0$, so the reference line is the mean line.

Calculation of $R_a$

$$\sum|y_i| = 2.0+1.5+3.5+2.5+1.0+3.0+2.5+1.0+3.0+4.0 = 24.0$$ $$R_a = \frac{24.0}{10} = 2.4\ \mu\text{m}$$

Calculation of $R_q$

$$\sum y_i^2 = 4+2.25+12.25+6.25+1+9+6.25+1+9+16 = 67.0$$ $$R_q = \sqrt{\frac{67.0}{10}} = \sqrt{6.7} = 2.59\ \mu\text{m}$$

Maximum peak-to-valley height

$$R_{max} = y_{max} - y_{min} = 3.5 - (-4.0) = 7.5\ \mu\text{m}$$

Factors causing roughness in machining

1. Feed marks: in turning the theoretical roughness is $R_{max} = f^2/(8r)$ where $f$ is the feed and $r$ the nose radius.
2. Built-up edge, tool wear, vibration (chatter) and machine tool rigidity.

Answer: $R_a = 2.4\ \mu$m, $R_q = 2.59\ \mu$m, $R_{max} = 7.5\ \mu$m.