--- title: "Surface roughness parameters Ra, Rq" subject: "Manufacturing and Production Processes" chapter: 3 key: "semester_4/manufacturing_and_production_processes:surface-roughness-ra-rq-numerical" source: https://www.bctengineer.com/q/semester_4/manufacturing_and_production_processes/surface-roughness-ra-rq-numerical site: BCT Engineering --- # Surface roughness parameters Ra, Rq Manufacturing and Production Processes · Chapter 3: Properties of Manufactured Products - Asked in: year not given, 5 marks - Video: [Surface Roughness Parameters: Ra, Ry, Rz, RMS (हिंदी में)](https://www.youtube.com/watch?v=WnKXj61YKKA) — Mr Mechanical Engineer - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/manufacturing_and_production_processes/surface-roughness-ra-rq-numerical - Whole chapter: https://www.bctengineer.com/md/semester_4/manufacturing_and_production_processes/chapter-3 ## Question Define centre-line average roughness $R_a$ and root-mean-square roughness $R_q$. The heights of 10 equally spaced points of a surface profile measured from the mean line within a sampling length are (in micrometre): +2.0, -1.5, +3.5, -2.5, +1.0, -3.0, +2.5, -1.0, +3.0, -4.0. Calculate $R_a$, $R_q$ and the maximum peak-to-valley height. Name two factors that cause surface roughness in machining. ## Short answer Ra is the mean absolute deviation from the mean line; Rq is the root mean square deviation. For the data, Ra = 24.0/10 = 2.4 um, Rq = sqrt(67/10) = 2.59 um, and peak-to-valley height = 3.5 - (-4.0) = 7.5 um. Roughness in machining comes mainly from feed marks, built-up edge, tool wear and vibration. ## Answer **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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Ra is defined as A. Square root of the mean of squared deviations B. Arithmetic mean of absolute deviations from the mean line C. Highest peak minus lowest valley D. Mean of signed deviations Answer: B. Ra = (1/n) sum |y_i|; the root-mean-square version is Rq. (Key point: Definition of Ra) 2. For the given 10 heights, the sum of absolute values is 24.0 um. Ra is A. 0 um B. 2.4 um C. 2.59 um D. 7.5 um Answer: B. Ra = 24.0/10 = 2.4 um (the signed sum is zero, so a signed mean would be useless). (Key point: Ra value) 3. Sum of squares is 67.0 um^2 for n = 10. Rq equals A. 6.7 um B. 3.35 um C. 2.59 um D. 2.4 um Answer: C. Rq = sqrt(67.0/10) = sqrt(6.7) = 2.59 um. (Key point: Rq value) 4. The maximum peak-to-valley height here is A. 7.5 um B. 4.0 um C. 3.5 um D. 0.5 um Answer: A. Rmax = highest peak - lowest valley = 3.5 - (-4.0) = 7.5 um. 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