--- title: "Six-speed machine tool gearbox design" subject: "Machine Design II" chapter: 5 key: "semester_7/machine_design_ii:machine-tool-speed-ladder-gear" source: https://www.bctengineer.com/q/semester_7/machine_design_ii/machine-tool-speed-ladder-gear site: BCT Engineering --- # Six-speed machine tool gearbox design Machine Design II · Chapter 5: System Design - Asked in: year not given, 8 marks - Video: [Design of Gear Box - Six output Speeds - Part 6](https://www.youtube.com/watch?v=a7CrWIw04Oo) — Prof Nawaz E Patshala - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/machine_design_ii/machine-tool-speed-ladder-gear - Whole chapter: https://www.bctengineer.com/md/semester_7/machine_design_ii/chapter-5 ## Question A machine tool spindle needs six speeds in geometric progression from 125 rpm to 400 rpm. The gearbox input shaft runs at 400 rpm, driven by a 1440 rpm motor through a belt. (a) Find the progression ratio and list the standard spindle speeds. (b) Select the structural formula 3(1) x 2(3) and draw the speed ray diagram. (c) Find suitable tooth numbers for each gear pair of the two stages (take the sum of teeth as 70 for the first stage and 60 for the second). ## Short answer φ = 1.26; spindle speeds 125, 160, 200, 250, 315, 400 rpm. Structural formula 3(1)×2(3). Stage‑1 gears (sum 70): 35/35, 31/39, 27/43. Stage‑2 gears (sum 60): 30/30, 20/40. ## Answer #### (a) Progression ratio and speeds For $z$ speeds in GP: $n_{max}/n_{min} = \varphi^{\,z-1}$. $$ \varphi = \left(\frac{400}{125}\right)^{1/5} = 3.2^{0.2} = 1.262 \approx 1.26 \ (\text{R10 series}) $$ Spindle speeds ($\times 1.26$ each step): 125, 160, 200, 250, 315, 400 rpm. #### (b) Structure and ray diagram Six speeds $= 3 \times 2$. Structural formula: **3(1) x 2(3)**, where the number in brackets is the characteristic $x$ (the step is $\varphi^{x}$). - Group 1 (3 speeds, $x=1$): ratios $1,\ 1/\varphi,\ 1/\varphi^{2} = 1,\ 0.794,\ 0.630$. - Group 2 (2 speeds, $x=3$): ratios $1,\ 1/\varphi^{3} = 1,\ 0.5$. Check of range: group 1 $\varphi^{2}=1.6$, group 2 $\varphi^{3}=2$, both below the limit of 8, and no gear ratio is smaller than 1/4. Belt drive: motor 1440 rpm to shaft I at 400 rpm, ratio $1440/400 = 3.6$. ```text Shaft I Shaft II Shaft III (rpm) 400 ----> 400 ---------> 400 (x1) \ \---> 200 (x0.5) \--> 315 --------> 315 \ \---> 160 \-> 250 -------> 250 \---> 125 ``` Shaft II runs at 400, 315, 250 rpm. Each speed of shaft II gives two speeds on shaft III: the same speed (ratio 1) or three steps lower (ratio 0.5), giving 400, 315, 250 and 200, 160, 125 rpm. #### (c) Gear tooth numbers Stage 1 (sum 70): ratio $= z_1/z_2$. | Ratio | $z_1$ | $z_2$ | Actual ratio | |---|---|---|---| | 1 | 35 | 35 | 1.000 | | 0.794 | 31 | 39 | 0.795 | | 0.630 | 27 | 43 | 0.628 | Stage 2 (sum 60): | Ratio | $z_1$ | $z_2$ | Actual ratio | |---|---|---|---| | 1 | 30 | 30 | 1.000 | | 0.5 | 20 | 40 | 0.500 | All pinions have more than 17 teeth (no undercutting for 20 degree teeth). **Check of actual spindle speeds:** 400, 317.9, 251.2, 200.0, 159.0, 125.6 rpm. Errors against the standard speeds are below 1%, within the permitted $\pm 10(\varphi-1)\% = 2.6\%$. **Answer:** $\varphi = 1.26$; speeds 125, 160, 200, 250, 315, 400 rpm; structure $3(1)\times2(3)$; teeth 35/35, 31/39, 27/43 and 30/30, 20/40. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For six speeds from 125 to 400 rpm, what is the GP ratio? A. 1.12 B. 1.58 C. 3.2 D. 1.26, from (400/125)^(1/5) Answer: D. n_max/n_min = phi^(z-1), so phi = 3.2^0.2 = 1.26 (R10 series). (Key point: Progression ratio) 2. In 3(1) x 2(3), what does the number in brackets mean, and what is the group 2 ratio? A. It is the characteristic x, the step is phi^x; group 2 ratios are 1 and 1/phi^3 = 0.5 B. It is the number of teeth; ratios are 1 and 3 C. It is the number of shafts; ratios are 1 and 2 D. It is the module; ratios are 1 and 0.794 Answer: A. The bracket is the characteristic x. For x = 3 the second group uses 1 and 1/phi^3, about 0.5. (Key point: Structural formula) 3. Shaft II runs at 400, 315 and 250 rpm. What do the two ratios 1 and 0.5 of group 2 give on shaft III? A. Only 400, 315, 250 rpm B. The same speed or three steps lower, giving 400, 315, 250 and 200, 160, 125 rpm C. Speeds twice as high D. Speeds one step lower, 315, 250, 200 rpm Answer: B. Each shaft II speed gives itself (ratio 1) or a speed 3 steps lower (ratio 0.5), producing six speeds. (Key point: Speeds on the last shaft) 4. With a tooth sum of 70, which pair gives the ratio 0.794 for stage 1? A. 35/35 B. 27/43 C. 31/39 D. 30/40 Answer: C. 31/39 = 0.795, close to 0.794. 35/35 gives 1 and 27/43 gives 0.628. 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