Automated Packaging, CNC Tool Changers & Rotary Turrets, External Geneva Mechanism
External Geneva Wheel Dwell, Velocity & Accel Formula
Calculate Dwell-to-Index Motion Ratio, Mid-Stroke Peak Wheel Angular Velocity (rad/s), Crank Angle at Peak Acceleration (deg), and Peak Angular Acceleration (rad/s²).
Dropping a rotary turret from a 4-slot to a 3-slot Geneva wheel for longer dwell multiplies peak angular acceleration 5.8-fold—shearing cam-follower pins at 90 RPM!
An **External `N`-Slot Geneva (`Maltese Cross`) Intermittent Indexing Mechanism** (`Slots N >= 3`, tangent slot entry so impact velocity is zero at `beta_0 = 90° - (180° / N)`, `Center-to-Crank Ratio k = 1 / sin(pi / N)`) converts continuous Driver Crank Rotation `N_c` (`RPM` -> `omega_c = 2 * pi * N_c / 60` rad/s) into `360° / N` turret steps separated by locked crescent dwell! Each crank revolution spends `(180° - 360° / N)` indexing and `(180° + 360° / N)` locked stationary, fixing **Dwell-to-Motion Ratio `(N + 2) / (N - 2)`** (`3.00` for `4`-slot; `5.00` for `3`-slot) and **Mid-Index (`beta = 0°`) Peak Angular Velocity `omega_{w,max} = omega_c / (k - 1)`** (`rad/s`)! Differentiating `omega_w(beta)` locates **Peak Angular Acceleration right at `cos(beta_p) = (-(1 + k^2) + sqrt((1 + k^2)^2 + 32 * k^2)) / (4 * k)`** (`beta_p = 11.46°` on a `4`-slot wheel!) where **`alpha_{max} = (omega_c^2) * (k * sin(beta_p) * (k^2 - 1)) / ((1 + k^2 - 2 * k * cos(beta_p))^2)`** (`rad/s²`)!
Verify on a `4`-Station Bottling Starwheel / CNC ATC Geneva Turret—Slot_Count `N` = 4 (B2) and Driver_Crank_Speed `N_c` = 90.0 RPM (`omega_c = 9.425` rad/s, B3): Dwell-to-Motion Ratio equals **`3.00`** (`270°` dwell / `90°` index! E2), Peak Driven-Wheel Velocity equals `22.75` rad/s (E3), Peak Acceleration occurs at Crank Angle **`11.46°`** before/after centerline (E4), reaching **`480.3` rad/s²** (`= 5.409 * omega_c^2`! E5)!
| External Geneva Indexing Kinematics KPI (Driver Crank Nc=90 RPM, wc=9.425 rad/s) | Formula | Verified Output |
|---|---|---|
| Dwell-to-Motion Ratio | (Slot_Count_N + 2) / (Slot_Count_N - 2) | 3.00 @ 4-slot (5.00 @ 3-slot) |
| Peak Wheel Velocity w_max (rad/s) | wc_rad_s / ((1 / SIN(PI()/N)) - 1) | 22.75 rad/s (60.92 rad/s @ 3-slot!) |
| Crank Angle @ Peak Accel (deg) | DEGREES(ACOS((-(1+k^2) + SQRT((1+k^2)^2 + 32*k^2)) / (4*k))) | 11.46° @ 4-slot (4.76° @ 3-slot) |
| Peak Wheel Accel a_max (rad/s²) | wc^2 * k * sin(bp) * (k^2 - 1) / (1 + k^2 - 2*k*cos(bp))^2 | 480.3 rad/s² (2,788.5 = 5.8x SHOCK!) |
Exact External Geneva Mechanism Dwell, Velocity & Peak Acceleration Formula
Validate integer-station `Slot_Count N` in `[3, 24]` and `Driver_Crank_Speed N_c` in `[1, 1,200]` RPM!
=LET(ns_g, B2, nc_r, B3, wc_r, (2 * PI() * nc_r) / 60, kr_g, 1 / SIN(PI() / ns_g), cb_p, (-(1 + (kr_g ^ 2)) + SQRT(((1 + (kr_g ^ 2)) ^ 2) + 32 * (kr_g ^ 2))) / (4 * kr_g), sb_p, SQRT(1 - (cb_p ^ 2)), am_x, (wc_r ^ 2) * (kr_g * sb_p * ((kr_g ^ 2) - 1)) / (((1 + (kr_g ^ 2) - 2 * kr_g * cb_p) ^ 2)), IF(OR(ns_g < 3, ns_g > 24, nc_r < 1, nc_r > 1200), "Invalid Geneva Input", ROUND(am_x, 1)))3-Slot Geneva Substitution (`N = 3` at `90` RPM) 5.8x Inertial Cam-Pin Shear Check
Drop station count to `B2 = 3` (`120°` turret index squeezed into just `60°` of crank rotation -> `5.00` dwell ratio in E2): Peak Wheel Speed E3 jumps `2.68x` to `60.92` rad/s while **Peak Angular Acceleration E5 explodes `5.81-fold` from `480.3` to `2,788.5` rad/s²** at `4.76°` crank angle (E4)—explaining why high-speed dial tables use `>= 6` Geneva slots or conjugate globoidal cams!
=IF(ISTEXT(E5), E5, LET(wc_r, (2 * PI() * B3) / 60, kr_g, 1 / SIN(PI() / B2), ROUND(wc_r / (kr_g - 1), 2)))Try the example
Use the original values or change them to test your understanding.