Vibratory Bowl & Pan Feeders, Packaging Automation & Granular Micro-Hopping Mechanics
Vibratory Feeder Throw Factor Gamma & Conveying Formula
Calculate Peak Trough Acceleration in Gs, Dimensionless Vertical Throw Factor Gamma = (a*omega²*SIN(beta))/g, Ideal Horizontal Peak Velocity (mm/s), and Effective Transport Speed (mm/s).
Vibratory feeders don’t slide parts—above Throw Factor Gamma > 1.0, parts micro-hop forward while the pan pulls back underneath!
Electromagnetic `50/60` Hz packaging pan feeders and unbalanced-motor quarry grizzly feeders oscillate along an inclined leaf-spring Line of Action `beta` (`~15°–30°` above the horizontal trough) with Peak-to-Peak Stroke `S` (mm, so Single-Amplitude `A = S / 2`). Peak Linear Acceleration in `g`s is `a_max / g = ((S / 2,000) * (2*PI()*f)^2) / 9.80665`, while the Dimensionless Vertical Throw Number (Machine Factor) `Gamma = (a_max / g) * SIN(beta_rad)` compares peak upward pan deceleration against `1 g` gravity! When `Gamma <= 1.0`, parts never leave the deck and merely shuffle by stick-slip friction; when `1.5 <= Gamma <= 3.3`, parts launch into repeatable micro-ballistic hops once per cycle while the trough retracts beneath them (`Gamma > 3.3` causes chaotic sub-harmonic bouncing!), achieving Net Transport Speed `v_actual = eta_tr * ((S / 2) * 2*PI()*f * COS(beta_rad))` (mm/s)!
Verify on a 50 Hz Electromagnetic Weigh-Scale Pan Feeder—Peak_to_Peak_Stroke `S` = 1.40 mm (`A = 0.70` mm, B2), Coil_Drive_Frequency `f` = 50.0 Hz (B3), Leaf_Spring_Drive_Angle `beta` = 20.0° (B4), and Material_Transport_Factor `eta_tr` = 0.78 (B5): Peak Deck Acceleration hits `7.04` g (E2), Vertical Throw Number `Gamma = 2.41` sits squarely inside the `[1.5, 3.3]` synchronous micro-flight window (E3), Ideal Horizontal Deck Peak Velocity equals `206.6` mm/s (E4), and Effective Conveying Speed hits `161.2` mm/s (E5)!
| Vibratory Pan Feeder KPI (Stroke=1.40 mm p-p @ 50 Hz, Angle=20°, eta=0.78) | Formula | Verified Output |
|---|---|---|
| Peak Deck Acceleration (g) | ((Stroke_mm/2000) * (2*PI()*Hz)^2) / 9.80665 | 7.04 g (2.26 g @ 0.45 mm dribble) |
| Vertical Throw Factor Gamma | Peak_Accel_g * SIN(Angle_deg * PI()/180) | 2.41 micro-hop (0.77 non-flight @ 0.45 mm!) |
| Ideal Horizontal Velocity (mm/s) | (Stroke_mm/2) * (2*PI()*Hz) * COS(beta) | 206.6 mm/s (66.4 mm/s @ 0.45 mm) |
| Effective Transport Speed (mm/s) | Ideal_Horiz_mm_s * Transport_Eff | 161.2 mm/s (51.8 mm/s @ 0.45 mm) |
Dimensionless Throw Number Gamma & Vibratory Trough Conveying Velocity Formula
Validate `Stroke_p_p > 0` mm, `Frequency` in `[5, 120]` Hz, `Drive_Angle beta` in `[5°, 45°]`, and `Transport_Efficiency` in `(0, 1]`!
=LET(st_m, B2, fq_h, B3, bt_d, B4, ef_t, B5, IF(OR(st_m <= 0, fq_h < 5, fq_h > 120, bt_d < 5, bt_d > 45, ef_t <= 0, ef_t > 1), "Invalid Feeder Input", LET(om_r, 2 * PI() * fq_h, rd_b, bt_d * (PI() / 180), vi_m, (st_m / 2) * om_r * COS(rd_b), ROUND(vi_m * ef_t, 1))))Fine-Dribble 0.45 mm Stroke Sub-Flight Sliding Transition Check (Set Stroke B2 = 0.45 mm)
Throttle SCR controller voltage for final checkweigher dribble feed (`S` B2 = 0.45 mm): Peak Deck Gs E2 drop to `2.26` g, Throw Factor `Gamma` E3 falls below unity to `0.77` (parts transition from micro-hopping to gentle friction-sliding without bouncing!), and Transport Speed E5 slows smoothly to `51.8` mm/s!
=IF(ISTEXT(E5), E5, LET(am_m, B2 / 2000, om_r, 2 * PI() * B3, ag_p, (am_m * (om_r ^ 2)) / 9.80665, ROUND(ag_p * SIN(B4 * (PI() / 180)), 2)))Try the example
Use the original values or change them to test your understanding.