Rotordynamics, Turbomachinery Shafts & Lateral Whirling Resonance
Dunkerley Shaft Critical Whirling Speed & Margin Formula
Calculate Dunkerley Cumulative Static Deflection Equivalent (mm), Conservative First Lateral Critical Speed N_cr (RPM), Operating Speed Separation Margin (%), and Sub/Supercritical Regime.
Reciprocal squared critical speeds add like capacitors in series—summing static deflections gives Dunkerley N_cr directly!
Every rotating pump, fan, or compressor shaft carrying impellers and couplings exhibits a First Lateral Critical Whirling Speed N_cr where synchronous unbalance excitation causes violent lateral bending resonance! For a single concentrated mass producing gravity static deflection delta_i (mm, with g = 9,810 mm/s²), isolated natural frequency is N_i = (30 / PI()) * SQRT(9,810 / delta_i) RPM. Under Dunkerley’s Theorem, multiple rotors plus distributed shaft self-weight combine conservatively (lower-bound safe estimate) via 1 / N_cr^2 = SUM(1 / N_i^2)—which simplifies directly to N_cr = (30 / PI()) * SQRT(9,810 / (delta_1 + delta_2 + delta_shaft / 1.27)) or using modal static deflection delta_s! API turbomachinery standards demand at least ±20% separation margin between N_op and N_cr!
Verify on an 1,800 RPM Two-Stage Overhung/Between-Bearings Pump Shaft—Impeller_1_Static_Defl = 0.040 mm (B2), Impeller_2_Static_Defl = 0.060 mm (B3), Effective_Shaft_Modal_Defl = 0.025 mm (B4 -> Combined Dunkerley Deflection = 0.125 mm in E2), and Operating_Speed = 1,800.0 RPM (B5): Dunkerley First Critical Speed hits (30 / PI()) * SQRT(9,810 / 0.125) = 2,675.17 RPM (E3), Separation Margin is +48.62% above running speed (E4), and Regime Status reads "STIFF SUBCRITICAL" (E5)!
| Dunkerley Rotordynamics KPI (d1=0.04, d2=0.06, ds=0.025 mm, 1,800 RPM) | Formula | Verified Output |
|---|---|---|
| Combined Dunkerley Deflection (mm) | d_rotor1 + d_rotor2 + d_shaft_modal | 0.125 mm (0.500 @ flexible shaft) |
| Dunkerley 1st Critical Speed (RPM) | (30 / PI()) * SQRT(9810 / Sum_Defl_mm) | 2,675.17 RPM (1,337.58 @ 0.50 mm) |
| Critical Speed Margin (%) | ((N_cr - Operating_RPM) / Operating_RPM) * 100 | +48.62% (-25.69% @ 0.50 mm) |
| API ±20% Resonance Regime | |Ncr-Nop|/Nop < 20% -> RESONANCE DANGER | STIFF SUBCRITICAL |
Dunkerley Lower-Bound First Critical Speed Formula
Validate d_1 >= 0, d_2 >= 0, d_shaft > 0 (total deflection > 0!), and Operating_RPM > 0!
=LET(d_r1, B2, d_r2, B3, d_sh, B4, n_op, B5, IF(OR(d_r1 < 0, d_r2 < 0, d_sh <= 0, n_op <= 0), "Invalid Shaft Input", LET(d_sm, d_r1 + d_r2 + d_sh, ROUND((30 / PI()) * SQRT(9810 / d_sm), 2))))Slender Shaft Supercritical Crossover & Resonance Check (Set Deflections = 0.15, 0.25, 0.10 mm)
Quadruple static deflections on a slender longer-span shaft (B2 = 0.15, B3 = 0.25, B4 = 0.10 -> 0.500 mm total): Dunkerley Critical Speed E3 halves to 1,337.58 RPM (-25.69% below 1,800 RPM in E4), transitioning to "FLEXIBLE SUPERCRITICAL" (requiring rapid ramp-up through 1,338 RPM during startup)!
=IF(ISTEXT(E3), E3, LET(nc_r, (30 / PI()) * SQRT(9810 / (B2 + B3 + B4)), IF(ABS(nc_r - B5) / B5 < 0.2, "RESONANCE DANGER", IF(nc_r > B5, "STIFF SUBCRITICAL", "FLEXIBLE SUPERCRITICAL"))))Try the example
Use the original values or change them to test your understanding.