| name | critical-speed |
| description | Critical speed analysis for rotating shafts — Dunkerley, Rayleigh-Ritz, Myklestad, whirl, gyroscopic effects, flexible rotor balancing criteria, API/AGMA limits. |
| metadata | {"priority":7,"promptSignals":{"phrases":["critical speed","whirling","rotor","Dunkerley","Rayleigh","flexible rotor","balancing"],"minScore":4}} |
Critical Speed — Complete Skill
Why Critical Speed Matters
At critical speed: deflection grows unboundedly (without damping)
Physical: centrifugal force exceeds shaft restoring force at that speed
Result: vibration, noise, bearing damage, potential catastrophic failure
Must NOT operate at or near critical speed for extended time
Single Mass on Massless Shaft
Simple Jeffcott rotor:
ω_cr = √(k/m) = √(48EI/mL³) [simply supported, center mass]
n_cr = 60×ω_cr/(2π) [rpm]
From static deflection:
n_cr = 946/√(δ_st[mm]) [rpm, where δ_st = static deflection at mass location]
Or: n_cr = 30/π × √(g/δ_st) [rpm]
Quick reference:
δ_st = 0.1mm → n_cr = 2990 rpm
δ_st = 1.0mm → n_cr = 946 rpm
δ_st = 10mm → n_cr = 299 rpm
δ_st = 25mm → n_cr = 189 rpm
Multiple Masses — Dunkerley's Equation
1/n_cr² ≈ 1/n₁² + 1/n₂² + ... + 1/n_shaft²
Where n_i = critical speed if only mass i were present (use static deflection formula)
n_shaft = critical speed of shaft alone (no attached masses) = π²/(L²) × √(EIg/ρA) for uniform shaft
Dunkerley is a lower bound estimate — actual n_cr is slightly higher
Safe for design: actual > Dunkerley estimate → conservative
Rayleigh-Ritz Method (Upper Bound)
Assume mode shape {X}, then:
ω² ≤ {X}ᵀ[K]{X} / {X}ᵀ[M]{X}
Using static deflection as mode shape (Rayleigh's method for shaft):
ω_cr² = g × Σ(W_i × y_i) / Σ(W_i × y_i²)
Where y_i = static deflection at mass i location under gravity
W_i = weight of mass i
This gives a better estimate than Dunkerley (upper bound vs. lower bound)
Actual value lies between Rayleigh and Dunkerley estimates.
Stiffness-Based Formula
For uniform shaft, simply supported:
n_cr = π²/(2πL²) × √(EI/ρA) [rad/s] = (π/L²) × √(EIg/w) rpm
Where w = weight per unit length [N/m]
Cantilever shaft (overhang):
n_cr = (1.875)²/(L²) × √(EI/(ρA)) (first mode)
For rigid disk at tip: n_cr = √(3EI/(mL³)) × 1/(2π) Hz
Design Rules (API/AGMA)
API 670 (machinery protection):
Operating speed ≤ 0.8 × n_cr1 (subcritical regime)
OR pass through n_cr during startup/shutdown quickly
AGMA 6014:
For gearboxes: n_operating < 0.75 × n_cr
Or n_operating > 1.4 × n_cr (supercritical, uncommon)
Separation margin:
SM% = (n_cr - n_op)/n_cr × 100% ≥ 20% (subcritical)
SM% = (n_op - n_cr)/n_op × 100% ≥ 25% (supercritical)
Damped Critical Speed
Undamped: ω_cr = √(k/m)
Damped: ω_d,cr = ω_cr × √(1-ζ²) (slightly lower)
At damped critical: amplitude is finite but still large
Peak amplitude at resonance: X = F₀/(2ζk) = F₀/(c×ω_cr)
Damping in rotating machinery: ζ = 0.01-0.05 (internal + bearing damping)
Gyroscopic Effects (Disk on Shaft)
For disk with large polar moment of inertia I_p:
Gyroscopic couple: M_g = I_p × Ω × dψ/dt
Where Ω = spin speed, ψ = precession rate = ω_whirl
Gyroscopic effect: forward whirl increases n_cr, backward whirl decreases n_cr
At high speeds: n_cr splits into two frequencies
For thin disk: I_p = I_d/2 (where I_d = diametral moment)
Gyroscopic parameter: G = I_p/(I_d)
Bearing Stiffness Effects
Rigid bearings: n_cr from pure shaft bending formula
Flexible bearings (common): effective stiffness k_eff = k_shaft×k_bearing/(k_shaft+k_bearing)
n_cr decreases with bearing flexibility
For oil film bearings: stiffness depends on speed, load → n_cr is speed-dependent (iterative solution)
Balancing
Static balance: center of mass on spin axis (ΣmR = 0)
Add balancing mass: m_b×R_b = |Σm_i×R_i| at any angular position
Dynamic balance: both static and couple balanced (ΣmR=0 AND ΣmRl=0)
Two balancing planes needed for 3D balance
Couple imbalance: I_xz ≠ 0 → requires two masses
ISO G balance grades:
G0.4: gas turbine rotors (high precision)
G2.5: automotive: crankshafts, aircraft gas turbines
G6.3: general industrial fans, pumps, electric motors
G16: agricultural equipment, cardan shafts
G40: rigid assembled crankshafts for car engines
Residual unbalance:
U_per = eW = e×m×g [g·mm] (e = eccentricity, W = weight, m = mass)
G_grade = e×ω [mm/s at max speed]
Output
Provide: n_cr [rpm] by Dunkerley and Rayleigh, operating speed safety margin %, balance grade required, bearing type recommendation.
