Critical Speed Calculator

Second Moment of Area, Solid Circular Shaft

I = π · d⁴ / 64

Variables: I = second moment of area of the shaft cross-section (m⁴), d = shaft outside diameter (m), π = 3.14159 (dimensionless). Source: Hibbeler, Mechanics of Materials, 10th ed., Appendix A (Geometric Properties of an Area — circular section)

Static Deflection at an Intermediate Concentrated Load (Simply Supported Shaft)

δᵢ = mᵢ · g · aᵢ² · bᵢ² / (3 · E · I · L) , with bᵢ = L − aᵢ

Variables: δᵢ = static deflection of the shaft at disk i, produced by disk i's own weight acting alone (m), mᵢ = mass of disk i (kg), g = gravitational acceleration, fixed at 9.81 m/s², aᵢ = distance from the left bearing to disk i (m), bᵢ = distance from disk i to the right bearing (m), E = Young's modulus of the shaft material (Pa), I = second moment of area of the shaft (m⁴), L = bearing-to-bearing span (m). Disk positions are clamped to the range 0.001 m to L − 0.001 m before use. Source: Hibbeler, Mechanics of Materials, 10th ed., Appendix C (Slopes and Deflections of Beams — simply supported beam, intermediate concentrated load), evaluated at the load point x = aᵢ

Dunkerley's Equation — Lower-Bound Critical Speed

1/ω_c² = Σ 1/ωᵢ² , ωᵢ² = g / δᵢ | ⇒ | ω_c = √( g / Σ δᵢ ) | N_c = 60 · ω_c / (2 π)

Variables: ω_c = first lateral (whirling) critical angular speed, lower-bound estimate (rad/s), ωᵢ = critical angular speed the shaft would have carrying disk i alone (rad/s), δᵢ = self-weight static deflection at disk i from the equation above (m), g = gravitational acceleration, fixed at 9.81 m/s², N_c = critical speed (RPM), π = 3.14159 (dimensionless). The shaft's own distributed mass is not included: this calculator sums only the disk terms, so a heavy shaft carrying light disks will be overestimated. Source: Shigley's Mechanical Engineering Design, 10th ed., Ch. 7 (Shafts and Shaft Components — Critical Speeds for Shafts, Dunkerley's equation)

Rayleigh's Energy Method — Upper-Bound Critical Speed

ω_c = √( g · Σ mᵢ·δᵢ / Σ mᵢ·δᵢ² ) | N_c = 60 · ω_c / (2 π)

Variables: ω_c = first lateral critical angular speed, upper-bound estimate (rad/s), mᵢ = mass of disk i (kg), δᵢ = assumed mode-shape ordinate at disk i (m), g = gravitational acceleration, fixed at 9.81 m/s², N_c = critical speed (RPM), π = 3.14159 (dimensionless). Important: this calculator uses for δᵢ the SELF-WEIGHT deflection from the equation above — the deflection produced by disk i's own weight alone. The classical Rayleigh method instead uses the total deflection at station i with all disks acting simultaneously (yᵢ = Σⱼ mⱼ·g·δᵢⱼ, from the influence coefficients). With two or more disks the values used here are smaller than the classical yᵢ, so the Rayleigh speed reported is higher (non-conservative) and no longer brackets the Dunkerley value; for a single disk both reduce to ω_c = √(g/δ) and agree exactly. Source: Shigley's Mechanical Engineering Design, 10th ed., Ch. 7 (Critical Speeds for Shafts — Rayleigh's method and influence coefficients)

Reported Critical Speed and Safety Margin

N_avg = (N_Dunkerley + N_Rayleigh) / 2 | SM = (N_avg − N_op) / N_avg × 100

Variables: N_avg = averaged critical speed reported as the headline result (RPM), N_Dunkerley = critical speed from Dunkerley's equation (RPM), N_Rayleigh = critical speed from Rayleigh's method (RPM), N_op = shaft operating speed entered by the user (RPM), SM = safety margin, the fraction by which the critical speed exceeds the operating speed (%). Averaging the two is a convenience of this tool, not standard practice: Dunkerley and Rayleigh are intended as bracketing lower and upper bounds, and design should be checked against the lower (Dunkerley) value. Source: Shigley's Mechanical Engineering Design, 10th ed., Ch. 7 (Critical Speeds for Shafts — bracketing of the first critical speed)