Mechanical Engineering Calculators: Rayleigh Multi-Mass Shaft Critical Speed Calculator

Rayleigh multi-mass critical-speed equation

Rayleigh’s method estimates the lowest lateral natural frequency by equating the maximum strain energy represented by a gravity-loaded deflection shape with the maximum kinetic energy of the vibrating lumped masses. For rotating machinery, this first lateral natural frequency provides a first synchronous whirl critical-speed estimate.

The same result can be expressed through an equivalent static deflection:

Input provenance is part of the equation

The most important input rule is that yᵢ is not the deflection created by Wᵢ alone. It is the total deflection at station i after all participating weights act simultaneously under gravity. The weights and deflections must use consistent signs or, as implemented here, nonnegative magnitudes referenced to one common direction.

Obtain the station deflections from a compatible beam, transfer-matrix, finite-element, measurement, or other verified static model. The model must represent the same rotor, supports, load directions, boundary conditions, and participating weights used in the Rayleigh sums. Mixing deflections from unrelated load cases invalidates the energy calculation even if every number has valid units.

Use zero weight and zero deflection for an unused station. A zero weight with a positive deflection is ignored and produces a warning. A positive weight requires a positive total deflection.

Worked example

Use the default active stations:

The calculation is:

1. Σ(Wᵢyᵢ) = 1000(1.5) + 1500(2.5) + 750(1.2) = 6,150 N·mm.
2. Σ(Wᵢyᵢ²) = 1000(1.5²) + 1500(2.5²) + 750(1.2²) = 12,705 N·mm².
3. δeq = 12,705 / 6,150 = 2.065854 mm.
4. ω₁ = √(9,806.65 / 2.065854) = 68.898626 rad/s.
5. f₁ = 10.965557 Hz and n₁ = 657.933414 rpm.
6. At 500 rpm, the speed ratio is 0.759955 and the operating point is 24.0045% below the estimate.
7. The default ±20% review band extends from 526.346731 to 789.520097 rpm, so 500 rpm lies outside it.

The contribution bars show the denominator fractions: 17.71% from station 1, 73.79% from station 2, 8.50% from station 3, and 0% from the disabled station. These fractions explain numerical influence; they are not physical shaft coordinates or predicted vibration amplitudes.

Relationship to the single-deflection calculator

When only one station has positive weight and deflection, the same weight cancels from numerator and denominator:

Calculator #025 therefore provides an exact algebraic cross-check when it receives the Rayleigh equivalent deflection. The related-action link transfers δeq, operating RPM, and the user-selected review band without duplicating the formula in the page layer.

Understanding the review band

The orange SVG band is generated from the percentage entered by the user. It is not a protected zone, prohibited zone, standard-mandated margin, or response curve. It only makes proximity visible and triggers an engineering warning when operating RPM lies inside the selected range.

Operating above the first estimate produces a separate warning. Supercritical operation may be intentional, but it requires evaluation of excitation orders, damping, unbalance response, acceleration through resonances, stress, clearances, bearing and seal behavior, vibration limits, controls, coast-down, and every relevant mode.

Engineering scope and limitations

This calculator handles exactly four input slots and up to four active lumped rotating weights. It excludes:

• generation or verification of the gravity static-deflection field;
• distributed shaft mass unless independently discretized and justified;
• bearing, seal, housing, pedestal, and foundation properties not already embedded in the static deflections;
• overhung, stepped, branched, coupled, anisotropic, cracked, or nonlinear rotor behavior unless the input deflection model validly represents it;
• higher bending modes, gyroscopic splitting, speed-dependent coefficients, cross-coupled forces, fluid-induced instability, rubs, and nonlinear clearances;
• unbalance-response amplitude, resonance stress, orbit, phase, amplification, damping ratio, run-up or coast-down response, and protection settings;
• universal separation criteria, permissible operating ranges, pass/fail classification, standards compliance, or engineering approval.

Use an applicable transfer-matrix, finite-element, modal, or full rotordynamic analysis when the rotor-support system cannot be represented by a verified first-mode gravity deflection shape. Confirm results with qualified engineering review and suitable test or operational evidence before releasing a design or operating through a predicted critical speed.