Rotor Dynamics Calculator — Critical Speeds, Campbell Diagram & ISO 1940 Balance (API 684 / ISO 1940-1)

Critical speeds are found by solving the generalised eigenvalue problem K·φ = ω²·M·φ for the shaft, where K is the global lateral stiffness matrix assembled from four-DOF beam elements and M is the consistent mass matrix. Shear deformation and the rotary inertia of the cross-section are both included (Timoshenko), which matters for the stubby shaft sections typical of machine-tool spindles — neglecting them over-predicts the critical speed, by 0.3 % at L/d = 20 but 31 % at L/d = 1.6. Bearing supports are modelled as undamped linear springs at their axial positions, added to the diagonal of K. Discrete rotor discs contribute a lumped mass to M and, through their polar inertia, to the gyroscopic coupling matrix G. The symmetric eigenproblem is solved by Cholesky transformation followed by a cyclic Jacobi rotation sweep.

Gyroscopic coupling splits each undamped natural frequency into a forward whirl (FW) branch that rises with spin speed and a backward whirl (BW) branch that falls. The gyroscopic ratio γ = (φᵀ G φ) / (2·φᵀ M φ) is computed from the FEM modal solution and applied analytically so that the Campbell diagram correctly separates FW and BW crossing points — a step that is critical for turbomachinery separation-margin compliance.

The Campbell diagram plots the rotor natural frequencies (FW and BW branches) against running speed, together with integer multiples of the running-speed excitation (1×, 2×, 3× engine orders). Where an excitation line crosses a natural-frequency branch, a resonance crossing occurs. API 684 §2.8.2.5 sets the required separation margin from the amplification factor rather than as a flat percentage: below AF 2.5 the response is critically damped and no margin is required; between 2.5 and 3.55 the requirement is 15 % above the maximum continuous speed and 5 % below the minimum operating speed; above 3.55 it rises towards 26 % and 16 % respectively. The calculator screens every computed critical against that requirement, in the standard's own denominator (a percentage of the operating speed).

This screening uses an undamped eigen-solution with an assumed modal damping ratio. API 684 §1.4.2 is explicit that an undamped critical-speed analysis should not be used to establish critical speeds and separation margins for compliance, because it omits bearing and seal damping and cross-coupling, which move the criticals — a damped unbalance-response analysis is required for that. Treat the result as a design screen, not as a compliance demonstration.

ISO 1940-1 defines balance quality grades G (in mm/s) as the product of the permissible residual eccentricity e_per and the maximum angular velocity ω. Rearranging, the permissible eccentricity for a chosen grade is e_per = G / ω, from which the permissible unbalance U_per = e_per × M (rotor mass) and the required correction mass at radius r follows as m_corr = U_per / r. The calculator evaluates this for every standard grade from G0.4 (precision spindles and gyroscopes) to G40 (assembled crankshafts) and highlights the achieved grade from the entered residual unbalance.