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

The natural frequencies of the non-rotating shaft come from the generalised eigenvalue problem K·φ = ω²·M·φ, where K is the global lateral stiffness matrix assembled from four-DOF beam elements and M is the consistent mass matrix; the critical speeds themselves add the gyroscopic coupling of the spinning rotor, described below. 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 36 % at L/d = 1.6. Bearing supports are modelled as linear springs at their axial positions, added to the diagonal of K; the critical speeds themselves come from the undamped system. Bearing damping can optionally be supplied, in which case the damped system is also solved and each mode's damping ratio and logarithmic decrement are reported — computed without seal damping or destabilising cross-coupled stiffness, so the log decrement is an upper bound on the true value and not an API 684 stability result. Discrete rotor discs contribute a lumped mass to M and, through their polar inertia, to the gyroscopic coupling matrix G. The non-rotating 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 calculator solves this exactly rather than estimating it: with P the polar-inertia matrix (each disc's polar inertia plus the shaft's own distributed polar inertia), the forward and backward synchronous critical speeds are the roots of det(K − Ω²(M − P)) = 0 and det(K − Ω²(M + P)) = 0, and every point on the Campbell diagram solves (K + ωΩP − ω²M)·φ = 0 at that speed, so on supports of equal stiffness the diagram crosses the running-speed line exactly at the listed critical speeds (the diagram draws the lowest two forward and backward branches; further roots inside the running range are listed and screened). The common shortcut — scaling each natural frequency by a single gyroscopic ratio γ — is accurate for a mode in which the disc barely tilts, but it can misplace the second critical by tens of percent where a disc tilts strongly, and it wrongly predicts that a disc-dominated rotor has no forward critical at all: on a flexible shaft, gyroscopic stiffening locks the disc's tilt and the shaft bends instead, and that mode does cross the running-speed line. The shaft element is the shear-deformable (Timoshenko) one throughout — stiffness, translational mass, rotary and gyroscopic inertia at the same shear parameter — so very short, thick sections are modelled as consistently as slender ones. The critical speeds and whirl frequencies have been checked against ROSS, an open-source rotordynamics code, on stepped, stubby and disc-carrying solid shafts, including thick rolls: the first critical speeds agree with it to within about 0.1 % and higher ones to within about 1 %, the remainder being the calculator's fixed mesh, which reads slightly high on the upper modes of very short, thick rotors. With unequal horizontal and vertical support stiffness the two planes are coupled by the spinning rotor, so the critical speeds are solved for both planes together and every one of them is screened.

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 (2nd ed.) 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 critical speed an unbalance can excite against that requirement — the forward-whirl ones on supports of equal stiffness, and every one of them when the horizontal and vertical support stiffness differ — 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 (2nd ed.) 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. The unbalance response itself is computed differently: as the damped synchronous forced response of the whole rotor model — every mode, both transverse planes and the gyroscopic coupling — with the modal damping ratio applied as a uniform loss factor. It reports the largest orbit anywhere on the shaft and where it occurs, with the unbalance on the entered plane or, by default, at whichever station along the shaft gives the worst case. It is still not the API 684 unbalance-response analysis, which prescribes the unbalance amounts and placements and includes bearing and seal cross-coupling.

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.