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

Governing standard: API 684 / ISO 1940-1· API 684:2005 (critical-speed separation margins) · ISO 1940-1:2003 (balance quality grades G0.4–G40) · multi-DOF Euler-Bernoulli beam-FEM eigensolver

How ISO 1940-1 works — the method explained

The MechanixCalc rotor dynamics calculator predicts the lateral critical speeds, Campbell diagram, and unbalance response of multi-segment stepped shafts to API 684 and ISO 1940-1. A multi-DOF Euler-Bernoulli beam finite-element model solves the governing eigenvalue problem directly, capturing gyroscopic splitting of forward and backward whirl modes and the influence of discrete bearing stiffnesses at each support. Enter the shaft geometry, rotor disc mass and position, bearing stiffnesses, and operating speed range, and the tool returns mode frequencies, a Campbell intersection map, an unbalance response curve, and a bearing orbit plot in a single pass.

It is built for machinery engineers sizing high-speed spindles, compressor rotors, pump shafts, and turbine stages who need a defensible separation-margin check to API 684 — or who need to specify a balance tolerance to ISO 1940-1 before sending a rotor to a balancing shop. The integrated ISO 1940 panel converts a residual unbalance directly to a permissible eccentricity and correction mass at any radius, with a grade chart for G0.4 through G40.

What this calculator does

  • Lateral critical speeds via multi-DOF Euler-Bernoulli beam-FEM eigensolver (validated against ROSS reference data)
  • Gyroscopic splitting into forward and backward whirl branches
  • Campbell diagram with 1×, 2×, and 3× engine-order excitation lines and API 684 separation-margin check
  • Unbalance response amplitude and phase vs. speed (SDOF magnification on modal mass)
  • Bearing orbit plot with keyphasor and clearance circle
  • ISO 1940-1 balance quality grade (G0.4 to G40) — permissible eccentricity, unbalance, and correction mass
  • Branded PDF engineering report with the full method and governing standards cited

Method & formulas

Lateral critical speeds — Euler-Bernoulli beam FEM

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 Euler-Bernoulli four-DOF beam elements and M is the consistent mass matrix. Bearing supports are modelled as 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 eigensolver uses inverse power iteration with modal deflation to extract the lowest modes sequentially — the same approach used by ROSS and Code_Aster for rotordynamic problems.

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.

Generalised eigenvalue problem (free vibration)
K · φ = ω² · M · φ

where K = global lateral stiffness matrix (beam elements + bearing springs); M = global consistent mass matrix (beam + lumped discs); ω = undamped natural frequency (rad/s); φ = mode shape vector

Gyroscopic critical speeds (forward / backward whirl)
N_cFW = (60·f_n) / (1 − γ) [rpm] ; N_cBW = (60·f_n) / (1 + γ) [rpm]

where f_n = undamped natural frequency (Hz); γ = gyroscopic ratio from modal solution; N_cFW = forward-whirl critical speed; N_cBW = backward-whirl critical speed

Campbell diagram and API 684 separation margin

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 requires the first critical speed to have a separation margin of at least 15–20 % from the maximum continuous operating speed (MCOS); the calculator flags the margin for each crossing so the engineer can evaluate compliance at a glance.

API 684 separation margin (first critical)
SM (%) = |N_c1 − N_op| / N_c1 × 100

where N_c1 = first lateral critical speed (rpm); N_op = maximum continuous operating speed (rpm); SM ≥ 15–20 % required by API 684 §2.7

ISO 1940-1 balance quality grade

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.

ISO 1940-1 balance quality grade
G = e · ω [mm/s]

where G = balance quality grade (mm/s); e = specific eccentricity = U / M (mm, where U = residual unbalance in g·mm and M = rotor mass in kg); ω = angular velocity at max operating speed (rad/s). Grade boundaries: G0.4, G1, G2.5, G6.3, G16, G40.

Worked example

Determine the ISO 1940-1 balance quality grade for a pump rotor operating at 6000 RPM, with a rotor mass of 10 kg and a residual unbalance of 1.0 g at an eccentricity radius of 50 mm. Then find the permissible eccentricity and correction mass required to achieve G2.5.

Given

  • Rotor mass M10 kg
  • Residual unbalance mass m_u1.0 g
  • Eccentricity radius r50 mm
  • Maximum operating speed N6000 RPM
  • Target gradeG2.5
  • Correction plane radius r_c50 mm

Result

  • Achieved gradeG6.3 (3.14 mm/s)
  • Permissible eccentricity for G2.53.98 µm
  • Permissible unbalance for G2.539.8 g·mm
  • Correction mass at 50 mm radius0.80 g
  1. Compute the angular velocity: ω = 2π × N / 60 = 2π × 6000 / 60 = 200π ≈ 628.3 rad/s.
  2. Compute the residual unbalance: U = m_u × r = (1.0 / 1000) kg × (50 / 1000) m = 5.0 × 10⁻⁵ kg·m.
  3. Compute the specific eccentricity: e = U / M = 5.0 × 10⁻⁵ / 10 = 5.0 × 10⁻⁶ m.
  4. Compute the achieved quality grade: G = e × ω × 1000 = 5.0 × 10⁻⁶ × 628.3 × 1000 = 3.14 mm/s → falls in the G6.3 bracket (2.5 < 3.14 ≤ 6.3).
  5. To achieve G2.5: permissible eccentricity e_per = G / ω = 2.5 / 628.3 = 3.98 × 10⁻³ mm = 3.98 µm.
  6. Permissible unbalance: U_per = e_per × M = 3.98 × 10⁻³ mm × 10 kg = 39.8 g·mm.
  7. Required correction mass at r_c = 50 mm: m_corr = U_per / r_c = 39.8 / 50 = 0.80 g.

This is an illustrative example. The achieved grade G6.3 exceeds the G2.5 target, so 0.80 g of material must be removed or added at 50 mm radius to bring the rotor to G2.5. The calculator also checks API 684 separation margins and generates the full Campbell diagram from your actual shaft geometry and bearing stiffnesses.

Frequently asked questions

Which standard does this rotor dynamics calculator use?

Critical-speed separation margins are checked to API 684:2005, which requires at least 15–20 % margin between the first lateral critical speed and the maximum continuous operating speed. Balance quality grades are calculated to ISO 1940-1:2003 (grades G0.4 through G40). The natural frequencies are computed with a multi-DOF Euler-Bernoulli beam finite-element model that has been validated against ROSS reference data.

What is a Campbell diagram and why does it matter?

A Campbell diagram plots the rotor's natural frequencies against running speed, overlaid with engine-order excitation lines (1×, 2×, 3× running speed). Where an excitation line crosses a natural-frequency branch, the rotor will resonate. The diagram lets an engineer see all potential resonance crossings across the full speed range at once, so operating ranges or rotor stiffnesses can be adjusted to keep crossings outside normal working speeds.

What is the difference between forward whirl and backward whirl?

Gyroscopic coupling caused by rotor disc inertia splits each undamped natural frequency into two branches. The forward whirl (FW) mode — where the shaft orbits in the same direction as rotation — rises in frequency with speed. The backward whirl (BW) mode — where the orbit is counter-rotational — falls. Both can be excited by unbalance and structural asymmetry, so the Campbell diagram shows both branches.

How do I use the ISO 1940 balance grade results?

Enter the residual unbalance mass and its radius from the rotor centre. The calculator computes the achieved quality grade G (in mm/s) and compares it against all standard grades. Pick the target grade for your application (e.g. G2.5 for turbomachinery or G6.3 for general industrial machinery), and the calculator shows the permissible eccentricity, permissible unbalance in g·mm, and the correction mass needed at your chosen balance-plane radius.

Is the rotor dynamics calculator free?

You can use it during a free 30-minute preview with no sign-up, and a free 14-day account trial unlocks every calculator with no credit card required. The branded PDF engineering report and saved calculations are part of a paid plan.

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