Campbell Diagram Simulator — Rotor Critical Speeds

Steam and gas turbine design: In thermal, nuclear and geothermal steam turbines, in jet engines and in industrial gas turbines, several critical speeds line up above and below the operating range (typically 3000 to 3600 RPM for utility turbines, tens of thousands of RPM for jet engines). Drawing a Campbell diagram early in the design and securing a 15 percent or larger separation margin (per API 612) is standard practice. Blade-passing frequencies (NPF) are added as additional excitation lines so blade-vibration interference is also evaluated.

Turbomachinery (compressors and pumps): In centrifugal and axial compressors, and in large boiler feed-water pumps, multiple impeller stages on a long shaft push natural frequencies close together. Because the operating speed has to pass through several critical speeds, the start-up and shutdown sequence is planned on the Campbell diagram so each critical speed is traversed quickly. Squeeze-film dampers and magnetic bearings are also chosen here when extra damping is needed.

Automotive engines and drivetrains: Torsional vibration of crankshafts and propeller shafts uses the same framework. The horizontal axis becomes engine speed and the vertical axis is the torsional natural frequency; for a 4-cylinder engine the dominant excitation is 2x (firing order), for a V8 it is 4x. Torsional vibration dampers — the rubber inside flywheels or dual-mass arrangements — are sized from this diagram to suppress amplitudes within the critical range.

Wind turbines and condition monitoring: In a large wind-turbine rotor and nacelle, the blade edgewise and flapwise modes, the tower bending mode, and the generator-shaft torsional mode all interact. Because the rotor speed varies, the Campbell diagram is used to identify "permanent resonance bands" and the variable-speed control is programmed to avoid sitting at dangerous speeds. Vibration data from condition-monitoring systems (CMS) is overlaid on the Campbell plot to detect new excitations or natural-frequency shifts early.

The most common misconception is to assume "there is one critical speed". Textbook examples of the Jeffcott rotor really do produce only one critical speed, but only because they consider one mode times one excitation. Real rotors have several modes, and orders other than 1x — such as 2x, 3x, and blade-passing frequencies — are often dominant, so several critical speeds always appear above and below the operating range. The twelve red dots produced by this simulator (two whirl branches times two modes times three orders) reproduce that situation. The correct workflow is to check every critical speed near the operating range, not only the single nearest one.

The next pitfall is to design while ignoring the gyroscopic effect. Setting g = 0 in the simulator turns the natural-frequency curves into horizontal lines and reduces the critical speed to $N_\text{crit}=60 f_n / k$. Real disks and impellers always have a gyroscopic effect, so the correct expression is $N_\text{crit}=60 f_n/(k-g)$. For example, with f_n = 80 Hz, k = 1, and g = 0.1, ignoring the gyroscope predicts 4800 RPM but the real value is 5333 RPM — a 533 RPM gap. This is one of the most common reasons predictions disagree with measurements, so the gyroscopic effect must always be included in the analysis.

Finally, beware of the oversimplification "if the operating speed is far from any critical speed, all is well". The Campbell diagram is a linear, viscously damped model. In real machines, non-linear and non-conservative forces such as seal forces, fluid forces, and internal friction can excite backward whirl or sub-critical modes. The critical speeds shown by this tool correspond to "synchronous excitation of the forward whirl", which alone is not enough to declare a design safe. Detailed designs require a full eigenvalue analysis (complex modal analysis and stability analysis), and this simulator should be used as a tool for concept-stage understanding and initial scoping.