Campbell Diagram

A Campbell diagram is a speed-frequency map for rotating machinery. The horizontal axis is usually running speed, often in rpm. The vertical axis is frequency, usually in Hz. Modal branches show how natural frequencies or damped rotor modes change with speed, while straight or curved excitation lines show running orders such as 1x, 2x, blade passing, vane passing or gear mesh.

For rotational speed n in rpm and excitation order m, the order-line frequency is:

A possible critical-speed crossing occurs when an order line intersects a mode branch:

The crossing is a candidate resonance, not an automatic failure point. The engineering decision still depends on damping, response amplitude, mode shape, forcing strength, dwell time, operating range, uncertainty and measured run-up or coast-down evidence.

Engineering Role

Campbell diagrams help engineers see whether a machine’s operating range intersects dangerous dynamic conditions. They are used for turbines, compressors, pumps, motors, generators, propeller shafts, gearboxes, aircraft engine rotors, flywheels and high-speed spindles.

In rotor dynamics, modal branches may split into forward and backward whirl modes because gyroscopic effects change the effective dynamics with speed. Bearing stiffness, damping, seal coefficients, support flexibility, thermal state and fluid loading can also move the branches. That is why a useful Campbell diagram should state the model configuration and operating condition.

A practical review overlays the intended operating speed range, transient run-up and coast-down paths, excitation orders, known mode branches, measured response peaks, acceptance limits and uncertainty bands. The diagram is a screening and communication tool; it does not replace unbalance response analysis, torsional response analysis or measured vibration acceptance.

Margins, Uncertainty and Evidence

A Campbell diagram becomes useful for decisions when each crossing is tied to a margin and an evidence plan. If a predicted crossing occurs at n_{crit} and a continuous operating speed is n_{op}, a simple speed separation margin is:

This margin is only a screen. A speed outside the normal operating point can still matter if the machine dwells there during start-up, shutdown, load rejection or control transients. If a restricted band extends from n_{low} to n_{high} and the ramp rate is \dot{n}, the exposure time during a run-up is:

A short exposure may be acceptable for a well-damped crossing, while a long dwell near a lightly damped mode can require avoidance logic, balancing, support changes, damping treatment or monitoring.

Uncertainty should be visible rather than hidden. Bearing stiffness, support flexibility, seal coefficients, thermal state, fluid loading, mass properties and mode identification can shift a branch. A practical diagram often plots a branch band:

where U_f is a defensible frequency uncertainty or correlation allowance. If an order line crosses the band, the crossing should be treated as a candidate until measured run-up, coast-down, waterfall spectrum, order tracking, phase and orbit evidence either confirm or dismiss the risk.

Whirl direction also matters. Forward and backward branches should be labelled consistently with the sign convention used by the rotor-dynamic model or test system. A forward-whirl crossing with a strong synchronous force can be more important than a plotted intersection that has weak modal participation at the measurement location.

Worked Example: Reading Order Crossings

A variable-speed compressor is reviewed from:

A simplified rotor model gives a first bending branch:

where f_A is in Hz and n is in rpm. The 1x order line is:

Set the branch equal to the order line:

At that speed:

Now check a second branch, representing a coupled support or torsional mode:

For a 2x order:

Set:

At that speed:

Engineering comment: both crossings lie inside the reviewed speed range, so neither should be ignored. The first crossing is a 1x candidate critical speed near 1532 rpm. The second is a 2x crossing near 2406 rpm. The next step is not simply to ban those speeds; it is to check response amplitude, damping, ramp rate, expected dwell time, balance condition, bearing loads and measured vibration evidence.

Distinction from Related Terms

Campbell diagram is not critical speed. Critical speed is a speed or crossing condition. The Campbell diagram is the plot that helps identify such crossings.

Campbell diagram is not a frequency response function. An FRF relates response to excitation frequency for a fixed configuration. A Campbell diagram maps modal branches and excitation orders across changing rotational speed.

Campbell diagram is not a waterfall spectrum. A waterfall spectrum shows measured spectral amplitude versus speed or time. A Campbell diagram may be compared with a waterfall plot, but it is primarily a modal and order-crossing map.

Campbell diagram is not proof of unacceptable vibration. A crossing with weak forcing or high damping may be acceptable; a crossing with strong forcing and low damping may require avoidance, redesign or monitoring.

Validation and Common Mistakes

A defensible Campbell diagram states the rotor model, bearing and support data, speed range, operating condition, mode identification, forward or backward whirl convention, excitation orders, measurement channels, speed reference, damping assumptions and uncertainty.

Common mistakes include:

• plotting only 1x and missing blade-passing, gear-mesh, vane-passing or electrical orders;
• treating every intersection as equally dangerous without checking forcing strength and damping;
• using stationary natural frequencies when gyroscopic effects shift the modes with speed;
• ignoring whether a mode is forward whirl, backward whirl, torsional or support dominated;
• comparing a predicted diagram with measured spectra without aligning speed reference and units;
• hiding uncertainty in bearing stiffness, seal coefficients or support flexibility;
• using the diagram as a pass/fail result without run-up, coast-down or operating vibration evidence.