Waterfall Spectrum

A waterfall spectrum is a vibration display made from many spectra collected across speed, time or operating condition. Each slice is a spectrum. The stack shows how spectral amplitude changes as the machine runs up, coasts down, changes load or passes through an operating transient.

For a speed-indexed waterfall, the displayed quantity can be written as:

where A is spectral amplitude, f is frequency and n is rotational speed. For a time-indexed waterfall:

where t is measurement time. The amplitude may be plotted as height, colour, contour level or trace intensity. The amplitude unit must be stated, for example acceleration, velocity in \text{mm/s RMS}, displacement, sound pressure or decibels.

Engineering Role

Waterfall spectra help engineers interpret vibration that changes with speed or load. They are common in rotating machinery commissioning, fan and pump troubleshooting, marine propulsion trials, drivetrain testing, gearbox diagnostics, turbine run-up reviews and structural resonance checks.

The display is useful because different phenomena leave different traces:

• a synchronous order moves in frequency as shaft speed changes;
• a structural resonance often appears as a high-amplitude ridge when an order crosses a natural frequency;
• electrical or environmental content may remain fixed in frequency;
• broadband noise growth can reveal flow instability, cavitation, rubbing, looseness or measurement problems;
• a transient event may appear only during acceleration, deceleration or a load step.

A waterfall spectrum is therefore an evidence display, not a fault diagnosis by itself. The interpretation still depends on sensor location, speed reference, load state, phase, direction, machine geometry, signal quality and repeatability.

Acquisition and Speed Reference

A useful waterfall starts with a clear indexing variable. If the display is speed-indexed, each spectrum must be tied to a measured shaft speed rather than an assumed motor command. The rotational frequency is:

and the order of a frequency component is:

That conversion is only as trustworthy as the speed reference. A tachometer, encoder or once-per-revolution pulse should be checked for missing pulses, electrical noise, wrong edge selection, torsional speed fluctuation and synchronisation delay relative to the vibration channels. If speed is interpolated between pulses, the interpolation method should be compatible with the ramp rate.

FFT settings also control what the waterfall can prove. Frequency spacing is:

where f_s is the sampling frequency and N is the FFT block length. A long block gives finer frequency spacing but averages over more speed change during a run-up or coast-down. The speed change during one block can be estimated as:

where \dot{n} is ramp rate and T_{block} is block duration. If \Delta n_{block} is too large, a narrow resonance or order ridge may smear, shift or split across slices. A defensible test therefore records ramp rate, block length, window, overlap, averaging and slice spacing, then confirms that the visible ridges persist under reasonable processing changes.

For acceptance work, the waterfall should not be the only evidence. Engineers usually pair it with order tracking, phase, dwell-point spectra, orbit plots, operating deflection shapes or repeat run-up and coast-down comparisons before assigning a root cause.

Worked Example: Reading a Run-Up Waterfall

A variable-speed fan is measured during a controlled run-up. The analyst extracts the 1x peak from three waterfall slices:

Speed	Rotational frequency	1x frequency	1x velocity amplitude
1200\ \text{rpm}	20\ \text{Hz}	20\ \text{Hz}	2.0\ \text{mm/s RMS}
1500\ \text{rpm}	25\ \text{Hz}	25\ \text{Hz}	6.8\ \text{mm/s RMS}
1800\ \text{rpm}	30\ \text{Hz}	30\ \text{Hz}	3.1\ \text{mm/s RMS}

The rotational frequency is:

At 1200\ \text{rpm}:

At 1500\ \text{rpm}:

At 1800\ \text{rpm}:

The 1x line passes through a strong response near 25\ \text{Hz}, where the amplitude rises to 6.8\ \text{mm/s RMS}. If the site action level is 7.1\ \text{mm/s RMS}, the peak is below the action limit but close enough to require review:

The peak is about 95.8\% of the action level. An engineer would not release the machine on that number alone. The next checks are repeatability, ramp rate, operating dwell time, phase trend, sensor mounting, possible structural natural frequency near 25\ \text{Hz} and whether the machine normally operates close to 1500\ \text{rpm}.

Now suppose the same waterfall also contains a 60\ \text{Hz} peak at roughly constant amplitude. Its order changes with speed:

At 1200\ \text{rpm}:

At 1500\ \text{rpm}:

At 1800\ \text{rpm}:

Engineering comment: the 60 Hz feature is not locked to one shaft order. It may be electrical, environmental, structural or from another machine. The waterfall view prevents the analyst from treating every peak near 60 Hz as the same rotating-machine fault.

Distinction from Related Terms

Waterfall spectrum is not order tracking. Order tracking organizes response by shaft order and needs a speed or phase reference. A waterfall may be frequency-based, order-based or both, depending on the acquisition and processing.

Waterfall spectrum is not a Campbell diagram. A Campbell diagram maps modal branches and excitation order lines against speed. A waterfall spectrum shows measured amplitude slices, often used as evidence to compare with a Campbell diagram.

Waterfall spectrum is not a single FFT. A single FFT describes one record or one operating point. A waterfall shows how many spectra evolve across speed, time or load.

Waterfall spectrum is not proof of a mode shape, root cause or acceptance decision. It shows where amplitude changes occur; causal interpretation needs corroborating evidence.

Validation and Common Mistakes

A defensible waterfall spectrum states the sensor type, location, mounting, calibration, amplitude unit, sampling rate, anti-alias filtering, window, averaging, overlap, record length, speed or time increment, speed reference source, ramp rate and operating state.

Common mistakes include:

• comparing waterfalls with different colour scales, amplitude units or window settings;
• using a waterfall without enough speed resolution to see a narrow resonance;
• ignoring aliasing, clipping, poor sensor mounting or low signal-to-noise ratio;
• treating a fixed-frequency component as a shaft-synchronous order;
• missing a tachometer or encoder reference when the speed axis is required;
• diagnosing a fault from a bright ridge without checking phase, repeatability, load state and machine geometry;
• comparing run-up and coast-down waterfalls without noting thermal state, load, control mode and dwell time.