Definition: What is a Harmonic?

In vibration analysis, a harmonic is a frequency that is an exact integer multiple of a fundamental frequency. In rotating machinery, the fundamental frequency is typically the shaft rotational speed, referred to as the 1st harmonic or 1×. The subsequent harmonics are integer multiples: 2× (twice shaft speed), 3× (three times), and so on. These frequencies are also called orders of running speed, or synchronous harmonics because they are precisely synchronized with shaft rotation.

For example, if a motor operates at 1,800 RPM (30 Hz), its harmonics appear at 60 Hz (2×), 90 Hz (3×), 120 Hz (4×), 150 Hz (5×), and so forth. The harmonic series is theoretically infinite, but in practice, amplitude decreases at higher orders and only the first several harmonics carry diagnostic information.

Harmonic Frequency Definition
fn = n × f1 = n × (RPM / 60)
where n = 1, 2, 3, 4… (harmonic order) and f₁ = shaft rotational frequency in Hz
Harmonics vs. Sub-Harmonics vs. Non-Synchronous Peaks

Harmonics are integer multiples of shaft speed (2×, 3×, 4×…). Sub-harmonics are fractional multiples (½×, ⅓×, ¼×) and always indicate severe mechanical problems. Non-synchronous peaks are frequencies unrelated to shaft speed — such as bearing fault frequencies, gear mesh frequencies, line frequency (50/60 Hz), or natural frequencies — and require different diagnostic approaches. A peak at 3.57× RPM is NOT a harmonic; it is likely a bearing fault frequency.

Why Are Harmonics Generated?

In a perfectly linear system excited by a pure sinusoidal force (such as a perfectly balanced, perfectly aligned rotor in perfect bearings), only the 1× fundamental would appear. Real machinery is never perfectly linear. Harmonics appear whenever the vibration waveform is distorted from a pure sine wave — whenever the system response is non-linear or the forcing function itself is non-sinusoidal.

The Mathematics: Fourier's Theorem

Fourier's theorem states that any periodic waveform — no matter how complex — can be decomposed into a sum of sine waves at the fundamental frequency and its integer multiples, each with a specific amplitude and phase. The FFT (Fast Fourier Transform) algorithm used by vibration analyzers performs this decomposition computationally, revealing the harmonic content of the signal.

A pure sine wave has only a single frequency component. A square wave contains all odd harmonics (1×, 3×, 5×, 7×…) with amplitudes decreasing as 1/n. A sawtooth wave contains all harmonics with amplitudes decreasing as 1/n. The specific shape of the distortion determines which harmonics appear — this is what makes harmonic analysis so diagnostically powerful.

Physical Mechanisms That Generate Harmonics

  • Waveform clipping / truncation: When shaft movement is physically constrained (bearing housing, rub contact), the resulting waveform is clipped, generating harmonics. More severe clipping produces more harmonics.
  • Asymmetric stiffness: If system stiffness differs between positive and negative halves of the vibration cycle (cracked shaft opening/closing, misalignment creating different tension/compression stiffness), even harmonics (2×, 4×, 6×) are generated.
  • Impact events: Periodic impacts (loose bolts, bearing defect impacts) create sharp, short-duration waveforms that are extremely rich in harmonic content — like how a drum stick produces many overtones.
  • Non-linear restoring forces: When stiffness changes with displacement (bearings under varying load, progressive-rate rubber mounts), the response to a sinusoidal force contains harmonics.
  • Parametric excitation: When system properties vary periodically at a frequency related to shaft speed, they can generate harmonics and sub-harmonics of the excitation frequency.
The Key Diagnostic Principle

The pattern of which harmonics are present, their relative amplitudes, and which are absent tells the analyst what physical mechanism generates the non-linearity. Experienced analysts examine the complete harmonic structure of the spectrum — not just overall vibration level — to identify specific fault mechanisms.

Detailed Fault Signatures — Harmonic Patterns

1× Dominant — Unbalance

A dominant peak at 1× with minimal higher harmonics is the classic signature of mass unbalance. The unbalance force is inherently sinusoidal (it rotates with the shaft at 1× frequency), producing a clean single peak in the frequency domain.

Diagnostic Details

  • Amplitude: Proportional to speed² (double speed → 4× amplitude) and proportional to unbalance mass
  • Phase: Stable, repeatable, single-valued. Changes predictably with trial weight addition — this is the foundation of all balancing procedures
  • Direction: Primarily radial; axial 1× is low unless rotor has significant overhang
  • Confirmation: Response to trial weights confirms unbalance. If 1× doesn't respond to trial weights, consider bent shaft, eccentricity, or resonance
Not All 1× Vibration is Unbalance

Several conditions produce high 1× that is NOT correctable by balancing: bent shaft, shaft eccentricity, electrical runout on proximity probes, rotor bow from thermal effects, coupling eccentricity, and resonance amplification. Always verify the diagnosis before attempting to balance.

2× Dominant — Misalignment

A strong 2nd harmonic, often comparable in amplitude to or exceeding the 1× peak, is the primary indicator of shaft misalignment. Misalignment forces the shaft through a non-sinusoidal path during each revolution, creating the distortion that generates 2× and sometimes higher harmonics.

Angular vs. Parallel Misalignment

  • Angular misalignment: Shaft centerlines intersect at an angle at the coupling. Produces high 1× axial vibration. Phase across coupling shows ~180° shift in the axial direction.
  • Parallel (offset) misalignment: Shaft centerlines are parallel but offset. Produces high 2× radial vibration, often with 2× ≥ 1×. Severe cases generate 3× and 4×. Radial phase across coupling shows ~180° shift.
  • Combined: In practice, both usually coexist, producing a mix of the signatures.

The 2×/1× Ratio as a Diagnostic Indicator

2×/1× RatioLikely ConditionAction
< 0.25Normal; 2× present at low level in most machinesNo action required
0.25 – 0.50Mild misalignment possible; normal for some coupling typesCheck alignment; compare with baseline
0.50 – 1.00Significant misalignment probablePerform precision laser alignment
> 1.00Severe misalignment; 2× exceeds 1×Urgent — realign; check coupling and pipe strain

Multiple Harmonics — Mechanical Looseness

A rich series of running speed harmonics (1×, 2×, 3×, 4×, 5×… to 10× or more) indicates mechanical looseness. The impacts, rattling, and non-linear contact/separation cycles generate extreme waveform distortion that decomposes into many harmonic components.

Three Types of Looseness

  • Type A — Structural: Loose machine-to-foundation connection (soft foot, cracked base, loose anchor bolts). Produces directional 1× (higher in the loose direction). Key test: tighten/loosen individual bolts while monitoring 1× amplitude.
  • Type B — Component: Loose bearing liner in cap, loose cap on housing, excessive bearing clearance. Produces a family of harmonics, often with sub-harmonics (½×). Sub-harmonics are the key differentiator from misalignment.
  • Type C — Bearing seat: Loose impeller on shaft, loose coupling hub, excessive bearing clearance allowing rotor to bounce. Produces many harmonics with broadband noise floor elevation.
Sub-Harmonics: The Looseness Fingerprint

The presence of sub-harmonics (½×, ⅓×) is the most reliable differentiator between looseness and misalignment. Misalignment generates 2× and 3× but rarely produces sub-harmonics. Looseness (Type B and C) characteristically generates ½× because the rotor contacts one side of the bearing on one half-revolution and bounces to the other on the next — creating a pattern that repeats every two revolutions, hence ½×.

Other Harmonic-Generating Conditions

Bent Shaft

Produces both 1× and 2× vibration with high axial component. Unlike misalignment, a bent shaft shows 1× that cannot be corrected by balancing (geometric eccentricity, not mass distribution) and ~180° axial phase difference between shaft ends. The 2× comes from asymmetric stiffness as the bend opens and closes during rotation.

Reciprocating Machinery

Engines, compressors, and reciprocating machines inherently generate rich harmonic spectra because piston/crankshaft motion is fundamentally non-sinusoidal. The harmonic pattern depends on cylinder count, firing order, and stroke type (2-stroke vs. 4-stroke).

Rotor Rub

A partial rub (contact for a portion of each revolution) produces many high-order harmonics — sometimes to 10×, 20×, or more. A full annular rub (continuous 360° contact) generates dominant sub-harmonics (½×, ⅓×, ¼×) through reverse precession mechanisms.

Electrical Issues in Motors

AC motors generate vibration at multiples of line frequency (50 or 60 Hz) independently of shaft speed. The most common is 2× line frequency (100 Hz in 50 Hz systems, 120 Hz in 60 Hz systems). This is NOT a harmonic of shaft speed — it's a harmonic of line frequency, which is the key to distinguishing electrical from mechanical vibration. The power cut test is definitive: electrical vibration drops instantly when power is removed, mechanical vibration persists during coast-down.

Rotor bar defects produce sidebands around 1× spaced at the pole pass frequency (slip frequency × number of poles). These sidebands are very close to 1× (within 1–5 Hz), requiring high-resolution zoom FFT analysis to resolve.

Non-Synchronous Frequencies — Not True Harmonics

Several important frequencies are sometimes confused with harmonics but are actually independent of shaft speed:

Frequency TypeFormulaRelationship to RPMNotes
Bearing fault frequenciesBPFO, BPFI, BSF, FTFNon-integer multiples (e.g. 3.57×, 5.43×)Always non-synchronous; depend on bearing geometry
Gear mesh frequencyGMF = #teeth × RPMInteger but very high orderTechnically a harmonic but analyzed separately
Blade/vane passBPF = #blades × RPMInteger multipleNormal; excessive amplitude indicates problem
Line frequencyFL = 50 or 60 HzNot related to RPMElectrical; disappears on power cut
Natural frequenciesfn = √(k/m)/2πFixed; not related to RPMConstant frequency regardless of speed changes
Belt frequenciesfbelt = RPM×π×D/LSub-synchronous (< shaft speed)Belt frequency and its harmonics 2×, 3×, 4× BF

Analysis Guide — How to Interpret Harmonic Patterns

Step 1: Identify the Fundamental (1×)

Locate the 1× peak corresponding to shaft rotational speed. Verify using a tachometer or motor nameplate. In variable-speed machines, 1× must be identified precisely for each measurement.

Step 2: Catalog All Peaks

For each significant peak, determine: is it an exact integer multiple of 1× (true harmonic)? A fractional multiple (sub-harmonic)? Unrelated to shaft speed (non-synchronous)? Use analyzer harmonic cursor features for efficiency.

Step 3: Examine the Amplitude Pattern

  • Which harmonic is dominant? → Points to specific fault
  • How many harmonics are present? → More = more severe distortion
  • Does 2× exceed 1×? → Likely misalignment
  • Are sub-harmonics present? → Looseness, rub, or oil whirl
  • Is amplitude decreasing with order (1/n decay)? → Typical for looseness

Step 4: Check Directionality

  • High radial, low axial: Unbalance or looseness
  • High axial: Misalignment (especially angular) or bent shaft
  • Directional radial: Structural looseness (higher in loose direction)

Step 5: Trend Over Time

  • Are harmonic amplitudes increasing? → Fault is progressing
  • Are new harmonics appearing? → New fault mechanism developing
  • Is the noise floor rising? → General wear or late-stage failure

Step 6: Correlate with Phase Data

  • Unbalance: 1× phase is stable and repeatable
  • Misalignment: 1× or 2× phase shows ~180° across coupling
  • Looseness: Phase is unstable, may shift randomly between measurements

Case Studies — Real-World Harmonic Analysis

Case 1: Motor-Pump — Is it Unbalance or Misalignment?

Machine: 30 kW motor driving centrifugal pump at 2960 RPM via flexible coupling. Overall vibration: 6.2 mm/s at motor drive-end bearing.

Spectrum: 1× = 4.1 mm/s, 2× = 3.8 mm/s, 3× = 1.2 mm/s. The 2×/1× ratio = 0.93.

Direction: High radial 2× at both drive-end bearings. Axial 1× at coupling: motor = 2.8 mm/s, pump = 3.1 mm/s with 165° phase difference.

Diagnosis: Combined angular and parallel misalignment. The 2×/1× ratio approaching 1.0, high axial readings, and ~180° phase across coupling all confirm. NOT unbalance — even though 1× is elevated, the 2× pattern is the real story.

Action: Laser alignment performed. Post-alignment: 1× = 0.8 mm/s, 2× = 0.3 mm/s. Overall dropped to 1.1 mm/s — an 82% reduction.

Case 2: Fan — Why Doesn't Balancing Work?

Machine: Centrifugal fan at 1480 RPM. Vibration: 8.5 mm/s. Previous balancing attempt reduced 1× but overall vibration remained high.

Spectrum: 1× = 2.1 mm/s (low after balancing), ½× = 1.8 mm/s, 2× = 3.2 mm/s, 3× = 2.5 mm/s, 4× = 1.8 mm/s, 5× = 1.1 mm/s, 6× = 0.7 mm/s.

Diagnosis: Mechanical looseness (Type B). The harmonic family with ½× sub-harmonic is the signature. Balancing corrected 1× but couldn't address the looseness-generated harmonics that dominate overall vibration.

Action: Inspection revealed bearing housing 0.08 mm loose in pedestal bore. Housing rebored and new bearing fitted. Post-repair: all harmonics dropped to baseline. Overall: 1.4 mm/s.

Case 3: Compressor Motor — Electrical or Mechanical?

Machine: 4-pole, 50 Hz induction motor at 1485 RPM driving a screw compressor. Vibration increased from 2.0 to 5.5 mm/s over 3 months.

Spectrum: Dominant peak at 100 Hz (= 2FL). Also: 1× at 24.75 Hz = 1.2 mm/s, sidebands around 1× at ±1.0 Hz spacing.

Key Test: Power cut — the 100 Hz peak dropped to zero within one revolution. The 1× sidebands persisted during coast-down.

Diagnosis: Two problems: (1) Electrical — stator eccentricity causing 2FL. (2) Mechanical — 1× sidebands at ±1.0 Hz (= pole pass frequency for 4-pole motor with 1.0% slip) suggest developing rotor bar defect.

Action: Motor sent for rewind. Confirmed: 2 broken rotor bars + stator eccentricity from base sag. After rewind and shimming: vibration 1.6 mm/s.

Vibromera Equipment for Harmonic Analysis

The Balanset-1A and Balanset-4 provide real-time FFT spectrum analysis with harmonic cursor tracking, enabling field identification of 1×, 2×, 3× patterns and fault diagnosis. The devices combine vibration analysis for diagnostics and precision balancing for correction — identifying the problem and fixing it with one instrument.

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