11.1 Vibration Analysis, Dynamic Balancing & Thermography

Condition monitoring through vibration analysis, dynamic rotor balancing, and infrared thermography allows millwrights to assess machine health non-intrusively, detect mechanical degradation at an early stage, and execute precision corrections before catastrophic failure occurs.

Quick Answer: Vibration units are tailored to frequency ranges: displacement (mils peak-to-peak) for low-speed shaft orbit tracking (<10 Hz), velocity (in/sec or mm/s RMS) for general machine health (10–1000 Hz), and acceleration (g's RMS) for high-frequency gear and bearing defects (>1000 Hz). FFT spectrum analysis pinpoints fault frequencies such as 1× RPM unbalance, 2× RPM misalignment, and bearing defect frequencies (BPFO, BPFI, BSF, FTF).

Vibration Amplitude Metrics and Measurement Units

Industrial vibration is periodic oscillatory motion about a reference position. Selecting the correct amplitude metric depends directly on the forcing frequency of interest.

+---------------------------------------------------------------------------------------------------+
|                                VIBRATION METRIC SELECTION MATRIX                                  |
+-----------------------+-----------------------+-----------------------+---------------------------+
| Metric & Unit         | Primary Measurement   | Frequency Range       | Target Machine Faults     |
+-----------------------+-----------------------+-----------------------+---------------------------+
| Displacement          | Peak-to-Peak (p-p)    | Low Frequency         | Shaft sleeve wear,        |
| (mils or µm)          | Total physical distance| (< 10 Hz / < 600 CPM) | oil whirl, structural     |
|                       | of shaft displacement |                       | looseness, foundation motion|
+-----------------------+-----------------------+-----------------------+---------------------------+
| Velocity              | Root Mean Square (RMS)| Mid Frequency         | Machine unbalance,        |
| (in/sec or mm/s)      | Peak velocity / 1.414 | (10 Hz to 1000 Hz /   | angular/parallel          |
|                       | Direct fatigue energy | 600 to 60,000 CPM)    | misalignment, bent shaft  |
+-----------------------+-----------------------+-----------------------+---------------------------+
| Acceleration          | RMS or Peak           | High Frequency        | Rolling element bearing   |
| (g's, where 1g =      | Rate of change of     | (> 1000 Hz /          | defects, gear mesh        |
| 9.81 m/s²)            | velocity; forces (F=ma)| > 60,000 CPM)         | impacts, blade pass       |
+-----------------------+-----------------------+-----------------------+---------------------------+

Mathematical Relationship of Vibration Motion

For a single sinusoidal vibration frequency ω = 2π f:

• Displacement: $x(t) = D \sin(\omega t)$
• Velocity: $v(t) = \frac{dx}{dt} = D\omega \cos(\omega t) = V \cos(\omega t)$, where V = Dω
• Acceleration: $a(t) = \frac{dv}{dt} = -D\omega^2 \sin(\omega t) = A \sin(\omega t + \pi)$, where A = Dω² = Vω

Because velocity is proportional to frequency (ω) and acceleration is proportional to frequency squared (ω²), acceleration amplitude amplifies high-frequency signals, making it ideal for detecting micro-impacts in bearings long before displacement shows any change.

Vibration Transducers & Signal Sensing

Selecting the correct sensor ensures accurate signal capture without mechanical attenuation or signal distortion.

1. Accelerometers (Piezoelectric Sensors)

Piezoelectric accelerometers contain a quartz crystal or ceramic mass element that generates an electrical charge proportional to applied acceleration forces. Internally Amplified (IEPE/ICP) accelerometers integrate signal conditioning electronics requiring constant-current DC power.

• Frequency Range: Broad response from 1 Hz to 15,000 Hz.
• Mounting Methods: Threaded stud mounting yields maximum flat frequency response (up to 10 kHz). Rare-earth magnetic bases reduce usable range to ~5 kHz. Handheld probe tips limit accurate response to <1 kHz due to operator resonance.

2. Velocity Pickups (Seismic Sensors)

Velocity pickups utilize a spring-suspended coil moving through a permanent magnetic field, generating a self-powered voltage proportional to velocity.

• Limitations: Internal moving mechanical parts suffer physical wear over time. Heavy physical weight lowers sensor resonance, restricting application to low-to-mid frequency ranges (10 Hz to 1000 Hz).

3. Eddy Current Proximity Probes

Non-contact eddy current probes measure relative dynamic displacement between the stationary bearing housing and the rotating conductive shaft surface.

                    PROXIMITY PROBE MOUNTING IN JOURNAL BEARING

                             +-----------------------+
                             |  Probe Driver / BNC   |
                             +-----------+-----------+
                                         |
                                    Coaxial Cable
                                         |
                                  +------+------+
                                  | Probe Body  |
                                  +------+------+
                                         |
                  +----------------------v----------------------+
                  |           BEARING HOUSING WALL              |
                  |    +-----------------------------------+    |
                  |    |          X-Probe (45°)            |    |
                  |    |               \                   |    |
                  |    |  Y-Probe (45°) \   Target Gap     |    |
                  |    |           \     \  (0.040" DC)    |    |
                  |    |            \     v                |    |
                  |    |             v  +---+              |    |
                  |    |                |   | Shaft        |    |
                  |    |                +---+              |    |
                  +----+-----------------------------------+----+

• Operating Principle: High-frequency RF excitation creates an alternating magnetic field at the probe tip. Shaft proximity induces eddy currents in the shaft surface, suppressing RF amplitude. The probe driver conditions this into a DC gap voltage (measuring static shaft position) and an AC voltage (measuring dynamic vibration displacement).
• Orthogonal Mounting: Proximity probes are installed in orthogonal pairs (X and Y probes mounted 90° apart, typically at 45° left and right of vertical) to generate shaft centerline position plots and 2D shaft orbit Lissajous patterns.

Fast Fourier Transform (FFT) & Machine Fault Signatures

Fast Fourier Transform algorithms convert complex time-domain vibration waveforms x(t) into a frequency spectrum displaying discrete amplitude peaks versus frequency (CPM or Hz).

+---------------------------------------------------------------------------------------------------+
|                                 MACHINERY FAULT DIAGNOSTIC CHART                                  |
+-----------------------+-----------------------+-----------------------+---------------------------+
| Mechanical Fault      | Dominant Peak (FFT)   | Phase & Direction     | Key Spectral Symptoms     |
+-----------------------+-----------------------+-----------------------+---------------------------+
| Mass Unbalance        | 1x RPM                | Radial (Horiz/Vert)   | Pure sinusoidal waveform; |
|                       |                       | 90° H-to-V phase shift| stable amplitude & phase  |
+-----------------------+-----------------------+-----------------------+---------------------------+
| Angular Misalignment  | 1x, 2x, 3x RPM        | Dominant Axial        | 180° phase shift across   |
|                       |                       |                       | coupling split line       |
+-----------------------+-----------------------+-----------------------+---------------------------+
| Parallel Misalignment | 2x RPM                | Dominant Radial       | High 2x RPM radial peak;  |
|                       |                       | 180° radial phase     | strong 1x RPM component   |
+-----------------------+-----------------------+-----------------------+---------------------------+
| Mechanical Looseness  | 1x, 2x, 3x... 10x RPM | Directional (Weakness)| Long harmonic series;     |
| (Rotating / Fit)      | + 0.5x sub-harmonics  | Truncated waveform    | non-repeatable phase      |
+-----------------------+-----------------------+-----------------------+---------------------------+
| Gear Mesh Fault       | Gear Mesh Frequency   | Radial & Axial        | Sidebands spaced at 1x    |
|                       | (Fgm = RPM/60 x N)    |                       | RPM of damaged gear shaft |
+-----------------------+-----------------------+-----------------------+---------------------------+
| Blade / Vane Pass     | Pass Frequency        | Radial                | High peak at Blade Pass;  |
|                       | (Fbp = RPM/60 x V)    |                       | flow turbulence sidebands |
+-----------------------+-----------------------+-----------------------+---------------------------+

Rolling Element Bearing Defect Frequencies

As rolling elements pass over raceway defects, specific impact frequencies are generated based on bearing geometry (N = number of elements, d = element diameter, D = pitch diameter, θ = contact angle):

1. Outer Race Fault (BPFO): $\text{BPFO} = \frac{N}{2} \cdot \text{RPM} \cdot \left(1 - \frac{d}{D}\cos\theta\right)$
2. Inner Race Fault (BPFI): $\text{BPFI} = \frac{N}{2} \cdot \text{RPM} \cdot \left(1 + \frac{d}{D}\cos\theta\right)$
3. Ball/Roller Spin Fault (BSF): $\text{BSF} = \frac{D}{2d} \cdot \text{RPM} \cdot \left(1 - \left(\frac{d}{D}\cos\theta\right)^2\right)$
4. Cage / Train Fault (FTF): $\text{FTF} = \frac{1}{2} \cdot \text{RPM} \cdot \left(1 - \frac{d}{D}\cos\theta\right)$

Bearing Degradation Stages

• Stage 1 (Ultrasonic): High-frequency peaks appear in ultrasonic ranges (20 kHz–60 kHz / gSE / HFE). No visible FFT velocity peaks.
• Stage 2 (Natural Resonance): Micro-cracks ring bearing component natural frequencies (500 Hz–2000 Hz). Sidebands begin developing.
• Stage 3 (Fundamental Defect Peaks): Discrete BPFO, BPFI, or BSF peaks and multiple harmonics become clearly visible in the velocity spectrum.
• Stage 4 (Terminal Failure): High discrete peaks collapse into a raised broadband noise floor ("haystack"). Shaft clearance expands, leading to catastrophic failure.

Dynamic Rotor Balancing (1-Plane & 2-Plane)

Unbalance exists when a rotor's principal mass axis does not coincide with its geometric axis of rotation.

Types of Unbalance

• Static Unbalance: Principal inertia axis is displaced parallel to the shaft center line. Correctable in a single plane passing through the center of gravity.
• Dynamic Unbalance: Principal inertia axis is both displaced and tilted relative to the shaft axis, creating a couple unbalance. Requires correction in two distinct planes.

Heavy Spot vs. High Spot (Phase Lag)

• Heavy Spot: The physical location of excess mass mass imbalance.
• High Spot: The point of maximum outward physical shaft displacement measured by the transducer.
• Phase Lag (φ): At low rotational speeds well below critical speed (resonance), phase lag is near 0° (high spot equals heavy spot). At critical speed, phase lag is exactly 90°. At high speeds well above critical speed, phase lag approaches 180°.

Influence Coefficient Vector Balancing Method

1. Measure Baseline Vector ($\vec{O}$): Record baseline 1× RPM vibration amplitude (O) and phase angle (θ_O).
2. Add Trial Weight (W_T): Install a known trial weight at a known angular radius and position θ_T.
3. Measure Combined Vector ($\vec{O+T}$): Record new 1× RPM amplitude and phase angle.
4. Calculate Net Trial Effect Vector ($\vec{T}$): $\vec{T} = \vec{O+T} - \vec{O}$
5. Determine Influence Coefficient ($\vec{\alpha}$): $\vec{\alpha} = \frac{\vec{T}}{W_T}$
6. Calculate Correction Weight Vector (W_C): $W_C = -\frac{\vec{O}}{\vec{\alpha}}$ The correction mass is installed at the calculated magnitude and angle (or 180° opposite to remove mass by drilling).

Infrared Thermography Inspection

Infrared thermography measures emitted thermal radiation to evaluate equipment condition non-destructively.

Emissivity (ε) Principles

Emissivity measures a surface's efficiency in emitting thermal infrared radiation compared to a perfect blackbody (ε = 1.0).

• High Emissivity (ε ≈ 0.95): Painted metal, rubber, electrical tape, oxidized steel. Yields accurate temperature readings.
• Low Emissivity (ε < 0.10): Polished copper, brass, and aluminum busbars act as thermal mirrors, reflecting ambient background radiation and producing false cold readings.
• Correction Technique: Apply black electrical tape (ε = 0.95) or high-emissivity matte paint to reflective targets prior to thermographic measurement.

+---------------------------------------------------------------------------------------------------+
|                             NETA / IEEE THERMAL SEVERITY GUIDELINES                               |
+-----------------------+-----------------------+---------------------------+-----------------------+
| Temperature Delta (ΔT)| Classification        | Recommended Action        | Target Applications   |
+-----------------------+-----------------------+---------------------------+-----------------------+
| 1°C to 10°C           | Minor / Normal        | Monitor at next routine   | Electrical contacts,  |
| (1.8°F to 18°F)       | Anomaly               | PM interval               | busbars, breakers     |
+-----------------------+-----------------------+---------------------------+-----------------------+
| 11°C to 20°C          | Intermediate /        | Schedule repair at next   | Motor bearings,       |
| (19.8°F to 36°F)      | Serious Anomaly       | planned shutdown          | couplings, steam traps|
+-----------------------+-----------------------+---------------------------+-----------------------+
| > 20°C                | Critical / Severe     | Immediate emergency       | Loose lugs, phase     |
| (> 36°F)              | Anomaly               | corrective action required| imbalance, hot spots  |
+-----------------------+-----------------------+---------------------------+-----------------------+