Vibration measurement and condition monitoring

Why it matters

Most rotating-machine faults, such as unbalance, misalignment, looseness, bearing wear and gear damage, show up in the vibration signature weeks or months before a breakdown. Measuring vibration and reading its spectrum is the core of predictive maintenance in plants, wind farms and fleets, and it is a natural mechatronics job: sensor selection, signal conditioning, sampling, FFT and decision rules.

Key ideas

What is measured. For harmonic motion x = X·sin ωt, velocity and acceleration have amplitudes ωX and ω²X. So at low frequency displacement is large and acceleration small, and at high frequency the reverse. Hence:

• displacement (µm) for low-frequency and shaft-motion problems (below about 10 Hz, and journal-bearing shafts);
• velocity (mm/s, usually RMS) for overall machine severity, roughly 10 Hz to 1 kHz, the basis of ISO severity charts;
• acceleration (m/s² or g) for high-frequency faults such as rolling bearings and gears. Amplitudes are quoted as peak, peak-to-peak or RMS; for a pure sinusoid RMS = peak/√2.

Seismic (inertial) instruments. A small mass on a spring and damper inside a case fixed to the vibrating body. The measured quantity is the relative displacement Z between mass and case. With r = ω/ω_n:

• Vibrometer (displacement pickup): very soft, low ω_n, used at r ≫ 1. The mass stays nearly still in space, so Z ≈ Y (the base displacement). Large and heavy.
• Accelerometer: very stiff, high ω_n, used at r ≪ 1. Then Z ≈ a/ω_n², proportional to base acceleration. With ζ ≈ 0.65–0.7 the amplitude error stays small up to about r = 0.2–0.4 with nearly linear phase.
• Modern sensors: piezoelectric accelerometers (the crystal is the spring; wide range, a few Hz to over 10 kHz; mounting method limits the usable top frequency), MEMS accelerometers (low cost, down to DC), velocity pickups (coil and magnet, self-generating), non-contact eddy-current proximity probes (shaft displacement relative to the bearing), and laser vibrometers.

Frequency-measuring instruments. Frahm reed tachometer (a set of reeds of graded natural frequency; the reed in resonance shows the frequency) and the Fullarton (variable-length reed) tachometer, classic examples of using resonance.

Signal processing. Sample above twice the highest frequency of interest (Nyquist), with an anti-aliasing filter before the ADC. The FFT of N samples at f_s gives lines spaced Δf = f_s/N. Windowing (e.g. Hanning) reduces leakage; averaging reduces noise. Envelope (demodulation) analysis extracts repetitive impacts from bearings out of high-frequency resonances. Order tracking follows the running speed in variable-speed machines.

Fault signatures (typical).

• Unbalance: high 1× (running speed) radial vibration; at each bearing the horizontal and vertical readings differ in phase by about 90°.
• Misalignment: high 2× and 1×, strong axial component, about 180° phase across the coupling.
• Mechanical looseness: many harmonics of running speed, sometimes ½× sub-harmonics.
• Rolling-element bearing defects: non-integer frequencies (BPFO, BPFI, ball spin, cage), often seen in the envelope spectrum.
• Gears: gear-mesh frequency (teeth × shaft speed) with sidebands spaced at shaft speed.
• Resonance: a high peak that changes greatly with small speed changes.

Condition monitoring programme. Baseline measurement, regular trending at fixed points and directions, alarm levels (from ISO 10816/20816 severity zones or the manufacturer; take values from the standard), diagnosis by spectrum and phase, then planned repair. Vibration is combined with oil and wear-debris analysis, infrared thermography, airborne ultrasound, motor current signature analysis and performance monitoring.

Formulas

v_peak = ω·X ; a_peak = ω²·X ; ω = 2πf (X in m, v in m/s, a in m/s²) RMS = peak / √2 (sinusoid) ; peak-to-peak = 2 × peak

Seismic instrument, relative to base motion Y: Z / Y = r² / √((1 − r²)² + (2ζr)²) (vibrometer reads Z ≈ Y for r ≫ 1) Z·ω_n² / (ω²·Y) = 1 / √((1 − r²)² + (2ζr)²) (accelerometer reads Z·ω_n² ≈ a for r ≪ 1)

f_s > 2·f_max (Nyquist) ; Δf = f_s / N

Bearing outer- and inner-race defect frequencies (n balls, ball diameter d, pitch diameter D, contact angle α, shaft frequency f_r): BPFO = (n/2)·f_r·(1 − (d/D)·cos α) ; BPFI = (n/2)·f_r·(1 + (d/D)·cos α)

Gear mesh frequency = T × f_shaft

Worked examples

Example 1 (standard): from a velocity reading to displacement, acceleration and a bearing frequency. Given: a motor runs at 1480 rpm; vibration velocity at running speed is 4.5 mm/s RMS. Its bearing has 9 balls of 7.94 mm on a 39 mm pitch diameter, α = 0°.

1. f_r = 1480/60 = 24.67 Hz → ω = 2π × 24.67 = 155.0 rad/s.
2. v_peak = 4.5 × √2 = 6.36 mm/s.
3. X_peak = v_peak/ω = 6.36/155.0 = 0.0411 mm = 41 µm (82 µm peak-to-peak).
4. a_peak = ω·v_peak = 155.0 × 6.36 × 10⁻³ = 0.986 m/s².
5. BPFO = (9/2) × 24.67 × (1 − 7.94/39) = 111.0 × 0.7964 = 88.4 Hz; BPFI = 111.0 × 1.2036 = 133.6 Hz. Answer: 41 µm peak, 0.99 m/s² peak; watch for 88.4 Hz (outer race) and 133.6 Hz (inner race) in the envelope spectrum.

Example 2 (GATE level): seismic instrument errors. (a) A vibrometer with f_n = 4 Hz and ζ = 0.2 reads a relative amplitude of 1.0 mm on a machine vibrating at 20 Hz. True amplitude?

1. r = 20/4 = 5.
2. Z/Y = 25 / √((1 − 25)² + (2 × 0.2 × 5)²) = 25 / √(576 + 4) = 25 / 24.08 = 1.038
3. Y = 1.0 / 1.038 = 0.963 mm. (b) An accelerometer with f_n = 2 kHz and ζ = 0.7 measures a 200 Hz vibration. Amplitude error?
4. r = 0.1 → 1 / √((1 − 0.01)² + (0.14)²) = 1 / √(0.9801 + 0.0196) = 1.00015. Answer: (a) true amplitude ≈ 0.963 mm (the reading is 3.8% high); (b) error ≈ +0.015%, negligible.

Common mistakes

• Mixing peak, peak-to-peak and RMS values when comparing with alarm limits.
• Using a vibrometer below its natural frequency or an accelerometer near its resonance.
• Forgetting ω = 2πf when converting between displacement, velocity and acceleration.
• Sampling too slowly, so high-frequency content aliases into false low-frequency peaks.
• Diagnosing unbalance from a 1× peak alone without checking phase; misalignment and resonance can also give 1×.
• Mounting an accelerometer on a magnet or thin bracket and trusting readings near the mount resonance.

For GATE ME

Questions here are usually about seismic instruments: the frequency-ratio condition for vibrometers and accelerometers, the amplitude ratio formula, and simple conversions between displacement, velocity and acceleration amplitudes. Practise the Z/Y expression for r ≫ 1 and r ≪ 1 and the role of ζ ≈ 0.7.

Quick check

1. A vibration of 10 µm peak at 100 Hz. Peak velocity?
2. Which instrument needs a natural frequency well above the measured frequency?
3. RMS of a sinusoid with 2 mm/s peak?
4. Signals up to 5 kHz are needed. Minimum sampling rate?
5. A 40-tooth gear on a 25 Hz shaft. Gear mesh frequency?

Answers: 1. 2π × 100 × 10 × 10⁻⁶ = 6.28 mm/s. 2. Accelerometer. 3. 1.41 mm/s. 4. Above 10 kHz (in practice about 2.56 × 5 kHz = 12.8 kHz). 5. 1000 Hz.