Vibration measurement and ground vibration testing

Why it matters

Analysis predicts natural frequencies and mode shapes, but damping cannot be predicted and stiffness of joints, fittings and composite parts is uncertain. Before first flight every new or significantly modified aircraft goes through a ground vibration test (GVT): the measured frequencies, mode shapes and damping are used to correct the finite-element model on which the flutter clearance is based. Knowing how transducers, excitation and frequency-response data work lets you trust (or question) those numbers.

Key ideas

Seismic transducers. Most vibration pickups are a small mass on a spring and damper inside a case fixed to the structure, i.e. a base-excited SDOF system. The output is the relative motion z of the mass, with Z/Y = r²/√((1 − r²)² + (2ζr)²), r = ω/ω_n of the transducer.

• Accelerometer: used far below its own natural frequency (r ≪ 1). Then Z ≈ ω²·Y/ω_n², i.e. the output is proportional to the case acceleration. The accuracy factor is 1/√((1 − r²)² + (2ζr)²); with ζ ≈ 0.7 it stays flat to a larger fraction of ω_n, while lightly damped piezoelectric accelerometers are simply used below about 20 % of their (very high, 20–50 kHz) natural frequency.
• Seismometer (vibrometer): used far above its natural frequency (r ≫ 1); then Z ≈ Y, the output is the case displacement. Needs a very soft, heavy instrument, so it is used for low-frequency motion.
• Piezoelectric accelerometers dominate in aerospace testing: small, light, wide band. Their mass must be small compared with the local structural mass (mass loading lowers the measured frequency). Non-contact laser Doppler vibrometers avoid mass loading entirely. Strain gauges measure strain, useful for mode shapes of curvature and for loads.

Excitation. Impact hammer (quick, broadband, good for small parts), electrodynamic shakers with stingers (controlled force for large structures), signals such as stepped or swept sine, random and burst random. A force transducer at the input measures F(t).

Frequency-response functions (FRF). H(ω) = X(ω)/F(ω): receptance (displacement/force), mobility (velocity/force) or accelerance (acceleration/force). Near a lightly damped mode the FRF peaks and its phase passes through 90° (for receptance). In practice H is estimated by averaging (H1 = cross-spectrum/input auto-spectrum), and the coherence function (between 0 and 1) shows whether the output is really caused by the measured input.

Extracting modal parameters.

• Natural frequency: peak of |H| or the 90° phase point (quadrature response).
• Damping: half-power bandwidth, ζ = (f₂ − f₁)/(2·f_n), where f₁ and f₂ are where |H| = peak/√2; or the logarithmic decrement of the free decay after the excitation is stopped; or curve fitting (circle fit, multi-mode fitting algorithms).
• Mode shape: relative amplitude and sign of the response at many points at the natural frequency.
• Model correlation: the modal assurance criterion (MAC), between 0 and 1, compares test and analysis mode shapes; frequencies are compared percentage by percentage and the FE model is updated (stiffness of joints, mass of non-structural items).

Ground vibration test of an aircraft. The aircraft is supported on soft air springs or bungees, or on deflated tyres, so that the suspension (rigid-body) frequencies are well below the lowest elastic mode (a factor of three or more is a common aim), representing free–free flight. Hundreds of accelerometers are fitted; several shakers excite symmetric and antisymmetric modes. Phase-resonance (normal-mode tuning) methods adjust shaker forces until a single mode responds in quadrature; phase-separation methods fit many modes from broadband FRFs. Fuel and payload states and control-surface actuator stiffness are tested because they change the modes that matter for flutter.

Signal processing basics. Sample at f_s > 2·f_max (Nyquist), with an anti-aliasing filter; frequency resolution Δf = 1/T for a record of length T; use windows to limit leakage for non-periodic signals.

Formulas

Z/Y = r²/√((1 − r²)² + (2ζr)²) — seismic transducer relative motion; r = ω/ω_n of the transducer.

Accuracy factor (accelerometer) = 1/√((1 − r²)² + (2ζr)²) — should be close to 1 (r ≪ 1).

H(ω) = X(ω)/F(ω) — receptance FRF (m/N); mobility (m/(N·s)), accelerance (m/(N·s²)).

ζ ≈ (f₂ − f₁)/(2·f_n) — half-power bandwidth, light damping; f₁, f₂ where |H| = |H|_max/√2.

ζ = δ/√(4π² + δ²), δ = (1/n)·ln(x₀/x_n) — from a free decay.

f_s > 2·f_max, Δf = 1/T — sampling rate (Hz) and frequency resolution (Hz) for record length T (s).

c = 2·ζ·M_r·ω_r — equivalent modal damping coefficient (N·s/m) for modal mass M_r (kg).

Worked examples

Example 1 (standard). A GVT frequency sweep on a tailplane shows a receptance peak at 12.00 Hz, and the amplitude falls to 1/√2 of the peak at 11.76 Hz and 12.24 Hz. The modal mass of this mode, from the updated model, is 40 kg. Find ζ and the modal damping coefficient.

1. ζ ≈ (f₂ − f₁)/(2·f_n) = (12.24 − 11.76)/(2 × 12.00) = 0.48/24 = 0.020.
2. ω_r = 2π × 12 = 75.40 rad/s.
3. c = 2·ζ·M_r·ω_r = 2 × 0.020 × 40 × 75.40 = 120.6 N·s/m.

Answer: ζ = 0.020 (2 %), c ≈ 121 N·s/m.

Example 2 (GATE level). A piezoelectric accelerometer has a mounted natural frequency of 30 kHz and negligible damping. (a) What is the amplitude error when it measures a 5 kHz vibration? (b) What is the highest frequency it can measure with at most 5 % error? (c) To analyse up to 5 kHz with 0.5 Hz resolution, what minimum sampling rate and record length are needed?

1. (a) r = 5/30 = 0.1667. Accuracy factor = 1/(1 − r²) = 1/(1 − 0.02778) = 1.0286: it reads 2.9 % high.
2. (b) 1/(1 − r²) = 1.05 gives r² = 1 − 1/1.05 = 0.04762, r = 0.218, f = 0.218 × 30 = 6.55 kHz.
3. (c) f_s > 2 × 5000 = 10 kHz (in practice about 12.8 kHz with an anti-aliasing filter); T = 1/Δf = 1/0.5 = 2 s.

Answer: (a) about +2.9 %; (b) about 6.5 kHz; (c) f_s above 10 kHz and T = 2 s.

Common mistakes

• Using an accelerometer near its own resonance, or a seismometer below its natural frequency.
• Ignoring mass loading of a heavy transducer on a light panel, which lowers the measured frequency.
• Reading damping from the half-power points of a coarse-resolution FRF; the peak is under-sampled and ζ comes out too large.
• Taking f₂ − f₁ divided by f_n (gives 2ζ) and forgetting the factor 2.
• Testing on a stiff suspension whose rigid-body frequencies interfere with low elastic modes.
• Sampling below twice the highest frequency present, so that high frequencies alias into the band of interest.

For GATE AE

Expect transducer questions (accelerometer versus seismometer frequency ranges, the base-excitation relative-motion formula), damping from half-power bandwidth or decay, conversion between Hz and rad/s and from ζ to c, and conceptual questions on why GVT is done and how the results are used. Practise the half-power formula and accuracy-factor calculations.

Quick check

1. Should an accelerometer's natural frequency be well above or below the measured frequency?
2. Half-power points at 49 Hz and 51 Hz around a 50 Hz peak: what is ζ?
3. Why is an aircraft supported on soft springs for a GVT?
4. For a 1 s record, what is the frequency resolution?
5. Which quantity shows whether the measured output is caused by the measured input?

Answers: 1. well above (r ≪ 1); 2. 2/(2 × 50) = 0.02; 3. to simulate free–free flight, with suspension modes well below the elastic modes; 4. 1 Hz; 5. the coherence function.