Interactive Lecture
Frequency Response Functions, Curve Fitting, and Modal Parameters
by Dr Arnaldo Delli Carri
Modal testing is an experimental technique used to identify the dynamic characteristics of structures. By measuring the structure's response to excitation, we can extract modal parameters: natural frequencies, damping ratios, and mode shapes. These parameters form the foundation for structural dynamics analysis, validation of finite element models, and troubleshooting vibration problems.
Key objectives of modal testing:
A complete modal test requires: (1) Excitation device (hammer or shaker), (2) Force transducer to measure input, (3) Accelerometers to measure response, (4) Data acquisition system, and (5) Modal analysis software for parameter extraction.
The Frequency Response Function (FRF) is the ratio of output (response) to input (force) in the frequency domain. It completely characterises the linear dynamic behaviour of a structure at a given location.
FRF Definition:
\[H(\omega) = \frac{X(\omega)}{F(\omega)}\]
Where:
For a SDOF system:
\[H(\omega) = \frac{1}{k - m\omega^2 + j c\omega} = \frac{1/k}{1 - r^2 + j(2\zeta r)}\]
Where \(r = \omega/\omega_n\) and \(\zeta\) is the damping ratio
FRF Representations:
Stiffness: \(k = m\omega_n^2 =\) 0 N/m
Damping coeff: \(c = 2\zeta m\omega_n =\) 0 Ns/m
Peak at \(f_n\); phase is −90° at resonance for the displayed FRF type; Nyquist traces a circle for SDOF.
Real structures have many modes. The measured FRF is a superposition of individual modal contributions, each with its own frequency, damping, and mode shape.
MDOF FRF (Modal Superposition):
\[H_{jk}(\omega) = \sum_{r=1}^{N} \frac{\Phi_{jr}\Phi_{kr}}{m_r(\omega_{r}^2 - \omega^2 + j2\zeta_r\omega_r\omega)}\]
Where:
Residue Form:
\[H_{jk}(\omega) = \sum_{r=1}^{N} \left[\frac{A_{jk}^r}{j\omega - \lambda_r} + \frac{\bar{A}_{jk}^r}{j\omega - \bar{\lambda}_r}\right]\]
Where \(A_{jk}^r\) is the modal residue and \(\lambda_r = -\zeta_r\omega_r + j\omega_r\sqrt{1-\zeta_r^2}\) is the complex pole
Peaks appear at each natural frequency. Well-separated modes (spacing > 2× bandwidth) are easier to identify accurately.
Modal density: 1.5 modes / 100 Hz
Modal parameters are extracted from measured FRFs using curve fitting algorithms. Different methods suit different situations based on mode separation and data quality.
Peak Picking (Simplest):
\[f_n = f_{\text{peak}}\]
Damping from the peak amplitude (quality factor \(Q\)):
\[\zeta \approx \frac{1}{2Q}, \quad Q = k\,|H(\omega_n)|\]
Half-Power Bandwidth:
\[\zeta = \frac{\Delta f}{2f_n} = \frac{f_2 - f_1}{2f_n}\]
Where \(f_1, f_2\) are half-power points (\(|H| = |H_{max}|/\sqrt{2}\))
Circle Fit (Nyquist Plot):
For well-separated modes, FRF traces a circle in Nyquist plot (Real vs Imaginary). Fitting circle provides:
Rational Fraction Polynomial (Advanced):
\[H(\omega) = \frac{b_0 + b_1(j\omega) + \cdots + b_m(j\omega)^m}{a_0 + a_1(j\omega) + \cdots + a_n(j\omega)^n}\]
Polynomial coefficients solved from measured data; poles yield natural frequencies and damping
Complex Mode Indicator Function (CMIF):
Uses SVD of FRF matrix to identify number of modes and separate closely spaced modes
True Frequency: 40.0 Hz
Extracted Frequency: -- Hz
Frequency Error: -- %
True Damping: 3.0 %
Extracted Damping: -- %
Damping Error: -- %
The coherence function quantifies the quality of FRF measurement by indicating how much of the output is linearly related to the input. Values near 1.0 indicate good measurements; values significantly below 1.0 suggest problems.
Coherence Definition:
\[\gamma^2(\omega) = \frac{|G_{XF}(\omega)|^2}{G_{XX}(\omega)G_{FF}(\omega)}\]
Where:
Causes of Poor Coherence:
Improving Coherence:
Avg coherence: 0.00 | Min: 0.00 | Quality: Excellent
γ²>0.95 Excellent · 0.90–0.95 Good · 0.80–0.90 Fair · <0.80 Poor
Modal testing and experimental modal analysis are everyday tools in several industries:
| Concept | Equation | Description |
|---|---|---|
| FRF Definition | \(H(\omega) = \frac{X(\omega)}{F(\omega)}\) | Ratio of response to force in frequency domain |
| SDOF Receptance | \(H(\omega) = \frac{1/k}{1 - r^2 + j(2\zeta r)}\) | Single degree of freedom FRF |
| Modal Superposition | \(H_{jk}(\omega) = \sum_{r=1}^{N} \frac{\Phi_{jr}\Phi_{kr}}{m_r(\omega_{r}^2 - \omega^2 + j2\zeta_r\omega_r\omega)}\) | Multi-mode FRF summation |
| Half-Power Damping | \(\zeta = \frac{\Delta f}{2f_n}\) | Damping from bandwidth at peak/√2 |
| Coherence | \(\gamma^2(\omega) = \frac{|G_{XF}(\omega)|^2}{G_{XX}(\omega)G_{FF}(\omega)}\) | Measurement quality indicator (0 to 1) |
An FRF shows a resonance peak at 45 Hz with magnitude 0.005 m/N. The half-power points occur at 43.5 Hz and 46.5 Hz. Calculate the damping ratio.
Show Solution
Solution:
Using the half-power bandwidth method:
\(\Delta f = f_2 - f_1 = 46.5 - 43.5 = 3.0\) Hz
\(\zeta = \frac{\Delta f}{2f_n} = \frac{3.0}{2 \times 45} = \frac{3.0}{90} = 0.0333\)
Converting to percentage: \(\zeta = 3.33\%\)
Answer: ζ = 3.33%
An FRF peak at 30 Hz has a receptance magnitude of 0.01 m/N. Given that \(\zeta = 2\%\), calculate the modal mass assuming the measurement was at maximum mode shape location (\(\Phi = 1\)).
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Solution:
At resonance for receptance FRF with \(\Phi = 1\):
\(|H(\omega_n)| = \frac{1}{2\zeta m \omega_n^2}\)
Rearranging: \(m = \frac{1}{2\zeta \omega_n^2 |H(\omega_n)|}\)
\(\omega_n = 2\pi f_n = 2\pi \times 30 = 188.5\) rad/s
\(m = \frac{1}{2 \times 0.02 \times (188.5)^2 \times 0.01}\)
\(m = \frac{1}{2 \times 0.02 \times 35532 \times 0.01} = \frac{1}{14.21} = 0.0704\) kg
Answer: Modal mass = 70.4 grams
A measurement shows coherence of 0.85 with 5 averages. Approximately how many averages are needed to achieve coherence of 0.95, assuming random noise is the limiting factor?
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Solution:
Coherence improves approximately with square root of number of averages for random noise:
\(\gamma_2^2 \approx 1 - (1 - \gamma_1^2)\frac{n_1}{n_2}\)
Rearranging: \(n_2 \approx n_1 \frac{1 - \gamma_1^2}{1 - \gamma_2^2}\)
\(n_2 \approx 5 \times \frac{1 - 0.85}{1 - 0.95} = 5 \times \frac{0.15}{0.05} = 5 \times 3 = 15\)
Answer: Approximately 15 averages needed
Note: This is an approximation. In practice, 20-25 averages recommended for γ² > 0.95