Frequency Response Analysis (FRA) — CAE Terminology
Frequency Response Analysis
What exactly is frequency response analysis? I heard it is a type of vibration analysis, but I would like to understand it better.
Simply put, it is an analysis where you apply a sinusoidal force to a structure "while gradually changing the frequency," and measure how much it vibrates at each frequency (amplitude) and how much the vibration lags behind the force (phase). In English it is called Frequency Response Analysis, or FRA for short.
But real loads aren't sinusoidal, are they? Engine vibration and other such forces have much messier waveforms, don't they?
Good question. Using Fourier transform, any complex waveform can be decomposed into a superposition of sinusoids. So if you know the response at each frequency, you can add them together to reconstruct the actual time-history response. That is the strength of frequency response analysis.
Definition and Equations of Motion
What form does the equation of motion take?
The fundamental equation of motion is this:
where $\mathbf{M}$ is the mass matrix, $\mathbf{C}$ is the damping matrix, and $\mathbf{K}$ is the stiffness matrix. Assuming a harmonic external force $\mathbf{F}(t)=\mathbf{F}_0 e^{i\omega t}$, the response also takes the form $\mathbf{u}(t)=\mathbf{U}(\omega)e^{i\omega t}$, so in the frequency domain:
$$\left[-\omega^2 \mathbf{M} + i\omega \mathbf{C} + \mathbf{K}\right]\mathbf{U}(\omega) = \mathbf{F}_0$$The expression inside the brackets on the left is called the dynamic stiffness matrix. By solving this system of equations at each frequency $\omega$, we can obtain the amplitude and phase.
FRF (Frequency Response Function)
I see FRF (Frequency Response Function) frequently in papers and experimental reports. What exactly does it represent?
FRF is the ratio of output (displacement, velocity, or acceleration) to input (force) expressed as a function of frequency. For a single-degree-of-freedom system:
When the output is displacement, it is called compliance (or receptance); when it is velocity, it is called mobility; when it is acceleration, it is called inertance (or accelerance). When plotted with a dB scale on the vertical axis and logarithmic frequency on the horizontal axis, it becomes a Bode diagram.
So the types of FRF differ depending on the physical quantity of the output. Which type is most commonly used in experiments?
In shaker tests, accelerometers are most commonly used, so inertance is the most frequently used. With an impact hammer, you strike the structure and pick up the acceleration with an accelerometer. The FFT analyzer then automatically calculates $H(\omega) = A(\omega)/F(\omega)$. In automotive NVH (Noise, Vibration, and Harshness) development, this type of data is acquired daily.
Modal and Direct Methods
In CAE software, frequency response analysis offers choices between "Modal Method" and "Direct Method." What is the difference?
The Modal Frequency Response Method first performs eigenvalue analysis to find mode shapes and natural frequencies, then calculates the response as a superposition of modes. The physical coordinates $\mathbf{u}$ are transformed using the mode matrix $\mathbf{\Phi}$:
This makes each mode an independent single-degree-of-freedom system, so computation is very fast.
Then why not just use the modal method all the time? If it is faster, that would be better, wouldn't it?
It is not that simple. The modal method has limitations. First, there is the problem of truncation modes. For accurate results, you must extract modes up to 1.5 to 2 times the upper limit of your analysis frequency range. Second, it cannot accurately handle frequency-dependent damping (such as viscoelastic materials). Additionally, the modal method struggles with non-symmetric matrices (such as fluid-structure interaction).
So we use the direct method in those cases, right?
Exactly. The Direct Frequency Response Method assembles the dynamic stiffness matrix $\mathbf{D}(\omega) = -\omega^2\mathbf{M}+i\omega\mathbf{C}+\mathbf{K}$ at each frequency step and solves the system of equations directly. There is no truncation error from mode cutoff, and frequency-dependent materials can be handled directly. However, with degrees of freedom in the tens or hundreds of thousands, solving large-scale complex systems of equations at each frequency becomes extremely computationally expensive. For example, in automotive body vibration analysis, the modal method might finish in a few hours, but the direct method can take an entire day or more.
Half-Power Bandwidth Method
I heard there is a method called "half-power bandwidth method" for determining the damping ratio from experimental FRF data. What is this method?
It is a method that focuses on the resonance peak of the FRF. Taking the peak maximum as $|H|_{\max}$, you identify the two frequencies where the magnitude drops to $|H|_{\max}/\sqrt{2}$ (about -3 dB) on either side. Call these $f_1$ and $f_2$. Then the damping ratio $\zeta$ is:
where $f_n$ is the resonance frequency. The reason for dividing by $\sqrt{2}$ is that the power (amplitude squared) reaches half the peak at this point, hence the name "half-power."
How accurate is this method? Are there any precautions to take?
There are two important considerations. First is frequency resolution. For small $\Delta f$ (low damping), the peak becomes sharp, and if the frequency step is coarse, you cannot read $f_1$ and $f_2$ accurately. As a rule of thumb, you want at least 5-10 points within the resonance peak width. Second is mode overlap. When adjacent resonances are close together, their skirts overlap, making it impossible to read the -3 dB points correctly. In such cases, it is practical to switch to a curve-fitting method (least-squares fitting to analytically identify modal parameters).
Physical Meaning of Transfer Function
I would like to delve deeper into the physical meaning of the transfer function $H(\omega)$. What is happening at the resonance point?
Let us look at $H(\omega)$ for a single-degree-of-freedom system again:
When $r=1$ (excitation frequency equals natural frequency), the denominator becomes just $2i\zeta$, and the amplitude jumps to $|H| = 1/(2k\zeta)$. For a damping ratio of $\zeta=0.01$ (1%), the vibration is **50 times** the static displacement. That is the danger of resonance.
50 times! That would be a serious design issue. What about the phase behavior?
The phase lags by exactly 90 degrees at resonance. At low frequencies ($r \ll 1$), the displacement and force are nearly in phase (0 degrees), while at high frequencies ($r \gg 1$), the lag is 180 degrees—meaning the structure moves opposite to the applied force. The fact that the phase is "90 degrees at resonance" is so clearly observed in experiments that phase information is often used to identify resonance frequencies. When you look at a Bode diagram showing both amplitude and phase side by side, the resonance point is immediately obvious.
Practical Notes
Are there key points to watch out for when performing frequency response analysis in CAE?
There are three major points. First is frequency step setting. With uniform spacing, you can miss resonance peaks. So it is safer to use "adaptive frequency stepping" that automatically refines the spacing near peaks, or manually reduce the step size by ±10% around natural frequencies.
What is the second point?
Damping model selection. Rayleigh damping ($\mathbf{C}=\alpha\mathbf{M}+\beta\mathbf{K}$) only fits at two frequencies, so over a broad band the high-frequency damping tends to be overestimated. Using structural damping (hysteretic damping with loss factor $\eta$) gives a frequency-independent constant damping ratio, which is convenient for broadband analysis. However, structural damping does not strictly satisfy causality mathematically, so caution is needed when converting back to time history.
What is the third point?
Residual mode correction. When truncating modes above a certain frequency in the modal method, the static contribution from higher-order modes is lost. Using features like Nastran's RESVEC (residual vector) or Abaqus's residual mode corrects for this significantly, and accuracy improves even with fewer modes. This effect is especially large for point FRFs (when load and response points are close), so be sure to enable this correction.
Related Terms
Natural Frequency, Mode Shape, Damping Ratio, FRF (Frequency Response Function), Bode Diagram, Compliance (Receptance), Mobility, Inertance (Accelerance), Transfer Function, Rayleigh Damping, Structural Damping (Hysteretic Damping), Modal Superposition Method, Residual Vector
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