What does the Q factor (quality factor) tell us?

The Q factor is defined as Q = 1/(2ζ), where ζ is the damping ratio. A high Q means a sharp, narrow resonance peak with little energy dissipation. Structural metals typically have Q = 20–200, while rubber or viscoelastic materials have Q = 2–10. From experiments, ζ can be extracted using the half-power bandwidth method: ζ = Δf/(2fn), where Δf is the frequency range between the -3dB points.

Why does the dynamic amplification factor (DAF) exceed 1?

When the excitation frequency approaches the natural frequency, resonance causes dynamic displacements to greatly exceed the static deflection under the same force magnitude. For an undamped system DAF → ∞. With damping, the peak DAF ≈ 1/(2ζ) = Q. This is why structural engineers aim to keep operating frequencies well away from natural frequencies, or add damping to limit the resonant amplification.

What happens to the FRF of a 2-DOF system?

A 2-DOF system exhibits two resonance peaks corresponding to its two natural frequencies. Between the peaks, an anti-resonance (amplitude minimum) may appear. This anti-resonance is exploited in tuned mass dampers (TMDs): by adding a secondary mass tuned to the primary frequency, the primary structure's vibration amplitude at resonance can be dramatically reduced — a key technique in tall building and machinery vibration control.

Frequency Response Function (FRF) Calculator — Bode Plot, Resonance, Damping

Q: What is a Frequency Response Function (FRF)?

A Frequency Response Function describes how a structure responds to sinusoidal excitation across a range of frequencies. Defined as H(ω) = X(ω)/F(ω), it expresses the ratio of output (displacement, velocity, or acceleration) to input force in the frequency domain. The receptance FRF (displacement/force, units m/N) is the most common form used in structural dynamics testing and finite element model validation.

Mode 1 Parameters

Natural frequency fn₁50 Hz

Damping ratio ζ₁3.0 %

Mass m₁10.0 kg

Add 2nd mode

Results (Mode 1)

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Resonance freq. fr (Hz)

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Q factor

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Half-power BW Δf (Hz)

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Dyn. amplif. factor DAF

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Peak receptance (μm/N)

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Stiffness k (kN/m)

Receptance FRF

$H(\omega) = \dfrac{1/k}{1 - r^2 + 2j\zeta r}$

$r = \omega/\omega_n,\quad Q = \dfrac{1}{2\zeta}$

Bode Plot — Magnitude (Receptance |H|)

Bode Plot — Phase (°)

What is a Frequency Response Function (FRF)?

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What exactly is a Frequency Response Function? I see the simulator plots "Amplitude" and "Phase" against "Frequency", but what does it actually tell me?

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Basically, an FRF is a system's ID card for vibration. It tells you how a structure, like a car chassis or a building floor, responds when you poke it at different frequencies. In practice, if you apply a force at a certain frequency, the FRF predicts how much it will shake and whether it lags behind the force. Try moving the "Natural Frequency fn₁" slider above—you'll see the big peak in the amplitude plot shift left or right. That peak is the resonance, where the system shakes the most.

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Wait, really? So the "Damping ratio ζ₁" controls how tall and sharp that peak is? What's the "Q factor" that's calculated?

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Exactly! The damping ratio is like the system's shock absorber. A low ζ (like 0.01) means very little damping, so the resonance peak is extremely tall and narrow—that's a high "Quality" or Q factor. Q is basically the "sharpness" of the resonance. For instance, a tuning fork has high Q, while a car's suspension has lower Q. The simulator calculates it as $Q = 1/(2\zeta)$. Try setting ζ₁ to 0.05 and see Q jump to 10. Then, drag it to 0.5 and watch the peak flatten and Q drop to 1.

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That makes sense! But what about the second set of parameters? When I add the second natural frequency (fn₂), I get a second, smaller hump in the plot. What's happening there?

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Great observation! Real structures rarely have just one resonance. They have many modes of vibration. The second set of parameters lets you model a 2-degree-of-freedom system. The "Mass ratio m₂/m₁" controls how strong that second mode is. A common case is a cantilever beam: its first bending mode is at a low frequency (fn₁), and its second bending mode is at a higher frequency (fn₂). Play with the frequency separation—set fn₂ to be 3 times fn₁—and you'll see two distinct peaks. This is crucial for predicting how a structure will behave under complex vibrations, like an airplane wing in turbulence.

Physical Model & Key Equations

The core model is a single-degree-of-freedom (SDOF) damped harmonic oscillator. The Receptance FRF, $H(\omega)$, describes the displacement output per unit force input as a function of excitation frequency $\omega$.

$$H(\omega) = \frac{1/k}{1 - r^2 + 2j\zeta r}$$

Where:
• $k$ is the static stiffness (N/m).
• $r = \omega / \omega_n$ is the frequency ratio.
• $\omega_n = 2\pi f_n$ is the natural angular frequency (rad/s).
• $\zeta$ is the damping ratio (dimensionless).
• $j$ is the imaginary unit. The denominator's imaginary part, $2j\zeta r$, is what creates the phase lag.

From the FRF, we derive key dynamic properties that describe the resonance behavior.

$$Q = \frac{1}{2\zeta}, \quad \Delta f = f_n \cdot 2\zeta$$

Where:
• $Q$: Quality factor. A higher Q means a sharper, more pronounced resonance peak.
• $\Delta f$: Half-power bandwidth. This is the width of the resonance peak (in Hz) measured at an amplitude of $1/\sqrt{2}$ of the peak value. It's a direct measure of damping in the frequency domain. In the simulator, you can see how increasing ζ widens $\Delta f$ and lowers the peak amplitude.

Real-World Applications

Structural Health Monitoring: Engineers regularly measure the FRF of bridges or wind turbine blades. A shift in the natural frequency or a change in damping can indicate developing cracks or fatigue, allowing for maintenance before failure.

Automotive NVH (Noise, Vibration, Harshness): Car manufacturers use FRF analysis to find and modify resonances in body panels, steering columns, and exhaust systems. For instance, they might add damping material to suppress a resonance at the engine's idle frequency, making the cabin quieter.

Aerospace Flutter Prediction: Aircraft wings and control surfaces are tested extensively for their FRF. Engineers must ensure that resonant frequencies do not align with excitation frequencies from airflow or engine rotation to avoid catastrophic flutter vibrations.

Consumer Electronics Design: The design of smartphones and hard drives heavily relies on FRF analysis. Engineers ensure the natural frequencies of internal components and the casing are away from the operating frequencies of motors or speakers to prevent annoying buzzes and potential damage.

Common Misconceptions and Points to Note

First, do not assume there is only one type of FRF. In reality, multiple FRFs are defined based on the input-output pair, such as displacement/force (receptance), velocity/force (mobility), and acceleration/force (inertance). The simulator shows the displacement-based "receptance." For instance, when using an accelerometer in a vibration test, you are dealing with inertance, which has different characteristics, such as more noticeable noise in the low-frequency range. Next, the damping ratio ζ and the resonance peak height do not have a simple proportional relationship. Even if ζ doubles from 0.1 to 0.2, the peak height decreases by slightly more than half. Specifically, the peak amplitude is approximately 50 (1/(2ζ)) for ζ=0.01 and about 5 for ζ=0.1. Finally, be careful when setting the "mass ratio" or "coupling spring" in a 2-degree-of-freedom system. For example, if you set an extremely large mass ratio (e.g., m2/m1=100), the lower resonance mode becomes a "local vibration" state where essentially only the smaller mass (m1) vibrates, while in the higher mode, the two masses vibrate in opposite phases. When setting parameters in practical work, you must accurately estimate the masses and stiffnesses of the physical components you are considering; otherwise, you risk analyzing modes that are completely different from reality.

Related Engineering Fields

The concept of FRF is directly applicable to acoustical engineering. The driver unit of a speaker or the diaphragm of headphones is precisely a mechanical vibration system. By measuring and designing its FRF (in this case, sound pressure/input voltage), flat frequency characteristics and enhanced bass are achieved. Furthermore, it is inextricably linked to control engineering. In the position control of robot arms or the servo control of machine tools, the FRF of the "plant" (the controlled object) is identified, and based on this, the parameters (gain, phase compensation) of the feedback controller are designed. The rapid phase shift near a resonance peak can cause instability (oscillation) in the control system, requiring particular attention. Moreover, in the field of materials mechanics, FRF measurement is used to evaluate the dynamic properties of composite materials or viscoelastic materials. For example, a material sample is attached to an oscillator, its FRF is measured, and the complex modulus (storage modulus and loss modulus) of the material is calculated from the changes in its resonance frequency and damping. Thus, the seemingly simple single-degree-of-freedom model functions as a "common language" across diverse engineering fields.

For Further Learning

The next step is to understand the concepts of "modal analysis" and "modal coordinates." The reason two peaks appear in the 2-DOF FRF is that the system has two "modes" (specific vibration patterns between masses that occur at particular frequencies). Each of these modes behaves almost like an independent single-degree-of-freedom system. In other words, the FRF of a complex multi-degree-of-freedom system can actually be expressed as a superposition (linear sum) of multiple single-degree-of-freedom FRFs. The key to understanding this is the "transformation to modal coordinates." Mathematically, this involves solving the equations of motion as a matrix eigenvalue problem. You find the eigenvalues (squares of the natural frequencies) and eigenvectors (mode shapes) from the mass and stiffness matrices. Using these, instead of solving simultaneous equations, you decompose them into a set of independent equations. After observing in the simulator how the two peaks move when you change the mass ratio or spring constants, try following this matrix-based formulation in a textbook. Once you grasp this, you will deeply understand the meaning of the "modal contribution ratio" output by CAE software, significantly advancing your ability in practical vibration troubleshooting and design improvement.