Vary damping ratio and natural frequency to visualize the Frequency Response Function (FRF) in real time. Intuitively understand resonance mechanism and dynamic magnification factor for structural dynamics.
Parameters
Natural frequency f₀
Hz
System resonance frequency
Damping ratio ζ
ζ=0.05 is a typical structural damping value
Frequency range f_max
Hz
Resonance peak condition
r = f/f₀ = 1 when response is maximum
Peak value ≈ 1/(2ζ)
Chart Interaction
Click/drag on the FRF graph to select the excitation frequency. Use the vertical cursor to check the amplification factor.
Results
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Resonance Freq f_r (Hz)
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Dynamic Amplification Q
—
Half-power BW Δf (Hz)
—
Damping Ratio ζ
Frequency Response H(f) = X / (F₀/k) f = — Hz | H = —
Phase Angle φ(f)
Mass-Spring-Damper Animation
Excitation Freq f:
Hz
H = —
Theory & Key Formulas
$$|H(\omega)| = \frac{1/k}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}, \quad r = \frac{\omega}{\omega_n}$$
Dynamic Amplification Factor (DAF): at \(r=1\) resonance, \(1/(2\zeta)\); for \(\zeta=0.05\), amplification is 10x
Natural angular frequency and natural frequency relationship
$$\phi(\omega) = \arctan\frac{2\zeta r}{1-r^2}$$
Phase lag: 0-90° for \(r<1\), 90-180° for \(r>1\)
What is a Frequency Response Function?
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What exactly is a "frequency response function" (FRF)? I see it's the main output of this simulator, but what does it tell us?
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Basically, it's a graph that shows how a vibrating system, like a mass on a spring, reacts to different shaking frequencies. The FRF tells you the system's "output" (like its displacement) for a given "input" (like a shaking force). In practice, it's the ultimate tool for predicting resonance. Try moving the "Natural frequency f₀" slider above—you'll see the peak of the curve shift, showing where the system vibrates most violently.
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Wait, really? So that big peak is the famous "resonance"? What determines how tall and sharp that peak is?
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Exactly! The height and width of the resonance peak are controlled almost entirely by the damping ratio ζ. A common case is a car's suspension: low damping gives a bouncy, tall peak (a rough ride on bumpy roads), while high damping flattens it out. Grab the "Damping ratio ζ" slider and watch. At ζ=0.01, the peak is huge and narrow. At ζ=0.5, it almost disappears. This is the dynamic amplification factor in action.
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That makes sense! So the horizontal axis is the shaking frequency. What's the deal with the "r" in the equation? It's not just frequency.
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Great observation! The horizontal axis here is the frequency ratio r, defined as $r = \omega / \omega_0$, where $\omega$ is the shaking frequency and $\omega_0$ is the natural frequency you set with the f₀ slider. This is key because resonance always happens at r=1, regardless of the actual natural frequency. When you change the "Frequency range f_max" parameter, you're zooming in or out on this r-axis to see the full behavior around that critical point.
Physical Model & Key Equations
The simulator solves the fundamental equation of motion for a damped, single-degree-of-freedom system being shaken by a sinusoidal force. This is Newton's second law applied to vibration.
$$ m\ddot{x}+ c\dot{x}+ kx = F_0\sin(\omega t) $$
Here, m is the mass, c is the damping coefficient, k is the stiffness, and $F_0\sin(\omega t)$ is the oscillating force. $\ddot{x}$ and $\dot{x}$ are acceleration and velocity. The solution to this equation gives the steady-state oscillating displacement $x(t)$.
From that equation, we derive the Frequency Response Function (FRF) in dimensionless form. It describes the amplitude of the system's response relative to the static displacement.
$H(r)$ is the dynamic amplification factor (the height of the curve). $r = \omega / \omega_0$ is the frequency ratio. $\zeta = c / (2\sqrt{mk})$ is the damping ratio, the key parameter you adjust in the simulator. When $r=1$ and $\zeta$ is small, the denominator gets tiny, making $H(r)$ very large—this is resonance.
Real-World Applications
Structural Engineering & Earthquake Design: Engineers use FRF analysis to design buildings and bridges to withstand seismic activity. By knowing the structure's natural frequency (f₀) and adding appropriate damping (ζ), they can ensure the building's resonance peak doesn't align with the dominant frequencies of expected earthquakes, preventing catastrophic amplification of ground motion.
Automotive Suspension Tuning: The ride comfort of a car is essentially a 1DOF vibration problem (the car body on its springs and shock absorbers). Tuners adjust spring stiffness (which changes f₀) and shock absorber damping (ζ) to flatten the FRF curve over the frequency range of typical road bumps, minimizing how much the car body shakes in response.
Microelectromechanical Systems (MEMS): Tiny sensors like accelerometers and gyroscopes in your phone are microscopic vibrating masses. Their sensitivity depends on a very sharp resonance peak. Engineers use this exact 1DOF model to design them, carefully controlling damping (often from air friction) to get the desired peak height and width for accurate signal measurement.
Machinery Vibration Analysis: When a pump, turbine, or fan rotates, it applies periodic forces at its rotation speed. Maintenance teams measure the FRF of the machine's housing or supports to predict vibration levels. If the machine's running speed (ω) gets close to the support's natural frequency (ω₀), the high amplification shown in the simulator warns of potential for noise, wear, or failure.
Common Misconceptions and Points to Note
When you start using this simulator, there are a few points that are easy to misunderstand. First, you might think "a damping ratio of ζ=0 is the most dangerous," but in reality, perfect zero damping (undamped) conditions do not exist. Air resistance and internal material friction always provide some damping. In practice, problems arise when ζ becomes extremely small, such as below 0.01. For example, with minute damping in a turbine blade, the amplitude at resonance can exceed 100 times the design value. Conversely, remember that increasing ζ too much (e.g., above 0.5) involves a trade-off where the system response becomes too sluggish, worsening controllability and energy efficiency.
Next, the assumption that 'there is only one natural frequency'. This tool deals with a single degree of freedom, so there is only one natural frequency, but real objects have infinite degrees of freedom. For example, an automobile body has multiple natural frequencies (modes) such as vertical bouncing, fore-aft pitching, and lateral rolling. Even if you avoid only the first mode (the lowest frequency), the possibility of resonance in higher-order modes remains.
Finally, pitfalls in parameter settings. In the simulator, you manipulate the natural frequency f_n and damping ratio ζ directly, without touching mass m and stiffness k. This is convenient, but when you need to think in practical terms like "what happens to the natural frequency if I double the stiffness?", you must return to the formula $\omega_n = \sqrt{k/m}$. Doubling stiffness k increases the natural frequency by a factor of √2 (approximately 1.414). It's important to develop an intuitive understanding that the influence of changes in mass and stiffness on frequency comes through the square root.
Resonance amplitude = static displacement × Q. For a static load F = 100 N: x_st = F/k = 100/9870 ≈ 10.1 mm, so resonance amplitude ≈ 101 mm
Design Criteria: ISO 10816 (vibration evaluation). To avoid resonance, it is recommended to separate the excitation frequency from f_n by at least 20%.