Change mass, spring, damping and driving frequency to explore amplitude and phase.
System Parameters
Mass m
kg
Spring stiffness k
N/m
Damping ratio ζ
Drive frequency f_d
Hz
Presets
Resonance Stats
Results
1.59 Hz
Natural freq. fₙ
5.0
Q factor
5.05
Maximum amplitude ratio
0.32 Hz
Bandwidth Δf
The maximum amplitude ratio is the peak of the one-degree-of-freedom response normalized by the static displacement F₀/k. For ζ ≥ 1/√2 there is no resonant peak; the low-frequency limit is 1.00. Δf = fₙ/Q is a bandwidth approximation for light damping. Steeneken: Forced vibrations
Mass
About this diagram
The mass, spring and force illustrate a one-degree-of-freedom model. Displacement is automatically scaled to fit the view, not shown at physical length. Numerical response and resonance gain are unchanged.
Tune mass, spring stiffness, and damping ratio to watch resonance curves reshape in real time. Q factor, half-power bandwidth, and phase lag calculated instantly — the core of CAE vibration analysis.
What is Resonance?
Explanation in dialogue
🙋
What exactly is resonance? I hear it can make bridges collapse, but the simulator just shows a wiggling mass and a graph.
🎓
Basically, it's when a system vibrates with maximum amplitude because you push it at just the right frequency. In this simulator, the "Drive frequency" slider controls how fast you push. When it matches the system's natural frequency—which depends on the Mass and Stiffness you set—the amplitude spikes. Try setting damping (ζ) very low and slide the drive frequency near the peak to see a huge response.
🙋
Wait, really? So the damping ratio (ζ) just controls how tall and pointy that spike is? What's the "Q factor" number that pops up?
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Exactly! The damping ratio tells you how much energy is lost per cycle. The Q factor, or Quality factor, quantifies the "sharpness" of the resonance peak. It's calculated as $Q = 1/(2\zeta)$. A high Q (low ζ) means a very tall, narrow peak—the system "rings" for a long time. In practice, try moving the ζ slider from 0.01 to 0.5 and watch the peak get lower and wider. The bandwidth shown is the frequency range around resonance where the vibration is strong.
🙋
So if I'm designing something, I want to avoid resonance, right? But sometimes engineers seem to use it, like in musical instruments.
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Great point! It's a double-edged sword. For a bridge or a turbine blade, you absolutely must ensure its natural frequency (which you can calculate here with $\omega_n = \sqrt{k/m}$) is far from any driving frequency, like wind vortices. But for a radio tuner or a guitar, you want a high-Q resonance to pick out or amplify a specific frequency. The phase lag curve in the simulator shows how the vibration timing shifts relative to the driving force, which is also crucial for control systems.
Physical Model & Key Equations
The core equation governing the steady-state amplitude X of a damped, driven harmonic oscillator is shown below. It tells you how much the mass moves for a given driving force.
$X$: Vibration amplitude [m] $F_0$: Amplitude of the driving force [N] $k$: Spring stiffness [N/m] $r$ : Frequency ratio $r = \omega / \omega_n$ $\zeta$: Damping ratio (dimensionless)
The denominator gets smallest when $r \approx 1$ (resonance), making $X$ largest, especially if $\zeta$ is small.
These related equations define the system's inherent properties and the sharpness of its resonance.
$\omega_n$: Natural angular frequency [rad/s]. Set by mass (m) and stiffness (k) in the simulator. $Q$: Quality factor. A single number for resonance sharpness. High Q = low damping = narrow bandwidth.
The bandwidth $\Delta \omega$ is approximately $\omega_n / Q$, which you can observe directly on the frequency response graph as you change $\zeta$.
Real-World Applications
Structural Engineering & Earthquake Design: Buildings have natural frequencies based on their mass and stiffness. Engineers use damping devices (like tuned mass dampers) to increase ζ and lower Q, preventing dangerous resonance during earthquakes or strong winds. The simulator's low ζ values (~0.02) mimic lightly damped steel frames.
Automotive Suspension Tuning: A car's suspension is a spring-mass-damper system. Engineers tune the damping ratio (ζ) to be near 0.2–0.3 for a balance of comfort (isolating road bumps) and handling (keeping wheels in contact with the road). The phase lag plot helps understand wheel response timing.
Electronics & Filter Design: Resonant circuits (RLC circuits) are mathematically identical to this mechanical model. High-Q circuits are used in radios to select a specific station frequency from the airwaves, while low-Q, broad-bandwidth filters are used for digital signal processing.
Musical Instruments & Acoustics: The body of a guitar or violin is designed to have high-Q resonances at specific frequencies to amplify sound beautifully. Conversely, microphone and speaker designers sometimes aim for low Q (high damping) to get a "flat" frequency response without coloration.
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 "the resonant frequency is determined solely by the spring constant", but in reality, it's based on the natural frequency $\omega_n = \sqrt{k/m}$, which is determined by the combination with mass. For example, if you double the stiffness of the spring but also double the mass simultaneously, the resonant frequency remains unchanged.
Next, the idea that "setting the damping ratio ζ to zero results in infinite vibration". Theoretically, that's true, but in the real world, friction and air resistance always exist, so a system with ζ=0 does not exist. In practical engineering when analyzing metal structures, estimating this "invisible damping" can be quite challenging. When using materials not listed in catalogs, you need to infer values from similar materials or measure them through experiments.
Also, make sure you firmly grasp the meaning of the graph's vertical axis, "Amplification Factor". This indicates how many times the static displacement ($F_0/k$) is multiplied . For instance, even if the amplification factor reads 10, the actual displacement can be minuscule if the external force $F_0$ is small. To assess the "danger" of resonance, you need to look at both this amplification factor and the magnitude of the actual applied force. Don't get overly excited or worried just by looking at the simulation result numbers.