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Sandbox Physics

Tuned-Mass Damper Design

Can a small tuned mass keep a structure from shaking hard at resonance? A tuned absorber can quiet the structure. Its reaction splits the resonance; suitable damping and enough travel protect the frequency band.

Watch a bridge deck and its sliding absorber move. Change tuning and damping, compare the split resonance peaks, then search for a design with enough travel.

Interactive modelTuned-Mass Damper Design
Deck displacementLoading\text{Loading}
Signed absorber travelLoading\text{Loading}
Acquired deck amplitudeLoading\text{Loading}
Reference band maximumLoading\text{Loading}

Let the small mass push back

BackgroundThe bridge deck pushes its absorber through a spring and dashpot. The absorber pushes back, and the timing of that reaction can keep the deck much quieter.

Why it mattersA quiet point can sit between two strong resonances. A useful design protects a frequency band and allows enough travel.

Focus question
Can a small tuned mass quiet a structure at resonance?
One-sentence intuition
The absorber changes the coupled motion. Tuning controls the peaks; damping and relative travel decide how useful the reduction is.

Core mathematical model

The two masses push on each other

Mx¨=F0cos⁡ωt+f0+faf0=−Kx−Cx˙my¨=−fafa=k(y−x)+c(y˙−x˙)\begin{gathered}M\ddot x=F_0\cos\omega t+f_0+f_a\\f_0=-Kx-C\dot x\\m\ddot y=-f_a\\f_a=k(y-x)+c(\dot y-\dot x)\end{gathered}

Both displacements are measured from ground. The connection depends on their displacement and velocity differences.

Mass and tuning ratios

μ=m/Mω0=K/Mra=k/mω0\begin{gathered}\mu=m/M\\\omega_0=\sqrt{K/M}\\r_a=\frac{\sqrt{k/m}}{\omega_0}\end{gathered}

The isolated absorber frequency sets the tuning ratio. The connected masses have coupled modes at different frequencies.

Energy passes between masses

T=12Mx˙2+12my˙2U=12Kx2+12k(y−x)2Ploss=Cx˙2+c(y˙−x˙)2E˙=F0cos⁡ωt x˙−Ploss\begin{gathered}T=\tfrac12M\dot x^2+\tfrac12m\dot y^2\\U=\tfrac12Kx^2+\tfrac12k(y-x)^2\\P_{\rm loss}=C\dot x^2+c(\dot y-\dot x)^2\\\dot E=F_0\cos\omega t\,\dot x-P_{\rm loss}\end{gathered}

The connecting dashpot removes energy only when the velocities differ. With no drive or damping, energy moves between the masses and springs.

Reduce the largest sampled response

J=max⁡ω∈G∣X(ω)∣max⁡ω∈G∣Y(ω)−X(ω)∣≤smax⁡\begin{gathered}J=\max_{\omega\in\mathcal G}|X(\omega)|\\\max_{\omega\in\mathcal G}|Y(\omega)-X(\omega)|\leq s_{\max}\end{gathered}

The search compares a finite tuning and damping grid. The maximum and travel condition are checked only at its frequency samples.

Common difficulties

A quiet point is not a quiet band

Typical misconceptionIf the deck stops at one frequency, the absorber works at every frequency.

Better mental modelThe quiet point lies between two coupled resonance peaks. Both peaks and the travel still need checking.

Run the experiment

  1. 01

    Detune the absorber

    Compare Protect the bridge with Miss the tuning, then sweep Drive frequency ratio.

    What to observe: The deck motion and split peaks change together.
  2. 02

    Prepare an energy exchange

    Choose Pass energy back and forth. Drag either mass to set a new release.

    What to observe: The drive is off, yet energy passes between the masses.
  3. 03

    Try the quiet point

    Choose Ideal quiet point and compare Late cycles with Startup.

    What to observe: The steady deck response has a notch, but a finite record can retain a startup transient.
  4. 04

    Allow enough travel

    Open Masses, release and sensors, lower Allowed relative travel, then choose Tune for the band.

    What to observe: The best tested design can change, or no tested design may meet the limit.
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