---
title: "Tuned-Mass Damper Design Interactive Lab | Sandbox Physics"
description: "Can a small tuned mass keep a structure from shaking hard at resonance? With careful tuning and damping, it can. Tuned near the structure's own frequency, the small mass splits the single tall resonance into two lower peaks, and the right damping in the absorber brings them to similar heights. Check the response across the whole frequency band, not at one frequency, and make sure the absorber has room to swing."
canonical_url: "https://sandboxphysics.com/labs/tuned-mass-damper/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/tuned-mass-damper/"
last_modified: "2026-10-22"
---

M067 · Vibration design

# Tuned-Mass Damper Design

**Can a small tuned mass keep a structure from shaking hard at resonance?** With careful tuning and damping, it can. Tuned near the structure's own frequency, the small mass splits the single tall resonance into two lower peaks, and the right damping in the absorber brings them to similar heights. Check the response across the whole frequency band, not at one frequency, and make sure the absorber has room to swing.

Two carts are joined by a spring and a dashpot in parallel, shown in three dimensions. Release the structure by hand, watch both masses move and compare the exact force-response curves with a measurement over three cycles. Then search for tuning and damping that keep the relative stroke within a limit.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The model has two linear horizontal degrees of freedom driven by a harmonic force. It does not model earthquakes or base motion. The absorber position is measured from the ground, and its stroke is measured relative to the primary mass. The exact curves are steady references reached after infinite time; a finite window can still contain startup transients. The search tries 21 tuning values and 18 damping values, with 101 frequencies across the displayed band. It returns the smallest feasible value on that grid, not a continuous or all-frequency optimum. With zero damping, the bare structure has an unbounded reference amplitude at resonance. The run stops if the motion leaves the five-metre display range; positions are never clipped. In the drawing the two masses rest one behind the other so that both absolute positions can be shown; the relative stroke is the difference of their displacements, not the gap between the blocks. The block sizes, chosen spring rest lengths and rigid drive linkage are schematic. Each record uses one fixed drawing scale that covers both the absolute and the relative travel. The force actuator acts only on the primary mass, and the absorber has no spring or dashpot to the ground. Ideal rails balance gravity.

Physics tutorial

## Protect the structure, budget the stroke

**Background** A small second mass is attached to the main structure by a spring and a dashpot. As it moves, it pushes back on the structure, changes how the structure responds and creates two resonances.

**Why it matters** Lecture 26 of MIT Engineering Dynamics derives the harmonic response of a two-degree-of-freedom system and demonstrates a dynamic absorber. This Lab adds a measurement over a finite time window and a finite grid search that keeps the absorber stroke within a limit.

Start with the essentials

- **Focus question**: Can taming one resonance create a worse peak nearby, or make the absorber travel too far?
- **One-sentence intuition**: Judge a damper by the response across the whole displayed frequency band and by the relative stroke, not by the structure amplitude at one frequency.

### Core mathematical model

#### Two coupled force balances

$$
M\ddot x+C\dot x+Kx+c(\dot x-\dot y)+k(x-y)=F_0\cos\omega t,\quad m\ddot y+c(\dot y-\dot x)+k(y-x)=0
$$

Both positions are measured from the fixed ground; only the connecting spring and dashpot depend on the relative motion.

#### Tuning and mass ratio

$$
\omega_0=\sqrt{K/M},\quad \omega_a=\sqrt{k/m},\quad \mu=m/M
$$

The absorber frequency is the frequency of the absorber on its own; once it is coupled to the structure, the actual modes are different.

#### Complex dynamic stiffness

$$
\begin{bmatrix}K+k-M\omega^2+i\omega(C+c)&-k-i\omega c\\-k-i\omega c&k-m\omega^2+i\omega c\end{bmatrix}\begin{bmatrix}X\\Y\end{bmatrix}=\begin{bmatrix}F_0\\0\end{bmatrix}
$$

Solving this system gives the steady harmonic response after infinite time, which serves as an independent reference.

#### Independent energy balance

$$
E=\tfrac12M\dot x^2+\tfrac12m\dot y^2+\tfrac12Kx^2+\tfrac12k(y-x)^2,\quad E-E_0=W-Q
$$

The dissipated energy is integrated from the power lost in both dashpots; it is not found by subtracting energies.

#### Design search on a grid with a stroke limit

$$
\min_{r_a,\zeta_a}\max_{\omega\in\mathcal G}|X(\omega)|,\quad \max_{\omega\in\mathcal G}|Y(\omega)-X(\omega)|\le s_{\max}
$$

The search checks only a finite grid of frequencies and parameters. The true maximum between grid points can be larger.

### Common difficulties

#### Tuning does not remove the two resonances

**Typical misconception** If the notch is perfect, the structure is safe at every frequency.

**Better mental model** The notch sits between the two split resonance peaks. Check both peaks and the stroke curve.

#### Position is not stroke

**Typical misconception** The absorber position tells you how far it travels on its mount.

**Better mental model** Travel is measured relative to the main structure, which is itself moving.

#### The grid search has limits

**Typical misconception** The search finds the best possible damper.

**Better mental model** It only tests one fixed frequency band, one finite grid and one convention for harmonic loading.

### Run the experiment

1. 01

   #### Compare tuning

   Use Acquire full record with Tuned absorber and again with Mistuned. Orbit the two carts and look at the equal and opposite forces where they are joined; compare the absolute displacement with the relative stroke.

   **What to observe:** The motion of both masses and the resonance peaks change together.

2. 02

   #### Find the antiresonance

   Choose Ideal antiresonance and use Acquire full record.

   **What to observe:** The exact steady response of the structure has a notch, yet the finite record can still show a startup transient that dies away.

3. 03

   #### Limit the absorber stroke

   Lower Allowed steady relative stroke and use Search feasible grid minimum.

   **What to observe:** The best grid point that meets the limit may change, or no grid point may meet it at all.

4. 04

   #### Check the calculation

   Choose Conservative exchange and open Energy & power without resetting the cursor. Compare the separate energy terms, then halve the integration step.

   **What to observe:** Energy keeps moving between the masses, while the energy-balance error shrinks.

Physical conclusion

Tuning, damping and the absorber mass together set the trade-off between how much the structure moves and how far the absorber travels.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)