Tuned Mass Damper System

Tuned mass damper systems have been developed in recent years to diminish the oscillations of buildings during earthquakes. A large mass is hung by springs at the top of the building. The oscillations of the mass dissipates the energy into heat by means of a damper.
This Demonstration shows a system with two masses, and . Let and be the positions of the masses, and be the lengths of the springs, and and be the spring constants. The effect of the damper can be represented by a damping coefficient in the system. The energy decreases whenever the second mass is moving relative to the first. It is said that the damper is "tuned" when the optimal value of is selected for a specific building. Hamilton's equations of motion are solved for the initial conditions and (positions and momenta). Their plots are shown for all three possible combinations of three variables. Below the plot, the state of the masses with the springs and the damper is shown as a function of time.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


The equations of motion are
[1] P. Blanchard, R. L. Devaney, and G. R. Hall, Differential Equations, 4th ed., Boston: Brooks/Cole, 2012 pp. 515–519.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+