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Introduction to Building Vibrations
According to the US Geological Survey, the Earth sustains around 500,000 earthquakes a year, with 100 causing significant damage.
This WEAVE module engages with the physics necessary to understand what happens to a building when an earthquake hits.
You will learn about structural dynamics and the vibrations of buildings in order to mathematically describe the behavior of a three-story building in terms of its oscillatory response to earthquakes.
This module uses imaginary and complex numbers, some linear algebra, and ordinary differential equations.
Module Objectives:
- To understand the relationship between the mass and stiffness of buildings and their dynamic characteristics.
- To understand some special physical properties of the dynamic characteristics.
- To examine how changes in damping and stiffness influence how buildings respond to earthquakes.
- To relate experimental measurements to computer simulations of a building's behavior during earthquake-like conditions.
You will find that for a three-story building, one can describe building vibrations using a set of three second-order differential equations, similar to the spring-mass-damper equation that you studied in your first-year physics course.
(The external link to that applet is here .)
The key difference between the single-mass system you studied in physics, and the three-mass system here, is that the three equations used in the three-mass system are inter-related, or coupled.
Physically, this means that the response and oscillations of one floor depends on the oscillations of all the other floors.
In coupled systems, it is often convenient to write differential equations in terms of matrices.
Matrix notation simplifies and unifies the mathematical description of dynamic systems like this three-story building. (If you need to review matrices, a recommended external site is http://www.morello.co.uk/matrixalgebra.htm.)
To get a general idea of what building vibrations are about, please view the videos of our small-scale building model oscillating at various frequencies.
Right now this unfortunately requires Internet Explorer and Windows Media Player.
Browser-independent versions of the video will be up soon.
The
following videos for select sine wave frequencies are available.
All files compressed
using Microsoft Windows Media Encoder using Microsoft MPEG-4 V3
codec.
Viewable using Windows Media
Player 6.4+.
Studying the following sections will give you a deeper understanding of structural dynamics in terms of:
- What defines mass, stiffness, and damping matrices.
- How the dynamic response of all the stories of a building are interrelated, or coupled.
- How differential equations are used to analyze building vibrations, and how to solve certain cases.
- How the stiffness and mass of each story are related to the natural frequencies of the entire building.
- How structures respond, in general, to sinusoidal and random loads (like earthquakes), and how resonant frequencies affect this response.
Subsections
Next: Single Story Vibrations
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Previous: m1index