A single-story building may be thought of as a single mass (the roof) supported by elastic walls that also have damping. This system behaves similarly to the single spring-mass-damper system you studied last year.
,
and the kinetic energy is the energy of the structure's mass (floors and roof) in motion,
.
During oscillatory vibration (like ground motion in an earthquake), motion is sinusoidal:
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and
.
Whenever the kinetic energy is zero, 
and
.
Setting the maximum kinetic energy equal to the maximum potential energy in order to describe free oscillation results in a solution for 
of the form
.
This solution
is the natural frequency.
When an undamped system with a single degree of freedom (one story building) is allowed to oscillate freely from some initial displacement or velocity,
it will always oscillate at its natural frequency,
.
| Resonance |
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| Resonance works like pushing someone on a swing. If you push at the right intervals, the person starts going higher and higher, even if you aren't pushing very hard. |
is sinusoidal with a frequency equal to the building's natural frequency, then the building will resonate and the amplitude of its dynamic response can become extremely large.
Resonance amplifies the vibration of a building, potentially to the point of breaking, because the external force is at the same frequency as the building already vibrates after an initial displacement.
The role of damping in building vibrations is primarily to limit the amplitude of the vibration during resonance.
is the amplitude, 
is the phase-shift of the oscillation.