where g is the acceleration due to gravity (about 9.8 m/s2 near the surface of Earth), and L is the length from the pivot point to the center of mass. (An elliptic integral yields a description for any displacement.) Note that, in this approximation, the frequency does not depend on mass.
Mechanical resonators work by transferring energy repeatedly from kinetic to potential form and back again. In the pendulum, for example, all the energy is stored as gravitational energy (a form of potential energy) when the bob is instantaneously motionless at the top of its swing. This energy is proportional to both the mass of the bob and its height above the lowest point. As the bob descends and picks up speed, its potential energy is gradually converted to kinetic energy (energy of movement), which is proportional to the bob's mass and to the square of its speed. When the bob is at the bottom of its travel, it has maximum kinetic energy and minimum potential energy. The same process then happens in reverse as the bob climbs towards the top of its swing.
Some resonant objects have more than one resonance frequency, particularly at harmonics (multiples) of the strongest resonance. It will vibrate easily at those frequencies, and less so at other frequencies. It will "pick out" its resonance frequency from a complex excitation, such as an impulse or a wideband noise excitation. In effect, it is filtering out all frequencies other than its resonance. In the example above, the swing cannot easily be excited by harmonic frequencies, but can be excited by subharmonics.
Resonance may cause violent swaying motions in improperly constructed structures, such as bridges and buildings. The London Millennium Footbridge (nicknamed the Wobbly Bridge) exhibited this problem. A faulty bridge can even be destroyed by its resonance (see "Angers Bridge"; that is why soldiers are trained not to march in lockstep across a bridge, although it is suspected to be a myth, see eg., MythBusters (season 2). Mechanical systems store potential energy in different forms. For example, a spring/mass system stores energy as tension in the spring, which is ultimately stored as the energy of bonds between atoms.
The United States Patent Office classifies devices that tests mechanical resonance under subclass 579, resonance, frequency, or amplitude study, of Class 73, Measuring and testing. This subclass is itself indented under subclass 570, Vibration. Such devices test an article or mechanism by subjecting it to a vibratory force for determining qualities, characteristics, or conditions thereof, or sensing, studying or making analysis of the vibrations otherwise generated in or existing in the article or mechanism. Devices include methods to cause vibrations at a natural mechanical resonance and measure the frequency and/or amplitude the resonance made. Various devices study the amplitude response over a frequency range is made. This includes nodal points, wave lengths, and standing wave characteristics measured under predetermined vibration conditions.
Nikola Tesla established a laboratory on Houston Street in New York at 46 E. There, at one point while experimenting with mechanical oscillators, he allegedly generated a resonance of several buildings causing complaints to the police. As the speed grew he hit the resonance frequency of his own building and belatedly realizing the danger he was forced to apply a sledge hammer to terminate the experiment, just as the astonished police arrived. The Discovery Channel's popular MythBusters show examined Tesla's claim that he had created an "Earthquake Machine" in their 60th episode. They tested the physical phenomenon known as mechanical resonance on a traffic bridge, which today are built to withstand such forces. While a single I-beam of steel was deflected several feet in each direction by their oscillator, and they reportedly felt the bridge shaking many yards away, there were no "earth shattering" effects. It is worth indicating that, in the time of the event undertaken by Tesla, buildings were not built to withstand such resonance.