# What is the phenomenon of mechanical resonance

## Forced vibration resonance

An oscillating system (spring pendulum, oscillating circuit) oscillates with its own when it is left to its own devices **Natural frequency** $ f_E $.

#### Natural frequencies of systems

We have already got to know the following formulas for the natural frequencies:

A spring pendulum oscillates with the natural frequency $ f_E = \ frac {1} {2 \ pi} \ sqrt {\ frac {D} {m}} $.

An oscillating circuit oscillates with the natural frequency $ f_E = \ frac {1} {2 \ pi} \ sqrt {\ frac {1} {LC}} $.

### External excitation of (electromagnetic) vibrations

A vibratory system can also be stimulated from the outside. With a spring pendulum you could z. B. an electric motor with the **Excitation frequency** Use $ f $ to set the spring pendulum to vibrate. In this case one speaks of **forced vibration**.

You can now force an oscillation in an oscillating circuit from the outside. One possibility would be, for example, to connect the resonant circuit to an alternating voltage $ U ~ $. If the excitation frequency of the alternating voltage is $ f $, the system oscillates with the frequency $ f $ and not with its natural frequency $ f_E $.

#### Resonance curves

If you vary the excitation frequency $ f $ of the alternating voltage and plot the amplitude of the current intensity $ I $ in a diagram, you get the following graphic result (**Resonance curves**). The **amplitude** turns out to be one **Function of****Excitation frequency** $ f $.

##### Discussion of the resonance curves

- If the excitation frequency $ f $ is equal to the so-called
**Resonance frequency**$ f_ {Res} $ ($ f = f_ {Res} $), the amplitude of the current intensity is maximum. In this case one speaks of**resonance**. It turns out that this resonance frequency is approximately equal to the natural frequency $ f_E $ of the resonant circuit ($ f_ {Res} \ approx f_E $). So one can use the equation $ f = f_E $ in the case of resonance. Resonance occurs when the exciter of the forced oscillation has the natural frequency of the system / resonant circuit or reaches this frequency range. - The size of the amplitude depends on the
**damping**of the system. This damping is due to an ohmic resistance. The smaller the resistance or the damping, the higher the amplitude in the case of resonance.

The resonance frequency $ f_ {Res} $ is in the range of the natural frequency $ f_E $ ($ f_ {Res} \ approx f_E $).

It should be noted that the same behavior (dependence of the amplitude on the excitation frequency, resonance, resonance curves, etc.) is also observed with forced mechanical vibrations.

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