# 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 \$.

Creation of a forced oscillation

#### 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 ofExcitation 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.