Resonance frequency

A reed frequency meter for electromechanical measurement of the mains frequency shows the measured value of 49.9 Hz. Each of the white points visible in the middle marks the end of a spring pendulum . When excited by an electromagnet, the pendulum is made to oscillate whose resonance frequency corresponds best to the frequency of the alternating current flowing through the coil of the magnet .

The resonance frequency is the frequency at which the amplitude of a forced oscillation becomes maximum (see amplitude resonance ). If a system has several natural frequencies, it has several resonance frequencies, i.e. H. (local) maxima of the forced amplitude.

A small exciting force is sufficient to produce vibrations of large amplitude when the frequency of the excitation is close to the resonance frequency.

In some cases, the resonance frequency is also understood to mean the frequency at which the resulting oscillation of the system has a phase angle of 90 ° to the exciting oscillation ( phase resonance ); this is the case with the undamped natural frequency .

In weakly damped systems, the difference between amplitude and phase resonance is small, as is the difference between natural frequency and resonance frequency.

There are several resonance frequencies depending on the number of degrees of freedom of the system.

With increasing damping of the system, the resonance frequency decreases.

properties

If the oscillation system is excited close to the resonance frequency, large amplitudes occur with little damping. In the vicinity of the resonance frequency, the phase between exciting and excited vibration changes particularly strongly. With increasing deviation of the excitation frequency from the resonance frequency, the amplitude is reduced. See also the magnification function .

Large amplitudes are often undesirable, e.g. B. in buildings, cable car cabins, overhead lines etc. and can lead to resonance disaster . Vibration absorbers are installed to avoid damage .

With electrical oscillating circuits or in acoustics for sound generation, the resonance effect is sometimes desirable if the amplitude is to be increased. With loudspeakers, on the other hand, resonance frequencies should not occur if possible, because some tones are reproduced particularly loudly.

Examples of undamped systems

Resonance frequencies occur in systems with at least two different types of energy storage. In simple (theoretical) systems without damping, the resonance frequency is equal to the undamped natural frequency (characteristic frequency) . In damped systems, the frequency at which the maximum amplitude occurs is always lower than the undamped natural frequency. ${\displaystyle f_{0}}$

• The Thomson oscillation equation applies in an LC oscillating circuit

${\displaystyle f_{0}={\frac {1}{2\pi {\sqrt {LC}}}}}$

where stand for the inductance of the coil and for the capacitance of the capacitor. The field energy of the capacitor is periodically converted into the magnetic energy of the coil. ${\displaystyle L}$ ${\displaystyle C}$

• A spring of hardness and a mass piece form a mechanical oscillation system of the natural frequency ${\displaystyle D}$ ${\displaystyle m}$

${\displaystyle f_{0}={\frac {1}{2\pi }}{\sqrt {\frac {D}{m}}}}$

• A thread pendulum of the length leads to oscillations of the frequency under the influence of the acceleration of gravity ${\displaystyle l}$ ${\displaystyle g}$

${\displaystyle f_{0}={\frac {1}{2\pi }}\cdot {\sqrt {\frac {g}{l}}}}$

out.

• The earth and the ionosphere, both of which are good electrical conductors, delimit a spherical cavity resonator whose Schumann resonances can be calculated:

${\displaystyle f_{n}={\frac {c}{2\pi a}}{\sqrt {n(n+1)}}}$

where is - there are multiple resonances. is the speed of light and the radius of the earth. ${\displaystyle n=1,2,3,\ldots }$ ${\displaystyle c}$ ${\displaystyle a}$

• A laser resonator of this length usually has a very large number of closely spaced resonance frequencies ${\displaystyle L}$

${\displaystyle f_{n}={\frac {nc}{2L}}}$

Quantum Mechanical Systems

Quantum mechanical systems are only to a limited extent classical oscillatory systems. Nevertheless, one speaks of resonance frequencies here too. In contrast to classical systems capable of vibrating, interactions can only take place at the respective resonance frequencies. At the same time, every frequency in such a system corresponds to a certain energy of a particle, and thus every resonance frequency corresponds to what is called a resonance energy .

The fact that every propagation can be described as the propagation of a wave , but every interaction as an interaction of particles , is called wave-particle dualism .

For example, light spreads in the form of electromagnetic waves , but interactions such as absorption and emission take place in the form of photons . Each photon corresponds to an amount of energy determined by the frequency of the radiation. If a photon is absorbed or emitted by an electron of an atom , it is said that the photon (or the electromagnetic field ) and the electron are "in resonance". An emission line forms in a spectrum at the corresponding frequency of the emitted photon .

See also

• Sound box
• Resonator
• Resonance absorption
• Vibration isolation

literature

• E. Meyer and D. Guicking: Schwingungslehre . Vieweg-Verlag, Braunschweig 1981, ISBN 3-528-08254-2 .
• Walter K. Sextro, Karl Popp , Kurt Magnus : Vibrations: An introduction to physical principles and the theoretical treatment of vibration problems . Vieweg + Teubner, 2008, ISBN 978-3835101937 .