Resonance

In physics and technology, resonance (from the Latin resonare "to reverberate") is the increased resonance of a vibratory system when it is subject to a time-varying influence. In this case, the system can oscillate many times more strongly than when it is subjected to the constant effect of the excitation with its maximum strength. In the case of periodic excitation, the excitation frequency or an integral multiple thereof must be close to a resonant frequency of the system. The phenomenon can occur in all oscillating physical and technical systems and is also common in everyday life. Resonances are often exploited in engineering to filter out or amplify a particular frequency. However, where amplification is not desired, unwanted resonances must be avoided.

The increasing deflections in the case of resonance are caused by the fact that the system absorbs and stores energy again with each oscillation. In order to prevent the system from leaving the oscillating amplitude range (resonance catastrophe) or being destroyed by excessive excursions, its damping can be increased, its natural frequency or the excitation frequency can be changed, or the strength of the excitation can be reduced. The initial growth of the excursions is limited by the fact that the energy supplied is increasingly consumed by the damping (e.g. friction), or by the fact that if the difference between the resonance frequency and the excitation frequency is too great, the energy flow is always reversed because the excitation and the oscillating system get "out of sync".

As a result, in the course of time the state of steady-state oscillation is established, in which the amplitude remains constant and the oscillation frequency coincides with the excitation frequency. The energy that continues to be supplied in each oscillation is then completely consumed by the damping. After the excitation is switched off, the system gradually comes to rest in the form of a damped oscillation with its natural frequency.

The phenomenon of resonance plays an important role in physics and technology in many fields, for example in mechanics, acoustics, structural dynamics, electricity, earth science, astronomy, optics and quantum physics. In modern quantum physics, the equation which assigns to each amount of energy by means of Planck's constant the frequency of an oscillation. Instead of the resonance at a certain frequency, one considers here the resonance at a certain energy, which corresponds to the difference of the energies of two different excited states of the system under consideration.

History

The term resonance originates from acoustics, where it has always referred to the clearly noticeable resonance of strings with tones of suitable pitch. The excitation of large oscillations by periodically acting forces of the right frequency was already described in Galileo's investigations in 1602 and 1638 on pendulums and strings, which were at the beginning of modern natural science. However, he also assumed that oscillations with frequencies other than the natural frequency could not be excited at all. A corresponding equation of motion for a point of mass (without damping of the motion) was first set up by Leonhard Euler in 1739. His general solution already contained the co-oscillation with the frequency of the exciting force in superposition with an oscillation with the natural frequency, as well as in the case of equality of both frequencies the unlimited increase of the oscillation width. However, he considered these results, which resulted from the calculation, as a "whimsical" theoretical prediction. In 1823, in connection with tides, Thomas Young treated mechanical resonance, including damping, and gave for the first time the complete calculation of resonance curve and phase shift. In connection with the generation and detection of electric and magnetic oscillations, Anton Oberbeck found the same phenomena for the electric oscillating circuit, whereupon he extended the meaning of the term "resonance" accordingly. The discovery of electromagnetic waves by Heinrich Hertz, as well as their use for wireless telegraphy by Guglielmo Marconi from 1895, then quickly gave electromagnetic resonance great importance in science and technology.

However, mechanical resonance was essentially only properly appreciated from the beginning of the 20th century, after the physicist and mathematician Arnold Sommerfeld - as the first professor of engineering mechanics who had not previously been an engineer - had pointed it out. At that time, suspension bridges with marching soldiers or fast-moving steam locomotives had already collapsed due to resonance, and the long drive shafts of larger steamships had already experienced unexpectedly strong vibrations at certain speeds, which had already led to destruction on several occasions.

Everyday examples

Resonance occurs frequently in everyday life. However, not all vibrations are the result of resonance.

When swinging a child's swing repeatedly, one always gives the swing a push when it swings forward. The excitation pushes occur periodically and obviously just at the frequency of the swing oscillation: this is therefore resonance. Note that the force applied to the exciting thrusts is by no means like a sinusoid; it is sufficient that it is periodic. The excitation frequency can also be a whole-number fraction of the oscillation frequency, for example, if you only push every second or third time.

It is different with a pendulum at rest if you give it a single shock. Even if the result is similar, namely that the pendulum now swings, there is no periodic excitation and it is not resonance.

Everyone knows the situation in the canteen: you carry a plate of soup on the tray. If the frequency with which the soup sloshes back and forth in the plate just matches your own step frequency, this oscillation builds up with every step until the soup spills over, or you walk slower or faster. But not all sloshing over is resonance: the frequency at which coffee sloshes back and forth in a coffee cup (the natural frequency of the coffee in the cup) is significantly higher than the usual step frequency, namely about two to three times as high. Nevertheless, it also happens that if someone suddenly comes around the corner, you have to stop abruptly and the coffee spills over. Here, there is no periodic excitation and thus no resonance. The coffee spills over - analogue to the pendulum which is bumped only once - based on conservation of momentum.

The rotary knob on a transistor radio may have been somewhat forgotten in the age of radios with automatic station selection and pre-programmed program knobs: it is used to change the variable capacitor in an LC resonant circuit so that the resonant circuit is set to a specific frequency. Radio waves of this frequency can now be amplified and the small amplitude or frequency changes modulated onto them (see amplitude modulation and frequency modulation) can be converted into the transmitted acoustic signal. The resonant frequency set in the LC resonant circuit filters out just those radio waves that were transmitted at a particular frequency.

The drum in a washing machine is suspended with springs that can oscillate at a certain frequency. If this oscillation is poorly damped, or if the washing machine - possibly due to overloading - remains too long in the frequency range of this oscillation with its speed when the spin cycle starts, then this oscillation builds up due to resonance and the entire washing machine begins to shake. Only when a higher speed is reached (and resonance is no longer present) does this shaking calm down (due to damping) until, at the end of the spin cycle, the corresponding frequency range is passed through again and the machine begins to shake again due to resonance. Typically, however, the laundry is drier at the end of the spin cycle, thus generating less imbalance, and the shaking at the end of the spin cycle is significantly weaker.

Loose parts in or on motors can also have a certain natural frequency. If the speed of the motor is just at this frequency, the wobbling of such parts is often very loudly audible, which disappears again at other speeds.