resonance
In physics and technology, resonance (from the Latin resonare "echo") is the increased resonance of a vibratory system when it is subject to a time-varying influence. The system can deflect many times more strongly than when the stimulus is constantly acting at its maximum strength. In the case of periodic excitation, the excitation frequency or an integral multiple thereof must be close to a resonance frequency of the system. The phenomenon can occur in all oscillatory physical and technical systems and also occurs frequently in everyday life. In technology, resonances are often used to filter out or amplify a certain frequency. However, where amplification is not desired, undesired resonances must be avoided.
The increasing deflections in the case of resonance arise from the fact that the system absorbs and stores energy again with each oscillation. To prevent the system from escaping from the oscillatable amplitude range ( resonance catastrophe ) or from being destroyed by excessive deflections , 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 increase in the deflections is limited by the fact that the supplied energy 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 reverses again and again, because excitation and oscillating system “out of step”.
As a result, the steady-state oscillation is established over time, in which the amplitude remains constant and the oscillation frequency corresponds to the excitation frequency. The energy that continues to be supplied in each oscillation is then completely consumed by the damping. After the excitation has been 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 areas, for example in mechanics , acoustics , structural dynamics , electricity , optics and quantum physics . In modern quantum physics, the equation applies that assigns the frequency of an oscillation to each amount of energy by means of Planck's constant . Instead of the resonance at a certain frequency, one considers here the resonance at a certain energy, which corresponds to the difference between the energies of two different excitation states of the system under consideration.
history
The term resonance comes from acoustics , where it has long been used to describe the clearly noticeable resonance of strings with tones of a suitable pitch. The excitation of large vibrations by periodically acting forces of the correct frequency was already described in Galileo's studies in 1602 and 1638 on pendulums and strings , which were at the beginning of modern science. However, he also assumed that vibrations with other than the natural frequency could not be excited at all. A corresponding equation of motion for a mass point (without damping the motion) was first established by Leonhard Euler in 1739 . His general solution already contained the oscillation with the frequency of the exciting force in superposition with an oscillation with the natural frequency, as well as in the case of the equality of both frequencies, the unlimited increase of the oscillation amplitude. However, he viewed these results, which resulted from the calculation, as “strange” theoretical predictions. In 1823 Thomas Young dealt with the mechanical resonance including damping in connection with the tides and for the first time gave the complete calculation of the resonance curve and phase shift. In connection with the generation and detection of electrical and magnetic vibrations, Anton Oberbeck found the same phenomena for the electrical oscillating circuit , whereupon he expanded the meaning of the term "resonance" accordingly. The discovery of electromagnetic waves by Heinrich Hertz and their use for wireless telegraphy by Guglielmo Marconi from 1895 onwards quickly made electromagnetic resonance very important in science and technology.
However, the 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 technical 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 collapsed due to resonance, and the long drive shafts of larger steam ships had unexpectedly strong vibrations at certain speeds, which had already led to destruction several times. For an overview of the sluggish development of the scientific understanding of resonance, which has so far gone unnoticed, see the article by Mark Buchanan, which is based on a detailed study that is only accessible as a preprint.
Everyday examples
Resonance occurs frequently in everyday life. However, not all vibrations are the result of a resonance.
When you repeatedly “give momentum” to a children's swing, you always give the swing a push when it swings forward. The excitation shocks occur periodically and obviously precisely with the frequency of the rocking oscillation: it is therefore a matter of resonance. Note that the force applied to the stimulating thrusts is by no means like a sine curve, it is sufficient that it occurs periodically. The excitation frequency can also be an integral fraction of the oscillation frequency if one z. B. only push every second or third time.
It is different with a stationary pendulum if you give it a single push. Even if the result is similar, namely that the pendulum is now swinging, there is no periodic excitation and it is not a question of resonance.
Everyone knows the situation in the canteen: you carry a plate of soup on a tray. If the frequency with which the soup sloshes back and forth in the plate just coincides with your own step frequency, this oscillation swings up with every step until the soup spills over, or you walk slower or faster. But not every spill over is a matter of resonance: The frequency with which coffee sloshes back and forth in a coffee cup (the natural frequency of the coffee in the cup) is significantly higher than the normal step frequency , namely about two to three times as high . However, it also happens that if someone suddenly comes around the corner, you have to stop abruptly and the coffee spills over. There is no periodic excitation and therefore no resonance. The coffee spills over - analogous to the pendulum that is only pushed once - due to conservation of momentum .
The rotary knob on a transistor radio may be somewhat forgotten in the age of radios with automatic station selection and preprogrammed program buttons: it changes the variable capacitor in an LC resonant circuit so that the resonant circuit is set to a certain frequency. Radio waves of this frequency can now be amplified and the small amplitude or frequency changes modulated on them (see amplitude modulation and frequency modulation ) can be converted into the transmitted acoustic signal. The resonance frequency set in the LC resonant circuit filters out the radio waves that were transmitted at a certain frequency.
The drum in a washing machine is suspended by springs that can vibrate at a certain frequency. If this oscillation is poorly dampened, or if the washing machine remains in the frequency range of this oscillation for too long - possibly due to overloading - when the spin cycle starts, this oscillates due to resonance and the entire washing machine begins to shake. Only when a higher speed is reached (and there is no longer any resonance) does this shaking calm down (due to the damping) until the corresponding frequency range is passed through again at the end of the spin cycle and the machine starts shaking again due to resonance. Typically, however, the laundry is drier at the end of the spin cycle, so it creates less imbalance and the shaking at the end of the spin cycle is significantly less.
Loose parts in or on motors can also have a certain natural frequency. If the speed of the motor is precisely at this frequency, the wobbling of such parts can often be heard very loudly, which disappears again at other speeds.
Resonance using the example of the harmonic oscillator
The phenomena associated with resonance can be viewed using the harmonic oscillator , for example a mechanical mass-spring-damper system as shown on the right.
The system is excited by a periodic force that acts on the mass. Different transient processes occur depending on the initial conditions. If the oscillation system was previously at rest, the amplitude initially increases and, if the excitation frequency is close to its natural frequency, it can reach higher values than if the maximum force was constant. If the oscillation system is not overloaded ( resonance catastrophe ) and the damping is not exactly zero, the oscillation gradually changes into a harmonic oscillation with constant values for amplitude, frequency and phase shift compared to the excitation oscillation. This behavior is shown to be perfectly consistent for each type of harmonic oscillator. In reality, most of the systems that can oscillate are only approximately harmonic, but they all show the resonance phenomena in at least a similar way (see anharmonic oscillator ).
Equation of motion
An external force is added to the homogeneous differential equation for a linearly damped harmonic oscillator . This makes the equation inhomogeneous.
This describes the momentary deflection from the rest position, the mass of the body, the spring constant for the restoring force, and the damping constant (see Fig. 1).
Without external force and damping, the system would oscillate freely with its natural angular frequency . With damping , the complex exponential approach quickly leads to , where is. The solution obtained is a free damped oscillation with the angular frequency , the amplitude of which decreases proportionally to .
Constant force
A static constant force of an exciter would result in a constant deflection from the rest position .
Steady state for periodic force
If the exciting force is sinusoidal with the amplitude and the angular frequency , it can be described as the imaginary part of
grasp.
As a stationary solution with constant amplitude d. H. for the steady state one obtains again from the complex exponential approach
The imaginary part of describes a harmonic oscillation
about the rest position . It has the angular frequency of the exciting force, the (real) amplitude
and a constant phase shift from the exciting force of
In it are:
- : the amplitude of the exciter, ie the deflection when the force acts statically .
- : the excitation frequency related to the natural frequency,
- : the related, dimensionless Lehr's damping , which is often also expressed by the quality factor . The quality factor means that it indicates the number of oscillations after which (in the absence of an external force) the amplitude has decayed to the initial value (after oscillations up ).
Amplitude resonance
The dependence of the amplitude on the excitation frequency is also referred to as the amplitude response of the system. The resonance curve is the graph of the amplitude response. Figure 2 shows the dimensionless amplitude ratio for typical value ranges of the parameters for excitation frequency (also shown dimensionlessly as ) and damping .
If the damping is sufficiently weak , a maximum is shown, the amplitude resonance . It is at the resonance frequency and shows the value for the maximum resonance amplitude
- .
The ratio is the resonance exaggeration . The resonance frequency is below the natural angular frequency of the undamped oscillation system and also below the angular frequency at which the free damped oscillation of the system takes place.
With low (but not negligible) damping , the resonance is a sharp maximum, which is almost exactly at the natural angular frequency. The resonance amplitude is then inversely proportional to . The amplitude can therefore reach a multiple of the static deflection in the steady state . During the transient process from the rest position, it can even temporarily increase to almost .
With strong damping, however, there is no resonance with increased amplitude. The maximum amplitude of the steady oscillation is fixed with the value in the static case .
Phase resonance and energy flow
With, the steady oscillation lags the exciting force by exactly 1/4 period (phase response −90 °, also known as phase resonance ). Therefore, speed and force are exactly in phase so that the force always acts in the direction of the current speed. The energy then flows continuously into the system, while at other frequencies it changes direction twice per period because the phase difference is smaller than 90 ° and greater than 90 ° (and up to 180 °) at higher frequencies. The kinetic energy of the steady state reaches its maximum in the phase resonance. It is then as large as the total energy input during the last oscillations.
Energy resonance
The greatest potential energy of an oscillation with amplitude is . The corresponding resonance curve is given by the square of the amplitude response and has its maximum at the frequency of the amplitude resonance .
The greatest kinetic energy in an oscillation with amplitude is . This function has its maximum exactly at .
In the case of the application, which is important for optics, to the emission and absorption of electromagnetic waves by oscillating dipoles, the radiation power is proportional to . The maximum of this function is slightly above .
In the case of sharp resonances, i.e. low damping, the differences between these three resonance frequencies are usually neglected and an approximation formula symmetrical about the natural frequency is used for the resonance range , which is called the Lorentz curve:
- .
This formula shows, in addition to the resonance and the characteristic of forced vibration long streamers and is therefore also suitable for high frequencies and useful.
The energy stored in the vibration system comes from the acceleration work by the stimulating force. The vibration energy is increased when the force acts in the direction of the speed. Otherwise the force draws energy from the system and thus acts as a brake. In the steady state, the energy input compensates for the energy loss due to the damping.
Half width and figure of merit
As a half-value width (engl. Full width at half maximum ) of the resonance of the range of frequencies around the resonant frequency referred to, the applicable in for the amplitude: . According to the approximation formula for the Lorentz curve, these limits are included in the low damping area of interest . Converted to the frequency axis results in the half width
- .
The sharpness of the resonance can be adjusted with the damping or with the quality factor
can be specified.
According to the meaning of the quality factor given above, a period of periods of the natural frequency can be seen as characteristic of the decay of a damped natural oscillation, i.e. also characteristic of the duration of the transient process or, in a figurative sense, of the "memory of the oscillator". If you analyze an oscillation with frequency with the help of a series of resonators at different resonance frequencies , then the determination of the resonance amplitude requires time and delivers the resonance frequency with the accuracy . If two oscillators differ in frequency , then the faster one makes one oscillation more than the slower one during this period . It follows : the more precisely the frequency of an oscillation is to be determined, the longer one has to let it act on a resonator. This is an early form of the frequency-time uncertainty relation .
Resonance with zero damping
Vanishing attenuation is only a theoretical limit case; real systems with very little damping come close to it, however, if one considers them for a period of time that is not too long , but which can include a large number of oscillations.
In the damping-free case, there is no transient process that would lead to a specific stationary oscillation regardless of the initial conditions. A possibly co-excited natural oscillation does not die away here, but remains unabated. With resonant excitation,, there is no stationary solution of the equation of motion, rather the amplitude varies linearly with time. Starting from the state at rest in the rest position , the amplitude increases z. B. proportional to the elapsed time:
In theory, a resonance catastrophe occurs here in any case. In practice, this can be avoided by limiting the amplitude in some other way, i.e. by changing the law of force (see anharmonic oscillator ).
Outside of the exact resonance frequency, however, there is a stationary oscillation under suitable initial conditions. It results from the above equations for . The amplitude ratio is greater at each excitation frequency than in the case with damping. In the case of resonance , the formula for the amplitude diverges and there is no state of stationary oscillation. The phase delay is for frequencies below the resonance, above, as can be seen from the above formula by the limit transition . (For further formulas and explanations see Forced oscillation # limit case of vanishing damping .)
Examples of the occurrence of resonance
mechanics
- In the case of a tongue frequency meter , the one of many flexural vibrators that is in resonance with the excitation frequency is excited to a particularly large vibration amplitude.
- Is a bridge in resonance with the cadence of marching pedestrian masses, the construction can be dangerous get progressively , as Millennium Bridge (London)
- Vehicle bodies tend to vibrate (roar) at certain engine speeds
- Orbital resonance can cause planets to put one celestial body on a collision course with another. This resonance can have a stabilizing effect at Lagrange points , for example the solar observation satellite SOHO has always stayed near the inner Lagrange point L _{1} since 1995 .
- To generate ultrasound for medical or technical applications, electromechanical, mostly piezoelectric , transducers are excited to produce resonant vibrations.
- An ultrasonic drill brings the rock to be drilled into resonance, causing the rock to crumble.
Hydromechanics
Acoustics
- the sound generation of musical instruments (string and wind instruments), see z. B. woodwind instrument , soundboard
- resonating an unplayed string when a same tuned instrument sounds (z. B. sympathetic string )
- In closed rooms, there can be disruptive room resonance at certain frequencies .
- A resonance exhaust enables a certain increase in performance in two-stroke engines at a specific speed.
Acoustic resonance, for example, plays a role in almost all musical instruments, often through the formation of a standing wave .
If you measure the sound pressure and sound velocity at the end of a cylindrical pipe that is open on both sides with suitable microphones , you can calculate the acoustic flow impedance if you know the pipe cross-section . This shows multiple resonances, as we know them as a special case λ / 2 when electromagnetic waves propagate along wires . The measurement result in the picture shows several sharp minima of the flux impedance at multiples of the frequency 500 Hz. A check with the pipe length of 325 mm and the speed of sound in air results in the target value 528 Hz.
Because the measured value of the lowest minimum of around 40,000 Pa · s / m³ deviates considerably from the characteristic acoustic impedance of the surrounding air (413.5 Pa · s / m³), there is a mismatch and the vibrating air column in the pipe is only faintly audible. This low energy loss is expressed in a high quality factor of the resonator.
Electrical engineering
Without resonance, there would be no radio technology with the well-known sub-areas of television , mobile phone , radar , radio remote control and radio astronomy , because without the ability to separate transmission frequencies, there could only be a few isolated transmitters worldwide with sufficient spacing. In the majority of all oscillator circuits and electrical filters , resonant circuits are used, which use Thomson's vibration equation for the resonance frequency
underlying. The efficiency of antennas and Tesla transformers is drastically increased by resonance.
The safety in the railway network is improved by inductive train control . Here, an oscillating circuit attached to the vehicle enters into resonant interaction with an oscillating circuit attached to the track, the frequency of which is different depending on the position of the next train signal; When the signal is in the "Halt" position, an emergency brake is triggered.
The large particle accelerators in elementary particle physics are based on resonance effects, as is nuclear magnetic resonance spectroscopy in chemistry and magnetic resonance tomography in medicine.
RFIDs , also known colloquially as radio tags, enable the automatic identification and localization of objects and living beings. The operating energy is transmitted to the RFID by resonance and the RFID sends its information back in the same way.
An absorption frequency meter acts like a selective voltmeter at resonance.
A magnetron only generates vibrations if the speed of rotation matches the natural frequency of the cavity resonators .
Atomic and Molecular Physics
In atomic and molecular physics , one speaks of resonance when a photon of energy ( h : Planck's quantum of action , ν : frequency of light) is absorbed in the shell of the atom . This is only possible if the energy difference between two states G and A of the electron shell is exactly the same . An electron is then raised from state G to state A. The excitation probability of such a transition is described by a Lorentz curve:
The process is called resonance absorption . For example, he explains the Fraunhofer lines in the spectrum of sunlight.
Usually the electron now falls back from the excited state to the ground state, whereby a photon of the energy is emitted again. This happens either spontaneously ( spontaneous emission , fluorescence , phosphorescence ) or by impact of a second irradiated photon of the same energy ( stimulated emission , exploited by the laser ).
The atom can now be excited again from the ground state. It can therefore carry out an oscillation of the population numbers between the states G and A, which is referred to as the Rabi oscillation . As mentioned, the oscillation only occurs when the incident photons are in resonance with the energy levels of an atom. Such resonances can e.g. B. can be used to identify gases in spectroscopy , as they allow the measurement of the atomic or molecular-typical energy levels.
There are three different types of cones (color receptors) in the human eye . The opsin molecules contained therein differ in their spectral sensitivity and, when they resonate with photons of a suitable wavelength, trigger intracellular signal cascades (see phototransduction ). Electrical signals are generated that are passed on to the brain via the ganglion cells . A color impression is created there from the transmitted signals (see color perception ).
Further resonance phenomena occur when the magnetic moment of an atom, atomic nucleus, molecule or electron ( spin ) is coupled to a magnetic field, for example electron spin resonance and nuclear magnetic resonance . A magnetic field oscillating at a suitable frequency stimulates the flipping of the spin between two discrete states of different energies. This effect can also be described according to the Rabi oscillations and is z. It is used, for example, in medical technology and for material examinations (see e.g. magnetic resonance imaging ).
Nuclear physics
In nuclear physics, resonance means that when there is a collision with certain kinetic energy, the two partners unite to form a temporarily bound system , the compound core , in one of its possible energy states . The cross section shows a maximum of the form of in this impact energy Breit-Wigner curve , which is typical for resonances Lorentz curve is similar.
Such a system cannot be stable, but rather disintegrates after a short time, e.g. B. in the two particles from which it was formed. But from the decay width of the curve it can be seen that it has existed much longer than would correspond to a reaction of the particles in the flyby.
All larger nuclei show the giant resonance , an excited state in which the protons vibrate together in relation to the neutrons .
The resonance absorption of gamma quanta enables the comparison of excitation energies with an accuracy of more than 10 ^{12} by utilizing the Doppler effect . Atomic nuclei are resonators with z. Some extremely high quality factors of 10 ^{12} and upwards (e.g. quality factor of ^{99 }Tc : 6.8 · 10 ^{24} ). ^{}^{}^{}^{}
Particle physics
Similar to the compound core formation, an unstable but comparatively long-lived bound system or even a single, different particle can arise from two collision partners if the collision energy in the center of gravity system is just sufficient. The excitation function of the impact process, i.e. its cross-section plotted as a function of the energy, then shows a maximum at this energy with the curve shape typical for a resonance. Systems formed in this way are often referred to as resonance or resonance particles. The half- width of the curve (see decay width ) can be used to determine the lifetime of the resulting particle, which is too short for a direct measurement .
See also
Individual evidence
- ↑ Andrea Frova, Mariapiera Marenzana: Thus spoke Galileo. The great scientist's ideas and their relevance to the present day. Oxford University Press, Oxford u. a. 2006, ISBN 0-19-856625-5 , pp. 133-137 .
- ^ Stillman Drake : Essays on Galileo and the history and philosophy of science. Volume 1. Selected and introduced by Noel M. Swerdlow and Trevor H. Levere. University of Toronto Press, Toronto u. a. 1999, ISBN 0-8020-7585-1 , pp. 41-42 .
- ↑ Leonhard Euler: Letter of May 5 (16) 1739 (No. 23) to Johann Bernoulli . In: Emil A. Fellmann, Gleb K. Mikhajlov (eds.): Euler: Opera Ommnia (= Correspondence Series Quarta A Commercium Epistolicum ). tape II . Birkhäuser, Basel 1998, p. 58, 303 ( online ).
- ↑ Thomas Young: Tides: From the Supplement to the Encyclopedia Britannica 1823: Printed in: Peacock, George (ed.) (1855). Miscellaneous works of the late Thomas Young. London: J. Murray. Vol. 2, pp. 291ff
- ↑ Anton Oberbeck: About a phenomenon similar to resonance in electrical oscillations . In: Annals of Physics . tape 262 , no. 10 , 1885, p. 245-253 .
- ^ Johann Friedrich Radinger: About steam engines with high piston speed . Printed and published by Carl Gerold's Sohn, 1892.
- ^ Arnold Sommerfeld: The scientific results and the goals of modern technical mechanics . In: Naturwissenschaftliche Rundschau . tape 49 , 1903, pp. 621-624 .
- ↑ Armin Hermann: The bridge between mathematics and technology: Arnold Sommerfeld's merits in a scientific technical mechanics . In: Physical sheets . tape 23 , no. 10 , 1967, p. 442-449 .
- ↑ Mark Buchanan: Going into resonance . In: Nature Physics . tape 15 , no. 3 , 2019, p. 203 . (English, accessed July 19, 2020 )
- ↑ Jörn Bleck-Neuhaus: Mechanical resonance: 300 years from discovery to the full understanding of its importance. Retrieved July 19, 2020 . (engl.)
- ↑ Measurement of the acoustic flow impedance (English; PDF; 856 kB)