Resonance

Resonance ( " echo " from the Latin resonare ) is in physics and engineering, the amplified oscillation of a vibratory system. With periodic excitation, the excitation frequency must be in the vicinity of a resonance frequency of the system. If the system is not too heavily damped, it can turn out more many times ( resonant peak ) than in the case that the same excitation is not periodic, but would act with constant strength. The phenomenon can occur in all oscillatory physical and technical systems and is also used in everyday life frequently, eg:

• " give Anschwung " when on a child's swing
• When spilling the coffee in the cup or soup in the dish, if you go a few steps with them,
• When tuning a radio to a particular station
• The sudden shaking or wobbling the washing machine, if at the beginning or end of the spin cycle, the speed goes through a bad value and the damp laundry causes a great imbalance.
• The audible vibration of the compressor on the fridge
• At certain speeds of the engine when it starts or expires.

Resonances are often exploited in the art or just deliberately avoided.

Increasing the fluctuations in the resonance caused by the fact that the system is again at each oscillation receives and stores energy. Thus, the system is not destroyed by excessive deflections ( disaster response ), its damping must be increased, changes its natural frequency or the excitation frequency, or the strength of the excitation can be reduced. The initial growth of the rashes is then limited by the fact that the supplied energy is increasingly on the attenuation (eg friction) consumed, or the fact that in too large difference between resonance and excitation frequency, the energy flow reverses again and again because excitation and oscillating system " out of step " advised. As a result is in the course of time of the steady state of the oscillation forth, in which the amplitude is constant and coincides with the oscillation frequency of the excitation frequency. The continuing supplied in each vibrational energy is then completely absorbed by the damping. After switching off the excitation, the system comes in the form of a damped oscillation at its natural frequency gradually to rest.

Galileo's investigations of vibrations and resonances of commuting and strings ( 1602) were the start of modern physical science. In modern quantum physics, the concept of resonance has experienced an expansion by being also applied to changes in the excitation -energy state of a system. Basis here is the quantum condition, the means of the Planck constant amount of energy attributed to each of the frequency of oscillation.

• 2.1 Equation of motion
• 2.2 Stationary solution: steady state
• 2.3 amplitude resonance
• 2.4 phase response and energy flow
• 2.5 Energy Resonance
• 2.6 FWHM of merit
• 2.7 resonance damping at zero

Examples of the occurrence of resonance

Everyday examples have already been mentioned above.

Mechanics

• With a tongue blade frequency will be the one of many bending transducers, which is connected to the exciting frequency at resonance is excited to vibration particularly large amplitude.
• If a bridge in resonance with the cadence of marching pedestrian masses, the design may swaying dangerous example Millennium Bridge ( London)
• Vehicle bodies tend at certain engine speeds, excessive vibration ( rumble )
• In the inner ear hair cells, there are about 100 having different resonance frequencies, which facilitates the separation of sounds or of human speech into individual audio frequencies.
• Orbital resonance with the planet can provide that a celestial body falls on a collision course with another. But on Lagrange points this resonance can have a stabilizing effect, because the sun observation satellite SOHO always remains in the vicinity of the inner Lagrangian point L1 since 1995.

Hydromechanics

• Tideresonanz
• Wave resonance

Acoustics

• The sound generation of musical instruments (string and wind instruments ), see, eg, woodwind
• The sympathetic vibration of a string is not played if a same -tuned instrument sounds
• In confined spaces, may cause disturbing room resonance at certain frequencies.
• A resonant exhaust allows for some increase in performance 2- stroke engines with a specific speed.

Acoustic resonance, for example, makes almost all instruments, a roller, often through the formation of a standing wave.

If you measure at the end of both sides open, cylindrical tube with suitable microphones sound pressure and particle velocity, one can calculate the acoustic impedance flow with knowledge of the pipe cross section. This shows multiple resonances, as they are known in the propagation of electromagnetic waves along wires as a special case λ / 2. The measurement result in the picture shows several sharp minima of the flow impedance at multiples of the frequency 500 Hz A review with the tube length of 325 mm and the speed of sound in air provides the setpoint 528 Hz

Because the measured value of the lowest minimum with about 40,000 Pa · s / m of the characteristic acoustic impedance of the surrounding air ( 413.5 Pa · s / m³) is significantly different, there is a mismatch, and the vibrating air column in the pipe is only quiet place. 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 known subdivisions television, mobile phone, radar, remote control and radio astronomy, because it could only be a few isolated stations with sufficient distances without the ability to separate transmission frequencies from each other, around the world. In the majority of all oscillator circuits and electrical filter resonant circuits are used, where the Thomson vibration equation

The safety on the rail network is improved by the inductive train control. In this case mounted on the vehicle, a resonant circuit in resonant interaction occurs with a track mounted on the resonant circuit, the frequency of which is different depending on the position of the next track signal; with signal set to "Halt " emergency brake is triggered.

The large particle accelerator particle physics based on resonance effects, as well as nuclear magnetic resonance spectroscopy in chemistry and magnetic resonance imaging in medicine.

RFIDs, colloquially known as radio frequency identification tags, enables automatic identification and localization of objects and living things. The operating energy is transferred by resonance to the RFID and this sends its information on the same route back.

An absorption rate monitor acts as a selective voltmeter at resonance.

A magnetron only generates vibration when the rotation speed coincides with the natural frequency of the cavity resonators.

Atomic and Molecular Physics

In atomic and molecular physics is called resonance, when a photon of energy (h: Planck's constant, ν: frequency of light) is absorbed in the shell of the atom. This is only possible if just is equal to the energy difference between two states of G and A is the electron shell. An electron is then raised from the state G to the state A. The probability of excitation of such a transition will also be described by a Lorentz curve (as above):

The process is called resonant absorption. He explains, for example, the Fraunhofer lines in the spectrum of sunlight.

Most now falls the electron from the excited state back to the ground state, again a photon of energy is emitted. This occurs either spontaneously (spontaneous emission, fluorescence, phosphorescence) or by collision of a second incident photon of the same energy ( stimulated emission, used in laser ).

For the ground state the atom can now be excited again. Thus, it can perform a Besetzungszahloszillation between states E and A, which is referred to as Rabi oscillation. The oscillation occurs, as mentioned above, only when the incident photons are in resonance with the energy levels of an atom. Such resonances can be used for example for the identification of gases in spectroscopy, as they allow the measurement of the atomic or molecular typical energy levels.

In the human eye there are three different types of pins ( color receptors ). The opsin molecules therein are distinguished by their spectral sensitivity and put in resonance with photons of appropriate wavelength intracellular signaling cascade in motion (see Fototransduktion ). Are formed electrical signals that are passed through the ganglion cells to the brain. There arises from the transmitted signals a perceived color (see color vision ).

Another resonance phenomena occur at the coupling of the magnetic moment of the atom, the atomic nucleus, the molecule or electron ( spin) of a magnetic field on, for example, electron spin resonance and nuclear magnetic resonance. In this case, an oscillating magnetic field with appropriate frequency excites the flipping of the spins between two discrete states of different energy. This effect can also be described according to the Rabi oscillations and is used eg in medical and material investigations ( see, eg, magnetic resonance imaging).

Nuclear physics

Resonance means in nuclear physics that when a collision process with finite kinetic energy, the two partners combine to form a system momentarily bound in a potential of the energy states of the compound nucleus. The cross section shows that the impact energy at a maximum of the form of a Breit-Wigner - curve which is similar to the typical resonances Lorentz curve. Such a system may not be stable, but decays after a short time, such as the two particles, from which it was formed. But can be inferred from the half- width of the curve that it has existed much longer than would correspond to a reaction of the particles in the flyby.

All larger cores show the giant resonance, an excited state in which the protons vibrate in unison against the neutrons.

By taking advantage of the Mößbauereffekts the resonance absorption of gamma quanta allows the comparison of excitation energy with more than 12 digits of precision. The excited nuclear state here corresponds to a resonator with extremely high Q factor.

Particle Physics

Similar to the compound nucleus formation can arise from two collision partners an unstable but relatively durable bonded system, or even a single, different type of particles when the collision energy in the center- just sufficient to do so. The excitation function of the collision process, that is to be applied cross-section as a function of energy, it is at this maximum power with the typical resonance waveform. Systems thus formed are often referred to as resonance or Resonanzteilchen. The half- width of the curve (see decay width ) can - be determined lifetime of the resulting particle - for a direct measurement is too short.

Resonance at the harmonic oscillator

In the harmonic oscillator, for example, a mechanical mass-spring -damper system as shown on the left, is associated with the resonance phenomena can be studied most easily.

The system is excited by a periodic force, acting on the mass. It is depending on the initial conditions at different transients. The vibration system was previously at rest, the amplitude first increases and, when the excitation frequency is close to its natural frequency, can reach values ​​larger than the maximum force at a constant exposure. Provided that the vibration system is not overloaded (resonance disaster ) and the attenuation is not exactly zero, the oscillation gradually changes into a harmonic oscillation with constant values ​​for amplitude, frequency and phase shift relative to the vibration exciter. This behavior is completely consistent for each type of harmonic oscillator. In reality though, most systems that can perform oscillations only approximately harmonic (examples as in the introduction ), but they all show the resonance phenomena in at least a similar manner (see anharmonic oscillator ).

Equation of motion

Of the homogeneous differential equation for a damped harmonic oscillator is linearly added to an external force. The equation is thus inhomogeneous.

This refers to the instantaneous displacement from the equilibrium position, the mass of the body, the spring constant for the restoring force and the damping constant (see figure). Without external force and damping, the system would oscillate freely at its natural angular frequency. With damping, it can run free damped oscillations with angular frequency whose amplitude decreases proportionally, which is. A static constant force would result in a shift in the position of rest to the result.

When the force has a sinusoidal with the amplitude and angular frequency, they can be described as the imaginary part of

Interpret.

Stationary solution: steady state

For the steady state with a constant amplitude of the complex Exponentialansatz, from and determined enough. It follows

The imaginary part of describing a harmonic oscillation

To the rest position. She is the angular frequency, the (real) amplitude

And the constant phase shift with respect to the exciting force

Therein:

• : The deflection under static action of the force,
• The related to the natural frequency of the excitation frequency,
• The Lehr to related dimensionless damping, which is often expressed by the quality factor. The quality factor has the meaning that it indicates the number of vibrations according to which (in the absence of an external force ), the amplitude has decayed to the initial value (after vibrations ).

The dependence of the amplitude of the excitation frequency is also referred to as the amplitude response of the system.

Amplitude resonance

The resonance curve is the graph of the amplitude response. The diagram opposite shows the dimensionless amplitude ratio for typical parameter ranges for excitation frequency and damping. For sufficiently weak damping, shows a maximum, the amplitude response. It is at the resonant frequency and the resonant amplitude, and for showing the value of

The ratio is the resonant peak. The resonant frequency is lower than the natural angular frequency of the undamped vibrating system and also the angular frequency, which takes place with the free damped oscillation of the system.

At low (but nonzero ) damping the resonance is a sharp maximum, which lies almost exactly at the natural angular frequency. The resonance amplitude is then inversely proportional. The amplitude may thus reach a multiple of the static deflection in the steady state. During the transient from the rest position they can even temporarily rise up to almost.

In case of strong damping, however, there is no resonance with increased amplitude. The maximum amplitude of the vibration is steady with the value determined in the static case.

Phase response and energy flow

In the steady state vibration of the exciting force is advanced by exactly 1/4 period afterwards ( -90 ° phase response, referred to as phase resonance). Therefore, speed and power are exactly in phase, so that the force always acts in the direction of the current velocity. The power then flows continuously into the system, while it changes direction twice per period at other frequencies, because the phase difference is at less than 90 ° and at a higher frequency is greater than 90 ° ( and 180 °). The kinetic energy of the steady state obtained in the stage resonance maximum. It is then as great as the total energy input during the last vibrations.

Energy resonance

The energy stored in an oscillation with amplitude potential energy. The corresponding resonance curve is given by the square of the amplitude response.

The energy stored in an oscillation with amplitude kinetic energy. This function has its maximum at exactly.

In the key for the optics application, the emission and absorption of electromagnetic waves by dipole radiation oscillating output is proportional. The maximum of this function is a little above.

For sharp resonance, that is low loss, the differences of these three resonance frequencies are neglected, and usually used in the field of resonance at the characteristic frequency symmetric approximation formula called a Lorentz curve:

This formula shows off resonance also the characteristic of the forced oscillation long streamers and is therefore also suitable for high frequencies or useful.

The stored energy in the vibration system comes from the acceleration work by the exciting force. The vibration energy is increased, when the force acts in the direction of the velocity. Otherwise, the power cut the system energy, ie acts as a brake. In the steady state, the energy input is like straight out of the energy loss due to damping.

Half-width of merit

A half value width ( full width at half maximum Sheet ) of the resonance is the range of frequencies around the resonant frequency referred to, in which is true for the amplitude. In the region of interest, low attenuation occur after the approximate formula for the Lorentz curve at these limits. Converted to the frequency axis gives the half-width

The sharpness of the resonance can with the damping or the quality factor

Be specified. After the above-mentioned importance of the quality factor can be a period of cycles of the natural frequency as characteristic view for the decay of a damped natural frequency, so also characteristic for the duration of the transient or in the figurative sense of the "memory of the oscillator ." An analysis of an oscillation with frequency using a series of resonators at different resonance frequencies, then therefore requires the determination of the resonance amplitude of the time and provides the resonance frequency with the accuracy. Are there differences between the two oscillators to the frequency, then makes in this period of rapid just a vibration more than the slower. It follows more accurately the frequency of vibration is to be determined, the longer you have to soak them in a resonator. This is an early form of the frequency-time uncertainty relation.

Resonance damping at zero

Disappearing damping is indeed a theoretically achievable limit case, real systems with very low attenuation come close to him but when you look at a not too long period of time (). This corresponds to a potentially large number of oscillations. In principle it should be noted that there is no transient in the damping -free case, the leads, regardless of the initial conditions of a particular stationary oscillation. A possibly with excited natural frequency sounds here that is not turned away, but remains undiminished presence. For resonant excitation, there are already mathematically no stationary solution of the equation of motion, but the amplitude varies linearly with time. Starting from the state at rest in the rest position, the amplitude increases, for example, in proportion to the elapsed time:

Theoretically, it comes here so definitely resonate disaster. This is useful to avoid otherwise caused only by a limitation of the amplitude, ie, generally speaking, by a change of the power law (see anharmonic oscillator ).

Outside the exact resonance frequency, however, exists to appropriate initial conditions a stationary oscillation. It results from the above equations for. The amplitude ratio is larger at each excitation frequency as in the case with damping. At resonance diverges the formula for the amplitude, and it is no condition of the stationary wave. The phase lag at frequencies below the resonance, above, as is apparent from the above formula by the border. ( For further explanation, see formulas and Forced Vibration # limit of vanishing damping. )