Contenido

Un ejemplo común es un columpio de parque, que actúa como un péndulo. Al empujar a una persona en un columpio en sincronía con el intervalo natural del columpio (su frecuencia de resonancia), el columpio sube cada vez más (amplitud máxima), mientras los intentos de empujar el columpio con un «tempo» más rápido o más lento producen que los arcos sean más pequeños. Esto se debe a que la energía que absorbe el columpio se maximiza cuando los empujones se emparejan con las oscilaciones naturales del mismo.

La resonancia ocurre extensamente en la naturaleza y es explotada en muchos dispositivos hechos por el hombre. Es el mecanismo por el que prácticamente todas las ondas sinusoidales y las vibraciones son generadas. Muchos sonidos que escuchamos, como cuando objetos duros de metal, vidrio, o la madera son golpeados, están causado por vibraciones resonantes breves en el objeto. La luz y otras radiaciones electromagnéticas de corta longitud de onda son producidas por la resonancia a escala atómica, tales como los electrones en los átomos. Otros ejemplos de resonancia:.

The Tacoma Narrows Bridge

The dramatically visible and rhythmic torsion that resulted in the collapse of the Tacoma Narrows Bridge in 1940 is erroneously characterized as an example of the phenomenon of resonance in certain texts.[4] The catastrophic vibrations that destroyed the bridge were not due to simple mechanical resonance, but to the interaction between the bridge and the wind passing through it, producing a phenomenon known as fluttering, which periodically pushed the bridge, causing it to move like a "self-sustaining oscillation"). Robert H. Scanlan), father of bridge aerodynamics, wrote an article about this misunderstanding.[5]​.

The International Space Station

The rocket engines for the International Space Station (ISS) are controlled by autopilot. Normally, the parameters installed to control the engine control system for the Zvezda module cause the rocket engines to propel the International Space Station into a higher orbit. Rocket engines are mounted on hinges; the crew is usually unaware of the operation. However, on January 14, 2009, the loaded parameters caused the autopilot to rock the rocket engines in increasingly wider oscillations, at a frequency of 0.5 Hz. These oscillations were captured on video and lasted 142 seconds.[6]​.

La resonancia se manifiesta en varios sistemas lineales y no lineales como oscilaciones alrededor de un punto de equilibrio. Cuando el sistema es impulsado por una entrada sinusoidal externa, una salida del sistema puede oscilar en respuesta. La razón entre la amplitud de las oscilaciones estables de la salida y las oscilaciones de entrada es llamada ganancia, y la ganancia puede ser una función de la frecuencia de la entrada sinusoidal externa. Los picos en la ganancia a ciertas frecuencias corresponden a resonancias, donde la amplitud de las oscilaciones de salida son desproporcionalmente largas.

Dado que la mayoría de sistemas lineales y no lineales son modelados como osciladores armónicos cerca de su equilibrio, esta sección comienza con la derivada de la frecuencia resonante de un oscilador armónico amortiguado. La sección continúa con un el uso de un circuito RLC para ilustrar las conexiones entre resonancia y la función de transferencia de un sistema, respuesta de frecuencia, polos y ceros. Partiendo del ejemplo del circuito RLC, la sección generaliza esta relación para sistemas lineales de orden superior con múltiples entradas y salidas.

Resonance of a damped and driven oscillator

Considering a mass anchored to a spring that is driven by an applied external sinusoidal force in which the system is damped. Newton's second law takes the form of:.

Where m is the mass, x the displacement of the mass from the equilibrium point, F the amplitude, ω is the angular frequency, k is the spring constant, and c is the viscocity coefficient. This can take the form of:.

Where.

ω is also known elsewhere as resonance frequency. However, as shown below, when we analyze the oscillations of the displacement x(t), the resonance frequency is close to but not equal to ω. The RLC circuit example in the next section provides examples of different resonant frequencies for the same system.

The general solution of equation (2) is the sum of a transient solution that depends on the initial conditions and a steady state solution that is independent of the initial conditions and depends only on the amplitude F, external angular frequency ω, the natural frequency ω, and the damping factor ζ. The steady state decays in a relatively short period of time, so to study the resonance it is enough to consider the solution in the steady state.

It is possible to write the steady-state solution for x(t) as a function proportional to the driving force with a change of Phase (wave) "Phase (wave)") φ,.

Where.

The phase value is normally between -180° and 0° so it represents a delay or offset for the positive and negative values ​​of the arctan argument.

Resonance occurs when, at certain conduction frequencies, the steady-state amplitude of x(t) is large compared to its amplitude at other frequencies. For mass in a spring, resonance physically corresponds to oscillations of the mass that have large displacements from the equilibrium position of the spring at certain activation frequencies. Observing the amplitude of x(t) as a function of the frequency ω, the amplitude is maximum at the natural frequency.

ω is the resonance frequency for this system. Again, note that the resonance frequency is not equal to the natural angular frequency ω of the oscillator. They are proportional, and if the damping factor reaches zero they are equal, but if it is not zero they are not the same frequency. As shown in the figure, resonance can also occur at other frequencies close to the resonance frequency, including ω, but the maximum response is the resonance frequency.

Note that ω is real and non-zero if , therefore this system can only resonate when the harmonic oscillator is significantly damped. For systems with a very small damping ratio and a driving frequency close to the resonance frequency, the steady-state oscillations can become very large.

For other controlled and damped harmonic oscillators whose equations of motion do not look exactly like that of the mass in a spring example, the resonant frequency remains.

But the definitions of ω and ζ change depending on the physics of the system. For a pendulum of length l and a small angular displacement θ <15°, Equation (1) becomes:.

and therefore.

See also: RLC circuit § Series circuit.

Consider a circuit consisting of a resistor with resistance R , an inductor with inductance L and a capacitor with capacitance C connected in series with current i ( t ) and driven by a source voltage "Voltage (electricity)") with voltage v ( t ). The voltage drop around the circuit is.

Instead of analyzing a proposed solution to this equation as in the spring mass of the previous example, this section will analyze the frequency response of this circuit. Taking the transformed Laplace equation ( 4 ).

where I (s) and V (s) are the Laplace transform of the input current and voltage, respectively, and s is a complex frequency parameter") in the Laplace domain. Rearranging the terms,.

Voltage resonance across a capacitor.

A series RLC circuit presents several options for locating a location to measure an output voltage. Suppose the output voltage of interest is the voltage drop across the capacitor. As shown above, in the Laplace domain this voltage is.

either.

Define for this circuit a natural frequency and a damping ratio.

The ratio between the output voltage and the input voltage becomes.

Where H ( s ) is the transfer function between the input voltage and the output voltage. Note that this transfer function has two poles")–roots of the polynomial in the denominator of the transfer function.

and without non-zero roots of the polynomial in the numerator of the transfer function. Also, note that for ζ ≤ 1, the magnitude of these poles is the natural frequency ω and that for ζ < 1 /, our condition for resonance in the harmonic oscillator example, the poles are closer to the imaginary axis than the real axis.

By evaluating H (s) along the imaginary axis s = iω, the transfer function describes the frequency response of this circuit. Equivalently, the frequency response can be analyzed by taking the Fourier transform of equation (4) instead of the Laplace transform. The transfer function, which is also complex, can be written as gain and phase.

A sinusoidal input voltage at frequency ω results in an output voltage at the same frequency that has been scaled by G (ω) and has a phase shift Φ (ω). Gain and phase can be plotted against frequency on a Bode plot. For the capacitor voltage of the RLC circuit, the transfer function gain H (iω) is.

Note the similarity between gain and amplitude in equation (3). Once again, the gain is maximized at the resonant frequency..

Here, resonance physically corresponds to having a relatively large amplitude for the steady-state oscillations of the voltage across the capacitor compared to its amplitude at other driving frequencies.

The resonant frequency does not always have to take the form given in the examples above. Suppose that for the RLC circuit, the output voltage of interest is the voltage across the inductor. As shown above, in the Laplace domain, the voltage across the inductor is.

using the same definitions for ω and ζ as in the previous example. The transfer function between V (s) and this new V (s) through the inductor is.

Note that this transfer function has the same poles as the transfer function in the previous example, but it also has two zeros in the numerator at s = 0. By evaluating H (s) along the imaginary axis, its gain becomes .

Compared to the gain in equation (6) which uses the capacitor voltage as the output, this gain has a factor of ω in the numerator and will therefore have a different resonance frequency that maximizes the gain. That frequency is.

So for the same RLC circuit but with the voltage across the inductor as the output, the resonant frequency is now higher than the natural frequency, although it still tends towards the natural frequency as the damping ratio goes to zero. That the same circuit can have different resonant frequencies for different output options is not contradictory. As shown in equation (4), the voltage drop in the circuit is divided among the three elements of the circuit, and each element has a different dynamic. The capacitor voltage grows slowly by integrating the current over time and is therefore more sensitive to lower frequencies, while the inductor voltage grows when the current changes rapidly and is therefore more sensitive to higher frequencies. While the circuit as a whole has a natural frequency at which it tends to oscillate, the different dynamics of each circuit element cause each element to resonate at a slightly different frequency.[7]​.

Assume that the output voltage of interest is the voltage across the resistor. In the Laplace domain, the voltage across the resistor is.

and using the same natural frequency and damping ratio as in the capacitor example, the transfer function is.

Note that this transfer function also has the same poles as the RLC circuit examples above, but it only has a zero in the numerator at s = 0. For this transfer function, its gain is .

The resonant frequency that maximizes this gain is.

and the gain is one at this frequency, so the voltage across the resistor resonates at the natural frequency of the circuit and at this frequency the amplitude of the voltage across the resistor is equal to the amplitude of the input voltage.

Mechanical and acoustic resonance

Mechanical resonance is the tendency of a mechanical system to absorb more energy when the frequency of its oscillations matches the natural frequency of vibration of the system than it does at other frequencies. It can cause violent swaying motions and even catastrophic failures in improperly constructed structures, including bridges, buildings, trains, and airplanes. When designing objects, engineers must ensure that the mechanical resonance frequencies of component parts do not coincide with the vibrational frequencies of motors or other oscillating parts, a phenomenon known as resonance disaster.

Avoiding resonance disasters is a major concern in every construction project, tower and bridge construction. As a countermeasure, floating mounting can be installed to absorb resonant frequencies and thus dissipate the absorbed energy. The Taipei 101 building relies on a 660 ton (727.5 ST; 660,000 kg)—mass damper—to cancel resonance. Additionally, the structure is designed to resonate at a frequency that does not normally occur. Buildings in seismic zones are often built to take into account the oscillating frequencies of expected ground motion. In addition, engineering design objects that have motors must ensure that the mechanical resonant frequencies of the component parts do not coincide with the vibrational frequencies of the motors or other strongly oscillating parts.

Clock keeps time by mechanical resonance in a balance wheel, pendulum or quartz crystal.

It has been hypothesized that runners' cadence is energetically favorable due to the resonance between the elastic energy stored in the lower extremity and the runner's mass.

Acoustic resonance is a branch of mechanical resonance that deals with mechanical vibrations across the frequency range of human hearing, in other words sound. For humans, hearing is normally limited to frequencies between approximately 20 Hz and 20,000 Hz (20 kHz),[8] Many objects and materials act as resonators with resonant frequencies within this range, and when struck they vibrate mechanically, pushing on the surrounding air to create sound. This is the source of many percussive sounds we hear.

Acoustic resonance is an important consideration for instrument makers, as most instruments use resonators, such as the strings and body of a violin, the length of tubing in a flute, and the shape and tension of a drum membrane.

Like mechanical resonance, acoustic resonance can lead to catastrophic failure of the resonating object. The classic example of this is breaking a wine glass with sound at the glass's precise resonant frequency, although this is difficult in practice.[9]​.

electrical resonance

Electrical resonance occurs in an electrical circuit at a particular resonant frequency when the circuit impedance is minimum in a series circuit or in a parallel circuit (usually when the transfer function reaches its absolute maximum value). Resonance in circuits is used to transmit and receive wireless communications, such as television, cell phones, and radio.[10]​.

Optical resonance

An optical cavity, also called an “optical resonator,” is a mirror arrangement that forms a standing wave resonator for light. Optical cavities are a major active medium component, surrounding the gain medium and providing laser light feedback. They are also used in optical parametric oscillator and some interferometer. The light confined in the cavity is reflected several times producing standing waves for certain resonant frequencies. The standing wave patterns produced are called "modes." Longitudinal modes differ only in frequency while transverse mode differ for different frequencies and have different intensity patterns in the cross section of the beam. Ring Resonators and Whispering Gallery are examples of optical resonators that do not form standing waves.

The different types of resonators are distinguished by the focal lengths of the two mirrors and the distance between them; Plane mirrors are not frequently used because of the difficulty in aligning them accurately. The geometry (type of resonator) must be chosen so that the beam remains stable, that is, the size of the beam does not continue to grow with each reflection. Resonator types are also designed to meet other criteria, such as minimum beam waist or not having a focal point (and therefore intense light at that point) within the cavity.

Optical cavities are designed to have a Q factor.[11]​ A beam reflects a large number of times with little attenuation - therefore, at frequency the line width of the beam is small compared to the frequency of the laser.

Additional optical resonances are guided mode resonances and surface plasmon resonances, which result in anomalous reflection and high evanescent fields at the resonance. In this case, the resonant modes are guided modes of a waveguide or surface plasmon modes of a dielectric-metallic interface. These modes are generally excited by a subwavelength grating.

Orbital resonance

In celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a periodic gravitational influence on each other, usually because their orbital period is related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of bodies. In most cases, this results in an "unstable" interaction, in which the bodies exchange momentum and change orbits until the resonance no longer exists. In some circumstances, a resonant system can be stable and self-correcting, so that bodies remain in resonance. Examples are the 1:2:4 resonance of Jupiter's moons Ganymede "Ganymede (moon)"), Europa "Europa (moon)"), and Io "Io (satellite)"), and the 2:3 resonance between Pluto "Pluto (dwarf planet)") and Neptune "Neptune (planet)"). Unstable resonances with Saturn's inner moons "Saturn (planet)") give rise to gaps in Saturn's rings. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large Solar System bodies to clear the neighborhood around their orbits by ejecting almost everything else around them. This effect is used in the current definition of a planet.

Atomic, particle and molecular resonance

Nuclear magnetic resonance (NMR) is the name given to a physical resonance phenomenon that involves the observation of quantum mechanical magnetic properties of an atom's nucleus in the presence of an applied external magnetic field. Many scientific techniques exploit NMR phenomena to study molecular physics, crystal and non-crystalline materials through nuclear magnetic resonance spectroscopy. NMR is also commonly used in advanced medical imaging techniques, such as magnetic resonance imaging (MRI).

All nuclei containing odd numbers of nucleons have a magnetic moment and intrinsic angular momentum. A key feature of NMR is that the resonance frequency of a particular substance is directly proportional to the strength of the applied magnetic field. It is this characteristic that is exploited in imaging techniques; If a sample is placed in a non-uniform magnetic field, the resonant frequencies of the nuclei in the sample depend on where in the field they are located. Therefore, the particle can be located quite precisely by its resonant frequency.

Electron paramagnetic resonance, also known as electron spin resonance (EER), is a spectroscopic technique similar to NMR, but uses unpaired electrons. The materials for which it can be applied are much more limited since the material must have an unpaired spin and be paramagnetic.

The Mössbauer effect is the resonant and recoil-free emission and absorption of gamma ray photons by atoms bound in solid form.

Resonance in particle physics "Resonance (particle physics)") appears in circumstances similar to classical physics at the level of quantum mechanics and quantum field theory. However, they can also be considered unstable particles, with the above formula valid if Γ is decay rate and Ω replaced by the mass of the particle M. In that case, the formula comes from the propagator ] of the particle, with its mass replaced by the complex number M + iΓ. The formula is more related to the disintegration of particles by the optical theorem.

A physical system can have as many resonant frequencies as there are degrees of freedom; each degree of freedom can vibrate as a harmonic oscillator. Systems with one degree of freedom, such as a mass on a spring, balance wheels, pendulum, and LC tuned circuits have a resonant frequency. Systems with two degrees of freedom, such as coupled pendulums and resonant transformers, can have two resonant frequencies. As the number of coupled harmonic oscillators grows, the time it takes to transfer energy from one to another becomes significant. The vibrations in them begin to travel through the coupled harmonic oscillators in waves, from one oscillator to the next.

Extended objects that can experience resonance due to vibrations within them are called resonators, such as organ pipe, vibrating string, quartz, microwave, and laser. As you can see that they are made of many coupled moving parts (like atoms), they can have many resonant frequencies. The vibrations within them travel like waves, at a roughly constant speed, bouncing back and forth between the sides of the resonator. If the distance between the sides is d , the length of a round trip is 2d . To cause resonance, the phase of a sine wave after a round trip must be equal to the initial phase, so the waves reinforce the oscillation. So the condition for resonance in a resonator is that the round trip distance, 2d , is equal to an integer number of wavelengths λ of the wave:.

If the speed of a wave is v, the frequency is: f = then the resonant frequencies are:.

Then, the resonant frequencies of the resonators,.

Called normal modes, they are equally spaced multiples of a lower frequency called the fundamental frequency. Multiples are often called harmonics. There may be several series of resonant frequencies, corresponding to different modes of oscillation.

The quality factor is a dimensionless quantity (see dimensionless magnitude), a parameter that describes a damped motion such as an oscillator or resonator.[12]​

A higher Q indicates a lower rate of energy loss relative to the stored energy of the oscillator, so the oscillations will stop slowly. A high-quality bearing pendulum suspended, oscillating in air has a higher Q, while a pendulum immersed in oil would have a lower Q. To maintain a system in resonance with constant amplitude, energy must be provided externally; the energy provided in each cycle must be less than the energy stored in the system (i.e., the sum of potential and kinetics) by a factor of . Oscillators with high quality factors have low damping, which tends to make them resonate longer.

Resonators with higher Q factors resonate with larger amplitudes (at the resonant frequency) but have a smaller range of frequencies around the frequency at which they resonate. The range of frequencies in which the oscillator resonates is called bandwidth. Therefore, a high Q tuned circuit in a radio receiver would be harder to tune, but would have greater selectivity, doing a better job of filtering out signals from other stations that are close to the spectrum. High "Q" oscillators operate in a smaller range of frequencies and are more stable. (See phase noise.).

The quality factor of oscillators varies substantially from one system to another. Systems for which damping is important (such as dampers that prevent a door from slamming) have Q = , on the other hand clocks, lasers and other systems that need strong resonance or high frequency stability need high quality factors, for example tuning forks

have a quality factor of around Q = 1000, while the quality factor of an atomic clock and a high laser can reach Q =10[13]​ and may even be higher.[14]​ Physicists and engineers use many alternative quantities to describe how damped an oscillator is that are closely related to its quality factor.

The exact response of a resonance, especially for frequencies far from the resonant frequency, depends on the details of the physical system and is usually not exactly symmetric with respect to the resonant frequency, as illustrated for the harmonic oscillator. For a lightly damped linear oscillator with a resonance frequency Ω, the intensity of the oscillations I when the system operates with a driving frequency ω is typically approximated by a formula that is symmetric with respect to the resonance frequency:[15]​.

Where susceptibility links the amplitude of the oscillator to the driving force in frequency space:[16]​.

Intensity is defined as the square of the amplitude of the oscillations. This is a Lorentzian function, or Cauchy distribution, and this response is found in many physical situations involving resonant systems. Γ is a parameter that depends on the damping of the oscillator, and is known as the linewidth of the resonance. Highly damped oscillators tend to have wide linewidths, and respond to a wider range of driving frequencies around the resonance frequency. The line width is inversely proportional to the quality factor, which is a measure of the sharpness of the resonance.

In broadcast engineering and electronic engineering, this approximate symmetric response is known as the universal resonance curve, a concept introduced by Frederick Terman in 1932 to simplify the approximate analysis of radio circuits with a range of center frequencies and Q values.[17][18]​.

Resonance occurs when a system is able to easily store and transfer energy between two or more different forms of storage (such as kinetic energy and potential energy in the case of a simple pendulum). However there are some losses from cycle to cycle, called damping. When the damping is small, the resonance frequency is approximately equal to the natural frequency of the system, which is a frequency of unforced vibrations. Some systems have multiple resonant frequencies.

Contenido

Un ejemplo común es un columpio de parque, que actúa como un péndulo. Al empujar a una persona en un columpio en sincronía con el intervalo natural del columpio (su frecuencia de resonancia), el columpio sube cada vez más (amplitud máxima), mientras los intentos de empujar el columpio con un «tempo» más rápido o más lento producen que los arcos sean más pequeños. Esto se debe a que la energía que absorbe el columpio se maximiza cuando los empujones se emparejan con las oscilaciones naturales del mismo.

La resonancia ocurre extensamente en la naturaleza y es explotada en muchos dispositivos hechos por el hombre. Es el mecanismo por el que prácticamente todas las ondas sinusoidales y las vibraciones son generadas. Muchos sonidos que escuchamos, como cuando objetos duros de metal, vidrio, o la madera son golpeados, están causado por vibraciones resonantes breves en el objeto. La luz y otras radiaciones electromagnéticas de corta longitud de onda son producidas por la resonancia a escala atómica, tales como los electrones en los átomos. Otros ejemplos de resonancia:.

The Tacoma Narrows Bridge

The dramatically visible and rhythmic torsion that resulted in the collapse of the Tacoma Narrows Bridge in 1940 is erroneously characterized as an example of the phenomenon of resonance in certain texts.[4] The catastrophic vibrations that destroyed the bridge were not due to simple mechanical resonance, but to the interaction between the bridge and the wind passing through it, producing a phenomenon known as fluttering, which periodically pushed the bridge, causing it to move like a "self-sustaining oscillation"). Robert H. Scanlan), father of bridge aerodynamics, wrote an article about this misunderstanding.[5]​.

The International Space Station

The rocket engines for the International Space Station (ISS) are controlled by autopilot. Normally, the parameters installed to control the engine control system for the Zvezda module cause the rocket engines to propel the International Space Station into a higher orbit. Rocket engines are mounted on hinges; the crew is usually unaware of the operation. However, on January 14, 2009, the loaded parameters caused the autopilot to rock the rocket engines in increasingly wider oscillations, at a frequency of 0.5 Hz. These oscillations were captured on video and lasted 142 seconds.[6]​.

La resonancia se manifiesta en varios sistemas lineales y no lineales como oscilaciones alrededor de un punto de equilibrio. Cuando el sistema es impulsado por una entrada sinusoidal externa, una salida del sistema puede oscilar en respuesta. La razón entre la amplitud de las oscilaciones estables de la salida y las oscilaciones de entrada es llamada ganancia, y la ganancia puede ser una función de la frecuencia de la entrada sinusoidal externa. Los picos en la ganancia a ciertas frecuencias corresponden a resonancias, donde la amplitud de las oscilaciones de salida son desproporcionalmente largas.

Dado que la mayoría de sistemas lineales y no lineales son modelados como osciladores armónicos cerca de su equilibrio, esta sección comienza con la derivada de la frecuencia resonante de un oscilador armónico amortiguado. La sección continúa con un el uso de un circuito RLC para ilustrar las conexiones entre resonancia y la función de transferencia de un sistema, respuesta de frecuencia, polos y ceros. Partiendo del ejemplo del circuito RLC, la sección generaliza esta relación para sistemas lineales de orden superior con múltiples entradas y salidas.

Resonance of a damped and driven oscillator

Considering a mass anchored to a spring that is driven by an applied external sinusoidal force in which the system is damped. Newton's second law takes the form of:.

Where m is the mass, x the displacement of the mass from the equilibrium point, F the amplitude, ω is the angular frequency, k is the spring constant, and c is the viscocity coefficient. This can take the form of:.

Where.

ω is also known elsewhere as resonance frequency. However, as shown below, when we analyze the oscillations of the displacement x(t), the resonance frequency is close to but not equal to ω. The RLC circuit example in the next section provides examples of different resonant frequencies for the same system.

The general solution of equation (2) is the sum of a transient solution that depends on the initial conditions and a steady state solution that is independent of the initial conditions and depends only on the amplitude F, external angular frequency ω, the natural frequency ω, and the damping factor ζ. The steady state decays in a relatively short period of time, so to study the resonance it is enough to consider the solution in the steady state.

It is possible to write the steady-state solution for x(t) as a function proportional to the driving force with a change of Phase (wave) "Phase (wave)") φ,.

Where.

The phase value is normally between -180° and 0° so it represents a delay or offset for the positive and negative values ​​of the arctan argument.

Resonance occurs when, at certain conduction frequencies, the steady-state amplitude of x(t) is large compared to its amplitude at other frequencies. For mass in a spring, resonance physically corresponds to oscillations of the mass that have large displacements from the equilibrium position of the spring at certain activation frequencies. Observing the amplitude of x(t) as a function of the frequency ω, the amplitude is maximum at the natural frequency.

ω is the resonance frequency for this system. Again, note that the resonance frequency is not equal to the natural angular frequency ω of the oscillator. They are proportional, and if the damping factor reaches zero they are equal, but if it is not zero they are not the same frequency. As shown in the figure, resonance can also occur at other frequencies close to the resonance frequency, including ω, but the maximum response is the resonance frequency.

Note that ω is real and non-zero if , therefore this system can only resonate when the harmonic oscillator is significantly damped. For systems with a very small damping ratio and a driving frequency close to the resonance frequency, the steady-state oscillations can become very large.

For other controlled and damped harmonic oscillators whose equations of motion do not look exactly like that of the mass in a spring example, the resonant frequency remains.

But the definitions of ω and ζ change depending on the physics of the system. For a pendulum of length l and a small angular displacement θ <15°, Equation (1) becomes:.

and therefore.

See also: RLC circuit § Series circuit.

Consider a circuit consisting of a resistor with resistance R , an inductor with inductance L and a capacitor with capacitance C connected in series with current i ( t ) and driven by a source voltage "Voltage (electricity)") with voltage v ( t ). The voltage drop around the circuit is.

Instead of analyzing a proposed solution to this equation as in the spring mass of the previous example, this section will analyze the frequency response of this circuit. Taking the transformed Laplace equation ( 4 ).

where I (s) and V (s) are the Laplace transform of the input current and voltage, respectively, and s is a complex frequency parameter") in the Laplace domain. Rearranging the terms,.

Voltage resonance across a capacitor.

A series RLC circuit presents several options for locating a location to measure an output voltage. Suppose the output voltage of interest is the voltage drop across the capacitor. As shown above, in the Laplace domain this voltage is.

either.

Define for this circuit a natural frequency and a damping ratio.

The ratio between the output voltage and the input voltage becomes.

Where H ( s ) is the transfer function between the input voltage and the output voltage. Note that this transfer function has two poles")–roots of the polynomial in the denominator of the transfer function.

and without non-zero roots of the polynomial in the numerator of the transfer function. Also, note that for ζ ≤ 1, the magnitude of these poles is the natural frequency ω and that for ζ < 1 /, our condition for resonance in the harmonic oscillator example, the poles are closer to the imaginary axis than the real axis.

By evaluating H (s) along the imaginary axis s = iω, the transfer function describes the frequency response of this circuit. Equivalently, the frequency response can be analyzed by taking the Fourier transform of equation (4) instead of the Laplace transform. The transfer function, which is also complex, can be written as gain and phase.

A sinusoidal input voltage at frequency ω results in an output voltage at the same frequency that has been scaled by G (ω) and has a phase shift Φ (ω). Gain and phase can be plotted against frequency on a Bode plot. For the capacitor voltage of the RLC circuit, the transfer function gain H (iω) is.

Note the similarity between gain and amplitude in equation (3). Once again, the gain is maximized at the resonant frequency..

Here, resonance physically corresponds to having a relatively large amplitude for the steady-state oscillations of the voltage across the capacitor compared to its amplitude at other driving frequencies.

The resonant frequency does not always have to take the form given in the examples above. Suppose that for the RLC circuit, the output voltage of interest is the voltage across the inductor. As shown above, in the Laplace domain, the voltage across the inductor is.

using the same definitions for ω and ζ as in the previous example. The transfer function between V (s) and this new V (s) through the inductor is.

Note that this transfer function has the same poles as the transfer function in the previous example, but it also has two zeros in the numerator at s = 0. By evaluating H (s) along the imaginary axis, its gain becomes .

Compared to the gain in equation (6) which uses the capacitor voltage as the output, this gain has a factor of ω in the numerator and will therefore have a different resonance frequency that maximizes the gain. That frequency is.

So for the same RLC circuit but with the voltage across the inductor as the output, the resonant frequency is now higher than the natural frequency, although it still tends towards the natural frequency as the damping ratio goes to zero. That the same circuit can have different resonant frequencies for different output options is not contradictory. As shown in equation (4), the voltage drop in the circuit is divided among the three elements of the circuit, and each element has a different dynamic. The capacitor voltage grows slowly by integrating the current over time and is therefore more sensitive to lower frequencies, while the inductor voltage grows when the current changes rapidly and is therefore more sensitive to higher frequencies. While the circuit as a whole has a natural frequency at which it tends to oscillate, the different dynamics of each circuit element cause each element to resonate at a slightly different frequency.[7]​.

Assume that the output voltage of interest is the voltage across the resistor. In the Laplace domain, the voltage across the resistor is.

and using the same natural frequency and damping ratio as in the capacitor example, the transfer function is.

Note that this transfer function also has the same poles as the RLC circuit examples above, but it only has a zero in the numerator at s = 0. For this transfer function, its gain is .

The resonant frequency that maximizes this gain is.

and the gain is one at this frequency, so the voltage across the resistor resonates at the natural frequency of the circuit and at this frequency the amplitude of the voltage across the resistor is equal to the amplitude of the input voltage.

Mechanical and acoustic resonance

Mechanical resonance is the tendency of a mechanical system to absorb more energy when the frequency of its oscillations matches the natural frequency of vibration of the system than it does at other frequencies. It can cause violent swaying motions and even catastrophic failures in improperly constructed structures, including bridges, buildings, trains, and airplanes. When designing objects, engineers must ensure that the mechanical resonance frequencies of component parts do not coincide with the vibrational frequencies of motors or other oscillating parts, a phenomenon known as resonance disaster.

Avoiding resonance disasters is a major concern in every construction project, tower and bridge construction. As a countermeasure, floating mounting can be installed to absorb resonant frequencies and thus dissipate the absorbed energy. The Taipei 101 building relies on a 660 ton (727.5 ST; 660,000 kg)—mass damper—to cancel resonance. Additionally, the structure is designed to resonate at a frequency that does not normally occur. Buildings in seismic zones are often built to take into account the oscillating frequencies of expected ground motion. In addition, engineering design objects that have motors must ensure that the mechanical resonant frequencies of the component parts do not coincide with the vibrational frequencies of the motors or other strongly oscillating parts.

Clock keeps time by mechanical resonance in a balance wheel, pendulum or quartz crystal.

It has been hypothesized that runners' cadence is energetically favorable due to the resonance between the elastic energy stored in the lower extremity and the runner's mass.

Acoustic resonance is a branch of mechanical resonance that deals with mechanical vibrations across the frequency range of human hearing, in other words sound. For humans, hearing is normally limited to frequencies between approximately 20 Hz and 20,000 Hz (20 kHz),[8] Many objects and materials act as resonators with resonant frequencies within this range, and when struck they vibrate mechanically, pushing on the surrounding air to create sound. This is the source of many percussive sounds we hear.

Acoustic resonance is an important consideration for instrument makers, as most instruments use resonators, such as the strings and body of a violin, the length of tubing in a flute, and the shape and tension of a drum membrane.

Like mechanical resonance, acoustic resonance can lead to catastrophic failure of the resonating object. The classic example of this is breaking a wine glass with sound at the glass's precise resonant frequency, although this is difficult in practice.[9]​.

electrical resonance

Electrical resonance occurs in an electrical circuit at a particular resonant frequency when the circuit impedance is minimum in a series circuit or in a parallel circuit (usually when the transfer function reaches its absolute maximum value). Resonance in circuits is used to transmit and receive wireless communications, such as television, cell phones, and radio.[10]​.

Optical resonance

An optical cavity, also called an “optical resonator,” is a mirror arrangement that forms a standing wave resonator for light. Optical cavities are a major active medium component, surrounding the gain medium and providing laser light feedback. They are also used in optical parametric oscillator and some interferometer. The light confined in the cavity is reflected several times producing standing waves for certain resonant frequencies. The standing wave patterns produced are called "modes." Longitudinal modes differ only in frequency while transverse mode differ for different frequencies and have different intensity patterns in the cross section of the beam. Ring Resonators and Whispering Gallery are examples of optical resonators that do not form standing waves.

The different types of resonators are distinguished by the focal lengths of the two mirrors and the distance between them; Plane mirrors are not frequently used because of the difficulty in aligning them accurately. The geometry (type of resonator) must be chosen so that the beam remains stable, that is, the size of the beam does not continue to grow with each reflection. Resonator types are also designed to meet other criteria, such as minimum beam waist or not having a focal point (and therefore intense light at that point) within the cavity.

Optical cavities are designed to have a Q factor.[11]​ A beam reflects a large number of times with little attenuation - therefore, at frequency the line width of the beam is small compared to the frequency of the laser.

Additional optical resonances are guided mode resonances and surface plasmon resonances, which result in anomalous reflection and high evanescent fields at the resonance. In this case, the resonant modes are guided modes of a waveguide or surface plasmon modes of a dielectric-metallic interface. These modes are generally excited by a subwavelength grating.

Orbital resonance

In celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a periodic gravitational influence on each other, usually because their orbital period is related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of bodies. In most cases, this results in an "unstable" interaction, in which the bodies exchange momentum and change orbits until the resonance no longer exists. In some circumstances, a resonant system can be stable and self-correcting, so that bodies remain in resonance. Examples are the 1:2:4 resonance of Jupiter's moons Ganymede "Ganymede (moon)"), Europa "Europa (moon)"), and Io "Io (satellite)"), and the 2:3 resonance between Pluto "Pluto (dwarf planet)") and Neptune "Neptune (planet)"). Unstable resonances with Saturn's inner moons "Saturn (planet)") give rise to gaps in Saturn's rings. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large Solar System bodies to clear the neighborhood around their orbits by ejecting almost everything else around them. This effect is used in the current definition of a planet.

Atomic, particle and molecular resonance

Nuclear magnetic resonance (NMR) is the name given to a physical resonance phenomenon that involves the observation of quantum mechanical magnetic properties of an atom's nucleus in the presence of an applied external magnetic field. Many scientific techniques exploit NMR phenomena to study molecular physics, crystal and non-crystalline materials through nuclear magnetic resonance spectroscopy. NMR is also commonly used in advanced medical imaging techniques, such as magnetic resonance imaging (MRI).

All nuclei containing odd numbers of nucleons have a magnetic moment and intrinsic angular momentum. A key feature of NMR is that the resonance frequency of a particular substance is directly proportional to the strength of the applied magnetic field. It is this characteristic that is exploited in imaging techniques; If a sample is placed in a non-uniform magnetic field, the resonant frequencies of the nuclei in the sample depend on where in the field they are located. Therefore, the particle can be located quite precisely by its resonant frequency.

Electron paramagnetic resonance, also known as electron spin resonance (EER), is a spectroscopic technique similar to NMR, but uses unpaired electrons. The materials for which it can be applied are much more limited since the material must have an unpaired spin and be paramagnetic.

The Mössbauer effect is the resonant and recoil-free emission and absorption of gamma ray photons by atoms bound in solid form.

Resonance in particle physics "Resonance (particle physics)") appears in circumstances similar to classical physics at the level of quantum mechanics and quantum field theory. However, they can also be considered unstable particles, with the above formula valid if Γ is decay rate and Ω replaced by the mass of the particle M. In that case, the formula comes from the propagator ] of the particle, with its mass replaced by the complex number M + iΓ. The formula is more related to the disintegration of particles by the optical theorem.

A physical system can have as many resonant frequencies as there are degrees of freedom; each degree of freedom can vibrate as a harmonic oscillator. Systems with one degree of freedom, such as a mass on a spring, balance wheels, pendulum, and LC tuned circuits have a resonant frequency. Systems with two degrees of freedom, such as coupled pendulums and resonant transformers, can have two resonant frequencies. As the number of coupled harmonic oscillators grows, the time it takes to transfer energy from one to another becomes significant. The vibrations in them begin to travel through the coupled harmonic oscillators in waves, from one oscillator to the next.

Extended objects that can experience resonance due to vibrations within them are called resonators, such as organ pipe, vibrating string, quartz, microwave, and laser. As you can see that they are made of many coupled moving parts (like atoms), they can have many resonant frequencies. The vibrations within them travel like waves, at a roughly constant speed, bouncing back and forth between the sides of the resonator. If the distance between the sides is d , the length of a round trip is 2d . To cause resonance, the phase of a sine wave after a round trip must be equal to the initial phase, so the waves reinforce the oscillation. So the condition for resonance in a resonator is that the round trip distance, 2d , is equal to an integer number of wavelengths λ of the wave:.

If the speed of a wave is v, the frequency is: f = then the resonant frequencies are:.

Then, the resonant frequencies of the resonators,.

Called normal modes, they are equally spaced multiples of a lower frequency called the fundamental frequency. Multiples are often called harmonics. There may be several series of resonant frequencies, corresponding to different modes of oscillation.

The quality factor is a dimensionless quantity (see dimensionless magnitude), a parameter that describes a damped motion such as an oscillator or resonator.[12]​

A higher Q indicates a lower rate of energy loss relative to the stored energy of the oscillator, so the oscillations will stop slowly. A high-quality bearing pendulum suspended, oscillating in air has a higher Q, while a pendulum immersed in oil would have a lower Q. To maintain a system in resonance with constant amplitude, energy must be provided externally; the energy provided in each cycle must be less than the energy stored in the system (i.e., the sum of potential and kinetics) by a factor of . Oscillators with high quality factors have low damping, which tends to make them resonate longer.

Resonators with higher Q factors resonate with larger amplitudes (at the resonant frequency) but have a smaller range of frequencies around the frequency at which they resonate. The range of frequencies in which the oscillator resonates is called bandwidth. Therefore, a high Q tuned circuit in a radio receiver would be harder to tune, but would have greater selectivity, doing a better job of filtering out signals from other stations that are close to the spectrum. High "Q" oscillators operate in a smaller range of frequencies and are more stable. (See phase noise.).

The quality factor of oscillators varies substantially from one system to another. Systems for which damping is important (such as dampers that prevent a door from slamming) have Q = , on the other hand clocks, lasers and other systems that need strong resonance or high frequency stability need high quality factors, for example tuning forks

have a quality factor of around Q = 1000, while the quality factor of an atomic clock and a high laser can reach Q =10[13]​ and may even be higher.[14]​ Physicists and engineers use many alternative quantities to describe how damped an oscillator is that are closely related to its quality factor.

The exact response of a resonance, especially for frequencies far from the resonant frequency, depends on the details of the physical system and is usually not exactly symmetric with respect to the resonant frequency, as illustrated for the harmonic oscillator. For a lightly damped linear oscillator with a resonance frequency Ω, the intensity of the oscillations I when the system operates with a driving frequency ω is typically approximated by a formula that is symmetric with respect to the resonance frequency:[15]​.

Where susceptibility links the amplitude of the oscillator to the driving force in frequency space:[16]​.

Intensity is defined as the square of the amplitude of the oscillations. This is a Lorentzian function, or Cauchy distribution, and this response is found in many physical situations involving resonant systems. Γ is a parameter that depends on the damping of the oscillator, and is known as the linewidth of the resonance. Highly damped oscillators tend to have wide linewidths, and respond to a wider range of driving frequencies around the resonance frequency. The line width is inversely proportional to the quality factor, which is a measure of the sharpness of the resonance.

In broadcast engineering and electronic engineering, this approximate symmetric response is known as the universal resonance curve, a concept introduced by Frederick Terman in 1932 to simplify the approximate analysis of radio circuits with a range of center frequencies and Q values.[17][18]​.