Resonance

Resonance is a phenomenon that occurs when an object or system is subjected to an external force or vibration whose frequency matches a resonant frequency of the system, defined as a frequency that generates a maximum amplitude response in the system. When this happens, the object or system absorbs energy from the external force and starts vibrating with a larger amplitude. Resonance can occur in various systems, such as mechanical, electrical, or acoustic systems, and it is often desirable in certain applications, such as musical instruments or radio receivers. However, resonance can also be detrimental, leading to excessive vibrations or even structural failure in some cases.
All systems, including molecular systems and particles, tend to vibrate at a natural frequency depending upon their structure; when there is very little damping this frequency is approximately equal to, but slightly above, the resonant frequency. When an oscillating force, an external vibration, is applied at a resonant frequency of a dynamic system, object, or particle, the outside vibration will cause the system to oscillate at a higher amplitude than when the same force is applied at other, non-resonant frequencies.
The resonant frequencies of a system can be identified when the response to an external vibration creates an amplitude that is a relative maximum within the system. Small periodic forces that are near a resonant frequency of the system have the ability to produce large amplitude oscillations in the system due to the storage of vibrational energy.
Resonance phenomena occur with all types of vibrations or waves: there is mechanical resonance, orbital resonance, acoustic resonance, electromagnetic resonance, nuclear magnetic resonance, electron spin resonance and resonance of quantum wave functions. Resonant systems can be used to generate vibrations of a specific frequency, or pick out specific frequencies from a complex vibration containing many frequencies.
The term resonance originated from the field of acoustics, particularly the sympathetic resonance observed in musical instruments, e.g., when one string starts to vibrate and produce sound after a different one is struck.

Overview

Resonance occurs when a system is able to store and easily transfer energy between two or more different storage modes. However, there are some losses from cycle to cycle, called damping. When damping is small, the resonant frequency is approximately equal to the natural frequency of the system, which is a frequency of unforced vibrations. Some systems have multiple and distinct resonant frequencies.

Examples

A familiar example is a playground swing, which acts as a pendulum. Pushing a person in a swing in time with the natural interval of the swing makes the swing go higher and higher, while attempts to push the swing at a faster or slower tempo produce smaller arcs. This is because the energy the swing absorbs is maximized when the pushes match the swing's natural oscillations.
Resonance occurs widely in nature, and is exploited in many devices. It is the mechanism by which virtually all sinusoidal waves and vibrations are generated. For example, when hard objects like metal, glass, or wood are struck, there are brief resonant vibrations in the object. Light and other short wavelength electromagnetic radiation is produced by resonance on an atomic scale, such as electrons in atoms. Other examples of resonance include:

Timekeeping mechanisms of modern clocks and watches, e.g., the balance wheel in a mechanical watch and the quartz crystal in a quartz watch
Tidal resonance of the Bay of Fundy
Acoustic resonances of musical instruments and the human vocal tract
Shattering of a crystal wineglass when exposed to a musical tone of the right pitch
Friction idiophones, such as making a glass object vibrate by rubbing around its rim with a fingertip
Electrical resonance of tuned circuits in radios and TVs that allow radio frequencies to be selectively received
Creation of coherent light by optical resonance in a laser cavity
Orbital resonance as exemplified by some moons of the Solar System's giant planets and resonant groups such as the plutinos
Material resonances in atomic scale are the basis of several spectroscopic techniques that are used in condensed matter physics
*Electron spin resonance
*Mössbauer effect
*Nuclear magnetic resonance
Linear systems

Resonance manifests itself in many linear and nonlinear systems as oscillations around an equilibrium point. When the system is driven by a sinusoidal external input, a measured output of the system may oscillate in response. The ratio of the amplitude of the output's steady-state oscillations to the input's oscillations is called the gain, and the gain can be a function of the frequency of the sinusoidal external input. Peaks in the gain at certain frequencies correspond to resonances, where the amplitude of the measured output's oscillations are disproportionately large.
Since many linear and nonlinear systems that oscillate are modeled as harmonic oscillators near their equilibria, a derivation of the resonant frequency for a driven, damped harmonic oscillator is shown. An RLC circuit is used to illustrate connections between resonance and a system's transfer function, frequency response, poles, and zeroes. Building off the RLC circuit example, these connections for higher-order linear systems with multiple inputs and outputs are generalized.

The driven, damped harmonic oscillator

Consider a damped mass on a spring driven by a sinusoidal, externally applied force. Newton's second law takes the form
where m is the mass, x is the displacement of the mass from the equilibrium point, F₀ is the driving amplitude, ω is the driving angular frequency, k is the spring constant, and c is the viscous damping coefficient. This can be rewritten in the form
where

is called the undamped angular frequency of the oscillator or the natural frequency,
is called the damping ratio.

Many sources also refer to ω₀ as the resonant frequency. However, as shown below, when analyzing oscillations of the displacement x, the resonant frequency is close to but not the same as ω₀. In general the resonant frequency is close to but not necessarily the same as the natural frequency. The RLC circuit example in the next section gives examples of different resonant frequencies for the same system.
The general solution of Equation is the sum of a transient solution that depends on initial conditions and a steady state solution that is independent of initial conditions and depends only on the driving amplitude F₀, driving frequency ω, undamped angular frequency ω₀, and the damping ratio ζ. The transient solution decays in a relatively short amount of time, so to study resonance it is sufficient to consider the steady state solution.
It is possible to write the steady-state solution for x as a function proportional to the driving force with an induced phase change, φ.
where
The phase value is usually taken to be between −180° and 0 so it represents a phase lag for both positive and negative values of the arctan argument.
Resonance occurs when, at certain driving frequencies, the steady-state amplitude of x is large compared to its amplitude at other driving frequencies. For the mass on a spring, resonance corresponds physically to the mass's oscillations having large displacements from the spring's equilibrium position at certain driving frequencies. Looking at the amplitude of x as a function of the driving frequency ω, the amplitude is maximal at the driving frequency
ω_r is the resonant frequency for this system. Again, the resonant frequency does not equal the undamped angular frequency ω₀ of the oscillator. They are proportional, and if the damping ratio goes to zero they are the same, but for non-zero damping they are not the same frequency. As shown in the figure, resonance may also occur at other frequencies near the resonant frequency, including ω₀, but the maximum response is at the resonant frequency.
Also, ω_r is only real and non-zero if, so this system can only resonate when the harmonic oscillator is significantly underdamped. For systems with a very small damping ratio and a driving frequency near the resonant frequency, the steady state oscillations can become very large.

The pendulum

For other driven, damped harmonic oscillators whose equations of motion do not look exactly like the mass on a spring example, the resonant frequency remains
but the definitions of ω₀ and ζ change based on the physics of the system. For a pendulum of length ℓ and small displacement angle θ, Equation becomes
and therefore

RLC series circuits

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 and driven by a voltage source with voltage v_in. The voltage drop around the circuit is
Rather than analyzing a candidate solution to this equation like in the mass on a spring example above, this section will analyze the frequency response of this circuit. Taking the Laplace transform of Equation,
where I and V_in are the Laplace transform of the current and input voltage, respectively, and s is a complex frequency parameter in the Laplace domain. Rearranging terms,

Voltage across the capacitor

An RLC circuit in series presents several options for where 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
or
Define for this circuit a natural frequency and a damping ratio,
The ratio of the output voltage to the input voltage becomes
H is the transfer function between the input voltage and the output voltage. This transfer function has two poles–roots of the polynomial in the transfer function's denominator–at
and no zeros–roots of the polynomial in the transfer function's numerator. Moreover, for, the magnitude of these poles is the natural frequency ω₀ and that for, our condition for resonance in the harmonic oscillator example, the poles are closer to the imaginary axis than to the real axis.
Evaluating H along the imaginary axis, the transfer function describes the frequency response of this circuit. Equivalently, the frequency response can be analyzed by taking the Fourier transform of Equation instead of the Laplace transform. The transfer function, which is also complex, can be written as a 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 and has a phase shift Φ. The gain and phase can be plotted versus frequency on a Bode plot. For the RLC circuit's capacitor voltage, the gain of the transfer function H is
Note the similarity between the gain here and the amplitude in Equation. Once again, the gain is maximized at the resonant frequency
Here, the resonance corresponds physically 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.