Resonance

Resonance is a phenomenon that occurs when an object or system is subjected to an external force or vibration whose frequency aligns with a resonant frequency (or resonance frequency) of the system, which is defined as a frequency generating a maximal amplitude response. Consequently, the object or system absorbs energy from the external force and exhibits oscillations of increased amplitude. Resonance can manifest in diverse systems, including mechanical, electrical, and acoustic configurations, and is frequently leveraged in specific applications such as musical instruments or radio receivers. However, resonance may also pose significant risks, potentially leading to excessive vibrational stress or catastrophic structural failure in certain scenarios.

All systems, encompassing molecular structures and individual particles, possess an inherent tendency to oscillate at a natural frequency determined by their structural properties. In conditions of minimal damping, this natural frequency closely approximates, yet slightly exceeds, the resonant frequency. When an external oscillating force, or vibration, is applied at a system's resonant frequency, this external excitation induces oscillations of significantly greater amplitude (and force) compared to the response elicited by the same force applied at non-resonant frequencies.

The resonant frequencies of a system are characterized by its response to an external vibration exhibiting a relative maximum in amplitude. Even modest periodic forces, when applied near a system's resonant frequency, can generate substantial amplitude oscillations within that system, a phenomenon attributed to the efficient accumulation of vibrational energy.

Resonance manifests across diverse forms of vibrations and waves, including mechanical resonance, orbital resonance, acoustic resonance, electromagnetic resonance, nuclear magnetic resonance (NMR), electron spin resonance (ESR), and the resonance of quantum wave functions. Such resonant systems find utility in both the generation of vibrations at precise frequencies (e.g., in musical instruments) and the selective extraction of specific frequencies from intricate, multi-frequency vibrations (e.g., in filters).

The term resonance (derived from Latin resonantia, meaning 'echo', and resonare, 'to resound') has its origins in the field of acoustics, as explored by Galileo Galilei in his seminal work Dialogues Concerning Two New Sciences. It specifically referred to the phenomenon of sympathetic resonance observed in musical instruments, where, for instance, one string begins to vibrate and emit sound subsequent to another being struck.

Overview

Resonance occurs when a system facilitates the efficient storage and interconversion of energy between two or more distinct modes (such as kinetic and potential energy in the context of a simple pendulum). Nonetheless, energy dissipation occurs with each cycle, a process termed damping. Under conditions of low damping, the resonant frequency closely approximates the system's natural frequency, which represents the frequency of its unforced oscillations. It is noteworthy that certain systems exhibit multiple, discrete resonant frequencies.

Examples

A common illustration of resonance is a playground swing, which functions as a pendulum. Applying force to a person on a swing in synchrony with its natural period (i.e., its resonant frequency) progressively increases the swing's amplitude, leading to higher arcs. Conversely, attempts to push the swing at tempos either faster or slower than its resonant frequency result in diminished arcs. This phenomenon occurs because the energy absorbed by the swing is maximized when the applied impulses align precisely with its natural oscillatory frequency.

Resonance is ubiquitous in natural phenomena and extensively utilized in numerous technological applications. It constitutes the fundamental mechanism underlying the generation of nearly all sinusoidal waves and vibrations. For instance, striking rigid materials such as metal, glass, or wood induces transient resonant vibrations within these objects. Furthermore, light and other forms of short-wavelength electromagnetic radiation arise from atomic-scale resonance, specifically involving electrons within atoms. Additional instances of resonance encompass:

• The timekeeping mechanisms found in contemporary clocks and watches, for example, the balance wheel in mechanical timepieces and the quartz crystal in quartz watches.
• The tidal resonance observed in the Bay of Fundy.
• Acoustic resonances present in musical instruments and the human vocal tract.
• The phenomenon of a crystal wineglass shattering when subjected to a musical tone precisely matching its resonant frequency.
• Friction idiophones, exemplified by inducing vibrations in a glass object (e.g., a glass, bottle, or vase) through rubbing its rim with a fingertip.
• Electrical resonance in tuned circuits of radios and televisions enables the selective reception of specific radio frequencies.
• Optical resonance within a laser cavity facilitates the generation of coherent light.
• Orbital resonance is exemplified by certain moons orbiting the Solar System's giant planets and by resonant groups like the plutinos.
• Material resonances at the atomic scale form the foundation for various spectroscopic techniques employed in condensed matter physics, including
  • electron spin resonance.
  • Mössbauer effect.
  • Nuclear magnetic resonance.

Linear Systems.

Resonance is observed in numerous linear and nonlinear systems as oscillations occurring around an equilibrium point. When such a system is subjected to a sinusoidal external input, its measured output may oscillate responsively. The gain, defined as the ratio of the amplitude of the output's steady-state oscillations to the input's oscillations, can vary as a function of the sinusoidal external input's frequency. Peaks in this gain at specific frequencies indicate resonances, characterized by disproportionately large amplitudes in the measured output's oscillations.

Given that many oscillating linear and nonlinear systems are modeled as harmonic oscillators near their equilibrium states, a derivation of the resonant frequency for a driven, damped harmonic oscillator is presented. An RLC circuit serves to exemplify the relationships between resonance and a system's transfer function, frequency response, poles, and zeros. Extending from the RLC circuit illustration, these relationships are then generalized for higher-order linear systems featuring multiple inputs and outputs.

The Driven, Damped Harmonic Oscillator.

Consider a damped mass-spring system actuated by a sinusoidal, externally applied force. Newton's second law for this system is expressed as:

where m denotes the mass, x represents the displacement of the mass from its equilibrium position, F§67§ signifies the driving amplitude, ω is the driving angular frequency, k stands for the spring constant, and c indicates the viscous damping coefficient. This equation can be reformulated as:

where:

• ω<!-- ω --> §1112§ = k / m {\textstyle \omega _{0}={\sqrt {k/m}}} is designated as the undamped angular frequency of the oscillator or, alternatively, the natural frequency.
• ${\displaystyle \zeta ={\frac {c}{2{\sqrt {mk}}}}}$ is referred to as the damping ratio.

While numerous sources identify ω§23§ as the resonant frequency, it is important to note that, as subsequently demonstrated, the resonant frequency for oscillations of the displacement x(t) approximates but does not precisely equal ω§1213§. Generally, the resonant frequency is similar to, yet not necessarily identical to, the natural frequency. The RLC circuit example presented in the subsequent section illustrates instances of distinct resonant frequencies within the same system.

The general solution for Equation (2) comprises a transient solution, which is contingent upon initial conditions, and a steady-state solution, which is independent of initial conditions and solely dependent on the driving amplitude F§45§, driving frequency ω, undamped angular frequency ω§1011§, and the damping ratio ζ. Since the transient solution dissipates within a comparatively brief period, focusing on the steady-state solution is adequate for the analysis of resonance.

The steady-state solution for x(t) can be expressed as a function directly proportional to the driving force, incorporating an induced phase shift, φ.

The phase, denoted by φ<!-- φ -->, is calculated using the following expression: = arctan ⁡<!-- ⁡ --> ( §2223§ ω<!-- ω --> ω<!-- ω --> §3233§ ζ ω §4647§ − ω §5859§ §6263§ ) + n π . This can also be represented as: {\displaystyle \varphi =\arctan \left({\frac {2\omega \omega _{0}\zeta }{\omega ^{2}-\omega _{0}^{2}}}\right)+n\pi .}

Typically, the phase value is constrained to the range of −180° to 0°, which signifies a phase lag regardless of whether the arctangent argument is positive or negative.

Resonance is defined as the phenomenon where the steady-state amplitude of x(t) becomes significantly elevated at specific driving frequencies, in contrast to its amplitude at other frequencies. In the context of a mass-spring system, resonance physically manifests as substantial displacements of the mass from the spring's equilibrium position when subjected to particular driving frequencies. Analyzing the amplitude of x(t) as a function of the driving frequency ω reveals that the amplitude reaches its maximum at the following driving frequency: ω<!-- ω --> r = ω<!-- ω --> §3233§ §3839§ − §4344§ ζ §5051§ . This is also expressed as: {\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}.}

The term ω_r designates the resonant frequency for this specific system. It is important to reiterate that the resonant frequency is distinct from the undamped angular frequency ω§1011§ of the oscillator. While these frequencies are proportional and converge when the damping ratio approaches zero, they remain unequal under conditions of non-zero damping. As illustrated in the accompanying figure, resonance can manifest at frequencies proximate to the resonant frequency, including ω§1415§; however, the peak response invariably occurs precisely at the resonant frequency.

Furthermore, ω_r maintains a real and non-zero value exclusively when _26§ , or equivalently, {\textstyle \zeta <1/{\sqrt {2}}}. Consequently, resonance in this system is contingent upon the harmonic oscillator being substantially underdamped. In scenarios involving systems characterized by a minimal damping ratio and a driving frequency close to the resonant frequency, the steady-state oscillations can achieve considerable magnitudes.

The pendulum

For other driven, damped harmonic oscillators, even if their equations of motion differ from the standard mass-on-a-spring model, the resonant frequency is consistently expressed as ω<!-- ω --> r = ω<!-- ω --> §2223§ §2829§ − §3334§ ζ<!-- ζ --> §4041§ , {\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}},} . However, the specific definitions of ω§5859§ and ζ are contingent upon the physical characteristics of the system under consideration. For instance, in the case of a pendulum with length ℓ and a small displacement angle θ, Equation (§6667§) transforms into: m ℓ<!-- ℓ --> d §8889§ θ<!-- θ --> d t §105106§ = F §118119§ sin ⁡ ( ω t ) − m g θ − c ℓ d θ d t {\displaystyle m\ell {\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}=F_{0}\sin(\omega t)-mg\theta -c\ell {\frac {\mathrm {d} \theta }{\mathrm {d} t}}}

Consequently,

• ω<!-- ω --> §1112§ = g ℓ , {\displaystyle \omega _{0}={\sqrt {\frac {g}{\ell }}},}
• ${\displaystyle \zeta ={\frac {c}{2m}}{\sqrt {\frac {\ell }{g}}}.}$

RLC Series Circuits

An RLC series circuit comprises a resistor (resistance R), an inductor (inductance L), and a capacitor (capacitance C) connected sequentially. This circuit is driven by a voltage source, v_in(t), which generates a current i(t). The total voltage drop across this circuit is determined as follows:

Instead of analyzing a candidate solution to this equation, as was done in the mass-on-a-spring example, this section will examine the frequency response of this specific circuit. By applying the Laplace transform to Equation (4), the following expression is derived: $${\displaystyle sLI(s)+RI(s)+{\frac {1}{sC}}I(s)=V_{\text{in}}(s),}$$ where I(s) and V_in(s) represent the Laplace transforms of the current and input voltage, respectively, and s denotes a complex frequency parameter within the Laplace domain. Rearranging these terms yields: I ( s ) = s s §119120§ L + R s + §135136§ C V in ( s ) . {\displaystyle I(s)={\frac {s}{s^{2}L+Rs+{\frac {1}{C}}}}V_{\text{in}}(s).}

Voltage Across the Capacitor

A series RLC circuit offers several potential points for measuring an output voltage. Consider the scenario where the output voltage of interest is the voltage drop across the capacitor. As previously demonstrated, this voltage in the Laplace domain is expressed as: V out ( s ) = §2425§ s C I ( s ) {\displaystyle V_{\text{out}}(s)={\frac {1}{sC}}I(s)} Alternatively, this can be written as: V out = §6869§ L C ( s §8182§ + R L s + §101102§ L C ) V in ( s ) . {\displaystyle V_{\text{out}}={\frac {1}{LC(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}})}}V_{\text{in}}(s).}

For this circuit, the natural frequency and damping ratio are defined by the following expressions: ω<!-- ω --> §1112§ = §1920§ L C , {\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}},} $${\displaystyle \zeta ={\frac {R}{2}}{\sqrt {\frac {C}{L}}}.}$$

The ratio of the output voltage to the input voltage is expressed as: H ( s ) ≜<!-- ≜ --> V out ( s ) V in ( s ) = ω<!-- ω --> §6263§ §6667§ s §7576§ + §8182§ ζ<!-- ζ --> ω<!-- ω --> §9192§ s + ω §104105§ §108109§ {\displaystyle H(s)\triangleq {\frac {V_{\text{out}}(s)}{V_{\text{in}}(s)}}={\frac {\omega _{0}^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}}

H(s) represents the transfer function relating the input voltage to the output voltage. This transfer function exhibits two poles, which are the roots of the polynomial in its denominator, located at:

It has no zeros, as the numerator's polynomial lacks roots. Furthermore, for ζ ≤ 1, the magnitude of these poles corresponds to the natural frequency ω§67§. When ζ < 1/{\displaystyle {\sqrt {2}}}, a condition that signifies resonance in the harmonic oscillator context, the poles are situated nearer to the imaginary axis than to the real axis.

By evaluating H(s) along the imaginary axis, where s = iω, the transfer function effectively characterizes the frequency response of this circuit. Alternatively, the frequency response can be determined by applying the Fourier transform to Equation (§1011§) instead of the Laplace transform. This complex transfer function can be expressed in terms of its gain and phase components: $${\displaystyle H(i\omega )=G(\omega )e^{i\Phi (\omega )}.}$$

When a sinusoidal input voltage at frequency ω is applied, the resulting output voltage maintains the same frequency but is scaled by G(ω) and exhibits a phase shift of Φ(ω). These gain and phase characteristics can be graphically represented against frequency using a Bode plot. For the capacitor voltage within an RLC circuit, the gain of the transfer function H(iω) is defined as:

A notable resemblance exists between this gain and the amplitude presented in Equation (3). The gain reaches its maximum value at the resonant frequency, which is expressed as: ω<!-- ω -->r=ω<!-- ω -->§2627§§3233§−§3738§ζ§4445§.{\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}.}

In this context, resonance physically manifests as a comparatively large amplitude in the steady-state oscillations of the capacitor voltage when contrasted with its amplitude at alternative driving frequencies.

Voltage Across the Inductor

The resonant frequency is not exclusively defined by the forms presented previously. For an RLC circuit, if the output voltage is defined as the voltage across the inductor, then in the Laplace domain, this voltage is expressed as:

Employing the identical definitions for ω§23§ and ζ established in the preceding example, the transfer function relating V_in(s) to the newly defined V_out(s) across the inductor is expressed as: H ( s ) = s §40_41§ s §49_50§ + §55_{56§ ζ ω §6566§ s + ω §7879§ §8283§} . {\displaystyle H(s)={\frac {s^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.}

This transfer function shares its poles with the transfer function from the preceding example; however, it additionally features two zeroes in the numerator located at s = 0. When H(s) is evaluated along the imaginary axis, its corresponding gain is determined to be: G ( ω<!-- ω --> ) = ω<!-- ω --> §3233§ ( §4243§ ω<!-- ω --> ω<!-- ω --> §5253§ ζ ) §6465§ + ( ω §7778§ §8182§ − ω §9394§ ) §101102§ . {\displaystyle G(\omega )={\frac {\omega ^{2}}{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.}

In contrast to the gain presented in Equation (6), which utilizes the capacitor voltage as the output, this particular gain incorporates a factor of ω§45§ in its numerator. Consequently, it exhibits a distinct resonant frequency at which the gain is maximized. This resonant frequency is given by: ω<!-- ω --> r = ω<!-- ω --> §3031§ §3536§ − §4041§ ζ §4748§ , {\displaystyle \omega _{r}={\frac {\omega _{0}}{\sqrt {1-2\zeta ^{2}}}},}

When considering the voltage across the inductor as the output for an identical RLC circuit, the resonant frequency becomes greater than the natural frequency; however, it converges to the natural frequency as the damping ratio approaches zero. The observation that a single circuit can exhibit distinct resonant frequencies based on the chosen output is not contradictory. As demonstrated in Equation (4), the total voltage drop across the circuit is distributed among its three constituent elements, each possessing unique dynamic characteristics. Specifically, the capacitor's voltage accumulates gradually through current integration over time, rendering it more responsive to lower frequencies. Conversely, the inductor's voltage increases in response to rapid current variations, making it more sensitive to higher frequencies. Although the entire circuit possesses a natural frequency at which it inherently oscillates, the disparate dynamics of its individual components cause each element to resonate at a marginally different frequency.

Voltage Across the Resistor

Let us consider the output voltage to be the voltage across the resistor. In the Laplace domain, the voltage across the resistor is expressed as: $${\displaystyle V_{\text{out}}(s)=RI(s),}$$ V out ( s ) = R s L ( s §8384§ + R L s + §103104§ L C ) V in ( s ) , {\displaystyle V_{\text{out}}(s)={\frac {Rs}{L\left(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}\right)}}V_{\text{in}}(s),}

Employing the identical natural frequency and damping ratio established in the capacitor example, the transfer function is given by: H ( s ) = §1920§ ζ<!-- ζ --> ω<!-- ω --> §2930§ s s §4142§ + §4748§ ζ<!-- ζ --> ω<!-- ω --> §5758§ s + ω §7071§ §7475§ . {\displaystyle H(s)={\frac {2\zeta \omega _{0}s}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.}

This transfer function shares the same poles as the previously discussed RLC circuit examples, but it features a single zero in the numerator located at s = 0. The gain for this specific transfer function is expressed as: G ( ω<!-- ω --> ) = §2223§ ζ ω §3233§ ω ( §4647§ ω ω §5657§ ζ ) §6869§ + ( ω §8182§ §8586§ − ω §9798§ ) §105106§ . {\displaystyle G(\omega )={\frac {2\zeta \omega _{0}\omega }{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.}

The resonant frequency at which this gain is maximized is given by: ω<!-- ω --> r = ω<!-- ω --> §2223§ , {\displaystyle \omega _{r}=\omega _{0},} At this frequency, the gain is unity, indicating that the voltage across the resistor resonates at the circuit's natural frequency, and its amplitude matches that of the input voltage.

Antiresonance

Certain systems demonstrate antiresonance, a phenomenon amenable to analysis using methods similar to those employed for resonance. In antiresonance, the system's response amplitude at specific frequencies is notably small, contrasting with the disproportionately large amplitudes observed in resonance. Within the context of an RLC circuit, this phenomenon becomes apparent through a combined analysis of the inductor and capacitor.

Consider the output voltage in the RLC circuit to be the series combination of the voltages across the inductor and the capacitor. Equation (4) established that the sum of voltages across the three circuit elements equals the input voltage. Consequently, defining the output voltage as the combined inductor and capacitor voltages is equivalent to v_in minus the voltage drop across the resistor. As demonstrated in the preceding example, at the system's natural frequency, the amplitude of the voltage drop across the resistor equals the amplitude of v_in. This implies that the combined voltage across the inductor and capacitor exhibits zero amplitude, a fact verifiable through the transfer function.

The total voltage across the inductor and capacitor is given by the following expression: V out ( s ) = ( s L + §3233§ s C ) I ( s ) , {\displaystyle V_{\text{out}}(s)=(sL+{\frac {1}{sC}})I(s),} . This relationship can also be expressed as: V out ( s ) = s §9192§ + §99100§ L C s §114115§ + R L s + §135136§ L C V in ( s ) . {\displaystyle V_{\text{out}}(s)={\frac {s^{2}+{\frac {1}{LC}}}{s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}}}V_{\text{in}}(s).}

By utilizing the identical natural frequency and damping ratios from the preceding examples, the transfer function is derived as: H ( s ) = s §2324§ + ω<!-- ω --> §3435§ §3839§ s §4849§ + §5455§ ζ<!-- ζ --> ω<!-- ω --> §6465§ s + ω §7778§ §8182§ . {\displaystyle H(s)={\frac {s^{2}+\omega _{0}^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.}

This transfer function exhibits poles identical to those observed in the preceding examples, but its zeroes are located at:

When the transfer function is evaluated along the imaginary axis, its gain is expressed as: G ( ω<!-- ω --> ) = ω<!-- ω --> §2526§ §2930§ − ω §4142§ ( §5253§ ω<!-- ω --> ω<!-- ω --> §6263§ ζ ) §7475§ + ( ω §8788§ §9192§ − ω §103104§ ) §111112§ . {\displaystyle G(\omega )={\frac {\omega _{0}^{2}-\omega ^{2}}{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.}

Instead of identifying resonance through gain peaks, it is notable that the gain approaches zero at ω = ω§45§. This observation aligns with the analysis of the resistor's voltage. This phenomenon is termed antiresonance, which fundamentally opposes the effects of resonance. Unlike resonance, which produces disproportionately large outputs at a specific frequency, this circuit configuration, with its chosen output, exhibits no response at all at this particular frequency. The frequency that undergoes filtering precisely corresponds to the transfer function's zeroes, which were previously presented in Equation (§89§) and located on the imaginary axis.

Interconnections between Resonance and Frequency Response in the RLC Series Circuit Example

The provided RLC circuit examples demonstrate the relationship between resonance and the system's frequency response. Specifically, these illustrations highlight:

• The methodology for identifying resonant frequencies by observing peaks in the transfer function's gain, which represents the relationship between system input and output, as exemplified in a Bode magnitude plot.
• The variability of the resonant frequency within a single system, contingent upon the selection of the system's output.
• The intrinsic relationship among the system's natural frequency, its damping ratio, and its resonant frequency.
• The correlation between the system's natural frequency and the magnitude of the transfer function's poles, as established in Equation (5), thereby indicating a link between these poles and the resonant frequency.
• The relationship between the transfer function's zeroes and the frequency-dependent gain profile, consequently linking these zeroes to the resonant frequency that yields maximum gain.
• The association between the transfer function's zeroes and the phenomenon of antiresonance.

The subsequent section expands upon these concepts, applying them to resonance within general linear systems.

Generalization of Resonance and Antiresonance for Linear Systems

To illustrate, consider an arbitrary linear system characterized by multiple inputs and outputs. For instance, a third-order linear time-invariant system with three inputs and two outputs can be expressed using state-space representation as follows:

The system is characterized by a transfer function matrix, where each element represents the transfer function linking specific inputs to corresponding outputs. For instance,

Each H_ij(s) represents a scalar transfer function, establishing a connection between a specific input and a particular output. For instance, the previously discussed RLC circuit examples featured a single input voltage and presented four potential output voltages: across the capacitor, across the inductor, across the resistor, and across the series combination of the capacitor and inductor. Each of these outputs possessed its own distinct transfer function. Consequently, if the RLC circuit were configured to measure all four of these output voltages, the resulting system would be characterized by a 4×1 transfer function matrix, correlating the sole input to each of the four outputs.

When evaluated along the imaginary axis, each H_ij(iω) can be expressed as a combination of gain and phase shift, as shown below: $${\displaystyle H_{ij}(i\omega )=G_{ij}(\omega )e^{i\Phi _{ij}(\omega )}.}$$

Assuming system stability, peaks observed in the gain at specific frequencies indicate resonances occurring between the input and output associated with that particular transfer function.

Furthermore, each transfer function H_ij(s) can be expressed as a rational function, where both the numerator and denominator are polynomials of s, as illustrated below: $${\displaystyle H_{ij}(s)={\frac {N_{ij}(s)}{D_{ij}(s)}}.}$$

The complex roots of the numerator are defined as zeroes, while the complex roots of the denominator are referred to as poles. For a stable system, the placement of these poles and zeroes within the complex plane offers insights into the system's potential for resonance or anti-resonance, and at which specific frequencies these phenomena might occur. Specifically, any stable or marginally stable complex conjugate pair of poles possessing imaginary components can be formulated in terms of a natural frequency and a damping ratio, as follows: s = −<!-- − --> ζ<!-- ζ --> ω<!-- ω --> §2122§ ± i ω §3536§ §4142§ − ζ §5152§ , {\displaystyle s=-\zeta \omega _{0}\pm i\omega _{0}{\sqrt {1-\zeta ^{2}}},} consistent with Equation (§6768§). The natural frequency ω§7172§ for that pole corresponds to the magnitude of the pole's position on the complex plane, while its damping ratio dictates the rate at which the associated oscillation attenuates. In general,

• Complex conjugate pairs of poles situated near the imaginary axis are associated with a peak or resonance in the frequency response, occurring proximate to the pole's natural frequency. Should these pole pairs reside directly on the imaginary axis, the system exhibits infinite gain at that specific frequency.
• Complex conjugate pairs of zeroes positioned near the imaginary axis are indicative of a notch or antiresonance in the frequency response, manifesting at a frequency approximating the zero's magnitude. When these zero pairs are situated precisely on the imaginary axis, the system's gain becomes zero at that particular frequency.

In the RLC circuit example, the first generalization linking poles with resonance is demonstrated by Equation (5). The second generalization connecting zeroes with antiresonance is observed in Equation (7). Specifically, within the contexts of the harmonic oscillator, the RLC circuit's capacitor voltage, and its inductor voltage, "poles near the imaginary axis" signifies a substantially underdamped condition where ζ < 1/§1213§ {\displaystyle {\sqrt {2}}} .

Standing waves

A physical system possesses a number of natural frequencies equivalent to its degrees of freedom, and is capable of resonating at frequencies proximate to each of these. For instance, a single mass-spring system, characterized by one degree of freedom, exhibits a solitary natural frequency. Conversely, a double pendulum, possessing two degrees of freedom, may manifest two distinct natural frequencies. With an escalating number of coupled harmonic oscillators, the temporal duration required for inter-oscillator energy transfer becomes substantial. Consequently, systems endowed with an exceptionally high number of degrees of freedom are often conceptualized as continuous entities, rather than as collections of discrete oscillators.

Energy propagates between oscillators through wave phenomena. For instance, both a guitar string and the surface of water within a container can be analytically represented as continua of interconnected small oscillators, along which waves are capable of propagation. Frequently, such systems exhibit the capacity to resonate at specific frequencies, thereby generating standing waves characterized by high-amplitude oscillations at stationary locations. This standing wave resonance mechanism is fundamental to numerous common phenomena, including the acoustic output of musical instruments, the operation of electromagnetic cavities in lasers and microwave ovens, and the quantized energy levels observed in atoms.

Standing waves on a string

Upon excitation of a string of finite length at a specific frequency, a wave initiates propagation along its length at that identical frequency. These waves subsequently reflect from the string's termini, ultimately establishing a steady state characterized by bidirectional wave propagation. The resultant waveform represents the superposition of these propagating waves.

At specific frequencies, the steady-state waveform exhibits no apparent longitudinal propagation along the string. At designated fixed points, termed nodes, the string remains perpetually undisplaced. Conversely, between these nodes, the string undergoes oscillation, with the maximal oscillation amplitude occurring precisely midway between nodes, at locations referred to as anti-nodes.

For a string of length ${\displaystyle L}$ with fixed boundary conditions, the transverse displacement ${\displaystyle y(x,t)}$, measured perpendicular to the ${\displaystyle x}$-axis at a given position x and time ${\displaystyle t}$, is mathematically expressed as: $${\displaystyle y(x,t)=2y_{\text{max}}\sin(kx)\cos(2\pi ft),}$$.

In this equation,

• ${\displaystyle y_{\text{max}}}$ represents the amplitude of the constituent left- and right-traveling waves that superimpose to generate the standing wave;
• ${\displaystyle k}$ denotes the wave number; and
• ${\displaystyle f}$ signifies the frequency.

The resonant frequencies capable of generating standing waves are determined by the string's length according to the following relationship: $${\displaystyle f={\frac {nv}{2L}},}$$, where n = §4647§ , §5051§ , §5455§ , …<!-- … --> {\displaystyle n=1,2,3,\dots } .

Here, ${\displaystyle v}$ represents the wave speed, and the integer ${\displaystyle n}$ designates the various modes or harmonics of oscillation. Specifically, the standing wave corresponding to n = 1 vibrates at its fundamental frequency, possessing a wavelength equivalent to twice the string's length. Consequently, the permissible oscillation modes constitute a harmonic series.

Categories of Resonance

Mechanical Resonance

Mechanical resonance describes a phenomenon where a mechanical system preferentially absorbs greater energy when the frequency of its applied oscillations coincides with its inherent natural frequency of vibration, compared to other frequencies. This effect can induce severe oscillatory movements and potentially lead to catastrophic structural failures in inadequately engineered constructions such as bridges, buildings, trains, and aircraft. Therefore, during the design process, engineers are obligated to verify that the mechanical resonance frequencies of individual components do not align with the driving vibrational frequencies originating from motors or other oscillating elements, a critical condition referred to as a resonance disaster.

Preventing catastrophic resonance is a critical consideration in the design and construction of all buildings, towers, and bridges. To mitigate this risk, shock mounts can be integrated to absorb resonant frequencies, thereby dissipating the absorbed energy. For instance, the Taipei 101 building incorporates a 660-tonne (730-short-ton) tuned mass damper, functioning as a large pendulum, to counteract resonant oscillations. Additionally, the structural design aims for a resonant frequency that is not commonly encountered in its operational environment. In seismically active regions, structures are frequently engineered to account for the anticipated oscillating frequencies of ground motion. Furthermore, engineers designing engine-equipped systems must ensure that the mechanical resonant frequencies of individual components do not coincide with the driving vibrational frequencies generated by motors or other highly oscillating elements.

Clocks maintain temporal accuracy through the principle of mechanical resonance, utilizing components such as balance wheels, pendulums, or quartz crystals.

The energetic efficiency of a runner's cadence has been theorized to stem from a resonant interaction between the elastic energy stored within the lower limbs and the runner's body mass.

International Space Station

The rocket engines of the International Space Station (ISS) are managed by an autonomous autopilot system. Typically, pre-programmed parameters for the Zvezda module's engine control system direct the rocket engines to propel the ISS into a higher orbital trajectory. These hinge-mounted engines usually operate without the crew's awareness. However, on January 14, 2009, an anomaly occurred when uploaded parameters caused the autopilot to induce progressively larger oscillations in the rocket engines, at a frequency of 0.5 Hz. These oscillations, which were recorded on video, persisted for 142 seconds.

Acoustic

Acoustic resonance constitutes a specialized domain of mechanical resonance, specifically addressing mechanical vibrations within the audible frequency range for humans, which is commonly referred to as sound. Human auditory perception typically spans frequencies from approximately 20 Hz to 20,000 Hz (20 kHz). Numerous objects and materials function as resonators, possessing resonant frequencies within this spectrum; when impacted, they vibrate mechanically, displacing the surrounding air to generate sound waves. This phenomenon is the fundamental source of many percussive sounds encountered in daily life.

Acoustic resonance is a crucial factor for instrument manufacturers, given that the majority of acoustic instruments incorporate resonators, exemplified by the strings and body of a violin, the tube length of a flute, and the configuration and tension of a drum membrane.

Similar to mechanical resonance, acoustic resonance can induce catastrophic structural failure in an object when its resonant frequency is excited. The archetypal illustration of this principle involves shattering a wine glass using sound precisely tuned to its resonant frequency, although achieving this in practical scenarios remains challenging.

Electrical

Electrical resonance manifests in an electric circuit at a specific resonant frequency, characterized by either a minimum impedance in a series circuit or a maximum impedance in a parallel circuit (typically corresponding to the peak absolute value of the transfer function). Circuit resonance is extensively employed in both the transmission and reception of wireless communications, including applications such as television, cellular phones, and radio.

Optical

An optical cavity, alternatively termed an optical resonator, comprises a configuration of mirrors that establishes a standing wave cavity resonator for light waves. These cavities serve as a fundamental component in lasers, enclosing the gain medium and facilitating feedback for the laser light. Furthermore, they are utilized in optical parametric oscillators and certain interferometers. Light confined within the cavity undergoes multiple reflections, generating standing waves at specific resonant frequencies. The resulting standing wave patterns are designated as "modes." Longitudinal modes are distinguished solely by their frequency, whereas transverse modes vary across different frequencies and exhibit distinct intensity patterns across the beam's cross-section. Examples of optical resonators that do not generate standing waves include ring resonators and whispering galleries.

Resonator types are differentiated by the focal lengths of their two mirrors and the distance between them; flat mirrors are seldom employed due to the challenges of precise alignment. The chosen geometry (resonator type) necessitates a configuration that ensures beam stability, preventing continuous expansion of the beam size with each reflection. Resonator designs also fulfill additional criteria, such as achieving a minimum beam waist or the absence of an internal focal point, thereby mitigating localized light intensity within the cavity.

Optical cavities are engineered to exhibit an exceptionally high Q factor. A beam undergoes numerous reflections with minimal attenuation; consequently, the frequency linewidth of the beam is significantly narrower relative to the laser's fundamental frequency.

Furthermore, optical resonances encompass guided-mode resonances and surface plasmon resonance, manifesting as anomalous reflection and pronounced evanescent fields at resonance. Specifically, these resonant modes correspond to guided modes within a waveguide or surface plasmon modes at a dielectric-metallic interface. These modes are typically instigated by a subwavelength grating.

Orbital

In celestial mechanics, an orbital resonance is observed when two orbiting bodies impose a consistent, periodic gravitational interaction on each other, typically stemming from their orbital periods exhibiting a ratio of two small integers. Orbital resonances significantly amplify the mutual gravitational influence between the bodies. Predominantly, this leads to an unstable interaction, in which celestial bodies exchange momentum and undergo orbital shifts until the resonant condition is abrogated. Conversely, certain conditions permit a resonant system to achieve stability and self-correction, thereby maintaining the bodies in resonance. Illustrative instances include the 1:2:4 resonance of Jupiter's moons Ganymede, Europa, and Io, and the 2:3 resonance between Pluto and Neptune. Unstable resonances with Saturn's inner moons contribute to the formation of gaps in the rings of Saturn. The special case of 1:1 resonance (between bodies with similar orbital radii) enables massive Solar System bodies to gravitationally clear their orbital vicinity through the expulsion of proximate celestial material; this phenomenon underpins the contemporary definition of a planet.

Atomic, particle, and molecular

Nuclear magnetic resonance (NMR) refers to a physical resonance phenomenon characterized by the observation of distinct quantum-mechanical magnetic properties of an atomic nucleus when subjected to an external magnetic field. Many scientific techniques leverage NMR phenomena for the investigation of molecular physics, crystals, and non-crystalline materials through NMR spectroscopy. NMR is also commonly employed in sophisticated medical imaging modalities, for instance, in magnetic resonance imaging (MRI).

All nuclei possessing an odd number of nucleons exhibit intrinsic angular momentum and a magnetic moment. A salient characteristic of NMR is the direct proportionality between the resonant frequency of a specific substance and the strength of the applied magnetic field. This attribute is leveraged in imaging methodologies; consequently, if a sample is positioned within a non-uniform magnetic field, the resonant frequencies of its constituent nuclei become spatially dependent. Thus, a particle's location can be determined with considerable precision based on its resonant frequency.

Electron paramagnetic resonance, also termed electron spin resonance (ESR), is a spectroscopic technique analogous to NMR, yet employing unpaired electrons. The applicability of this technique is considerably more restricted, as the material must possess both an unpaired spin and paramagnetic properties.

The Mössbauer effect describes the resonant, recoil-free emission and absorption of gamma ray photons by atoms constrained within a solid matrix.

In particle physics, resonance manifests under conditions analogous to those observed in classical physics, extending to the realms of quantum mechanics and quantum field theory. These resonances are often conceptualized as unstable particles, with the formula for the Universal resonance curve applying if Γ represents the particle's decay rate and ω<!-- ω --> §1314§ {\displaystyle \omega _{0}} denotes its mass M. Under these conditions, the formula is derived from the particle's propagator, where its mass is substituted with the complex number M + iΓ. Furthermore, this formula's relationship to the particle's decay rate is established through the optical theorem.

Disadvantages

When a column of soldiers marches in synchronized step across a narrow, structurally flexible bridge, the structure can be induced into oscillations of dangerously large amplitude. A notable incident occurred on April 12, 1831, when the Broughton Suspension Bridge, located near Salford, England, collapsed as a contingent of British soldiers traversed it. Consequently, the British Army instituted a permanent directive requiring soldiers to break their synchronized stride when crossing bridges, thereby mitigating the risk of resonant vibrations induced by their regular marching cadence.

Vibrations originating from a motor or engine are capable of inducing resonant oscillations within their supporting structures, particularly when the natural frequency of these structures approximates that of the engine's vibrations. A prevalent illustration of this phenomenon is the rattling noise emanating from a bus body when its engine operates at idle.

Wind-induced structural resonance in suspension bridges can precipitate catastrophic failure. Historically, numerous early suspension bridges across Europe and the United States succumbed to destruction due to structural resonance triggered by even moderate wind conditions. The catastrophic failure of the Tacoma Narrows Bridge on November 7, 1940, is frequently cited in physics as a quintessential illustration of resonance. However, Robert H. Scanlan and other researchers have posited that the bridge's destruction was, in fact, attributable to aeroelastic flutter. This phenomenon involves a complex interaction between the bridge structure and the airflow, representing a form of self-oscillation, or a 'self-sustaining vibration' as defined within the nonlinear theory of vibrations.

Q factor

The Q factor, also known as the quality factor, is a dimensionless metric that quantifies the under-damping of an oscillator or resonator and defines the resonator's bandwidth in relation to its central frequency. An elevated Q value signifies a reduced rate of energy dissipation compared to the energy stored, implying that the system exhibits light damping. This parameter is formally expressed by the following equation: Q = §1617§ π maximum energy stored total energy lost per cycle at resonance {\displaystyle Q=2\pi {\text{ }}{\frac {\text{ maximum energy stored}}{\text{total energy lost per cycle at resonance}}}} .

An elevated Q factor correlates with a larger amplitude at the resonant frequency and a narrower bandwidth, which defines the frequency range where resonance is observed. Within the context of electrical resonance, a high-Q circuit in a radio receiver presents challenges in tuning but offers enhanced selectivity, thereby improving its capacity to isolate desired signals from extraneous transmissions. Oscillators characterized by a high Q factor exhibit superior stability.

Illustrative examples of systems typically possessing a low Q factor include door closers, with a Q value of approximately 0.5. Conversely, systems exhibiting high Q factors encompass tuning forks (Q=1000), atomic clocks, and lasers (Q≈10¹¹).

Universal resonance curve

The precise response of a resonant system, particularly at frequencies significantly divergent from the resonant frequency, is contingent upon the specific characteristics of the physical system. This response typically lacks exact symmetry around the resonant frequency, a phenomenon previously demonstrated for the simple harmonic oscillator.

For a lightly damped linear oscillator characterized by a resonance frequency ω<!-- ω --> §1112§ {\displaystyle \omega _{0}} , the intensity of its oscillations, denoted as ${\displaystyle I}$, when subjected to a driving frequency ${\displaystyle \omega }$, is commonly approximated by the following formula, which exhibits symmetry around the resonance frequency:I ( ω<!-- ω --> ) ≡<!-- ≡ --> | χ<!-- χ --> | §9192§ ∝<!-- ∝ --> §100101§ ( ω − ω §116117§ ) §124125§ + ( Γ §139140§ ) §147148§ . {\displaystyle I(\omega )\equiv |\chi |^{2}\propto {\frac {1}{(\omega -\omega _{0})^{2}+\left({\frac {\Gamma }{2}}\right)^{2}}}.}

In this context, the susceptibility ${\displaystyle \chi (\omega )}$ establishes the relationship between the oscillator's amplitude and the driving force within the frequency domain, as expressed by:$${\displaystyle x(\omega )=\chi (\omega )F(\omega )}$$

The intensity is formally defined as the square of the oscillation amplitude. This characteristic response, often described by a Lorentzian function or Cauchy distribution, is prevalent in numerous physical systems exhibiting resonance. The parameter Γ, which is directly related to the oscillator's damping, is termed the linewidth of the resonance. Oscillators with significant damping typically exhibit broad linewidths, indicating a responsiveness across a broader spectrum of driving frequencies near the resonant frequency. Conversely, the linewidth demonstrates an inverse proportionality to the Q factor, which quantifies the sharpness of the resonance.

Within the fields of radio engineering and electronics engineering, this approximate symmetric response is recognized as the universal resonance curve. This concept was established by Frederick E. Terman in 1932 to streamline the approximate analysis of radio circuits possessing diverse center frequencies and Q values.

Notes

Notes

References

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