Oscillation

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Oscillation

"Oscillator" redirects here. For other uses, see Oscillator (disambiguation).

"Vibrating" redirects here. For other uses, see Vibrations (disambiguation) and Vibrate (disambiguation).

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An undamped spring–mass system is the quintessential example of an oscillatory system, a concept so basic it's almost insulting to have to explain it. It’s the idealized, frictionless version of something we see everywhere.

Oscillation, in its most fundamental sense, is the repetitive, almost predictable, variation of some measure. Think of it as a system’s way of fidgeting around a central point, usually one of equilibrium, or perhaps swinging between two distinct states. It’s like a pendulum’s relentless swing, or the ebb and flow of alternating current. In physics, these predictable patterns are often exploited to approximate incredibly complex interactions, like the subtle dance of atoms.

The phenomenon isn't confined to mechanical systems, oh no. Oscillations are the secret pulse of dynamic systems across nearly every scientific discipline. Consider the rhythmic beating of the human heart, essential for circulating lifeblood; the ebb and flow of business cycles in economics, a more chaotic, less elegant form of oscillation; the predictable cycles of predator–prey populations in ecology, a stark reminder of nature’s balancing act; the dramatic, periodic eruptions of geothermal geysers in geology; the resonant hum of strings in a guitar or any string instrument; the synchronized firing of nerve cells in the brain, a symphony of electrical pulses; and, out in the vastness of space, the periodic swelling and shrinking of Cepheid variable stars in astronomy, cosmic metronomes. The term vibration, by the way, is merely the specific, and rather pedestrian, term for mechanical oscillation.

Now, oscillation, especially when it’s rapid, can be a real nuisance. In process control and control theory, the goal is usually stability, a steady state. When oscillations interfere with that, they’re often called chattering or flapping – think of a faulty valve constantly opening and closing, or the infuriating route flapping in network infrastructure. It’s the system’s inability to settle, a sign of underlying instability.

Simple Harmonic Oscillation

The bedrock of oscillation theory is the simple harmonic oscillator. Imagine a weight tethered to a linear spring, subject only to its own weight and the spring's tension. It’s an idealized scenario, perhaps best approximated on a frictionless surface like an air table or a sheet of ice. The system finds its equilibrium when the spring is at rest, unextended.

But displace it, even slightly, and a restoring force emerges, a relentless tug pulling it back towards equilibrium. Yet, as it rushes back, it gains momentum, overshooting the mark. This overshoot then triggers a restoring force in the opposite direction. Introduce a constant force, like gravity, and the equilibrium point simply shifts; the oscillation continues, just around a new center. The duration of one complete cycle, the oscillatory period, is a key characteristic.

Systems where this restoring force is directly proportional to the displacement – that’s the hallmark of simple harmonic motion. The spring-mass system is the classic illustration: kinetic energy at equilibrium is converted to potential energy stored in the spring at the extremes of motion, and back again. It’s a perpetual exchange, a perfect dance of energy. This illustrates the fundamental features of oscillation: an equilibrium point and a restoring force that intensifies with deviation.

For the spring-mass system, Hooke's law dictates this restoring force:

$F = -kx$

Where $k$ is the spring constant and $x$ is the displacement from equilibrium. Applying Newton's second law, we derive the governing differential equation:

$\ddot{x} = -\frac{k}{m}x = -\omega^2x,$

Here, $\omega = \sqrt{k/m}$ is the angular frequency. The solution to this elegant equation is a sinusoidal position function:

$x(t) = A\cos(\omega t - \delta)$

Where $\omega$ is the frequency of the oscillation, $A$ is the amplitude (the maximum displacement), and $\delta$ is the phase shift, all determined by the initial conditions. In this perfect, frictionless world, the cosine function’s infinite oscillation means the spring-mass system would oscillate forever. A grim thought, really, if you consider the implications of perpetual motion.

Two-Dimensional Oscillators

Extending harmonic oscillators into two or three dimensions doesn’t fundamentally alter their nature, though it adds layers of complexity. The simplest case is the isotropic oscillator, where the restoring force is uniformly proportional to the displacement from equilibrium in all directions, with a single restorative constant, $k$.

$\vec{F} = -k\vec{r}$

The solution mirrors the one-dimensional case, but with distinct equations for each directional component.

$x(t) = A_x\cos(\omega t - \delta_x),$ $y(t) = A_y\cos(\omega t - \delta_y),$ $\vdots$

Anisotropic Oscillators

When the restorative constants differ in different directions – an anisotropic oscillator – the frequencies also diverge. The resulting motion can become quite intricate. If, for instance, the frequency in one direction is precisely double that of another, the path traces a figure-eight pattern. When the ratio of frequencies is irrational, the motion enters the realm of quasiperiodic behavior. It’s a state where each axis oscillates periodically, but the overall motion in relation to the origin never repeats itself. It’s a fascinating dance of near-predictability that never quite closes the loop. [1]

Damped Oscillations

The pristine world of undamped oscillation is a theoretical construct. In reality, all oscillator systems are subject to thermodynamically irreversible processes. Friction, electrical resistance – these are the dissipative forces that relentlessly convert stored energy into heat, bleeding it into the environment. This is damping. Without an external energy source, oscillations inevitably decay.

The simplest model for this decay involves adding a resistive force, dependent on velocity, to the harmonic oscillator equation. Newton's second law now includes a damping term, characterized by a constant $b$:

$m\ddot{x} + b\dot{x} + kx = 0$

Rewriting this, we get:

$\ddot{x} + 2\beta\dot{x} + \omega_0^2x = 0,$

where $2\beta = b/m$. The general solution reveals an exponential decay superimposed on the oscillation:

$x(t) = e^{-\beta t}(C_1e^{\omega_1 t} + C_2e^{-\omega_1 t}),$

with $\omega_1 = \sqrt{\beta^2 - \omega_0^2}$. The term $e^{-\beta t}$ is the decay function, and $\beta$ is the damping coefficient. The behavior falls into three categories: under-damped ($\beta < \omega_0$), where oscillations persist but decay; over-damped ($\beta > \omega_0$), where the system returns to equilibrium slowly without oscillating; and critically damped ($\beta = \omega_0$), the fastest return to equilibrium without overshoot. It’s a delicate balance between the restoring force and the dissipative forces.

Driven Oscillations

Then there are driven oscillations, where an external force continuously injects energy into the system. Think of an AC circuit connected to a power source. The simplest model is a spring-mass system subjected to a sinusoidal driving force:

$\ddot{x} + 2\beta\dot{x} + \omega_0^2x = f(t),$

where $f(t) = f_0\cos(\omega t + \delta)$.

The solution is a combination of a steady-state oscillation and a transient response:

$x(t) = A\cos(\omega t - \delta) + A_{tr}\cos(\omega_1 t - \delta_{tr}),$

where the amplitude $A$ is given by:

$A = \sqrt{\frac{f_0^2}{(\omega_0^2 - \omega^2)^2 + 4\beta^2\omega^2}}$

And the phase shift $\delta$ by:

$\delta = \tan^{-1}\left(\frac{2\beta\omega}{\omega_0^2 - \omega^2}\right)$

The second term, the transient solution, eventually fades, leaving the system to oscillate at the driving frequency.

Some systems can absorb energy from their surroundings in a more complex fashion, often involving fluid dynamics. Flutter, a phenomenon in aerodynamics, occurs when a slight displacement of an aircraft wing alters its angle of attack to the airflow, increasing lift, which causes a further displacement. Eventually, the wing's stiffness provides the restoring force, leading to oscillation. It's a dangerous feedback loop, where instability breeds more instability.

Resonance

Resonance is the dramatic amplification of oscillation that occurs when the driving frequency ($\omega$) matches the system's natural frequency ($\omega_0$). In a damped driven oscillator, this alignment minimizes the denominator in the amplitude equation, sending the amplitude soaring. It’s the principle behind tuning a radio or the shattering of a glass with a specific sound frequency.

Coupled Oscillations

When systems possess more than a single degree of freedom, their oscillations become interconnected. Imagine two masses linked by three springs. The motion of one mass inevitably affects the others. This coupling can lead to complex emergent behaviors, such as the synchronization of two pendulum clocks mounted on a shared wall, a phenomenon first observed by Christiaan Huygens in 1665. [2] While the combined motion can appear intricate, it can be simplified by resolving it into normal modes, fundamental patterns of collective oscillation.

The simplest coupled system involves two masses ($m_1, m_2$) and three springs ($k_1, k_2, k_3$). Applying Newton's second law yields a system of differential equations:

$\begin{cases} m_1\ddot{x}_1 = -(k_1+k_2)x_1 + k_2x_2 \\ m_2\ddot{x}_2 = k_2x_1 - (k_2+k_3)x_2 \end{cases}$

These can be expressed in matrix form: $M\ddot{x} = kx$, where:

$M = \begin{bmatrix} m_1 & 0 \\ 0 & m_2 \end{bmatrix}$, $x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$, and $k = \begin{bmatrix} k_1+k_2 & -k_2 \\ -k_2 & k_2+k_3 \end{bmatrix}$

For the common case where $m_1 = m_2 = m$ and $k_1 = k_2 = k_3 = k$:

$M = \begin{bmatrix} m & 0 \\ 0 & m \end{bmatrix}$, $k = \begin{bmatrix} 2k & -k \\ -k & 2k \end{bmatrix}$

Plugging these into the general solution $(k - M\omega^2)a = 0$ leads to the eigenvalue problem:

$\begin{bmatrix} 2k-m\omega^2 & -k \\ -k & 2k-m\omega^2 \end{bmatrix} = 0$

The determinant yields a quadratic equation for the squared frequencies:

$(3k-m\omega^2)(k-m\omega^2) = 0$

This gives two distinct natural frequencies:

$\omega_1 = \sqrt{\frac{k}{m}}$, $\omega_2 = \sqrt{\frac{3k}{m}}$

The system's behavior depends on the initial conditions. If the masses start in phase, they oscillate at the lower frequency, as the middle spring remains unstretched. If they move out of phase, the higher frequency dominates. [1]

Other special cases include the Wilberforce pendulum, where oscillation energy transfers between vertical stretching and torsional rotation. Coupled oscillators can also describe situations where one oscillation influences another without being reciprocally affected, leading to complex synchronization phenomena within Arnold Tongues, potentially including chaotic dynamics.

Small Oscillation Approximation

Near an equilibrium point, many systems governed by conservative forces can be approximated as harmonic oscillators. Consider the Lennard-Jones potential:

$U(r) = U_0\left[\left(\frac{r_0}{r}\right)^{12}-\left(\frac{r_0}{r}\right)^{6}\right]$

The equilibrium point $r_0$ is found where the derivative is zero:

$\frac{dU}{dr} = 0 = U_0\left[-12r_{0}^{12}r^{-13}+6r_{0}^{6}r^{-7}\right] \Rightarrow r \approx r_0$

The second derivative at this point, $\gamma_{\text{eff}}$, provides the effective potential constant:

$\gamma_{\text{eff}} = \left.{\frac {d^{2}U}{dr^{2}}}\right|_{r=r_{0}} = U_0\left[12(13)r_{0}^{12}r^{-14}-6(7)r_{0}^{6}r^{-8}\right] = \frac {114U_{0}}{r^{2}}$

This leads to an effective force $F = -\gamma_{\text{eff}}(r-r_0)$, and when combined with an effective mass $m_{\text{eff}}$, it forms the simple harmonic oscillator equation:

$\ddot{r} + \frac{\gamma_{\text{eff}}}{m_{\text{eff}}}(r-r_0) = 0$

Thus, the frequency of small oscillations is:

$\omega_0 = \sqrt{\frac{\gamma_{\text{eff}}}{m_{\text{eff}}}} = \sqrt{\frac{114U_{0}}{r^{2}m_{\text{eff}}}}$

Or, more generally:

$\omega_0 = \sqrt{\left.{\frac {d^{2}U}{dr^{2}}}\right\vert _{r=r_{0}}}$ [3]

This approximation is intuitively understood by visualizing the potential energy curve as a "well." A system displaced from the minimum will oscillate back and forth within this well, much like a ball rolling in a bowl. This concept is also applicable to understanding Kepler orbits.

Continuous Systems – Waves

As the number of degrees of freedom approaches infinity, a system becomes continuous. Think of a vibrating string or the surface of water. These systems exhibit oscillations in the form of waves, capable of propagating through the medium. They possess an infinite number of normal modes in the classical limit.

Mathematics

The mathematical study of oscillation quantifies the extent to which a sequence or function fluctuates. Several related concepts exist: the oscillation of a sequence of real numbers, the oscillation of a real-valued function at a specific point, and its oscillation over an interval or open set. The oscillation of a sequence, for instance, is formally defined as the difference between its limit superior and limit inferior.

Examples

Mechanical

• Double pendulum
• Foucault pendulum
• Helmholtz resonator
• Solar oscillations (helioseismology), stellar oscillations (asteroseismology), and Neutron-star oscillations.
• Quantum harmonic oscillator
• Playground swing
• String instruments
• Torsional vibration
• Tuning fork
• Vibrating string
• Wilberforce pendulum
• Lever escapement

Electrical

• Alternating current
• Armstrong (or Tickler or Meissner) oscillator
• Astable multivibrator
• Blocking oscillator
• Butler oscillator
• Clapp oscillator
• Colpitts oscillator
• Delay-line oscillator
• Electronic oscillator
• Extended interaction oscillator
• Hartley oscillator
• Oscillistor
• Phase-shift oscillator
• Pierce oscillator
• Relaxation oscillator
• RLC circuit
• Royer oscillator
• Vačkář oscillator
• Wien bridge oscillator

Electro-mechanical

• Crystal oscillator

Optical

• Laser (oscillation of electromagnetic field at frequencies around $10^{15}$ Hz)
• Oscillator Toda or self-pulsation (pulsation of laser output power between $10^4$ Hz and $10^6$ Hz in the transient regime)
• Quantum oscillator can refer to an optical local oscillator or a model in quantum optics.

Biological

• Circadian rhythm
• Bacterial Circadian Rhythms
• Circadian oscillator
• Lotka–Volterra equation
• Neural oscillation
• Oscillating gene
• Segmentation clock

Human

• Neural oscillation
• Insulin release oscillations
• Gonadotropin releasing hormone pulsations
• Pilot-induced oscillation
• Voice production

Economic and Social

• Business cycle
• Generation gap
• Malthusian economics
• News cycle

Climate and Geophysics

• Atlantic multidecadal oscillation
• Chandler wobble
• Climate oscillation
• El Niño-Southern Oscillation
• Pacific decadal oscillation
• Quasi-biennial oscillation

Astrophysics

• Neutron stars
• Cyclic Model

Quantum Mechanical

• Neutral particle oscillation, e.g., neutrino oscillations
• Quantum harmonic oscillator

Chemical

• Belousov–Zhabotinsky reaction
• Mercury beating heart
• Briggs–Rauscher reaction
• Bray–Liebhafsky reaction

Computing

• Cellular Automata oscillator

See Also

• Antiresonance
• Beat (acoustics)
• BIBO stability
• Critical speed
• Cycle (music)
• Dynamical system
• Earthquake engineering
• Feedback
• Fourier transform (for analyzing periodicity in evenly spaced data)
• Frequency
• Hidden oscillation
• Madden–Julian oscillation
• Least-squares spectral analysis (for analyzing periodicity in unevenly spaced data)
• Oscillator phase noise
• Periodic function
• Phase noise
• Quasiperiodicity
• Reciprocating motion
• Resonator
• Rhythm
• Seasonality
• Self-oscillation
• Signal generator
• Squegging
• Strange attractor
• Structural stability
• Tuned mass damper
• Vibration
• Vibrator (mechanical)

References

• 1. a b Taylor, John R. (2005). Classical mechanics. Mill Valley, California. ISBN) 1-891389-22-X. OCLC) 55729992. {{cite book}}: CS1 maint: location missing publisher (link)
• 2. Strogatz, Steven (2003). Sync: The Emerging Science of Spontaneous Order. Hyperion Press. pp. 106–109. ISBN) 0-786-86844-9.
• 3. "23.7: Small Oscillations". Physics LibreTexts. 2020-07-01. Retrieved 2022-04-21.

External links

• Oscillation in Wiktionary, the free dictionary.
• Vibrations/04%3A_Harmonic_Motion/4.02%3A_Vibrations) Archived 2010-12-14 at the Wayback Machine – a chapter from an online textbook.

Authority control databases

• International: GND
• National: United States, France, BnF data, Japan, Czech Republic, Israel
• Other: Yale LUX