Vibration - Wikiparency

# Vibration ## Overview Structural resonance and modal analysis—how buildings, bridges, and machines oscillate. ## Acoustics bridge Modal analysis, structural resonance ## Hub connections ### Engineering Center of Excellence (primary) This article bridges [[PORTAL_Acoustics]] to the [[WT!Engineering_Center_of_Excellence]]. The acoustic signature of vibration is central to engineering operations. ### Advanced Manufacturing Center of Excellence Secondary acoustic application context (to be developed). ### Transportation Center of Excellence Secondary acoustic application context (to be developed). ### Energy Center of Excellence Secondary acoustic application context (to be developed). ### IT Center of Excellence Secondary acoustic application context (to be developed). ### HealthForce Center of Excellence Secondary acoustic application context (to be developed). ### Northern Agricultural Center of Excellence Secondary acoustic application context (to be developed). ### Southern Agricultural Center of Excellence Secondary acoustic application context (to be developed). ### Space Mining In Minnesota — Center Circle Secondary acoustic application context (to be developed). --- *Bridge article, scaffolded from [[PORTAL_Acoustics]]. Minimal content; awaiting expansion.*  ## Microsims — p5.js · added 2026-08-05 <div class="microsim-player"> <iframe src="https://editor.p5js.org/sciencenibber/full/TFprpY6vu" width="100%" height="620" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Vibration — p5.js microsim"></iframe> </div> <div class="microsim-fallback"><em>Live p5.js microsim (desktop) · <a href="https://editor.p5js.org/sciencenibber/sketches/TFprpY6vu">open / fork the sketch in the p5.js editor</a></em></div> **Vibration (p5.js).** A live single-degree-of-freedom mass-spring-damper integrated by [[Velocity|velocity]]-Verlet under the equation `m x'' + c x' + k x = F0 sin(omega_d t)`. Five sliders sweep mass, stiffness, damping, drive amplitude, and drive frequency; a "toggle force" button drops F0 to zero so the reader can isolate free vibration. Four panels: the rig (wall, spring, mass, dashpot, force arrow), the (x, v) phase portrait, the x(t) and v(t) scope, and a steady-state amplitude curve `X = (F0/k) / sqrt((1 - r^2)^2 + (2 zeta r)^2)` with `omega_0` and the live `omega_d` marked. Slide `omega_d` past `omega_0` and watch resonance. ### Overview (expanded from legacy GENERATIVE lane) Vibration is the [[Oscillation|oscillation]] of a mechanical [[System|system]] about an equilibrium configuration: any [[Structure|structure]] with mass and elasticity will, when disturbed, return some of its energy as motion that ripples back and forth until losses bring it to rest. The canonical model is the single-degree-of-freedom mass-spring-damper, governed by `m*x_ddot + c*x_dot + k*x = F(t)`, whose three regimes — undamped (`c = 0`), underdamped (`c < 2*sqrt(m*k)`), and overdamped — exhaust the qualitative behaviour of nearly every linear vibrating system before nonlinearities, multiple modes, or distributed inertia complicate the picture. Free vibration releases stored strain energy at the natural frequency `omega_0 = sqrt(k/m)`; forced vibration drives the system at a chosen frequency and exposes resonance, where the steady-state amplitude diverges in proportion to `1/(2*zeta)` as the [[Damping|damping]] ratio `zeta = c/(2*sqrt(m*k))` shrinks. Engineers chase resonance to harvest energy in piezoelectric scavengers and tune mass dampers atop skyscrapers, and they flee it after Tacoma Narrows, broken turbine blades, and brittle aircraft skin. The same equation runs underneath musical instruments, atomic-[[Force|force]]-microscope cantilevers, vehicle suspensions, MEMS gyroscopes, and seismic isolation pads — the universal first chapter of every applied dynamics course because it is the universal first chapter of every real moving thing. **On the spine:** [[Sound]] · [[Wave]] · [[Oscillation]] · [[Acoustic_wave]] · [[Mechanical_wave]] · [[Plane_wave]] · portal [[PORTAL_Acoustics]]. *Transferred microsim-first · p5 sciencenibber/TFprpY6vu · append-only, 0 deletions.*  ## Types **Vibration** is the repeated [[Oscillation|oscillation]] of a mechanical system about an equilibrium position. Engineers sort it three ways. By cause, *free vibration* follows a single disturbance, such as a struck bell or a [[String_vibration|plucked string]], and rings at the system's own natural frequencies until [[Damping|damping]] removes the energy; *forced vibration* is driven by a continuing input, such as an unbalanced rotor in an [[Electric_motor|electric motor]], an engine's firing pulses or road roughness, and follows the [[Frequency|frequency]] of that input. By time history, vibration is *deterministic*, a sine or a periodic waveform that can be predicted, or *random*, like the buffeting of a vehicle on gravel or a rocket at launch, which can only be described statistically by its [[Fourier_analysis|spectrum]]. By energy balance, it is *damped*, losing energy each cycle, or *self-excited*, drawing energy from a steady flow of air, water or friction faster than damping removes it. The self-excited class holds the famous failures. The Tacoma Narrows Bridge, which tore itself apart in a steady wind of about 19 m/s on November 7, 1940, is often described in textbooks as a case of [[Resonance|resonance]], but Billah and Scanlan showed that it was an aeroelastic flutter, a self-excited torsional oscillation whose energy came from the wind's interaction with the deck's own motion.[^billah1991] A related mechanism, [[Vortex-induced_vibration|vortex-induced vibration]], occurs when the vortices shed alternately from a chimney, cable or riser, at a frequency `f_s = St U/D` with Strouhal number `St ≈ 0.2` for a circular cylinder, fall into step with a natural frequency of the structure.[^blevins1990] A 0.2 m pole in a 5 m/s wind sheds vortices at about 5 Hz. The Wikitube framework microsim *Vibration: the normal modes of a two-mass chain* puts the central fact of the subject in front of the reader: two masses on three springs can vibrate freely at only two natural frequencies, its [[Normal_mode|normal modes]], one per degree of freedom, and the reader selects either mode or plucks one mass and watches the energy trade between them, while a second panel shows how a spring-and-damper mount keeps a vibration from reaching the floor. Vibration is also the source of most [[Sound|sound]]: a vibrating panel, string or loudspeaker cone pushes on the surrounding air and radiates a pressure [[Wave|wave]], which is why the subject opens the Acoustics spine. ## Isolation [[Vibration_isolation|Vibration isolation]] places a flexible mount, a spring with some damping, between a vibrating source and whatever must be protected from it. The quantity that matters is the transmissibility `T`, the ratio of the [[Force|force]] or motion that gets through the mount to the force or motion applied. For a single-degree-of-freedom mount with damping ratio `zeta` and frequency ratio `r = f_drive / f_mount`, `T = sqrt((1 + (2 zeta r)^2) / ((1 - r^2)^2 + (2 zeta r)^2))` which is the curve on the right of the microsim, after Rao's standard treatment.[^rao2017] Three features of the curve are the whole of isolation practice. Near `r = 1` the mount amplifies: with `zeta = 0.1` the transmitted force is about 5.1 times the applied force. At `r = sqrt(2)`, `T = 1` for every value of damping. Above `sqrt(2)`, `T < 1` and the mount isolates, and the less damping it has the better it isolates: at `r = 2.5`, `T = 0.21` (13.5 dB of isolation) with `zeta = 0.1` but only 0.58 with `zeta = 0.7`. Damping is what keeps the mount safe while a machine runs up through [[Resonance|resonance]]; above it, damping works against isolation. A worked design shows how the numbers are used. A pump running at 1,800 rpm forces its base at 30 Hz. To pass only about 10 percent of that force to the floor, a lightly damped mount needs `r^2 - 1 ≈ 10`, so `r ≈ 3.3` and a mount frequency near 9 Hz. Because a spring's natural frequency is set by how far the machine's weight compresses it, `f = (1/2 pi) sqrt(g/delta)`, a 9 Hz mount must deflect about 3 mm under the static load. Lower target frequencies call for proportionally softer springs, which is why isolators for slow machinery are large coil springs or air springs rather than rubber pads. When the forcing sits at one frequency, a second approach is to cancel it rather than filter it. Hermann Frahm patented the dynamic vibration absorber in 1911, a small auxiliary mass on a spring tuned to the troublesome frequency, which vibrates in opposition to the main structure and holds it nearly still.[^frahm1911] Den Hartog's analysis of the damped absorber became the basis for the tuned mass dampers of [[Structural_engineering|structural engineering]], now built into tall buildings, bridges and power-line conductors.[^denhartog1985] Active isolation replaces the passive spring with sensors, a [[Control_engineering|feedback controller]] and actuators, and is used where soft passive mounts are impractical, such as in semiconductor lithography and precision optics. *Try: set **drive ratio r** below 1.4 and watch the orange marker climb above T = 1; move it past sqrt 2 into the shaded isolation band; then raise **isolator damping zeta** and see the resonance peak fall while the isolation at high r gets worse.* ## Testing Vibration testing does one of two things: it measures how a structure responds, or it proves that a product survives the vibration it will meet in service. The first is modal testing. A structure is excited with an instrumented impact hammer or an electrodynamic shaker, its response is measured with [[Piezoelectricity|piezoelectric]] accelerometers, and the ratio of response to force is computed as a [[Frequency_response|frequency response]] function; the peaks of that function give the natural frequencies, their widths give the damping, and the relative amplitudes at many points give the [[Normal_mode|mode shapes]].[^ewins2000] The computation relies on the [[Fast_Fourier_transform|fast Fourier transform]], whose 1965 algorithm by Cooley and Tukey made spectral analysis of measured signals routine.[^cooley1965] The second is qualification testing on a shaker table, with sine sweeps that pass slowly through every frequency to find resonances, random-vibration tests with a specified spectrum that mimics a road or a launch, and shock tests. In [[Mechanical_engineering|mechanical engineering]] practice the same measurements run continuously on rotating machinery. Condition monitoring tracks the vibration velocity of bearings and casings and compares it with severity zones such as those of ISO 20816-1; a rising level at once or twice running speed points to unbalance or misalignment, and energy at bearing defect frequencies warns of wear long before failure.[^iso20816] Because repeated stress cycles drive [[Fatigue_(material)|fatigue]] cracking, the number of cycles a component sees at a given amplitude is itself a design quantity. ## Analysis The analysis of vibration reduces a structure to masses, springs and dampers and solves [[Newton's_laws_of_motion|Newton's second law]] for them. One mass gives one [[Ordinary_differential_equation|ordinary differential equation]]; many masses give a matrix equation whose solutions are the modes. ### Free vibration without damping A mass `m` on a spring of stiffness `k` obeys `m x'' + k x = 0`. Its solution is [[Simple_harmonic_motion|simple harmonic motion]], `x = A cos(omega_n t + phi)`, at the natural [[Angular_frequency|angular frequency]] `omega_n = sqrt(k/m)`.[^openstax15-5] The microsim's masses are 1 kg on 100 N/m springs, so `omega_n = 10 rad/s` and `f_n = 1.59 Hz`. For a spring loaded by gravity, `k/m = g/delta`, so the natural frequency follows directly from the static deflection: 1 mm of sag means about 15.8 Hz, 10 mm about 5.0 Hz. ### Free vibration with damping Adding a viscous damper `c` gives `m x'' + c x' + k x = 0`. The damping ratio `zeta = c/(2 sqrt(k m))` sorts the behavior: below 1 the system is underdamped and oscillates inside a decaying exponential envelope at `omega_d = omega_n sqrt(1 - zeta^2)`; at 1 it is critically damped and returns to rest fastest without overshoot, as a car's shock absorbers are designed to do; above 1 it is overdamped and creeps back.[^openstax15-5] With `zeta = 0.05` each cycle keeps `e^(-2 pi × 0.05) ≈ 73` percent of the previous amplitude, so ten cycles leave about 4 percent. Measuring that loss of [[Energy|energy]], the logarithmic decrement, is the simplest way to find a structure's damping. ### Forced vibration with damping With a harmonic force `F_0 cos(omega t)`, the steady-state amplitude is `X = (F_0/k) / sqrt((1 - r^2)^2 + (2 zeta r)^2)` with `r = omega/omega_n`. At low `r` the response follows the static deflection; near `r = 1` it peaks at about `1/(2 zeta)` times the static value, the [[Resonance|resonance]] peak whose height is the [[Q_factor|Q factor]], and the phase lag passes 90 degrees; well above `r = 1` the mass barely moves.[^openstax15-6] OpenStax gives the London Millennium Footbridge as the cautionary case: pedestrians' footsteps fell into step with the bridge's sway, the motion grew, and the bridge was closed for about two years to add damping.[^openstax15-6] This single-degree-of-freedom [[Harmonic_oscillator|driven harmonic oscillator]] is the subject of section 2 of the Acoustics spine and of the legacy p5.js sketch above. ### Multiple degrees of freedom systems and mode shapes A structure with `n` independent coordinates has `n` natural frequencies, its [[Normal_mode|normal modes]], and at each one it moves in a fixed pattern, a mode shape. The two-mass chain is the smallest example. With `k_1` and `k_3` tying the masses to the walls and `k_2` coupling them, the equations of motion are `M x'' + K x = 0` with a diagonal mass matrix and a stiffness matrix `K` whose rows are `(k_1 + k_2, -k_2)` and `(-k_2, k_2 + k_3)`. In the first mode the two masses swing together; with equal masses and outer springs the coupling spring is never stretched, so `f_1 = sqrt(k/m)/(2 pi) = 1.59 Hz` whatever `k_2` is. In the second they swing in opposition and stretch the coupling spring on every stroke, so `f_2 = sqrt((k + 2 k_2)/m)/(2 pi)`, 1.81 Hz with `k_2 = 15 N/m` and 4.21 Hz with `k_2 = 300 N/m`. Doubling the second mass breaks the symmetry and lowers both frequencies, to 1.19 Hz and 1.72 Hz, and the mode shapes stop being simply "together" and "apart." ### Eigenvalue problem Seeking solutions `x = phi cos(omega t)` turns the equations into the generalized [[Linear_algebra|eigenvalue]] problem `(K - omega^2 M) phi = 0`, which has non-zero solutions only where `det(K - omega^2 M) = 0`. The roots `omega^2` are the squared natural frequencies and the vectors `phi` are the mode shapes; this is exactly the 2 × 2 problem the microsim solves each time a slider moves. For a finite-element [[Simulation|simulation]] of a car body or an airframe the matrices have millions of rows, and only the lowest few hundred eigenpairs are computed, but the equation is the same. ### Illustration of a multiple DOF problem Pluck only the left mass of the chain and neither mode is excited alone; the motion is the sum of both. With `f_1 = 1.59 Hz` and `f_2 = 1.81 Hz` the two modes drift in and out of step every `1/(f_2 - f_1) ≈ 4.5 s`, so the motion passes entirely to the right mass about 2.2 s after the pluck and back to the left mass 2.2 s later. The same beating between two close modes is what makes a pair of coupled pendulums, or two strings tuned to nearly the same pitch, hand energy back and forth. *Try: set **motion** to mode 1, then mode 2, and read f1 and f2; choose both and watch the x1(t) and x2(t) traces trade amplitude; raise **coupling spring k2** and see only f2 rise; change **mass ratio m2/m1** to break the symmetry; press **reset** to pluck again.* ### Multiple DOF problem converted to a single DOF problem The mode shapes are orthogonal with respect to the mass and stiffness matrices, so the change of coordinates `x = Phi q`, with the mode shapes as the columns of `Phi`, turns the coupled equations into `n` independent single-degree-of-freedom equations, one per modal coordinate `q_i`, each with its own modal mass, stiffness and frequency. If the damping is assumed proportional to the mass and stiffness matrices, the damped equations decouple too. Each mode can then be treated with all the single-degree-of-freedom results above, and by [[Superposition_principle|superposition]] the full response is the sum of the modal responses. This modal superposition is how most structural dynamics software works, and it is also exactly how the microsim draws the chain's motion.[^rao2017] ### Rigid-body mode If the chain is detached from both walls (`k_1 = k_3 = 0`), one of its two eigenvalues becomes zero. The corresponding mode, both masses moving together with no spring stretched, is a rigid-body mode: the whole system translates without any restoring force. A free body in three dimensions has six rigid-body modes, three translations and three rotations, and an unrestrained model of an [[Aircraft|aircraft]] or a spacecraft always shows six zero-frequency modes before its first elastic one. Finding exactly six is a routine check that a finite-element model has been assembled correctly. ## See also - [[Vortex-induced_vibration]] - [[Vibration_isolation]] - [[Harmonic_oscillator]] - [[String_vibration]] - [[Soundproofing]] - [[Infrasound]] ## References [^billah1991]: Billah, K. Yusuf; Scanlan, Robert H. (1991). "Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks." *American Journal of Physics* 59 (2): 118–124. https://doi.org/10.1119/1.16590 [^blevins1990]: Blevins, Robert D. (1990). *Flow-Induced Vibration* (2nd ed.). Van Nostrand Reinhold. Ch. 3, vortex-induced vibration (Strouhal number about 0.2 for a circular cylinder over a wide range of Reynolds number). [^rao2017]: Rao, Singiresu S. (2017). *Mechanical Vibrations* (6th ed.). Pearson. Chapters on harmonically excited vibration (vibration isolation and transmissibility) and multidegree-of-freedom systems (modal analysis). ISBN 9780134361307. [^frahm1911]: Frahm, Hermann (1911). "Device for damping vibrations of bodies." U.S. Patent 989,958, filed October 30, 1909, issued April 18, 1911. https://patents.google.com/patent/US989958A/en [^denhartog1985]: Den Hartog, J. P. (1985). *Mechanical Vibrations* (reprint of the 4th ed., McGraw-Hill, 1956). Dover. ISBN 9780486647852. Ch. 3, the damped vibration absorber. [^ewins2000]: Ewins, D. J. (2000). *Modal Testing: Theory, Practice and Application* (2nd ed.). Research Studies Press. ISBN 9780863802188. [^cooley1965]: Cooley, James W.; Tukey, John W. (1965). "An algorithm for the machine calculation of complex Fourier series." *Mathematics of Computation* 19 (90): 297–301. https://doi.org/10.1090/S0025-5718-1965-0178586-1 [^iso20816]: International Organization for Standardization (2016). ISO 20816-1:2016, *Mechanical vibration — Measurement and evaluation of machine vibration — Part 1: General guidelines*. https://www.iso.org/standard/63180.html [^openstax15-5]: Ling, Samuel J.; Sanny, Jeff; Moebs, William (2016). *University Physics Volume 1*. OpenStax. §15.1 "Simple Harmonic Motion" and §15.5 "Damped Oscillations," pp. 710–737 (natural angular frequency, Eq. 15.25; underdamped, critically damped and overdamped motion, Fig. 15.27; the car shock absorber as a critically damped system). https://openstax.org/details/books/university-physics-volume-1 (Portal Books 077.) [^openstax15-6]: Ling, Sanny & Moebs (2016). *University Physics Volume 1*. OpenStax. §15.6 "Forced Oscillations," pp. 737–741 (resonance curves and their width; the London Millennium Footbridge, "closed for roughly two years," Fig. 15.33). (Portal Books 077.)  **Microsim — three.js (Wikitube framework):** *Vibration* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/acoustics/Vibration.html" data-title="Vibration"></div> *Built from `MICROSIM_GUIDE/specs/acoustics/sims/Vibration.json`; part of the [[PORTAL_Acoustics|Acoustics portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Vibration) : [Wikitube](https://en.wikitube.io/wiki/Vibration) - skeleton pinned to revision 1348429654 (2026-09-11).