Vibration_isolation - Wikiparency

# Vibration isolation **Vibration isolation** is the practice of interposing a resilient element between a source of [[Vibration|vibration]] and the structure around it, so that only a fraction of the oscillating force or motion crosses the join. It is used in both directions: to keep a running [[Machine|machine]] from shaking the building it stands in, and to keep a shaking building from reaching an instrument that must stay still. The resilient element — a rubber mount, a steel [[Spring_(device)|spring]], an air bag, a rope of woven wire — is chosen not for softness as such but for the [[Natural_frequency|natural frequency]] it gives the mounted mass, because that frequency is what decides which disturbances are cut and which are made worse. The governing result is compact and slightly counterintuitive. For a mass on a [[Stiffness|spring]] and damper, the fraction of the source's force that reaches the floor, the transmissibility, depends only on the damping ratio ζ and the ratio `r = f / f_n` of the driving frequency to the mount's natural frequency: `T = √((1 + (2ζr)²) / ((1 − r²)² + (2ζr)²))`. Below `r = √2` the mount transmits more force than the machine produces, worst of all at [[Resonance|resonance]]; only above `r = √2` does it isolate at all. A mount is therefore always soft enough that the disturbance lands well up the right-hand side of that curve, and the designer's real question is not how much damping to use but how low the natural frequency can be pushed before the static deflection becomes impractical. [[Damping|Damping]] is the second surprise. It is indispensable near resonance, where it is the only thing limiting the peak, and it is a liability in the isolation region, where every mechanism that couples the machine to the floor — including the damper — carries force across the join. The two requirements point in opposite directions, and the semi-active and active systems described below exist largely to escape that compromise. The framework microsim *Vibration isolation: a soft mount isolates, a stiff one transmits* puts a machine on springs and a dashpot above a log-log transmissibility chart. The reader sets the mass, the mount stiffness, the damping ratio and the driving frequency, and watches the operating point travel along its curve while the transmitted-force arrow under the floor grows or shrinks — through the amplifying region, over the resonant peak, across the `r = √2` crossing where every curve meets, and out into isolation. ## Passive isolation Passive isolation uses only springs, dampers and mass: nothing is sensed, powered or controlled, and the mount's behaviour is fixed once it is installed. It is by far the most common approach, because it is cheap, needs no power, cannot fail into an unstable state and works at frequencies where no actuator could keep up. The disturbance it works against is usually periodic and usually comes from rotation. A rotor whose centre of mass sits a distance e off its axis of spin throws a rotating unbalance force of magnitude `F = m_u e ω²`, so the excitation grows with the square of running speed while its frequency rises in step. That is why a machine can be quiet at idle and violent at speed, and why [[Balancing_machine|balancing]] is the first remedy and isolation the second. The same [[Harmonic_oscillator|second-order response]] describes the rotor itself on its own flexible shaft: the Jeffcott model gives the whirl amplitude as `X/e = r² / √((1 − r²)² + (2ζr)²)`, which peaks at the critical speed and then falls back toward unity.[^jeffcott1919] Above the critical speed the shaft's deflection and the unbalance point in opposite directions, the rotor spins about its own centre of mass rather than about the bearing axis, and it runs quieter the faster it goes — the rotating counterpart of the same `r > √2` rule that governs the mount beneath it. The See-also variant for [[Rotordynamics|rotordynamics]] opens in exactly that regime, at `r` = 3.97, where the whirl amplitude has settled to 1.07 times the eccentricity and the heavy spot lags the deflection by 177°, nearly half a turn: the rotor has centred itself. Its companion variant for [[Critical_speed|critical speed]] opens instead on the peak, at `r` = 0.993 with ζ = 0.05, where the amplitude is 9.85 times the eccentricity and the lag has swung to 83°.[^pack-rotor] Machines that must run above their critical speed therefore have to pass through it, which is why large [[Turbine|turbines]] accelerate briskly through the band rather than dwelling in it. *Try: in the rotordynamics variant, sweep the speed ratio up through `r` = 1 and watch the orbit swell to nearly ten times the eccentricity and then collapse back toward 1.07 as the lag rolls from 0° through 90° to 177°; raise the damping ratio and the peak flattens while the far end of the curve barely moves.* ### Common passive isolation systems The practical families differ mainly in the static deflection they can deliver, and therefore in the [[Natural_frequency|natural frequency]] they can reach. **Elastomeric mounts** — moulded [[Natural_rubber|rubber]] or synthetic [[Elastomer|elastomer]] bonded to metal fittings — are compact, cheap, carry load in several directions at once and bring useful internal damping with them, but they deflect little, so their natural frequencies are high; they also creep under sustained load and stiffen in cold weather. **Steel [[Coil_spring|coil springs]]** deflect far more for the same load and are dimensionally stable over decades and over a wide temperature range, at the cost of almost no inherent damping, which is why they are usually fitted with a separate damper or a friction pad. **[[Air_suspension|Air springs]]** carry load on a column of gas whose pressure is set by the load itself, reaching the largest deflections of all and, with a levelling valve, holding a constant ride height as load changes; they need a supply of air and they leak. **[[Wire_rope|Wire rope]] isolators**, loops of stranded cable clamped between bars, dissipate energy by [[Friction|friction]] between the strands and tolerate shock, heat and vacuum, which suits them to military and aerospace mountings. **Pads** of cork, felt, ribbed rubber or bonded fibre are the simplest of all and are used in bulk under presses and compressors where the disturbance frequency is high. ### How passive isolation works The model behind every passive mount is a [[Harmonic_oscillator|driven damped oscillator]]: a mass on a spring and a damper, forced harmonically and standing on a rigid floor. Its natural frequency is `f_n = √(k/m) / 2π`, its [[Damping|damping ratio]] is `ζ = c / (2√(k m))`, and the steady-state force reaching the floor, as a fraction of the force the machine produces, is `T = √((1 + (2ζr)²) / ((1 − r²)² + (2ζr)²))`, with `r = f / f_n`.[^hallauer] The machine's own motion follows the same second-order form, with amplitude `(F₀/m) / √((ω₀² − ω²)² + (Γω)²)` and phase `atan2(Γω, ω₀² − ω²)`;[^cline] it is that displacement, carried through the spring and the damper in parallel, that becomes the force on the floor. Three features of that expression decide everything. At `r` well below 1 the mount is stiff compared with the disturbance, the machine and floor move together, and `T` → 1: nothing is isolated. Near `r = 1` the response is [[Resonance|resonant]] and `T` reaches `√(1 + 4ζ²) / 2ζ`, which for light damping is large. And at `r = √2` the numerator and denominator become equal for every value of ζ, so every curve in the family passes through `T = 1` at the same point regardless of damping — the crossing that divides amplification from isolation.[^pack-transmissibility] The microsim's default makes the arithmetic concrete. A 100 kg machine on mounts totalling 100 kN/m has `ω_n = √(10⁵/100)` = 31.62 rad/s, so `f_n` = 5.03 Hz. The same number follows from the static deflection, since the mount settles by `δ = m g / k` = 9.81 mm under its own load and `f_n = 15.76 / √δ` with δ in millimetres gives 5.03 Hz — the identity that lets a mount be specified from a deflection alone. Driven at 20 Hz, the machine sits at `r` = 3.97 and the readout gives `T` = 0.0862: of the 100 N the [[Machine|machine]] generates, 8.6 N reaches the floor and 91 per cent is stopped. Isolation begins at `√2 × 5.03` = 7.1 Hz, so everything the machine does below about 7 Hz is passed on or amplified. Damping is where the design becomes a compromise, and the sim shows both halves of it at once. More damping means worse isolation above the crossing, because the damper is itself a path for force; less damping means a violent passage through resonance on the way up. Three values of ζ at the sim's own two operating points make the trade plain: | ζ | peak `T` at `r` = 1 | `T` at `r` = 3.97 | |---|---|---| | 0.01 | 50 | 0.068 | | 0.1 | 5.1 | 0.086 | | 0.3 | 1.9 | 0.173 | Nothing in the sim's physics is simplified and nothing in it is marked ILLUSTRATIVE: transmissibility, the machine's amplitude and phase, and the transmitted force all come from the portal's library, and the drawn curves cross `T` = 1 at `r = √2` exactly.[^pack-transmissibility] The display uses stated conventions that the sheet prints on its face: the animation runs at a fixed 0.8 Hz whatever frequency is set, the bob's travel is not to scale, and the transmitted-force arrow is clamped at 1.2 input-arrow lengths, with a note on screen when it is. *Try: with the machine at 100 kg on 100 kN/m mounts, drag the driving frequency down from 20 Hz and watch the point climb its curve, cross `T` = 1 at 7.1 Hz and rear up to 5.1 at 5.03 Hz while the floor arrow outgrows the input arrow; then raise the damping ratio and watch the peak collapse while the point at 20 Hz slides the wrong way, 0.086 to 0.173.* #### Factors influencing the selection of passive vibration isolators Selection starts with the item to be isolated. Its **weight** must fall inside the isolator's rated load range, since a mount loaded far below its rating is too [[Stiffness|stiff]] and one loaded above it bottoms out; its **size and layout** decide whether one mount will do or a set of four or six is needed, and where they sit relative to the centre of mass, because a mount set that is not balanced about that point couples vertical motion into rocking. **Moving parts** inside the machine matter in their own right: their mass, speed and travel determine the disturbance the mount must handle, and a machine whose centre of mass shifts as it runs presents a moving target. **Side effects** count too, since some instruments are sensitive to magnetic or electromagnetic fields that an isolator's components may produce. The operating environment narrows the field further. Industrial settings bring strong, broadband vibration and dust; laboratories bring specific building disturbances from nearby machinery, foot traffic and ventilation airflow. Outdoor service adds water, salt and a wider temperature range, and corrosive atmospheres attack metal parts and some [[Elastomer|elastomers]]. Cleanroom use rules out anything that sheds particles, vacuum service rules out air mounts that leak and most oils, and low-magnetism work rules out steel where a non-magnetic alternative exists. [[Noise_control|Acoustic noise]] is a two-way concern: some instruments respond to airborne sound directly, and some isolation systems can be excited by it, so an acoustic enclosure may be needed alongside the mounts. Because the object of the exercise is often human comfort rather than machine survival, the relevant criteria may be those of the whole-body vibration standard rather than a structural limit.[^iso2631] #### Comparison of passive isolators The honest way to compare families is by the natural frequency their static deflection implies, since that single number fixes where isolation begins. For a linear mount, `f_n = 15.76 / √δ` with δ in millimetres, and isolation starts at `√2 f_n`: | Static deflection δ | `f_n` | isolation begins above | |---|---|---| | 1 mm | 15.8 Hz | 22.3 Hz | | 3 mm | 9.1 Hz | 12.9 Hz | | 10 mm | 5.0 Hz | 7.1 Hz | | 25 mm | 3.2 Hz | 4.5 Hz | | 100 mm | 1.6 Hz | 2.2 Hz | The rows follow from the deflection identity alone and are ILLUSTRATIVE, not manufacturer data for any product; a real mount departs from them as its rubber stiffens with frequency or its air spring follows a gas law rather than a straight line. Read down the table, though, and the ranking of the families falls out. Pads and bonded elastomers occupy the stiff top rows and suit high-frequency sources such as compressors and presses; steel springs occupy the middle and reach the few-hertz region a building's floor vibration demands; air springs occupy the bottom, which is why they carry vehicle bodies and [[Optical_table|optical tables]]. [[Damping|Damping]] runs the other way: elastomers and wire rope bring their own, air springs bring some through their orifices, and steel springs bring almost none. ### Negative-stiffness vibration isolator The table above states the difficulty of passive isolation in one line. A natural frequency of 0.5 Hz would need about a metre of static deflection from an ordinary spring, which is impossible in a machine mount. The negative-stiffness isolator escapes the limit by adding, in parallel with an ordinary load-carrying spring, an element whose restoring force acts the wrong way — most often a column or a pair of flexures loaded just short of [[Buckling|buckling]], where a small lateral displacement releases stored axial energy and pushes the structure further from centre. The two [[Stiffness|stiffnesses]] subtract, so the assembly can carry its load on a stiff spring while presenting a very small net stiffness to small motions, and the [[Natural_frequency|natural frequency]] can be tuned toward a fraction of a hertz without any corresponding sag. The same trick is applied axis by axis. Vertical isolation uses a stiff coil spring in series with a negative-stiffness flexure that cancels most of its rate; horizontal isolation uses beam-columns that act as pendulums whose effective length is lengthened by the same axial preload; and stacking the two arrangements with a tilt mechanism gives isolation in all six degrees of freedom, three translations and three rotations. The cost is sensitivity: because the net stiffness is a small difference between two large numbers, the system must be adjusted for its load and it drifts with temperature, so these mounts are used where the payoff is worth the care — gravitational-wave and precision-measurement [[Optical_table|optical tables]], electron microscopes, and instruments that must reject [[Seismic_noise|seismic noise]] in the band below a few hertz. ### Vibration isolation of supporting joint Pipework, ducting and cable trays carry vibration away from a machine as surely as its feet do, and isolating the machine alone simply redirects the path. A supporting joint is one that carries load: a hanger, a clamp or a bracket that holds the pipe's weight while connecting it to the structure. Isolating such a joint means replacing the rigid connection with a resilient one that still carries the load — a [[Spring_(device)|spring hanger]], a rubber-in-shear clamp, a lined saddle — chosen by the same deflection rule as a machine mount, since the pipe run's mass on the hanger's [[Stiffness|stiffness]] sets a [[Natural_frequency|natural frequency]] that must sit well below the disturbance. The constraint is that the mount's static deflection changes the run's alignment and can load the machine's flange, so the isolation available at a supporting joint is usually less than at a free mount. ### Vibration isolation of unsupporting joint An unsupporting joint carries no weight and exists only to pass fluid or to close a gap, which makes it far easier to isolate: it can be made as soft as the pressure allows. Flexible connectors, [[Elastomer|rubber]] and metal bellows, braided hose and expansion joints all work this way, interrupting the structural path between the machine's outlet and the fixed run while remaining sealed against the line pressure. A branch pipe taken off at such a joint gains the same protection, and the effectiveness depends on the connector being genuinely soft compared with what it joins and on the run beyond it being restrained, since a flexible element between two unsupported masses isolates nothing. Because pressure tends to [[Stiffness|stiffen]] a bellows and to push its ends apart, these joints are specified by pressure and movement together, not by flexibility alone. ### Subframe isolation A subframe is an intermediate structure that carries the machine and is itself mounted resiliently, so that the disturbance crosses two soft joints in series instead of one. Above both natural frequencies the transmissibilities multiply, and a pair of stages that each reach 0.1 together reach 0.01, a far steeper roll-off than a single stage of the same total deflection could give. The automotive case is the familiar one: an engine and transmission sit on mounts on a [[Subframe|subframe]], which sits on [[Bushing_(isolator)|bushes]] in the body, and the arrangement carries [[Car_suspension|suspension]] loads as well. Instrument tables, precision machine tools and ship machinery seatings use the same two-stage arrangement. The price is an extra [[Resonance|resonance]]. The intermediate mass adds a second mode, and if the two stages are tuned too close together the pair is worse than either alone; the intermediate mass must also be large enough to be useful, which is why an isolated subframe is deliberately heavy. Tuning the two stages apart, and damping the lower one, is the whole of the design. ## Semi-active isolation A semi-active mount changes its own properties in real time but adds no energy to the system: it can only modulate the force a passive element already produces, by varying [[Damping|damping]] or, less often, [[Stiffness|stiffness]]. That restriction is a virtue, because a semi-active mount cannot drive the structure unstable and fails, on loss of power, into an ordinary passive mount. The usual variable is damping. A controllable orifice, a solenoid valve or a [[Magnetorheological_fluid|magnetorheological fluid]] whose apparent [[Viscosity|viscosity]] rises in a magnetic field lets the damping coefficient be changed within milliseconds, and the standard control law, skyhook damping, commands damping as though the mass were connected to an imaginary fixed point in the sky rather than to the moving base. Because the fluid device can only resist motion, the law is clipped: full damping when the actual damper force and the desired skyhook force agree in sign, minimum damping when they do not. The payoff is precisely the compromise that passive design cannot escape. A semi-active mount can be heavily damped while passing through resonance and lightly damped once the disturbance is well above `√2 f_n`, approaching the resonant behaviour of a `ζ` = 0.3 mount and the isolation of a `ζ` = 0.01 mount within the same hardware. Semi-active dampers are used in [[Car_suspension|vehicle suspension]], in seat and cab mounts, and in building dampers, and they cost a [[Sensor|sensor]], a controller and a few watts rather than the kilowatts an [[Active_suspension|active system]] would need. ## Active isolation Active isolation measures the vibration and cancels it with a powered [[Force|force]], which frees the design from the `r > √2` rule entirely: a well-designed active system can attenuate at frequencies below its own natural frequency, where every passive mount amplifies. The general arrangement is a passive mount to carry the static load and handle high frequencies, an inertial or relative-motion [[Sensor|sensor]], a [[Control_theory|controller]], and an actuator working in parallel with the spring. Two control topologies are used, usually together. **Feedback** measures the residual motion of the isolated platform and drives the actuator to null it, which makes the payload behave as though attached to an inertial reference; its bandwidth is limited by the phase lag of the loop, and structural [[Resonance|resonances]] in the platform set the ceiling on loop gain, since gain beyond that point turns the loop unstable. **Feedforward** measures the incoming disturbance at the base, or takes a reference signal directly from the machine causing it, and injects an opposing force before the effect arrives, which performs very well on periodic disturbances of known frequency and not at all on unpredictable ones. A practical system uses feedback for broadband floor motion and feedforward for the tonal component at a known running speed. The limits are power, noise and stability. Cancelling a large low-frequency motion requires real work; the sensors' own noise sets a floor below which the system cannot push the residual, and outside its bandwidth an active system must not make matters worse, so the passive mount underneath it still has to be right. Active isolation is therefore found where passive performance is genuinely insufficient: [[Photolithography|semiconductor lithography]] stages, electron and [[Scanning_tunneling_microscope|scanning-probe microscopes]], metrology frames and gravitational-wave interferometers. Related techniques that add energy to a structure rather than to a mount are treated under [[Active_vibration_control|active vibration control]]. ### Sensors for active isolation The sensor sets what the controller can know, and the choice turns on whether an inertial reference is available. **Accelerometers** are the common instrument: a [[Piezoelectricity|piezoelectric]] [[Accelerometer|accelerometer]] is rugged and wide-band but loses sensitivity and phase accuracy at low frequency, while a force-balance servo accelerometer works down to a fraction of a hertz at the cost of size and price. **[[Geophone|Geophones]]**, moving-coil velocity sensors originally built for seismology, give an excellent noise floor above their own natural frequency of a few hertz and are widely used in isolation platforms because velocity is the quantity skyhook and feedback laws want. **Relative-displacement sensors** — capacitive, inductive or optical — measure across the mount rather than against inertia, and they are what a levelling loop needs to hold a platform at a set height. Since the residual motion to be measured may be nanometres, the sensor's own noise, not the actuator, is usually what limits an active system, and sensors are often combined so that each covers the band it is best in. ### Actuators for active isolation The actuator must produce a controlled [[Force|force]] with little [[Friction|friction]], little hysteresis and no stiffness of its own to spoil the passive mount. **Voice-coil actuators**, a coil moving in a permanent-magnet field, are the standard choice: the force is proportional to current, the moving part touches nothing, and the stroke is millimetres, which covers building motion. **[[Piezoelectricity|Piezoelectric]] stacks** give very high stiffness and bandwidth over a stroke of micrometres, which suits nanometre-scale correction and rules them out for large motion; they are [[Hysteresis|hysteretic]] and generally run inside their own position loop. **Pneumatic actuators**, usually the isolator's own [[Air_suspension|air spring]] driven through a servo valve, deliver large forces cheaply at low bandwidth and are the natural choice when the passive mount is already an air spring. **Magnetostrictive and hydraulic** actuators occupy the high-force end. In every case the actuator's reaction force has to go somewhere, so it is mounted either on a reaction mass or on the base whose motion is being rejected, and that choice is as consequential as the actuator type. ## See also - [[Vibration]] - [[Damping]] - [[Resonance]] - [[Rotordynamics]] - [[Critical_speed]] - [[Shock_absorber]] - [[Shock_mount]] - [[Seismic_base_isolation]] - [[Active_vibration_control]] - [[Tuned_mass_damper]] - [[Bushing_(isolator)]] - [[Noise,_vibration,_and_harshness]] - [[Soundproofing]] - [[Car_suspension]] (section 16) - [[Friction]] (section 14) ## References [^hallauer]: Hallauer, William L., Jr. *Introduction to Linear, Time-Invariant, Dynamic Systems for Students of Engineering*. Virginia Tech open textbook. pp. 185–217: the second-order system, undamped and damped natural frequency, the damping ratio `ζ = c / (2√(k m))`, and the steady-state frequency response of `m x″ + c x′ + k x = F₀ sin(ω t)` with its magnitude and phase. Portal Book 021. [^cline]: Cline, Douglas. *Variational Principles in Classical Mechanics*. University of Rochester. pp. 84–89: the driven damped linear oscillator, amplitude `(F₀/m) / √((ω₀² − ω²)² + (Γω)²)`, phase `atan2(Γω, ω₀² − ω²)`, and resonance. Portal Book 073. [^pack-transmissibility]: Portal engineering pack, `design.vehicle.transmissibility(r, ζ)` and `design.vehicle.isolationStartsAt`, with `mech.oscillation.forcedAmplitude` and `forcedPhase` for the machine's own motion and `f_n = √(k/m)/2π`; sim spec `specs/sims/Vibration_isolation.json`, sub-manual `02_mechanics` §2.2. The transmitted force `k x + c x′` has amplitude `T F₀`, and the library's identity that every curve crosses `T` = 1 at `r = √2` is checked by the framework's own tests. Nothing in the physics is ILLUSTRATIVE; the display conventions printed on the sim's sheet are that the motion is animated at a fixed 0.8 Hz, the bob is not to scale (0.12 world units per unit of `X k / F₀`, capped at 0.5), and the floor arrow is clamped at 1.2 input-arrow lengths with an on-screen note when clamped. Hand check recorded in the build report: `ω_n` = 31.62 rad/s, `f_n` = 5.033 Hz, `r` = 3.974, `T` = 0.0862, `F_T` = 8.62 N. Build report, engineering run, September 18, 2026. [^jeffcott1919]: Jeffcott, H. H. (1919). "The lateral vibration of loaded shafts in the neighbourhood of a whirling speed — the effect of want of balance." *Philosophical Magazine*, series 6, vol. 37. (Volume and pages as recalled; the single-disc rotor model named for this paper is the standard form.) [^pack-rotor]: Portal engineering pack, See-also variants `Rotordynamics` and `Critical_speed` (`specs/variants/`), built on the `Vibration_isolation` sim with `design.vehicle.rotorResponse` for `X/e` against `r`. Values as built: `r` = 3.97 → `X/e` = 1.07 with a 177° lag (self-centred, above the critical); `r` = 0.993 at ζ = 0.05 → `X/e` = 9.85 with an 83° lag, the drawn orbit clamped and labelled as clamped. Build report, engineering run, September 18, 2026. [^iso2631]: International Organization for Standardization. ISO 2631-1, *Mechanical vibration and shock — Evaluation of human exposure to whole-body vibration — Part 1: General requirements* (frequency-weighted evaluation of vibration as it affects people, used where comfort rather than machine survival sets the isolation target).  **Microsim — three.js (Wikitube framework):** *Vibration isolation* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/engineering/Vibration_isolation.html" data-title="Vibration isolation"></div> *Built from `MICROSIM_GUIDE/specs/sims/Vibration_isolation.json`; part of the [[PORTAL_Engineering|Engineering portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Vibration_isolation) : [Wikitube](https://en.wikitube.io/wiki/Vibration_isolation) - skeleton pinned to revision 1341415069 (2026-09-18).