Controlled linear-dynamics worksheet

Linear SDOF Transmissibility Worksheet

Calculate magnitude and phase for one declared, steady-state, harmonically excited, linear single-degree-of-freedom system with viscous damping. The magnitude and phase now come from the same complex transfer function.

Base motion or force transmissionCorrect complex phaseNo ISO compliance claimNo isolator approval

Boundary: this is general linear-dynamics arithmetic, not a formula issued by ISO 2017, ISO 2041 or ISO 10846. A result does not establish that a real mount is linear, SDOF, viscously damped, stable, safe, correctly installed or suitable for service.

Declared model and evidence

Case / calculation IDTraceable identifier; no default case is assumed.

Vibration source / asset

Receiver / supported item

Axis, point and positive-direction conventionFor force work, state whether the recorded support reaction has been polarity-reversed.

Operating and measurement condition

Transmissibility model

Matched input/output quantityMotion ratios require the same derivative quantity; the force model requires force.

Harmonic excitation frequency f (Hz)Finite and greater than zero.

Undamped natural frequency f_n (ჰც)ვ_n = (1 / 2π) √(k / m); do not substitute a damped resonance peak without derivation.

Viscous damping ratio ζDimensionless, finite and non-negative. The numerical ceiling is only a software guard.

Natural-frequency basisModel, test, fit, date and source.

Damping-ratio basisMethod, condition, uncertainty and source.

Controlled calculation / model reference

Reviewer and review date

Eight applicability and decision gates

Linearity and time invariance are adequate over the stated amplitude and condition. One mode/DOF is an adequate representation in the evaluated frequency region. The input is steady-state harmonic and transients/non-harmonic content are out of scope. A linear viscous damper model is adequate; friction, hysteresis and nonlinear mounts are addressed separately. The undamped natural frequency and excitation frequency refer to the same axis/configuration. The damping estimate is applicable to this amplitude, temperature, preload and frequency region. Input/output quantity, amplitude convention, axis and polarity are matched. Clearance, travel, load, stability, durability and acceptance decisions use a separate controlled procedure.

No result

Enter a traceable case and submit explicitly.

Frequency ratio r = f / f_n

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Magnitude T = |H|

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Output phase relative to input

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Signed amplitude change (T − 1) × 100%

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Non-zero crossover comparison

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Evidence gates recorded

0 of 8

Select the physical input/output ratio. A dimensionless number without that definition is ambiguous.

Equations, variables and scope

For the declared base-motion model, H = X/Y. For the declared force-transmission model, H = F_{გადაცემული}/ფ_applied under the stated scalar positive-direction convention. In the same ideal spring–mass–viscous-damper arrangement they have the same complex form:

r = f / f_nf and f_n are in the same frequency unit; r is dimensionless.

H(i r) = (1 + i 2 ζ r) / (1 − r² + i 2 ζ r)This numerator is essential to both magnitude and phase.

T = |H| = √[1 + (2 ζ r)²] / √[(1 − r²)² + (2 ζ r)²]T is an amplitude ratio, not energy efficiency.

φ = atan2(2 ζ r, 1) − atan2(2 ζ r, 1 − r²)Wrapped to −180°…+180°; output is relative to input under the stated polarity.

amplitude change = (T − 1) × 100%Negative means modelled amplitude reduction; positive means amplification.

T = 1 at r = √2 (non-zero crossover)This follows algebraically from this H and does not shift with ζ. At ζ = 0 and r = 1 the ideal model is singular.

General engineering model, not an ISO formula. NASA sources below provide the spring–mass–damper equation, transfer function and magnitude relationship. The ISO records establish vocabulary, information-exchange and laboratory-measurement scope; public abstracts do not establish this calculator or an approval criterion.

Source and lifecycle register

იდენტიფიკატორი	Source / exact status	Used for	Not used for
S1	NASA, Fundamentals of Microgravity Vibration Isolation, MEIT-2004, Section 17, pp. 10 and 15	Base-excited SDOF equation and P(s) = (d s + k)/(m s² + d s + k). The displayed magnitude and phase are derived from the same P(iω).	No real-isolator acceptance, clearance, load, stability or durability criterion.
S2	Niebuhr & Hagen, Development of the Vibration Isolation System for the Advanced Resistive Exercise Device, NASA NTRS 20110024049, publication record 2011, proceedings 2012, p. 4	Published transmissibility magnitude equation and a rounded application table. The Z-axis 0.11/0.09 Hz, ζ=0.10 case gives model T≈1.868 versus reported 1.9.	No universal damping range or approval threshold.
S3	ISO 2017-1:2005, Edition 1 — Published; last confirmed 2019; currently stage 90.60 (under review)	Public scope: information exchange among users, manufacturers and suppliers for isolation-system applications.	The public record does not issue this SDOF formula or prove suitability. Exact requirements: NEEDS_LICENSED_SOURCE.
S4	ISO 2041:2018, Edition 4 — Published; confirmed 2024; stage 90.93	Current mechanical-vibration vocabulary lifecycle and scope.	Protected definitions/clauses are not reproduced or inferred: NEEDS_LICENSED_SOURCE.
S5	ISO 10846-1:2008, Edition 2 — Published; confirmed 2022; stage 90.93	Public scope: principles/guidance for laboratory determination of resilient-element transfer properties and selection of the relevant series part.	No claim that this ideal analytical model is an ISO 10846 laboratory result. Exact procedure: NEEDS_LICENSED_SOURCE.
S6	ISO 10846-3:2002, Edition 1 — Published; confirmed 2022; stage 90.93	Public scope: indirect laboratory method for dynamic transfer stiffness of resilient supports, including measurement of vibration transmissibility under stated test conditions.	No protected apparatus, validity-band or uncertainty detail is embedded: NEEDS_LICENSED_SOURCE.

Source access and technical review: 17 July 2026. Recheck ISO 2017-1 before controlled future use because its official record is at review stage 90.60.

What was corrected

1. Phase now belongs to the displayed transfer function

The former page used the numerator-inclusive magnitude but φ = atan2(2ζr, 1−r²), which is the phase of a different denominator-only response. The worksheet uses the complex numerator and denominator for both outputs. For f=50 Hz, f_n=15 Hz and ζ=0.05. The former display was about +178.1°; the declared H gives −159.68°.

2. Force and motion ratios are no longer an unused switch

The former selector never changed the calculation or defined the numerator and denominator. The selected model now changes the physical definition and enforces a compatible quantity. Relative displacement, insertion loss, power efficiency and multi-DOF transfer paths are not silently substituted.

3. The crossover contradiction was removed

For this H, the non-zero solution of T=1 is exactly r=√2 for any finite ζ. The former FAQ said damping shifts it while the theory section said it is independent of damping.

4. r=1 is not labelled as the exact damped peak

The former plot labelled T(r=1) as “Peak”. For numerator-inclusive transmissibility and ζ>0 the exact maximum is generally below r=1. The worksheet reports the requested point and does not invent a peak claim.

5. “Isolation efficiency” was replaced by an amplitude statement

(1−T)×100% is an amplitude reduction under the declared model, not thermodynamic, power or installation efficiency. The worksheet reports signed amplitude change and keeps engineering approval separate.

6. Defaults, auto-calculation and detached result state were removed

No presets, implicit defaults, URL parameters, local storage, clipboard export, dynamic markup insertion, external formula CDN or automatic calculation remains. Full-string decimal point/comma parsing, explicit submit, stale-state invalidation and error clearing are tested.