Linear SDOF relation · controlled target · no isolation claim

Idealized SDOF Spring-Stiffness Relation Worksheet

Calculate the ideal total stiffness, equal per-spring stiffness and weight-induced static deflection for one documented translational natural-frequency target.

k = m(2πfₙ)²Point or commaNo design presets

Not a spring selector or isolation-performance calculator. The arithmetic assumes one linear, undamped translational degree of freedom, a rigid supported mass and identical parallel springs sharing load equally. It does not model damping, forcing spectrum, startup/shutdown, rocking modes, centre-of-mass offset, base/floor flexibility, isolator mass, nonlinear travel limits or stability.

Documented model inputs and traceability

Effective supported mass m (kg) positive mass for the documented mode

Target natural frequency fₙ (Hz) positive controlled design input—not inferred from RPM

Number of identical parallel springs N positive integer; equal stiffness/load idealization

Calculation / model record ID

Machine, payload and support boundary

Effective-mass source, method and revision

Target-frequency source, requirement and revision

Translation direction, coordinates and SDOF mode evidence

Spring count, positions and equal-load assumption

Centre of mass, inertia and rocking-mode review

Damping and isolator dynamic-property source

Forcing spectrum, harmonics and startup/shutdown review

Supplier load, travel, stability and environmental limits

Input uncertainty and reporting precision

Responsible engineer / reviewer

Review date

Notes and exclusions Optional: preload, isolator self-mass, coupling, floor dynamics, nonlinear rate, snubbers and restraint conditions

The effective mass and target natural frequency apply to the same documented translational mode and boundary; neither value was inferred from a generic machine preset.
The N springs are idealized as linear, identical and parallel with equal deflection/load; unequal locations or centre-of-mass offset require individual reactions and coupled modes.
The static deflection is weight-only under conventional standard acceleration gₙ = 9.80665 m/s²; required travel must also include preload, tolerances and dynamic motion.
No transmissibility, isolation percentage, damping adequacy, spring capacity, stability or acceptance verdict will be inferred from this idealized relation.

Idealized SDOF quantities

Angular natural frequency ωₙ

—

Required total stiffness k

—

Equal nominal stiffness per spring

—

Weight-induced static deflection δ

—

Formula, units and classification

ω_n = 2πf_n [rad/s]
k_total = mω_n² [N/m]
k_cada = k_total/N [N/m], only for N identical parallel springs
δ_weight = mg_n/k_total = g_n/ω_n² [m], g_n = 9.80665 m/s²

Símbolo	Grandeza e unidade	Limite
m	effective supported mass for the mode, kg	not automatically total machine nameplate mass
f_n	target natural frequency, Hz	controlled design input, not operating RPM or forcing frequency
N	number of ideal identical parallel springs	equal-rate/equal-deflection allocation only
k	linear stiffness, N/m and N/mm	tangent/secant and frequency dependence must match supplier data
δ	weight-induced static deflection, mm	not total required travel or dynamic displacement

This is a rearranged linear SDOF relation, not a standards-compliance formula and not a complete isolator design. The exact conventional value g_n is used instead of the former rounded 9.81 m/s².

Model boundary and removed unsupported claims

No “one-third frequency” design rule. The former page prescribed f_n ≤ one third of the lowest operating frequency. Real forcing contains orders, harmonics, transients and multiple directions; acceptance must evaluate the full spectrum and each relevant mode.
No fixed 87.5% isolation claim. The former 12.5% transmissibility at frequency ratio 3 is the undamped scalar result 1/|1−r²|. NASA’s published transmissibility relation includes damping ratio; input/output quantity and forcing/base-excitation model must also be defined.
No equal-load assumption hidden as spring selection. Dividing total stiffness by N is valid only for identical parallel springs with compatible deflection. Actual reactions depend on mount positions, centre of mass and rotational modes.
No universal frequency, deflection or layout ranges. “2–8 Hz,” “5–25 mm,” four-corner/three-point and equal-spring recommendations were removed as unsourced application-dependent advice.
No capacity or travel verdict. Weight-only deflection excludes preload, isolator self-mass, dynamic motion, tolerances, nonlinear stiffness, snubbers and stability.
No hidden state or unit ambiguity. RPM equivalence, imperial approximations, defaults, presets, auto-calculation, URL/storage history, clipboard output, dynamic FAQ HTML and external KaTeX were removed.

Source traceability

Claim	Classificação	Evidence
For an ideal spring-mass mode, f_n = (1/2π)√(k/m); damping and forcing frequency enter the transmissibility relation, and different degrees of freedom can require different stiffness/damping.	NASA engineering paper; formula page visually inspected	NASA NTRS 20110024049, p. 4
For an ideal gravity-loaded SDOF system, natural frequency may be related to static deflection by √(g/δ).	MIT OpenCourseWare engineering dynamics demonstration	MIT OCW: Predicting natural frequency by √(g/δ)
Conventional standard acceleration of free fall g_n = 9.80665 m/s².	Joint Committee for Guides in Metrology, VIM3 2.12	JCGM VIM, conventional quantity value
Weight is a force equal to mass times acceleration due to gravity; conventional g_n is 980.665 cm/s².	Official CGPM declaration	3rd CGPM (1901), Declaration 2

Accessed: 15 July 2026. No ISO or manufacturer acceptance requirement is claimed. Supplier dynamic stiffness, damping, load/travel limits and the project-specific multi-degree-of-freedom model remain controlled external inputs.

Reference checks

For m = 300 kg, f_n = 4 Hz and N = 4, ω_n = 25.132741229 rad/s, k_total = 189.496404501 N/mm, k_cada = 47.374101125 N/mm and δ = 15.525334149 mm using g_n = 9.80665 m/s². Doubling frequency quadruples stiffness and quarters the ideal weight deflection.

Perguntas

Does this select a purchasable spring?

No. Match the result to supplier dynamic-rate, load, preload, travel, stability, environment, tolerance and fatigue data, then verify the assembled system.

Does a target at one third of running speed guarantee isolation?

No. Transmissibility depends on the defined input/output, damping and frequency ratio; machinery also produces harmonics, orders and transients. Evaluate the actual spectrum and relevant modes.

Why is RPM input absent?

Operating speed is not automatically the target natural frequency. Convert and classify excitation orders in a separate controlled spectrum analysis, then supply the approved natural-frequency target in hertz.