Linear SDOF Natural-Frequency Worksheet
Calculate the undamped natural frequency of one translational single-degree-of-freedom model from documented effective mass and stiffness. Optional damping and excitation order are evaluated separately;the result is not automatically a machine critical speed.
Define the linearized coordinate
Mass and stiffness must refer to the same displacement coordinate,direction,equilibrium/preload and boundary conditions. For mounts or elastomers,use applicable dynamic/tangent stiffness at the frequency,amplitude,temperature and preload of interest.
Undamped natural frequency
Undamped natural angular frequency
Undamped period
Damped natural frequency
1× cyclic-speed equivalent
Entered-order speed crossing
Effective mass in SI
Effective stiffness in SI
Coordinate record
Equations,units and scope
| Quantité | Equation | Applicabilité |
|---|---|---|
| Undamped natural frequency | m x¨+kx=0 ωn=√(k/m) fn=ωn/(2π) | Linear translational SDOF about an equilibrium;positive effective m and k in the same coordinate. |
| Underdamped free frequency | fd=fn√(1−ζ²) | Linear viscous damping with0≤ζ<1.Not the forced-response peak in every response measure. |
| Excitation-order crossing | n=60fn/q | Only when q cycles/revolution represents the documented forcing mechanism. |
Exact conversions used are1lb=0.45359237kg,1N/mm=1000N/m and1lbf/in=175.12683524647638N/m.The radian is dimensionless in these relations. Static deflection was removed because δ=F/k needs an explicitly supported force,direction and the applicable static stiffness;it is not a universal output of any mass-stiffness pair.
Sources and standards boundary
MIT OpenCourseWare18.03,Class13 derives the linear mass-spring-damper equation,ωn=√(k/m),and distinguishes the lower underdamped oscillation frequency. This is university mechanics,not an ISO formula.
ISO10846-2:2008,Edition2,confirmed2022 and current,defines a laboratory direct method for dynamic transfer stiffness of resilient supports under specified preload. ISO10846-4:2003,Edition1,confirmed2023 and current,addresses dynamic transfer stiffness of other resilient elements for translation. Their scopes show why a catalogue static spring rate is not automatically the dynamic k of this model.Exact procedures and clauses beyond the public scopes are NEEDS_LICENSED_SOURCE.
Le BIPM SI Brochure,9th edition,updated2026 and exact international pound/inch definitions provide the unit basis.Standard acceleration of free fall is9.80665m/s²,but no gravity-derived deflection is calculated without an entered supported force.
Model safeguards
- For a rigid machine translating on parallel mounts,k is the sum only if the mounts share that coordinate and the rigid-body/equal-motion assumptions hold.Rotation,coupling and center-of-mass offset introduce more degrees of freedom.
- Effective/modal mass is coordinate and normalization dependent;total machine mass is not always the correct m.
- Elastomer,pneumatic and hydraulic mount stiffness may depend on preload,frequency,amplitude,temperature and history.Use a compatible dynamic property.
- Resonance is tied to forcing frequency,not shaft RPM alone.Excitation can occur at1×,blade/vane pass,gear mesh,electromagnetic orders or transient sweeps.
- No universal20–30% separation rule is applied.Acceptance needs applicable codes,manufacturer limits,uncertainty,damping/response and all operating states.
Corrections apportées
The former page used the valid SDOF equation but prefilled generic machine presets,stored history/units/URL data,used rounded lb and lbf/in conversions,reported 60fn as equivalent RPM without an excitation order,and advised a universal20–30% speed separation.It also reported mg/k static deflection for every model using rounded g=9.81.
The replacement starts blank,requires coordinate/property evidence,uses exact conversions and strict point/comma parsing,and labels the result as a linearized SDOF reference.Optional damping and excitation order are explicit inputs.Static deflection,presets,persistence,automatic resonance/safety recommendations,prefix parsing and innerHTML were removed.
