Does this result prove that a machine foundation avoids resonance?

No. It is an undamped one-degree-of-freedom arithmetic check. A resonance assessment must include excitation orders,all relevant modes,damping,frequency-dependent impedance and acceptance criteria for the actual system.

When may area-normalized stiffness be multiplied by area?

Only when the source explicitly defines that coefficient for the same direction,mode,foundation geometry,frequency range and K equals coefficient times area model.

Linear undamped model — one degree of freedom

Equivalent-Stiffness SDOF Natural Frequency Check

Calculate one undamped translational natural frequency from a documented equivalent stiffness and participating mass. This is not a complete machine-foundation dynamic analysis.

ωₙ=√(K/m)fₙ=ωₙ/(2π)No resonance margin inferred

Undamped SDOF result

Equivalent stiffness

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Natural angular frequency

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Tần số tự nhiên

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Cycles per minute

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Undamped period

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Implemented model

mẍ+Kx=0
ω_N=√(K/m)
f_N=ω_N/(2π)
T_N=1/f_N

K is the equivalent translational stiffness in N/m for the stated mode,m is its participating mass in kg,ω_N is in rad/s,f_N is in Hz and T_N is in seconds. Dimensional check:√[(N/m)/kg]=√(s⁻²)=s⁻¹.

Two input bases

Direct K uses a documented modal/equivalent stiffness. Area-normalized uses K=k_AA only when the cited source explicitly defines k_A in MN/m³ and that linear area model for the same direction,geometry and conditions. The calculator does not treat k_A as an intrinsic soil property.

Traceability and soil-model warning

MIT OpenCourseWare,Dynamics and Control I,Lecture20 gives mẍ+cẋ+Kx=0 and ω_N²=K/m for an SDOF system. FHWA NHI-05-037 §5.4.6 states that the modulus of subgrade reaction is not a true soil property because it also depends on footing/slab size and stiffness. It further distinguishes dynamic and traditional static k-values.

No automatic resonance criterion

This result alone cannot label an operating speed safe or unsafe. Compare all excitation orders with all relevant coupled modes using documented damping,forcing and project-specific acceptance criteria. No universal30% separation rule,soil range or machine preset is applied.