Bounded continuous-system model
Uniform Spring Axial-Wave Frequency Estimate
Estimate the first axial-wave mode of an ideal uniform distributed spring from measured or traceably calculated axial stiffness and active vibrating mass. Two ideal end conditions are reported separately.
Idealized axial-wave modes
For a uniform one-dimensional distributed elastic member, k=EA/L and ms=ρAL, so its wave speed divided by length is c/L=√(k/ms). The first fixed–fixed and first non-rigid free–free wavelengths are 2L; the first fixed–free wavelength is 4L. Therefore the fixed–free value is one half—not the fixed–fixed value.
MIT OpenCourseWare: Wave Propagation provides the classical one-dimensional wave and longitudinal-rod framework. NASA report KSC-DM-3538 / NASA-TM-103824, Appendix B, tabulates uniform longitudinal-vibration mode parameters by support condition.
This is a mechanics estimate, not a claim that an ISO standard defines a universal spring “critical speed.” No ISO acceptance factor was identified for this generic calculation. A design limit must come from the applicable product standard, validated manufacturer method, purchaser specification, modal analysis or test for the actual assembly.
No automatic safety decision: a single excitation ratio cannot prove avoidance of resonance because harmonics and transient or parametric excitation may govern. The former 13× and 15–20× universal rules are intentionally not used.
