Autoparametric Resonance in Two Systems: Spring–Pendulum and T-Screw Interpreted Through Δf Decay (msf:48210)

This document presents a unified interpretation of autoparametric resonance in two seemingly different systems: (1) the spring–pendulum, and (2) the freely rotating T-shaped screw in microgravity. Using both classical mechanics and USP Field perspectives, the analysis shows that both systems can be understood through a decaying frequency mismatch Δf(t). As Δf relaxes, the system crosses resonance windows where energy transfers from a dominant mode into a latent mode. Classically, the spring–pendulum is described by Hill/Mathieu-type equations, while the T-screw wobble arises from Euler’s rigid-body dynamics. The unified interpretation treats both as slow–fast systems: fast oscillatory variables modulated by a slow drift in Δf(t). Key relations (Word-friendly format): Frequency mismatch: Δf(t) = f₁(t) – f₂(t) Exponential relaxation law: dΔf/dt = –γ Δf(t) Δf(t) = Δf₀ · exp(–γ t) Coupling conditions: |Δf(t)| >> Δf_crit → modes decoupled |Δf(t)| ≤ Δf_crit → resonance window The paper constructs minimal coupled ODE models, demonstrates mode-transfer behavior, and outlines microgravity and laboratory experiments capable of testing Δf-driven activation. This work provides a bridge between classical vibration theory and the USP Field concept of resonance-corridor mismatch, showing how both viewpoints naturally describe the same underlying phenomenon.