Energy-Consistent and Mechanically Constrained Speed-Ratio Selection for a Dual-Gear Speed-Amplified Viscous Damper

This study develops a speed-ratio selection framework for a dual-gear speed-amplified viscous damper under small structural motions. The method combines multi-gear kinematics, virtual-work mapping of rotor-side power-law resistance, reflected rotational inertia, nonlinear cycle-energy equivalence, backlash loss, and gear efficiency/strength constraints. The principal device-specific result is that the ideal structural-side damping force scales with the (1 + α)-th power of the kinematic amplification factor, while the reflected inertance scales with the square of the speed ratio. For the specified single-story frame example, including a total output-side rotational inertia of 0.20 kg·m2 changes the acceleration-sensitivity boundary from √2 to 1.402 at the selected practical ratio, while the theoretical minimum ratio for the displacement target changes only from 2.422 to 2.422 within three decimals. A standard-tooth combination gives i = 53/21 = 2.524. A manufacturer-based lower-bound mesh efficiency of 98% per spur-gear mesh and explicit tooth-root/contact checks replace the former assumed efficiency and torque limits. For α = 0.3–0.7, direct nonlinear time-history solutions agree with the iterative cycle-energy linearization within 0.34% in response amplitude over the tested harmonic cases. The framework is therefore presented as a generic device-design method; independent prototype testing remains necessary before project-specific application.