Frequency-domain energy transformation characteristics of nonlinear inertia in vibration isolation systems

Nonlinear inertia in vibrating systems redistributes incoming kinetic energy among multiple frequency components. This paper examines two coupled aspects: the frequency-domain transformation of nonlinear inertial kinetic energy and its influence on the response and harmonic kinetic-energy distribution of the system output. Two metrics—Energy Transformation Ratio ( R _{E . T .} ) and Energy Distribution ( Dist _e )—are proposed to quantify this spectral redistribution. For a general nonlinear inertia coupled to a single-degree-of-freedom isolator, the energy distribution is nearly independent of excitation frequency at high frequencies but strongly frequency-dependent in the low-to mid-frequency range. Higher excitation amplitudes increase the proportion of higher-frequency energy, while a smaller minimum value of the prescribed nonlinear inertia further amplifies this effect. The type of coupled isolator also reshapes the energy spectrum: the X-structure results in a lower high-frequency energy proportion than linear or nonlinear spring–damper isolators. Nonlinear inertia can significantly enhance the high-frequency components in the system output, and its effects vary across different coupled systems. For even-power nonlinear inertia, the relative energy proportions are shown by mathematical induction to follow a squared Catalan-triangle law under first-harmonic approximation, independent of the coupled system. Experimental results confirm the predicted inertia forms, energy spectra, frequency and amplitude effects, and partial squared Catalan-triangle characteristics. Overall, this study provides an in-depth understanding of the energy distribution patterns and nonlinear phenomena induced by nonlinear inertia.