Author(s): Lukas Meier and Anna Kowalska

Abstract: Single-degree-of-freedom (SDOF) systems form the fundamental basis for understanding structural and mechanical vibrations. Classical linear vibration theory provides closed-form solutions and clear physical interpretations; however, many practical engineering systems exhibit mild nonlinear stiffness due to geometric effects, material behavior, boundary conditions, or large deformation responses. Such nonlinearities, though weak, can significantly influence dynamic characteristics, including resonance frequency shifts, amplitude-dependent behavior, and stability of motion. This research examines the dynamic response of SDOF systems incorporating mild nonlinear stiffness under harmonic excitation. The governing equation of motion is formulated by introducing a cubic stiffness term in addition to the linear restoring force, representing weak nonlinearity commonly encountered in engineering applications. Analytical approaches, including perturbation-based methods, are emphasized to derive approximate solutions that describe steady-state and transient responses. The influence of nonlinear stiffness on frequency-response curves, jump phenomena, and softening or hardening behavior is systematically discussed. Special attention is given to the transition from linear to nonlinear response regimes and the conditions under which linear approximations become inadequate. The research further highlights the relevance of mild nonlinearity in predicting realistic system behavior, particularly near resonance conditions where small deviations from linearity may produce disproportionately large dynamic effects. By synthesizing theoretical insights with established vibration concepts, this work aims to clarify the physical implications of nonlinear stiffness in SDOF systems. The findings are intended to support improved modeling accuracy and more reliable dynamic analysis in mechanical and structural engineering practice, where simplified linear models may fail to capture essential response characteristics under operational loading conditions.

DOI: 10.22271/2707806X.2026.v7.i1a.61

Pages: 44-48 | Views: 31 | Downloads: 7

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