Krylov-Bogolyubov Averaging Based Solution: Nonlinear Mode Superposition in a 2-DOF System with Coulomb Damping

• Published in: Journal of Vibration Engineering & Technologies Vol. 13; no. 8; p. 546
• Main Authors: Sreesha, Rakshith B., Karthik, B. K., Shrikanth, Venkoba
• Format: Journal Article
• Language: English
• Published: Singapore Springer Nature Singapore 01.12.2025; Springer Nature B.V
• Subjects: Acoustics; Asymmetry; Coefficient of friction; Control; Coordinate transformations; Coulomb friction; Damping; Dry friction; Dynamical Systems; Energy dissipation; Engineering; Engineering Acoustics; Equilibrium; Exact solutions; Friction; Harmonic oscillators; Image processing; Initial conditions; Krylov-Bogoliubov method; Mass-spring systems; Mathematical analysis; Original Paper; Simulation; Symmetry; Vibration; Vibration tests; Power spectral density; Energy dissipation; Free vibration; Coulomb friction oscillator; Nonlinear mode superposition; Dry friction damping
• ISSN: 2523-3920; 2523-3939
• DOI: 10.1007/s42417-025-02143-7

Purpose This work investigates the applicability of nonlinear mode superposition and the accuracy of the approximate analytical solution for a 2-DOF Coulomb friction oscillator system. A recent study presented a novel analytical solution for a harmonic oscillator subject to Coulomb damping, which is often simplistically modeled using a signum function due to its high non-linearity and discontinuity. This new solution incorporates the dissipation of energy and the resulting decrease in the amplitude of the dry friction oscillator, as represented within the phase plane. Methods An in-house customized 2-DOF tribo-vibratory test rig is designed and fabricated to support this analysis. The displacement of the inertial masses is captured videographically and analyzed by means of image processing. Results A comparison is made between numerical simulations, superposed analytical solutions, and experimental results. The relevance of the proposed solution for a 2-DOF system is demonstrated by analyzing both symmetric and asymmetric configurations of the spring-mass system. Additionally, the convenience of using this analytical solution for estimating energy in systems with extreme asymmetry in inertial masses, spring stiffness, and friction coefficients is also discussed. Conclusion A desirable working region, determined by the initial conditions, is identified for the applicability of individual solutions to yield favorable results. This region is based on the -norm of the displacement error.