• Romes Antonio Borges +5564999711575 https://orcid.org/0000-0002-7370-2092
  • Marcos N. Rabelo Federal University of Catalão
  • Alcione B. Purcina Federal University of Catalão
  • Marcos L. Henrique Federal University of Pernambuco, Campus Agreste, Caruaru-PE




nonlinear mechanical systems, primary resonances, multiple scales method, Lyapunov exponent, Poincaré map, stability analysis and chaos


In recent years, there has been growing interest in the study of nonlinear phenomena. This is due to the modernization of structures related to the need of using lighter, more resistant and flexible materials. Thus, this work aims to study the behavior of a mechanical system with two degrees of freedom with nonlinear characteristics in primary resonance. The structure consists of the main system connected to a secondary system to act as a Nonlinear Dynamic Vibration Absorber, which partially or fully absorbs the vibrational energy of the system. The numerical solutions of the problem are obtained using the Runge-Kutta methods of the 4th order and approximate analytical solutions are obtained using the Multiple Scales Method. Then, the approximation error between the two solutions is analyzed.

Using the aforementioned perturbation method, the responses for the ordinary differential equations of the first order can be determined, which describe the modulation amplitudes and phases. Thus, the solution in steady state and the stability are studied using the frequency response. Furthermore, the behavior of the main system and the absorber are investigated through numerical simulations, such as responses in the time domain, phase planes and Poincaré map; which shows that the system displays periodic, quasi-periodic and chaotic movements. The dynamic behavior of the system is analyzed using the Lyapunov exponent and the bifurcation diagram is presented to better summarize all the possible behaviors as the force amplitude varies. In general, the main characteristics of a dynamic system that experiences the chaotic response will be identified.


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Author Biographies

Marcos N. Rabelo, Federal University of Catalão

Mathematics and Technology Institute – School of Industrial Mathematics,

Alcione B. Purcina, Federal University of Catalão

Mathematics and Technology Institute – School of Industrial Mathematics

Marcos L. Henrique, Federal University of Pernambuco, Campus Agreste, Caruaru-PE

Interdisciplinary Nucleus of Exact Sciences and Technological Innovation


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DOI: 10.31686/ijier.vol8.iss3.2245

How to Cite

Antonio Borges, R., Rabelo, M. N., Purcina, A. B. ., & Henrique, M. L. . (2020). DYNAMIC MODELING AND STABILITY ANALYSIS OF A NONLINEAR SYSTEM WITH PRIMARY RESONANCE. International Journal for Innovation Education and Research, 8(3), 391–414. https://doi.org/10.31686/ijier.vol8.iss3.2245
Received 2020-02-19
Published 2020-03-01