Study of Primary and Internal Resonance on 3D Free-Free Double-Section Beam


  • Yi-Ren Wang Department of Aerospace Engineering, Tamkang University, New Taipei City, Taiwan
  • Yun-Shuo Chang Department of Aerospace Engineering, Tamkang University, New Taipei City, Taiwan



nonlinear vibration, internal resonance, method of multiple scales


This work investigates the primary resonance and internal resonance of a double-section beam with cubic nonlinearities. This model can be applied in a wide range of engineering problems, such as rocket and missile structures. Even space technology has been developed for decades; several nonlinear properties deserve further study, especially, for the internal resonance. The method of multiple scales (a perturbation technique) is employed to analyze this nonlinear problem. This study focuses on finding the forcing conditions of this 3D double-section beam to trigger the often-ignored internal resonance or prime resonance in rocket structures. A primary resonance is found on a uniform free-free beam at certain flight speed. The three-to-one internal resonance of the double-section beam occurs within the first and the second modes in the diameter ratio of 1/0.75 with the length ratio of 0.33 or 0.51. The semi-analytical results are verified by the time marching numerical method.


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How to Cite

Y.-R. Wang and Y.-S. . Chang, “Study of Primary and Internal Resonance on 3D Free-Free Double-Section Beam”, Adv. technol. innov., vol. 5, no. 4, pp. 270–291, Sep. 2020.