Coupled Bending–Bending–Axial–Torsional Vibrations of Rotating Blades

  • Feng LiangEmail author
  • Zhen Li
  • Xiao-Dong YangEmail author
  • Wei Zhang
  • Tian-Zhi Yang


In this paper, the coupled bending–bending–axial–torsional free vibrations of rotating blades are investigated based on the Euler–Bernoulli beam model. The coupled partial differential equations governing flapwise, edgewise, axial and torsional motions are derived by the Hamilton’s principle, wherein three types of velocity-dependent terms, namely static centrifugal terms, dynamic centrifugal terms and gyroscopic coupling terms, are focused. The ordinary differential equations are acquired by the Galerkin truncation, and the natural frequencies in all directions and complex mode shapes of the rotating blades are analyzed in detail. It is revealed that the three types of velocity-dependent terms have different effects on the natural frequencies. The natural frequencies are noticeably dependent on the rotating speed and preset angle, except for the axial vibration, which is almost immune to the preset angle. The complex modal motions are displayed by a series of positions of the central line and free-end cross section for different time instants, showing the coupled vibrations among different directions.


Rotating blades Coupled vibrations Gyroscopic coupling Complex modes Preset angle 



This work was supported in part by the National Natural Science Foundation of China (Project Nos. 11672007, 11672189), the Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University (VCAME201601) and Beijing Natural Science Foundation (Project No. 3172003).

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© The Chinese Society of Theoretical and Applied Mechanics 2019

Authors and Affiliations

  1. 1.College of Mechanical EngineeringYangzhou UniversityYangzhouChina
  2. 2.Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Engineering, College of Mechanical Engineering and Applied ElectronicsBeijing University of TechnologyBeijingChina
  3. 3.Department of Engineering MechanicsShenyang Aerospace UniversityShenyangChina

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