Nonlinear dynamic modeling and response analysis of a rotor–blade system with whirling motion

  • Jaewon Kim
  • Jun-Gi Hong
  • Jintai ChungEmail author
Original paper


In this paper, we propose a new nonlinear dynamic model of a rotor–blade system with whirling motion, in which a rigid rotor and blades are modeled as a Jeffcott rotor and Euler–Bernoulli beams, respectively. The stiffening effects of the rotating blades are considered using a hybrid set of deformations, which consist of the stretch and chordwise deformations. After the nonlinear partial differential equations of motion are derived using Hamilton’s principle, they are discretized using the Galerkin method. From the discretized equations, the nonlinear dynamic responses are computed using the generalized-\(\upalpha \) time integration method. Based on the dynamic responses and the frequency spectra, the proposed model is verified both for the case where the blade is considered to be a rigid body and for the case where there is no whirling motion of the rotor. In this study, the nonlinear dynamic responses of the rotor–blade system are analyzed in terms of the natural frequencies for whirling motion and the natural frequencies of the deformation of the rotating beam. In addition, the effects of the supporting stiffness, rotating speed and blade stiffness on the dynamic responses of the radial displacement and stretch/chordwise deformations are also analyzed.


Stiffening effects Nonlinear dynamic response Stretch deformation Chordwise deformation Supporting stiffness Rotating speed Blade stiffness 

List of symbols


Inertia reference frame fixed in space


Rotating reference frames fixed to the q-th blade

\(\Omega \)

Rotating speed


Geometric center of the rotor


Mass center of the rotor


Mass, radius, and eccentricity of the rotor


Equivalent damping coefficient and equivalent radial spring constant

\(\phi \)

Angle of the blade axis from the X-axis

\(\phi _{0}\)

Angle of the mass center from the blade

\(\varPsi \)

Angle from the r-axis to the mass center

\(r, \theta \)

Radial and angular displacements

\(\rho , A, L, E\)

Mass density, cross-sectional area, total length, and Young’s modulus of the blade


Mass moment of inertia for the rotor mass


Area moment of inertia of the blade

\(u_{q}, s_{q}, v_{q}\)

Axial, stretch, and chordwise deformations


Total number of basis functions

\(S_{j}, V_{n}\)

The basis functions for the stretch and chordwise deformations



M, C, G, K

Mass, damping, and gyroscopic, stiffness matrices

N, F, T

Nonlinear internal force, and force, unknown vectors

\(\mathbf{d}_{n}, \mathbf{v}_{n}, \mathbf{a}_{n}\)

Displacement, velocity, and acceleration vectors at time \(t=t_{n}\)

\(\Delta t\)

Time step size between \(t_{n}\) and \(t_{n+1}\)

\(\Omega _{m}\)

Maximum rotating speed

\(r_{e}, \varPsi _{e}\)

Equilibrium positions in the Jeffcott rotor model

\(\Delta r, \Delta \varPsi \)

Perturbed displacements in the Jeffcott rotor model

\(\Omega _{c}\)

Critical speed of the Jeffcott rotor


Spring constant corresponding to the critical speed



This work was supported by a Grant from the National Research Foundation of Korea (NRF), funded by the Korean government (MEST) (NRF-2018R1D1A1B07050187).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringHanyang UniversityAnsanRepublic of Korea

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