Nonlinear vibration and dynamic stability analysis of rotor-blade system with nonlinear supports

  • Bingqiang Li
  • Hui MaEmail author
  • Xi Yu
  • Jin Zeng
  • Xumin Guo
  • Bangchun Wen


A dynamic model of a rotor-blade system is established considering the effect of nonlinear supports at both ends. In the proposed model, the shaft is modeled as a rotating beam where the gyroscopic effect is considered, while the shear deformation is ignored. The blades are modeled as Euler–Bernoulli beams where the centrifugal stiffening effect is considered. The equations of motion of the system are derived by Hamilton principle, and then, Coleman and complex transformations are adopted to obtain the reduced-order system. The nonlinear vibration and stability of the system are studied by multiple scales method. The influences of the normal rubbing force, friction coefficient, damping and support stiffness on the response of the rotor-blade system are investigated. The results show that the original hardening type of nonlinearity may be enhanced or transformed into softening type due to the positive or negative nonlinear stiffness terms of the bearing. Compared with the system with higher support stiffness, the damping of the bearing has a more powerful effect on the system stability under lower support stiffness. With the increase in rubbing force and support stiffness, the jump-down frequency, resonant peak and the frequency range in which the system has unstable responses increase.


Main resonances Rotor-blade system Stability Nonlinear vibration 


A, \(A'\)

Cross-sectional area of the shaft and blade

\({{A}}_{1}, {{A}}_{2}\)

The complex functions of the dimensionless displacements to be solved

c, \( c_{\mathrm{blade}}\), \( c_{\mathrm{bearing}}\)

Damping coefficients of the shaft, blade and bearing

\({{c}}_{\mathrm{b}{{y}}{1}}, {{c}}_{\mathrm{b}{{z}}{1}}, {{c}}_{\mathrm{b}{{y}}{ 2}}, {{c}}_{\mathrm{b}{{z}}{2}}\)

The damping coefficients of bearing 1 and bearing 2 along Y and Z directions


Partial derivative with respect to \({{T}}_{0}\) and \({{T}}_{2}\)


Torsional and flexural stiffness

\(e_{y}, e_{z}\)

Eccentricity with respect to y and z axes


Young’s modulus

\({{E}}_{{{y}}}, {{E}}_{{{z}}}\)

Misalignment along Y and Z directions

\(f_\mathrm{f}\) , \(f_\mathrm{b}\)

Forward and backward whirl mode frequencies

\(F_{\mathrm{n}}, F_{\mathrm{t}}\)

Normal and tangential rubbing forces at the tip of blade


Maximum normal rubbing force

\(F_{\mathrm{t}{{i}}},{{F}}_{\mathrm{t}{{xi}}},{{F}}_{ \mathrm{t}{{yi}}},{{F}}_{\mathrm{n}i}\)

Rubbing forces on the ith blade


First-order forward and backward natural frequencies


First-order torsional natural frequency


Shear modulus


Cross section inertia moment of the blade

\(I_{1}, I_{2}, I_{3}\)

Polar and diametral mass moments of inertia


Area moment of inertia of the blade


Diametral and polar mass moment of inertia of the disk


Cross section inertia moment of the shaft


Linear support stiffness

\(k_{1}, k_{2}, k_{3}\)

Shaft curvatures

L, l

Length of the blade and shaft


The distances of the disk to the left and right end


Mass per unit length of the shaft


Density of the blade


The mass, the polar and diametral mass moment of inertia of bladed disk


The mass of the disk


Longitudinal stiffness


The number of blades


The center of rotating blade

\({{O}}_{{ 2}},{{O}}'_{{ 2}}\)

The center of static and rubbed casing


Dimensionless vibration displacement of the blade in the complex plane


The ratio of excitation frequency to rotating frequency


The radius of the blade-tip orbit

\({{R}}_{{ 0}}\)

The radius of the casing


The radius of the disk


Second-order forward and backward natural frequencies




Contact time


Rotating period


Start time of the rubbing


Kinetic energy, kinetic energies of shaft, blades and disk

\({{T}}_{{ 0}},{{T}}_{{ 2}}\)

Components of time on large scale and second-order small scale


Longitudinal and transverse displacements of the shaft


The displacement of the casing


Potential energy, potential energies of shaft, blades and bearings

\({{W}},W_{F_{ni}} \)

The work, the work done by rubbing force applied on ith blade


The location of the point along the flapwise and chordwise directions


The location of the disk


Dimensionless displacement of the shaft in the complex plane

Greek symbols

\(\alpha \)

strain along the neutral axis of the shaft

\(\alpha _{ni}, \beta _{ni}, c_{1i},c_{2i},c_{3i},c_{4i }(i=1,2)\)

The coefficients of the mode shape of the shaft to be solved

\(\beta _{\mathrm{c}}\)

Contact angle

\(\beta _\mathrm{f},\beta _\mathrm{b}\)

Dimensionless forward and backward whirl mode frequencies

\(\gamma \)

Stagger angle of the blade

\(\delta \)

Variational operator

\(\delta ({{x}})\)

Dirac delta function

\(\varepsilon \)

Non-dimensional small-scale parameter

\(\zeta \)

The distance of the point from the blade root

\({{\theta }}_{{{i}}},{{\theta }}_{{{yi}}},{{\theta }}_{{{zi}}}({{i=1,2}})\)

Angular displacements of the ith part of the shaft

\(\varTheta _\mathrm{f},\varTheta _\mathrm{b}\)

Forward and backward mode shape coefficients of the blades

\(\varTheta _{ni}(x)\,(i=1,2)\)

Piecewise nth-order mode shapes of angular displacements in the complex plane


Shear correction factor

\(\lambda _{i},r_{i} (i=1,2)\)

The coefficients of free vibration differential equation of the shaft to be solved

\(\varLambda \)

Vibration amplitude of the blade

\(\mu \)

Friction coefficient

\(\xi ,\eta \)

Coleman transformation parameters

\(\rho \)

Density of the shaft

\(\sigma \)

Detuning parameter

\(\psi ,\theta ,\beta \)

Euler angles

\({{\psi }}_{{\mathrm{f}}},{{\psi }}_{{\mathrm{b}}}\)

Forward and backward mode shapes of the rotating blades

\(\omega \)

The frequency of harmonic motion

\(\omega _{1},\omega _{2},\omega _{3}\)

Angular velocities of the rotating shaft

\(\varOmega \)

Rotating speed

\(\vartheta \)

Duffing term coefficient

\(\vartheta _{i}\)

The azimuth angle of the ith blade on the disk

\(\phi \)

Torsional deformation

\(\phi _\mathrm{f},\phi _\mathrm{b}\)

Forward and backward whirl mode shapes of the shaft

\(\phi _{i}\)

Angular position of the ith blade

\(\phi _{ni}\)

Piecewise mode shapes of the shaft at nth-order critical speed

\({{k}}_{\phi }\)

Torsional stiffness


Compliance with ethical standards


This project is supported by the National Natural Science Foundation (Grant No. 11772089), the Fundamental Research Funds for the Central Universities (Grant Nos. N170308028) and Program for the Innovative Talents of Higher Learning Institutions of Liaoning (LR2017035).

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Bingqiang Li
    • 1
  • Hui Ma
    • 1
    • 2
    Email author
  • Xi Yu
    • 1
  • Jin Zeng
    • 1
  • Xumin Guo
    • 1
  • Bangchun Wen
    • 1
  1. 1.School of Mechanical Engineering and AutomationNortheastern UniversityShenyangPeople’s Republic of China
  2. 2.Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of EducationNortheastern UniversityShenyangPeople’s Republic of China

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