Archive of Applied Mechanics

, Volume 85, Issue 1, pp 133–148 | Cite as

Stability and steady-state response analysis of a single rub-impact rotor system

  • Xingyu Tai
  • Hui MaEmail author
  • Fuhao Liu
  • Yang Liu
  • Bangchun Wen


Using a lumped mass model of a single rub-impact rotor system considering the gyroscopic effect, the stability and steady-state response of the rotor system are investigated in this paper. The contact between the rotor and the stator is described by the simple Coulomb friction and piecewise linear spring models. An algorithm combining harmonic balance method with pseudo arc-length continuation is adopted to calculate the steady-state vibration response of a nonlinear system. Meanwhile, Hill’s method is used to analyze the stability of the system. The nonlinear dynamic characteristics of the system are investigated when the gap size, stator stiffness and unbalance are regarded as the control parameters. The results show that the gap size determines the location of the rub-impact; besides, the smaller gap can improve the stability of the system. The unsteady motion can be found as the stator stiffness increases. Moreover, the unbalance directly affects vibration amplitude, which becomes greater with the increasing imbalance.


Steady-state response Stability Single rub-impact Rotor system Harmonic balance method 


A, B

Derivative matrixes


Fourier transformed displacement of q

\({{\tilde{ \varvec {a}}}}\)

Augmented unknown vector

a0, ai (i = 1, 2, …,n)

Fourier coefficients of constant and sine term

bi (i = 1, 2, …,n)

Fourier coefficient of cosine term

\({{\tilde{\varvec {C}}}}\)

Augmented damping matrix


Gap size

clx, crx, cly, cry

Dampings of the bearings in x and y directions


Viscous damping


Supporting damping

e1, e2

Eccentric distances of two disks

Fe, Fnon

Exciting force vector, nonlinear force vector

\({{\tilde{\varvec {F}}_e, \, {\tilde{\varvec {F}}}_{\rm non}}}\)

Nondimensional exciting force and nonlinear force vector


Fourier component vector of nonlinear function

fn, ft

Normal contact force, tangential contact force

\({\tilde{f}_n, {\tilde{f}}_t }\)

Nondimensional normal and tangential contact forces


Gyroscopic matrix


Constraint equation of pseudo arc-length


Single-sided contact function

\({{\tilde{\varvec {J}}}}\)

Augmented Jacobi matrix

Jdi, Jpi (i = 1, 2, 3, 4, 5)

Diametral and polar moment of inertia


Stiffness matrix

\({{\tilde{\varvec {K}}}}\)

Augmented stiffness matrix


Number of harmonic components

klx, kly, krx, kry

Stiffnesses of the bearings in x and y directions


Stator stiffness


Mass matrix

\({{\tilde{\varvec {M}}}}\)

Augmented mass matrix

mi(i = 1, 2, 3, 4, 5)

Lumped mass


Sampling points per cycle in HBM


Dimension of system

\({\varvec {q}, { \dot {\varvec q}}, { \ddot {\varvec q}}}\)

Displacement, velocity, acceleration vector of the system

\({{\tilde{\varvec {q}}}, {\dot{\tilde{\varvec {q}}}}, {\ddot {\tilde{\varvec {q}}}}}\)

Nondimensional displacement, velocity, acceleration vector of the system


Orthonormal square matrix


Fourier transformed residual






Nondimensional operating time


Operating time


Fourier component vector of exciting force

xi, yi (i =  1, 2, 3, 4, 5)

Displacements in x and y directions

\({{\tilde{x}}_i, \, {\tilde{y}}_i (i = 1,2,3,4,5)}\)

Nondimensional displacements in x and y directions

Greek symbols




Downhill factor


Piecewise function

ξ1, ξ2

The first and second modal damping ratios

θxi, θyi (i = 1, 2, 3, 4, 5)

Angles of orientation associated with the x and y axes


Control factor


Friction coefficient





\({{\bf {\kappa}}_p}\)

Periodic term in the perturbation


Tangent vector

φ1, φ2

Phase angles of the unbalanced force


Nondimensional time


Rotating frequency

ωn1, ωn2

The first and the second natural frequencies


Subharmonic ratio


Upper triangular matrix


DFT matrix


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Xingyu Tai
    • 1
  • Hui Ma
    • 1
    Email author
  • Fuhao Liu
    • 2
  • Yang Liu
    • 1
  • Bangchun Wen
    • 1
  1. 1.School of Mechanical and AutomationNortheastern UniversityShenyangPeople’s Republic of China
  2. 2.Department of Automotive EngineeringYancheng Institute of TechnologyYanchengPeople’s Republic of China

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