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


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.

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A, B :

Derivative matrixes

a :

Fourier transformed displacement of q

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

Augmented unknown vector

a 0, a i (i = 1, 2, …,n):

Fourier coefficients of constant and sine term

b i (i = 1, 2, …,n):

Fourier coefficient of cosine term

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

Augmented damping matrix

c :

Gap size

c lx , c rx , c ly , c ry :

Dampings of the bearings in x and y directions

D 1 :

Viscous damping

D 2 :

Supporting damping

e 1, e 2 :

Eccentric distances of two disks

F e , F non :

Exciting force vector, nonlinear force vector

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

Nondimensional exciting force and nonlinear force vector

f :

Fourier component vector of nonlinear function

f n , f t :

Normal contact force, tangential contact force

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

Nondimensional normal and tangential contact forces

G :

Gyroscopic matrix

g :

Constraint equation of pseudo arc-length

g s :

Single-sided contact function

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

Augmented Jacobi matrix

J di , J pi (i = 1, 2, 3, 4, 5):

Diametral and polar moment of inertia

K :

Stiffness matrix

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

Augmented stiffness matrix

k :

Number of harmonic components

k lx , k ly , k rx , k ry :

Stiffnesses of the bearings in x and y directions

k n :

Stator stiffness

M :

Mass matrix

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

Augmented mass matrix

m i (i = 1, 2, 3, 4, 5):

Lumped mass

N :

Sampling points per cycle in HBM

n :

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

Q :

Orthonormal square matrix

R :

Fourier transformed residual

r :


s :


T :

Nondimensional operating time

t :

Operating time

u :

Fourier component vector of exciting force

x i , y i (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



η :

Downhill factor

δ :

Piecewise function

ξ 1, ξ 2 :

The first and second modal damping ratios

θ xi , θ yi (i = 1, 2, 3, 4, 5):

Angles of orientation associated with the x and y axes

σ :

Control factor

μ :

Friction coefficient

ɛ :


ɛ p :


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

Periodic term in the perturbation

\({{\tau}}\) :

Tangent vector

φ 1, φ 2 :

Phase angles of the unbalanced force

ϕ :

Nondimensional time

ω :

Rotating frequency

ω n1, ω n2 :

The first and the second natural frequencies

ν :

Subharmonic ratio

\({\mathfrak{R}}\) :

Upper triangular matrix

Γ :

DFT matrix


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Correspondence to Hui Ma.

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Tai, X., Ma, H., Liu, F. et al. Stability and steady-state response analysis of a single rub-impact rotor system. Arch Appl Mech 85, 133–148 (2015).

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