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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
Original

Abstract

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.

Keywords

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

Nomenclature

A, B

Derivative matrixes

a

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

c

Gap size

clx, crx, cly, cry

Dampings of the bearings in x and y directions

D1

Viscous damping

D2

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

f

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

G

Gyroscopic matrix

g

Constraint equation of pseudo arc-length

gs

Single-sided contact function

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

Augmented Jacobi matrix

Jdi, Jpi (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

klx, kly, krx, kry

Stiffnesses of the bearings in x and y directions

kn

Stator stiffness

M

Mass matrix

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

Augmented mass matrix

mi(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

Residual

s

Arc-length

T

Nondimensional operating time

t

Operating time

u

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

λ

Eigenvalue

η

Downhill factor

δ

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

σ

Control factor

μ

Friction coefficient

ɛ

Tolerance

ɛp

Perturbation

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

Periodic term in the perturbation

\({{\tau}}\)

Tangent vector

φ1, φ2

Phase angles of the unbalanced force

ϕ

Nondimensional time

ω

Rotating frequency

ωn1, ωn2

The first and the second natural frequencies

ν

Subharmonic ratio

\({\mathfrak{R}}\)

Upper triangular matrix

Γ

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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