Nonlinear Dynamics

, Volume 94, Issue 4, pp 2937–2956 | Cite as

Nonlinear dynamics of a flexible rotor on tilting pad journal bearings experiencing rub–impact

  • Ebrahim Tofighi-Niaki
  • Pouya Asgharifard-Sharabiani
  • Hamid AhmadianEmail author
Original Paper


Rub–impact phenomenon occurring in hydrodynamic journal bearings is one of the main malfunctions in rotating machines and causes undesirable dynamic behavior. In order to investigate such a phenomenon, nonlinear dynamics due to rub–impact within tilting pad journal bearings supporting a flexible rotor is studied. In simulating rub–impact between the journal and the associated pads, this paper employs mixed lubrication theory along with elasto-plastic asperity contact model between pads and journal. Periodic, quasiperiodic and chaotic vibrational behaviors of system are studied by varying the unbalanced load magnitude and the rotational speed of the rotor as control parameters. Phase plane orbits, waterfall frequency response spectra and bifurcation diagrams are used to show various dynamic responses of rotor when the control parameters are changed. The Poincaré maps are used to determine the onset of irregular motions. Presented results provide better understanding of strongly nonlinear vibrations occurring due to rub–impact in TPJBs supporting industrial rotating machines.


Asperity contact Nonlinear dynamics Oil film force Rub–impact Tilting pad journal bearings 

List of symbols


Oil specific heat capacity


Shaft diameter

\({D}_{\mathrm{b}} = 2 {R}_{\mathrm{b}}, \, {D}_{\mathrm{j}} =2 {R}_{\mathrm{j}}, {D}_{\mathrm{p}} =2 {R}_{\mathrm{p}}\)

Diameters of bearing, journal and pads

\({d}_{\mathrm{e}, {k}}^{*}\)

Separation between the kth pad and journal

\({E}^{\prime }\)

Equivalent young modulus


Young modulus of shaft

\({e} = \sqrt{{X}_{\mathrm{j}}^{2} + {Y}_{\mathrm{j}}^{2}}\)

Journal eccentricity

\({\vec {{{e}}}}_{1}\)

Unit vector in the direction of journal velocity


Friction coefficient between journal and pads


Fractional rotational position of pivot


Normalized frequency

\(\{{{F}_{\mathrm{g}.}}\}, \, \{{{F}_{\mathrm{nl}.}}\}, \, \{{{F}_{\mathrm{unb}.}}\}\)

Force vectors due to gravity, nonlinearities and mass unbalance

\({F}_{{x}}, \, {F}_{{y}}\)

Components of oil film forces in horizontal and vertical directions

\({[{G}]}, \, {[{G}]}^{*}\)

Physical and modal gyroscopic matrices


Hardness of the softer material

\({H}_{\mathrm{j}}, \, {H}_{\mathrm{p}1}\)

Heat convection coefficients of oil film over surfaces of journal and pads

\({H}_{{k}} = {h}_{{k}} /{\sigma }\)

Non-dimensional oil film thickness on the kth pad

\({h}_{{k}}, \, {\bar{{h}}}_{\mathrm{T},{k}}\)

Nominal and average oil film thickness on the kth pad


Moment of inertia of each tilting pad

\({[{K}]}, \, {[{K}]}^{*}\)

Physical and modal stiffness matrices

\({L}_{\mathrm{b}}, \, {L}_{\mathrm{s}}\)

Lengths of bearing and shaft

\({[{M}]}, \, {[{M}]}^{*}\)

Physical and modal mass matrices


Moment applied to the kth pad

\({m}_{\mathrm{d}},\, {m}_{\mathrm{j}}\)

Masses of disk and journal




Number of pads

\({P}_{{k}}, \, {\bar{{P}}}_{{k}}\)

Local and mean hydrodynamic pressure distributions on the kth pad

\({P}_{\mathrm{a}, {k}}\)

Asperity contact pressure on the kth pad

\({P}_{\mathrm{e}, {k}}, \, \hbox {P}_{\mathrm{p}, {k}}\)

Asperity contact pressures due to elastic and plastic deformations on the kth pad

\(\{{{q} ({t})}\}\)

Vector of generalized coordinates


Journal surface temperature

\({T}^{{k}}, \, {T}_{\mathrm{Sp}}^{{k}}\)

Temperature distributions in oil film and on the inner surface of kth pad

\({t}_{\mathrm{d}}, {t}_{\mathrm{P}}\)

Thicknesses of disk and pads

\(\{{{U} ({t})}\}\)

Time-dependent displacement vector

\({V}_1 = {R}_{\mathrm{j}} {\omega }\)

Velocity of journal

\({V}_{2, {k}}\)

Velocity of the kth pad projected on journal velocity direction

\({w}_{\mathrm{p}}^{*} = ({\beta }^{\prime }/{\sigma }^{*}) (2 {{H}}_{\mathrm{a}}/ {E}^{\prime })^{2}\)

Plasticity index

\({x} = {R}_{\mathrm{j}} {\theta }\), y, z

Horizontal, vertical and axial coordinates

\({X}_{{\mathrm{j}}}, \, {Y}_{{\mathrm{j}}}, \, {X}_{{\mathrm{d}}}, \, {Y}_{{\mathrm{d}}}\)

Horizontal and vertical displacements of the journal and the disk


Thermo viscosity index

\({\beta }^{\prime }\)

Mean radius of asperities

\({\delta }_{{k}}\)

Tilt angle of the kth pad

\({\eta }\)

Asperity density

\({\theta }\)

Bearing circumferential coordinate

\({\theta }_{\mathrm{p}}^{{k}}, \, {\theta }_{\mathrm{l}}^{{ k}}, \, {\theta }_{\mathrm{t}}^{{k}}\)

Pivoting, leading and trailing angles of the kth pad

\({\mu }, \, {\mu }_0\)

Oil viscosities at temperatures T and \({T}_0\)

\({\sigma }_1, \, {\sigma }_2\)

Asperity height standard deviations of pads and journal

\({\sigma } = ({{\sigma }_1^2 + {\sigma }_2^2})^{0.5}\)

Root mean square (rms) of roughness standard deviations of surfaces of pads and journal

\({\sigma }^{{*}}\)

Asperity summit height standard deviation

\({\tau }_{\mathrm{a}, {{k}}^{\prime }}, \, {\bar{{\tau }}}_{{k}}\)

Tangential asperity contact stress and mean hydrodynamic shear stress on the kth pad

\(\nu \)

Poisson ratio

\({[\varPhi ]}\)

Modal matrix

\(\phi ({s})\)

Probability density function

\(\phi _{\mathrm{s}}\)

Couette shear flow factor

\(\phi _{\theta }, \, \phi _{{z}}\)

Pressure flow factors in circumferential and axial directions

\(\phi _{\mathrm{fs}}, \, \phi _{\mathrm{fp}}\)

Shear stress factors

\({\rho }, \, {\rho }_{\mathrm{s}}\)

Densities of oil and shaft

\({\omega }\)

Rotational speed of the rotor


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical EngineeringIran University of Science and TechnologyNarmak, TehranIran

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