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

Ebrahim Tofighi-Niaki; Pouya Asgharifard-Sharabiani; Hamid Ahmadian

doi:10.1007/s11071-018-4535-0

Nonlinear Dynamics

pp 1–20 | Cite as

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

Authors
Authors and affiliations

Ebrahim Tofighi-Niaki
Pouya Asgharifard-Sharabiani
Hamid Ahmadian

Original Paper

First Online: 15 September 2018

Abstract

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.

Keywords

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

List of symbols

${\bar{{C}}}_{\mathrm{P}}$: Oil specific heat capacity
${D}_{\mathrm{s}}$: 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
${E}_{\mathrm{s}}$: 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
F: Friction coefficient between journal and pads
${f}_{\mathrm{p}}$: Fractional rotational position of pivot
${f}^{*}$: 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
${H}_{\mathrm{a}}$: 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
J: 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
${M}_{{k}}$: Moment applied to the kth pad
${m}_{\mathrm{d}},\, {m}_{\mathrm{j}}$: Masses of disk and journal
M: Preload
${N}_{\mathrm{pads}}$: 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
${T}_{\mathrm{J}}$: 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
B: 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

This is a preview of subscription content, to check access.

Appendix

The contact between the journal and the kth pad at the point C and their corresponding absolute velocities at this point are displayed in Fig. 22. Since ${e}\ll {R}_{\mathrm{j}}$, the journal absolute velocity can be approximated as:

$$\begin{aligned} {\vec {V}}_{1}\cong {\vec {\omega }} \times {\overrightarrow{{O_{\mathrm{J}}C}}}. \end{aligned}$$

(A.1)

Also, the absolute velocity of the kth pad at the contact point C can be expressed as:

$$\begin{aligned} {\vec {V}}_{2,k} = \frac{\mathrm{d}{\vec {\delta }}_{k}}{\mathrm{d}t}\times {\overrightarrow{PC}}, \end{aligned}$$

(A.2)

where the components of vectors ${\overrightarrow{{O_{{\mathrm{J}}} C}}}$ and ${\overrightarrow{PC}}$ in planar Cartesian coordinates are represented as:

$$\begin{aligned} {\overrightarrow{O_{{\mathrm{J}}}C}}= & {} R_{j} \left( \cos \theta \vec {\imath } + \sin \theta \vec {{{J}}}\right) , \end{aligned}$$

(A.3)

$$\begin{aligned} {\overrightarrow{PC}}= & {} |{\overrightarrow{PC}}| \left( \cos (\gamma -\theta _{p}) {\vec {\imath }} + \sin (\gamma -\theta _{p}) {\vec {\imath }}\right) . \end{aligned}$$

(A.4)

Fig. 22
A display of the contact between the journal and the kth pad

Applying ${e}\ll {R}_{\mathrm{j}}$, the magnitude of vector ${\overrightarrow{{PC}}}$ can be obtained as follows:

$$\begin{aligned} |{\overrightarrow{{PC}}}|= & {} \sqrt{(|{\overrightarrow{{HP}}}|)^{2} + (|{\overrightarrow{{HC}}}|)^{2}}, \end{aligned}$$

(A.5)

$$\begin{aligned} |{\overrightarrow{{HP}}}|= & {} |{\overrightarrow{O_{{\mathrm{J}}} P}}| - |{\overrightarrow{O_{{\mathrm{J}}} H}}| \cong R_\mathrm{b} + t_\mathrm{p} - R_\mathrm{j} \cos ({\theta - \theta _\mathrm{p}}), \end{aligned}$$

(A.6)

$$\begin{aligned} |{\overrightarrow{HC}}|= & {} R_\mathrm{j} \sin ({\theta - \theta _\mathrm{p}}). \end{aligned}$$

(A.7)

The sine law is applied to the triangle ${O}_{\mathrm{J}} {PC}$, as shown in Fig. 22, and the direction of vector ${\overrightarrow{{PC}}}$ may be determined by:

$$\begin{aligned} \frac{|{\overrightarrow{{PC}}}|}{\sin ({\theta - \theta _\mathrm{p}})} = \frac{|{{\overrightarrow{O_{{\mathrm{J}}} C}}}|}{\sin (\gamma )}. \end{aligned}$$

(A.8)

In order to obtain the direction of friction force applied to the journal, the sign of algebraic difference between the journal absolute velocity and the projected pad absolute velocity on the journal velocity direction has to be determined. The unit vector along the journal absolute velocity at point C is represented as:

$$\begin{aligned} {\vec {e}}_{1} = -\sin \theta {\vec {\imath }} + \cos \theta \vec {{{J}}}. \end{aligned}$$

(A.9)

Finally, exerting forces on the journal and the moment on the kth pad resulting from the hydrodynamic and asperity contact forces in normal and tangential directions are expressed by:

$$\begin{aligned} F_{x}= & {} -\sum \limits _{k=1}^{N_{\mathrm{pads}}} \int _{-\frac{L}{2}}^{\frac{L}{2}} \int _{\theta _{\mathrm{s}}}^{\theta _{\mathrm{e}}} ((P_{\mathrm{a},k}+{\bar{P}}_{k} ){\mathrm{cos}}\theta \nonumber \\&\quad -\,{\mathrm{sign}}( ( {\vec {V}}_{1}-( {\vec {V}}_{2,k}\cdot {\vec {e}}_{1} ){\vec {e}}_{1} )\cdot {\vec {e}}_{1} )( \tau _{\mathrm{a},k}+{\bar{\tau }}_{k} ){\mathrm{sin}}\theta ) \nonumber \\&\qquad R_{j}\mathrm{d}\theta \mathrm{d}z, \end{aligned}$$

(A.10)

$$\begin{aligned} F_{y}= & {} -\sum \limits _{k=1}^{N_{\mathrm{pads}}} \int _{-\frac{L}{2}}^{\frac{L}{2}} \int _{\theta _{\mathrm{s}}}^{\theta _{\mathrm{e}}} ( ( P_{\mathrm{a},k}+{\bar{P}}_{k} ){\mathrm{sin}}\theta \nonumber \\&\quad +\,{\mathrm{sign}}( ( {\vec {V}}_{1}-( {\vec {V}}_{2,k}\cdot {\vec {e}}_{1} ){\vec {e}}_{1} )\cdot {\vec {e}}_{1} )( \tau _{\mathrm{a},k}+{\bar{\tau }}_{k} ){\mathrm{cos}}\theta ) \nonumber \\&\qquad R_{j}\mathrm{d}\theta \mathrm{d}z, \end{aligned}$$

(A.11)

$$\begin{aligned} M_{k}= & {} -\int _{-\frac{L}{2}}^{\frac{L}{2}} \int _{\theta _{\mathrm{s}}}^{\theta _{\mathrm{e}}} \left[ -( R_{\mathrm{j}}+t_{\mathrm{p}} )( \theta -\theta _{\mathrm{p}}^{k} )( P_{\mathrm{a},k}+{\bar{P}}_{k} ) \right. \nonumber \\&\quad \left. -\,{\mathrm{sign}}( ( {\vec {V}}_{1}-( {\vec {V}}_{2,k}\cdot {\vec {e}}_{1} ){\vec {e}}_{1} )\cdot {\vec {e}}_{1}) t_{\mathrm{p}}( \tau _{\mathrm{a},k}+{\bar{\tau }}_{k} ) \right] \nonumber \\&\qquad {\mathrm{cos}}( \theta -\theta _{\mathrm{p}})R_{j}\mathrm{d}\theta \mathrm{d}z; \nonumber \\&\qquad \, k=1,\ldots ,\, N_{\mathrm{pads}}. \end{aligned}$$

(A.12)

References

1.
Chang-Jian, C.-W., Chen, C.-K.: Chaos and bifurcation of a flexible rub-impact rotor supported by oil film bearings with nonlinear suspension. Mech. Mach. Theory 42, 312–333 (2007)CrossRefGoogle Scholar
2.
Chang-Jian, C.-W., Chen, C.-K.: Non-linear dynamic analysis of rub-impact rotor supported by turbulent journal bearings with non-linear suspension. Int. J. Mech. Sci. 50, 1090–1113 (2008)CrossRefGoogle Scholar
3.
Chang-Jian, C.-W., Chen, C.-K.: Chaos of rub-impact rotor supported by bearings with nonlinear suspension. Tribol. Int. 42, 426–439 (2009)CrossRefGoogle Scholar
4.
Chang-Jian, C.-W., Chen, C.-K.: Nonlinear analysis of a rub-impact rotor supported by turbulent couple stress fluid film journal bearings under quadratic damping. Nonlinear Dyn. 56, 297–314 (2009)CrossRefGoogle Scholar
5.
Chang-Jian, C.-W., Chen, C.-K.: Couple stress fluid improve rub-impact rotor–bearing system—nonlinear dynamic analysis. Appl. Math. Model. 34, 1763–1778 (2010)MathSciNetCrossRefGoogle Scholar
6.
Wang, J., Zhou, J., Dong, D., Yan, B., Huang, C.: Nonlinear dynamic analysis of a rub-impact rotor supported by oil film bearings. Arch. Appl. Mech. 83, 413–430 (2013)CrossRefGoogle Scholar
7.
Xiang, L., Gao, X., Hu, A.: Nonlinear dynamics of an asymmetric rotor–bearing system with coupling faults of crack and rub-impact under oil-film forces. Nonlinear Dyn. 86, 1057–1067 (2016)CrossRefGoogle Scholar
8.
Cao, J., Ma, C., Jiang, Z., Liu, S.: Nonlinear dynamic analysis of fractional order rub-impact rotor system. Commun. Nonlinear Sci. Numer. Simul. 16, 1443–1463 (2011)CrossRefGoogle Scholar
9.
Abu Arqub, O.: Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions. Comput. Math. Appl. 73, 1243–1261 (2017)MathSciNetCrossRefGoogle Scholar
10.
Arqub, O.A., El-Ajou, A., Momani, S.: Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations. J. Comput. Phys. 293, 385–399 (2015)MathSciNetCrossRefGoogle Scholar
11.
El-Ajou, A., Arqub, O.A., Momani, S.: Approximate analytical solution of the nonlinear fractional KdV–Burgers equation: a new iterative algorithm. J. Comput. Phys. 293, 81–95 (2015)MathSciNetCrossRefGoogle Scholar
12.
Spikes, H.A.: Mixed lubrication—an overview. Lubr. Sci. 9, 221–253 (1997)CrossRefGoogle Scholar
13.
Patir, N., Cheng, H.S.: An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication. J. Lubr. Technol. 100, 12 (1978)CrossRefGoogle Scholar
14.
Patir, N., Cheng, H.S.: Application of average flow model to lubrication between rough sliding surfaces. J. Lubr. Technol. 101, 220 (1979)CrossRefGoogle Scholar
15.
Greenwood, J., Williamson, J.B.P.: Contact of nominally flat surfaces. Proc. R. Soc. A Math. Phys. Eng. Sci. 295, 300–319 (1966)CrossRefGoogle Scholar
16.
Johnson, K.L., Greenwood, J.A., Poon, S.Y.: A simple theory of asperity contact in elastohydro-dynamic lubrication. Wear 19, 91–108 (1972)CrossRefGoogle Scholar
17.
Yamaguchi, A., Matsuoka, H.: A mixed lubrication model applicable to bearing/seal parts of hydraulic equipment. J. Tribol. 114, 116 (1992)CrossRefGoogle Scholar
18.
Ai, X., Cheng, H.S., Hua, D., Moteki, K., Aoyama, S.: A finite element analysis of dynamically loaded journal bearings in mixed lubrication. Tribol. Trans. 41, 273–281 (1998)CrossRefGoogle Scholar
19.
Kazama, T., Narita, Y.: Mixed and fluid film lubrication characteristics of worn journal bearings. Adv. Tribol. 2012, 1–7 (2012)CrossRefGoogle Scholar
20.
Sander, D.E., Allmaier, H., Priebsch, H.H., Witt, M., Skiadas, A.: Simulation of journal bearing friction in severe mixed lubrication—validation and effect of surface smoothing due to running-in. Tribol. Int. 96, 173–183 (2016)CrossRefGoogle Scholar
21.
Leighton, M., Morris, N., Rahmani, R., Rahnejat, H.: Surface specific asperity model for prediction of friction in boundary and mixed regimes of lubrication. Meccanica 52, 21–33 (2017)CrossRefGoogle Scholar
22.
Varney, P., Green, I.: Rotordynamic analysis of rotor–stator rub using rough surface contact. J. Vib. Acoust. 138, 021015 (2016)CrossRefGoogle Scholar
23.
Asgharifard-Sharabiani, P., Ahmadian, H.: Nonlinear model identification of oil-lubricated tilting pad bearings. Tribol. Int. 92, 533–543 (2015)CrossRefGoogle Scholar
24.
Whitehouse, D.J., Archard, J.F.: The properties of random surfaces of significance in their contact. Proc. R. Soc. A Math. Phys. Eng. Sci. 316, 97–121 (1970)CrossRefGoogle Scholar
25.
Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1985)CrossRefGoogle Scholar
26.
Greenwood, J.A., Tripp, J.H.: The contact of two nominally flat rough surfaces. Proc. Inst. Mech. Eng. 185, 625–633 (1970)CrossRefGoogle Scholar

Copyright information

Authors and Affiliations

Ebrahim Tofighi-Niaki
- 1
Pouya Asgharifard-Sharabiani
- 1
Hamid Ahmadian
- 1
Email authorView author's OrcID profile

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

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

Abstract

Keywords

List of symbols

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite article