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

Received: 13 August 2017

Accepted: 23 August 2018

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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

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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)

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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

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References

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