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Nonlinear transient analysis of rigid rotor mounted on externally pressurized double-layered porous gas journal bearings accounting velocity slip

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Abstract

The present work aims at estimating the journal center trajectories of a rigid rotor mounted on a hybrid, finite gas lubricated journal bearings with double-layered porous bushing using nonlinear transient analysis. To consider velocity slip in the film at the interface between film and porous region, Reynolds equation is modified using the Beavers–Joseph boundary condition. The governing equations of flow at clearance, porous regions are discretized using finite volume method. The variable values at cell-face centers are obtained using interpolation scheme of third order. Those discretized equations coupled with equations of rigid rotor motion are solved by third-order total variation diminishing Runge–Kutta scheme. Influence of velocity slip and design parameters on critical mass parameter values were explored. The observations are presented in the form of graphs that serves as a reference during the design of such bearings.

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Abbreviations

C:

Radial clearance

D :

Diameter of the bearing

\(e\) :

Bearing eccentricity

\(\bar{F}_{\text{r}}\) :

Dimensionless film force along radial direction, \(F_{\text{r}} /LDp_{{\rm a}}\)

\(\bar{F}_{\Phi }\) :

Dimensionless film force along \(\Phi\) direction, \(F_{\Phi } /LDp_{{\rm a}}\)

\(h\) :

Local film thickness

\(\overline{h}\) :

Dimensionless film thickness (h/C)

\(H\) :

Thickness of the porous bushing

\(H_{1} ,H_{2}\) :

Thickness of the fine and coarse layers respectively

\(k_{x1} ,\,k_{y1} ,\,k_{z1}\) :

Fine layer permeability coefficients along x, y, z directions respectively

\(k_{x2} ,\,k_{y2} ,\,k_{z2}\) :

Coarse layer permeability coefficients along x, y, z directions respectively

\(\bar{K}_{x1} ,\bar{K}_{z1}\) :

Dimensionless permeability coefficients, \(k_{x1} /k_{y1} ,k_{z1} /k_{y1} \,\) respectively

\(\bar{K}_{x2} ,\bar{K}_{z2}\) :

Dimensionless permeability coefficients, \(k_{x2} /k_{y2} ,k_{z2} /k_{y2}\) respectively

\(\bar{K}_{y2}\) :

Dimensionless interlayer permeability coefficient, \(k_{y2} /k_{y1}\)

\(L\) :

Length of the bearing

\(\bar{M}\) :

Mass parameter, \(MC\omega^{2} /(LDp_{{\rm a}} )\)

\(\bar{M}_{{\rm c}}\) :

Critical mass parameter, \(M_{{\rm c}} C\omega^{2} /(LDp_{{\rm a}} )\)

\(O_{{\rm b}}\) :

Bearing center

\(O_{{\rm j}}\) :

Journal center

\(p\,\) :

Film pressure

\(\bar{p}\,\) :

Dimensionless film pressure, \(p/p_{{\rm a}}\)

\(p_{{\rm a}}\) :

Ambient pressure

\(p_{{\rm s}}\) :

Supply pressure

\(\bar{p}_{{\rm s}}\) :

Dimensionless supply pressure, \(p_{{\rm s}} /p_{{\rm a}}\)

\(p\,,p_{0}\) :

Film pressure and steady-state film pressure respectively

\(\bar{p}\,,\bar{p}_{0}\) :

Dimensionless film pressure, \(p/p_{{\rm a}} ,p_{0} /p_{{\rm a}}\)

\(p^{\prime}_{1}\) :

Pressure at fine layer

\(p^{\prime}_{2}\) :

Pressure at coarse layer

\(\bar{p}^{\prime}_{1}\) :

Dimensionless pressures at the fine layer, \(p'_{1} /p_{{\rm a}}\)

\(\bar{p}^{\prime}_{2}\) :

Dimensionless pressures at the coarse layer, \(p'_{2} /p_{{\rm a}}\)

\(R\) :

Journal radius

\(W_{0}\) :

Load capacity

\(\overline{{W_{0} }}\) :

Dimensionless load capacity

\(\overline{{W_{{\rm r}} }} ,\overline{{W_{t} }}\) :

Dimensionless components of load

x, y, z :

Cartesian coordinates

\(\theta ,\bar{y},\bar{z}\) :

Dimensionless coordinates, x/R, y/H, 2z/L

\(\theta^{ *}\) :

Coordinate in circumferential direction

\(\chi\) :

Ratio of thickness of coarse layer by fine layer, \(H_{2} /H_{1}\)

α :

Slip coefficient

σ x, y, z :

Dimensionless permeability factors in x, y and z direction, (C (Kx, y, z)−1/2)

\(\zeta_{x,z}\) :

Slip function in x and z direction defined by \(\frac{{3\left( {\overline{h} \sigma_{x,z} + 2\alpha } \right)}}{{\sigma_{x,z} \overline{h} \left( {1 + \alpha \overline{h} \sigma_{x,z} } \right)}}\)

\(\zeta_{ox}\) :

Slip function defined by \(\frac{1}{{\left( {1 + \alpha \overline{h} \sigma_{x} } \right)}}\)

\(\lambda\) :

Whirl ratio, \(\omega_{p} /\omega\)

\(\beta\) :

Bearing feeding parameter, \(12R^{2} k_{y1} /HC^{3}\)

\(\phi , \phi_{o}\) :

Attitude angle, steady-state attitude angle (in degrees)

\(\varepsilon , \varepsilon_{o}\) :

Eccentricity ratio \({e \mathord{\left/ {\vphantom {e C}} \right. \kern-0pt} C}\), steady-state eccentricity ratio

\(\eta\) :

Coefficient of absolute viscosity of fluid

\(\mu_{1} ,\mu_{2}\) :

Porosities of the fine and the coarse layer

\(\mu_{{\rm f}}\) :

Coefficient of friction

\(\gamma_{p1,2}\) :

Porosity parameters of the fine and the coarse layers, \(\gamma_{p1,2} = {{\mu_{1,2} C^{2} H^{2} } \mathord{\left/ {\vphantom {{\mu_{1,2} C^{2} H^{2} } {6R^{2} k_{y1,2} }}} \right. \kern-0pt} {6R^{2} k_{y1,2} }}\)

τ :

Non-dimensional time, \(\omega t\)

\(\omega\) :

Journal rotational speed

\(\omega_{p}\) :

Frequency of journal vibration

\(\Lambda\) :

Bearing number, \(6\eta \omega /p_{{\rm s}} \left( {{C \mathord{\left/ {\vphantom {C R}} \right. \kern-0pt} R}} \right)^{2}\)

\({\Re }\) :

Gas constant

TVD:

Total variation diminishing

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Acknowledgements

The authors would like to thank Prof. Harish Hirani, Director, CSIR-CMERI for his encouragement and permission to publish the paper. The authors also would like to thank Dr. Sudipta De, CSIR-CMERI for his prolific discussions.

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No financial support is received from any funding agency to carry this research work.

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Authors and Affiliations

  1. Academy of Scientific and Innovation Research (AcSIR), Anusandhan Bhawan, 2 Rafi Marg, New Delhi, 110001, India

    Phani Kumar Mallisetty

  2. Surface Engineering and Tribology Group, CSIR-Central Mechanical Engineering Research Institute, Durgapur, 713209, India

    Phani Kumar Mallisetty, P. Samanta & N. C. Murmu

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  1. Phani Kumar Mallisetty
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Correspondence to P. Samanta.

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Mallisetty, P.K., Samanta, P. & Murmu, N.C. Nonlinear transient analysis of rigid rotor mounted on externally pressurized double-layered porous gas journal bearings accounting velocity slip. J Braz. Soc. Mech. Sci. Eng. 42, 530 (2020). https://doi.org/10.1007/s40430-020-02616-8

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