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Static and Dynamic Analyses of a Rotor with Aerostatic Ceramic Porous Journal

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Proceedings of the 9th IFToMM International Conference on Rotor Dynamics

Part of the book series: Mechanisms and Machine Science ((Mechan. Machine Science,volume 21))

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Abstract

This work presents the numerical model of an experimental set-up where it was made an evaluation on the static characteristics of the shaft and the identification of the dynamic characteristics for an aerostatic radial porous bearing. The use of ceramic porous as journal for aerostatic bearings can improve its perform related to the wear, thermal stability, stiffness and load capacity allowing that spindles work with precision at speed above 20,000 rpm with small clearances (40 μ). In order to investigate these bearings were developed static analyses to obtain the deflection and stiffness of the support shaft, stiffness of the aerostatic porous bearing and dynamic identification for experimental set-up. The static analysis indicated stiffness of shaft and aerostatic porous bearing of 20.1 and 2.6 kN/mm, respectively. The dynamic analysis indicated that the first natural frequency of the rotor is close to 1365.9 Hz, which is much higher than the first natural frequency of the aerostatic ceramic porous bearing whose value is 775.0 Hz. One can concluded that geometrical configuration and support conditions choosen allow a robust condition to proceed the experimental tests in order to obtain dynamic characteristic of the aerostatic porous bearing.

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Abbreviations

(u,v,w) :

Fluid velocity (m/s)

[K rb ], [K ac ] :

Ball bearing, angular contact ball bearing stiffness matrices

[M g ], [K g ], [C g ], [G g ] :

Global mass, stiffness, damping, gyroscopic matrices

{q} :

Displacement vector

c :

Aerostatic bearing clearance (m)

D yy , D zz , D yz , D zy :

Damping coefficients of the aerostatic ceramic porous bearing (N s/mm)

F y :

Tangential direction force (N)

F z :

Radial direction force (N)

H :

Porous matrix thickness (ro – ri)(m)

K 1 :

Porous matrix permeability (m2)

K ac :

Angular contact ball bearing AC stiffness (N s/m)

K rb :

Ball bearing BB stiffness (N s/m)

K yy , K zz , K yz , K zy :

Stiffness coefficients of the aerostatic ceramic porous bearing (kN/mm)

Ks :

Static stiffness of the rotor system (kN/mm)

\( \varGamma \) :

Porous matrix non-dimensional parameter (12 K1 L 2t /c3 ri ln(ri/ro)

[M d ], [N d ], [G d ] :

Disk translational mass, rotational mass, gyroscopic matrices

m :

Aerostatic bearing mass (kg)

m d , I d , I p :

Disk mass, diametric inertia, polar inertia

Ps :

Supply pressure (Pa)

R :

Rotor radius (m)

r i :

Aerostatic bearing inner radius (m)

r o :

Aerostatic bearing outer radius (m)

U :

Rotational speed (rad/s)

η :

Air viscosity (N s/m2)

ρ :

Rotor and rigid disk density (kg/m3)

ω :

Excitation frequency (rad/s)

[D g ] :

Final damping matrix ([Cg + Ω[Gg])

p :

Air film pressure (N/m2)

ω sI , ω sII :

First natural frequencies of the rotor in different situations

[K cp ], [D c p] :

Aerostatic porous bearing stiffness, damping matrices

[Frf] :

Frequency response matrix

µ :

Element mass per unit of length (m)

[M], [N], [K], [G] :

Element translational mass, rotational mass, stiffness, gyroscopic matrices

x :

Axial direction coordinate (m)

z :

Radial direction coordinate (m)

t :

Time (s)

\( \tau \) :

Non-dimensional time

\( \overline{x} \) :

Non-dimensional coordinate (2x/L)

\( \overline{y} \) :

Non-dimensional coordinate (y/Lt)

ω nb :

First natural frequency of the aerostatic bearing (Hz)

V :

Linear speed of the rotor in the gap (m/s)

\( \psi \) :

Excitation non-dimensional parameter (12 η ω L 2t /c2 Ps)

\( \varLambda \) :

Rotating velocity non-dimensional parameter (6 η U Lt/c2 Ps)

r :

Element radius (m)

L t :

Aerostatic bearing perimeter (m)

\( \overline{h} \) :

Non-dimensional clearance (h/c)

Pa :

Ambient pressure (Pa)

l :

Element length (m)

\( \overline{p} \) :

Non-dimensional pressure (p/Ps)

h :

Bearing clearance (m)

L :

Aerostatic bearing width (m)

X,Y,Z :

Global inertial coordinates (m)

y :

Tangential direction coordinate (m)

References

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Acknowledgments

The Brazilian research foundations CAPES and FAPESP are gratefully acknowledge for the support given to this project. The authors would like to acknowledge Prof. Rodrigo Nicoletti from University of Sao Paulo to provide the program dedicated to rotor dynamics analysis where it was included data of the aerostatic porous bearing.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, School of Sao Carlos, University of Sao Paulo, Sao Carlos, Brazil

    Luis Renato Chiarelli & Zilda de Castro Silveira

Authors
  1. Luis Renato Chiarelli
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  2. Zilda de Castro Silveira
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Corresponding author

Correspondence to Zilda de Castro Silveira .

Editor information

Editors and Affiliations

  1. Mechanical Engineering, Politecnico di Milano, Milano, Italy

    Paolo Pennacchi

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Chiarelli, L.R., de Castro Silveira, Z. (2015). Static and Dynamic Analyses of a Rotor with Aerostatic Ceramic Porous Journal. In: Pennacchi, P. (eds) Proceedings of the 9th IFToMM International Conference on Rotor Dynamics. Mechanisms and Machine Science, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-319-06590-8_99

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  • DOI: https://doi.org/10.1007/978-3-319-06590-8_99

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-06589-2

  • Online ISBN: 978-3-319-06590-8

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