Investigations on characteristics of micro/meso scale gas foil journal bearings for 100–200

TECHNICAL PAPER

Investigations on characteristics of micro/meso scale gas foiljournal bearings for 100–200 W class micro power systems usingfirst order slip velocity boundary conditions and the effectiveviscosity model

Skylab P. Bhore • Ashish K. Darpe

Received: 7 June 2012 / Accepted: 21 July 2012 / Published online: 11 August 2012

� Springer-Verlag 2012

Abstract In this paper, a modified compressible Rey-

nolds equation for micro/meso scale gas foil journal bear-

ings considering first order slip and effective viscosity

under rarefied flow conditions is presented. The influence

of rarefaction effect on the load carrying capacity, attitude

angle, speed and frequency dependent stiffness and

damping coefficients, modal impedance, natural frequen-

cies and unbalance response is studied. From numerical

analysis, it has been found that there is significant change

in all the static and dynamic characteristics predicted by

the no-slip model and model with effective viscosity. There

is also a considerable difference between the values pre-

dicted by a model with effective viscosity and a model

without effective viscosity. For a given eccentricity ratio,

the influence of effective viscosity on load carrying

capacity and attitude angle is more significant for the

typical operating speed range of micro/meso scale gas

turbines. The influence of effective viscosity decreases

with increase in compliance of bearing structure. Similarly,

the influence of effective viscosity on frequency dependent

stiffness and damping coefficients increases with excitation

frequency ratio. Significant change in natural frequency,

modal impedance and unbalance response for model with

no slip and slip with effective viscosity is observed. The

influence of effective viscosity is found to be significant

with increase in Knudsen number.

List of symbols

a Accommodation coefficient

c Damping (Ns/m)

C Bearing radial clearance (m)

CB Dimensionless bump damping coefficient (gKB/

m)

d Gas molecule diameter (m)

e Unbalance eccentricity (m)

E Young’s modulus for bump foil (N/m2)

f (Kn) Effective viscosity coefficient

feX, feY External force in X and Y direction (N)

fX, fY Bearing reaction force in X and Y direction (N)

h Dimensionless film thickness (h/C)

h Gas film thickness (m)

k Stiffness (N/m)

ks_modal Critical modal stiffness (N/m)

Kf Bump foil stiffness (N/m3)

K Dimensionless bearing stiffness coefficient

KB Dimensionless bump stiffness coefficient (1/a)

Kn Knudsen number

lo Bump half length (m)

L Bearing length (m)

mr Rotor mass (kg)

NA Avogadro’s number

Ns Natural frequency (rpm)

pa Atmospheric pressure (N/m2)

p Dimensionless pressure (p/pa)

R Bearing radius (m)

R* Universal gas constant

So Bump pitch (m)

tb Bump foil thickness (m)

ts Dimensionless time (xt)

T Temperature (oK)

U0 Velocity (m/s)

z Dimensionless axial coordinate (z/R)

a Dimensionless compliance of bump foil [(2paSo/

CE)(lo/tb)3(1-t2)]

S. P. Bhore (&) � A. K. Darpe

Department of Mechanical Engineering, Indian Institute

of Technology Delhi, New Delhi 110016, India

e-mail: [email protected]

123

Microsyst Technol (2013) 19:509–523

DOI 10.1007/s00542-012-1639-1

d Dimensionless foil deflection

x Angular velocity of shaft (rad/s)

xs Excitation frequency (rad/s)

ø Attitude angle

øp Rarefaction coefficient

eX, eY Eccentricity ratios in X and Y direction

f Dimensionless bearing load parameter

g Loss factor for bump foil material

K Bearing number [(6lx/pa)(R/C)2]

q Density of gas (kg/m3)

l Viscosity of gas (Ns/m2)

m Excitation frequency ratio (xs/x)

k Molecular mean free path (m)

t Poisson’s ratio for bump foil

w Surface correction factor

Subscript

0 Static

X and Y Direction

eff Effective

1 Introduction

Recent developments in micro/meso scale gas turbines

have given new promises to the world. These gas turbines

can be used in many potential applications of micro/meso

scale power generation units. These days, the use of por-

table electronic devices has tremendously increased in our

society. Several smart and small size machines have been

developed to run on batteries. The batteries are highly

ineffective and having low energy density (energy per unit

mass of the system). It cannot supply continuous power as

it needs frequent charging and time of charging can be

considerable. It is envisaged that the micro and meso scale

gas turbines can be used as continuous power supply for

several applications viz. micropower generation for robots,

micro and nano satellites, micro systems, hybrid energy

system with fuel cells, unmanned air vehicles (UAVs), etc.

(Bhore and Darpe 2012).

The silicon based micro gas turbine was developed in

MIT with lithographic manufacturing technologies. The

design of micro gas bearings used in such turbine is chal-

lenging and inherently it requires a very small length to

diameter (L/D) ratio due to structural properties of silicon.

Silicon is a brittle material and has limited life during rotor

vibration and crashes. It is not a very desirable bearing

material and exhibits high wear rates, high coefficients of

friction and poor stiction characteristics (Epstein 2003;

Breuer 2001). To circumvent the problems associated with

silicon and to enhance the reliability of system, few

attempts to manufacture such systems in metals was car-

ried out (Isomura et al. 2005, 2006; Peirs et al. 2003).

Micro-milling, micro-elctro discharge machining, micro-

molding and more recently X-ray lithography and LIGA

processes have been used to manufacture the gas turbine in

metals (ceramics) for 100–200 W class micro power sys-

tem (Lee et al. 2007; Salehi et al. 2007; Kim et al. 2009).

This gives more freedom in manufacturing and helps to

reduce the operating speed below one million rpm with

little increased impeller size of 10–15 mm and bearing size

of 5–7 mm. The gas turbine with the above mentioned size

is called meso scale gas turbine or palm-sized gas turbine.

The support for the high speed rotor also needs special

consideration compared to conventional gas turbine; the

operating speed of meso scale gas turbine is very high and

ranges between 500,000 and 1,000,000 rpm. The temper-

ature inside the turbine usually reaches to 800–1,300 K.

For such critical operating conditions, gas foil bearing is

considered as the best candidate. The foil bearing com-

prises compliant structure and its shape and size can be

designed to achieve desired dynamic properties. Recently,

Kim et al. (2009) designed and used nonconventional

methods such as X-ray lithography, ultraviolet lithography,

electroplating and molding to manufacture the meso scale

foil bearings in metals. They carried out orbit simulation by

solving the Reynolds equation, bump dynamics and rotor

dynamic equations of motion simultaneously. Subsequent

rotordynamic analysis is carried out to investigate the

influence of preload offset distance and loss factor. Salehi

et al. (2007) have reported the operation of a meso scale

gas turbine simulator supported on foil bearings at speed of

700,000 rpm. The test was carried out with specially

designed meso scale gas foil journal and thrust bearings.

Lee et al. (2007) have performed the rotordynamic analysis

and stability analysis of a meso scale rotor system sup-

ported on gas foil bearings. They operated the rotor up to

300,000 rpm and found the foil journal and thrust bearings

had enough load capacity, stiffness and damping charac-

teristics to maintain the stability of the system. Vleugels

et al. (2006) presented stability analysis using a simple foil

bearing model without any damping effects due to inter-

action between foil and bearing housing. The linear per-

turbation method is used to evaluate stiffness and damping

coefficients and subsequently stability analysis is carried

out resulting in the stability maps.

The main drawback of these investigations is that the

researchers have not considered slip velocity in the math-

ematical model of the bearings. In all the above investi-

gations, they have neglected the possibility of having

higher Knudsen numbers when the characteristic film

thickness which influences the slip effect becomes very

small. Knudsen number Kn (the ratio between gas molec-

ular mean free path and gas lubrication film thickness) is

important parameter and used for identifying the flow

regimes. During the operating temperature (800–1,300�K)

510 Microsyst Technol (2013) 19:509–523

123

of micro/meso scale gas turbine, the molecular mean free

path of the gas molecules increases and gas becomes rare.

Thus, velocity slip occurs between journal surface and

gaseous fluid in contact with it. As per the Knudsen

number, the flow regimes are categorized as Kn \ 0.001

no slip (continuum flow); 0.001 \ Kn \ 0.1 first order

slip boundary condition (rarefied flow); 0.1 \ Kn \ 10

2nd order or higher order slip boundary condition (tran-

sition flow region); Kn [ 10 free molecular flow. The

bearing characteristics viz. load capacity, stiffness and

damping coefficients at high temperature differ from those

at room temperature (Dellacorte 1997). When heavy load

is applied at high temperature, the slip flow effect can be

large as the molecular mean free path increase with

temperature (Gad-el-Hak 2001). In case of micro/meso

scale bearing, the preload is essential for stable operation

of rotor therefore rotor rotates at extreme eccentricity and

thus characteristic dimension of the flow geometry

(minimum film thickness) becomes small. Similarly, by

virtue of its use in applications such as micro scale

unmanned air vehicle (MUAV), micro satellites and

micro power generation unit, the rotor supported on

MMGFBs may be exposed to the large impulsive forces.

Such forces may be encountered during maneuvering, due

to impact or free fall on the ground. Resulting minimum

film thickness during such transient operation will be

comparable to the mean free path of the gas molecules at

its operating temperature. The fluid flow in MMGFB is

different from those in macro scale foil bearing and

cannot be directly predicted from the conventional con-

tinuum flow compressible Reynolds equation. Therefore,

the use of continuum modeling of compressible Reynolds

equation for the prediction of MMGFB may be ques-

tionable. The compressible Reynolds equation needs to be

extended to include the slip effect or the effect of gas

rarefaction.

The predictions and analysis on rarefied gas or slip

velocity boundary condition has been reported in the lit-

erature. Shen et al. (2007) reported that the 2nd order and

1.5th order slip velocity boundary condition models are

based on mathematical approach and concluded that the

first order slip velocity boundary condition represents

actual physical mechanism. According to gas kinetic

theory, Veijola and Turowski (2001) proposed an

approximate expression of effective viscosity in modeling

the damping for laterally moving microstructures with gas

rarefaction, compared with experimental data and found

good agreement. Chan and Sun (2003) implemented the

gas rarefaction effect with the effective viscosity formula

proposed by Veijola and Turowski (2001) in modeling

ultra-thin gas film bearings. Zhang et al. (2009) have

studied the static characteristics of plain micro gas journal

bearings based on effective viscosity model. They studied

the variation of load carrying capacity and attitude angle

with bearing number for assumed range of Knudsen

numbers. They found that the model of first order slip

with effective viscosity is in better agreement with FK

model (Fukui and Kaneko 1988). The FK model is con-

sidered as accurate model which is derived from linear

Boltzmann equation to describe the kinetics of rarefied

gas. However, actual operating molecular mean free path

at higher temperature and characteristic film thickness that

defines the Knudsen number has not been considered. In

addition, the influence of effective viscosity model on

stiffness and damping coefficients have not been addres-

sed. For macro scale gas foil bearings, Lee et al. (2002)

have studied the influence of first order slip flow on

steady state load capacity, speed dependent stiffness and

damping coefficients. However, the gas foil bearing

stiffness and damping coefficients are amplitude and fre-

quency dependent because of its compliant structure.

Therefore influence of rarefaction effect on amplitude and

frequency dependent stiffness and damping coefficients

need to be studied. In addition, the modal analysis and

unbalance response of rotor bearing system considering

rarefaction effect is necessary. Lee et al. (2002) have

focused on macro scale gas foil bearing. For micro/meso

scale turbomachinery applications and its critical operat-

ing conditions an updated rarefaction model is essential.

For designing MMGFB, the influence of rarefaction effect

on frequency dependent dynamic characteristics needs to

be investigated.

In this paper, a modified compressible Reynolds equa-

tion for MMGFB considering the effective viscosity or

change in viscosity based on first order slip velocity

boundary condition is derived. Using this modified Rey-

nolds equation, perturbation equations for speed and fre-

quency dependent dynamic coefficients are derived. The

modal analysis of rotor bearing system using linear

dynamic coefficients with rarefaction effect is carried out.

The unbalance response of rotor is obtained by simulta-

neously solving modified unsteady Reynolds equation,

unsteady foil deflection equation and equations of motion,

using time marching technique. The modified Reynolds

equation and perturbed equations are solved by control

volume formulation (CVF). The influence of effective

viscosity on static as well as dynamic characteristics of

bearing as a function of Knudsen number Kn, compliance

of bearing compliant structure a, speed N and excitation

frequency ratio t are studied. The variations in load

capacity, attitude angle, speed and frequency dependent

dynamic coefficients, modal impedance, unbalance

response with Fast Fourier Transforms (FFT) are analyzed.

Comparative study for three different models namely, no

slip, first order slip without effective viscosity and first

order slip with effective viscosity is carried out.

Microsyst Technol (2013) 19:509–523 511

123

2 Modified Reynolds equation with first order slip

velocity and effective viscosity

For a typical flow geometry of a gas bearing shown in

Fig. 1, under the assumptions of smaller film thickness

with comparison to other dimensions and small inertia

effect the reduced Navier–Stokes equations (Hamrock et al.

1994) can be written as,

op

om¼ o

onl

ou

on

� �ð1Þ

op

oz¼ o

onl

ow

on

� �: ð2Þ

The gas flows with velocity u and w in m and z direction

respectively (see Fig. 1).

For incorporating slip or rarefaction effect, a first order

slip velocity boundary conditions (Burgdorfer 1959) are.

uðn ¼ hÞ ¼ �wkou

on

��n¼h

uðn ¼ 0Þ ¼ U0 þ wkou

on

��n¼0

wðn ¼ hÞ ¼ �wkow

on

��n¼h

wðn ¼ 0Þ ¼ wkow

on

��n¼0

ð3Þ

In the above equations, w is the surface correction factor

and defined as (2-a)/a. In this a is accommodation coefficient

which represents the tangential momentum exchange of the

gas molecules impinging on the journal surface. k is the

molecular mean free path and is the function of temperature

and pressure. The molecular mean free path k is defined as,

k ¼ R�Tffiffiffi2p

pd2NApð4Þ

Integrating Eqs. 1, 2 twice, with boundary conditions

given in Eq. 3, the velocity distributions can be obtained as,

u ¼ 1

2lop

omðn2 � nh� wkhÞ þ U0 1� wkþ n

hþ 2wk

� �;

w ¼ 1

2lop

ozðn2 � nh� wkhÞ:

ð5Þ

Substituting velocity distribution u and w in following

continuity equation

Zh

0

oqotþ o

omðquÞ þ o

ozðqwÞ

� �dn ¼ 0: ð6Þ

Solving Eq. 6, modified Reynolds equation with first

order slip velocity can be written as

o

om

op

omqðh3 þ 6wkh2Þ

� �þ o

oz

op

ozqðh3 þ 6wkh2Þ

� �

¼ 6lU0

oðqhÞomþ 12l

oðqhÞot

: ð7Þ

The effective viscosity coefficient given by Veijola and

Turowski (2001) is defined as

leff ¼ f ðKnÞl ¼ l

1þ 2Knþ 0:2Kn0:788e�Kn=10: ð8Þ

Substituting Eq. 8 into Eq. 7, dimensionless modified

Reynolds equation with first order slip and effective

viscosity for ideal and isothermal gas assumption can be

written as,

o

oh/p�p�h3 o�p

oh

� �þ o

o�z/p�p�h3 o�p

o�z

� �

¼ f ðKnÞK o

ohð�p�hÞ þ f ðKnÞ2Km

o

otsð�p�hÞ; ð9Þ

where,

�p ¼ p

pa; h ¼ m

R; �z ¼ z

R; �h ¼ h

C; ts ¼ xst;

m ¼ xs

x; K ¼ 6lx

pa

R

C

� �2

:

ð10Þ

The rarefaction coefficient /Pcan be written as,

/p ¼ 1þ 6wKn: ð11Þ

The Knudsen number Kn is defined as,

Kn ¼ kh¼ R�Tffiffiffi

2p

pd2NAph: ð12Þ

For co-ordinate system of MMGFB in Fig. 2, the film

thickness is

�h ¼ 1� eX cosðhÞ � eY sinðhÞ þ dðh; �zÞ: ð13ÞFor steady state, the deflection of foil is written as,

dðh; �zÞ ¼ a �pðh; �zÞ � 1ð Þ: ð14Þ

The dimensionless compliance of the bump foil given by

Heshmat et al. (1983) is

a ¼ 2paSo

CE

lotb

� �3

1� t2�

: ð15Þ

The compliance of foil bearing is a function of material

properties and geometrical dimensions of bump foil.

For unsteady state, the deflection of foil with equivalent

viscous damping given by Kim (2007) is written as,Fig. 1 Typical flow geometry of gas bearing


123

dts ¼ CB

CB þ KBDtsdts�Dts þ Dts

CB þ KBDts�ptsðh; �zÞ; ð16Þ

where,

KB ¼1

a; CB ¼

gm

�KB: ð17Þ

It may be noted that the bearing is open to the

atmosphere at both ends and in addition the bump foil

and top foil weld point is under atmospheric pressure.

Therefore, the appropriate boundary conditions are,

�pð�z ¼ �L=2; hÞ ¼ 1;

�pð�z; h ¼ pÞ ¼ 1;

�pð�z; h ¼ 0Þ ¼ �pð�z; h ¼ 2pÞ:ð18Þ

From the pressure field �pðh; zÞ, the load carrying

capacities FX and FY and dimensionless load parameters

fX and fY along X and Y directions are obtained as,

fX ¼ �paR2

Z2p

0

ZL=R

0

�pðh; zÞ cosðhÞdhdz;

fY ¼ �paR2

Z2p

0

ZL=R

0

�pðh; zÞ sinðhÞdhdz;

fX ¼fX

2paRL; fY ¼

fY

2paRL:

ð19Þ

The resultant load parameter and attitude angle are given as,

f ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif2

X þ f2Y

q; / ¼ tan�1 � fY

fX

� �: ð20Þ

3 Stiffness and damping coefficients

The speed and frequency dependent dynamic coefficients

are obtained by using linear perturbation method in the

approach of Kim and San Andres (2008). The method is

extended by adding rarefaction effect and effective vis-

cosity in the perturbed equations. In the perturbation

method, the rotor is excited with small amplitude (DeX,

DeY) of harmonic motion with frequency xs around a static

equilibrium point (eX0, eY0). The perturbed pressure, film

thickness and rarefaction coefficient can be written as,

�p ¼ �p0 þ D�peits ; ð21Þ�h ¼ �h0 þ D�heits ; ð22Þ

/p ¼ /p0 þ

o/p

o�p

� �0

D�peits þ o/p

o�h

� �0

D�heits ;

where;

�p0 ¼ p0=pa;

�h0 ¼ 1� eX0 cosðhÞ � eY0 sinðhÞ þ d�p0a;

/p0 ¼ 1þ 6aKn;

D�p ¼ �pXDeX þ �pYDeY ;

D�h ¼ DeX½� cosðhÞ þ �ad�pX� þ DeY ½� sinðhÞ þ �ad�pY �;

a ¼ pa

CKf; �a ¼ pa

C½Kf ð1þ igÞ� ;

d�p0 ¼1

L

ZL

0

ð�p0 � 1Þdz; d�pX ¼1

L

ZL

0

�pXdz; d�pY ¼1

L

ZL

0

�pY dz;

ð23Þ

where, a and �a represents compliance coefficients for static

and dynamic conditions respectively. Eqs. 21–23 are

substituted into modified Reynolds Eq. 9. After neglecting

higher order terms, yields zeroth and two first order

equations for static and perturbed pressure fields.

Zeroth-order equation,

o

ohf ðKn0ÞK�p0

�h0 � /p0 �p0

�h30

o�p0

oh

� �

þ o

o�z�/p

0�p0�h3

0

o�p0

o�z

� �¼ 0: ð24Þ

First-order equations along X and Y direction,

o

ohf Knð ÞK�pC

�h0 � /p0�p0

�h30

o�pC

oh

� �þ o

o�z�/p

0 �p0�h3

0

o�pC

o�z

� �

¼ o

oh/p

0�pC�h3

0

o�p0

oh� /p

0�p03�h20PC

o�p0

ohþ /p

0�p03�h20�ad�pC

o�p0

oh

�

þ o/P

o�p

� �0

�pC�p0�h3

0

o�p0

oh� o/P

o�h

� �0

PC � �ad�pCð Þ�p0�h3

0

o�p0

oh

þf Knð ÞK�p0 PC � �ad�pCð ÞÞ

þ o

o�z/p

0�pC�h3

0

o�p0

o�z� /p

0�p03�h20PC

o�p0

o�zþ /p

0�p03�h20�ad�pC

o�p0

o�z

�

þ o/P

o�p

� �0

�pC�p0�h3

0

o�p0

o�z� o/P

o�h

� �0

PC � �ad�pCð Þ�p0�h3

0

o�p0

o�z

�

� f Knð Þ2Kmi ��p0PC þ �p0�ad�pC þ �pC�h0ð Þ;

ð25Þ

Fig. 2 Co-ordinate system for MMGFB


123

Where, C ¼ X; Y and PX ¼ cos h; PY ¼ sin h:For solving Eqs. 24–25, the boundary conditions are,

�pð�z; h ¼ 0Þ ¼ �pð�z; h ¼ 2pÞ;�pð�z ¼ �L=2; hÞ ¼ 1;

�pð�z; h ¼ pÞ ¼ 1;

�pXð�z ¼ �L=2; hÞ ¼ �pYð�z ¼ �L=2; hÞ ¼ 0;

�pXð�z; h ¼ pÞ ¼ �pYð�z; h ¼ pÞ ¼ 0:

ð26Þ

For a given rotating speed N and eccentricity ratio e,zeroth order equation is solved and pressure �p0 and film

thickness �h0 are obtained. Substituting �p0 and �h0 in first

order Eq. 25, the perturbed pressure �pX and�pY are obtained.

Both zeroth order and first order perturbed equations are

solved by using control volume formulation (CVF)

discussed in Sect. 6. The perturbed pressure �pX and �pY

are in complex number form and comprises real and

imaginary parts. The stiffness and damping coefficients are

determined by integrating perturbed pressures �pX and �pY

over bearing surface as follows,

where, WC

KXX KXY

KYX KYY

� ¼ kxx kxy

kyx kyy

� ;

where, WCxs

CXX CXY

CYX CYY

� ¼ cxx cxy

cyx cyy

� :

4 Modal analysis of rotor bearing system

The eigenvalue problem for rotor supported on foil bear-

ings is formulated similar to the approach of Pan and Kim

(2006). The stiffness and damping coefficients obtained by

perturbation method in previous section are used in the

modal analysis. The natural frequencies are predicted for

three different models viz. no- slip, slip without leff and

slip with leff.

For a Jeffcott rotor having two degrees of freedom and

with no external force, the equations of motion in matrix

form can be written as,

mr 0

0 mr

� D €XD €Y

� �þ BXX BXY

BYX BYY

� DXDY

� �¼ 0

0

� �ð29Þ

where, BIJ ¼ kIJ þ jcIJxs is impedance of foil bearing in

which stiffness and damping coefficients varies with

excitation frequency.

With assumption of nonzero eigen modes fDXm

DYmgTG¼bacward=forward , BG ¼ �mrk and k ¼ ða� jxsÞ2, a

eigenvalue problem can be written as,

BXX � BG BXY

BYX BYY � BG

� DXm

DYm

� �G¼backward=forward

¼ 0

0

� �: ð30Þ

After solving Eq. 30, modal impedance for different

modes can be written as,

BG¼backward=forward ¼BXX þ BYY

2

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBXX � BYY

2

� �2

þBXYBYX

s: ð31Þ

The modal impedance is a complex number; the real and

imaginary part represents modal stiffness and modal damping

respectively. When modal damping becomes zero, the

corresponding excitation frequency ratio and modal

stiffness is called as meigen and critical modal stiffness

respectively. Thus from Eq. 31, it can be written as,

mrx2s eigen ¼ ks modal: ð32Þ

For a given speed x and rotor mass mr, the modal

impedances for different values of excitation frequency

ratio m are obtained from Eq. 31 and plotted as shown in

kxx kxy

kyx kyy

� ¼ � WR

2CL

Z 2p

0

Z L=R

0

Reð�pXÞ cosðhÞ Reð�pYÞ cosðhÞReð�pXÞ sinðhÞ Reð�pYÞ sinðhÞ

� dhdz; ð27Þ

cxx cxy

cyx cyy

� ¼ � WR

2CLxs

Z 2p

0

Z L=R

0

Imð�pXÞ cosðhÞ Imð�pYÞ cosðhÞImð�pXÞ sinðhÞ Imð�pYÞ sinðhÞ

� dhdz; ð28Þ


123

Fig. 11a, b. The critical modal stiffness ks modal

corresponding to meigen are obtained from Fig. 11a, b. The

natural frequency of rotor bearing system can be obtained

from Eq. 32.

5 Unbalance response of rotor bearing system

Generally, some amount of unbalance in the rotor bearing

system is always present due to the manufacturing con-

straints in micro/meso scale turbomachinery.

For a Jeffcott rotor with two degrees of freedom model

including unbalance, the equations of motion can be writ-

ten as,

€eX ¼paR2

mrCx2

� �Z2p

0

ZL=R

0

�pðh; �zÞ cosðhÞdhd�zþ feX

paR2þ mrex2 cosðtsÞ

paR2

0B@

1CA;

ð33Þ

€eY ¼paR2

mrCx2

� �Z2p

0

ZL=R

0

�pðh; �zÞ sinðhÞdhd�zþ feY

paR2þ mrex2 sinðtsÞ

paR2

0B@

1CA:

ð34Þ

The bearing reaction forces (integral terms) in the above

equation are obtained by solving unsteady Reynolds Eq. 9

and unsteady foil deflection Eq. 16 at every time instant. The

unsteady Reynolds equation for three different models viz.

no- slip, slip without leff and slip with leff are considered.

The velocity and displacement of journal center at each

time step is obtained by using following relations,

_eXðts þ DtsÞ ¼ _eXðtsÞ þ €eXðts þ DtsÞDts; ð35Þ_eYðts þ DtsÞ ¼ _eYðtsÞ þ €eYðts þ DtsÞDts; ð36Þ

eXðts þ DtsÞ ¼ eXðtsÞ þ _eXðts þ DtsÞDts þ1

2€eXðts þ DtsÞDt2

s ;

ð37Þ

eYðts þ DtsÞ ¼ eYðtsÞ þ _eYðts þ DtsÞDts þ1

2€eYðts þ DtsÞDt2

s :

ð38Þ

6 Control volume formulation

Following the procedure given in Patankar (2004) for

solving convection and diffusion type problem, a two

dimensional grid for solving Eqs. 9, 24 and 25 is consid-

ered (see Fig. 3). For the grid point P, points E and W are

its h direction neighbors, while N and S are the Z direction

neighbors. The control volume around point P is shown by

dashed lines. Its thickness along the normal plane of the

paper is assumed to be unity. The generalized form for

Eqs. 9, 24 and 25 can be written as

oJh

ohþ oJz

o�zþ Source term ¼ 0 ð39Þ

where, Jh and Jz are fluxes in h and Z directions

respectively. They are defined by,

Jh ¼ f ðKnÞK�p�h� /p�p�h3 o�p

oh

� �; Jz ¼ �/p�p�h3 o�p

o�z

� �:

The integration of Eq. 39 over the control volume

shown in Fig. 3, can be written as,

Je � Jw þ Jn � Js þ SP ¼ 0 ð40Þ

where,

Je ¼ f ðKnÞK�p�h� /p�p�h3 o�p

oh

� �e

D�z;

Jw ¼ f ðKnÞK�p�h� /p�p�h3 o�p

oh

� �w

D�z;

Jn ¼ �/p�p�h3 o�p

o�z

� �n

Dh; Js ¼ �/p�p�h3 o�p

o�z

� �s

Dh:

The source term SP for different equation can be written

as,

SP ¼ f ðKnÞ 2KmDhD�z

Dtsð�pP

�hPÞts � ð�pP�hPÞts�Dts

�for Eq: 9;

SP = 0 for Eq. 24, SP = (right hand side for Eq. 25).

Detail discretizations of Eq. 40 for solving steady,

unsteady and perturbed equations are given in Appendix.

7 Results and discussion

For numerical validation of the developed formulation and

Matlab code, the comparison of load carrying capacity with

different eccentricity ratios is carried with Peng and Car-

pino (1993). It is found that the results are in close

Fig. 3 Control volume grid for 2D Reynolds equation


123

agreement (see Fig. 4). In order to perform a numerical

study, the parameters of MMGFB compliance a = 0.1, loss

factor g = 0.25, clearance C = 25 lm, eccentricity ratio

e = 0.7, speed range N = 100,000–500,000 rpm, air

molecular mean free path k = 0.3 lm at 1023 K and

accommodation coefficient a = 1 are used.

7.1 Variation of load carrying capacity and attitude

angle with Knudsen number

The dimensionless load carrying capacity and attitude

angle in degrees for various models are shown in Fig. 5. It

is observed that the load carrying capacity significantly

decreases with increasing Kn. The difference in the load

carrying capacity for the two models increases with

increase in Kn. The attitude angle increases with increasing

Kn and difference between the two models is larger for

higher value of Kn. As the attitude angle relates to the

stability of bearing, the significant difference in the results

from the two models must be considered while studying the

stability of rotor bearing system at high Kn number.

The high value of Kn and resulting effective viscosity

may affect the stability of MMGFB system.

7.2 Variation of load carrying capacity with foil

bearing compliance

Foil bearing comprises compliant structure in the form of

top foil and bump foil. The lubrication film thickness

depends on the compliance of the bearing structure.

Therefore the variation of load carrying capacity of bearing

with compliance a is obtained for different models and

shown in Fig. 6. The load carrying capacity decreases with

increasing value of compliance. The difference in load

carrying capacity between no slip model and the other two

models is relatively more and the difference decreases with

increasing compliance. The effective viscosity model

shows significant change in load capacity for lower values

of compliance. At higher values of compliance, minimum

film thickness increases due to increase in deflection of top

foil and bump foil thereby decreasing the load capacity. In

addition, Knudsen number becomes less at higher value of

compliance and influence of rarefaction effect reduces.

This is reflected in the variation observed in Fig. 6.

7.3 Variation of load carrying capacity and attitude

angle with speed

Figure 7 shows load carrying capacity and attitude angle as

a function of speed. It is observed that for model with no

slip, model with first order slip without leff and model with

first order slip with leff the load carrying capacity increases

with increase in speed. For speeds ranging from 250,000 to

500,000 rpm, the model with effective viscosity leff shows

considerable difference in values of load capacity com-

pared to model with no slip. Since the typical micro/meso

scale gas turbine rotors are expected to work with the speed

range of 500,000–700,000 rpm and operating temperature

Fig. 4 Validation of load carrying capacity

Fig. 5 Variation of f and / with Knudsen number Kn Fig. 6 Variation of f with compliance a


123

of 1,000–1,300 K. Hence, the viscosity effect as brought

out in the present study should be considered. The attitude

angle decreases with increasing speed, and the attitude

angle for no slip model is the smallest among other models

(see Fig. 7). For effective viscosity model, attitude angle

becomes little larger; however for higher speeds there is

little influence of effective viscosity on attitude angle.

7.4 Variation of dynamic bearing coefficients

with speed

For rotordynamic analysis of MMGFB, the estimation of

dynamic bearing coefficients is required. The stability of

rotor bearing system, critical speeds and vibration

responses to different excitation forces of rotor bearing

system are strongly influenced by these coefficients. At

higher temperature, because of the rarefaction effect and

effective viscosity, these coefficients may change and

influence all the above dynamic response characteristics. In

order to study the influence of rarefaction effect on

dynamic bearing coefficients, speed and excitation fre-

quency dependent dynamic coefficients are obtained by

using linear perturbation method as explained in Sect. 3.

For obtaining these coefficients, excitation frequency ratio

m ¼ 1 is substituted in perturbation Eq. 25 and synchronous

coefficients are obtained.

Similar to load carrying capacity, direct as well as cross

coupled stiffness coefficients increases with speed (see

Fig. 8a–d). A difference in the values of stiffness coeffi-

cient obtained for no slip model and model with first order

slip with leff increases with speed. However, the difference

in the values predicted by model with first order slip

without leff and model with first order slip with leff is small

over studied speed range. Unlike the stiffness, the varia-

tions of damping coefficient for three models are quite

different. The damping coefficients shown by model with

first order slip with leff are greatly influenced at higher

speed (see Fig. 8a–d) and there is considerable rise up in

damping at higher speed. As speed increases, the stiffness

of gas film is also increases and resulting dissipation of

energy decreases which leads to decrease of damping

coefficients. However, the slip velocity boundary condition

reduces the velocity of gas in contact with journal surface

and in turn the resulting stiffness of gas film becomes less.

Therefore, it is observed that with comparison to no-slip

model the decrease of damping coefficients for model with

first order slip with leff and without leff are less and there is

significant difference in the values of damping coefficients

predicted by model with no slip and model with first order

slip with leff.

7.5 Variation of dynamic bearing coefficients

with excitation frequency ratio

The influence of rarefaction effect on dynamic coefficients

with excitation frequency ratio m at lower and higher speed

of rotor is studied. Fig. 9a–d depicts variation of stiffness

and damping coefficients obtained at lower speed

N = 100,000 rpm and m = 0.1–2.0. The difference in val-

ues obtained for model with no slip and model with first

order slip with leff is more significant for stiffness and

damping coefficient except at low excitation frequency

ratios m. At m = 1.8, it is found that compared to no-slip

model all stiffness and damping coefficients except CXY

predicted by model with first order slip without leff and

model with first order slip with leff decreases. Table 1

demonstrates the percentage change for different coeffi-

cients. Similarly, Fig. 10a–d depicts variation of stiffness

and damping coefficients obtained at higher speed

N = 500,000 rpm and m = 0.1–2.0. It is seen that there is a

significant difference between the values predicted with no

slip model and model with first order slip with leff. The

change in values predicted by no-slip model and model

with first order slip with leff becomes more significant at

higher values of excitation frequency ratios. Table 1 details

percentage increase/decrease of all the coefficients with

respect to no-slip model for m = 1.8 and two different

speeds. It is observed that for lower speed of

N = 100,000 rpm the direct damping coefficients decrea-

ses for model with first order slip without leff and model

with first order slip with leff respectively. Similarly, at

speed N = 500,000 rpm and m = 1.8, with comparison to

no-slip model, the direct damping coefficients increases for

model with first order slip without leff and model with first

order slip with leff respectively. For higher speed

(N = 500,000 rpm), the increase in cross coupled damping

coefficient is significantly large as compared to direct

damping coefficient. Thus the influence of rarefaction

effect is observed to be very significant at high operating

speed range typically encountered in MMGFB on the cross

coupled damping coefficient.

Fig. 7 Variation of f and / with speed N


123

In case of no-slip model, the gas molecules around the

journal are assumed as dense and the gas flows with the

linear velocity profile. On the other hand, in case of model

with first order slip without leff and with leff, the gas

molecules around the journal are assumed as rare and the

gas flows with slip velocity profile. Therefore, because of

the change in velocity profiles, it is observed that there is

considerable difference in bearing characteristics predicted

by no-slip model and model with first order slip without leff

and with leff. Considering the results presented, it is seen

that the rarefaction effect with effective viscosity leff

influences the static as well as dynamic characteristics of

MMGFB system.

7.6 Variation of modal impedance with excitation

frequency ratio

The stiffness and damping coefficients obtained for dif-

ferent excitation frequency ratios are used to calculate

modal stiffness and natural frequency of rotor bearing

Fig. 8 Variation of dynamic

coefficients for different speed

N at m = 1 and e = 0.7 a KXX

and CXX b KYX and CYX c KXY

and CXY d KYY and CYY


coefficients for different

excitation frequency ratio m at

N = 100,000 rpm and e = 0.7

a KXX and CXX b KYX and CYX

c KXY and CXY d KYY and CYY


123

system. The variation of modal impedance with excitation

frequency ratio m for rotor speed N = 100,000 rpm and

N = 500,000 rpm are shown in Fig. 11a, b. It is observed

that there is significant difference between the value of

modal stiffness predicted by model with no slip and model

with first order slip with leff. The difference between the

two models is more at lower excitation frequency ratios.

The small change in the value of modal stiffness between

the model with first order slip without leff and with leff is

observed at lower and higher speed. At N = 100,000 rpm,

significant change in modal damping for model with no slip

and model with first order slip with leff is observed

at higher excitation frequency ratios. The natural fre-

quencies are obtained for three different models at

N = 100,000 rpm and N = 500,000 rpm using method

discussed in Sect. 4 and given in Table 2. It is observed

that the modal stiffness and natural frequencies are

decreasing with slip effect. There is a significant change in

natural frequency and modal stiffness for model with no

slip and model with first order slip with leff.

7.7 Unbalance response and FFT for different model

The unbalance response of rotor in vertical X and hori-

zontal Y direction for three different models namely no

slip, first order slip without leff and first order slip with

leff are obtained. Fig. 12a–d shows the response and

corresponding Fast Fourier transform (FFT) for no-slip

model. The subsynchronous vibration amplitude of rotor

center in X and Y direction is 2 and 0.7 lm respectively.

Fig. 13a–d shows the response and corresponding FFT for

model with first order slip without leff. The subsynchro-

nous vibration amplitude of rotor center in X and

Y direction is 0.02 and 0.006 lm respectively. Similarly,

Fig. 14a–d shows the response and corresponding FFT for

model with first order slip with leff. The subsynchronous

vibration amplitude of rotor center in X and Y direction is

0.7 and 0.75 lm respectively. The synchronous vibration

amplitude in X and Y direction for all the three models is

0.1 lm and depicting the uniform rotational speed of


coefficients for different

excitation frequency ratio m at

N = 500,000 rpm and e = 0.7

a KXX and CXX b KYX and CYX

c KXY and CXY d KYY and CYY

Table 1 % decrease and increase of dynamic coefficients compared

to no-slip model at excitation frequency ratio m = 1.8 for rotor speed

N = 100,000 and 500,000 rpm

N = 100,000 rpm N = 500,000 rpm

% decrease

slip without

leff

% decrease

slip with leff

% increase

slip without

leff

% increase

slip with leff

KXX 20.85 25.21 6.18a 8.52a

KYX 56.52 63.75 2.4 4.77

KXY 14.21 17.65 5.57a 7.82a

KYY 15.17 19.46 2.9a 10a

CXX 9 10.91 8.43 12.68

CYX 7.8 9.85 23.72 28.77

CXY 2b 2.5b 54.73 90.73

CYY 3 4.5 6.6 7.8

a % Decrease, b % increase


123

journal. However, as discussed in Sect. 7.4, the reduction

of damping in the gas film is less for model with first

order slip with leff and without leff due to the presence of

slip velocity boundary condition. Therefore, the amplitude

of subsynchronous vibration for model with first order slip

with leff and without leff is significantly reduced. Since

the presence of rarefaction effect influences the static as

well as dynamic characteristics of rotor bearing system, it

must be incorporated in the Reynolds equation of

MMGFB.

8 Conclusions

For a rotor supported on MMGFB, understanding dynamics

of the rotor with more accurate modeling of bearing sup-

port is important due to typical operating conditions

encountered such as very high speed, high temperature, etc.

In this paper a comparative study of load carrying capacity,

attitude angle, stiffness and damping coefficients, modal

impedance and unbalance response for three different

bearing models, namely, model with no slip, model with

first order slip without effective viscosity and model with

first order slip with effective viscosity is carried out.

It is observed that the effective viscosity strongly influ-

ences the static and dynamic bearing characteristics of

MMGFB. For a given eccentricity ratio e = 0.7 and speed

ranging from N = 100,000 to 500,000 rpm, the model with

first order slip with effective viscosity shows significant dif-

ference in the values of load capacity and attitude angle

compared to model with no slip. Rarefaction effect becomes

more significant at higher Knudsen number. For frequency

dependent stiffness and damping coefficients the influence of

effective viscosity increases with excitation frequency ratios.

At speed N = 100,000 rpm and excitation frequency

ratio m = 1.8, it is found that compared to ‘‘Model with no

slip’’, KXX decreases by 25 and 21 % for the ‘‘Slip with

leff’’ model and ‘‘Slip without leff’’ model respectively.

Similar drop is observed for other stiffness and damping

values; KYY drops by 20 and 15 %; CXX drops by 11 and

9 % while CYY drops by 5 and 3 %.

At speed N = 500,000 rpm and excitation frequency

ratio m = 1.8, it is found that compared to ‘‘Model with no

slip’’, KXX decreases by 9 and 6 % for the ‘‘Slip with leff’’

model and ‘‘Slip without leff’’ model respectively. Similar

drop is observed for other stiffness values; KYY drops by 10

and 3 %. But the value of CXX rises by 13 and 9 %; CYY

rises by 8 and 7 %.

At speed N = 100,000 rpm, it is found that compared to

‘‘Model with no slip’’, natural frequency decreases by 12

and 10 % for the ‘‘Slip with leff’’ model and ‘‘Slip without

leff’’ model respectively. Similar drop in values of natural

frequency at N = 500,000 rpm by 6 and 4 % is observed

for the two models.

The amplitude of subsynchronous vibration is signifi-

cantly reduced for model with first order slip with leff and

without leff due to the presence of slip velocity boundary

condition.

Strong influence on load capacity, attitude angle, stiff-

ness and damping coefficients, modal impedance, natural

frequencies and subsynchronous vibration amplitude indi-

cates that the model with first order slip with effective

viscosity would greatly influence the dynamic behavior of

the miniaturized rotors supported on MMGFB. Therefore

the slip effect should be considered in the rotordynamic

Fig. 11 Variation of modal impedance with excitation frequency

ratio a speed N = 100,000 rpm b speed N = 500,000 rpm

Table 2 Modal stiffness and Natural frequencies for three different

models at rotor speed N = 100,000 and 500,000 rpm

N = 100,000 rpm

Model meigen ks_modal (N/m) Ns (rpm)

No-slip 0.1807 6229 7536

Slip without leff 0.1818 5097 6817

Slip with leff 0.1804 4838 6642

N = 500,000 rpm

No-slip 0.24 30050 16553

Slip without leff 0.2 28080 16001

Slip with leff 0.22 26850 15647


123

Fig. 12 Unbalance response

and FFT for no slip model at

e = 0.7 and N = 500,000 rpm

a response in X direction b FFT

in X direction c response in

Y direction d FFT in Y direction


and FFT for model with first

order slip without leff at e = 0.7

and N = 500,000 rpm





123

analysis through the use of more accurate bearing coeffi-

cients and unsteady bearing reaction forces evaluated in

this study.

Appendix

Continuity equation for the control volume shown in Fig. 3

can be written as,

Je � Jw þ Jn � Js þ SP ¼ 0 ð41Þ

Using neighboring coefficients of control volume,

Eq. 41 becomes,

aP�pP þ aE �pE þ aW �pW þ aN �pN þ aS�pS � bP ¼ 0

for Eqs. 9 and 24

aPð�pCÞP þ aEð�pCÞE þ aWð�pCÞW þ aNð�pCÞN þ aSð�pCÞS � bP ¼ 0;

for Eq. 25

where,

aE ¼ DeAðjPeejÞ; aW ¼ �DwðAðjPewjÞ þ PewÞ;aN ¼ �Dn; aS ¼ �Ds;

bP ¼ ½½0;�Fe þ Fw��pP þ2KmDhD�z

Dtsð�hP�pPÞts�Dts

aP ¼ �aE � aW � aN � aS þ ½½0;Fe � Fw�� þ2KmDhD�z

Dtsð�hPÞ

9>>=>>;;

for Eq. 9

bP ¼ ½½0;�Fe þ Fw��pP

aP ¼ �aE � aW � aN � aS þ ½½0;Fe � Fw��

);

for Eq. 24

bP ¼ ½½0;�Fe þ Fw��ð�pCÞP þ ðright hand side of Eq: 22ÞPDhD�z

aP ¼ �aE � aW � aN � aS þ ½½0;Fe � Fw��

)

for Eq. 25.

The convection and diffusion terms are coupled by Pe-

clet number Pe and the function of Pe defined using power

law scheme,

Pee ¼Fe

De; Pew ¼

Fw

Dw; Fe ¼ f ðKnÞK�heD�z;

Fw ¼ f ðKnÞK�hwD�z;

De ¼ ð/p�p�h3ÞeD�z

Dh; Dw ¼ ð/p�p�h3Þw

D�z

Dh;

Dn ¼ ð/p�p�h3ÞnDhD�z; Ds ¼ ð/p�p�h3Þs

DhD�z

AðjPejÞ ¼ ½½0; ð1� 0:5jPejÞ5��:

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Documents

Investigations on characteristics of micro/meso scale gas foil journal bearings for 100–200