dynamic properties of plain journal bearings · dynamic properties of plain journal bearings ......

Dynamic properties of plain journal bearings


With still growing speed of rotors it is not sufficient to calculate static bearing characteristics, as eccentricity and attitude angle of journal centre, friction losses and oil flow, eventually maximum temperature in oil film. Increasing importance gains knowledge of bearing dynamic properties, i.e. coefficients of the oil film stiffness and damping, which determine not only position of critical speeds and magnitude of its damping, but also about rotor stability. Dynamic characteristics of bearings used in a unit should be known already in phase of machine design, because later modifications are always very expensive and need not be effective in all cases. It is therefore inevitable to have at ones disposal data needed for calculation of bearing dynamic properties, but no less important is information about credibility and accuracy of these data. Not only methods of stiffness and damping coefficients are therefore described further on, but also methods of their experimental verification.

1.0 Reynolds equation of hydrodynamic lubrication

In Newtonian fluids flow the basic relations between mass forces, exterior forces and inertial forces are expressed by Navier –Stokes equations, which can be in vector form expressed as

∇∇+∇∇+∇∇+∇−= vvvpKdt

vd rrrrr

... µµµρ - for incompressible medium. (1-1)

[ ] vvpKdt

vd rrrr

∆+∇∇+∇−= µµρ .31 - for compressible medium. (1-2)

where wkvjuivrrrr ++= … vector of flow velocity,

ZkYjXiKrrr

++= … vector of mass forces,

z

ky

jx

i∂∂+

∂∂+

∂∂=∇

rrr,

222

2

zyx ∂∂+

∂∂+

∂∂=∇=∆ ,

( )vvt

v

dt

dv rr∇+∂∂= .ρρ .

Continuity equation can be in vector form expressed by relation

( ) 0. =∇+∂∂

vt

rρρ. (1-3)

By omitting mass forces and through simplification for thin layers, where film thickness is several order smaller in comparison with other dimensions, for small Reynolds numbers

νdu.

Re= ,

where u … flow velocity, d … characteristic dimension, ν … kinematic viscosity of lubricant, we get the system of simultaneous equations


∂∂

∂∂=

∂∂

y

u

yx

p µ , 0=∂∂

y

p,

∂∂

∂∂=

∂∂

y

w

yz

p µ , (1-4)

0)()()( =

∂∂+

∂∂+

∂∂

z

w

y

v

x

u ρρρ.

By modification of equations (1-4) we get general Reynolds equation in the form

( )

+∂

∂+=

∂∂

∂∂+

∂∂

∂∂

221

33

2)(

6.. Vx

hUU

z

ph

zx

ph

xρρ

µρ

µρ

, (1-5)

where U1, U2 … velocity of sliding surfaces in direction x (usually U1=0), V2 … velocity of sliding surface in direction y. For incompressible medium ρ is constant and equation (1-5) can be rewritten to

( )

+∂∂+=

∂∂

∂∂+

∂∂

∂∂

221

33

26.. Vx

hUU

z

ph

zx

ph

x µµ (1-6)

Hydrodynamic film thickness is usually a function of x and z coordinates, velocity U1 is usually zero and V2 is generally a time function of h. Reynolds equation for incompressible medium can be then expressed as

t

h

x

h

z

ph

zx

ph

x ∂∂+

∂∂=

∂∂

∂∂+

∂∂

∂∂

126..33

µµ. (1-7)

Equation (1-7) is valid incompressible medium, for compressible medium would get relation

( )t

hpph

xU

z

pph

zx

pph

x ∂∂+

∂∂=

∂∂

∂∂+

∂∂

∂∂

126.. 33 µ . (1-8)

1.1 Reynolds equation for journal bearings

In a journal bearing, for which it holds

ϑRx = , ω.RU = , R … bearing radius,

it is useful to introduce dimensionless quantities

Lz=ζ … dimensionless coordinate in direction of bearing width,

L … bearing width,

RhH = … dimensionless film thickness,

=R

cpP

ωµ..6… dimensionless pressure,

c … radial clearance,

so that equation (1-7) changes to

t

HHPH

L

RPH

∂∂+

∂∂−=

∂∂

+

∂∂

∂∂

ωϑζϑϑ2

2

23

23 . (1-9)

In case of compressible media it useful to introduce dimensionless parameters

2

6

=Λc

R

pa

µω,

appP =

pa … ambient pressure,


after which Reynolds equation takes the form

( ) ( )

∂∂+

∂∂Λ=

∂∂

∂∂+

∂∂

∂∂

t

HPHPPH

PH

ωϑζζϑϑ2

22

32

3 . (1-10)

Equation (1-10) is evidently non-linear in P and linearization is needed for its solution. Suitable method is introduction of substitution

Q=(PH)2, through which we get

t

Q

QHQ

H

H

Q

Q

H

H

QQ

∂∂Λ=

∂∂−

∂∂

Λ+∂∂−

∂∂+

∂∂

ωϑϑϑζϑ421

2

2

2

2

2

2

, (1-11)

Although equation (1-11) is still non-linear, it can be easily solved numerically by iteration methods.

2.0 Non-stationary flow through journal bearing gap with small harmonic motion of the journal

With small harmonic deviations of journal the film thickness is a function of time and can be expressed by relation

titi eeHH ΩΩ −−= ...sincos),( 1010 ϑεϑϑεϑε , (2-1)

where ( ) ϑεϑε cos1, 000 −== HH …clearance corresponding to steady journal position,

c

e00 =ε … relative eccentricity of steady journal position,

. ε1, ε0ϑ1 … complex amplitudes of deviations in normal and tangent directions,

Ω … circular frequency of harmonic journal vibrations.

For linearized case (small deviations) we get after substitution into (7)

+

−∂∂

+

∂∂

∂∂+

∂∂−=

∂∂

+

∂∂

∂∂ Ωtie

PH

L

RPH

HPH

L

RPH 12

22

0

22

02

23

0

23

0 sincos3cos3 εϑζ

ϑϑ

ϑϑϑζϑϑ

(2-2)

( ) titi eieP

HL

RPH ΩΩ −Ω−

+∂∂

+

∂∂

∂∂+ 101102

22

0

2

2

0 sincos2cossin3sin3 ϑϑεϑεω

ϑεϑζ

ϑϑ

ϑϑ

Solution of equation (2-2) can be get by superposition of stationary solution and non-stationary solution for the case of small deviations, thus

( ) ( ) titi eP

iPePiPPtPP ΩΩ

Ω++

Ω++==0

02103110 22,,,

εωϑε

ωεζϑζϑ , (2-3)

where P1, P2, P3… amplitudes of alternating component of dimensionless pressure.

Steady-state solution of equation (14) we get for elements without factor eiΩt

ϑζϑϑ ∂

∂−=

∂∂

+

∂∂

∂∂ 0

2

0

23

0

2

03

0

HPH

L

RPH (2-4)

For complex alternating components of pressure we get relations

..2

23

0

2

3

0 SRP

HL

RPH kk =

∂∂

+

∂∂

∂∂

ζϑϑ, k=1,2,3 (2-5)


pro k=1 ϑζ

ϑϑ

ϑϑ

sincos3cos3..2

0

22

0

2

02

0 −∂∂

+

∂∂

∂∂= P

HL

RPHSR , (2-6)

pro k=2 ϑζ

ϑϑ

ϑϑ

cossin3sin3..2

02

2

0

2

02

0 +∂∂

+

∂∂

∂∂=

PH

L

RPHSR , (2-7)

pro k=3 ϑcos.. −=SR . (2-8)

Similar system of equations we get also for compressible medium

2.1 Matrices of stiffness and damping

The forces acting on journal of the bearing can be expressed by means of so called matrices of stiffness and damping according to relation

Ω+Ω+

Ω+Ω+−=

10

1

ϑωωωω

e

e

BiKBiK

BiKBiK

c

W

F

F

tttttntn

ntntnnnn

t

n (2-9)

where W … static bearing load

W

ckK ijij = …dimensionless elements of stiffness matrix,

ωW

cbB ijij = … dimensionless elements of damping matrix,

1st index – direction of force, 2nd index – direction of deviation, n … direction of normal – to bearing centre, t … direction of tangent.

For practical purposes it is more suitable to transpose dynamic bearing forces to vertical and horizontal directions. Usual orientation of anti-clockwise coordinate system is as follows:

x axis – in direction of static load,

y axis is oriented to the right.

Dynamic forces acting on journal are then given by relation

Ω+Ω+

Ω+Ω+−=

y

x

BiKBiK

BiKBiK

c

WF

F

yytyyyxyx

xyxyxxxx

y

x

ωωωω (2-10)

Bearing stiffness a damping must be unconditionally considered in rotor dynamic calculation. Rotor dynamic analysis considering stiffness and damping of bearing oil film is essential condition of reliable and safe operation of modern high-speed machines.

As most computational data, also accuracy of calculated stiffness and damping coefficients should be experimentally verified. Unlike static characteristics, as e.g. friction losses, oil flow and temperatures in oil film, determining of stiffness and damping coefficients by measurement is a very complex matter and that is why the whole chapter (sect. 5.0) is dedicated to this subject.


3.0 Some methods of Reynolds equation numeric solution

Although possibilities of analytic solution of Reynolds equation exist for special cases (infinitely short or infinitely long bearing), numeric solutions are almost exclusively used at present. Due to relative simplicity of sliding surface shape, rectangular mesh is usually sufficient. Only in special cases, e.g. bearings with hydrostatic pockets, we use finite elements. Principle of solution consists in replacing derivatives by finite differences according to relations

ϑϑ ∆

−=

∂∂ −+

21,1, jiji PPP

,

( )2

1,,1,

2

2 2

ϑϑ ∆

+−=

∂∂ −+ jijiji PPPP

, (3-1)

( )2

1,,1,

2

2 2

ζζ ∆

+−=

∂∂ −+ jijiji PPPP

.

Naturally it is necessary to respect pertinent boundary conditions – pressure at the bearing or pad periphery is equal to ambient pressure, periodicity of solution in bearings of circular cross section.

As a sample of differential equation transform to difference one we will take relation (1-11) for compressible medium. Using relations (3-1) we get

( ) ( ) ..2

.2

1.2.22

21.1,

,

2

,1,,1

2

1,,1, SRH

H

QQ

Q

H

H

QQQQQQ jiji

ji

jijijijijiji =−∆−

Λ−−+−

++− −+−+−+

∂θ∂

θ∂θ∂

∂ς∂θ (3-2)

For steady state solution of equation (3-2) R.S.= 0. Boundary conditions are as follows

( ) 2, HDlQ =θ … pressure at bearing periphery is equal to ambient pressure, i.e. P= p/pa = 1

( )∂∂ς

θQ,0 0= … distribution of pressure is symmetrical relative to central plane of the bearing

(therefore usually only one halve of bearing length is calculated)

and ( ) ( )ζπϑζϑ ,2, += QQ … for bearing with one sliding surface – periodical solution,

or ( ) ( ) 221 ,, HQQ == ςθςθ … for bearing with more sliding surfaces; pressure at inlet and

outlet edge of sliding surface is equal to ambient pressure. Using boundary conditions for equation (3-2) we arrive at matrix equation

[ ] [ ] [ ] A Q B Q C Q Rj j j j j j j. . .+ + =− +1 1 , (3-3)

where [A], [B], [C] square matrix of order m (m … number of mesh points in ζ, direction)

Q … column vectors of dependent variable of order m,

R … column vectors of right hand side of order m.

Boundary conditions are

Q1,j=Hj2,

QM+1,j=QM-1,j, (3-4)

Qi,0=QiM, Qi,1=QiM+1 .. for periodic solution,

or Qi,1=H12 and Qi,N=HN

2… for bearing with more sliding surfaces.

Individual elements of matrices [A], [B], [C] and those of vector R are given by relations


( ) ( ) ( )AH

Hi i j, ,

.= − + +

21 1 1

2 2

2

2∆ ∆θ ς

∂∂θ

, for i M∈⟨ ⟩2, ,

( )A j1 1 1, , = , ( ) ( )Ai i j, ,+ =

1 2

1

∆ς for i M∈⟨ ⟩2, , ( )A

j1 20

, ,= ,

( ) ( )Ai i j, ,− =

1 2

1

∆ς for i M∈⟨ − ⟩2 1, , ( ) ( )A

M M j, ,− =1 2

2

∆ς,

( )A i l j, , = 0 for 1 ≤ l ≤ i-2, i+2 ≤ l ≤ M,

( ) ( )BH

H

Qi i j, , .= + +

1 1 1

22∆

Λ∆ς

∂∂θ θ , for i M∈⟨ ⟩2, ,

( )B j1 1 0, , = , ( )B i l j, , = 0 pro i ≠ l,

( ) ( )CH

H

Qi i j, , .= − +

1 1 1

22∆

Λ∆ς

∂∂θ θ , for i M∈⟨ ⟩2, ,

( )C j1 1 0, , = , ( )Ci l j, ,

= 0 for i ≠ l,

R Hj j12

, = , Ri.j=0.

Equation (3-3) can be solved by so called column method, which is very effective and rapid procedure for small values of m. Solution of equation (3-3) is assumed in the form

[ ] Q E Q Fj j j j− = −= +1 1 1 , (3-5)

where [Ej] … generally complex square matrix of order m,

Fj … generally complex vector of order m.

Qj … generally complex vector of dependent variable of order m. In case of steady state solution matrices [Ej] and vectors Fj,Qj are real. By substitution into (3-3) we get recurrent relations

[ ] [ ] [ ] [ ]( ) [ ]E A B E Cj j j j j= + −

−. 1

1

, (3-6)

[ ] [ ][ ]( ) [ ] ( )F A B E R B Fj j j j j j j= + −−

−

1

1

. (3-7)

Boundary condition, ambient pressure at the sliding surface inlet edge, is fulfilled by boundary conditions

[E1]=0, Fj=Q1. As first step calculation of matrices [E1] and vectors Fj for j N∈⟨ − ⟩2 1, is carried out, followed

by calculation of vectorsQj for j N∈⟨ − ⟩1 2, . Above described procedure usually converges very rapidly. Sufficient density of the mesh for circular and tilting pad bearing respectively is 7x30 and 7x15 points. Present day personal computer enable to select much higher number of mesh points, but the difference in calculated results is less than 5 %. Calculation of static and dynamic characteristics for one value of load and several speeds takes several second on common PC. In case, that energetic equation is solved simultaneously with Reynolds equation (inclusion of lubricant variable viscosity), the computation is prolonged to several tens of seconds.


4.0 Eigenvalues of stiffness and damping, problems of stability Eigenvalues of stiffness and damping are defined as roots of characteristic equation

( )( ) 0,

,21 =−−=

−−

γγγ

γZZ

ZZ

ZZ

yyyx

xyxx , (4-1)

where 2,12,12,12,1 VVV iBKZ +==γ ,

KV1,2 … eigenvalue of stiffness, BV1,2 … eigenvalue of damping.

Stability limit can be evaluated from relation

0,

,=

−−

KZZ

ZKZ

yyyx

xyxx , (4-2)

where K … eigenvalue of stiffness at stability limit. By solving of (3-4) we get relations for eigenvalue of stiffness and ratio of exciting to rotational frequency at the stability limit

yyxx

xyyxyxxyxxyyyyxxm BB

BKBKBKBKK

+−−+

= , (4-3)

( )

yxxyyyxx

yxxyyyxxyyyymm

m BBBB

KKKKKKKK

−−+−−

=

Ωω

. (4-4)

It is necessary to warn, that relations (4-3) and (4-4) are valid for mass point supported on the oil or gas film. Determination of real rotor eigenvalues is much more complicated and will not be solved here. Magnitudes and signs of individual elements of stiffness and damping are decisive for stiffness at stability limit. Due to the fact, that so called direct terms Kxx, Kyy, Bxx, Byy are always positive, it is important to judge namely magnitudes and signs of cross-coupled elements Kxy, Kyx, Bxy, Bxy. As an example of concrete values we will present table of stiffness and damping elements of different bearings with the same diameter and identical clearance, which operate at the same conditions (speed, load, lubricant and its inlet temperature).

Table 1 Comparison of stiffness and damping elements of various journal bearing types 90 mm in diameter, L/D=0,7, relative clearance 1,5.10-3, speed 15.000 rpm, specific load 0,5 MPa, oil VG46, inlet temperature 50°C (hydrodynamic bearings) air with ambient temperature, inlet pressure 0,5 MPa (aerodynamic and aerostatic bearings)

bearing type

stiffness (N.m-1) damping (N.s.m-1)

Kxx Kxy Kyx Kyy Bxx Bxy Byx Byy

circular 4,47e7 1,47e8 -5,25e7 3,54e7 1,85e5 1,97e4 1,97e4 7,04e4

lemon 2,68e8 6,59e7 -1,45e8 2,59e7 2,44e5 -8,16e4 -8,16e4 5,27e4

offset halves 1,61e8 2,60e8 4,14e7 2,42e8 1,72e5 9,09e4 9,09e4 1,12e5

four lobbed sym. 1,26e8 6,51e7 -6,45e7 1,14e8 9,38e4 6,34e2 6,34e2 9,06e4

four lobbed unsym. 2,16e8 8,03e7 -8,03e7 2,15e8 1,11e5 5,10e2 5,10e2 1,11e5

tilting pad 2,45e8 -1,76e6 1,70e6 2,39e8 1,56e5 5,90e2 -5,73e2 1,55e5

aerodyn. 90mm * 7,34e6 3,23e5 6,26e5 2,54e7 2,24e3 1,42e2 -4,87e2 6,69e3

aerostat. 90mm * 1,05e7 1,76e6 4,62e4 6,19e6 1,62e3 -3,89e2 4,38e2 1,25e3

aerodyn. 200mm 2,62e7 3,08e6 3,72e6 9,66e7 8,56e3 9,01e2 -4,19e3 2,51e4

aerostat. 200mm 6,19e7 1,11e7 9,04e6 4,25e7 1,31e4 -6,93e3 5,47e3 1,20e4


* aerodynamic and aerostatic bearing can not achieve comparable load carrying capacity as hydrodynamic bearings of the same dimensions; that is why also values for bearings 200 mm in diameter (L/D=1) are given, which could carry roughly the same load as hydrodynamic bearings 90 mm in diameter,

For simple comparison of different bearing types from stability point of view it is possible to use rigid symmetrical rotor supported in two identical bearings. On the basis of the rotor eigenvalues it is possible to construct diagram presented in Fig. 1, taken over from [1].

Fig. 1 Diagram of stability of various bearing types

Stability limit of rigid symmetrical rotor is expressed by relation

c

Mn R

R

8,9= ,

where MR is a value read from the diagram for given So number (equation 4-1).

The values below curve are stable, the values above curve they are unstable. Numbers at the curves designate bearing type with certain preload δ:

curve 1 … circular δ=0 curve 4 … three lobed δ=0,8 curve 2 …lemon δ=0,6 curve 5 … three lobed unidirectional δ=0,8 curve 3 …four lobed δ=0,6 curve 6 … offset halves δ=0,6

Merits of bearings with more sliding surfaces as concerns stability are clearly apparent from the diagram. When comparing lemon and offset halves bearings, the manufacturing demands of which is roughly the same, much higher stability limit of offset bearings is apparent, which though exists only in relatively limited range of operational parameters. Geometry of three lobed and four lobed bearings is relatively complex as regards manufacture, so that it almost equals that of tilting pad bearings. Stability properties of tilting pad bearings are much better than of those with fixed geometry, but not even tilting pad bearings can ensure rotor stability in any conditions. “Oil whirl“ instability of high-speed rotor supported in tilting pad bearings was caused by external excitation, because the rotor had low stability reserve.


5.0 Methods of experimental investigation of sliding bearings dynamic properties

As was already mentioned earlier, dynamic properties of sliding journal bearings are usually expressed by coefficients of stiffness and damping, or stiffness and damping matrix

=

yyyx

xyxx

KK

KKK , [ ]

=

yyyx

xyxx

BB

BBB ,

where Kxx, Kxy, Kyx, Kyy … coefficients of stiffness,

Bxx, Bxy, Byx, Byy … coefficients of damping,

1st index … force direction, 2nd index … direction of deviation.

Great number of computational methods with different precision exists for determination of stiffness and damping coefficients. That is why calculated values should be experimentally verified at least at limited region of operational parameters. Experimental method can be divided into several categories from the simplest to the most complex and costly.

5.1 Indirect verification of dynamic properties through measured amplitude-frequency characteristics

Relatively simple stand with rigid rotor supported in two identical journal bearings is needed for application of this method. This methodology was used in 80ies for experimental investigation of aerostatic bearings at the State research institute for machine design (SVÚSS) Běchovice [2]. Scheme of the used stand is shown in Fig. 2.

Fig. 2 Stand for experimental investigation of aerostatic journal bearings

Rotor 1 is supported in two identical aerostatic journal bearings 3 and its axial movement is restricted by thrust bearings 5. The drive is ensured by two air turbines 2, located symmetrically at both rotor ends. The rotor mass and thus also bearing load can be changed by discs 4 mounted on the shaft. Deviations of the rotor were traced by 4 relative sensors 6 working on capacitive


principle. Capacitive sensors suffer from significant zero shifts, even with small temperature changes. Therefore it was impossible to evaluate static journal position in the bearing, but measured rotor dynamic deviations were relatively accurate. For each combination of parameters (bearing clearance, rotor mass, air inlet pressure) so called amplitude-frequency characteristics was recorded, from which it was possible to determine both critical speed and stability limit. Sample of measured amplitude-frequency (A-F) characteristics is shown in Fig. 3

Fig. 3 Measured A-F characteristics of rotor in aerostatic bearings

As is apparent from Fig. 3, measured A-F characteristics enable to find critical speed frequency and also stability limit, which is characterized by very steep increase of vibration amplitude (around 580 Hz). The data were recorded on coordinate writer, with voltage proportional to speed as input for the x coordinate. With regard to air drive the signals from capacitive sensors were not disturbed and the record is very smooth.

Isotropy of aerostatic bearings is documented in Fig. 3 by very small differences in signals from sensors oriented in vertical (designated y) and horizontal direction (designated x). Rotor with disc shows two resonance peaks for cylindrical and conical vibration modes. Measured critical speed and stability limits were in quite good agreement with values calculated from stiffness and damping coefficients, determined by numeric solution of Reynolds equation. It demonstrates relatively good accuracy of calculated coefficients of stiffness and damping, which follows among others from weak dependence of gas viscosity on temperature. In hydrodynamic bearings the greatest differences between calculation and measurement result from inaccurate determination of stationary position of the journal in bearing, which is strongly dependent on accuracy of assessment of temperature field in oil film.


5.2 Determination of stiffness and damping coefficients through response to additional static loading and unbalance

This method was presented e.g. in [3]. Coefficient of stiffness are evaluated in such a way, that given static load of the bearing is increased by small value and response to this impulse is measured, i.e. deviations of journal are measured with influence of additional load at two different directions. By introducing so called influence coefficients the stiffness coefficients can be expressed as

( )γα yyxxk = , ( )γα xyxxk −= ,

( )γα yxyxk −= , ( )γα xxyyk = , (5-1)

where yxxyyyxx ααααγ .. −= ,

( )xxx Fx ∆= 1α , ( )xyx Fy ∆= 1α ,

( )yxy Fx ∆= 2α , ( )yyy Fy ∆= 2α , (5-2)

x1, y1 and x2, y2 …static deviations in horizontal and vertical directions with force variations ∆Fx and ∆Fy .

Even with significant progress in development of relative sensors measurement static position of journal is still suffers from relatively big errors, because it is influenced e.g. by changes of ambient sensor temperature. Through false evaluation of static position is of course influenced also accuracy of stiffness coefficients determination.

Coefficients of damping are according to [3] evaluated from response to known unbalance. The four damping coefficients are determined from measured deviations and phase shifts relative to unbalance position in two mutually perpendicular directions using earlier established coefficients of stiffness. Inaccuracies in determination of stiffness coefficients naturally affect also coefficients of damping.

The whole series of methods for determination of stiffness and damping coefficients by means of measuring the response to unbalance exists. Similar methodology is used by the firm Bently Nevada Dynamic Research Corporation (BNDRC) for establishing so called modal stiffness and modal damping [4] (BNDRC does not use classical stiffness and damping coefficients).

Recently a new method was elaborated, enabling improvement of stiffness and damping elements by means of measurement of response to known unbalance on standard balancing machine [12]. Results of experiments needed for verification of this method were not up to now published.

5.3 Determination of stiffness and damping coefficients by means of response to harmonic excitation

This method is apparently the most accurate, but also the most demanding as regards experimental basis. The principle of the method is measurement of response to harmonic excitation acting subsequently in two mutually perpendicular directions. Two variants are possible: a) harmonic force acts on test bearing, b) harmonic force acts on test shaft.

Variant sub b) was used at Škoda Plzeň research institute for testing of big diameter bearings – up to 600 mm [5, 6], stand for method sub a) was realized e.g. at Technical university of Karlsruhe [7] or in SVÚSS Běchovice [10, 11]. Test stand designed and built by professor Glienicke in Karlsruhe is shown in Figs. 4 and 5.


Fig. 4 Cross section of the stand for experimental evaluation of stiffness and damping coefficients

Fig. 5 Longitudinal section of the stand for experimental evaluation of stiffness and damping coefficients


Test shaft 120 mm in diameter is supported in two sliding journal bearings, the test bearing is located between them. Static load is generated by three bellows, two with horizontal axis and one with vertical axis. By combination of pressures in bellows it is possible to change not only magnitude, but also direction of bearing load. Dynamic harmonic force is generated by special kind of exciter, which enables continuous change of excitation force amplitude without change of its frequency. The most important part of exciter is the shaft, the elastic part of which is supported in eccentrically rotating bushings. Magnitude of dynamic force, transferred to rolling bearing connected with test bearing body is varied by shift of middle rolling bearing inner bushing. To suppress undesirable force component (so that only force in one direction was transferred to test bearing), the rolling bearing casing is mounted into elastic suspension and joint is inserted between the casing and test bearing. Joint is used at the same time as a load cell for measurement of dynamic force amplitude. Special gearbox provides drive of exciters and of test shaft, which ensures coupling between both exciters. Exciters are located in such a way, that directions of relevant dynamic forces contain angle of 90°. Simultaneous excitation of test bearing in both directions can thus ensure practically arbitrary amplitude of exciting force. In experiments carried out, although one dynamic force was always zero. Above described test stand enabled to carry out experiments with maximum speed of 10.000 rpm.

At the beginning of 80ies was in SVÚSS Běchovice designed and manufactured similar test stand for experimental investigation of ČKD Compressors bearings (see scheme in Fig. 6).

Fig. 6 Scheme of test stand SVÚSS Fig. 7 Test stand Škoda Plzeň

Test bearing 5 had diameter of 90 mm, maximum speed corresponding to sliding speed of about 80 m/s was 20.000 rpm. However, high-speed gearbox drive with gear coupling enabled to reach speeds up to 40.000 rpm, which was utilized in tests of smaller diameter bearings. Regarding ČKD requirements the stand was designed only for vertical direction of static load and was therefore equipped with only one loading member constituted by rubber compensator 6. Maximum static load was about 12 kN (corresponding specific load 2 MPa). With regard to necessity of experimental investigation of tilting pad bearings, the stiffness and damping


coefficients of which are dependent on excitation frequency, the exciters were equipped with individual drive by high-frequency motor with maximum speed of 17.500 rpm. Deviations of test bearing relative to shaft and frame were followed by induction sensors Hottinger. Exciters were equipped by optic sensors of phase marker, which enabled to determine phase shift between dynamic force and individual components of deviations. Signals from relative sensors and strain gauge bridges, for dynamic force evaluation, were processed by amplifiers of the same type (apparatus Hottinger KWS 3073), which ensured, that analogue part of apparatus will not cause relative changes of phase shifts. The phase shifts were evaluated by means of special apparatus designed in SVÚSS, the basic circuits of which were identical with balancing apparatus, because it was necessary to consider only components with given frequency. Program for HP9845 computer was elaborated for stiffness and damping evaluation, which determined stiffness and damping coefficients from measured deviations and their phase shift relative to excitation force. Later evaluations were carried out with modified program and automatic measurement system.

Test stand designed on similar principle in Škoda Works Plzeň [5] is shown in Fig.7. As differs from above described stands, the shaft is supported not in sliding but in rolling bearings. Dynamic excitation force does not act on the test bearing, but on the shaft, which to great extent complicates stand design. The principle of determining stiffness and damping coefficients by means of response to excitation in two different directions was preserved. The stand presented in Fig. 7 was destined for bearings 160 mm in diameter and speeds up to 3500 rpm. Harmonic force was generated by mechanical exciter with two counter rotating unbalances. For change of excitation force direction the exciter should be dismantled and oriented into other direction, which not only prolonged test times, but also made very difficult to adjust the same operating conditions for both directions of excitation.

In 90ies was in research institute Škoda built similar stand for tests of bearings up to 560 mm, destined for 1000 MW turbine of nuclear power plant Temelín [6]. As well as in stand for bearings 160 mm in diameter, dynamic force generated by mechanical exciter acts on test shaft.

Firm Mitsubishi Heavy Industry built stand similar to that of professor Glienicke for bearings up to 280 mm in diameter and speeds up to 10.000 rpm [8], which is schematically shown in Fig. 8. Hydraulic exciters 6 were used for dynamic harmonic force generation instead of mechanical ones.

Fig. 8 Test stand Mitsubishi Heavy Industry

Rotor Kit (RK) Bently Nevada, with which are equipped Czech technical universities, was utilized for identification of aerostatic bearings dynamic properties. RK enables controlled drive of test shaft up to 10.000 rpm. Super-structure of RK was designed (Fig. 9), with rigid shaft 1


between two precision rolling bearings 18, mounted in bearing bodies 2, 3 fastened to the frame of RK. Test bearing head 4 with tested aerostatic bearing 5 is situated between rolling bearings. Parallelism of test bearing axis with shaft axis is ensured by suspensions 35, embedded in circular plates 31 fastened to the frame of RK. Static and harmonic dynamic loads are generated by a pair of piezoactuators 12, which are connected with test head through joint hinges 7. Piezoactuators, oriented in vertical and horizontal directions, enable not only excitation of test bearing by dynamic harmonic force, but also very precisely controlled static shift, by which static load of test bearing can be realized. Deviations of test bearing relative to shaft are scanned by two pairs of relative sensors S3, S4 and S1, S2 (in horizontal direction). Components of dynamic forces are evaluated by load cells 13, inserted between piezoactuators and test bearing. Detailed description of test stand can be found in [12].

Fig. 9 Super-structure of Rotor Kit for identification of aerostatic bearings dynamic properties

5.4. Evaluation of stiffness and damping coefficients determined by measuring the response to harmonic excitation in two directions

The basis for evaluation of stiffness and damping coefficients constitute equations of movements of test bearing and shaft according to general dynamic model in Fig. 10.

The model considers apart from stiffness and damping of test bearing also stiffness and damping of bearings, in which the shaft is supported, and stiffness and damping of elastic support of the frame (e.g. in silent-blocks).


Fig. 10 Dynamic model for evaluation of stiffness and damping elements

Equation of test bearing movement:

[ ][ ] [ ][ ] [ ]kr fZxM =+ χ222 . && , k=1,2, (5-3)

where [ ]

=

2

22 ,0

0,

M

MM , M2 … mass of test bearing body,

[ ]

=

2

22 y

xχ , [ ]

−−

=

=

12

12

yy

xx

y

x

r

rrχ ,

[ ]

=

yyyxx

xyxx

ZZ

ZZZ 22

222

,

, … matrix of test bearing complex stiffness,

jkjkjk BiKZ Ω+=2 , 1−=i ,

[ ] ti

d

d eF

Ff Ω

=

1

11 2

2, [ ] ti

d

d eF

Ff Ω

−=

2

22 2

2.

Equation of shaft movement:

[ ][ ] [ ] [ ] [ ]( ) [ ][ ] 02 231

111 =−−+ rZZxM χχχ&& , (5-4)

where [ ]

=

1

11 ,0

0,

M

MM , M1 … shaft mass,

[ ] [ ] [ ]ry

xχχχ −=

= 2

2

22 , [ ] [ ] [ ] [ ] [ ]32

31

3131 χχχχχ −−=

−−

=− ryy

xx,

[ ]

=

yyyxx

xyxx

ZZ

ZZZ 11

111

,

, … matrix of support bearing complex stiffness,

Equation of frame movement:

[ ][ ] [ ] [ ] [ ]( ) [ ][ ] [ ]kfZZM −=−−− 32

311

33 ..2. χχχχ&& , k=1,2, (5-5)


where [ ]

=

3

33 ,0

0,

M

MM ,

[ ]

=

3

33 y

xχ ,

[ ]

=

y

x

Z

ZZ 3

33

,0

0, ... matrix of the frame complex stiffness.

The solution is assumed in the form

[ ] [ ] tijj e Ω= χχ , j=1,2,3, (5-6)

[ ] [ ] tijj e ΩΩ−= χχ 2

&& ,

and after substitution into (4-1), (4-2) a (4-3) we get

[ ][ ] [ ]kfZ =χ. , k=1,2, (5-7)

where [ ][ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ]

++Ω−−−+Ω−−−Ω

Ω−=

ZZMZZ

ZZMZZM

MZ

Z31

3211

111

2211

22

22

2,2,2

2,2,2

0,,

,

[ ][ ][ ][ ]

=

3

2

χχχ

χr

,

[ ]

−−

=

1

1

0

0

1

1

2

211 dFf , [ ]

−−

=

1

1

0

0

1

1

2

222 dFf .

To evaluate complex stiffness of test bearing it is sufficient to solve equation (5-3) for vectors complex amplitudes [χ2], [χr] measured at two different directions of excitation force Fd1, Fd2.

Drawback of above described method is great scatter of measured values of cross-coupled elements of damping matrix, caused by measurement errors. Cross-coupled elements of stiffness matrix, which had fundamental influence on rotor stability, though exhibit relatively good agreement with results of calculation [e.g. 11]. When comparing calculated stability limits with values measured on real rotors it was established, that calculated values are generally lower than experimental ones. Rotors thus operate with greater safety, which is favourable. However, in some cases it is necessary to choose bearings with better dynamic properties, which are more demanding as concerns manufacture (mostly tilting pad) even there, where it may not by necessary.


5.5 Some results of bearing dynamic properties measurement

Extensive measurement of dynamic characteristics of journal bearing 90 and 350 mm in diameter was carried out in 80ies at SVÚSS. Results from the test stand of bigger diameter were not much satisfactory due to unsuitable stand design. Test stand for bearings 90 mm in diameter, which was designed especially for this purpose, provided quite good results enabling to verify accuracy of theoretical solution.

Stiffness element Kxx vers. speed

1,0E+08

1,5E+08

2,0E+08

2,5E+08

3,0E+08

3,5E+08

4,0E+08

4,5E+08

0 5000 10000 15000 20000 25000speed (rpm)

Kxx

(N

/m)

theor. 0,5 MPatheor. 1,0 MPatheor. 1,5 MPaexp. 0,5 MPaexp. 1,0 MPaexp. 1,5 MPa

Stiffness element Kyy vers. speed

1,0E+07

2,0E+07

3,0E+07

4,0E+07

5,0E+07

6,0E+07

0 5000 10000 15000 20000 25000speed (rpm)

Kyy

(N

/m)


Fig. 11 Calculated and measured main stiffness elements vers. speed - lemon bore bearing 90 mm in diameter, l/D=0,7, preload 0,67

Damping element Bxx vers. speed

0,0E+00

1,0E+05

2,0E+05

3,0E+05

4,0E+05

5,0E+05

6,0E+05

7,0E+05

0 5000 10000 15000 20000 25000speed (rpm)

Bxx

(N

.s/m

)

theor. 0,5 MPa

theor. 1,0 MPa

theor. 1,5 MPa

exp. 0,5 MPa

exp. 1,0 MPa

exp. 1,5 MPa

Damping element Byy vers. speed

1,0E+04

3,0E+04

5,0E+04

7,0E+04

9,0E+04

1,1E+05

1,3E+05

0 5000 10000 15000 20000 25000

speed (rpm)

Byy

(N

.s/m

)

theor. 0,5 MPa

theor. 1,0 MPa

theor. 1,5 MPa

exp. 0,5 MPa

exp. 1,0 MPa

exp. 1,5 MPa

Fig. 12 Calculated and measured main damping elements vers. speed - lemon bore bearing 90 mm in diameter, l/D=0,7, preload 0,67


Figures 11 and 12 present calculated and measured main stiffness and damping elements of lemon bore bearing 90 mm in diameter with l/D=0,7, relative clearance of 2.10-3 and preload 0,67. Bearing was lubricated by VG46 oil with inlet temperature of 35ºC, specific load varied from 0,5 to 1,5 MPa. Measurement was carried out in speed range from 5.000 to 20.000 rpm, to which corresponds to sliding speed from 20 to 80 m.s-1. It can be seen from comparison of calculated and experimental values, that differences increase with growing specific load and that agreement with Kxx and Bxx elements, which are higher, is better than that with lower elements Kyy and Byy. The same holds for cross-coupling stiffness elements Kxy and Kyx, calculated and measured values of which are presented in Fig.. 13.

Stiffness element Kxy vers. speed

3,0E+07

5,5E+07

8,0E+07

1,1E+08

1,3E+08

1,6E+08

1,8E+08

0 5000 10000 15000 20000 25000

speed (rpm)

Kxy

(N

/m)

theor. 0,5 MPatheor. 1,0 MPatheor. 1,5 MPaexp. 0,5 MPaexp. 1,0 MPaexp 1,5 MPa

Stiffness element Kyx vers. speed

-2,0E+08

-1,5E+08

-1,0E+08

-5,0E+07

0,0E+00

5,0E+07

1,0E+08

0 5000 10000 15000 20000 25000speed (rpm)

Kyx

(N

/m)


Fig. 13 Calculated and measured cross-coupling stiffness elements vers. speed - lemon bore bearing 90 mm in diameter, l/D=0,7, preload 0,67

The worst agreement between calculated and measured values is encountered with cross-coupling damping elements Bxy and Byx. Due to complexity of identification and satisfactory agreements between theory and experiment of the main elements the differences ascertained with cross-coupling damping elements evidence rather measuring errors than inaccuracy of computer program.

Stiffness and damping elements of tilting pad bearings depend not only on speed, but also on excitation frequency, resp. i.e. on the ratio of excitation and rotation frequencies. Experimental research of tilting pad bearings 90 mm in diameter this influence was therefore considered too. Fig. 14 and 15 show dependence of main stiffness and damping elements on excitation frequency for bearing 90 mm in diameter, with 5 pads, l/D ratio 0,4 and load directed on pad. Bearing with relative clearance of 2.10-3 and preload of 0,5 was lubricated by VG46 oil with inlet temperature of 35ºC. Bearing specific load was 1 MPa, excitation ration was varied between about 0,1 and 2,4.

As can be both seen from Fig. 14, which presents main stiffness elements, and Fig. 15, which shows main damping elements, dependence of dynamic elements on excitation frequency is not much significant. Better agreement between calculation and experiment is achieved with stiffness element Kxx and damping element Byy. Somewhat paradoxical is better agreement at higher speeds, which however can be caused by smaller range of excitation ratio at this speed (excitation frequency was limited by mechanical vibrator).


Stiffness element Kxx vers. excitation frequency ratio

1,0E+07

1,0E+08

1,0E+09

0,00 0,50 1,00 1,50 2,00 2,50

fb/fo

Kxx

(N

/m)

exper. 3000 rpm

exper. 5000 rpm

exper. 10000 rpm

calcul. 3000 rpm

calcul. 5000 rpm

calcul. 10000 rpm

Stiffness element Kyy vers. excitation frequency ratio

1,0E+07

1,0E+08

1,0E+09

0,00 0,50 1,00 1,50 2,00 2,50fb/fo

Kyy

(N

/m) exper. 3000 rpm

exper. 5000 rpm

exper. 10000 rpm

calcul. 3000 rpm

calcul. 5000 rpm

calcul. 10000 rpm

Fig. 14 Calculated and measured main stiffness elements vers. excitation ratio – tilting pad bearing, relative clearance 2.10-3, preload 0,5

Damping element Bxx vers. excitation frequency ratio

1,0E+04

1,0E+05

1,0E+06

0,00 0,50 1,00 1,50 2,00 2,50

fb/fo

Bxx

(N

.s/m

)

exper. 3000 rpm

exper. 5000 rpm

exper. 10000 rpm

calcul. 3000 rpm

calcul. 5000 rpm

calcul. 10000 rpm

Damping element Byy vers. exritation frequency ratio

1,0E+04

1,0E+05

1,0E+06

0,00 0,50 1,00 1,50 2,00 2,50

fb/fo

Byy

(N

.s/m

)

exper. 3000 rpm

exper. 5000 rpm

exper. 10000 rpm

calcul. 3000 rpm

calcul. 5000 rpm

calcul. 10000 rpm

Fig. 15 Calculated and measured main damping elements vers. excitation ratio – tilting pad bearing, relative clearance 2.10-3, preload 0,5

With the use of Rotor Kit Bently Nevada (sect. 5.3) were in terms of grant project of Czech Science foundation identified dynamic elements of aerostatic bearings [15]. In comparison with above-mentioned hydrodynamic bearings, with stiffness and damping elements at least 2 orders higher, identification with aerostatic bearings was much more difficult. Cross-coupling terms of stiffness and damping, which are due to maximum achievable speed of 6.000 rpm in aerostatic bearings very small, could not be identified at all. Main damping elements were identified, but their scatter was relatively big. Main elements of stiffness could be determined with relatively small scatter, which is documented by Table 2. The table presents identified values of main stiffness for 2 values of inlet pressure and a series of rotating speeds and excitation frequencies.


Table 2 Measured and calculated main stiffness elements of aerostatic bearing 30 mm in diameter, l/D=1,5, relative clearance 2,67.10-3, 2 rows of orifices 0,2 mm in diameter, 8 orifices along periphery

Inlet pressure (MPa)

speed (rpm)

excitation frequency

(Hz)

measurement calculation

Kxx (N.m-1)

Kyy (N.m-1)

Kxx (N.m-1)

Kyy (N.m-1)

0,2 0 5 9,52E+05 9,65E+05 1,02E+06 1,00E+06

1000 16,7 / 1 9,88E+05 9,79E+05 1,02E+06 1,01E+06

2000 66,7 / 2 9,30E+05 9,79E+05 1,02E+06 1,01E+06

3000 25 / 0,5 1,00E+06 9,66E+05 1,02E+06 1,01E+06

4000 33,3 / 0,5 9,84E+05 9,77E+05 1,02E+06 1,02E+06

5000 41,7 /0,5 9,51E+05 9,71E+05 1,02E+06 1,01E+06

0,4 0 5 1,42E+06 1,41E+06 1,52E+06 1,51E+06

1000 8,3 / 0,5 1,43E+06 1,45E+06 1,52E+06 1,51E+06

2000 16,7 / 0,5 1,43E+06 1,44E+06 1,52E+06 1,51E+06

3000 25 / 0,5 1,42E+06 1,42E+06 1,53E+06 1,51E+06

4000 33,3 /0,5 1,44E+06 1,45E+06 1,53E+06 1,51E+06

5000 166,7 / 2 1,27E+06 1,72E+06 1,53E+06 1,51E+06

Experimentally determined values of main stiffness elements quite well agree with values calculated after revision of computer program, which are also given in the table. They are in accord also with the so called „quasi-static“ stiffness, which is obtained from dependence of deviation on load for very low excitation frequency (0,5 Hz).

Conclusions

With growing speeds of rotors, which results from effort to decrease machine dimensions, bearing dynamic properties gain greater importance. Still more frequent are cases of rotor instability, caused by destabilizing effect of labyrinth seals. Assessing of bearing dynamic characteristics by means of experimental values is directly related to these problems. Work [13] contains almost 300 references to publications dealing exclusively with problematic of sliding bearings dynamic properties. Apart from already mentioned methods, identification methods with the help of impact hammer or unknown excitation are used, which are however even more demanding from the standpoint of evaluation methodology than above-described methods.

Present computational methods, considering variations of lubricant viscosity in bearing gap and eventually respect even deformation of sliding surfaces, provide creditable values of stiffness and damping coefficients. Identification of stiffness and damping elements on test stand is very demanding as concerns accuracy of measurement and evaluation methods. Nevertheless it constitutes the only possibility of computational methods verification and therefore it is necessary to pursue these activities further.


References:

[1] Garner, D.R.- Lee, C.S.-Martin, F.A.: Stability of profile bore bearings: influence of bearing type selection, Tribology International, Oct. 1980, p.204

[2] Šimek, J.: Guide for calculation of hybrid gas journal bearings (in Czech). Research report No. SVÚSS 80-03005

[3] Woodcock, J. S. – Holmes, R.: Determination and application of the dynamic properties of turbo-rotor bearing oil film.

[4] Muszynska, A.: Synchronous dynamic stiffness testing. Bently Nevada

[5] Zmeko, J.: Experimental investigation of static and dynamic properties of oil layer of circular and lemon journal bearings (in Czech).

Technical report No. SV 3574

[6] Zmeko, J.: Experimental investigation of 4-pad bearing 560 mm in diameter; L/D=0,82 for TG - 1000 MW machine. (in Czech) Technical report No. VZVU 0357

[7] Glienicke, J.: Feder- und Dämfungskonstanten von Gleitlagern für Turbomaschinen und deren Einfluss auf das Schwingungsverhalten eines einfachen Rotors

Dissertation, Technischen Hochschule Karlsruhe, 1966

[8] Matsumoto, I. at al.: Oil film and vibration characteristics of offset-halves journal bearing

[9] Šimek, J.: Modification of Rotor Kit Bently Nevada for identification of aerostatic bearings dynamic properties. Technical report TECHLAB No. 06-409, 2006 (in Czech)

[10] Šimek, J.: Dynamic model of test stand for journal bearings 90 mm in diameter. Methodology of measurement and evaluation (in Czech). Research report No. SVÚSS 82-03004

[11] Šimek, J. – Pelnář, I.: Summary of results of experimental investigation of journal bearings 90 and 350 mm in diameter (in Czech). Research report No. SVÚSS 89-03005

[12] Hlaváč, Z.- Zeman,V.: Contribution to identification of stiffness and damping coefficients of oil-film bearings. Colloquium Dynamic of Machines 2005

[13] Tiwari, R.- Lees, A. V.- Friswell M. I.: Identification of Dynamic Bearing Parameters: A Review. The Shock and Vibration Digest, March 2004

[14] Šimek, J.: Verification of Rotor Kit Bently Nevada super-structure function as a means for identification of aerostatic bearings dynamic properties.

Technical report TECHLAB No. 07-407, 2007 (in Czech).

[15] Šimek, J. – Kozánek, J.: Identification of aerostatic bearings dynamic Properties. Part 2: Comparison of measured static and dynamic characteristics with calculation. Description of solution methodology and computer program revision.

Technical report TECHLAB No. 08-411, 2008 (in Czech).

dynamic properties of plain journal bearings · dynamic properties of plain journal bearings ......

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