Vatta, Furio

Vigliani, Alessandro

English

VATTA, Furio

VIGLIANI, Alessandro

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

05 August 2020POLITECNICO DI TORINORepository ISTITUZIONALELubricated bearings: determination of dynamic coefficients according to Warner’s theory / VATTA F; VIGLIANI A.. - In:MECCANICA. - ISSN 0025-6455. - 34 (4)(1999), pp. 291-294.OriginalLubricated bearings: determination of dynamic coefficients according to Warner’s theoryPublisher:PublishedDOI:10.1023/A:1004700724760Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/1406808 since:Post print (i.e. final draft post-refereeing) version of an article published on Meccanica. Beyond the journalformatting, please note that there could be minor changes from this document to the final published version.The final published version is accessible from here:http://dx.doi.org/10.1023/A:1004700724760This document has made accessible through PORTO, the Open Access Repository of Politecnico di Torino(http://porto.polito.it), in compliance with the Publisher’s copyright policy as reported in the SHERPA-ROMEO website:http://www.sherpa.ac.uk/romeo/issn/0025-6455/Lubricated bearings: determination of dynamic coefficientsaccording to Warner’s TheoryF. Vatta, A. ViglianiDipartimento di Meccanica - Politecnico di TorinoC.so Duca degli Abruzzi, 24 - 10129 Torino - ITALYE-mail: alessandro.vigliani@polito.itKeywords Journal bearing, Dynamic coefficients, Mechanics of machinesAbstract1NomenclatureD = Journal diameter n = Eccentricity ratio δ = Radial clearanceL = Journal length n˙ = Squeeze speed ε = EccentricityW = Load z = Axial co–ordinate µ = Dynamic viscosityh = Fluid film thickness β˙ = Whirl speed ω = Angular speed()0 = Stationary configuration1 IntroductionIt is well known that rotors supported by lubricated bearings can undergo instability due, inmost cases, to the forces exerted by the fluid. Stability is generally analyzed using two differentapproaches. One approach consists of numerical integration of motion equations so that the rotorbehavior can be analyzed as a function of time, while the other approach is characterized by thelinearization of the fluid forces acting on the bearings in the neighborhood of a well determinedstationary equilibrium position so that asymptotic stability conditions can be determined. In thelatter case it is necessary to know the analytical expressions of the stiffness and damping coefficientsto linearize the hydrodynamic forces.It is evident that the first approach can provide more information to study the rotor dynamicbehavior, e.g. the possibility to compute minimum fluid film thickness; moreover, this method isnot affected by the consequences of linearization. However, it is worth noting that the resultingnumerical code can be complex. It is well known the argument arisen between Trumpler andPoritsky [1] about the linearization necessary to determine the dynamic coefficients. Howeverthe authors agree with Poritsky when he says that “while the effect of nonlinearity of the oil–film forces may limit the whirl amplitude to a finite value, ( ..) nonlinearity by itself can neverrestore complete stability in a range where the linear theory indicates instability” [1]. Therefore, ifdetermination of the system’s stability is the only goal, it is undoubtedly more convenient to usedynamic coefficients.The analytical expressions of dynamic coefficients can be determined from Ocvirk’s theory,(short bearing), or from Sommerfeld’s solution (infinitely long bearing) or from Warner’s solution(finite bearing), consisting of Sommerfeld’s solution corrected through a suitable “end leakagefunction”.In the literature it is possible to find the expressions for the dynamic coefficients for Ocvirk’ssolution [2] and for Sommerfeld’s solution [3], but, to the authors’ knowledge, expressions of thedynamic coefficients for Warner’s solution have never been published. This fact is probably due tothe tedious nature of the analysis more than to numerical difficulties. Hence, the aim of the presentwork is to determine, in a stationary reference frame, the analytical expressions of the dynamiccoefficients according to Warner’s theory, giving a further tool for stability analysis.2 Warner’s modelWarner’s model [4] is based on the hypothesis that the oil pressure inside the bearing can be writtenasp(ϑ, z) = p∞(ϑ)[1− cosh(2zLA)/cosh (A)](1)where A = λL/D, z is the axial co–ordinate with the origin located at the middle of the bearingandλ2 =∫ ϑ2ϑ1h3(dp∞dϑ)2dϑ/∫ ϑ2ϑ1h3p2∞ dϑ. (2)The angular displacements ϑ1 and ϑ2 identify the bearing arc where pressure is positive andcan be determined by solving the systemn sinϑ1(ω − 2β˙)− n˙ (2 + n2) cosϑ1 = 0n cosϑ1(ω − 2β˙)+ n˙(2 + n2)sinϑ1 ≥ 0ϑ2 = ϑ1 + pi(3)2In eq.(2), p∞(ϑ) represents the pressure corresponding to an infinitely long bearing, given byp∞ − p(0) = 6µ(D2δ)2 [(ω − 2β˙) n sinϑ2 + n2− n˙ cosϑ]2 + n cosϑ(1 + n cosϑ)2 (4)The radial and tangential components of the oil force ~W acting on the journal are:Wr = −3µD(D2δ)2 [(ω − 2β˙) n2 + n2Ir1 − n˙Ir2]F (λ) (5)Wϑ = −3µD(D2δ)2 [(ω − 2β˙) n2 + n2Iϑ1− n˙Iϑ2]F (λ) (6)whereIr1= Iϑ2=∫ ϑ2ϑ1sinϑ cosϑ2 + n cosϑ(1 + n cosϑ)2 dϑ∂ Ir2 =∫ ϑ2ϑ1cos2 ϑ2 + n cosϑ(1 + n cosϑ)2 dϑ (7)Iϑ1=∫ ϑ2ϑ1sin2 ϑ2 + n cosϑ(1 + n cosϑ)2 dϑ F (λ) = L[1− tanh (A)A]= LG0 (8)The forces components given in Eqs. (5) and (6), in a stationary reference system (x, y), assumethe following expressions:Wx = Wr cos γ −Wϑ sin γ and Wy = Wr sin γ +Wϑ cos γ (9)It is then possible to compute the stiffness and damping dynamic coefficients, Kij and Cijrespectively, in any given equilibrium configuration. It is worth noting that the parameter λ is afunction of n, n˙ and β˙; here we give expressions of λ and of its partial derivatives.(λ)0 =√2n2 (2 + n2)(1− n2)1/2 (2 + 3n2)− 2 (1− n2)(∂λ∂β˙)0= 0 (10)(∂λ∂n)0=4(1− n2)1/2 (n4 − 2n2 − 2)+ 8 + 4n2 + 6n4 − 3n6[2 (2 + n2)]1/2(1− n2)5/4[2 + 3n2 − 2 (1− n2)1/2]3/2 = G1 (11)(∂λ∂n˙)0= 22 + n21− n24n2 + n4 + λ2n2(1− n2) [23+2n2+1n3ln1− n1 + n]ωλpi[2 + 3n2 − 2 (1− n2)1/2] = G3 (12)In order to simplify the coefficient expressions, let us define:G2 =G1λ2[Acosh2 (A)− tanh (A)]and G4 =G3λ2[Acosh2 (A)− tanh (A)](13)K¯ij = Kij4µDω(δL)3and C¯ij = Cij4µD(δL)3(14)3Finally, the following expressions hold for the dynamic coefficients:K¯xx = 3(DL)2G02n[pi2(2− n2)+ 4n2](1− n2) (2 + n2) [4n2 + pi2 (1− n2)] (15)K¯xy = 3(DL)2G0 pipi2(1− n2)2 − 4n4(1− n2)3/2 (2 + n2) [4n2 + pi2 (1− n2)](16)K¯yx = 3(DL)2 {DLG2pi n(1− n2)1/2 (2 + n2)+ (17)−G0 pi8n2(2 + n4)+ pi2(2− 3n2 + 3n4 − 2n6)(1− n2)3/2 (2 + n2)2 [4n2 + pi2 (1− n2)]}K¯yy = 3(DL)2 {−DLG22n2(1− n2) (2 + n2)+ (18)+G016n3(2 + n4)+ 2npi2(2− 3n2 + 3n4 − 2n6)(1− n2)2 (2 + n2)2 [4n2 + pi2 (1− n2)]}C¯xx = 3(DL)2G0pi(pi2 − 4n2)(1− n2)1/2 [4n2 + pi2 (1− n2)](19)C¯xy = 3(DL)2G02n(4n2 − pi2)(1− n2) [4n2 + pi2 (1− n2)] (20)C¯yx = 3(DL)2{DLωG4pi n(1− n2)1/2 (2 + n2)+ (21)−G02n(8n2 − 2pi2 − 2pi2 n2 + pi2 n4)(1− n2) (2 + n2) [4n2 + pi2 (1− n2)]}C¯yy = 3(DL)2 {− DLωG42n2(1− n2) (2 + n2)+ (22)+2piG02n2(6− 2n2 − n4)+ pi2 (1− n2)2(1− n2)3/2 (2 + n2) [4n2 + pi2 (1− n2)]}References[1] Gunter E. J., Dynamic Stability of Rotor–Bearing Systems, NASA SP 113, 1966, p.23.[2] Kra¨mer E., 1993, Dynamics of Rotors and Foundations, Springer - Verlag, Berlin, p.88.[3] Myers C. J., 1984, “Bifurcation Theory Applied to Oil Whirl in Plain Cylindrical JournalBearings”, J. Applied Mechanics, 51, p.250.[4] Warner P. C., 1963, “Static and Dynamic Properties of Partial Journal Bearings”, J. BasicEngineering, 85, p.247.4

Lubricated bearings: determination of dynamic coefficients according to Warner’s theory

PORTO Publications Open Repository TOrino

Post print (i.e. final draft post-refereeing) version of an article published on Meccanica. Beyond the journal
formatting, please note that there could be minor changes from this document to the final published version.
The final published version is accessible from here:
http://dx.doi.org/10.1023/A:1004700724760
This document has made accessible through PORTO, the Open Access Repository of Politecnico di Torino
(http://porto.polito.it), in compliance with the Publisher’s copyright policy as reported in the SHERPA-
ROMEO website:
http://www.sherpa.ac.uk/romeo/issn/0025-6455/
Lubricated bearings: determination of dynamic coefficients
according to Warner’s Theory
F. Vatta, A. Vigliani
Dipartimento di Meccanica - Politecnico di Torino
C.so Duca degli Abruzzi, 24 - 10129 Torino - ITALY
E-mail: alessandro.vigliani@polito.it
Keywords Journal bearing, Dynamic coefficients, Mechanics of machines
Abstract
1
Nomenclature
D = Journal diameter n = Eccentricity ratio δ = Radial clearance
L = Journal length n˙ = Squeeze speed ε = Eccentricity
W = Load z = Axial co–ordinate µ = Dynamic viscosity
h = Fluid film thickness β˙ = Whirl speed ω = Angular speed
()0 = Stationary configuration
1 Introduction
It is well known that rotors supported by lubricated bearings can undergo instability due, in
most cases, to the forces exerted by the fluid. Stability is generally analyzed using two different
approaches. One approach consists of numerical integration of motion equations so that the rotor
behavior can be analyzed as a function of time, while the other approach is characterized by the
linearization of the fluid forces acting on the bearings in the neighborhood of a well determined
stationary equilibrium position so that asymptotic stability conditions can be determined. In the
latter case it is necessary to know the analytical expressions of the stiffness and damping coefficients
to linearize the hydrodynamic forces.
It is evident that the first approach can provide more information to study the rotor dynamic
behavior, e.g. the possibility to compute minimum fluid film thickness; moreover, this method is
not affected by the consequences of linearization. However, it is worth noting that the resulting
numerical code can be complex. It is well known the argument arisen between Trumpler and
Poritsky [1] about the linearization necessary to determine the dynamic coefficients. However
the authors agree with Poritsky when he says that “while the effect of nonlinearity of the oil–
film forces may limit the whirl amplitude to a finite value, ( ..) nonlinearity by itself can never
restore complete stability in a range where the linear theory indicates instability” [1]. Therefore, if
determination of the system’s stability is the only goal, it is undoubtedly more convenient to use
dynamic coefficients.
The analytical expressions of dynamic coefficients can be determined from Ocvirk’s theory,
(short bearing), or from Sommerfeld’s solution (infinitely long bearing) or from Warner’s solution
(finite bearing), consisting of Sommerfeld’s solution corrected through a suitable “end leakage
function”.
In the literature it is possible to find the expressions for the dynamic coefficients for Ocvirk’s
solution [2] and for Sommerfeld’s solution [3], but, to the authors’ knowledge, expressions of the
dynamic coefficients for Warner’s solution have never been published. This fact is probably due to
the tedious nature of the analysis more than to numerical difficulties. Hence, the aim of the present
work is to determine, in a stationary reference frame, the analytical expressions of the dynamic
coefficients according to Warner’s theory, giving a further tool for stability analysis.
2 Warner’s model
Warner’s model [4] is based on the hypothesis that the oil pressure inside the bearing can be written
as
p(ϑ, z) = p∞(ϑ)
[
1− cosh
(
2z
L
A
)
/cosh (A)
]
(1)
where A = λL/D, z is the axial co–ordinate with the origin located at the middle of the bearing
and
λ2 =
∫ ϑ2
ϑ1
h3
(
dp∞
dϑ
)2
dϑ/
∫ ϑ2
ϑ1
h3p2∞ dϑ. (2)
The angular displacements ϑ1 and ϑ2 identify the bearing arc where pressure is positive and
can be determined by solving the system
n sinϑ1
(
ω − 2β˙
)
− n˙ (2 + n2) cosϑ1 = 0
n cosϑ1
(
ω − 2β˙
)
+ n˙
(
2 + n2
)
sinϑ1 ≥ 0
ϑ2 = ϑ1 + pi
(3)
2
In eq.(2), p∞(ϑ) represents the pressure corresponding to an infinitely long bearing, given by
p∞ − p(0) = 6µ
(
D
2δ
)2 [(
ω − 2β˙
) n sinϑ
2 + n2
− n˙ cosϑ
]
2 + n cosϑ
(1 + n cosϑ)
2 (4)
The radial and tangential components of the oil force ~W acting on the journal are:
Wr = −3µD
(
D
2δ
)2 [(
ω − 2β˙
) n
2 + n2
Ir1 − n˙Ir2
]
F (λ) (5)
Wϑ = −3µD
(
D
2δ
)2 [(
ω − 2β˙
) n
2 + n2
I
ϑ1
− n˙I
ϑ2
]
F (λ) (6)
where
I
r1
= I
ϑ2
=
∫ ϑ2
ϑ1
sinϑ cosϑ
2 + n cosϑ
(1 + n cosϑ)
2 dϑ∂ Ir2 =
∫ ϑ2
ϑ1
cos2 ϑ
2 + n cosϑ
(1 + n cosϑ)
2 dϑ (7)
I
ϑ1
=
∫ ϑ2
ϑ1
sin2 ϑ
2 + n cosϑ
(1 + n cosϑ)
2 dϑ F (λ) = L
[
1− tanh (A)
A
]
= LG0 (8)
The forces components given in Eqs. (5) and (6), in a stationary reference system (x, y), assume
the following expressions:
Wx = Wr cos γ −Wϑ sin γ and Wy = Wr sin γ +Wϑ cos γ (9)
It is then possible to compute the stiffness and damping dynamic coefficients, Kij and Cij
respectively, in any given equilibrium configuration. It is worth noting that the parameter λ is a
function of n, n˙ and β˙; here we give expressions of λ and of its partial derivatives.
(λ)0 =
√
2n2 (2 + n2)
(1− n2)1/2 (2 + 3n2)− 2 (1− n2)
(
∂λ
∂β˙
)
0
= 0 (10)
(
∂λ
∂n
)
0
=
4
(
1− n2)1/2 (n4 − 2n2 − 2)+ 8 + 4n2 + 6n4 − 3n6
[2 (2 + n2)]
1/2
(1− n2)5/4
[
2 + 3n2 − 2 (1− n2)1/2
]3/2 = G1 (11)
(
∂λ
∂n˙
)
0
= 2
2 + n2
1− n2
4n2 + n4 + λ2n2
(
1− n2) [2
3
+
2
n2
+
1
n3
ln
1− n
1 + n
]
ωλpi
[
2 + 3n2 − 2 (1− n2)1/2
] = G3 (12)
In order to simplify the coefficient expressions, let us define:
G2 =
G1
λ2
[
A
cosh2 (A)
− tanh (A)
]
and G4 =
G3
λ2
[
A
cosh2 (A)
− tanh (A)
]
(13)
K¯ij = Kij
4
µDω
(
δ
L
)3
and C¯ij = Cij
4
µD
(
δ
L
)3
(14)
3
Finally, the following expressions hold for the dynamic coefficients:
K¯xx = 3
(
D
L
)2
G0
2n
[
pi2
(
2− n2)+ 4n2]
(1− n2) (2 + n2) [4n2 + pi2 (1− n2)] (15)
K¯xy = 3
(
D
L
)2
G0 pi
pi2
(
1− n2)2 − 4n4
(1− n2)3/2 (2 + n2) [4n2 + pi2 (1− n2)]
(16)
K¯yx = 3
(
D
L
)2 {
D
L
G2
pi n
(1− n2)1/2 (2 + n2)
+ (17)
−G0 pi
8n2
(
2 + n4
)
+ pi2
(
2− 3n2 + 3n4 − 2n6)
(1− n2)3/2 (2 + n2)2 [4n2 + pi2 (1− n2)]
}
K¯yy = 3
(
D
L
)2 {
−D
L
G2
2n2
(1− n2) (2 + n2)+ (18)
+G0
16n3
(
2 + n4
)
+ 2npi2
(
2− 3n2 + 3n4 − 2n6)
(1− n2)2 (2 + n2)2 [4n2 + pi2 (1− n2)]
}
C¯xx = 3
(
D
L
)2
G0
pi
(
pi2 − 4n2)
(1− n2)1/2 [4n2 + pi2 (1− n2)]
(19)
C¯xy = 3
(
D
L
)2
G0
2n
(
4n2 − pi2)
(1− n2) [4n2 + pi2 (1− n2)] (20)
C¯yx = 3
(
D
L
)2{
D
L
ωG4
pi n
(1− n2)1/2 (2 + n2)
+ (21)
−G0
2n
(
8n2 − 2pi2 − 2pi2 n2 + pi2 n4)
(1− n2) (2 + n2) [4n2 + pi2 (1− n2)]
}
C¯yy = 3
(
D
L
)2 {
− D
L
ωG4
2n2
(1− n2) (2 + n2)+ (22)
+2piG0
2n2
(
6− 2n2 − n4)+ pi2 (1− n2)2
(1− n2)3/2 (2 + n2) [4n2 + pi2 (1− n2)]
}
References
[1] Gunter E. J., Dynamic Stability of Rotor–Bearing Systems, NASA SP 113, 1966, p.23.
[2] Kra¨mer E., 1993, Dynamics of Rotors and Foundations, Springer - Verlag, Berlin, p.88.
[3] Myers C. J., 1984, “Bifurcation Theory Applied to Oil Whirl in Plain Cylindrical Journal
Bearings”, J. Applied Mechanics, 51, p.250.
[4] Warner P. C., 1963, “Static and Dynamic Properties of Partial Journal Bearings”, J. Basic
Engineering, 85, p.247.
4


Bifurcation Theory Applied to Oil Whirl in

Dynamic Stability of Rotor{Bearing Systems,

Dynamics of Rotors and Foundations,

Static and Dynamic Properties of

Lubricated bearings: determination of dynamic coefficients according to Warner's theory

https://iris.polito.it/retrieve/handle/11583/1406808/45936/1999%20Meccanica%20Warner_PORTO.pdf

Lubricated bearings: determination of dynamic coefficients according to Warner's theory

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