Complete modeling of hydrodynamic bearings with a

boundary parameterization approach

J.A. Motaa, D.J.G. Maldonadob, J.V. Valerioa, T.G. Rittob

aInstitute of Computing, UFRJ, Rio de Janeiro, BrazilbDepartment of Mechanical Engineering, UFRJ, Rio de Janeiro, Brazil

Abstract

The present work aims to revisit the simplifications made in the Navier-Stokes equations for the flow between two cylinders with a small thickness oflubricating oil film. Through a dimensionless analysis, the terms of theseequations are mapped and ordered by importance for the hydrodynamicbearing application. An effective parameterization of the geometry is pro-posed, enabling a more detailed description of the problem and its adapta-tion to other contexts. At the end, an elliptical partial differential equation isreached and solved by the centered finite difference method, whose solution isthe pressure field between the cylinders. To illustrate the effectiveness of theproposed approach, the model is applied to hydrodynamic bearings, wherethe pressure field and some parameters resulting from it, such as stiffnessand damping coefficients, are computed. Based on the facilities offered bythe parameterization of the geometry, two different configurations are pre-sented: (1) elliptical and (2) worn bearings. Their responses are evaluatedand a comparative analysis is performed. The modeling exposed in this text,as well as all its simulations were developed to integrate Ross-Rotordynamics,an open library in Python, available on the GitHub platform.

Keywords: Lubrication Theory, Geometry parameterization, NumericalSimulation, Hydrodynamic Bearings, Dynamic Coefficients

1. Introduction

The dynamics of rotating machinery has been actively studied since thebeginning of the last century. Many industries rely on the performance ofsuch machines. Any failure or malfunction affects the productivity of the

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process. Thus, it is of great importance to know in advance the behaviorof the machine under different operating conditions. In that study, specialattention has been drawn to elements with fluid-induced forces. This isthe case of hydrodynamic bearings and annular seals. These components arecomposed of a stator, a rotor and a fluid between them that reacts to externalloads applied to the rotor. Depending on the geometry and the operatingconditions, those forces may induce high vibration on the rotor. In general,the prediction of the rotor vibration under a certain operating condition isthe main concern of the investigations.

Journal bearings play an important role in rotating machines since theysupport the rotor and the external loads applied to it. Since the reactionforces of the fluid may increase the vibration of the machine, it is neces-sary to understand the bearings forces in order to know the behavior of themachine under different conditions. Based on this information, predictionsabout malfunctions or different observed phenomena can be understand andcorrected in advance.

One of the main concerns of the rotordynamics are the high vibrationsdue to instabilities and resonances. Rankine [1], Dunkerley [2], Foppl [3] andJeffcott [4] were the pioneers that developed investigations on the criticalspeeds of a spinning rotor supported by bearings. The objective of theirstudies was the stability of the rotor’s motion beyond the first critical speed.Since then, stability has been one of the most important characteristics ofrotating machines; specially today, where there is a need of high performancemachines with reliable operating conditions. A more detailed review of bib-liography can be found in the books of Childs [5], Muszynska [6], Tiwari [7],Vance [8] and Ishida and Yamamoto [9].

Regarding hydrodynamic bearings, the research began in the 1880s, withthe experimental investigations of Beauchamp Tower (1845-1904) and NicolaiP. Petrov (1836-1920). However, the theoretical basis for their experimentalobservation was only presented by Osborne Reynolds (1842- 1912). Thethree of them can be considered the founding fathers of the hydrodynamiclubrication and their findings are still used these days. Sommerfeld [10] foundan analytical expression for infinitely long bearings, but only Swift [11] andStieber [12] used realistic boundary conditions to the Reynolds equation.Ocvirk [13] developed a detailed solution to infinitely short bearings, whichis still widely used today. For the finite bearing, a strong effort was givenby Cameron and Wood [14] with numerical calculations without the aid ofelectronic devices. After the arrival of computers, Pinkus [15] was the first

2

researcher to use the computer for the numerical calculations of the solutionsfrom circular bearings, elliptical bearings and a three lobes bearing. In 1981,Singhal [16] applied finite differences to the Reynolds equation.

In addition to the lubrication theory, the studies carried out by Pinaand Carvalho [17], Andrade [18] and Queiroz [19] are relevant because theirwork has inspired the parameterization used in the description of the geome-try. The well-known collection of lecture notes recorded by San Andres [20],especially the 1 to 5 notes, were essential for understanding the context of hy-drodynamic bearings. The works of Frene et al [21], Hamrock [22] and Ishidaand Yamamoto [23] stand out, for this text, in what concerns the approxima-tions of the Reynolds equation. The three authors present basic concepts ofhydrodynamic bearing geometry and use the approximate equation for shortand infinitely long bearings. Finally, the research of Machado and Cavalca[24] presents solutions for bearings of different geometries. Machado [25] an-alyzes dynamic and operational characteristics of different configurations ofhydrodynamic radial bearings: a cylindrical, an elliptical and a trilobular.Already Machado and Cavalca [24] expose a numerical model that character-izes the bearing with wear on its wall, analyzing its influence on the dynamicresponse of the rotor.

In this article, a study of a fluid film flow between two surfaces is carriedout in a novel way. A new and effective parameterization of the geometryis presented to describe the journal bearing, allowing a more detailed de-scription of the problem. Instead of describing the geometry in terms of agap function, this work describes the geometry of both the rotor and sta-tor. Thus, different type of bearings can be modeled by simply defining thedesired shape for each surface. Once the domain is defined, the pressure dis-tribution of the fluid is obtained by means of the lubrication theory. First,the governing equations of a differential fluid element are determined by theNavier-Stokes equations and they are simplified to obtained the Reynoldsequation, whose solution is the pressure field of the fluid flow. This equa-tion is discretized by means of the finite differences method. The pressurefield is then obtained and verified with results available in the literature.Afterwards, the fluid forces are linearized around the equilibrium point ofthe rotor and the dynamic coefficients (stiffness and damping) are computedby the least squares method. This methodology is tested for three differentgeometries: (1) a cylindrical bearing, (2) a lemon bearing and a (c) bearingwith a worn stator. More details about the modeling presented in this textcan be found in Mota [26].

3

Figure 1: Schematic drawing of an eccentricity journal bearing

This article is organized as follows. In Section 2, the description of thefluid flow problem is stated and detailed. In Section 3, the model of the fluidflow is described and the governing equation relating the pressure field andthe geometrical characteristics is obtained. In Section 4, the pressure fieldis obtained by using the finite differences method. Then, in Section 5 thereaction forces of the fluid film are computed by a numerical integration ofthe pressure field in the whole domain. Section 6 shows the stiffness anddamping coefficients obtained with the least-square method. In Section 7,the model is explored and three geometries are analyzed: a short bearing, anelliptical bearing and a worn bearing. Finally, in Section 8, the conclusionsare thrown.

2. Problem description

In this article, the dynamics of the fluid flow in the annular space betweentwo cylinders is studied. This is a representation of a hydrodynamic bearing.The system is shown in Fig. 1. The external cylinder, with radius Ro iscalled stator and the internal cylinder, with radius Ri, is the rotor whichspins with a speed ω.

With the rotation speed, the rotor moves to an eccentric position. Anypoint on the rotor’s surface can be described as (see Fig. 1)

4

Rθ =√R2i − e2 sin2 α + e cosα , (1)

with

α =

3π

2− θ + β, if

π

2+ β ≤ θ <

3π

2+ β (I)

−(

3π

2− θ + β

), if

3π

2+ β ≤ θ <

5π

2+ β (II)

,

where e is the eccentricity, β is the attitude angle, Ro is the stator’s radiusand θ is the angle used to define Rθ. Note that θ is measured from an axispassing through the points of maximum and minimum fluid width. Also, itis important to note that the difference between the radius of the stator andthe rotor is very small compared to the radius of the stator.

3. Theoretical Modeling

The dynamic of the fluid between the rotor and stator is modeled usingthe Navier-Stokes equations and the continuity equation. For this problem,the equations are

ρ(∂v

∂t+ v · ∇v

)= ∇ · σ, (2a)

∂ρ

∂t+∇ · (ρv) = 0, (2b)

where ρ is the specific mass of the fluid; v is the velocity field vector in cylin-drical coordinates with components u (axial), v (radial) and w (tangential);and σ = −pI + τ is Cauchy’s tensor in terms of the pressure field p, thetension tensor τ and the identity tensor I.

Assuming an incompressible Newtonian fluid in a steady-state laminarflow, Eqs. (2) can be simplified [27]. First, ρ can be considered constant.Also, the shear stress is proportional to its deformation rate, i.e., τ = µ(∇v).Moreover, the fluid is considered in a steady state, where the quantities do

5

not vary over time. Thus, Eq. (2) can be rewritten as:

ρ(v · ∇v) = −∇p+ µ∇2v, (3a)

∇ · v = 0 , (3b)

which are expressed in cylindrical coordinates to preserve the curvatureeffects, following [17, 18]. In most references it is assumed that the radiusof curvature of the boundaries is large compared to the thickness of thelubricating film, and these effects are neglected. Developing Eq. (3) yieldsthe following equations.Axial direction, z:

ρ

(u∂u

∂z+ v

∂u

∂r+w

r

∂u

∂θ

)= −∂p

∂z+ µ

(1

r

∂

∂r

[r∂u

∂r

]+

1

r2

∂2u

∂θ2+∂2u

∂z2

). (4a)

Radial direction, r:

ρ

(v∂v

∂r+ u

∂v

∂z+w

r

∂v

∂θ− w2

r

)=

−∂p∂r

+ µ

(∂

∂r

[1

r

∂(rv)

∂r

]+

1

r2

∂2v

∂θ2− 2

r2

∂w

∂θ+∂2v

∂z2

). (4b)

Tangent direction, θ:

ρ

(v∂w

∂r+ u

∂w

∂z+w

r

∂w

∂θ+vw

r

)=

−1

r

∂p

∂θ+ µ

(∂

∂r

[1

r

∂(rw)

∂r

]+

1

r2

∂2w

∂θ2+

2

r2

∂v

∂θ+∂2w

∂z2

). (4c)

Continuity:

1

r

∂(rv)

∂r+

1

r

∂w

∂θ+∂u

∂z= 0. (4d)

This system of partial differential equations is nonlinear, and the veloc-ity and pressure fields are unknown. After applying the proper boundaryconditions, it can be approximated numerically. However, additional simpli-fications can be applied when a dimensional analysis is performed.

6

3.1. Dimensionless Navier-Stokes equations

The fact that the radial clearance F = Ro − Ri is very small comparedto the radii Ro and Ri, and the length L is used to decide which termscan be neglected. First, consider U a typical speed and L a characteristiclength with the same order of magnitude as Ro and Ri. Then, the followingdimensionless variables are introduced:

u = Uu, v = Uv, w = Uw, p = P p, P =µUL

F 2,

∆z = L∆z, ∆r = F∆r, ∆(rθ) = L∆(rθ) , (5)

where the hat symbol means dimensionless variable. Note that the ex-pression for P was used since the pressure exhibits a higher variation nearthe smallest fluid width. Now, using the dimensionless variables (5) in (4a)yields:

ρ

(Uu

∂Uu

∂Lz+ Uv

∂Uu

∂F r+Uw

Lr

∂Uu

∂θ

)=

− ∂P p

∂Lz+ µ

(1

Lr

∂

∂F r

[Lr∂Uu

∂F r

]+

1

L2r2

∂2Uu

∂θ2+∂2Uu

∂L2z2

).

(6)

After some algebraic manipulations one obtains

Re

((F 2

L2

)u∂u

∂z+

(F

L

)v∂u

∂r+

(F 2

L2

)w

r

∂u

∂θ

)=

− ∂p

∂z+

(1

r

∂

∂r

[r∂u

∂r

]+

(F 2

L2

)1

r2

∂2u

∂θ2+

(F 2

L2

)∂2u

∂z2

),

(7)

where the Reynolds number

Re =ρUL

µ(8)

is used. Following the traditional theory of lubrication, the terms multiplied

7

by (F/L)2 and (F/L) are neglected and Eqs. (4a), (4b) and (4c) become:

−∂p∂z

+1

r

∂

∂r

(r∂u

∂r

)= 0, (9a)

−∂p∂r

= 0, (9b)

∂

∂r

(1

r

∂(rw)

∂r

)− 1

r

∂p

∂θ= 0. (9c)

After this process, the number of terms was reduced considerably. Forthe equation along the radial direction (9b), the radial velocity disappearedand the pressure gradient along this direction is zero.

3.2. Velocities

Equations (9a) and (9c) can now be integrated in order to compute thevelocities u and w:

u =

(∂p

∂z

)1

4r2 + c1 ln r + c2 , (10a)

w =1

2

∂p

∂θr

(ln r − 1

2

)+ c3r +

c4

r, (10b)

where c1, c2, c3 and c4 are constants. Since the fluid speed is zero atthe stator and equal to the tangential speed W at the rotor’s surface, theboundary conditions are: u(Ro) = 0, u(Rθ) = 0, w(Ro) = 0 and w(Rθ) =ωRi = W . Thus, the description of the velocities u and w is obtained as afunction of the pressure gradient:

u =∂p

∂z

R2θ

4

[(r

Rθ

)2

− (R2o −R2

θ)

R2i ln(Ro/Rθ)

ln

(r

Rθ

)− 1

], (11a)

w =1

2

∂p

∂θ

[r

(ln r − 1

2

)+Kr − R2

o

r

(lnRo +K − 1

2

)]+

WRθ

(R2θ −R2

o)

(r − R2

o

r

), (11b)

where

K =1

R2o −R2

θ

[R2θ

(lnRθ −

1

2

)−R2

o

(lnRo −

1

2

)]. (12)

8

3.3. Continuity Equation

In order to obtain the pressure field, the continuity equation (4d) is inte-grated in the annular region of interest,

∫ Ro

Rθ

(∂(rv)

∂r

)dr︸ ︷︷ ︸

I

+

∫ Ro

Rθ

(∂w

∂θ

)dr︸ ︷︷ ︸

II

+

∫ Ro

Rθ

(∂(ru)

∂z

)dr︸ ︷︷ ︸

III

= 0 . (13)

Each term is detailed as follows:

(I)

∫ Ro

Rθ

(∂(rv)

∂r

)dr = Rov(Ro)−Rθv(Rθ) ,

(II)

∫ Ro

Rθ

(∂w

∂θ

)dr =

∂

∂θ

∫ Ro

Rθ

w dr −[w(Ro)

∂Ro

∂θ− w(Rθ)

∂Rθ

∂θ

],

(III)

∫ Ro

Rθ

(∂(ru)

∂z

)dr =

∂

∂z

∫ Ro

Rθ

(ru) dr −[u(Ro)

∂Ro

∂z− u(Rθ)

∂Ri

∂z

].

Some remarks about the boundary conditions should be registered. Theradial speed at the stator v(Ro) is zero; although the speed at the rotor’ssurface is tangential to it, it has a projection along the radial direction in thecoordinate system with origin at the stator’s center (see Fig. 2); since the

external radius does not vary along the tangential direction,∂Ro

∂θis zero; due

to the eccentricity,∂Rθ

∂θis not zero; the speeds u(Ro) and u(Rθ) are zero.

After applying the aforementioned boundary conditions, Eq. (13) isrewritten as:

∂

∂θ

∫ Ro

Rθ

w dr + w(Rθ)∂Rθ

∂θ+

∂

∂z

∫ Ro

Rθ

ru dr −Rθv(Rθ) = 0 (14)

The remaining boundary conditions w(Rθ) and v(Rθ) are obtained by kine-matics. First, consider a point A at the surface of the rotor, as shown in Fig.2. The poition vector of this point, with respect to a cylindrical coordinate

9

Figure 2: Graphical description of the speed of point A.

system with origin at the stator’s center, can be described as

pA = Rθeθ, (15)

and its time derivative as

vA = Rθ er +Rθ er = ω∂Rθ

∂θ︸ ︷︷ ︸vrad

er + ωRθ︸︷︷︸vtan

eθ, (16)

where

vrad = u(Rθ) = ω∂Rθ

∂θ, (17a)

vtan = w(Rθ) = ωRθ. (17b)

The boundary conditions in Eq. (17) are substituted in Eq. (14), whichyields:

∂

∂θ

∫ Ro

Rθ

w dr +∂

∂z

∫ Ro

Rθ

ru dr = 0 (18)

Finally, the speeds u and w in Eqs. (11a) and (11b) are substituted in

10

Eq. (18) and integrated:

∂

∂θ

(C1∂p

∂θ

)+

∂

∂z

(C2∂p

∂z

)=

∂

∂θC0 (19)

where:

C0 = −WRθ

[ln

(Ro

Rθ

)(1 +

R2θ

(R2o −R2

θ)

)− 1

2

](20)

C1 =Rθ

2

[1

2Rθ

[R2o lnRo −R2

θ lnRθ −(R2o −R2

θ

)(1 +K)]

]−

R2θ

2µ

[(lnRθ −

1

2+K

)ln

(Ro

Rθ

)] (21)

C2 =−R2

θ

8

[R2o −R2

θ −(R4

o −R4θ)

2R2θ

]+(

R2o −R2

θ

R2r ln(Ro/Rθ)

)[R2o ln

(Ro

Rθ

)− (R2

o −R2θ)

2

] (22)

The original Navier-Stokes equation, Eq. (4), was reduced to an ellipticaldifferential equation, Eq. (19), which is known as the Reynolds equation thatcan be approximated numerically.

4. Numerical Approximation for the pressure field

In this section, an approximate solution to Eq. (19) is obtained for thepressure field p. A central finite difference approximation method is used(see Ref. [28] for more details).

4.1. Discretization of the domain

The cylindrical region of interest is opened up and the discretization isperformed in a rectangular region with Nz ×Nθ nodes. Thus, the fluid filmhas Nz divisions along the axial direction and Nz division along the tangentialdirection, both equally spaced. Fig. 3 shows this procedure.

11

Figure 3: Rectangular mesh used for discretization of the equation and graphic translationof the centered finite difference method; adapted from [18].

4.2. Central finite differences formula

The partial derivatives of Eq. (19) are dicretized according to the follow-ing formulas:

∂f(i, j)

∂θ=f(i, j − 1)− f(i, j + 1)

2∆θ,

∂f(i, j)

∂z=f(i− 1, j)− f(i+ 1, j)

2∆z(23)

After the substitution of Eq. (23) in Eq. (19), the following algebraicequation is obtained:

pi−1,j

(C2(i−1,j))

∆z2+ pi,j−1

(C1(i,j−1))

∆θ2− pi,j

((C1(i,j) + C1(i,j−1))

∆θ2+

(C1(i,j) + C1(i,j−1))

∆z2

)+

pi,j+1

(C1(i,j))

∆θ2+ pi+1,j

(C2(i,j))

∆z2=

1

∆θ

[C0W (i,j) − C0W (i,j−1)

], (24)

where the pressure at the node (i, j) is written as p(i, j). The boundaryconditions of Eq. (24) are:

p(z = 0) = p(1, j) = Pin,

p(z = L) = p(NZ , j) = Pout,

p(θ = 0) = p(θ = 2π) = p(i, 1) = p(i, Nθ).

12

Eq. (24) can be expressed in a matrix form as

Mp = f , (25)

where

p =[p0,0 p0,1 . . . pNZ ,Nθ

]T, (26)

f =[f0,0 f0,1 . . . fNZ ,Nθ

]T, (27)

and M is a sparse matrix. Eq. (25) is assembled in Python and solved withthe SymPy1 library.

5. Fluid Film Forces

Integrating the pressure field one obtains the fluid film force that is ap-plied to the rotor. The decomposition of this force in the orthogonal axisN-T shown in Fig. 1 are given by [23]:

fN = −Ri

∫ L

0

∫ π

0

p(θ, z) cos θdθdz (28a)

fT = Ri

∫ L

0

∫ π

0

p(θ, z) sin θdθdz (28b)

or in a X-Y axis by:

fx = fT cos (β)− fN sin (β) (29a)

fy = fT sin (β) + fN cos (β) (29b)

Since the pressure field is discrete, the double integrals are computednumerically using the Simpson’s method.

5.1. Equilibrium position

During the normal operation of a rotating machine supported by hydro-dynamic bearings, external loads are applied to the rotor and the fluid film

1https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.

linalg.spsolve.html

13

forces oppose to them. If only the weight of the rotor is considered, the cen-ter of the spinning rotor stays in an equilibrium position where the weightis balanced with the fluid film forces. This position can be described by theeccentric displacement e and the attitude angle β, as shown in Fig. 1. Atthis position, the following condition must be satisfied:

fx0 =0, (30a)

fy0 =W, (30b)

where W is the weight of the rotor.An iterative method is used to find the position of the rotor that meets the

condition of Eq. (30). Starting with an initial (guess) position, the differencebetween the current forces and the forces shown in Eq. (30) is calculated.Next, the rotor center is slightly moved, and the forces are recalculated.When a tolerance condition is met, the method stops. More details can befound in the Python library optimize.least squares

The static performance characteristics of the bearing are listed in a nondi-mensional number known as Sommerfeld Number (S) [29]:

S =µωLR3

o

πWF 2, (31)

where F = Ro −Ri is the radial clearance.This nondimensional number is directly linked to the equilibrium position.

Fig. 4 illustrates this relation.

Figure 4: Path taken by the center of the rotor related to the Sommerfeld Number; adaptedfrom [30].

14

6. Dynamic Coefficients

In the previous section, the fluid film forces calculated are valid only at thestatic equilibrium of the rotor’s center. If the rotor performs small movementsaround such position, the fluid film forces vary. One way to characterize therelation between the forces and small displacements around the equilibriumposition is to determine the stiffness and damping coefficients.

6.1. Radial and tangential speeds

Here, it is considered that the rotor is not only spinning but also movingalong the horizontal / vertical directions. Thus, the speeds w(Rθ) and v(Rθ)in the continuity Eq. (14) must be determined with this new dynamic state.

Let us consider again a point A belonging to the surface of the rotor. Ifthe rotor is spinning and the velocity of its center is being perturbed, itstotal velocity is

vA = vr + vp (32)

where vr is the velocity of A due to the rotation speed and vp is the velocityacquired by the perturbation. In order to derive an equation for vp, thesmall displacements around the equilibrium position (x0, y0) are assumed ofthe following form:

x = x0 + xp sin(ωpt), (33a)

y = y0 + yp sin(ωpt), (33b)

where xp and yp are the maximum amplitudes of the perturbations and ωp isthe perturbation frequency along each axis. The time derivative of Eq. (33)gives the two terms of vp:

(vp)x = ωpxp cos(ωpt)ex, (34a)

(vp)y = ωpyp cos(ωpt)ey, (34b)

where (vp)x and (vp)y stand for the speeds when perturbations along thehorizontal and vertical directions are applied, respectively. In cylindricalcoordinates, and using Eq. (34), one arrives to:

15

(vA)x =

(ω∂Rθ

∂θ+ ωpxp cos (ωpt) cos (θ)

)︸ ︷︷ ︸

(vrad)x

er + (ωRθ − ωpxp cos (ωpt) sin (θ))︸ ︷︷ ︸(vtan)x

eθ

(vA)y =

(ω∂Rθ

∂θ+ ωpyp cos (ωpt) sin (θ)

)︸ ︷︷ ︸

(vrad)y

er + (ωRθ − ωpyp cos (ωpt) cos (θ))︸ ︷︷ ︸(vtan)y

eθ

(35)

where

(vrad)x = u(Rθ)x = ω∂Rθ

∂θ+ ωpxp cos (ωpt) cos (θ), (36a)

(vtan)x = w(Rθ)x = ωRθ − ωpxp cos (ωpt) sin (θ), (36b)

(vrad)y = u(Rθ)y = ω∂Rθ

∂θ+ ωpyp cos (ωpt) sin (θ), (36c)

(vtan)y = w(Rθ)y = ωRθ − ωpyp cos (ωpt) cos (θ). (36d)

6.2. Additional terms in the continuity equation

Equations (36)a-d are used as the new boundary conditions in the conti-nuity equation, Eq. (14). This consideration affects only the coefficient C0

of Eq. (14), which is

(C0)x = C0 +Rθωpxp cos (ωpt) sin (θ), (37a)

(C0)y = C0 +Rθωpyp cos (ωpt) cos (θ). (37b)

Thus, for an excitation along the x direction, the fluid film forces are

16

computed according to the equation:

(fx(t))x =Ri

∫ L

0

∫ π

0

p(θ, z, t;xp, ωp) sin θ cos βdθdz

+Ri

∫ L

0

∫ π

0

p(θ, z, t;xp, ωp) cos θ sin βdθdz (38a)

(fy(t))x =Ri

∫ L

0

∫ π

0

p(θ, z, t;xp, ωp) sin θ sin βdθdz

−Ri

∫ L

0

∫ π

0

p(θ, z, t;xp, ωp) cos θ cos βdθdz, (38b)

where p(θ, z, t;xp, ωp) is the pressure field computed at time t with perturba-tion amplitude and frequency xp and ωp, respectively. Similar equations arevalid for a perturbation along the y direction.

6.3. Stiffness and damping coefficients

The forces of the fluid film can be considered general functions of thedisplacements and speeds of the center of the rotor. Thus, if small displace-ments around the center are applied, the expansions of the fluid forces inTaylor series yield

fx = fx0 +Kxx∆x+Kxy∆y + Cxx∆x+ Cxy∆y, (39a)

fy = fy0 +Kyx∆x+Kyy∆y + Cyx∆x+ Cyy∆y. (39b)

where

Kxx =∂fx∂x

,Kxy =∂fx∂y

,

Kyx =∂fy∂x

,Kyy =∂fy∂y

,

Cxx =∂fx∂x

, Cxy =∂fx∂y

,

Cyx =∂fy∂x

, Cyy =∂fy∂y

.

The terms Kxx, Kyy, Cxx and Cyy are called the direct stifness coeffi-cients and direct damping coefficients, respectively. Kxy, Kyx, Cxy and Cyx

17

are called the cross-coupled stiffness coefficients and cross-coupled dampingcoefficients, respectively.

By including the effects of the perturbation in the simplified equation(19), it is possible to rewrite the Taylor Series expansion of the forces in Eq.(39) as a function of the time t. Thus, for each direction of disturbance thefluid film forces are:

(fx(t))x = fx0 + kxx∆x(t) + cxx∆x(t),

(fy(t))x = fy0 + kyx∆x(t) + cyx∆x(t),

(fx(t))y = fx0 + kxy∆y(t) + cxy∆y(t),

(fy(t))y = fy0 + kyy∆y(t) + cyy∆y(t). (40)

Note that the ∆x(t), ∆x(t), ∆y(t) and ∆y(t) are known by Eq. (33); fx0 andfy0 are the forces at the equilibrium position from Eq. (30); and (fx(t))x,(fy(t))x, (fx(t))y and (fy(t))y are obtained from Eq. (38).

Equation (40) can be written in matrix form as:

• Towards x:

∆x[0] ∆x[0]∆x[1] ∆x[1]

......

∆x[N − 1] ∆x[N − 1]

︸ ︷︷ ︸

Ax

[kxx kyxcxx cyx

]=

(fx[0])x − fx0 (fy[0])x − fy0(fx[1])x − fx0 (fy[1])x − fy0

......

(fx[N − 1])x − fx0 (fy[N − 1])x − fy0

︸ ︷︷ ︸

Fx

.

(41)

• Towards y:

∆y[0] ∆y[0]∆y[1] ∆y[1]

......

∆y[N − 1] ∆y[N − 1]

︸ ︷︷ ︸

Ay

[kxy kyycxy cyy

]=

(fx[0])y − fx0 (fy[0])y − fy0(fx[1])y − fx0 (fy[1])y − fy0

......

(fx[N − 1])y − fx0 (fy[N − 1])y − fy0

︸ ︷︷ ︸

Fy

.

(42)

18

Or, in order to simplify the notation:

Ax

[kxx kyxcxx cyx

]= Fx e Ay

[kxy kyycxy cyy

]= Fy. (43)

In order to calculate the values of the coefficients, the Least Squaresmethod is employed. For that, one uses the Moore-Penrose pseudo inverseof the matrices Ax and Ay.

A+x =

(ATxAx

)−1ATx e A+

y =(ATy Ay

)−1ATy . (44)

And finally, the stiffness and damping coefficients are obtained:[kxx kyxcxx cyx

]= A+

x Fx e

[kxy kyycxy cyy

]= A+

y Fy. (45)

7. Results and Discussion

The choice of parameterization of the model presented in this work allowseasy adaptation to different geometries and operational conditions of rotatingmachine components. This section analyzes three different cases: (1) shortjournal bearing, (2) elliptical bearing, and (3) worn bearing.

The algorithm used was developed in Python, and is available on theGitHub platform, through the software Ross-Rotordynamics2 [31], an opensource library for rotodynamic analysis.

7.1. Short journal bearing

In the literature, it is common to use the Reynolds equation approxi-mated for bearings with short or infinitely long length. In this way, there ananalytical resolution is possible.

In order to verify the computational model developed in the present work,comparisons will be made with the results obtained by the current literatureapproximations. According to [21], a bearing can be considered short if theratio L/D (D = 2Ro is the diameter) is less than or equal to 1/8 and longif it is greater than 4. However, according to [20], the model for the shortbearing provides accurate results (in the case of simple cylindrical bearings

2https://github.com/ross-rotordynamics/

19

with small to moderate values of the bearing eccentricity, e ≤ 0.75F ) alreadywith the ratio L/D ≤ 1/2.

To analyze the short journal bearing, the pressure variation in the zdirection is considered much larger than in the θ direction, i.e. ∂p/∂θ �∂p/∂z. Thus, the first term in the Eq. (11a) is neglected, and one obtains[23]:

p =−3µεω sin θ

(Rθ −Ri)2 (1 + ε cos θ)3

[(z − L

2

)2

− L2

4

], (46)

where ε =e

Ro −Ri

is the eccentricity ratio and ω the rotor speed.

Element ValueRotor radius (Ri) 0.2 [m]

Gap (F ) 0.0001 [m]Load (W ) 50 [N ]

Ratio L/D 1/8Viscosity (µ) 0.015 [Pa.s]

Rotation speed (ω) 10.472 [rad/s]

Table 1: Parameters used for the simulation on a short cylindrical bearing.

The numerical simulation, using data shown in Table 1, generates a rel-ative error E ≈ 10−3. Figure 5 shows the pressure field computed togetherwith the reference result [23]. This result indicates that the present modelgenerate consistent results.

The forces can be obtained integrating the pressure field given in Eq. (46)using Eq. (28). The relative errors, between the forces computed numericallyand analytically (pressure fields shown in Fig. 5), are EN ≈ 0.02 for the radialforce and ET ≈ 0.005 for the tangential force. Concerning the equilibriumposition, the are were also small: fx ≈ |fy −W | ≈ 10−6[N ].

The errors in the coefficients are obtained comparing the numerical resultswith the ones presented in [30]. The following relative errors are obtained:

[Ekxx EkxyEkyx Ekyy

]≈[0.017 0.0030.001 0.04

]e

[Ecxx EcxyEcyx Ecyy

]≈[0.005 0.040.06 0.002

](47)

20

3 4 5 6 7 8 9 [rad]

0.0

0.2

0.4

0.6

0.8

1.0

Pres

sure

[Pa]

1e4Pressure along in the middle of the bearing

Ishida and Yamamoto (2021)Present work

Figure 5: Pressure field obtained for a short journal bearing; Table 1. Comparison of thecurrent results with the one found in [23].

Now that the proposed computational model was verified with the shortbearing approximation found in the literature, the new geometry parameter-ization, developed in Section 2, can be explored in other bearing configura-tions, or other machine elements.

7.2. Elliptical bearing

Cylindrical geometry bearings, at high rotation speeds and with small ex-ternal loads applied, are prone to instability or high vibrations [25]. This isbecause, in these situations, the machine tends to operate centered, reducingthe variation in the thickness of the lubricating film and, consequently, theability to generate hydrodynamic pressure from the bearing. Several con-figurations of bearings were developed in order to circumvent this problem,producing preloads that make the shaft work outside the new establishedcenters, thus creating the desired wedge effect.

A feature commonly found in high-speed machines is the elliptical bearingor ”lemon bearing”, as it is also known [32]. This is a variation of the cylindri-cal bearing with axial groove and reduced clearance in one direction. Ostayen

21

and Beek [32] claim that this bearing configuration is relatively cheap andeasy to manufacture, as it is made from adaptations in the cylindrical bearingitself, being composed of two circular arcs (cuts from the original structure),with aligned centers, generating two horizontal slits, as shown in Fig. 6.

Figure 6: Schematic drawing of an elliptical bearing.

To consider this new geometry, stator radius is is no longer constant inθ. As seen in Fig. 6, the new stator is composed of the arc C1, with centerin O1, joined to the arc C2, centered in O2, both with radius Ro. In this newconfiguration, the centers are at a distance ε from the origin, called ellipticity.Now, the stator radius is described from the origin. This new distance willbe called R∗

o and it varies along the angular position.Using the cosine law, the following relation is obtained:

R∗o =

√R2o − ε2 sin2 α + ε cosα, (48)

where α =

π/2 + θ, se θ ∈ 1st quadrant

3π/2 + θ, se θ ∈ 2nd quadrant

θ − π/2, se θ ∈ 3rd quadrant

5π/2− θ, se θ ∈ 4th quadrant

.

Another important parameter to be defined is the preload m which, in

22

Element ValueStator radius (Ro) 0.015 [m]

Gap (F ) 9 · 10−5 [m]Length (L) 0.02 [m]Load (W ) 100 [N ]

Viscosity (µ) 5.449 · 10−2 [Pa.s]Rotation speed (ω) 261.8 [rad/s]

Table 2: Numerical simulation for the elliptical bearing.

Fonte: Adapted from [34]

this text, will be established as:

m =ε

F(49)

where ε is the ellipticity and F = Ro −Ri is the radial clearance.For m = 0, the bearing becomes cylindrical, while when m tends to 1

the stator arcs tend to touch the axis. Garner et al [33] suggest that themaximum preload adopted should be 0.6. According to the authors, valuesabove 0.7 can be used in an effort to improve stability, however, greaterlateral clearances are generated, causing the horizontal dynamic coefficientsto decrease, in addition to requiring a greater flow of oil.

Given the data shown in Table 2, the results for the pressure on theelliptical bearing, using the modeling presented in this work, are compatiblewith those exposed in [34]. Figures 7 (a-b) illustrate the pressure field fora bearing described as elliptical, but with m = 0, that is, cylindrical. Incontrast, the graphs shown in Figs. 8 (a-b) and 9 (a-b), indicate the pressurebehavior for preloads equal to 0.4 and 0.6, respectively.

It can be seen that Figs. 8 and 9 show two pressure peaks: one for theupper bearing arch (0 to π) and one for the lower arc (π to 2π). This phe-nomenon can be explained due to the configuration of the elliptical bearing.Even if the shaft operates centered, there will be variations in the thicknessof the fluid, causing the oil film to be compressed, both in the upper andlower arc.

Analyzing also the graph of Fig. 10, it is understood that the reactionforce increases when the preload increases, and, hence the pressure. Form = 0.8, the pressures generated are considerably higher than for other

23

L [m] (×1e-2)

0.00.5

1.01.5

2.0 [rad]0 2 4 6Pr

essu

re (×

1e5)

0123

(a) Pressure distribution.

0 2 4 6 [rad]

0

1

2

3

4

Pres

sure

[Pa]

1e5

(b) Side view.

Figure 7: Cylindrical bearing (m = 0).

L [m] (×1e-2)

0.00.5

1.01.5

2.0 [rad]0 2 4 6

Pres

sure

(×1e

5)

01234

(a) Pressure distribution.

0 2 4 6 [rad]

0

1

2

3

4

5

Pres

sure

[Pa]

1e5

(b) Side view.

Figure 8: Elliptical bearing with (m = 0.4).

24

L [m] (×1e-2)

0.00.5

1.01.5

2.0 [rad]0 2 4 6

Pres

sure

(×1e

5)

02468

(a) Pressure distribution.

0 2 4 6 [rad]

0.0

0.2

0.4

0.6

0.8

1.0

Pres

sure

[Pa]

1e6

(b) Side view.

Figure 9: Elliptical bearing with (m = 0.6).

preloads, being an option for cases in which a greater demand is sought torotor stability.

0 1 2 3 4 5 6 [rad]

0

1

2

3

Pres

sure

[Pa]

1e6Pressure along in the middle of the bearing

m = 0.0m = 0.2m = 0.4m = 0.6m = 0.8

Figure 10: Pressure along θ in the middle of the elliptical bearing for different values ofthe preload m.

25

0

/2

3 /2

d0 = 10 m

d0 = 80 m

Posição do centro do rotor

(a) Rotor center.

0.0 0.2 0.4 0.6 0.8preload m

0.1

0.2

0.3

0.4

ecce

ntri

city

rat

io e

/F

Posição do centro do rotor

(b) Variation of eccentricity.

Figure 11: Variation of the equilibrium position on the elliptical bearing for differentpreloads.

With regard to the equilibrium position, Fig. 11a illustrates a differ-ent behavior comparing to the cylindrical bearing. That is, the SommerfeldNumber used in the cylindrical bearing is not valid to analyze the static per-formance characteristics of the elliptical bearing. For the elliptical bearing,increasing the preload, first the rotor moves vertically and, after m = 0.3, itwalks in the horizontal direction, approaching the center of the stator, alwaysreducing the eccentricity (Fig. 11b).

The forces of the fluid film can be seen in Fig. 12, in which it is evidentthat the tangential force is always much higher than the radial force for anypreload applied to the elliptical bearing.

It is also noticeable, when analyzing Figs. 13 (a-b), that the dynamiccoefficients of the elliptical bearing suffer great variation between their terms.For the preload m up to 0.2, the terms kxx, kxy and kyy are very close to eachother. From m = 0.3 it is observed that the difference between these termsstarts to increase. The cross term kxy is increasingly larger than the directterm kxx, which might indicate instability, as pointed out by [34]. Concerningthe damping coefficients, all terms increase as the preload increases. It isworth remembering that the main function of damping is to mitigate the

26

0.0 0.2 0.4 0.6 0.8Preload m

0

20

40

60

80

100

Forc

es [N

]

Fluid film forces

NT

Figure 12: Behavior of radial and tangential forces on the elliptical bearing for differentpreloads.

effects of vibration on the system, and that high values of these coefficientsare desired.

The difference between the results of the cylindrical and elliptical bearingis clear. In the first configuration, all the factors that reduce the eccentricityimply a decrease in the pressure field, forces and variation of the dynamic co-efficients. For the elliptical bearing configuration, the exact opposite occurs.By increasing the preload, rotor and stator become closer to concentricityand, this yields an increase in all the results just mentioned. However, theincrease of the stiffness cross coefficient indicates that the system risks to beinstable.

7.3. Wear bearing

Although lubrication reduces the friction between the metallic surfacesof the bearing, these structures usually suffer wear after a long operatingperiod or due to a certain number of starting cycles. One of the most commoncauses of interruptions in rotating systems is the occurrence of failures relatedto hydrodynamic bearings [24]. As the wear suffered by these componentsaffects the radial clearance, they certainly influence the results of the pressurefield and, consequently, the reaction forces and dynamic coefficients. For this

27

0.0 0.2 0.4 0.6 0.8Preload m

0

2

4

6

8

Stiff

ness

1e7kxx

kxy

kyx

kyy

(a) Stiffness coefficients.

0.0 0.2 0.4 0.6 0.8Preload m

0

1

2

3

Dam

ping

1e5cxx

cxy

cyx

cyy

(b) Damping coefficients.

Figure 13: Dynamic coefficients of the elliptical bearing for different preloads.

reason, it is relevant to investigate the effect of wear, under certain operatingconditions, on the dynamic response of the system.

The wear geometry that will be used in this text was adapted from [24].The description was initially proposed by [35] and validated experimentally[36].

Dufrane et al [35] consider the wear thickness uniform in the axial direc-tion, and located at the bottom of the bearing, symmetrically around the Yaxis. On the other hand, Machado and Cavalca [24] states that, for practicalcases, most wear and tear occurs in the fourth quadrant, the location of theaxis balance position for an anti-clockwise rotation. For this reason, takinginto account that the spent region is symmetrical in relation to the greatestdepth d0, an angular displacement γ will be considered from the vertical axisto the deepest part, as shown in Fig. 14.

To include wear in the geometry, it is necessary to make some adaptationsto the stator radius. Considering that the fault starts at the angular positionθ = θs and ends at θ = θf , the stator description from the origin is definedas

R∗o = Ro + dθ (50)

28

Figure 14: Schematic drawing of a worn bearing.

Element ValueStator radius (Ro) 0.015 [m]

Gap (F ) 9 · 10−6 [m]Length (L) 0.02 [m]Load (W ) 18.9 [N ]

Viscosity (µ) 0.1044 [Pa.s]Rotation speed (ω) 104.72 [rad/s]

Table 3: Numerical simulation for the worn bearing.

Fonte: Adapted from [24]

where dθ =

{0, se 0 ≤ θ ≤ θs, θf ≤ θ ≤ 2π

d0 − F (1 + cos (θ − π/2)) , se θs < θ < θf.

In θs and θf , the wear depth is zero, so the location of the edges can bedefined as follows:

θs = π/2 + cos−1 (d0/F − 1) + γ

θf = π/2− cos−1 (d0/F − 1) + γ (51)

With the geometry described as above, and data shown in Table 3, theresults for the pressure field in the bearing with wear are verified with [24].

29

L [m] (×1e-2)

0.00.5

1.01.5

2.0 [rad]0 2 4 6

Pres

sure

(×1e

4)

0246810

(a) Pressure distribution.

0 2 4 6 [rad]

0.0

0.2

0.4

0.6

0.8

1.0

Pres

sure

[Pa]

1e5

(b) Side view.

Figure 15: Bearing without wear.

For a cylindrical bearing without wear, the pressure field is represented byFigs. 15 (a-b). Figs. 16 (a-b) illustrate the pressure field for the bearingwith maximum depth wear d0 = 10[µm] and an offset γ = 10◦ ≈ 0.17[rad].In Figs. 17 (a-b) d0 = 50[µm] is considered for the same displacement and,finally, Figs. 18 (a-b) show the pressure for the same wear with displacementγ = 30◦ ≈ 0.52[rad].

It is possible to see that, even for minor wears, the pressure field is affecteddrastically. In addition to presenting two peaks, as occurred in the ellipticalbearing, the magnitude of the pressure is also changed. Fig. 19 shows that,as the depth d0 increases, the shape of the pressure field changes considerably.

30

L [m] (×1e-2)

0.00.5

1.01.5

2.0 [rad]0 2 4 6Pr

essu

re (×

1e4)

0246810

(a) Pressure distribution.

0 2 4 6 [rad]

0.0

0.2

0.4

0.6

0.8

1.0

Pres

sure

[Pa]

1e5

(b) Side view.

Figure 16: Wear bearing (d0 = 10[µm], γ = 10◦).

L [m] (×1e-2)

0.00.5

1.01.5

2.0 [rad]0 2 4 6

Pres

sure

(×1e

4)

02468101214

(a) Pressure distribution.

0 2 4 6 [rad]

0.00.20.40.60.81.01.21.4

Pres

sure

[Pa]

1e5

(b) Side view.

Figure 17: Wear bearing (d0 = 50[µm], γ = 10◦).

31

L [m] (×1e-2)

0.00.5

1.01.5

2.0 [rad]0 2 4 6

Pres

sure

(×1e

4)

0246810

(a) Pressure distribution.

0 2 4 6 [rad]

0.0

0.2

0.4

0.6

0.8

1.0

Pres

sure

[Pa]

1e5

(b) Side view.

Figure 18: Wear bearing (d0 = 50[µm], γ = 30◦).

0 1 2 3 4 5 6 [rad]

0.00

0.25

0.50

0.75

1.00

1.25

Pres

sure

[Pa]

1e5Pressure along in the middle of the bearing

d0 = 0 md0 = 20 md0 = 40 md0 = 60 md0 = 80 m

Figure 19: Pressure along θ in the middle of the bearing with wear to different depths d0(γ = 10◦).

The depth of wear on the structure also influences the equilibrium positionof the rotor, as seen in Figs. 20. As the depth d0 increases, the center of therotor moves away from the center of the stator, increasing the eccentricity. Inaddition to the direction opposite to the center of the stator, the path taken

32

0

/2

3 /2

d0 = 80 m

d0 = 10 m

(a) Rotor center.

0 20 40 60 80Wear depth d0 [ m]

0.2

0.4

0.6

0.8

Ecc

entr

icity

rat

io e

/F(b) Variation of eccentricity.

Figure 20: Variation of the balance position in the bearing with wear for different depthsd0 (γ = 10◦).

also differs in behavior when compared to the cylindrical bearing withoutwear. As the wear depth increases, the center of the rotor moves furtherdown, describing a course close to a straight line.

The radial and tangential forces for the worn bearing develop a differentbehavior, comparing to the elliptical bearing. Figure 21 indicates that theradial component of the force increases rapidly at greater depths of wear,while the value of the tangential component decreases.

Finally, with regard to the dynamic coefficients, Figs. 22 (a-b) describethe behavior of stiffness and damping for different wear depths. There isan oscillation in the stiffness coefficients, with all the terms distinct fromeach other and only negative kyx values. For maximum depth wear up tod0 = 30[µm], the cross term kxy is greater than the direct term kxx, indicatingpossibility of instability. From that value, kxx grows, distancing itself fromthe cross term, but then decreases again shortly thereafter. The coefficientsrelated to the vertical forces do not show very relevant variations, the directterm being always much higher than the cross term. The damping coefficientsdevelop positive values in their direct terms, and negative and similar valuesin their crossed coefficients.

Thus, it can be concluded that wear on the cylindrical bearing are vari-

33

0 20 40 60 80Wear depth d0

5.0

7.5

10.0

12.5

15.0

17.5Fo

rces

[N]

Fluid film forces

NT

Figure 21: Behavior of radial and tangential forces on the bearing with wear to differentdepths d0 (γ = 10◦).

0 25 50 75Wear depth d0 [ m]

1

0

1

Stiff

ness

[N/m

]

1e6

kxx

kxy

kyx

kyy

(a) Stiffness coefficients.

0 25 50 75Wear depth d0 [ m]

0

1

2

Dam

ping

[N-s

/m]

1e4

cxx

cxy

cyx

cyy

(b) Damping coefficients.

Figure 22: Dynamic bearing coefficients with wear for different depths d0 (γ = 10◦).

34

Figure 23: Schematic drawing listing the three bearing configurations shown.

Element ValueStator radius (Ro) 0.015 [m]

Gap (F ) 9 · 10−5 [m]Length (L) 0.02 [m]Load (W ) 100 [N ]

Viscosity (µ) 0.05449 [Pa.s]Rotation speed (ω) 50 ≤ ω ≤ 550 [rad/s]

Table 4: Numerical simulation for the comparison of bearing geometries.

ations in geometry that directly affect the dynamic response of the system.The stiffness coefficients show intervals in which the risk of instabilities ismore critical, which can serve to indicate operational conditions that shouldbe avoided.

7.4. Comparison between geometries

Nowadays, many rotating machines operate at high rotation speeds. Thisspeed influences the entire dynamic response of the bearing, from the equi-librium position to the stiffness and damping coefficients. For this reason,it is essential to understand how the model’s responses behave for differentrotation speeds.

35

0

/2

3 /2

= 50 rad/s= 550 rad/s

Posição do centro do rotor

cylindricalellipticalwear

(a) Rotor center.

200 400Rotor speed [rad/s]

0.2

0.4

0.6

0.8

1.0

Ecc

entr

icity

rat

io e

/F

Posição do centro do rotor

cylindricalellipticalwear

(b) Variation of eccentricity.

Figure 24: Balance position for different bearing configurations, varying the rotation speed.

In order to compare the results, the data shown in Table 4 is used, adapt-ing only the specifics of each geometry and varying the rotation speed inthe range of 50 ≤ ω ≤ 550 [rad/s]. For the elliptical bearing the preloadm = 0.4 was considered and, for the bearing with wear, the maximum depthd0 = 50[µm] with an angular displacement γ = 10◦. The schematic draw-ing that relates the three bearing configurations is illustrated in Fig. 23, inwhich the geometries are indicated by the letters C, E and D, representingcylindrical, elliptical and worn bearings, respectively.

Figures. 24 (a-b) illustrate the variation of the equilibrium position for thecylindrical, elliptical and worn bearing. In the three cases, the eccentricitydecreases with the increase of the rotation speed. However, it is possibleto observe the different paths. While the cylindrical bearing moves in asemicircle-like path, the elliptical bearing generates a steeper curve, in whichmost points are aligned with the horizontal axis. On the other hand, theworn bearing has much higher eccentricities for all speeds. It is interestingto highlight the difference between the bearing balance positions with andwithout wear. Despite having very similar geometry, wear alters the positionof the rotor center and the way it behaves at different speeds.

As already shown, the pressure behavior is changed for each geometry.Fig. 25 demonstrates these changes in order to facilitate comparison. It is

36

0 1 2 3 4 5 6 [rad]

0

1

2

3

4

5Pr

essu

re [P

a]

1e5Pressure along in the middle of the bearing

CylindricalEllipticalWear

Figure 25: Pressures along θ in the middle of the bearing for different geometry configu-rations, varying the rotation speed.

noticed that, in addition to the new peak, the maximum pressure value isalso modified. The cylindrical bearing has the lowest pressure value, whilethe elliptical reaches the greatest magnitude, even though the configurationwith the least eccentricity. It is also possible to notice that the wear causes anincrease in the pressure comparing to the cylindrical bearing, also modifyingthe place where the pressure reaches its highest value.

The radial and tangential components of the forces of the fluid film (Fig.26), as well as the pressure fields, are different for the three cases presented.In cylindrical and elliptical bearings, the radial force N presents a cleardecrease when the rotation speed increases, even reaching negative values,for the case of the elliptical bearing. For wear, the decrease of this componentis moderate, with values much higher than the tangential force T , which isalso growing smoothly. The tangential force T for the elliptical bearing islittle influenced by the rotation speed. From 150 rad/s up, the this force isalmost constant for the rotation speed range analyzed. On the other hand,for cylindrical geometry, varying the rotation speed has a higher impact onthe forces.

Finally, Figs. 27 (a-d) and 27 (a-d) show the variations of each termof the stiffness and damping coefficients for the different geometries in the

37

100 200 300 400 500rotor speed [rad/s]

0

20

40

60

80

100

Forc

es [N

]Fluid film forces

f_N (cylindrical)f_T (cylindrical)f_N (elliptical)f_T (elliptical)f_N (wear)f_T (wear)

Figure 26: Forces for different bearing configurations, varying the rotation speed. The con-tinuous line represents the radial component and the dashed line represents the tangentialcomponent.

rotation speed range adopted. Some characteristics stand out in these resultsand are commented below.

Regarding the stiffness coefficients, it is noticed that, in the ellipticalbearing, the terms kxy and kyy have rapid growing up to ω ≈ 300[rad/s] and,then there is a drastic reduction in their values, which does not occur in theother two geometries. The term kxx shows little variation with the rotationspeed for the cylindrical bearing. It decreases for the elliptical bearing anddevelops much higher values for the wear geometry. The terms kxy, kyx andkyy show similar growth and decrease their values in cylindrical and wornbearings, with the greatest difference between them in the crossover termkyx.

In order to analyze the stability conditions in the different bearings, itis necessary to compare the direct and crossed stiffness coefficients for thesame direction. Such comparison becomes more understandable in Figs. 28(a-b), which illustrate the differences (kxx−kxy) and (kyy−kyx) in the appliedrotation speed range. If these differences are negative, they mean that thecross term has a higher value than the direct term, indicating possibility ofrotor instability. It is desired that these values are high and positive. In

38

200 400Rotor speed [rad/s]

0

1

2

3

4

k xx

[N/m

]

1e6CylindricalElliptical

Wear

(a) kxx

200 400Rotor speed [rad/s]

0

2

4

6

k xy

[N/m

]

1e6CylindricalElliptical

Wear

(b) kxy

200 400Rotor speed [rad/s]

7

6

5

4

k yx

[N/m

]

1e6CylindricalElliptical

Wear

(c) kyx

200 400Rotor speed [rad/s]

0

2

4

6

k yy

[N/m

]

1e6CylindricalElliptical

Wear

(d) kyy

Figure 27: Stiffness coefficients for different bearing configurations, varying the rotationspeed.

39

200 400Rotor speed [rad/s]

4

2

0

2

4

k xx

k xy

[N/m

]

1e6CylindricalElliptical

Wear

(a) Direction x.

200 400Rotor speed [rad/s]

0.6

0.8

1.0

k yy

k yx

[N/m

]

1e7CylindricalElliptical

Wear

(b) Direction y.

Figure 28: Difference between the direct and cross terms of stiffness in each direction.

this way, it is possible to notice that the differences, for the three bearingconfigurations in the x direction, tend to decrease. However, the ellipticalbearing presents a different behavior from the others, with the cross termgreater than the direct one for rotation speeds in the range 100 < ω <300 [rad/s]. The difference in the terms in the y direction is positive for theentire variation, for the three geometries, also showing much less uniformityin the elliptical bearing.

Unlike what happens in stiffness, damping is not as sensitive to the pres-ence of faults. All coefficients develop close values for cylindrical and wornbearings, except for the direct terms cxx, which is different for lower speeds,but end up getting closer as the speed increases. The elliptical geometryresults in damping coefficients that follow the same growth and decreasepatterns as the others, but always with higher values. The cross terms cxyand cyx show increasing and negative values for all cases. However, withhigher rotation speeds, they become positive only on the elliptical bearing.Direct terms decrease in all cases without reaching negative results.

40

200 400Rotor speed [rad/s]

2

4

6

8

c xx

[N-s

/m]

1e4CylindricalElliptical

Wear

(a) cxx

200 400Rotor speed [rad/s]

6

4

2

0

2

c xy

[N-s

/m]

1e4CylindricalElliptical

Wear

(b) cxy

200 400Rotor speed [rad/s]

6

4

2

0

c yx

[N-s

/m]

1e4CylindricalElliptical

Wear

(c) cyx

200 400Rotor speed [rad/s]

0.5

1.0

1.5

c yy

[N-s

/m]

1e5CylindricalElliptical

Wear

(d) cyy

Figure 29: Damping coefficients for different bearing configurations, varying the speed.

41

8. Conclusions

Considering the flow of a small thickness of lubricating oil film betweentwo cylinders, this work presents a review of the whole Lubrication Theory,starting from the Navier-Stokes Eq. (2a) until reaching an elliptical partialdifferential Eq. (19) that describes the pressure field in the annular space,corresponding to the well-known Reynolds equation normally used in theliterature on the subject.

Through a dimensionless analysis, it was possible to promote a mappingof the terms of the equations, ordering them by relevance in the physicalphenomenon for this type of situation. By revisiting this formulation, it wasalso possible to choose a new and effective parameterization of geometry,which enables a more detailed description of the problem, allowing to adaptthe modeling of this work to other contexts.

Without changing the proposed model, simulations were carried out indifferent configurations of hydrodynamic bearings. This parameterizationalso allows the model to be adapted in the future for many other geometries,including changes not only in the tangential direction, but also in the axialdirection. In addition, the equations were written in cylindrical coordinatesin order to preserve the effects of curvature, which may be interesting forfuture works.

The pressure field was obtained numerically through the centered finitedifferences method. This result was verified from its application in simula-tions for cylindrical hydrodynamic bearings, comparing the responses of thismodel with those obtained through classic approaches in the literature.

A new methodology was presented, different from the one commonly usedin the literature, to compute the stiffness and damping coefficients. Thisalternative, through a Least Squares solution obtained by the Moore-Penrosepseudo inverse, making the process computationally effective.

Another relevant result is the construction of the algorithm in Python,developed by the authors of this research and used in all the simulationspresented. Through Fluid Flow it is possible to choose geometric and opera-tional parameters according to analyzes of the desired pressure performance,forces, equilibrium position, dynamic coefficients, etc. In addition, the lowcomputational cost of the implemented modeling allows to generate differentsimulations, which enables its use in machine learning algorithms, a functionalready explored in Ross, which contains a stochastic module to provide thistype of analysis. This module also explores uncertainties about the model

42

presented in this work, such as, for example, the impact of the equilibriumposition on the dynamic coefficients.

Two other configurations of hydrodynamic bearings were also presented:the elliptical and wear bearing. These new structures, obtained throughchanges only in the geometric description of the problem, without changingthe modeling, portray the gain resulting from the chosen parameterization.In each situation, the impact of these changes on the dynamic response ofthe system was analyzed and, finally, comparisons were made between thethree geometries, exploring the changes in the responses for different rotationspeeds.

The results obtained make clear the impact of the variation of the bearingtype on the fluid model’s responses. The elliptical bearing, introduced as analternative to correct the instabilities generated in the cylindrical bearing,showed the high stiffness coefficients, which might lead to rotor instability.As for the bearing with wear, it is clear that even small changes compromisethe entire response of the model. The pressure field changes dramatically, asdo the forces and stiffness. On the other hand, the worn does not seem toinfluence the system’s damping so much.

It is concluded that the description of geometry is fundamental for thesuccess of the modeling. It is worth noting that the parameterization chosenfor the description of the problem allows such changes to be made naturally,without the need to modify the model. Furthermore the proposed modelmay constitute a tool for choosing the best geometry, in order to optimizethe operation of rotating machines within their specific contexts.

9. Acknowledgements

The code used for the simulations presented in this paper is part ofROSS3, an open source library written in Python for rotordynamic analy-sis, by [31].

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