Stochastic dynamic behaviour of hydrodynamic ...

Stochastic dynamic behaviour of hydrodynamic journal bearings including the effect of surface roughness K. Maharshia, T. Mukhopadhyayb,*, B. Roya, L. Roya, S. Deya a

Mechanical Engineering Department, National Institute of Technology Silchar, India Department of Engineering Science, University of Oxford, Oxford, United Kingdom * Corresponding author’s e-mail: [email protected]

b

Abstract This paper investigates the stochastic behaviour of hydrodynamic journal bearings by solving the Reynolds equation with random parameters numerically based on finite difference method. The steady state and dynamic characteristics are quantified considering random variabilities in eccentricity and surface roughness that can closely simulate the uncertain service conditions and inevitable manufacturing imperfections. Based on efficient radial basis function, a Monte Carlo simulation (MCS) algorithm is developed in conjunction with the governing equations for quantifying stochastic characteristics of the crucial performance parameters concerning hydrodynamic bearings. Relative sensitivity to stochasticity for different performance parameters is analysed. Physically insightful new results are presented in a probabilistic framework, wherein it is observed that the stochasticity has pronounced influence on the performance of bearing.

Keywords: hydrodynamic journal bearing; stochasticity; relative sensitivity; RBF based MCS

Nomenclature

C

radial clearance (m)

CS

roughness variation C S  

CS

non-dimensional roughness variation

D

diameter of the bearing (m)

Dij

damping coefficient (N-s/m)

DXX , DXY , DYX , DYY

non-dimensional damping coefficient Dij  Dij C LDp



3



1

e

eccentricity (m)

F

non-dimensional resultant force

H

film thickness

h

local film thickness

K ij

stiffness coefficient (N/m)

K XX , K XY , KYX , KYY

non-dimensional stiffness coefficients K ij  K ij C LDp

L

length of bearing (m)

M

mass parameter

p

supply pressure (Pa)

p

fluid film pressure p  pC 2 6UR

P1, P2

perturbed pressures

Q

non-dimensional flow coefficient

t

time (s)

W

steady state load W  WC 2 6R 2UL



eccentricity ratio  e



coefficient of absolute viscosity (Pa-s)

,Z

non-dimensional co-ordinates,



   whirl ratio    p   



friction variable   R C 



non-dimensional time,    P t 



attitude angle (rad)



journal rotational speed (rad/s)

P

frequency of journal vibration (rad/s)



expectancy operator

 ~

standard deviation

g

A set of the crucial performace parameters (attitude angle, load bearing capacity, Sommerfeld number, mass parameter, whirl ratio and minimum film thickness)

Re, Im

Real part, Imaginary part









C





measured from centre-line

stochastic characterization

2

1. Introduction The modern mechanical industries mostly work on the automatic machineries, which in turn comprises mostly of the rotating types. These rotating machines often need to operate under a heavy load and higher operating speed for prolonged periods with high efficiency. In order to achieve this, engineers have to design such machines, which could operate under such dynamic conditions and include stability, high-speed, and enhanced load bearing capacity, lower friction and noise. To endow such needs, designers have engineered the concept of journal bearings, which works on the principle of hydrodynamic lubrication. The journal (or rotating shaft) and the bearing are detached by a thin film of lubricant that allows the shaft to rotate freely without any physical contact with the bearing. With more efficient and realistic design of the bearings, designers can more accurately predict the dynamic behaviour of the journals. The present article aims to address the aspect of more realistic analysis of journal bearing by means of incorporating inevitable uncertainties into the system. The study of full journal bearing is of significant importance to assess the load bearing capacity and subsequently reduce friction between the relatively moving surfaces. By virtue of hydrodynamic action exerted in the form of wedge shaped film of lubricant, the dynamic performances of such bearings are always subjected to variability due to randomness in the system parameters. However, due to wedge action in the clearance space, the developed pressure supports the load without metal-to-metal contact. The whirling flow of lubricant through bearing depends on various input parameters and boundary conditions such as bearing clearance, viscosity and supply system of lubricant. Due to uncertain variability of bearing load and its inherent complexity of hydrodynamic action, the behaviour of such bearings are difficult to map precisely as per its true design specifications. Hence, dynamic characteristics of such system results in unavoidable uncertainty in the output responses. It involves various both known and unknown sources of uncertainty linked with geometry, material properties, environmental and operational conditions. Thus, to comprehensively understand this problem, it is important to study the uncertain dynamic

3

behaviour of such bearings. In this context, the damping and stiffness of the oil film in conjunction to the stability of the rotor bearing system are also required to be investigated to cushion the whirl instability. A plenty of experiments are carried out during the incubation periods of tribology to formulate the mechanism of hydrodynamic lubrication, which laid the foundation of tribology. As a pioneering work, Petroff [1] postulated two cardinal things; first being that viscosity is a fluid property with regards to its friction and second that the nature of friction is the viscous shearing between the two rotating surfaces. Tower [2] conducted a series of experiments on railroad axles while Reynolds [3] portrayed the theoretical basis through the differential equation which were not apparent in Tower’s experiments. However, the phenomenon of hydrodynamic lubrication subjected to compressible fluids was studied by Kingsbury [4] and the analytical solution for infinitely long journal bearings is investigated by Sommerfeld [5]. Harrison [6] derived the Reynolds equation for compressible fluid films while Swift [7] formulated the Reynolds equation applied to the dynamic loading. Ocvirk[8] proposed a closed form solution using the short bearing approximation to develop the detailed and full solution to the problem. Later, Cameroon and Wood [9] employed Southwell relaxation technique to solve the Reynolds equation for full finite oil journal bearing for a wide range of aspect ratios while in contrast, Ebrat et al. [10] calculated for dynamic characteristics of journal bearing including misalignment and bearing structural deformation. Pinkus [11] and Raimondi and Boyd [12] solved the design parameters numerically for full and partial journal bearings using the modern computers. The effect of variation of viscosity in conjunction to Generalised Reynolds equation is investigated by Dowson [13]. Pinkus and Sternlicht [14] performed the theoretical analysis on plain journal bearing, providing significant data for various aspect ratios. A plenty of deterministic research work is carried out on static as well as dynamic characteristics of journal bearing in past [15-34]. Majumdar et al. [35] investigated on the higher load capacity and pressure development of three axial grooved bearings while Kini et al. [36] suggested the use of smaller groove angles for better stability based on the mass

4

parameter and whirl ratios while Roy [37-39] exhaustively studied on the effect of groove location for higher eccentricity ratio. Brito et al. [40] presented an analysis on the effect of grooves in single and twin axial groove journal bearings under varying load direction whileHajishafieeet al.[41]studied on coupled finite-volume CFD solver for elasto-hydrodynamic lubrication of rolling contact bearings. Some more studies on this topic includes the investigation on the performance under severe operational conditions [42], effect of ring misalignment on the fatigue life of cylindrical roller bearing [43], bearing coefficients identification of operating flexible rotor-bearing [44], bearing clearance effects in dynamics of turbochargers [45] and the effect of surface texturing [46]. A further description of the works reported concerning the aspect of surface roughness in journal bearings id provided in the following paragraph. The inculcation of the effect of surface roughness on the hydrodynamic lubrication is initially contributed by Patir and Cheng [47-48] dealing with Reynolds equation in terms of pressure and shear flow stress. The analysis of dynamic coefficients for the effect of isotropic surface roughness is presented by Guha [49]. A stochastic finite element analysis of the hydrodynamic bearing with rough surface is performed by Turaga et al. [50-51], while some researchers studied on radial basis function based modelling concerning the analyses of bearing [52-53]. In contrast, Chiang et al. [54] and Hsu et al. [55] evaluated the apparent effect of couple stress and surface roughness on the performance of oil blended lubricants on the static characteristics. Bhaskaret al. [56] studied the effect of surface roughness on the steady state characteristics for various aspect ratios by using the flow factor method. A significant volume of scientific literatures exists on hydrodynamic journal bearings. The steady state and dynamic characteristics of journal bearings have been analysed deterministically and significant amount of results are reported to constitute the design data for journal bearings. It is observed that the parameters of a journal bearing have significant effect due to surface roughness of the bearings. Subsequently, several researches are carried out including the roughness parameter, which showed considerable improvements in various steady and dynamic characteristics. Another

5

important aspect that still needs proper attention is the dynamic behaviour of journal bearings under the compound effect of surface roughness and inevitable stochasticity in the system parameters. The present article aims to analyse this problem in a Monte Carlo simulation based stochastic framework. However, as indicated in various previous literatures [57-67], Monte Carlo simulation is normally a computationally intensive procedure that requires thousands of simulations to obtain the probabilistic descriptions of the parameters of interest. To mitigate this lacuna we have developed a radial basis function [59-60] based algorithm in conjunction with the finite difference method for hydrodynamic journal bearings that is capable of reducing the required number of actual model evaluations to a significant extent. Hereafter, this article is organised as follows: section 2 provides the theoretical formulation for hydrodynamic journal bearings; the radial basis function based Monte Carlo simulation algorithm in context to the analysis of journal bearings is described in section 3; probabilistic numerical results are presented in section 4; finally, section 5 provides a brief summary of the contribution of this work and concluding remarks.

Fig. 1 Full journal bearing and the co-ordinate system (x-x’ is the axis of bearing and y-y’ is the axis of journal) 2. Theoretical formulation The simplest configuration of a bearing is the plain cylindrical journal bearing. It consists of upper and lower semi cylindrical arcs bolted very closely around the shaft. Typically, lubricating oil is supplied into the bearing at the joint under pressure and fills the clearance space between the shaft and bearing due to the rotation of the shaft. The oil leaves the bearing axially. As the shaft rotates, there is 6

a converging region in the direction of rotation which is the wedge shape oil film supporting the shaft during the operation. In the figure 1, the line, which passes through the centres of the shaft and bearing, is called the line of centres. During the operation, the attitude angle is measured from the line of centres. The shaft is found to be offset from the centre of the bearing, which is the main reason for the development of the converging region within the bearing. The more the eccentric shaft is, the more will be the magnitude of pressure. This, in turn would provide the higher load carrying capacity. The load acts on the centre of the journal. Because of this nature of the fluid film, it acts as a system of springs and dashpots, often idealized by four stiffness coefficients and four damping coefficients. It is to be noted that if pressure becomes less than zero then it is to be assumed as zero i.e., atmospheric pressure as the bearing is submerged and there will be no cavitation. The z-axis is considered to be perpendicular to the plane of bearing cross-sectional view and the finite volume grids are formed on the basis of z and θ. At any point of rotation of the journal, the thickness of the film is considered as C which is mentioned as hmin (minimum film thickness at a particular value of eccentricity) in Figure 1. The governing equation is the Reynolds equation in two dimensions for an incompressible fluid, given by

  3 P    3 P  h h  12 h  h   6U X  X  Z  Z  X t

(1)

Using the following substitutions

x z h pC 2 D   ,z  ,h  , p  ,   p t , R  R L2 C 6UR 2 the equation (1) can be written in non-dimensional form as

  3 P   D    3 P  h h  h      h    2      L  Z  Z    2

(2)

where, h  1   cos  . Considering the effect of surface roughness, Reynolds equation can be expressed as [52]

7

  D    P    P  H   H     H 3      H 3   1  2   2        L  Z  Z 

 

2

 

(3)

where  ( ) is the expectancy operator, defined by 

 x    xf x dx

(4)



andf(x) is the probability density function for the variable x. The film thickness for surface roughness in general form is

H  h  , T   h  ,  

(5)

where, h ( , T ) denotes smooth part of the geometry and h ( ,  ) for surface roughness both measured at the nominal level. For isotropic surface roughness, the expectancy operator for film thickness transforms into

 H   h ,  H 3   h 3 

h cs 3

2

(6)

where, cs  cs / C and cs   / 3 , c s is roughness variation and  is the standard deviation of roughness. In the above governing equations, a variation parameter ~ is introduced to represent the stochasticity. This ensures the randomness of the journal motion under stochastic randomness effect. 2.1 Steady State Characteristics 2.1.1 Smooth hydrodynamic journal bearing The variation parameter ( ~ ) considered for the randomness in the dynamic characteristics of the hydrodynamic journal bearing is introduced in the Reynolds equation. Therefore, a modified form of the equation is obtained. Also, in the process, the film thickness equation also gets modified and takes the form as

~   1   ~ cos  h  Under steady state conditions, the equation is modified to

8

(7)

~   D 2   ~   3 ~ P  ~  P   h    h        h 3       L  Z  Z  

(8)

Furthermore, it is assumed that the pressure drop is linear from the feeding end to the exit end. The above equation is solved numerically by the finite difference method using the Successive OverRelaxation Method (SOR) scheme satisfying the boundary conditions of the equation. After calculating the pressure, the load carrying capacity can be calculated by the following components of force along and perpendicular to the line of centre as 2

~   W  ~ cos   ~    L p ~ ~ WX  0  cos   .Rd

(9)

0 2

~   W  ~  sin   ~    L p ~ ~ Wz  0  sin   .Rd

(10)

0

After non-dimensioning these two equations, we get

~  W X 

2

 p cos   .d ~

~

(11)

0

~  Wz 

2

 p sin   .d ~

~

(12)

0

Thus, the magnitude of load carrying capacity can be obtained as ~   W 2  ~   W 2  ~ W  X Z

where W 

(13)

WC 2 . The attitude angle is calculated as 6R 2UL ~ 1  WZ    ~  0    tan  ~    W   X 

(14)

The non-dimensional flow coefficient can be calculated as follows ~   1  D  Q  2 L 

~ 3 ~ p     h  d 0 z

2 2

9

(15)

~  can be found from ~  and friction variable   The friction force F  2 ~  1 ~ ~  p  d F      ~  3h  h     0

~  1 ~  p   d    3 h    h ~     ~   0  ~ 6W 

(16)

2

(17)

Since no cavitation is observed, the integration was carried out in the [0 2π] range. Due to stochastic variation, a set of steady state parameters are obtained corresponding to the random inputs. 2.1.2 Hydrodynamic journal bearing including surface roughness By coupling, the roughness variation parameter for stochasticity and the isotropic surface roughness by substituting the expression for the expectancy operator from equation (6) deduce the Reynolds equation into

 

 h cs2 3   h  3 

 P   D  2   3 h cs2      h      L  Z  3   

 P     1  2  h  2 h  Z     

(18)

Under the steady state conditions, the equation further reduces to

 

~ c 2  ~   P  ~ c 2  ~   P  ~   D  2   ~  h  ~  3 ~ h  h  3 ~ s s      h         L  Z  h     Z    3 3      

(19)

The above equation is solved for pressure distribution and further using the equations for load capacity, friction variable, Sommerfeld number and other static characteristics using the same equations as defined before. Here the boundary conditions can be expressed as

p( , z)

z 0

 p( , z)

z 1

p ( , z )  0 z 1 z

2.2 Dynamic Characteristics 10

0 (20)

2.2.1 Smooth hydrodynamic journal bearing The Reynolds equation in the dynamic condition is given by equation (1). In the dynamic characteristic equations, the variation parameter

is introduced in the eccentricity ratio and film

thickness. The journal performs a harmonic motion of small amplitude along and perpendicular to the line of centres around its steady state position,

and

. It is assumed that at the onset of instability,

the position of journal centre can be defined in terms of its steady state values with the vibration of frequency  p ; thus ~    ~    ~ e i   0 1

(21)

 ~   0 ~   1 ~ ei

(22)

Therefore, the pressure and film thickness can be expressed using first order perturbation method as ~   p  ~    ~ ei p  ~    ~   ~ e i p  ~ p 0 1 1 0 1 2

(23)

~   h  ~    ~ e i cos  ~    ~   ~ e i sin   ~ h  0 1 0 1

(24)

Substituting the above equations into the Reynolds equation and retaining the first linear terms, gives ~  and p  ~  .The equations for p and p are solved ~  p  the three differential equations in p0  1 2 1 2

numerically satisfying the modified boundary conditions and known values of

.

2 2 2 ~ ~ ~ ~ ~   p0    3   ~ sin   ~ h 2  ~   p0     D  h 3  ~   p0    h0   h03  0 0    2 Z 2 L

(25)

2 ~ ~  2 p1 ~  2 ~  p1   2 ~  p1   ~ ~ ~ ~ h    3   sin   h0    3 cos   h0     2  2 2 2 ~  p0 ~   D  3 ~  p1 ~  2 ~ p 0   ~ ~ ~ ~ ~                6   cos   sin   h0   3 sin   h0     h0    Z 2 L 2  2 p0 ~  D  3  cos  ~ h02 ~   sin  ~   i2 ~ cos  ~   0 2 L  Z  

(26)

3 0

11

2 ~ ~  2 p 2 ~  ~ sin  ~ h 2 ~   p 2    3 sin  ~ h 2 ~   p0    h03 ~   3   0 0   2  2 2 2 ~ p0 ~   D  3 ~  p 21 ~  2 2 ~ p 0   ~ ~ ~ ~  6   sin   h0    3 cos   h0      h0     Z 2 L 2  2 p0 ~  D  3  sin  ~ h02 ~   cos  ~   i2 ~ sin  ~   0 2 L  Z  

(27)

The dynamic pressures are obtained due to the journal displacement parallel and perpendicular to the line of centres. Knowing these pressures, the components of the dynamic loads W1 and W2 can be expressed as

W ~  1

2

X

~ cos   ~ d   P1 

(28)

0

W ~  1

2

X

~ sin   ~ d   P1 

(29)

0

W ~  2

2

X

~ cos   ~ d   P2 

(30)

0

W ~  2

2

X

~ sin   ~ d   P2 

(31)

0

Since p1 and p2 are complex by nature, the dynamic loads take the form as

~   ReW  ~   i ImW  ~ W1  1 1

(32)

~   ReW  ~   i ImW  ~ W2  2 2

(33)

In the dynamic stage, where the fluid film supports the rotor, it is equivalent to a system of spring and dashpot executing harmonic motion. Thus, the stiffness and damping characteristics for the film can be expressed in the form as the components of the dynamic load components ~   ReW  ~  , K  ~   ReW  ~  K XX  1 ZX 1 X Z ~   ReW  ~  , K  ~   ReW  ~  K ZZ  2 XZ 2 Z X

12

(34)

~   ReW  ~  , D  ~   ReW  ~  DXX  1 ZX 1 Z Z ~   ReW  ~  , D  ~   ReW  ~  DZZ  2 XZ 2 Z X

The above stiffness and damping coefficients are the spring and the dashpot systems, which bounds the motion of the journal in operation. They have the significant effect on the overall dynamic performance of the rotating machineries. The natural frequencies of the machines strongly depend upon the bearing stiffness, thus making it an important criterion for the designer to tune the dynamic performance. The damping effect obtained by the fluid film is the only damping source available for the bearing. Therefore, they are the adherent coefficients for overall performance of the bearing. During the running conditions, the bearing forces produce unbalance in the bearing known as whirling, even without the application of any external exciting forces. The iniquitous “half frequency whirl” phenomenon of the bearing is a result of the film cross coupling stiffness terms K XZ and K ZX . Because of this reason, an important relation called the mass parameter is calculated by combining the equations of motion and bearing film forces.   ~ ..K ~   K  ~ .D  ~   D   XX XX XX XX   1 ~  ~ .D  ~   D  ~ .K  ~       M  K  ZX XZ ZX XZ  ~ 2 DZZ ~   D XX ~    ~ ~ cos   ~   D  ~ sin   ~   W   D  0 ZX 0 ~  XX 0     W ~  cos 0 ~   2 4 2 M ~   ~    ~   M ~   K ZZ ~   K XX ~   D XX ~ .DZZ ~   DZX ~ .D XZ ~  ~ 0       ~ W   K XX ~ cos 0 ~   K ZX ~ sin 0 ~   0  K XX ~ .K ZZ ~   K ZX ~ .K XZ ~    ~ 

(35)

(36)

0

~) , which are also stochastic in ~) and  ( The above equations are linear algebraic equations in M ( nature due to initial consideration of variability of input condition. The mass parameter obtained above can be called as the “critical” value of shaft mass for the initiation of the whirl instability. The speed of journal calculated by M is the threshold speed, above which the bearing system will be unstable.

13

2.2.2 Hydrodynamic journal bearing including surface roughness Using the same substitution from equations (20) - (23), the modified equation takes the shape as 2 ~ 2 ~ ~  ~  h0 ~ C S2 ~    D    P1    P1  3 ~ ~   h0  C S     h0         h03         3 3   L  Z  Z   2 2 ~ 2 ~ 2 ~  ~   P0  ~   C S    cos  ~    D   3h 2 ~   C S    cos   ~   P0    sin  ~   3h02      0     3  3  Z 2   L   ~ cos  ~   0  i2 

2   P2 ~   3 ~ h0 ~ C S2 ~    D    P2 ~   3 ~ h0 ~ C S2 ~    h0         h0           3 3   L  Z  Z   2 2 ~ 2 ~   P0 ~   2 ~ C S2 ~   ~    D   3h 2 ~   C S    sin  ~   P0    cos  ~        3 h   sin     0  0     3  3  Z 2   L    i2 ~ sin  ~   0

In a similar manner, as discussed in the preceding sections, the dynamic pressures

and

(37)

(38)

and other

dynamic characteristics are obtained. 3. Stochastic formulation 3.1. Radial Basis Function (RBF) based surrogate modelling Modern engineering analyses use approximation of costly experiments or simulations, wherein surrogate models are often employed to save the cost and time by eliminating the expensive simulation model/experiment by efficient mathematical models. In the present analysis of hydrodynamic journal bearings, Monte Carlo simulation is employed for probabilistic assessment of the system that involves plenty of realizations being carried out. In general, for complex systems such as hydrodynamic journal bearings, the performance functions or parameters of interest are unavailable as an explicitly closedform mathematical function of random input parameters [68-76]. Hence, the entire solution often becomes computationally quite intensive. Carrying out thousands of such simulations becomes practically impossible unless the system is coupled with an efficient surrogate model [77-82]. The radial basis function (RBF) model is utilised to form a representative surrogate of the true model 14

Fig.2 Flow diagram for stochastic analysis of hydrodynamic bearing (The original simulation model is replaced by RBF model, which establishes an efficient mathematical model between the stochastic input parameters ( x( ) ) and output parameters of interest ( y( x( )) )) pertaining to the output parameters of interest to random inputs. The RBF model is sufficient to find the predictive output values of the stochastic system encountering each possible combination of input parameters. In the present study, the surrogate relates the true values of input parameters vector of d xˆ  ( xˆ1 , xˆ2 ,......., xˆd ) with the true value of output parameter of y . Assuming a finite number n of

training observations ( xˆ(i ) , yˆ(i ) ), i  1,2,....., n the aim is to form a generic equation F that allows in 15

forecasting the output parameters from unknown set of input parameters as close as possible. The radial basis function is utilized by Hardy [83] for the interpolation of geographical scattered data and later on, is utilised for solving the partial differential equation by Kansa [84]. In general, RBF is employed to interpolate the scattered multivariate data [85-90]. The significance of such function allows their monotonic responses to increase or decrease from the mean central point with sparsity. The quadratic surrogates are smooth to implement while modelling the curvature of the underlying function. Alternatively, it is carried out by considering interpolating surrogates, wherein linear combinations of nonlinear basis functions are compensated for the interpolation points. The RBF model is reckoned only upon the distance to a centre point xˆ j and is of the form g ( xˆ  xˆ j ) and RBF is calculated from the shape parameter ( c ), in which case g ( xˆ  xˆ j ) may be

reconstituted by g ( xˆ  xˆ j , c ) [84, 88 - 93]. Assuming a set of nodes xˆ1, xˆ2 ,......., xˆN    Rˆ n , the radial basis functions centred at xˆ j can be expressed as g j ( xˆ )  g ( xˆ  xˆ j )  Rˆ n , j  1,2,......N

(39)

Here xˆ  xˆ j is the Euclidian norm. The radial function for RBF model can be expressed as,

g j ( xˆ )  e

 c 2 xˆ  xˆ j



2

g j ( xˆ )  ( xˆ  xˆ j )  c 2



(For Gaussian)



1 2

g j ( xˆ )  ( xˆ  xˆ j )  c 2

(For multi-quadratic)





1 2

(For inverse multi-quadratic)

2

g j ( xˆ )  xˆ  xˆ j log xˆ  xˆ j

(For thin plate splines)

k

g j ( xˆ )  xˆ  xˆ j , k  1,3,5...... k

g j ( xˆ )  xˆ  xˆ j log xˆ  xˆ j , k  2,4,6.....

16

(For Biharmonic)

(40)

A remarkable behaviour of radial basis function is that it does not require the grid. The geometric properties are needed to measure the pairwise distances between points. It is also not complex while dealing with higher dimensional problems as the distances are easy to calculate in any number of space dimensions. In the present approach, the basis function is chosen by comparing relative performance with respect to original Monte Carlo simulation and the RBF is employed with the fixed parameter c 2 = 1. The accuracy of the results may depend on the shape parameter (c). It is to be noted that an RBF exactly passes through all the sampling points. This denotes that the approximate function values are equal to the true function values at the sampling points. This is also observed while the coefficients are measured. The local deviation at an unknown point ( x ) is amplified using stochastic processes. Finally, the performance of the RBF model is validated with respect to traditional Monte Carlo simulation. 3.2. RBF based Monte Carlo simulation algorithm for stochastic analysis The crucial performace parameters ( g ( ) ) for the function of a hydrodynamic journal bearing are investigated for the effect of stochasticity. These parameters are attitude angle, load bearing capacity, Sommerfeld number, mass parameter, whirl ratio and minimum film thickness. The attitude angle represents the angle between the maximum pressure line and vertical axis of the bearing. Therefore, lesser the attitude angle, more is the vertical component of the maximum pressure and hence more is the vertical load carrying capacity. The load bearing capacity indicates the amount of load, the fluid film in bearing can withstand without making surface contact between the journal and bearing. Therefore, the load carrying capacityincreases with the decrease of wearing of the bearing and hence longer is the bearing life.The Sommerfeldnumber, also known as the bearing characteristic number, depicts a non-dimensionalparameter that gives the characteristics of bearing as it contains the variables required for design of bearing. In general, Sommerfeld Number (S) is defined by the following equation.

17

 r  N S   C  P 2

(41)

where r, C, µ, N and P denotethe shaft radius and radial clearance, coefficient of absolute viscosity of the lubricant, speed of the rotating shaft in rev/s and load per unit of projected bearing area, respectively. The mass parameter indicates the square root of critical mass that is related to stability.If the mass of the journal is found greater than the critical mass,the system becomes unstable. Therefore for a stable condition, square root of the journal mass should be less than the value of mass parameter, otherwise the system will become unstable. In contrast, the whirl ratio is defined as the ratio of angular frequency of vibration of journal to the angular speed of the journal. Therefore, lesser the whirl ratio, higher is the stability of the bearing system. The minimum film thickness represents the minimum clearance between the journal and bearing. As the load increases the minimum film thicknessdecreases. Hence, it implies that higher the minimum film thickness more is the load carrying capacity of the bearing. Three different forms of stochasticity are considered in the present Monte Carlo simulation based analysis encompassing the individual and combined variation of the random input variables such as eccentricity ratio (  ) and roughness parameter ( cs )

g 1 ( )   ( ) g 2 ( )  cs ( )

(42)

g ( )   ( ), cs ( ) 3

Monte Carlo simulation [94, 95] furnishes a range of prospective outcomes along with their respective probability of occurrence. This technique performs uncertainty quantification by forming probabilistic models of all possible results accounting a range of values from the probability distributions of any factor that has inherent uncertainty. It simulates the outputs over and over, each time using a different set of random values from the probability distribution of stochastic input parameters. Depending upon the nature of stochasticity, a Monte Carlo simulation could involve thousands or tens of thousands of recalculations before it can provide a converged result depicting the distributions of possible outcome values of the response quantities of interest. Each set of samples is called an iteration or realization, 18

and the resulting outcome from that sample is recorded. In this way, Monte Carlo simulation provides not only a comprehensive view of what could happen, but how likely it is to happen i.e. the probability of occurrence. The mean or expected value of a function f ( x) of a n dimensional random variable vector, whose joint probability density function is given by  ( x) , can be expressed as

 f  E  f  x    f  x   x  dx

(43)



Similarly the variance of the random function f ( x) is given by the integral below,

 2f  Var  f  x     f  x    f    x  dx 2

(44)



The above multidimensional integrals, as shown in equation (43) and (44) are difficult to evaluate analytically for many types of joint density functions and the integrand function f ( x) may not be available in analytical form for the problem under consideration. Thus the only alternative way is to calculate it numerically. The above integral can be evaluated using MCS approach, wherein N sample points are generated using a suitable sampling scheme in the n-dimensional random variable space. The N samples drawn from a dataset must follow the distribution specified by  ( x) . Having the N samples for x, the function in the integrand f ( x) is evaluated at each of the N-sampling points xi of the sample set    x1 ,............, xN  . Thus, the integral for the expected value takes the form of averaging operator as shown below (45)

N

1

 f  E  f  x     f  xi  N i 1 Similarly, using sampled values of MCS, the equation (43) leads to

 2f  Var  f  x     f  xi    f  N  1 i 1 1

N

19

2

(46)

Thus the statistical moments can be obtained using a brute force Monte Carlo simulation based approach, which is often computationally very intensive due the evaluation of function f ( xi ) corresponding to the N-sampling points xi , where N ~ 103.The noteworthy fact in this context is the adoption of surrogate based Monte Carlo simulation approach in the present study that reduces the computational burden of traditional (i.e. brute force) Monte Carlo simulation to a significant extent. The RBF based algorithm is developed to obtain computational efficacy as furnished in figure 5. For developing the RBF surrogate models, Sobol sequence [96] is adopted as previous investigations have reported it to perform better than other random and quasi-random sequences [97]. In the present approach, the surrogate model (RBF) is initially formed using few optimally chosen design points based on Sobol sequence. Hence, identical number (design point numbers) of simulations is carried out. The RBF model, in turn, reconstitutes the real upscale simulation model by the mathematical model. After formation of the surrogate model, a plenty of virtual simulation is exercised for all possible combinations of input variables (as shown in figure 2).

4.Results and discussion Based on the RBF based stochastic framework developed in the preceding section, probabilistic results are presented for hydrodynamic journal bearings. The code for hydrodynamic journal bearing is validated with the deterministic results available in literature considering the flow to be laminar, assuming the fluid viscosity to be constant, no slip at the boundary and lastly neglecting the cavitation phenomenon and hence carrying out the integration in the range [0 2π]. In this article, the minimum film thickness is expressed in a non-dimensional form dividing it by clearance of the bearing (C). In case of plain journal bearings with L/D ratio as 1 (all other results in this article are presented considering the same value unless otherwise mentioned), a comparison of attitude angle, friction variable, Sommerfeld number and mass parameter obtained from the developed computer code without involving variability parameter (i.e.

= 0) is made with that of Pinkus [14], as presented in

20

Table 1 Comparison of results for plain journal bearings with literature [14] Eccentricity ratio (ϵ) Parameters 0.2

0.4

0.6

0.8

Present study

73.95

62.52

50.37

35.98

Pinkus [14]

74.00

62.00

50.00

36.00

Present study

13.22

5.93

3.26

1.70

Pinkus [14]

12.90

5.80

3.21

1.71

Present study

0.6563

0.2694

0.1241

0.0451

Pinkus [14]

0.6320

0.2610

0.1200

0.0448

Present study

0.119

0.351

1.000

2.394

Pinkus [14]

0.248

0.509

1.071

1.231

φ

µ

S

M

Fig. 3 Non-dimensional load carrying capacity with respect to eccentricity ratio corresponding to L/D ratios

21

Probability density function plot

(a)

(d)

(b)

(e)

(c)

(f)

Mass parameter

Attitude angle

Load bearing capacity

Scatter plot

Fig. 4 Scatter plot and probability density function plot for validating the RBF models corresponding to the load bearing capacity, attitude angle (in degree) and mass parameter

22

Probability density function plot

(a)

(d)

(b)

(e)

(c)

(f)

Sommerfeld number

Minimum film thickness

Whirl ratio

Scatter plot

Fig. 5 Scatter plot and probability density function plot for validating the RBF models corresponding to the Whirl ratio, Minimum film thickness and Sommerfeld number

23

the Table 1. Close proximity of the values indicates the validity of the developed computer code. Based on the validated code, new results are presented in figure 3 for the deterministic variation of load bearing capacity with eccentricity ratio for different values of L/D ratios. A second set of validation is required in the present analysis to ensure the performance of RBF model in making accurate predictions for further probabilistic analysis. Scatter plot and probability density function plots for the six crucial performace parameters of interest (g) are shown in figure 4-5 considering compound variation of the stochastic input parameters (g3( )). The scatter plots show the relative deviation between the values of g obtained from the original simulation model and the RBF model constructed with different sample size. From the scatter plots, it is observed that a satisfactory level of accuracy is achieved for all the parameters corresponding to the sample size of 32, even though the nature of mass parameter and whirl ratio is evident to be quite nonlinear compared to the other parameters. The probability density function plots are presented considering the converged sample size of 32 along with the results obtained using the original simulation model. Minimal deviation among the plots corroborates the accuracy of the RBF models for carrying out further analyses. In this context it can be noted that all the probability density function plots in this articles are obtained using 10,000 samples of Monte Carlo simulation based on the constructed RBF models. Thus, a computational efficiency of more than 300 times is achieved in the present problem. The objective of the present study is to simulate the actual working conditions of the journal or shaft inside the bearings as closely as possible when acted upon by any vibratory or any self-exciting forces that occur even in the absence of any external agent. For the purpose of imitating actual field condition, different level of stochasticity is considered with the corresponding variation parameter ( ) of 10%, 20% and 30% with respect to the mean values. The stochastic input parameters are drawn from a random uniform distribution. Realistic values of eccentricity ratios ranging from 0.4 to 0.6 are considered in this study which is suitable from practical point of view. The effect of stochasticity in the eccentricity ratio (g1( )) is mapped on the crucial performance parameters of interest (g) and the

24

Mass Parameter

Whirl ratio

Load bearing capacity

 10%

(a)

(d)

(g)

(b)

(e)

(h)

(c)

(f)

(i)

 20%

 30%

Fig. 6 Effect of variations of eccentricity ratio on (a,b,c) mass parameter (d,e,f) whirl ratio (g,h,i) load capacity for smooth journal bearing corresponding to degree of stochasticity ( )(nature of stochasticity considered: g1( )) 25

Attitude angle

Minimum film thickness

Sommerfeld number

(a)

(d)

(g)

(b)

(e)

(h)

(c)

(f)

(i)

 10%

 20%

 30%

Fig. 7 Effect of variations of eccentricity ratio on (a,b,c) Attitude angle in degree (d,e,f) Minimum film thickness in non-dimensional form (g,h,i) Sommerfeld number for smooth journal bearing corresponding to degree of stochasticity ( )(nature of stochasticity considered: g1( )) 26

(a)

(b)

(c)

(d)

(e) (f) Fig. 8 Stochastic variation of (a) mass parameter (b) Whirl ratio (c) Load capacity (d) Attitude angle (in degree) (e) Sommerfeld number for corresponding to degree of stochasticity ( ) (nature 2 of stochasticity considered: g ( ))

probabilistic results are presented in Figure 6-7. Figure 6(a,b,c) show the probability density function plots of the mass parameter corresponding to different eccentricity ratios showing the variation for 10%, 20% and 30% respectively. It is evident from the plots that with the increase in the eccentricity ratio, the sparsity increases and the variation gets more uniformly distributed. This, in turn indicates that the stability of the system increases with the increase in the eccentricity ratio that correlates with the common notion of a typical characteristic of the journal bearings. Figure 6(d,e,f) shows the probability density function plots of the whirl ratio corresponding to different eccentricity ratios

27

showing the variation for 10%, 20% and 30% respectively. Whirl ratio is a parameter which gives the indication about the frequency at which the machine would operate under the operating conditions. This is an important parameter since the system would fail if it reaches the resonant condition. In the plots, it is apparent that with the increase in the eccentricity ratio,themagnitude of the operating frequency is more than the system natural frequency. Also, when the percentage variation is increased, the sparsity and the magnitude are increased, as expected from a deterministic point of view. This clearly indicates that with the higher eccentricity ratio, the system is more stable and has higher operating characterisitics. Figure 6(g,h,i) shows the probability density function plots of the load carrying capacity corresponding to different eccentricity ratios showing the variation for 10%, 20% and 30% respectively. With the increase in the eccentricity ratio, the loadcarrying capacity of the system increases, indicating that the journal is more stable at higher eccentricity ratio. Figure 7(a,b,c) presents the plots for the attitude angle, wherein it can be noticed that there is a decrease in the magnitude of the angle subtended between the line of centres of the journal and bearing and the load line. This implies that with the increase in the eccentricity ratio, the line of centres tends to move along the load line, thus, improving the load carrying capacity of the bearing. With the increase in the percentage variation, the trend follows a decrease in a smooth transition and the magnitude of the variation reduces significantly. Also, the same trend is observed when the increase in the roughness variation parameter. The amount of thickness of the film ensures minimal contact of the journal and the shaft and longer operating life of the bearing. Sommerfeld number is a number which is also known as the bearing characteristic number. It is inversely related with the load per unit bearing area and is directly related to the friction variable. In figure 7(g,h,i), the plots show a decrease in the magnitude of the Sommerfeld number with increasing eccentricity value that is an indication of the increase in the load capacity as well as decrease in the amount of friction between the shaft and the bearing.

28

Figure 8 shows the probabilistic variation of the stochastic performance parameters (g) obtained due to the individual variation of the roughness parameter (g2( )). The mean values of different parameters are found to be affected marginally due to the variation in roughness parameter. However, the sparcity of different performance parameters (g) are noticed to increase with increasing degree of variability in the roughness parameter, except for the case of whirl ratio, which is found to be insensitive of the roughness parameter.The individual effects of stochasticity in eccentricity ratio and roughness parameter (g1( ) and g2( )) are described in the preceding paragraphs. However, a combined influence of both these stochasric effects can be crucial for the performance of a practical system. Thus, we have investigated the combined effect (g3( )); the probabilistic descriptions of the performance parameters of interest (g) are depicted in Figure 9. For a better understanding, the results are presented in a comparative form including the probabilistic descriptions for both smooth and rough surface bearings. The plots show a coherence with the deterministic behaviour and connotes the impact of surface roughness on the journal bearings. As minimum film thickness is not dependent on the roughness parameter, the probability density function plots nearly overlap in case of minimum film thickmess. Figure 10 shows the relation of coefficient of variation for the performance parameters of interest (g) with different degree of stochasticity for the compound effect g3( )).From the figure it can be clearly discerned that the mass parameter is most sensitive to the stochastic effect,followed by whirl ratio, load carrying capacity, Sommerfeld number, attitude angle and minimum film thickness, respectively. Such sensitivity analysis results can provide the designer a certain perspective regarding the relative influence of stochasticity on various parameters and thus, the requirement of the relative degree of required control can be assessed. Figure 11 represents therelationship of pressure distribution with respect to angular variation and the axial length due to combined stochasticity. The graph points out the information about the angle at which peak pressure responsible for bearing the load of the journal would be generated. The band of peak pressure explores the working zone of the bearing and also gives the angle at which the wedge film is supposed to form. It can be inferred from 29

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 9 Comparative results for the stochastic output parameters in case of smooth (nature of stochasticity considered: g1( )) and rough surface bearing (nature of stochasticity considered: g3( ))with and corresponding to degree of stochasticity

the graph that the magnitude of pressure decreases towards the end indicating the exit of the lubricant as per the assumption. The figure of stochastic pressure diagram along the entire peripheral domain of the journal bearing can bring about a complete perspective on the distribution of pressure including the stochastic effect of the input parameters.

30

Fig. 10 Coefficient of variation for the output parameters due to combined stochasticity (g3( )) (The figure reveals a decreasing order of sensitivity as: mass parameter, whirl ratio, load carrying capacity, Sommerfeld number, attitude angle, film thickness)

Fig. 11 Variation of pressure distribution with respect to angular positions due to combined stochasticity (g3( )) 5. Conclusion This article provides an in-depth analysis on the stochastic dynamic behaviour of full cylindrical hydrodynamic journal bearings, wherein important performance parameters such as load carrying

31

capacity, whirl ratio, mass parameter, Sommerfeld number, attitude angle and minimum film thickness are investigated. The individual as well as compound effects of stochasticity arising due to the random variation of eccentricity ratio and roughness parameter are accounted in the analysis in a Monte Carlo simulation based probabilistic framework. Such stochastic input parameters can be regarded as the uncertain service conditions and manufacturing imperfections, which are practically inevitable in case of hydrodynamic journal bearings. Physically insightful probabilistic descriptions of different performance parameters are presented following an efficient radial basis function based algorithm, wherein it is found that the effect of stochasticity has significant influence of the dynamic behaviour of journal bearings. Development of the radial basis function based algorithm for stochastic analysis of hydrodynamic journal bearing has enabled us to achieve a computational efficiency more than 300 times compared to traditional Monte Carlo simulation. The relative sensitivity of various performance parameters rating the stochastic effect are quantified that provides a clear perspective about the necessary degree of control required to achieve satisfactory performance of a journal bearing. The novelty of the present study includes the stochastic analysis of hydrodynamic journal bearing including the effect of surface roughness and development of the efficient radial basis function based uncertainty quantification algorithm for journal bearings. This paper can bring about a comprehensive perspective on the performance of hydrodynamic journal bearings in the practically inevitable stochastic regime. Even though the present study includes two important parameters (eccentricity ratio and roughness parameter) as the source of stochasticity, future investigations will follow inclusion of various other stochastic parameters in the analysis and various types of bearing including the thermal and elastohydrodynamic aspects. The radial basis function based approach can be utilized to deal with such computationally intensive future researches in the field of tribology.

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Acknowledgement The first and third authors would like to acknowledge the financial support received from Ministry of Human Resource and Development, Govt. of India, during the period of this research work. References [1]

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