Global sensitivity analysis of static characteristics of tilting-pad journal bearing to manufacturing tolerances

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Global sensitivity analysis of static characteristics of tilting-pad journal bearing to manufacturing tolerances Mingyang Loua,b,∗, Olivier Bareillea, Wenqi Chaia, Mohamed Ichchoua, Wei Chenb a

Laboratoire de Tribologie et Dynamique des Systemes, Ecole Centrale de Lyon, 36 Avenue Guy de Collongue, 69134, Ecully Cedex, France School of Mechanical Engineering Key Laboratory of Ministry of Education for Modern Design &Rotor-Bearing System, Xi'an Jiaotong University, Xi'an, Shannxi, 710049, China

b

ARTICLE INFO

ABSTRACT

Keywords: Global sensitivity analysis Fourier amplitude sensitivity test Tilting-pad journal bearing Manufacturing tolerance Static characteristics

A tiny difference between nominal dimension and actual processing dimension in the range of manufacturing tolerance can vary the performance of journal bearing. For minimizing this variation, sensitivity analysis is usually conducted. In present paper, the variance-based global sensitivity analysis was introduced to deal with the uncertainty quantification problem for a five-pad journal bearing. By using Fourier Amplitude Sensitivity Test (FAST) method, the first-order sensitivity indices of static characteristics including capacity, power loss and inlet flow were analyzed and evaluated for five normal distributed processing parameters: pad internal radius, shaft radius, pivot radius, pivot position and pad angular extent. The FAST method was proved perfectly suited to the bearing model. The quantitative sensitivities of output to each input parameter were obtained.

1. Introduction In a practical machining process of journal bearing, the inevitable error between the real dimension and nominal dimension leads to a variation of performance. These errors come from the manufacturing tolerance and the distribution of studied parameters. Tong and Graziani [1] noted that the appropriate specification of the ranges and shapes of the distributions can dramatically affect the output of the sensitivity analysis. Shin et al. [2] stated that reducing or expending the ranges will affect the results, thereby causing insensitive parameters to become sensitive and vice versa. As a key part of mechanical system, bearing characteristics also affect a lot the whole rotor-bearing system, thus the importance of conducting a sensitivity analysis (SA) is self-evident to understand and determine which parameters have effect on the characteristics most and which that have no effect so that conclusions drawn in bearing studies are robust and stable. Iwamoto and Tanaka [3] studied the case of cylindrical journal bearing. They confirmed that the manufacturing error of roundness affects the minimum oil film thickness, friction coefficient and non-dimensional stability threshold. Iwamoto and Oishi [4] studied the case of four-lobe bearing. After comparing the manufacturing errors of studied parameters, they pointed out that in order to obtain the better characteristics, bearing design was found to demand the smaller value of the allowable error. Fillon et al. [5] studied the case of five pads tiling-pad journal bearing. They

defined the sensitivity parameters of bearing characteristics to manufacturing tolerances. The main bearing characteristics are the bearing radial clearance, the geometrical preload ratio, the relative pivot position on the pad, the pad angular extent, and the pad length. In addition the influence of the load direction, the lubricants characteristics, and the operating conditions were studied. At last, the author concluded that for predicting the static characteristics accurately, the influence of the manufacturing tolerances must be considered. All these references above use the local SA in which one parameter is varied while others are fixed at default nominal values. In a practical local SA, usually, one calculates the parameters’ importance based on the derivative of the model output with respect to that parameter, which means the derivative is taken at the nominal value point in parameter space. Although amount of papers using the local SA in all fields have notably increased over the last decade, its shortcomings were unveiled in lots of published literature [6]. The main drawback was summarized that local SA: 1) is unable to explore the complete parameter space; 2) ignores the complexity of the model, namely, relies on unjustified assumptions of model linearity and additivity; 3) cannot express the interaction between parameters. Global SA as a more reasonable approach has gained considerable attention among practitioners over the last decade due to its advantages over local SA methods [7]. Many reviews of global SA methods have been published in different fields. Saltelli et al. [8] focused on global SA

∗ Corresponding author. Key Laboratory of Education Ministry for Modern Design and Rotor-Bearing System, School of Mechanical Engineering, Xi'an Jiaotong University, Shannxi, Xi'an, 710049, PR China. E-mail address: [email protected] (M. Lou).

https://doi.org/10.1016/j.triboint.2019.04.018 Received 17 July 2018; Received in revised form 31 March 2019; Accepted 6 April 2019 0301-679X/ © 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Mingyang Lou, et al., Tribology International, https://doi.org/10.1016/j.triboint.2019.04.018

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Nomenclature

Cb Cp d e fr ft fx fy ff h l mp Nm Np Ob Oi Oj p

Qin r Rp Rb W z

bearing radial clearance Cb = Rb r , (m ) pad radial clearance Cp = Rp r , (m) shaft diameter (m) eccentricity (m) resultant force in radial direction (N ) resultant force in tangent direction (N ) resultant force in horizontal direction (N ) resultant force in vertical direction (N ) friction (N ) oil film thickness (m) bearing length (m) geometrical preload ratio shaft speed (rpm) number of pad geometric center of bearing curvature center of pad geometric center of shaft pressure distribution (Pa)

s, i e, i

i

i

in chemical models; Borgonovo [9] investigated sensitivity and uncertainty measures without reference to a specific moment of the output; Song et al. [10] reviewed the global SA methods which were used in hydrologic models; Perz et al. [11] reviewed the global SA and uncertainty analysis methods applying to ecological resilience. Some of these reviews explicitly highlighted the advantages and disadvantages of various methods, and thereby provide very good summaries of these topics. For conducting a global SA, many analytical techniques were developed including screening method, regression analysis, variancebased method, meta-modeling method [12] and others [13,14]. Variance-based methods use a variance ratio to estimate the importance of parameters [15,16]. This method was firstly employed by Cukier and his colleagues. They put forward using conditional variances to conducted the first-order effects for sensitivity analysis and started working on higher-order terms [17–20]. This is known as Fourier Amplitude Sensitivity Test (FAST). Since it is efficiently fitting for non-linear and non-monotonic models, FAST has been applied to different models, such as vibroacoustic transmission models [21], porous media models [22,23]. There is no doubt that this method will be more widely used. While few comprehensive and up-to-date papers have been conducted using the global SA on tilting-pad journal bearing. In present paper, the global SA and the calculation of the static characteristics of tilting-pad journal bearing were combined aiming to introduce the global SA to bearing performance analysis, to identify key machining dimensions and to determine which dimension affect the model sensitivity in a quantitative and robust way. A five-pad tilting-pad journal bearing was taken as an example. The studied parameters were sampled using a search-curve algorithm in parameter space. The global SA was conducted to study the sensitivity of bearing characteristics including capacity, power loss and inlet flow to five studied parameters: pad internal radius, shaft radius, pivot radius, pivot position and pad angular extent.

inlet folw (m3 / s ) shaft radius (m) pad internal radius (m) pivot radius (m) applied load (N ) coordinate in axial direction (m) pivot position on the pad ( ) pad angular extent ( ) leading edge of pad i ( ) trailing edge of pad i ( ) pivot location of pad i ( ) angle between two adjacent pivot ( ) pad tilting angle ( ) eccentricity ratio attitude angle ( ) dynamic viscosity of lubricant (Pa . s ) coordinate in circumferential direction ( ) shaft speed (rad / s ) pad subscribe

single pad in local coordinate system:

1 Rp2

h3 p

+

h3 p =6 z z

h (1)

where h is oil film thickness for each pad in titling-pad journal bearing and can be expressed as:

hi = Cp

(Cp

Cb)cos(

i

) + ecos (

)+

i Rpsin( i

)

(2)

The film pressure generates so as to satisfy the moment balances of pads around the pivot [27]. Assume that the film pressure of each pad is always positive over the entire pad surface [28]. The following pressure boundary conditions for each pad are employed:

pi ( i ± 0.5 , l/2) = 0 pi ( , l/2) = pi ( , l/2) = 0

(3)

Under the given design parameters and eccentricity ratio, the maximum tilting angle of each pad can be calculated:

2. Model and operating conditions Fig. 1 shows the configuration of the studied five pads tilting-pad journal bearing [24,25], five pads are located at the same intervals on the circumference of bearing, the load applied to the bearing is mainly supported by the lower pad (pad 3). Each pad in bearing is tilting-free around its pivot and the moment applied on pad equals to 0 to obtain an equilibrium state. Reynolds equations [26] governing the steady, laminar incompressible fluid in lubricant films are given in terms of a

Fig. 1. Geometry of tilting pad journal bearing. 2

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1 + (1

=

max , i

mp)cos(

)

e, i

sin(

mp (

i

preload is given in Table 5. The range of all studied parameters is shown in Table 4 and Table 5. Assume that the bearing operates at 2000 rpm and the lubricant viscosity is 0.018 Pa. s when 50 C (ISO VG32).

e, i )

(4)

e, i )

i

In order to simplify the solving process, the effect of inertia mass of pad is neglected [29]. The model is discretized by finite difference method and solved by Newton-Raphson root-finding algorithm. After the pressure distribution p is obtained, integrate the pressure distribution and oil film force is given by:

3. Variance-based global sensitivity analysis The variance-based global SA attempts to distribute the variance of out output to each input parameter [32]. For a square integrable model y = f (x ) with n input parameters, the parameter space is defined in a ndimension unit hypercube:

5

fx =

fr , i ·sin

i

(5)

i=1 5

fy =

fr , i ·cos

n x

i

(6)

i=1

n

pi cos(

i

) rd dz

5

f f ,i

(8)

i=1

n

V (y ) =

where f f , i donates the friction force for a single pad: l /2

e, i

r2

f f ,i =

l /2 s, i

d dz 1 + (f e x , i h 2r y, i

fx , i e y , i )

Qin, i

1=

Qin, i = l /2

2

s, i

h 3s, i 12 r

p

dz s, i

Cb/Cp

Sij + …+ S12… n i

(16)

j>i

where the first term in the right of Eq. (16) is the first-order sensitivity index, the second term is the second-order sensitivity index and so on. Assume that the first-order sensitivity index can be written as:

Si =

Vxi (Ex i (y|x i )) V (y )

(17)

where y donates the output variable, x i donates the certain input parameter, Ex i (y|xi ) donates the conditional expectation of y when x i is fixed, the variance operator Vx (·) is taken over all the possible values of xi. The first-order index also represents the main effect contribution of each input parameter to the variance of the output which indicates how much one could reduce the output variance if x i is fixed. Therefore, the first-order sensitivity analysis allow us to detect and rank those factors which need to be better machined in order to reduce the output variance, as well as to detect the factors that have a better chance of being estimated in a subsequent numerical or experimental estimation process. As an effective algorithm, Fourier Amplitude Sensitivity Test (FAST) method is based on the variance decomposition that has been developed and published by lots of investigators. FAST can quantitation apportions the output variance to the variance of input parameters. By using FAST, the individual contribution of each input parameter to the output variance can be estimated-what is known as the main effect. It should also be mentioned that the FAST appears to be computationally cheaper and good for additive mode (model with no importance

(11)

And h s,i is the oil film thickness at the leading edge of pad i , ( p / )|( s, i ) is the pressure gradient in circumferential direction at the leading edge of pad i . Before conducting the global sensitivity analysis, the numerical studies on the static characteristics of the studied bearing have been compared with the benchmark values in Ref. [30]. The bearing's parameters are listed in Table 1 and the numerical calculations and benchmark value are listed in Table 2.Where the bold type in Table 2 denotes the benchmark value by Ref. [30], the normal type denotes the numerical calculation. The agreement between these two parts is good, which supports a strong basis to the following sensitivity analysis. For conducting sensitivity analysis, the nominal size of the bearing and the manufacturing tolerance should be determined firstly. Table 3 gives the nominal size of the studied bearing. The associated tolerance of pad internal radius, shaft radius and pivot radius was chosen based on the standard international tolerance grade IT6 [31]. For the pivot position, the half of the tolerance corresponding to mechanical processing size in general tolerance was chosen. While for pad angular extent, the general tolerance was chosen. Pivot radius changes with the geometrical preload ratio mp according to its definition:

mp = 1

(15)

j> i

n

Si + i

(10)

rh

Vij + …+ V12 … n i

n

(9)

where Qin is the inlet flow for a single pad: l /2

n

Normalizing the above equation by dividing V (y ) on both sides:

5 i=1

(14)

j> i

Vi + i

And ex , i , e y, i are the projections of eccentricity in horizontal and vertical direction, respectively. The volume rate of inlet flow Qin is:

Qin =

fij + …+f12… n i

Sobol’ proved that this representation is unique if the expectance of each term in the representation is zero, then all components of the decomposition are orthogonal in pairs. By square integrating each term of the decomposition over, the so called analysis of variance (ANOVA) decomposition is obtained:

Friction force can be obtained from:

ff =

fi + i

(7)

l /2 s, i

n

y = f0 +

e, i

fr , i =

(13)

f (xi ), i = 1,2, …, n}

A high-dimensional model representation of f can be expressed as:

where fr , i donates the resultant oil film force in radial direction and can be expressed as: l /2

= {x1, x2, …, xn |x i

Table 1 Bearing's parameters by Ref. [30].

(12)

where Cb = Rb r , Cp = Rp r . The range of pivot radius for each corresponding geometrical 3

Load type

LOP

Aspect ratio, l/ d Geometrical preload ratio, m Pad angular extent, ( ) Pivot position on pad, /

1.0 0.5 60 0.5

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search-curve to explore the parameter space xn systematically. If all parameters are identically and uniformly distributed, the complex computation of the integral in n-dimensional space in Eq. (19) can be simplified into a mono-dimensional Fourier decomposition and the quantities can be obtained along the curve:

Table 2 Validation of the static characteristics of tilting-pad journal bearing.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Sn

Sn

f/

f/

Qin

Qin

0.8423 0.3995 0.2425 0.1578 0.1030 0.0646 0.0372 0.0186

0.835 0.399 0.243 0.158 0.103 0.0650 0.0376 0.0186

28.0644 13.5013 8.4021 5.6743 3.9101 2.6513 1.7170 1.0257

27.9 13.5 8.65 5.76 3.97 2.76 1.73 1.13

4.2769 4.2725 4.2651 4.2548 4.2416 4.2254 4.2041 4.1689

4.27 4.27 4.26 4.26 4.25 4.24 4.23 4.22

T

Value

Nominal diameter, d (m) Aspect ratio, l/ d Number of pads, Np

0.2004 0.5 5

Pad angular extent, ( ) Pivot position on pad, / Pivot Position on bearing,

i(

D=

60 0.5 36, 108, 180, 252, 324

)

1 2

( )

2

1 2

f (x (s )) ds

(22)

(Aj2 + Bj2)

(23)

j=1

where the Fourier coefficients Aj and Bj are defined as:

Lower Tolerance

Mean Value

Upper Tolerance

0.100189

0.1002

0.100211

0.099989 179.83 149.67

(21)

Expanding f (x (s )) in Fourier series and applying Parseval's theorem to the compute Eq. (22):

Table 4 Machining precision of the studied parameters.

Shaft radius, r (m) Pivot position, ( ) Pad angular extend,

f r (x1 (s ), x2 (s ), …, xk (s )) ds T

f 2 (x (s )) ds

D=2

Pad internal radius, Rp (m)

T

If the elements in frequency set {wi} are linearly independent, when s varies from to , the search-curve in Eq. (20) can effectively and fully explore the n-dimension parameter space xn . During this transformation, the search-curve is no longer a space-filling one, but becomes a periodic curve of period 2 , and approximate numerical integrations can be made. Extending this idea to the computation of variance, the variance of f ( ) is available:

Table 3 Nominal size of studied bearing. Parameters

1 2T

y¯(r ) = lim

0.1 180 150

0.100011 180.16 150.33

Lowe Tolerance

Mean Value

Upper Tolerance

0.10016|(0.2) 0.10014|(0.3) 0.10012|(0.4) 0.10010|(0.5) 0.10008|(0.6)

0.100149 0.100129 0.100109 0.100089 0.100069

0.10016 0.10014 0.10012 0.10010 0.10008

0.100171 0.100151 0.100131 0.100111 0.100091

1 2

f (x (s ))cos(js ) ds

Bj =

1 2

f (x (s ))sin(js ) ds

(24)

j Z={ , …, 1,0,1, …, }. The spectrum of the Fourier series expansion is defined as j = Aj2 + Bj2 with j Z . For a real–value function f (x (s )) , the portion of the output variance D coming from a certain parameter x i can be estimated by calculating the spectrum for the fundamental frequency wi and its higher harmonics component pwi , namely:

Table 5 Machining precision of pivot radius. Pivot radius, Rb (m) | (mp )

Aj =

+

Dˆ i =

pw i

=2

p Z

pwi

(25)

p=1

= Z {0} indicates the set of all relative integer numbers where except the zero. Then the estimation of total variance is calculated by summing all j: Z0

significant interactions among factors). These advantages make it fully competent in sensitivity analysis of journal bearing. Rewrite the square integrable model y = f (x ) into:

y = f (x1, x2, …, x n)

(18)

n

f r (x1, x2, …, x n) P (x1, x2, …, xn) d (x1, x2, …, xn)

=2

j

(26)

j=1

The definition of the first-order sensitivity index of x i on y is obtained:

Dˆ FAST Sˆi = i Dˆ

(27)

Considering the present model y = f (x1, x2 , x3 , x5, x5) . Obviously, x1, x2 , x3 , x 4 and x5 represent the five studied parameters, which are defined in parameter space x5 . These five parameters are independent each other and obey the given distribution respectively. To provide a uniformly distributed sample for the above five parameters in the parameters space x5, various search-curves have been proposed [34]. The following sampling scheme is suggested to obtain a more even distribution of the sample points in the parameter space:

(19)

By applying the ergodic theorem, a search-curve which explores the space nx is introduced:

x i = Gi (sin(wi s ))

j p Z0

The input parameters are still defined in the n-dimensional hypercube unit as described in Eq. (13). The function f ( ) can be obtained either by an analytical representation of y or directly as the output of a computer program. Assume that the random vector (x1, x2 , …, xn) obeys a certain probability density function P (x ) = P (x1, x2, …, xn) , the r-th moment of y can be expressed as:

y (r ) =

+

Dˆ =

(20)

, ) , Gi is where s is a variable scalar varying over the range ( transformation function, {wi} is a properly selected frequency set with different values [33]. As s varies, all the parameters change at the same time along the

xi =

1 1 + arcsin (sin ( i s + 2

i ))

(28)

where i is a random phase-shift distributed uniformly in [0, 2 ], s varies in ( /2, /2) . The variation of x i and its histogram of the 4

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empirical distribution are plotted with i = 11 is shown in Fig. 2: Considering the mechanical processing is a normal process, processing sizes that fall in the range between the lower tolerance and upper tolerance are converted into a normal distribution using the parameter's inverse cumulative distribution function, where µ equals to the mean value and equals to one third of upper tolerance, which is known as 3 rule [35]. This can guarantee each processing size falls in the range between the lower and upper tolerance in a probability of 99.73% as shown in Table 6. The converted normal distribution and the corresponding empirical distribution are shown in Fig. 3. After sampling, the samples were put into the numerical program to calculate the static characteristics of the bearing. Then the FAST method based on variance decomposed was conducted to obtain the first-order sensitivity indices. Fig. 4(a) shows the whole procedure of above instruction.

Table 6 Normal distribution of studied parameters. µ Pad internal radius, R (m) Shaft radius, r (m)

Pivot radius, Rp (m)

Pivot position, k ( ) Pad angular extend, ( ) 150

0.1002 0.1

0.10008–0.10016

180 0.1111

3.6667 × 10

06

3.6667 × 10

06

3.6667 × 10 0.0556

06

avoided when the preload ratio is chosen. The heavier the bearing load is, the larger the preload should be. In fact, to avoid the pad unloading or even pad accidentally inverted, a stop or elastic element is installed at the trailing edge of the pad normally as Fig. 6 shows. When is greater than 0.44, with the increase of , the numerical program falls into an infinite loop because of the negative oil film thickness, thus no numerical results are obtained. This can explain the absence of the numerical value of sensitivity analysis between threshold 2 and threshold 3. In order to effectively conduct the numerical calculation, should meet with the condition followed:

4. Results and discussions When a geometrical preload ratio mp is given, the range of the eccentricity ratio should be calculated to avoid the condition of greater 1) causes the pad 1 and pad 5 (upper pads) become unloaded and 2) causes a negative oil film thickness between the shaft and the pad 3. Meantime, because the present paper studied the tolerance range of manufacturing, 3) the probability of these two conditions above should be taken into consideration. Fig. 5 shows an example when geometrical preload ratio mp equals to 0.4. The horizontal axis represents the eccentricity ratio , the vertical axis represents the first sensitivity index (main effect) for capacity. In this figure, the vertical line 1 indicates the threshold value of that probably causes pad 1 and pad 5 unloaded, line 2 indicates the threshold value of that probably causes a negative oil film for pad 3, line 3 shows the threshold value of that geometrically causes pad 1 and pad 5 unloaded. When eccentricity ratio value is greater than threshold 1, there is a big probability that pad 1 and pad 5 become unloaded, which causes these two pads fluttering. As a results, there is not an only tilting angle value where the pads can seek an equilibrium state, the resultant unstable flutter could causes a huge mutation in sensitivity analysis. So this range is not recommended for the sake of robustness even if the (0.38, 0.44) . curve shows an accidental consistency when The unloaded state of the pad also can result in pad accidentally inverted. If this happened, the leading edge of the pad and the shaft contact, then the negative pressure formed in the diverging gap at the leading part of the pad as well as the friction between the leading edge of the pad and the shaft may make the leading edge of the pad tightly against the shaft. So it is obvious that the unloaded state should be

(1) should be smaller than the value which causes upper pad unloaded in probability. (2) + m < 1, which can make sure that the shaft will not contact with or through the inner face of pad 3 to avoid a negative oil film thickness; (3) should be smaller than the value which causes upper pads unloaded; As to threshold value 3, if the manufacturing tolerance is not taken into consideration, when the effective preload ratio decreases up to zero, the pad reaches the unloaded state, the geometrical condition of unloading can be written as:

mp

cos

i

0

(29)

In the present case, mp = 0.2 0.6, 1 = 36 , 5 = 324 , subscribe these parameters into inequality (29) and are obtained to correspond the threshold value 3 for each mp , respectively. However, when the term “probably” is used, the variation of processing size in manufacturing tolerance is considered. To avoid any probability of pad unloading and negative oil film thickness, the minimum should be calculated. In order to simulate the actual machining situation more realistically, statistical methods are used to explain the probability of pad unloading and negative oil film thickness. Take the mp = 0.2 as an example. The five parameters are sampled.

Fig. 2. Transformation and the corresponding empirical distribution of studied parameters. 5

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Fig. 3. Converted normal distribution and the corresponding empirical distribution of studied parameters.

Fig. 4. Flowchart of the numerical procedure.

Rd N (100.16, (3.6667 × 10 3) 2)[unit: mm], Where r∼N (100.00, (3.6667 × 10 3)2 )[unit: mm]. At the same time, the distribution of the difference between the pivot radius and shaft radius is obtained and (Rd r ) N (0.16, (5.1855 × 10 3)2) [unit: mm]. Fig. 7 shows the histogram of the difference. Then the minimum eccentricity can be calculated by:

(

0.024 )/cos(36 ) = 0.14 0.2

(30)

Others threshold values can be obtained by following this procedure. According to the above analysis, the scope of for each preload can be determined by choosing the minimum value from the threshold value of , which meeting with these three conditions above for an effective calculation. The final value (bold) shown in Table 7. Figs. 8–10 show the first-order sensitivity indices of studied parameters to static characteristics including capacity, power loss and inlet flow when preload ratio equals to 0.2. In Fig. 8, for capacity under given processing precision, the sensitivity indices of pivot position and angular extend of pad 3 almost come to be 0, which means these two parameters have no significant

Fig. 5. The first sensitivity index of studied parameters for capacity (mp = 0.4 ).

6

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Fig. 8. First-order sensitivity indices of studied parameters for capacity.

Fig. 6. Schematic of pad unloading and accidental invert.

of pad 3 determine the oil thickness in radial dimension from both sides. The oil film clearance is just a dozen of micrometers. Obviously, these three parameters have the same processing precision due to the tiny difference in radial size, while pivot radius affects the whole performance of the bearing. Shaft radius also has a universal influence but not stronger than pivot radius. Internal radius of pad 3 varies in manufacturing tolerance just affects the main support area so the influence is local and limited, which could be the reason that the sensitivity of the internal radius of pad 3 shows a relative less important influence than the other two. Also, pivot position and angular extent of pad 3 have no influence on oil film thickness, which leading to a non-significant result to static characteristics. The sensitivity indices is calculated when preload ratio equals to 0.3, 0.4, 0.5 and 0.6. All capacity outputs is put in one figure for comparing, the same as power loss and inlet flow. Fig. 11 shows the first-order sensitivity indices of capacity to the studied parameters when preload ratio equals to 0.2–0.6. Because the variation of pivot position and angular extent of pad 3 have non-influence on capacity, only the first-order sensitivity indices of capacity to pivot radius, shaft radius and internal radius of pad 3 were shown. Generally, as increases, the trends of the sensitivity of capacity to pivot radius and internal radius of pad 3 show a small increase, while that of shaft radius show a decreased trend. It is also can be noticed that

Fig. 7. Histogram of the difference between the pivot radius and shaft radius. Table 7 Scope of

for each preload.

Preload ratio

Threshold value 1

Threshold value 2

Threshold value 3

0.2 0.3 0.4 0.5 0.6

0.14 0.26 0.38 – –

– – 0.44 0.36 0.24

0.2472 0.3708 0.4944 0.6180 0.7416

influence on capacity. Pivot radius is the most important and shaft radius shows a relatively small importance than pivot radius. While internal radius of pad 3 has a smaller influence. Fig. 9 shows the first-order sensitivity indices of studied parameters for power loss. Pivot radius is the most sensitive to power loss. Shaft radius also has an important influence on power loss but smaller than pivot radius. The rest three parameters have no significant effect. Fig. 10 shows the first-order sensitivity indices of studied parameters for inlet flow. Shaft radius is more pronounced than pivot radius. Internal radius of pad 3 has a smaller influence. Pivot position and angular extent of pad 3 are non-significant. In the view of design, pivot radius, shaft radius and internal radius

Fig. 9. First-order sensitivity indices of studied parameters for power loss. 7

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Fig. 10. First-order sensitivity indices of studied parameters for inlet flow.

Fig. 12. First-order sensitivity indices of studied parameters for power loss at different preload.

Fig. 11. First-order sensitivity indices of studied parameters for capacity at different preload. Fig. 13. First-order sensitivity indices of studied parameters for inlet flow at different preload.

the sensitivity of capacity to pivot radius and internal radius of pad 3 almost have the same growth rate. This is because pad 3 act as a main support pad, the greater means the greater proportion of load applied on pad 3. Meanwhile the pivot radius determines the oil film between the shaft and the pad by affecting radial position of pad (including pad 3). While the oil force between the shaft and others pad decreases, namely, the greater reduces the impact on capacity to the variation of shaft radius. This explains the increase of the sensitivity of capacity to pivot radius and internal radius of pad 3 and the decrease of that to shaft radius. It also can be clearly seen that there is a small difference when different preload ratios were given, but the trends and the order of the sensitivity have no change. Fig. 12 shows the first-order sensitivity indices of the studied parameters for power loss when different preload ratios are given. Pivot radius and internal radius of pad 3 increase as increases. However, shaft radius shows an opposite trend for each preload ratio. The trends can be explained by the same reason which has been unveiled in capacity part. Greater results in the greater proportion of applied load and pivot radius dominates the radial position of pads, which cause an almost same growth rate for the sensitivity of power loss to pivot radius and internal radius of pad 3. Fig. 13 shows the first-order sensitivity indices of the studied

parameters for inlet flow when different preload ratios are given. The sensitivity indices of inlet flow to pivot radius and shaft radius remain constant with the increase of and internal radius of pad 3 become nonsensitive to the inlet flow for the range of . Because pad 3 supports the main applied load, the oil film thickness between pad 3 and shaft is in an order of micrometer, which means only a small part of oil flow into pad 3 comparing with others pads. In fact, most inlet flow comes from others pads. This is the reason that the sensitivity of inlet flow to internal radius of pad 3 shows no importance. The sensitivity of pivot radius and shaft radius do not change with the increases. The proportion of inlet flow of these two parameters does not change because both of them dominate the oil film in radial direction in a same extent. 5. Conclusions In present paper, the FAST method is used with considering the actual processing condition, five studied parameters are calculated. Through theoretical analysis and numerical calculation, the first-order sensitivity indices show the manufacturing tolerance has a significant influence on static characteristics of the bearing. Based on the results obtained, the following conclusions can be summarized: 8

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(1) The FAST method is perfectly suited to tilting pad journal bearing model for sensitivity analysis. The efficient and robust of the method have been proved in calculation. (2) For the first order sensitivity index of pad pivot position and pad angular extent, due to adopting the sufficient machining precisions, these two parameters varying in the given manufacturing tolerance have no influence on the static characteristics. (3) For a given geometrical preload ratio, if the five studied parameters obey normal distribution, as the eccentricity ratio increases, the variation of pivot radius, shaft radius and pad internal radius have important influence on the static characteristics, while the pivot radius is relative important. (4) For a given geometrical preload ratio, the load applied on the shaft (namely eccentricity ratio ) does not change the order of the sensitivity of studied parameters, which means that for a practice machining process, one can choose a proper processing precision without taking consideration of the load applied on the shaft. This leads to a meaningful result for the bearing design and manufacturing to obtain a good economy.

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