ISSN:2250-0138(Online) ISSN : 0976-2876 (Print) ANALYSIS OF A TILTING PAD JOURNAL BEARING'S CLEARANCE ALI ABASABAD ARAB 1a, MOHAMMAD ABASABAD ARAB b 1ab Msc of Mechanical Engineering, PayvaranParsian engineering co ABSTRACT Bearing clearance is one of the most important parameters in the operation of a bearing. Therefore, it s important to determine the installed clearance space along with the bore contour and concentricity to the outside fit diameter.a method of calculating the clearance space in a tilting padjournal bearing is presented, together with supporting measureddata.three distinct stages of journal position have been identified. Inthe first stage, the journal can move outwards towards the gap between any two of the stationary, untitled pads until it makes contact with their running surfaces. In the second stage, the shaft can move further outwards, as the pads tilt without sliding until the pad pivot, the journal center and the contact point between pad and journal become collinear. In the third and final stage, the pads slide circumferentially around the bearing homing without further tilting, until contacting their pad stops, thereby allowing further outward motion of the journal. Any or all of these three stages can co-exist to provide the complete clearance space.at the heart of the method of calculation is an expression for the clearance space as a linear combination of the pad tilting angle, pre-set and journal center coordinates. For a large scale model of a typical bearing, the shaft excursion distance at the end of the second stage was calculated lo be 32 mm, which is very close to the observed value of 33 mm. Comparitions are also made between predicted and measured clearance space for a 60 mm diameter shaft in a tilting pad bearing. The ability of a pad to slide in the circumferential direction suggests that, if sufficiently large, padstop clearance could have some influence on bearing performance. KEYWORDS : Clearance Space, Tilting Pad Journal Bearing, Oil Film Thickness, Pad Tilt Angle The measurement of the clearance space inside a tilting pad journal bearing (TPJB) is more complex than that of its plain counterpart. This is mainly because the pads can tilt about their own pivots and, in the non-rotating condition. The shaft can only attain a stable equilibrium position when it is in an inter-pad gap. If a clearance circle is defined for a TPJB as one with radius equal to the clearance (C r ) at the pivotal location, then the shaft center can be moved beyond this circle in the inter-pad gap direction by a certain fraction of C r. As the factors that affect the static and dynamic behavior of the bearing, e.g. the maximum stress on a pivot, depend on the pad tilt angle which in turn depends on the size of the shaft excursion distance, it is desirable to be able to estimate it with OIL FILM THICKNESS Figure 1 shows a journal, whose center has translated a distance Y j, i.e. from 0 to J in the direction (+Y) of an inter-pad gap, whereas X j remains zero at all times. Figure 1 also shows the i th pad tilted to φ i. The vector JQ, from J to any point Q' located by the angle α + β on the surface of the i th pad is given by: a fair degree of accuracy. Armentrout and Paquette (1)presented a very simple method of calculation for a TPJB with non-sliding pads, but without taking into account the free movement of the shaft before it touched the pads. This paper presents details of a more rigorous method of calculation, together with supporting measured data. The new method will also be applicable to pads which can slide along the bearing housing as well as tilting about their pivots. Analysis of the clearance space is facilitated by separating it into three stages of possible combinations of journal center and pad positions. All three stages can co-exist in an operational bearing. Y 2 = R p.cosγ PA.cosα Where: X 1 = R p.cosγ OP.cosα X 2 = PA.sinα R p.sinγ Y 1 = R p.sinγ OP.sinα 1 Corresponding author JQ ' = X i+x 2 i -X j Y i +Y 2 i -Y j (1) Figure 1.The geometry of two adjactment pads In a tilting pad Journal bearing, showing the tilt angle of the i th pad (exaggerated).
And the distance AA' is neglected because it is very small compared to all 'the other dimensions involved, and i is assumed to be small enough that putting does not introduce significant errors. The oil film thickness H at any point Q' is defined as: This can be shown to have the following non-dimensional form (2): This equation is used in the following sections to develop expressions and dataa for the three stages involved in calculating the clearance space. MATHEMATICAL ANALYSIS First Stage of Journal Motion As regards the motion of a journal towards an inter-pad gap starting from the concentric position, by symmetry it is only necessary to concentrate on one of the two pads, the i th pad. Assuming that, for this initial stage, i = 0 and that i remains zero throughout, during this first stage of journal translation the oil film thickness is given by: Using a typical value, f =1, corresponding to P r = 1/2, film thickness profiles are obtained and shown in Fig.2 for a range of values of y j. It can be seen that h is positive everywhere for y j 1.1. Moreover, h would become negative for some values of β when y j is much greater than 1.1, which is clearly inadmissible. In fact, as shown below, the shaft comes into first contact with a pad at a point on the pad given by β = 18.9" when y j = 1.103. Any further advancement of the shaft will cause the pad to tilt, i.e. i 0, and Eq. [4] will no longer be applicable. The condition for this initial contact is given by the minimization of y, under the constraints of h = 0 and i =0, i.e. when: Equating d y /d β to zero gives: Figure2. Calculated clearance profiles (Stage 1). Hence: At this value of β: Cos(α+β) = P r cosα β = β 1 = arcos(p r cosα) - α [5a] [5b] Where: Some results for β 1 and γ l are given in Table 1, obtained with a typical value of α = 54, σ = 0.5 mm, R B = 42.0 mm and R G = 38.0 mm, for various values of the pre-set. It should be noted that Eq. [6] depends only upon the angle α and pre-set ratio P r Since: 166
d 2 y J dβ 2 = γ 1 + f cos β 1 sin γ 1 > 0 [6a] If P r <P r *, then β 1 > 30, and Eqs. [5a]-[6a] are no longer valid because the minimum occurs outside the range of β. In this case the journal can, in fact, move a little more towards the pads before touching their surfaces. The results y l, y 2 and P r =P r * = cos (α+30 )/ cos α=0.1778 The value of y l given by Eq. [6] is, indeed, the minimum. Table 1 shows that contact will occur at one edge of the pad, i.e. at β 1 = 30, when: y 3 in Table 1, for P r =0.0 and 0.1, have been adjusted to allow for this slight complication. For the sake of simplicity, the case of P r <P r * will be omitted in the following analysis. Table 1. Calculated results for a typical tilting pad Journal Bearing α = 30, P r = ½, R B = 42mm P r f β 1 y 1 y 2 y 3 0.0 0.00 30.00 1.006 1.166 1.168 0.1 0.11 30.00 1.020 1.199 1.201 0.1778 0.22 30.00 1.034 1.230 1.231 0.2 0.25 29.25 1.039 1.230 1.232 0.3 0.43 25.84 1.059 1.232 1.234 0.4 0.67 22.40 1.081 1.233 1.236 0.5 1.00 18.91 1.103 1.233 1.238 0.6 1.5 15.35 1.126 1.234 1.240 0.7 2.33 11.70 1.150 1.234 1.241 0.8 4.00 7.97 1.177 1.234 1.242 0.9 9.00 4.06 1.205 1.234 1.244 SECOND STAGE OF JOURNAL MOTION For the next stage of the journal motion, i.e. for y J >y l, and i 0, Eq. [3] must be used. As the journal moves further outward, it causes the pad to rotate, and hence the point of contact to shift to a new position given by the value of β at which h = 0, i.e. (f + y j sinα) cosβ = 1+f - (r B i + y J cosα)sinβ [7] Putting 1 1 and substituting: (cos β= cos β 1 -ξ sin β 1-1 2 ξ2 cos β 1 + ( sin β= sin β 1 -ξ cos β 1-1 2 ξ2 sin β 1 + ) Into Eq.[7] then gives: fcos β 1 +y 1 sinαcos β 1 -η sinαcos β 1 -ξ(f+y 1 sinα )sinβ 1-1 2 ξ f+y 1 sinα cos β 1 -ηξ sin α sin β 1 + =1+f-y 1 cosα sin β 1 -r B i sin β 1 -η cos α sin β 1 -ξ i r B cos β 1 + 1 2 ξ2 y 1 cos α sin β 1 -ηξ cos α cos β 1 + Equating zeroth order terms gives: y 1 = 1+ 1- cos β 1 f sin γ 1 9 Which is exactly the same as Eq[6]. Equating first order terms gives: r B i = -Cη [9a] Where C is a constant. Here, for the same typical example, given that α = 54, P r = 18.9 and P, = 0.5, then since C > and η > 0, one has i < 0, i.e. the i th pad tilts in the clockwise direction as expected. Equating second order terms gives: -2η sin α ξ= [10] 1+f sin β 1 Again, for the same typical bearing, the above becomes ξ= -2.5η for the i th pad. Hence ξmust be negative, i.e. the point of contact moves towards the pivot. Under this condition, the clearance space is therefore given by: h=1+[1- cos β-ξ f-y J sin α+β-ξ +Cη sin (β-ξ) [11] - 167 -
Where the transformation β β - ξ has been made to allow for the shift of the contact point, thus ensuring that, when β = β 1 + ξ, then h = 0. Some typical clearance profiles for a range of values of q are shown in Fig. 3 for the same bearing with P r = ½. Thus the pad can be made to tilt in a manner depicted above to accommodate the further radial movement of the journal toward an inter-pad gap. Clearly, however, there is a limit beyond which the pad cannot be made to tilt any more. It is observed that, as q increases, increases, as illustrated by Eq. [9]. The pad pivot moves away from the line AP', whereas the contact point between the journal and the pad shifts toward it, as shown in Fig. 1. Eventually the journal center, contact point and pad pivot become collinear. When this happens, the transverse components of the applied forces on the ith and (i+1) th pads cancel out, while the radial components pass through the pad pivots. Further tilting of the pads is then impossible. Figure 4 shows the condition at the end of this second stage, in which: Since, ξ is very small compared with β 1. Thus, ξ can be neglected with only a small loss of accuracy. Assuming that the i th and (i + 1) th pads can roll along the bearing housing without sliding, then: θ i = S i [13] Where: S = R G /(R B - R G ) Equations [9], [10], [12] and [13] then give: Results for y 1 = y 2 = y 1 + η 2 are shown in Table 1 for a = 54. In particular, when P r = 1/2, one has: Figure 1. Calculated clearance profiles (Stage 2). And the point of contact between journal and pad is almost back to being adjacent to the pivot point, i.e. β=0. THIRD STAGE OF SHAFT MOTION In a typical TPJB there is a pad-stop between each adjacent pair of pads. To allow free play of the pads, there is some circumferential clearance (σ) of the order of 10 C r. It follows that a pad can slide circumferentially around the bearing housing, thus enabling the journal to travel further outwards. This is the characteristic feature of Stage 3. A large scale model comprising two pads, a journal and bearing outer ring, was constructed to facilitate understanding of the possible stages of journal motion within the clearance space. For Stage 3, the observations are that: - Pads i and i + 1 can only slide circumferentially outwards - There is very little additional tilting of the pads - The shaft remains in contact with each pad very near the pad center, i.e. β = 0 - The axis of the ith pad moves clockwise (α decreases), whereas that of the (i + l) th pad moves counterclockwise (α increases) - there is a very short transitional stage during which the contact point between the shaft, and the pad moves from β 2 to p 0, and this transitional stage can be neglected with only a small loss of accuracy. Considering the i th pad and using the above observations, it follows that the clearance profile should be basically the same as it was at the end of the second stage, except that α should be modified to become K + α, where K is to be found by the - 168 -
condition that R B K is the circumferential sliding distance. If λ is the extra non-dimensional excursion distance starting from y 2, then: For ease of observation, the clearance ratio C r /R J was chosen as 0.1 which is approximately 100 times that of a typical bearing. However, the bearing housing radius, R B, is approximately 10 times a typical value. In the case of the large scale model, it was possible to deliberately put the journal and pads into the conditions defined by each of Stages 1, 2 and 3, a circumstance which may not be readily obtainable in an operational bearing. Figures 5 and 6 show comparisons between measured and calculated film thickness profiles for two values (η=0.04 and 0.07) of the non-dimensional journal center translation in Stage 2. There is particularly good agreement for the upper half of the pad (β > 0), but the calculated values for the lower half are consistently smaller by up to 0.15 C r. Together, Figs. 5 and 6 show the increase in corresponding to increase in q in accordance with Eq. [10]. Figure 2. Condition at the end of Stage 2 (not to scale). When λ = 0, one has K= 0, and Eq. [15] becomes: Which is simply Eq. [11] evaluated at the end of Stage 2. From observation (c), one has h = 0 at β= 0, and Eq. [15] therefore becomes: Figure 3. Compartion of meusred and calculated clearance profile at λ=0.04 in stage 2, λ<λ max From the zero th order terms, Eq. [16] for h = 0, β = 0 is reproduced, whereas when allowing for λ <<y 2, the first order terms give: The end of Stage 3 is determined by the magnitude of σ, the circumferential clearance of the pads. Taking a typical value of σas 0.5 mm, for the hitherto used example, gives: Figure 4.Compartion of meusred and calculated clearance profile at λ=0.07 instage 2, λ <λ max Then, for the case of P r =1/2 and R B =42 mm: Results for y 3 = y 2 + λ max are shown in Table 1 P r values, where R B and σ are the same as before. for a range of VALIDATION Large Scale Model - 169 -
the journal and pad contact point deviates little from β = 0 in Stage 3, for either value of λ. Very good agreement is shown between the calculated and measured values in the upper half of the pad (β > 0). Again, as in Stage 2, the calculated values are up to 0.075 C r less than their measured counterparts in the lower half of the pad (β < 0). Figure 5.Compartion of meusred and calculated clearance profile at λ=0.04 in stage 3, λ <λ ma ax Figures 7 and 8 show comparisons for the two values of the non-dimensional journal center translation in Stage 3(λ= 0.04, 0.07 with σ/cr > 0.07). Both Figs. 7 and 8 illustrate that Figure 6. Comparison of measured and calculated clearance profile at λ=0.07 in stage 3, λ <λ max Stages End of Stage 1 End of Stage 2 Data for the conditions at the end of Stages 1 and 2 were also measured and compared with the calculated values in Table 2. Very good agreements are obtained. In particular, the shaft excursion distance at the end of Stage 2 is predicted to be 32 mm, which differs from the observed value by just over three percent. In Figs. 5-8, the different data points shown for a given β represent independent measurements which can differ by as much as 2 mm, particularly along the lower half of the pad. This can be explained by the fact that the various components of the test apparatus are made of plywood to a nominal accuracy of ± 0.5 mm, but the lower half of the pad is found to have manufacturing errors twice this amount. However, this uncertainty in the film thickness of up to 2 mm, which occurs at locations far from the pivot, does not appear to be significant, as far as the shaft excursion distance is concerned, as shown in Table 2. TYPICAL TPJB Attempts to measure the clearance space in a typical TPJB were made using an installation in a test rig that has been extensively described elsewhere (3). Table 2. Comparison Calculated Measured Values Parameters Calculated Measured β 1 18.9 20.0 y 1 1.103 1.10 y 1 cr 30.10 mm 30.0 mm η 2 0.070 mm 0.10 mm y 2 1.233 1.20 y 2 cr 32.0 mm 33.0 mm This test facility allows static forces to be exerted onto the non-rotating shaft in almost any desired direction. Thus, by forcing the non-rotating journal radially to contact the pad surface in a large number of different radial directions, readings of shaft center coordinates, to an accuracy of ± 2µm, provide evidence of the clearance space(4). Due to the thing freedom of the pads, it was found to be almost impossible to obtain a sufficiently stable journal position other than in an intr-pad gap. Moreover, in such a real situation, the separate pad positions specified in Stages 2 and 3 cannot be guaranteed. It follows that in the circumstances arising in such a test, pad tilting will occur and some pad sliding of Stage 3 (i.e. K 0) will almost certainly exist, although K may not be the same for the two adjacent pads, and neither be at K max. Figure 9 shows the results of such a test, wherein nearly all the data points lie within the range of 1.0 to 1.3 C r. The measured value of C r = 46µm agrees well with the value of C r = 45 µm supplied by the bearing designers. The five groups of data points represent results of independent observations. Each group is found to lie close to the central axis of the pad-stop, except for the upper left pad, where the much greater scatter along both the radial and transverse directions arises from the difficulty of pulling the shaft in that particular direction - 170 -
because of the configuration of the test rig which was built several years ago for completely different purposes. In compartion, the range predicted for this typical bearing, with α = 54 and P r = ½, is 1.0 to 1.238 C r, as listed in Table 1. Assessment of the quality of this agreement must take account of the fact that the difference between 1.3 and 1..238 C r is only 2.8 pm. possible effects which are not incorporated in the theoretical treatment, e.g. manufacturing imperfections and distortion of the various elements under load, become significant at this order of magnitude. The maximum possible journal center translation, y 3, depends on the pad-stop clearance. In the typical bearing examined, this pad-stop clearance is relatively small, and its contribution to the third stage translation is therefore rather insignificant. However, designers should note that if a larger pad-stop clearance is employed, then a correspondingly larger Stage 3 contribution would arise, and so would have a greater influence on the bearing characteristics. COMPARISON DATA WITH OTHER PUBLISHED Armentrout and Paquette (1) describe a TPJB wherein the pads are supported on flexural pivots. Their paper contains a short section in which the largest possible pad tilt angle is evaluated in order to determine the maximum stress on the pivot. They give a very simple geometrical derivation of the maximum pad tilt angle and journal translation distance. The free movement of the shaft before it touches the pads has not been taken into account. The new method can be applied to the situation in Ref. (1) except that Stage 3 does not exist, and there is no movement of the pivot point as the pad tilts, i.e. K = 0, and S=0. Applying these conditions, a comparison between the results calculated by the two different methods, where possible, is shown in Table 3. Goodagreement is shown for Y 2 (or y 2 ). However, Eq. [9] in the present method, when evaluated at η = η 2, gives a much lower pad tilt angle than that given in Ref. (1). Table 3. Comparison of different Calculation method Parameters Present Method Ref (4) Y 2 y 2 0.10 mm 0.10 mm 1.97 2.00 0.35 0.53 Application of the new technique to the situation Ref. (1) demonstrates the versatility engendered by the incorporation there in of a multi-stage approach. Figure 7.The measured clearance space in a typical TPJB with five pads (R B =42mm, Cr=45µm, P r =0.5) CONCLUSION Amethod for calculating the clearance space in a tilting pad journal bearing has been presented. In doing so, three possible stages of journal radial motion have been identified and characterized. They are: Stage 1-pads remain stationary. Stage 2-pads tilt but do not slidecircumferentially. - 171 -
Stage 3-pads slide circumferentially without further tilting. The validity and accuracy of the quantities provided by the theoretical method were specifically checked against readings taken from a large two-dimensional scale model. Furthermore, results obtained from a typical bearing mounted in a test rig also supported the findings of the mathematical treatment. The pad circumferential sliding clearance could, if sufficiently large, have an influence on the between-pad performance. Designers could usefully incorporate Eqs. [6], [14] and those leading to λ max to quickly determine the maximum increase in magnitude of the clearance space beyond C r (nominal clearance radius in the inter-pad direction), thecorresponding maximum pad tilt, and hence the maximumstress likely to act on a pad pivot. Table 4. Nomenclature A, A' point of contact between pad and outer ring, i = 0, i 0 respectively R p radius, pad machined running surface Cr clearance at pad pivot point = R P - OP R J R G radius, pad outer surface f P r /(1 - P r ) S R G / (R B R G ) h non-dimensional oil film thickness= H x J, non-dimensional coordinates of J, x J = / Cr y J X J /Cr, y J = YJ/Cr H oil film thickness X J, Y J coordinates of J J Journal center α angular coordinate, pad central axis K modification to a in Stage 3 β angle subtended at P by A and Q P r pre-set ratio = (R p - R j - Cr)/(R p - R j ) γ α + β P, P' center, pad machined running surface, non-dimensional journal translation in η i = 0, i 0 respectively Stage 2 Q, Q' arbitrary point on pad running surface, angular change in pad pivot point during θ i = 0, i 0 respectively Stage 2 r B non-dimensional bearing housing radius= R B /Cr λ non-dimensional journal translation in Stage 3 R B radius, bearing housing σ pad-stop clearance R J radius, journal i tilt angle of i th pad Acknowledgements.First of all,i must my gratitudefor my parents and family for their efforts against me. I would like to express my appreciation for our direct management Mr.Jafar Maher and our Technical managementmr.ahmad REFERENCES ArmentroutR. W. and Paquette. D. J., 1993. Rotor Dynamic Characteristics of Flexure Pivot tilting Pad Journal Bearing," Trib. Trans, 36,3:443-451. [FuW. B. and ParkinsD. W.; 1992. Mathematical Analysis of the Performance of Tilting Pad Journal Rearing under Static Load" Institute of Physic. Journal of Physic, D: Appl. Phys., 25:A108-A115. ParkinsD. W. 1979. Theoretical and Experimental Determination of the Dynamic Characteristics of a Hydrodynamic Journal Bearings," ASME Journal of Lubrication Technology, 101, P129. ParkinsD. W. 1990. Description and Appraisal of some Techniques for the Measurement of the Static and Dynamic Hasanzadehbecause of selfless effortsand for their support and encouragement and also all the my staff in PayvaranParsian Engineering co (PTEDco) and their valuable comments. Characteristics of Hydrodynamic Journal Bearings," Proc. of the Intl. Conf. on Hydrodynamic Bearing Rotor System Dynamics, Fouad Y., Zeidan and Bernard S., Herbage.; 1994. Fluid Film Fundamental and Failure Analysis", Proceedings of the twenty-third turbomachinery symposium, Dallas, Texas, USA. Nicols J. C and Kirk R.G. 1982. Four Pad Tilting pad Bearing Design and Application for Multi-Stage Axial Compressors" ASME Journal of Lubrication Technology, 104(4):523-532(October 1982). BrockwellK., DmochowskiW. and DeCamillo S.; 2004. An Investigation of the Steady-State Performance of a Pivoted Shoe Journal Bearing with ISO VG 32 and VG 68 Oils, STLE Tribology Transactions, 47:480-488. - 172 -