PART A GUIDELINES ON SHAFTING ALIGNMENT TAKING INTO ACCOUNT VARIATION IN BEARING OFFSETS WHILE IN SERVICE Fig. 10.3 FE model integrated main engine structure with hull used to calculate the displace ment at each bearing support point. 10.2 Discontinuity in Slope of Total Deflection Curve over Entire Length of Shafting Line Figure 10.4 is a close-up view of the deflection of the aft engine part of the structure obtained from the model shown in Fig. 10.3. As can be seen from the result, while the deflection curve of the top of the double bottom beneath the shafting line is quite smooth, there is clearly a discontinuity at the aft engine end where the deflection curve consists of the point on the top of the double bottom in the shaft portion and the supporting points of the engine bearing in the engine portion of the double bottom. 3.00 3.50 4.00 4.50 5.00 600080001000012000140001600018000 Distance from foremost main bearing (mm) Displacement (mm) Center Line Bearing Line Aftermost end of the engine structure Fig. 10.4 FEM results showing discontinuity between slopes of deflection curves of double bottom and main bearing centers. A discontinuity in the slope of the total deflection curve over the entire shafting line could cause advers e effects on bearing reactions. Fig. 10.5 shows two sets of bearing heights that differ from each other only in the presence of a slope continuity at the longitudinal boundary between the hull and engine structures. While the deflection in the engine portion of the curve in red smoothly joins the deflection in the shaft portion of the curve, the slope of the curve representing the engine portion of the deflection in blue is discontinuous at the longitudinal boundary between the hull and engine structures. The bearing reaction forces are calculated for the two sets of bearing offsets, and are shown in Fig. 10.6. The shafting model used in this calculation is the same as that used in previous chapters. 46 PART A GUIDELINES ON SHAFTING ALIGNMENT TAKING INTO ACCOUNT VARIATION IN BEARING OFFSETS WHILE IN SERVICE 47 As can be seen from the results shown in Fig. 10.6, the bearing reactions will change significantly if the discontinuit y in the slope of the deflection curve exists despite the magnitude of the total hog that remains unchanged. In this example, the first aftmost engine bearing becomes overloaded and the second aftmost engine bearing becomes loaded in the wrong direction. Therefore, it is desirable to ascertain whether there is any discontinuity in the slope of the deflection curve, using an integarated engine-hull FE model. Then let's consider why such a discontinuity in the slope of deflection cuve is generated. As mentioned before in section 10.1.1, a smooth curve in an elastic body will remain smooth after the body is deformed. This can be explained by the fact that the deflection curve at the center line on the double bottom top plate is relatively smooth, as shown in Fig. 10.6. It can be easily understood that when assuming there is a straight line directly beneath the double bottom top, then the sraight line will become a smooth curve after hull deformation. Bearing reaction force (kgf), (-) Upward and (+) Downward -200000 -150000 -100000 -50000 0 50000 No.1 No.2 No.3 No.4 No.5 No.6 No.7 No.8 No.9 Bearing number Reaction force (kgf) Original Smooth deflection Deflection with breaking point Fig. 10.6 The effect of slope discontinuity to bearing reactions continuity at the longitudinal boundary between the hull and engine structures. Fig. 10.5 Two sets of bearing heights that differ from each other only in slope -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 0 5000 10000 15000 20000 25000 30000 Bearing location, distance from left end (mm) Offset (mm) Original offset (mm) Final smooth offset Smooth hull deflection (mm) Straight engine portion Final breaking offset Quadratic polynomial (Smooth hull deflection) PART A GUIDELINES ON SHAFTING ALIGNMENT TAKING INTO ACCOUNT VARIATION IN BEARING OFFSETS WHILE IN SERVICE Furthermore, the deflection line of the center line on the top plate of the double bottom will also become a smooth line, neglecting small local deformations between the double bottom top plate and the assumed straight line. On the other hand, the deflection at the supporting points of the engine bearings in the engine portion of the line can not be regarded as coming from the same straight line as the center line on the top plate of the double bottom, as shown in Fig. 10.7. While the change in offsets of the intermediate bearing and stern tube bearings comes from the center line on the top plate of the double bottom, the change in offsets of the engine bearings comes from the engine seating lines on the top plate of the double bottom. It is noted from the FE analysis results that the displacemet of the top plate of the double bottom varies in the transverse direction, although the magnitude of the fluctuation depends on the location of the cross section. Therefore, the discontinuity in the slope of the deflection curve is considered to arise from these variations in displacement in the transverse direction. Fig. 10.7 Illustration showing how the slope discontinuity is developed Schematic sectional deflection curve of the top plate of the double bottom showing variation over the width Center line on the top plate of the double bottom Engine seating lines on the top plate of the double bottom Shafting line 10.3 Determination of Final Bearing Offsets in Shafting Installation Based on the background described above, the final bearing offsets in a shafting installation should be determined so that the shafting alignment can meet all relavent requirements, even after adding both static and dynamic changes in bearing offsets to it. Therefore, specifically, the first step is to determine the bearing offsets as in usual practice. Then the next step is to re-calculate the shafting alignment taking into account the predicted changes in bearing offset to check whether the alignment still meets the relevant requirements. If the re-calculated result is satisfactory, then the initially obtained bearing offsets can be set as the final parameters for the shafting installation. If the re-calculated result is not satisfactory, then the initially obtained bearing offsets should be readjusted until the relevant requirements are met. 48 PART A GUIDELINES ON SHAFTING ALIGNMENT TAKING INTO ACCOUNT VARIATION IN BEARING OFFSETS WHILE IN SERVICE 11. Confirmation of Bearing Reactions 11.1 Jack-up Method The Jack-up method is a method in which the reaction of a bearing in question is measured from the jack load that is set as close to the bearing as possible through a separately determined modification factor (see Fig. 11.1). The jack load is estimated from the relationship between the jack load and the jack displacement which is recorded during the jack up and jack down process. In the jack up test, the dial gauge used to measure the jack displacement should be properly secured so that it is affected by neither the rise of the shaft nor the deformation of the floor plate. H y draulic j ack Dial gauge Fig. 11.1 Jack-up measurement of the bearing load. 11.1.1 Theoretical Jack up Process The jack up process can be theoretically simulated through calc u lation. It is demonstrated using a shafting model shown in Fig. 11.2. The simulation is performed by calculating the relationship between the lift up and the reaction at the jack point. The main steps in the simulation are as follows. 500 m m Jac k Fig. 11.2 A shafting model used to demonstrate the theoretical jack-up process. - An additional supporting point is generated by setting the jack directly under the shaft at a point 500 mm 49 PART A GUIDELINES ON SHAFTING ALIGNMENT TAKING INTO ACCOUNT VARIATION IN BEARING OFFSETS WHILE IN SERVICE away from the intermediate bearing of which the reaction to be measured. The reaction of this newly added supporting point is, however, zero at this point. - The jack is gradually l ifted, in increments of 0.002 mm in this case, until the bearing load becomes zero. The total lift is 0.01587 mm in this case. - Because the bearing has been co mpletely unloaded, it is no longer a supporting point. Therefore, the calculation described below is performed using the beam model without the supporting point as shown in Fig. 11.3. Since the number of supporting points is reduced by one, the relationship between the jack lift and the jack load (the slope) is also changed. The result of the simulation is shown in Fig. 11.4. 500 m m Jac k Fig. 11.3 A state where the load of the bearing in question is just beco m ing zero. 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0 5000 10000 15000 20000 25000 30000 Jack load (kgf) Shaft lift at jack point (mm) Intermediate bearing loaded Intermediate bearing unloaded Linear extrapolation (Intermediate bearing unloaded) P 0 R' J R J Fig. 11.4 Theoretically calculated jack-up result of the shafting model. Point P 0 in the above figure represents a state in which the reaction of the intermediate bearing has just become zero. The jack load at this point is represented by R' J . By fitting the relationship between the jack lift and the jack load after this point to a linear curve and then extrapolating the line to where the jack lift is zero, the jack load when the jack lift is zero, R j , can be obtained. This load from which the bearing load is usually calculated is, in other words, the jack load when the jack is being set and the bearing in question is removed. 50 PART A GUIDELINES ON SHAFTING ALIGNMENT TAKING INTO ACCOUNT VARIATION IN BEARING OFFSETS WHILE IN SERVICE In this example, kgf.'R J 524488= kgf R J 24450= 11.1.2 Determination of Bearing Load from Jack up Test The bearing load R B can be calculated from the obtained jack load RB J using the following equation (11.1). JB RCR × = (11.1) where, C is the correction factor calculated as follows: BJ BB I I C −= where, I BB : The reaction influence number of the bearing when the jack is regarded as a supporting point. I BJ : The reaction influence number of the bearing to the jack supporting point. In order to better understand the jack up method，Eq. (11.1) is derived below. Figure 11.5 shows three important states during the jack up process using the calculated shafting lines. 0.016 mm δ h Fig. 11.5 Deflection curves of the shaft line at three im portant stat es during the jack-up process. 51 PART A GUIDELINES ON SHAFTING ALIGNMENT TAKING INTO ACCOUNT VARIATION IN BEARING OFFSETS WHILE IN SERVICE State 1： The initial state, at this point the reaction of the bearing is R B . The jack is set underneath the shaft but the jack load remains zero. This state is represented by the solid black line in Fig. 11.5. B State 2： After gradually lifting the jack up by a lift of δ, the bearing load becomes zero while the bearing remains in contact with the shaft. This state is represented by the solid red line in Fig. 11.5. At this point the jack load is R' J . State 3： Supposing that the bearing has been removed at State 2, the jack is lowered by an amount δ back to the original setting position. This state is represented by the dotted red line in Fig. 11.5. During this process the bearing supporting point is supposed to have lowered by an amount h. At this point the jack load is R J . State 4： Supposing that the bearing has been set again at the supporting point, the bearing is lifted by an amount h, until the jack load becomes zero. During this process the jack, of course, remains in contact with the shaft. The bearing load at this point is exactly the load R B that is to be measured. In other words, state 4 is identical to state 1. B In the process of transition from state 3 to state 4, the change in the bearing and jack loads can be expressed as follows: Bearing: 0 + I BB h = R B B Jack: R J ＋I BJ h = 0 Therefore， J BJ BB B R I I R −= Since the reaction influence numbers for the shafting model shown in Fig. 11.5 are: I BJ = -1538261 (kgf/mm) I BB = 1535661 (kgf/mm) Therefore, the jack correction factor is: 99830. I I C BJ BB =−= In addition, R J is 24450 kgf as mentioned in 11.1.1, thus the bearing load R B is obtained as follows: B kgf 2440824450.R B =×= 99830 This result completely agrees with that directly calculated using the bea m model described earlier. The bearing load can also be expressed in Eq. (11.2), if we consider the process of transition from state 1 to state 2 in the same way as above. J JJ JB B 'R I I R −= (11.2) 52 PART A GUIDELINES ON SHAFTING ALIGNMENT TAKING INTO ACCOUNT VARIATION IN BEARING OFFSETS WHILE IN SERVICE where, I JJ : The reaction influence number of the jack when the jack is regarded as a supporting point. I JB : The reaction influence number of the jack to the baring. Equation (11.2) theoretically leads to the same result as that obtained from Eq. (11.1). In practice, however, R' J is difficult to determine accurately from the measured relationship between the jack lift and jack load, because the real jack load-lift curve will form a loop referred to as an hysteresis curve due to the unavoidable friction in the hydraulic jack. Equation (11.2) is thus less used in practice. 11.1.3 Analysis Method of Jack up Recording Curves The hydraulic pressure is higher in the lifting process than in the lowering process for a given external load, because fricti on between the cylinder and the rod is unavoidable. Due to this reason, a loop called an hysteresis curve is generated during the jack up and jack down process as schematically shown in Fig. 11.6. To eliminate the effect of the friction, the middle line of the loop should be used to determine the jack load. 2 DI J RR R + = (11.3) Jack load Increasing Dec r easing Average line R D R J R I Vertical lift of shaft at jack point Fig. 11.6 H ysteresis curve of actual jack-up result due to unavoidable friction in hydraulic jack. It is im portant to confirm the sudden change, known as 'break point', that appears in the slope of the jack load-lift curve. The break point comes about because the number of the supporting points in the shafting decreases by one when the bearing load becomes zero. If such a break point does not appear and the jack load-lift curve looks like that shown in Fig. 11.7, there is a high possibility that the bearing load is nearly zero at the original position. 53 PART A GUIDELINES ON SHAFTING ALIGNMENT TAKING INTO ACCOUNT VARIATION IN BEARING OFFSETS WHILE IN SERVICE Jack load Increasing Decreasing Vertical lift of shaft at jack point Fig. 11.7 Jack-up result showing no break point is a sign that the bearing being unloaded. As noted above, the break point comes about because the num ber of the supporting points in the shafting decreases by one when the bearing load becomes zero. The load of the bearing immediately next to the jack first becomes zero and the first break point appears. When the jack continues to rise, then another bearing close to the jack will also become unloaded and the slope of the jack load-lift curve will become steeper, as shown in Fig. 11.8. If the jack continues to rise further, at some point, the upper clearance of the nearest bearing to the jack will finally disappear, causing a downward bearing reaction, and slope of the jack lift-load curve will suddenly decrease. Therefore, in order to avoid excessive jack up, it is important to finish the jack-up test as soon as sufficient data for determining R J has been gathered, after the first break point has appeared. Jack load N o bearing unloaded Vertical lift of shaft at jack point First bearing unloaded Second bearing unloaded Shaft touched first bearing upper limit Fig. 11.8 Each slope in the jack-up result corresponds to a different shafting model. 54 PART A GUIDELINES ON SHAFTING ALIGNMENT TAKING INTO ACCOUNT VARIATION IN BEARING OFFSETS WHILE IN SERVICE 11.2 Gauge Method 11.2.1 Mechanism of Gauge Method The bending moment along the shafting will change when the reactions of t h e bearings change due to the change in the bearing offsets. Since the change in bending moment at any cross section is linearly proportional to the change in bearing offsets or bearing reactions, the change in bearing offsets or bearing reactions can be reversely calculated from the change in bending moment at several cross sections which are measured. Supposing that the No. 4，No. 5 and No . 6 bearings i n Fig. 11.9 rise by 1.0 mm, respectively, the change in the bending moment at cross sections 10,000 mm, 17,000 and 18,000 mm from the left end, respectively, is calculated and shown in Table 11.1. These values are called moment influence numbers, and the matrix composed of these moment influence numbers as its elements is referred to as the moment influence number matrix. N o.1 N o.2 N o.3 N o.4 N o.5 N o.6 M 1 M 2 M 3 Fig. 11.9 A shafting model for demonstrating gauge method T able 11.1 Bending Moment Influence Number Matrix Offset ( mm ) Moment (kgf-mm) δ 4 (1mm) δ 5 (1mm) δ 6 (1mm) M 1 (at 10000 mm) -28154234 17284736 -12732321 M 2 (at 17000 mm) 18606641 -187185 -6315427 M 3 (at 18000 mm) 5815648 44923370 -43329799 Therefore, the relationship between the change in bending m oment and the change in bearing offsets can be given in Eq. (11.4). -28154234 17284736 -12732321 18606641 -187185 -6315427 5815648 44923370 -43329799 δ 4 δ 5 δ 6 M1 M2 M3 = (11.4) 55 . in Fig. 10 .6. The shafting model used in this calculation is the same as that used in previous chapters. 46 PART A GUIDELINES ON SHAFTING ALIGNMENT TAKING INTO ACCOUNT VARIATION IN BEARING. hull deflection (mm) Straight engine portion Final breaking offset Quadratic polynomial (Smooth hull deflection) PART A GUIDELINES ON SHAFTING ALIGNMENT TAKING INTO ACCOUNT VARIATION IN BEARING. magnitude of the fluctuation depends on the location of the cross section. Therefore, the discontinuity in the slope of the deflection curve is considered to arise from these variations in displacement