PART B AMENDMENT OF THE GUIDANCE TO THE RULES AND ITS EXPLANATORY NOTES 23 3 Hull Deflection that results in Engine Bearings Becoming Unloaded Using the reaction influence number S i , connecting with hull deformation, we can determine the amount of relative displacement of the hull that would cause the i th engine bearing to become unloaded. There are two cases in which either elastic support (spring support) or rigid support (simple support) are used as the support condition of the bearings to calculate the reaction influence numbers, C m, n . Each case is examined below. 3.1 In the case of elastic support In this case, δ B , which is the relative displacement at which the engine bearings become unloaded, can be obtained using a simplified calculation method. Let R i be the reaction of engine bearing No. i (from aft) before hull deformation and let δ B i be the relative displacement of the hull (at the position of the aftmost bulkhead of the engine room) at which engine bearing No. i becomes unloaded after hull deflection, then Hence, Since unloading of engine bearings due to hull deflection is often limited to the second or third aftmost bearing in most cases, both δ B2 and δ B3 are to be calculated and whichever is less is to be compared with the lower limit. 3.2 In the case of rigid support Figure 3.1 shows how to deal with the stiffness of an engine bearing support in the state before hull deflection. In cases where rigid support is considered in the calculation, bearing reaction R i corresponding to the given offset is easily obtained for each respective bearing, and the displacement of the support point due to the load is zero, as shown in Figure 3.1(a). However, if the effect of the stiffness (spring) is considered here, the support points are in fact displaced downward slightly by the load, as shown in Figure 3.1(b), which actually involves a change in the bearing reaction R i for each engine bearing. Figure 3.1 Initial condition of engine bearings. Assuming that reaction R i does not change even if the effect of the stiffness (spring) is considered, the displacement of each support point after loading is expressed with the stiffness K (= constant) of each support point using the following equation: In the discussion below, we examine the change in bearing reaction when the hull deflects from the initial condition shown in Figure 3.1(b) due to an increase of draught. Figure 3.2 shows the change in the support point (contact point with the shaft) that occurs before and after hull deflection. As can be seen from this figure, it is assumed that the change in the vertical position of the support point shown here is not caused by the deflection of hull structure below the engine, but by the elastic deformation of the engine bed due to the increase in the bearing load. ∆ R i = 0 – R i = δ Bi S i . δ Bi = – R i / S i . ( 6 ) R 1 R 2 R 3 R 4 R 5 (a) Rigid support (b) Elastic support K h 1 h 2 h 3 h 4 h 5 h i = R i / K h i = R i / K . ( 7 ) PART B AMENDMENT OF THE GUIDANCE TO THE RULES AND ITS EXPLANATORY NOTES 24 Figure 3.2 Displacement of support points due to elastic deformation of the engine bed. In paragraph 3.1 above, a simple calculation could be used to give the bearing reactions of each bearing after hull deflection. However, in order to obtain the bearing reactions after hull deflection using reaction influence numbers based on rigid support, the equations that include the reactions and displacements of the support points as variables need to be solved for specific conditions. Setting the number of engine bearings at five and denoting the relative displacement of the hull at the position of the engine room aftmost bulkhead by δ B , changes in the reactions of each engine bearing can be expressed as follows: where, i : number of support point (engine bearings) counted from the aft of the engine, R i : reaction of support point i (counted from the aft of the engine) before hull deflection, R′ i : reaction of support point i (counted from the aft of the engine) after hull deflection, δ i : elastic displacement of support point i (counted from the aft of the engine), C i, j : amount of increase of reaction (reaction influence number) at support point i when support point j is displaced downward by 1 mm. ( i, j : number of support point counted from the aft of the engine). (Note that the meaning of the subscripts in Equation (8) differ from those in Equation (2).) On the other hand, from the relationship: Equation (8) can be rewritten as: R′ 1 – R 1 = S 1 δ B + C 1, 1 δ 1 + C 1, 2 δ 2 + C 1, 3 δ 3 + C 1, 4 δ 4 + C 1, 5 δ 5 , R′ 2 – R 2 = S 2 δ B + C 2, 1 δ 1 + C 2, 2 δ 2 + C 2, 3 δ 3 + C 2, 4 δ 4 + C 2, 5 δ 5 , R′ 3 – R 3 = S 3 δ B + C 3, 1 δ 1 + C 3, 2 δ 2 + C 3, 3 δ 3 + C 3, 4 δ 4 + C 3, 5 δ 5 , R′ 4 – R 4 = S 4 δ B + C 4, 1 δ 1 + C 4, 2 δ 2 + C 4, 3 δ 3 + C 4, 4 δ 4 + C 4, 5 δ 5 , R′ 5 – R 5 = S 5 δ B + C 5, 1 δ 1 + C 5, 2 δ 2 + C 5, 3 δ 3 + C 5, 4 δ 4 + C 5, 5 δ 5 , ( 8 ) ( 9 ) R ′ i – R i = K δ i , S 1 δ B + ( C 1, 1 – K ) δ 1 + C 1, 2 δ 2 + C 1, 3 δ 3 + C 1, 4 δ 4 + C 1, 5 δ 5 = 0, S 2 δ B + C 2, 1 δ 1 + ( C 2, 2 – K ) δ 2 + C 2, 3 δ 3 + C 2, 4 δ 4 + C 2, 5 δ 5 = 0, S 3 δ B + C 3, 1 δ 1 + C 3, 2 δ 2 + ( C 3, 3 – K ) δ 3 + C 3, 4 δ 4 + C 3, 5 δ 5 = 0, S 4 δ B + C 4, 1 δ 1 + C 4, 2 δ 2 + C 4, 3 δ 3 + ( C 4, 4 – K ) δ 4 + C 4, 5 δ 5 = 0, S 5 δ B + C 5, 1 δ 1 + C 5, 2 δ 2 + C 5, 3 δ 3 + C 5, 4 δ 4 + ( C 5, 5 – K ) δ 5 = 0. ( 10 ) h i Bearing offset input δ i R i R ′ i Reference line (After hull deflection) (Before hull deflection) Tank top of engine room double bottom PART B AMENDMENT OF THE GUIDANCE TO THE RULES AND ITS EXPLANATORY NOTES 25 In Equation (10), we consider the situation that the second aftmost engine bearing becomes unloaded after hull deflection, which is expressed by the following equation: This indicates that the support point of the second aftmost engine bearing moves upward so that the displacement, h 2 , which is due to the bearing load R 2 before hull deflection return to the original state, in other words, the restoring force of the spring may be zero. Let δ B2 be the relative displacement, δ B , which causes the second aftmost engine bearing to become unloaded. Then, the following simultaneous equations with five variables, i.e., δ B2 , δ 1 , δ 3 , δ 4 , and δ 5 , need to be solved in order to find δ B2 , since δ 2 is not a variable: where, K = 5,000 kN/mm (constant value). Figure 3.3 illustrates the state of the shafting and relative displacement of the engine bearings before and after hull deflection. Figure 3.3 Reactions and displacements of support points when stiffness of engine bearings is considered. Simultaneous Equations (12) can be expressed in matrix form as follows: where, δ 2 = – h 2 = – R 2 / K . ( 11 ) S 1 δ B2 + ( C 1, 1 – K ) δ 1 + C 1, 3 δ 3 + C 1, 4 δ 4 + C 1, 5 δ 5 = C 1, 2 R 2 / K , S 2 δ B2 + C 2, 1 δ 1 + C 2, 3 δ 3 + C 2, 4 δ 4 + C 2, 5 δ 5 = ( C 2, 2 – K ) R 2 / K , S 3 δ B2 + C 3, 1 δ 1 + ( C 3, 3 – K ) δ 3 + C 3, 4 δ 4 + C 3, 5 δ 5 = C 3, 2 R 2 / K , S 4 δ B2 + C 4, 1 δ 1 + C 4, 3 δ 3 + ( C 4, 4 – K ) δ 4 + C 4, 5 δ 5 = C 4, 2 R 2 / K , S 5 δ B2 + C 5, 1 δ 1 + C 5, 3 δ 3 + C 5, 4 δ 4 + ( C 5, 5 – K ) δ 5 = C 5, 2 R 2 / K , ( 12 ) (a) Before hull deflection R 1 R 2 R 3 R 4 R 5 (b) After hull deflection δ B2 R ’ 2 = 0 : δ 2 = – h 2 R ’ 1 R ’ 2 R ’ 3 R ’ 4 R ’ 5 [D 2 ] [ δ B2 ] = h 2 [ C 2 ] , ( 13 ) PART B AMENDMENT OF THE GUIDANCE TO THE RULES AND ITS EXPLANATORY NOTES 26 Hence, where, [D 2 ] INV denotes the inverse matrix of [D 2 ]. Thus, we can obtain δ B2 as follows: where, ([D 2 ] INV ) L1 denotes the elements (vector) of the first row of [D 2 ] INV . Similarly, the relative displacement δ B3 which results in the third aftmost engine bearing to become unloaded can be obtained by solving the following simultaneous equations with five variables (the matrix representation is omitted): In a shafting alignment with constant offsets of the engine bearings, the calculation of δ B2 is sufficient to check the strength of all engine bearings, because the load on the second aftmost engine bearing decreases as draught increases. However, in some cases where the engine bearing offsets are unequal, the load on the third aftmost engine bearing could decrease. Consequently, both δ B2 and δ B3 , in general, should be calculated and whichever is less is to be compared with the lower limit. [D 2 ] = , S 1 C 1, 1K C 1, 3 C 1, 4 C 1, 5 S 2 C 2, 1 C 2, 3 C 2, 4 C 2, 5 S 3 C 3, 1 C 3, 3K C 3, 4 C 3, 5 S 4 C 4, 1 C 4, 3 C 4, 4K C 4, 5 S 5 C 5, 1 C 5, 3 C 5, 4 C 5, 5K δ B2 δ 1 δ 3 δ 4 δ 5 [ δ B 2 ] = , C 1, 2 C 2, 2K C 3, 2 C 4, 2 C 5, 2 [C 2 ] = , h 2 = R 2 / K , C i, i K = C i. i – K ( i = 1, 2, 3, 4, 5 ) . [ δ B2 ] = h 2 [D 2 ] INV [ C 2 ] , (14) δ B2 = h 2 ([D 2 ] INV ) L1 [ C 2 ] , (15) S 1 δ B3 + ( C 1, 1 – K ) δ 1 + C 1, 2 δ 2 + C 1, 4 δ 4 + C 1, 5 δ 5 = C 1, 3 R 3 / K , S 2 δ B3 + C 2, 1 δ 1 + ( C 2, 2 – K ) δ 2 + C 2, 4 δ 4 + C 2, 5 δ 5 = C 2, 3 R 3 / K , S 3 δ B3 + C 3, 1 δ 1 + C 3, 2 δ 2 + C 3, 4 δ 4 + C 3, 5 δ 5 = ( C 3, 3 – K ) R 3 / K , S 4 δ B3 + C 4, 1 δ 1 + C 4, 2 δ 2 + ( C 4, 4 – K ) δ 4 + C 4, 5 δ 5 = C 4, 3 R 3 / K , S 5 δ B3 + C 5, 1 δ 1 + C 5, 2 δ 2 + C 5, 4 δ 4 + ( C 5, 5 – K ) δ 5 = C 5, 3 R 3 / K . ( 16 ) PART B AMENDMENT OF THE GUIDANCE TO THE RULES AND ITS EXPLANATORY NOTES 27 4 Calculation Example Figure 4.1 and Table 4.1 show the dimensions of the shafting prepared to test and verify the calculations and the reaction influence numbers, respectively. The values of the reaction influence numbers were obtained based on the assumption of rigid support for the bearings, therefore paragraph 1.3.3-1.(2) of the Annex to the Guidance was applied in the calculation of δ B2 and δ B3 . The reactions and offsets of the bearings under the hot condition are shown in Table 4.2. Figure 4.1 Dimensions of shafting. Table 4.1 Reaction Influence Numbers (kN /mm) Table 4.2 Bearing Reactions and Offsets (Hot Condition) Results of Calculations Table 4.3 shows the results of the calculations of S i ( i =1 to 5). It can be understood from this table that the engine bearings most notably affected by hull deflection are the aftmost and second aftmost engine bearings, and further, that the load on the second aftmost engine bearing decreases whereas the load on the aftmost engine bearing increases. The values of δ B2 and δ B3 are shown in Table 4.4. This table shows that the second aftmost engine bearing will first become unloaded when the ship’s draught increases and δ B reaches 7.26 mm. Figure 4.2 shows a comparison between δ B2 and the allowable limit in the case of L = 10,400 mm. Figure 4.2 Comparison between δ B2 and the allowable limit. 1370 2555 5665 5540 870 1290 1290 1290 SB1 SB2 SB3 IB1 1 2 3 4 5 E/R aft bulkhead 10400 SB1 SB2 SB3 IB1 1 2 3 4 5 SB1 -1094.4 1760.31 -710.86 60.88 -58.32 44.00 -2.00 0.50 -0.08 SB2 1760.31 -2956.05 1338.87 -193.96 185.79 -140.17 6.38 -1.59 0.27 SB3 -710.86 1338.87 -778.71 217.35 -244.91 184.77 -8.41 2.10 -0.35 IB1 60.88 -193.96 217.35 -175.85 436.06 -357.35 16.26 -4.06 0.68 1 -58.32 185.79 -244.91 436.06 -3498.3 4574.60 -1761.8 440.43 -73.40 2 44.00 -140.17 184.77 -357.35 4574.60 -7194.4 4005.52 -1340.2 223.35 3 -2.00 6.38 -8.41 16.26 -1761.8 4005.52 -3991.1 2353.52 -618.22 4 0.50 -1.59 2.10 -4.06 440.43 -1340.2 2353.2 -2283.1 832.48 5 -0.08 0.27 -0.35 0.68 -73.40 223.35 -618.22 832.48 -364.72 SB1 SB2 SB3 IB1 1 2 3 4 5 Reaction (kN) 446.69 15.16 69.89 95.33 226.39 131.83 256.28 314.09 90.80 Offset (mm) 0.35 0.00 -0.15 0.20 0.40 0.40 0.40 0.40 0.40 S 1 S 2 S 3 S 4 S 5 73.91 -66.79 3.04 -0.75 0.14 Table 4.3 S i (kN/mm) δ B2 δ B3 7.26 19.14 Table 4.4 δ B2 and δ B3 (mm) 5 7 9 11 13 15x10 3 10 8 6 4 2 0 L (mm) δ B2 ( m m ) Allowable limit Copyright © 2006 All rights reserved NIPPON KAIJI KYOKAI NIPPON KAIJI KYOKAI . NOTES 25 In Equation (10) , we consider the situation that the second aftmost engine bearing becomes unloaded after hull deflection, which is expressed by the following equation: This indicates. in order to obtain the bearing reactions after hull deflection using reaction influence numbers based on rigid support, the equations that include the reactions and displacements of the support. (= constant) of each support point using the following equation: In the discussion below, we examine the change in bearing reaction when the hull deflects from the initial condition shown