#### Abstract

The shafting of large steam turbine generator set is composed of several rotors which are connected by couplings. The computing method of shafting with different structure under specific installation requirement is studied in this paper. Based on three-moment equation, shafting alignment mathematical model is established. The computing method of bearing elevations and loads under corresponding installation requirements, where bending moment of each coupling is zero and there exist preset sag and gap in some couplings, is proposed, respectively. Bearing elevations and loads of shafting with different structure under specific installation requirement are calculated; calculation results are compared with installation data measured on site which verifies the validity and accuracy of the proposed shafting alignment computing method. The above work provides a reliable approach to analyze shafting alignment and could guide installation on site.

#### 1. Introduction

The shafting of large rotating machinery is a multibearing rotor system. Shafts are manufactured separately and then connected by rigid or flexible couplings. All shafts have some form of catenary due to their own weight; thus, shafts are not straight. So in the installation, each bearing elevation should be adjusted along the vertical direction to keep the catenary curve smooth when the machine is running. Unreasonable bearing elevation will bring about extra bearing load which causes the vibration of shaft. When the bearing load is light, the formation of bearing oil film is difficult which induces self-excited oil film oscillation. When the bearing load is heavy, grinding pad phenomenon is prone to happen. Misalignment is a condition where the centerlines of coupled shafts do not coincide. It causes over 70% of rotating machinery vibration problems. Misalignment causes a decrease in motor efficiency, and misaligned machine is more prone to failure due to increased loads on bearing and couplings [1]. Misalignment conditions are generally classified as angular, parallel, and the combination of these two. Shaft misalignment effects negatively influence the rolling, sealing, and coupling parts and can also produce eccentricity in the air gap.

Misalignment of rotor causes generation of reaction forces in the coupling which are the major cause of machine vibration. Gibbons [2] first evaluated the misalignment forces generated in various types of coupling. Patel and Darpe [3] studied coupling characteristics under misalignment condition experimentally. They measured excitations forces due to parallel misalignment and angular misalignment, respectively. Force variation for angular misalignment case shows significant 1x and 3x harmonic vibration. Parallel misalignment on the other hand has significant 1x, 2x, and 3x components. Sekhar and Prabhu [4] numerically presented the effect of coupling misalignment on rotor vibration. It was shown that the location of coupling with respect to the bending mode shape has a strong influence on vibration. Research on dynamic response of misalignment rotor is limited. More importantly, there are discrepancies in the findings of different investigators. For example, vibrations at twice the rotational speed (i.e., 2x) and its harmonics are reported [5, 6] as a characteristic misalignment feature. Contrary to this, Al-Hussain and Redmond [7] reported coupled lateral and torsional 1x vibration for parallel misalignment. It is also reported that different types of couplings reveal different frequencies composition in the vibration response under the same misalignment condition.

The main investigations of shaft alignment date from the late 1960s. Early studies demonstrated the feasibility of aligning two shafts using an arrangement of strain gages and load cells [8, 9]. However, despite many advantages of such methods, including quickness and accuracy, this technique has a few limitations in its application for long shafts with several inaccessible bearings. In general rotating mechanical system, shaft alignment is traditionally achieved using some form of dial indicator alignment method [10], with the most common techniques being the rim and face method and the reverse indicator method [11]. The dial indicator alignment method ensures bearing elevations according to dial indicator readings measured at predetermined locations. In 1980s, laser shaft alignment technique became a hot research topic. Perry [12] presented a high-precision laser alignment system based on two position sensing detectors (PSDs) and showed that the system enabled a simpler and more accurate alignment of two shafts than using traditional dial indicator alignment method. As a result of their inherent accuracy, laser alignment systems have been widely applied for a diverse range of shaft alignment applications, including rotor systems [13] and gas turbines [14]. Simmons et al. [15] developed laser instrumentation system for reliably measuring casing, thermal distortion, and alignment deviation on a large combustion turbine. This system could accurately measure deflection and misalignment which are important for cold condition shafting alignment. Liao [16] constructed a measurement system comprising a laser light source and a detector to facilitate the alignment of two rotating shafts. The main purpose of these shafting alignment methods mentioned above is to decrease the gap and sag of every coupling. These methods are efficient to deal with the alignment of small rotating machinery which does not pay attention to the influence of bearing elevation for bearing load. However, for the shafting alignment of large rotating machinery, such as steam turbine generator set or ground-based heavy-duty gas turbine generator set, bearing elevations significantly influence bearing loads distribution. So these methods above are inefficient and may bring about irrational bearing load distribution for large rotating machinery.

For large rotating machinery alignment, bearing elevations should firstly be ensured through calculation. There are mainly two assumptions in calculation corresponding to two different installation requirements: the first assumption is that the bending moment of every bearing is zero and the second assumption is that the bending moment of every coupling is zero. Taking a 600 MW turbo-generator set with double bearing support as the study object, He et al. [17] calculated bearing elevations and loads based on these two assumptions, respectively. It was shown that bearing elevations and loads gained from the second assumption are more reasonable through comparison. The latter calculation method based on the second assumption does not apply to shafting in addition to double bearing support, because the catenary curve of every double bearing supported rotor is calculated firstly and each bearing elevation is adjusted to make each coupling fulfill the zero to zero condition. With the rapid development of unit capacity, shafting structures become more compact and more complex. Besides shafting with double bearing support, shafting with single bearing support and shafting with synchro-self-shifting (3S) clutch are widely introduced in the design of turbo-generator unit. There are various factors such as thermal distortion, fluctuation of bearing oil film thickness, and unequal settlement of foundation, which disturb the alignment condition and rotor position in operation compared with cold condition. In cold condition, there exist a preset gap and sag in rigid couplings before connection to compensate for the change in rotor position caused by the above factors in thermal condition. Study for new structure shafting alignment computing method under specific installation requirements is meaningful and necessary.

Based on three-moment equation, shafting alignment mathematical model was established by using transfer matrix method in this paper. Corresponding mechanical boundary conditions of different installation requirements, where the bending moment in couplings is zero and there exist a gap and sag in couplings, were determined. Detailed mathematical derivation of new structure shafting under different mechanical boundary conditions was conducted. Through calculation by means of programming, bearing elevations and loads of shafting with single bearing support, with double bearing support, and with synchro-self-shifting clutch under different installation requirements, were obtained, respectively.

#### 2. Shafting Alignment Computing Method

##### 2.1. Shafting Alignment Mathematical Model

The shafting of large rotating machinery is a multibearing rotor system. Several rotors are coupled together, which are supported on bearings. The following assumptions are made to simplify the model: (1) ignoring the fluctuation caused by bearing oil film, shafting elevations at the bearings are regarded as bearing elevations; (2) compared with the entire shafting, the length and weight of overhang are negligibly small so two shafting overhangs are not taken into account in mechanical equation derivation; (3) bearing support is viewed as a hinge support in modeling. Figure 1 shows a typical multisupport bearing system which is composed of rotors and bearings. It has bearings and beam spans ranked from the 2nd to the . The weights of rotors and blades are viewed as distributed load .

The multibearing rotor system is a redundant structure that is difficult to be solved directly. In order to facilitate solving, it is separated into several simply supported beams. There exists unknown value bending moment on each beam as Figure 2 shows.

Deflection angles at bearings are caused by three factors, respectively: bending moments at bearings, shafting weight, and elevation difference between adjacent bearings. Take the bearing as study object. The left deflection angle of the bearing caused by bending moment is given bywhere and are the bending moments at the and bearing. Flexibility coefficient is the left deflection angle of the bearing caused by unit moment at the bearing; is the left deflection angle of the bearing caused by unit moment at the bearing.

Similarly the right deflection angle of the bearing caused by bending moment is given byTaking the influence of elevation difference between adjacent bearings and shafting weight into consideration, the total deflection angles at the left and right side of the bearing are given, respectively, bywhere is the total deflection angle at the left side of the bearing and is the total deflection angle at the right side of the bearing. is the deflection angle at the left side of the bearing caused by the weight of the span, and is the deflection angle at the right side of the bearing caused by the weight of the span. and are the deflection angles caused by elevation difference at the left and right sides of the bearing as shown in Figure 3.

The left deflection angle of the bearing caused by elevation difference in the span is given bywhere and are the elevations of the and bearing. is the length of the span.

Similarly, the right deflection angle of the bearing caused by elevation difference in the span is given by Since shafting catenary curve is smooth when the machine is running, total deflection angles at the left and right sides of every bearing are equal:Thus, the moment equilibrium equation can be expressed asThis equation for bearings is ranked from 2nd to and can be written in matrix formand simplified formwhere is the flexibility coefficient matrix, is the moment vector, and is the deflection angle vector.

Moment equilibrium equations express the relationship between moments, deflection angles, and elevations at every bearing which lay the foundation of shafting alignment computation. However, only moment equilibrium equations are not enough for calculating the bearing elevations and loads, since the number of unknown quantities exceeds the number of equations. Thus, every bearing moment and other mechanical boundary conditions should be calculated firstly by using transfer matrix method.

##### 2.2. Transfer Matrix Method

The transfer matrix method is a fast and easy way to get the parameters such as moment, shear force, and deflection which are important in shafting alignment calculation. This method could guarantee high computational accuracy and save time. Although finite element method is versatile, it takes more time to build geometry model and requires more computational resources. This is the reason for using transfer matrix method in shafting alignment calculation.

Usually, shafting is divided into several segments with different diameters in calculation. The mechanical relationship between two arbitrary nodes could be represented by using transfer matrix method. Take the span as the study object. It is divided into segments and node as shown in Figure 4.

Figure 5 shows the forces and moments of the shaft segment on the span. From the equilibrium of forces and moments, an equation is derived by Timoshenko and Gere [18]:In matrix form,whereIf there is no coupling within the span, the relationship between the bearing and the bearing can be written asTransformation matrix

If there exists coupling within the span and the coupling locates at the node , the relation between the bearing, the coupling, and the bearing meets the following equations:where

##### 2.3. Bearing Moment and Load Calculation

In moment equilibrium equations, the expression of deflection angle vector has the elevation of every bearing which is unknown in value. With moment vector, the number of unknown qualities is more than equations. Thus, boundary conditions such as bending moment of every coupling being zero, are introduced to make equations solvable. During the installation of shafting at site, bearing elevations are determined by using the assumption that bending moment of each coupling is zero. This method reduces the bending moment on each coupling, thus enhancing the stability and reliability of turbo-generator set. Under this assumption, bending moment of each bearing is derived in a certain order by using transfer matrix method. Every bearing elevation can be solved by plugging bearing bending moment vector into moment equilibrium equations. Due to structure difference, mechanical boundary conditions of shafting in different configuration are distinct. So bending moment calculation order is diverse. Take the single bearing support shafting as example to illustrate bearing moment and load calculation procedure based on transfer matrix method.

The shafting of a 1000 MW ultra supercritical turbine with single bearing support is shown in Figure 6. This shafting is composed of a high pressure rotor (HP), intermediate pressure rotor (IP), two low pressure rotors (LP_{1}, LP_{2}), generator (GEN), and exciter rotor (EXC). High pressure rotor is supported with two bearings. Intermediate pressure rotor and low pressure rotors are supported with single bearing. Generator and exciter rotor are supported with three bearings. Single bearing support shafting is compact in structure which decreases the entire length of shafting and reduces the negative influence of base/foundation deformation on alignment.

Without considering the influence of two overhangs, simplified configuration diagram of single bearing support shafting is shown in Figure 7. Because of intermediate pressure rotors, low pressure rotor and exciter rotors are single bearing supported; there exists shear force in couplings* C*1,* C*2,* C*3, and* C*5 to support the rotor. Since generator rotor is double bearing support, shear force in coupling* C*4 is zero.

From the 8th span, spans with couplings are analyzed firstly in reverse order. In the 8th span, the bending moment and shear force in bearing 8 and coupling* C*5 meet the following relation: With the boundary condition that and , the expression of shear force in coupling* C*5 and bearing 8 can be obtained:When calculating bearing load, shear force in the right and left sides of bearing should be ensured firstly. It should be noted that when analyzing the span and the span, there exist two different shear forces in bearing . To distinguish these two different shear forces, we define to express the shear forces in bearing when analyzing the span.

The bending moment and shear force in bearing 7 and coupling* C*5 meet the following equation:Substituting and (18) into (19), moment and shear force in bearing 7 are obtained:Applying the same method to the 6th, 5th, 4th, and 3rd span, bending moments and shear forces in bearings 6, 5, 4, 3, and 2 are obtained successively. So far, the unknown moment vector in equilibrium equations has been determined. Value of every bearing elevation can be solved by plugging into equations.

There is no coupling with the 2nd and 7th span. We take the 7th span as example to illustrate the computing method of shear force on bearing. Bending moments and shear forces in bearing 6 and bearing 7 meet the following equation: and have been solved in the analysis of spans with couplings. Substituting them into (3), shear forces are obtained:Figure 8 shows the shearing forces and bearing reaction force of bearing . According to force balance, the bearing reaction force is given by

#### 3. Calculation of Results and Analysis

##### 3.1. Zero Bending Moment in Couplings

Based on the above mathematical modeling and derivation work, shafting alignment program is developed to calculate bearing elevations and loads of shafting system with different structure under the condition that the bending moment value of each coupling is zero. Bearing elevations and loads of single support shafting and shafting with 3S clutch are calculated, respectively, and compared with the installation data to verify the accuracy and validity of the computing method proposed above.

The first step of shafting alignment calculation is to divide each rotor into discrete elements. Bending stiffness, mass, and length of shafting should remain unchanged before and after modeling. The mass diameter represents the mass of each element which could be obtained from mass conservation. The bending stiffness diameter represents the bending stiffness of each element. The bending stiffness will be impaired where the cross section of rotor abruptly changes. 45° method and strain energy method have been widely used in engineering to calculate the stiffness diameter of rotor. Figure 9 shows the schematic diagram of the single bearing support 1000 MW turbine generator shafting with both mass diameter and bending stiffness diameter. The shafting is divided into 328 segments and has 329 nodes.

**(a) Rotor system with mass diameter**

**(b) Rotor system with bending stiffness diameter**

Giving the elevations of the 5th and 6th bearing as 0 mm, bearing elevations and loads distribution of single bearing support shafting are calculated and compared with installation data measured on site. The calculation and comparison results are shown in Table 1. The maximum relative deviation is 3.47% at the 2nd bearing.

Figure 10 shows the catenary curve calculation result of single bearing support shafting. Section bending stress calculation result under corresponding elevations is shown in Figure 11. Bending stress of each coupling is 0 MPa which meets the installation requirement that bending moment of every coupling is zero. The maximum bending stress is 26.68 MPa at the end of generator rotor.

The configuration diagram of shafting with synchro-self-shifting clutch is shown in Figure 12. High pressure rotor and low pressure rotors are supported by two bearings. Intermediate pressure rotor is supported by one bearing. Generator rotor and exciter rotor are three-bearing supported. Bending moment and shearing force of each coupling are shown in Figure 13. The bearing moment and load computing method of single bearing support shafting applies to shafting with 3S clutch as well.

Figure 14 is the schematic diagram of shafting with 3S clutch. The shafting is divided into 331 segments and has 332 nodes. Giving the elevations of the 7th and 8th bearing as 0 mm, bearing elevations and loads distribution of single bearing support shafting are calculated and compared with installation data from the company. The calculation and comparison results are shown in Table 2. The maximum relative deviation is −6.14% at the 1st bearing.

**(a) Rotor system with mass diameter**

**(b) Rotor system with bending stiffness diameter**

Figure 15 shows the catenary curve calculation result of shafting with 3S clutch. Section bending stress calculation result under corresponding elevations is shown in Figure 16. Bending stress of each coupling is 0 MPa which meets the installation requirement that bending moment of every coupling is zero as well. The maximum bending stress is 26.82 MPa at the end of generator rotor. The elevation relative error of both shafting structures is within acceptable range which verifies the validity and accuracy of shafting alignment computing method proposed above.

##### 3.2. Gap and Sag in the Coupling

In consideration of various factors which influence the rotor position in thermal condition such as thermal distortion, fluctuation of bearing oil film thickness, and unequal settlement of foundation, there exist preset gap and sag in rigid coupling in cold condition alignment to compensate for the position change in thermal condition. The gap and sag of coupling are parameters to describe shafting alignment condition in terms of angularity and offset in the horizontal and vertical view. Angularity can be expressed directly as the angle difference between two coupling flanges. The coupling gap itself has no alignment meaning; it must be divided by the coupling flange diameter to describe the angle difference between coupling flanges. Offset describes the distance between rotation axes. For each rotor, the catenary bow caused by weight is negligible and, therefore, the rotation axis is not straight. Coupling sag actually represents the elevation difference between two coupling flanges.

Since there exist preset gap and sag, shafting catenary curve is no longer continuous and smooth. Thus, moment equilibrium equation does not apply to the calculation of bearing elevation and load under this installation requirement. For a rotor with single support, an auxiliary support is added in the front-end of rotor as temporary support in installation. Every rotor in shafting becomes double bearing support through adding auxiliary bearing in installation.

The alignment procedure under this installation requirement will be described in the following passage, and an example of shafting with six bearings and three rotors as shown in Figure 17 will then illustrate this computing method.

Firstly when the elevation of each bearing is zero, the catenary curve of each double bearing support rotor is calculated, respectively. Before connection, bearing elevations of adjacent rotors are adjusted to make couplings fulfill geometry boundary condition caused by preset sag and gap. The geometry relationship between two coupling flanges can be written aswhere .

The elevation of every bearing including auxiliary bearing is ensured in turn/order according to the geometrical relationship, namely, deflection and angle, between two coupling flanges. Tighten bolts of each coupling to connect shafts after adjusting every bearing to assigned altitude along the vertical direction.

Figure 18 shows the schematic diagram of the double bearing support 600 MW turbine generator shafting. Since the exciter rotor is single support structure, auxiliary support will be added in the front-end. The preset sag and gap of each coupling are shown in Table 3.

Giving the elevations of the 8th and 9th bearing as 0 mm, bearing elevations and loads distribution of single bearing support shafting are calculated and compared with installation data. The calculation and comparison results are shown in Table 4. The maximum relative deviation is −6.48% at the 7th bearing.

Figure 19 shows the catenary curve calculation result of double bearing support shafting. Section bending stress calculation result under corresponding elevations is shown in Figure 20. As shown in Table 3, preset sag of couplings* C*1 and* C*2 is 0 mm; bending stress value of* C*1 and* C*2 approaches zero as well. The maximum value of bending stress is 26.70 MPa at the end of generator rotor. Different from previous analysis about single bearing support shafting or shafting with 3S clutch, bending moment stress at the 1st bearing varied intensely. The maximal bending stress is 24.15 MPa.

#### 4. Summary and Conclusions

This paper has established the mathematical model of shafting alignment, and computing methods of bearing elevations and loads under corresponding installation requirements, where bending moment of each coupling is zero and there exist preset sag and gap in every coupling, are proposed, respectively. Based on the above work, shafting alignment program aimed at bearing elevation and load calculation is developed. Bearing elevations and loads of shafting with different structure under specific installation requirements are calculated by using this alignment calculation program. Compared with installation data measured on site, the validity and accuracy of proposed model and computing method have been verified. Computing method presented in this study provides a comprehensive and reliable approach for analyzing shafting alignment and guiding the installation on site.

#### Appendix

The deflection angle vector is composed of two parts:The first part is the deflection angle caused by weight: ; the mathematical derivation is presented in Figure 21.

The relationship of force, moment, bending angle, and deflection between adjacent nodes isIn matrix form,whereIn this span, which is divided into segments and nodes, the relationship between node and node 1 iswhereIn matrix form,Substituting the boundary condition in this case,Shear force and deflection angle caused by weight could be obtained:This part caused by weight in deflection angle vector could be presented as

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work is supported by the National Natural Science Foundation of China (no. 11372234).