# Shafting Alignment Calculation And Validation Criteria

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5th International Congress of Croatian Society of Mechanics September, 21-23, 2006 Trogir/Split, Croatia

SHAFTING ALIGNMENT CALCULATION

AND VALIDATION CRITERIA

N.Vulić, A. Šestan, V. Cvitanić

Keywords: ship, propulsion, shaft line, alignment, transfer matrix method

1. Introduction

The main propulsion shaft line is the essential part of a modern ship propulsion system, exposed to various conditions throughout the ship’s lifetime. It has to function properly under all possible operating conditions. Consequently, the shaft line preliminary and final design, its static and dynamic behaviour shall be carefully considered by the designer and by the Classification society.

The shaft line preliminary design is based upon simple formulae from the Classification Societies Rules, which are used to determine shaft diameters and to help layout its basic concept from the point of view of shaft diameters, bearing positions and bearing distances. The final shaft line design is based upon its response to torsional vibrations. Additionally, this design shall also comply with the criteria for the static loading of the bearings, the shafting parts (propeller shaft, intermediate shaft/s and thrust shaft), reduction gearbox shafts (if any), the engine crankshaft and the criteria for dynamic response to axial, transverse and whirling vibrations.

Shafting alignment procedure considers static and pseudo-static loading of the shafting in order to determine its static response. This procedure consists of three phases: calculation, assembly and validation of the assembled shaft line on board the ship. The main goal of this procedure is to determine and ensure achievement onboard of the bearings designed positions in athwart direction in order to comply with the loading criteria for propulsion system and shafting parts. For this purpose the shaft line is usually modelled as a continuous multi-span beam on several supports. They may be modelled as absolutely stiff or linearly elastic (in the case of static and pseudo-static response), or even as real radial journal bearings (in the case of dynamic response).

The aim of this paper is to present the conventional shafting alignment calculation procedure and the validation of the designed situation onboard. Calculation presumptions, modelling of shafting parts, material properties and loading are given in detail, in order to help the designer understand the whole calculation process. The advantage of the transfer matrix method over the finite element method in this particular case is briefly described. The important part is to establish the designed shafting elastic line onboard the ship, during the outfitting phase in the shipyard. The validation procedure of the achieved shafting position is described, with a real life example.

The presented approach is of static and/or pseudo-static type. In cases when calculation of dynamic response is necessary the reader is directed to reference [2].

2. Shafting Alignment Calculation Procedure

The shafting alignment calculation comprises evaluation of the shafting elastic line and the reaction forces of supports for the pre-determined offsets of supports. In case of propulsion systems with gearboxes (mainly small, medium and high-speed four-stroke Diesel engines) the scope of the analysis is restricted to the shaft line from the propeller to the output shaft of the gearbox, together with its bearings and the bull gear. The remaining shaft line parts (clutches, input shaft, as well as

1

the engine itself) need not be taken into account. A typical shaft line layout of this kind is schematically shown in Figure 1.

In the case of directly coupled engines (mainly large slow-speed two-stroke Diesel engines) the shafting alignment analysis takes into account the model and the static behaviour of the engine crankshaft. The complete crankshaft need not be modelled in detail, as almost every slow-speed Diesel engine manufacturer is ready to provide the drawing describing this model as a girder system available to the shaft line designers.

Figure 1. A typical marine shaft line involving a gearbox [1]

2.1 Input data and modelling of the system

The data describing dimensions, material and loading of the shafts, together with the data describing the bearings concept (slide or roller), bearing clearances and lubrication means are to be available for shafting alignment calculations. This real system is modelled as a statically indeterminate system of variable section beams with multiple supports. The shaft line elements are modelled by means of circular section model elements, and the shaft line bearings are modelled by means of absolutely stiff or linearly elastic supports. In general, the cross-section varies from one beam to another.

In general, model elements are of conical shape. A special case of conical element is the element of cylindrical shape, as a cone with equal diameters on both ends.

Elements are made of homogenous material, of specific density ρ, submerged (completely, partially, or not at all) into sea-water of specific density ρw. The shaft material elastic properties are described by means of Young modulus of elasticity E and shear modulus G.

As the calculation presumes the ship afloat, after assembling all the parts of shaft line, loading of elements consists of: self-weight of the element; buoyancy in sea water (for submerged elements); external concentrated force F in the centre of the cross section of the left element end, [N]; external concentrated moment T in the centre of the cross section of the left element end, [Nm]; external uniformly distributed load q along the element (owing to other possible forces, addi-

tional to the shaft self weight and buoyancy), [N/m]. A general element model, together with the support at its right end is shown in figure 2.

F

dv, left

T

ρ, E, G

x

l

ρw

Figure 2. General element model [3]

2

q

du

dv, right

R

All the calculations are to be performed for the vertical plane, where the influence of self-weight and buoyancy shall be taken into consideration within the loading of the model. In the case of propulsion systems with gearboxes, where gearing forces in horizontal direction have a significant influence, the separate calculations for the horizontal plane are also needed.

2.2 Calculation presumptions

The calculations are based upon the real element dimensions, and the following presumptions: Propeller is completely or semi-submerged into water; Volumetric forces (self-weights and buoyancy) are uniformly distributed along each element; All the bearings may be modelled by means of absolutely rigid or linearly elastic supports; The influence of shear forces and deformations is to be taken into account; The axial position of each reaction force is on the half way of the bearing length.

If necessary, the inclination of shafting with respect to the ship water line (horizontal plane)

may be taken into account by calculating of components (for concentrated forces) and correction of

gravity constant (for volumetric forces).

2.3 Element transfer matrices and selection of initial parameters

For calculation purposes the whole shaft line is modelled as a system of multi-span beams, sup-

ported in rigid (absolute stiff) or linearly elastic supports. Each beam has a uniform circular cross-

section (solid or hollow). Conical shafting elements are modelled as cylindrical with mid-section

diameters, for the evaluation of stiffness and loading by volumetric forces.

The most appropriate modelling and calculation procedures are the methods of initial parame-

ters in its matrix form (so called: transfer-matrix method) or finite element method. Though equiva-

lent results are obtained by means of either of these two methods, the transfer matrix method is

chosen and preferred, as it requires linear systems of significantly smaller ranges to be solved. Par-

ticularly, finite element method requires solving of 2m equations (where m is the number of shaft

line elements). On the other hand, transfer matrix methods requires solving only of z+2 equations

(where z is the total number of stiff supports between the system ends). In addition to this, the

transfer matrix method is purely analytical, implementing the solutions to differential equations for

beams in bending and shear. So, the calculation model based on transfer matrices in a single (e.g.

vertical) plane is described further on.

The basic goal of the transfer matrix method is to determine the state vectors vi in each section

of the whole system. It is necessary to determine these vectors at each end section of each element:

vi = [− wi βi M i Qi 1]T

(1)

Considering the system element (i), the state vector (vi+1) at the right section of the element right

end is related with the state vector (vi) at the right section of the element left end as follows:

v(right ) i+1

=

Li,support

⋅

v

(left i+1

)

=

L i ,support

⋅ Li,elem

⋅

v

(left i

)

=

Li

⋅

v

(left i

)

(2)

In the equation (2) Li = Li,support ·Li,elem denotes the total transfer matrix of the element i (including the support at its right end). It may also be written in the expanded form (3):

⎡

λ2

⎢1 λi

i

λ3i − κ iλi

−

λ4i

⋅ ⎜⎜⎛ Ti

+ Fiλi

+

qi λ2i

⎟⎟⎞

+

κ iλi

⋅ ⎜⎛ F i

+

qiλi

⎞⎤ ⎟⎥

⎢

2EIi 6EIi GAi EIi ⎝ 2 6 24 ⎠ GAi ⎝

2 ⎠⎥

⎢

λ

λ2

⎢0 1 i

i

− λi ⋅ ⎜⎜⎛Ti + Fiλi + qiλ2i ⎟⎟⎞

⎥ ⎥ (3)

L =⎢

EIi

2EIi

EIi ⎝

2 6⎠

⎥

i⎢

⎛

q λ2 ⎞

⎥

⎢0 0 1

λi

⎢

− ⎜⎜Ti + Fi ⋅ λi + ⎝

ii

2

⎟⎟ ⎠

⎥ ⎥

⎢0 0 0

1

−

(F i

+

qiλi

+

) Ri+1

⎥

⎢⎣0 0 0

0

1

⎥ ⎦

3

The quantities in the equations (1) to (3) have the following meaning: λi – element length, [mm] EIi – element bending stiffness, [Nm2] GAi – element shear stiffness, [N] κi = f (du/dv) – shear form factor for the circular (solid or hollow) section, κi =1,11 … 1,45 qi – uniform distributed external loading along the element (see figure 2) [N/m], Fi – concentrated force at the element left end (see figure 2), [N] Ti – concentrated force at the element left end (see figure 2), [Nm] Ri+1 –reaction of the support (if any), at the element right end, positive downwards, [N]. w, β – displacement components (deflection, [m] and slope, [m/m]),

M, Q – internal forces (bending moment, [Nm] and shear force, [N]).

Note: In case there is no support at the element right end, the transfer matrix Li,elem = Li for the sole element is obtained from (3), taking Ri+1=0.

The initial parameters to be selected are the two additional unknowns at the whole system left end. They are finally determined from the two known parameters at the system right end, together with the reactions in all rigid supports. The system ends may either be free, propped, or fixed. Any case may be chosen, however, the most common situation is that both of the ends are free. In the case of free left end of the system the unknown initial parameters are:

w1 – deflection of the system left section; β1 – slope of the system left section.

These parameters, together with all the reaction forces in rigid supports (R1, R2, … , Rz) are determined from the known boundary conditions at the right end of the system, i.e.

Mm+1=0; Qm+1=0 – in case of free right end; wm+1=0; Mm+1=0 – in case of propped right end; wm+1=0; βm+1=0 – in case of fixed right end. The total number of equations to be solved is thus z+2 only.

2.4 Calculation of influence coefficients, initial reactions of supports and system response

The whole elastic system is described by means of the system matrix A, and the system vector

b. Both of them are assembled on the basis of the boundary conditions at each fixed support and at

the rightmost end of the system, by means of span transfer matrices. For each span these span

transfer matrices are simply matrix products of transfer matrices that relate the state vector in the

section of one stiff support (or system leftmost end) to the next one:

v (jR+1)

=

L span, j

⋅

v

(L j

)

=

Lk

⋅ L k−1 ⋅Κ

⋅

L1

⋅

v

(L) j

(4)

where:

k – number of elements in the present span.

In case of both the left and right ends free, the vector of unknowns k consists of the two initial

parameters (w0 and β0) and of the reaction forces in stiff supports (Rj), as follows:

k = [− w0 β 0 R1 R2 Κ Rz ]T

(5)

In this particular case, the boundary conditions used to assemble matrix A and vector b are: The zero displacements of the fixed supports (forming the first z equations). This condition may

also be expressed by the null-vector p0 of initial offsets of supports (p0=0). Mm+1=0 and Qm+1=0, used to form the remaining 2 equations (i.e. the 2 rows of A and the 2

components of b).

The best practice is to calculate the influence coefficients prior to the calculation of components

of k. That is the essential part of the complete analysis. The influence coefficient hij quantitatively

expresses the change of reaction force (in N) in the movement direction of the support i, when the support j moves for 1 mm in that direction.

4

The matrix of influence coefficients H, that is independent of the actual support offsets, is de-

termined as:

H = A −1

(6)

The vector of unknowns k0, containing the initial reactions in the stiff supports (i.e. those for

zero support offsets) then becomes:

k0 = H⋅b0

(7)

Once the components of the vector k0 are known, the state vectors in each section of the system

may be easily found by a simple matrix multiplication, beginning from the known state vector at

the leftmost end of the system. This is the system bending and shear response in terms of deflec-

tion, slope, bending moment and shear force at both ends of each element.

2.5 Calculation of bearing reactions and the system response for designed support offsets

Designed support offsets are to be determined in advance so that the system response satisfies

certain criteria for the final calculated case. This final case may be the static response of the assem-

bled shaft line during the ship outfitting, or even the pseudo-static response of the shaft line in op-

eration. If the external forces have not changed meanwhile, transferring from the system with zero

support offset to the present one, then the vector of unknowns k will be:

k = H ⋅ (p − p0 )+ k 0

(8)

Bending and shear response of the present system with the designed support offsets is deter-

mined according to the same procedure described in 2.4 for the system with zero support offsets.

2.6 Design acceptance criteria and their verification

Detailed description of the design acceptance criteria would be beyond the scope of this paper, so they will be briefly described here. These criteria are to be met for the pseudo-static response of the shaft line in operation, both for cold and hot working conditions, as follows [3]: The stresses in shafts are to be below the prescribed permissible limits. This criterion may be

applied either to the normal stresses or the equivalent stresses. Loading of the bearings is to be within prescribed limits. In case of vertical plane calculations,

bearing reactions are to be directed upwards (to avoid overloading of the neighbouring bearings) with the rule of thumb criterion for the minimal reaction value as 20% of the left and right total load of the span. Maximal reaction values shall not exceed the ones allowable by the specific pressure in the bearings, dependent upon the bearing material in question. Shaft line slope in the bearings are to be within allowable limits dependent upon the bearing pre-selected clearances, to avoid metal contact between the bearing and the shaft at the bearing ends. Otherwise, slope boring of the bearings will be unavoidable. The rule of thumb states that the slope may “spend” up to 50% of the bearing clearance. The shaft line is not to overload the gearbox itself, in case of propulsion systems with gearboxes. The gearbox manufacturers usually prescribe the maximal allowable load transmitted by the shaft line to the gearbox. In the absence of this data, the rule of thumb will be to limit the difference in reaction forces in the two bearings of the gearbox output shaft to maximum 20% of the weight of the bull gear. The shaft line is not to overload the main propulsion engine crankshaft or thrust shaft, in case of propulsion systems with directly coupled main engines. As a rule, the engine manufacturers usually prescribe the maximal allowable load transmitted by the shaft line to the engine flange in terms of shear forces and bending moments allowable range.

The stated design acceptance criteria shall be explicitly verified in the calculation phase, after all the results (system response values) have been obtained.

3. Shafting Alignment Validation On Board

The most common and widespread method to establish the elastic line and the designed position of the shaft line itself is to obtain, tune in and verify the measured differences on the flanges prior

5

to the joining of the shaft line parts during the assembly phase. After the shaft line is once assembled it is important to obtain, tune in and verify the bearing reactions onboard. Both these measurements (at the flanges and of the reactions) are always to be performed for the ship afloat.

The designer shall calculate the values of slope and deflection on each shaft line flange for the designed support offsets. This is to be done for each disassembled shaft and their combination during assembly phase (e.g. propeller shaft and aft intermediate shaft flanges joined together, prior to joining the fore intermediate shaft and the engine flanges). This will enable the calculation of the so-called SAG and GAP values (figure 3).

GAP1

SAG GAP2

Figure 3. Definition of SAG and GAP

The pre-calculated SAG and GAP values are to be tuned up within a certain tolerance (normally

0,05 mm) during the assembly phase onboard the ship afloat. They are based upon calculated de-

flections (w1, w2) and slopes (β1, β2) at the flanges of the shaft line parts (with diameter D) prior to

joining them together:

( ) SAG = w − w − D / 2 ⋅ (cos β − cos β ) ≈ w − w

(9)

GAP

=

2

GAP

1

− GAP

=

D ⋅ (sin

β

2

− sin

β

1

)≈

2) D⋅ β

1) −β

(10)

2

1

2

1

2

1

Figure 4. Measurement of bearing reactions Measurement of bearing reactions is shown in figure 4. As it is usually rather impractical or impossible to measure them onboard at the exact position (central section) of the bearing in question,

6

they are to be measured at a certain axial distance from the bearing. Consequently, the obtained measurement results are to be corrected by means of the bearing load correction factors. These correction factors are to be calculated from the system response at the position at which the measurements have been performed.

4. Calculation Example

The presented calculation procedure has been implemented in the computer program MarShAl (Marine Shafting Alignment), coded in MS Excel/VBA, specially developed for these purposes. For illustration a few characteristic diagrams obtained by this computer code, which have been implemented and verified on an inland navigation ship are presented hereafter. The propulsion system consists of a four-stroke engine (279 kW), connected to the shafting by a reduction gearbox.

System Elements Layout

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

-0,06 -0,04 -0,02

0 0,02 0,04 0,06 0,08

0,1 0

Deflection [mm]

1000

2000

3000

4000

5000

6000

7000

8000

9000

0,3 0,2 0,1

0 -0,1 -0,2 -0,3 -0,4

0

Bending Moment [kNm]

1000

2000

3000

4000

5000

6000

7000

8000

9000

7

1 0,5

0 -0,5

-1 -1,5

0

Shear Force [kN]

1000

2000

3000

4000

5000

6000

7000

8000

9000

Normal S tress [MPa]

12

10

8

6

4

2

0

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Figure 5. Example of calculated results

5. Conclusion

Shaft line is to be properly aligned in order to ensure its reliable functioning throughout the complete ship lifetime. Careful calculation, as well as setting up of its results onboard (for the ship afloat) and their validation during the assembly and the testing phase is essential. This paper describes details of the calculation procedure, with a brief description how to validate it onboard.

Details of the design acceptance criteria, real model of radial journal bearings and the review of classification society requirements are beyond the scope of this paper. However, from the authors’ long-term experience with this matter, it should be important to introduce the requirement for shafting alignment into classification societies technical rules even in the case of smaller ships. The proposal for this will be the matter of further work.

References

[1] Kozousek, W.M., Davies, P. G., “Analysis and Survey Procedures of Propulsion Systems: Shafting Alignment”, LR Technical Association, Paper No. 5, London 2000

[2] Vulić, N., “Advanced Shafting Alignment: Behaviour of Shafting in Operation”, Brodogradnja, Vol. 52, No. 3, 2004, pp 203-212.

[3] Vulić, N., “Shaft Alignment Computer Calculation (in Croatian)”, Bulletin Jugoregistar, No. 1, 1988, pp. 1-13.

Prof. dr. sc. Nenad Vulić, Croatian Register of Shipping, Marasovićeva 67, HR-21000 Split, Croatia, Telephone: +385 21 408 163, Telefax +385 21 358 159, e-mail: [email protected] Prof. dr. sc. Ante Šestan, University of Zagreb - FSB, Ivana Lučića 5, HR-10000 Zagreb, Telephone: +385 1 6168 378, Telefax +385 1 615 6940, e-mail: [email protected] Dr. sc. Vedrana Cvitanić, University of Split - FESB, Ruđera Boškovića bb, HR-21000 Split, Telephone: +385 21 305 865, Telefax +385 21 463 877, e-mail: [email protected]

8

SHAFTING ALIGNMENT CALCULATION

AND VALIDATION CRITERIA

N.Vulić, A. Šestan, V. Cvitanić

Keywords: ship, propulsion, shaft line, alignment, transfer matrix method

1. Introduction

The main propulsion shaft line is the essential part of a modern ship propulsion system, exposed to various conditions throughout the ship’s lifetime. It has to function properly under all possible operating conditions. Consequently, the shaft line preliminary and final design, its static and dynamic behaviour shall be carefully considered by the designer and by the Classification society.

The shaft line preliminary design is based upon simple formulae from the Classification Societies Rules, which are used to determine shaft diameters and to help layout its basic concept from the point of view of shaft diameters, bearing positions and bearing distances. The final shaft line design is based upon its response to torsional vibrations. Additionally, this design shall also comply with the criteria for the static loading of the bearings, the shafting parts (propeller shaft, intermediate shaft/s and thrust shaft), reduction gearbox shafts (if any), the engine crankshaft and the criteria for dynamic response to axial, transverse and whirling vibrations.

Shafting alignment procedure considers static and pseudo-static loading of the shafting in order to determine its static response. This procedure consists of three phases: calculation, assembly and validation of the assembled shaft line on board the ship. The main goal of this procedure is to determine and ensure achievement onboard of the bearings designed positions in athwart direction in order to comply with the loading criteria for propulsion system and shafting parts. For this purpose the shaft line is usually modelled as a continuous multi-span beam on several supports. They may be modelled as absolutely stiff or linearly elastic (in the case of static and pseudo-static response), or even as real radial journal bearings (in the case of dynamic response).

The aim of this paper is to present the conventional shafting alignment calculation procedure and the validation of the designed situation onboard. Calculation presumptions, modelling of shafting parts, material properties and loading are given in detail, in order to help the designer understand the whole calculation process. The advantage of the transfer matrix method over the finite element method in this particular case is briefly described. The important part is to establish the designed shafting elastic line onboard the ship, during the outfitting phase in the shipyard. The validation procedure of the achieved shafting position is described, with a real life example.

The presented approach is of static and/or pseudo-static type. In cases when calculation of dynamic response is necessary the reader is directed to reference [2].

2. Shafting Alignment Calculation Procedure

The shafting alignment calculation comprises evaluation of the shafting elastic line and the reaction forces of supports for the pre-determined offsets of supports. In case of propulsion systems with gearboxes (mainly small, medium and high-speed four-stroke Diesel engines) the scope of the analysis is restricted to the shaft line from the propeller to the output shaft of the gearbox, together with its bearings and the bull gear. The remaining shaft line parts (clutches, input shaft, as well as

1

the engine itself) need not be taken into account. A typical shaft line layout of this kind is schematically shown in Figure 1.

In the case of directly coupled engines (mainly large slow-speed two-stroke Diesel engines) the shafting alignment analysis takes into account the model and the static behaviour of the engine crankshaft. The complete crankshaft need not be modelled in detail, as almost every slow-speed Diesel engine manufacturer is ready to provide the drawing describing this model as a girder system available to the shaft line designers.

Figure 1. A typical marine shaft line involving a gearbox [1]

2.1 Input data and modelling of the system

The data describing dimensions, material and loading of the shafts, together with the data describing the bearings concept (slide or roller), bearing clearances and lubrication means are to be available for shafting alignment calculations. This real system is modelled as a statically indeterminate system of variable section beams with multiple supports. The shaft line elements are modelled by means of circular section model elements, and the shaft line bearings are modelled by means of absolutely stiff or linearly elastic supports. In general, the cross-section varies from one beam to another.

In general, model elements are of conical shape. A special case of conical element is the element of cylindrical shape, as a cone with equal diameters on both ends.

Elements are made of homogenous material, of specific density ρ, submerged (completely, partially, or not at all) into sea-water of specific density ρw. The shaft material elastic properties are described by means of Young modulus of elasticity E and shear modulus G.

As the calculation presumes the ship afloat, after assembling all the parts of shaft line, loading of elements consists of: self-weight of the element; buoyancy in sea water (for submerged elements); external concentrated force F in the centre of the cross section of the left element end, [N]; external concentrated moment T in the centre of the cross section of the left element end, [Nm]; external uniformly distributed load q along the element (owing to other possible forces, addi-

tional to the shaft self weight and buoyancy), [N/m]. A general element model, together with the support at its right end is shown in figure 2.

F

dv, left

T

ρ, E, G

x

l

ρw

Figure 2. General element model [3]

2

q

du

dv, right

R

All the calculations are to be performed for the vertical plane, where the influence of self-weight and buoyancy shall be taken into consideration within the loading of the model. In the case of propulsion systems with gearboxes, where gearing forces in horizontal direction have a significant influence, the separate calculations for the horizontal plane are also needed.

2.2 Calculation presumptions

The calculations are based upon the real element dimensions, and the following presumptions: Propeller is completely or semi-submerged into water; Volumetric forces (self-weights and buoyancy) are uniformly distributed along each element; All the bearings may be modelled by means of absolutely rigid or linearly elastic supports; The influence of shear forces and deformations is to be taken into account; The axial position of each reaction force is on the half way of the bearing length.

If necessary, the inclination of shafting with respect to the ship water line (horizontal plane)

may be taken into account by calculating of components (for concentrated forces) and correction of

gravity constant (for volumetric forces).

2.3 Element transfer matrices and selection of initial parameters

For calculation purposes the whole shaft line is modelled as a system of multi-span beams, sup-

ported in rigid (absolute stiff) or linearly elastic supports. Each beam has a uniform circular cross-

section (solid or hollow). Conical shafting elements are modelled as cylindrical with mid-section

diameters, for the evaluation of stiffness and loading by volumetric forces.

The most appropriate modelling and calculation procedures are the methods of initial parame-

ters in its matrix form (so called: transfer-matrix method) or finite element method. Though equiva-

lent results are obtained by means of either of these two methods, the transfer matrix method is

chosen and preferred, as it requires linear systems of significantly smaller ranges to be solved. Par-

ticularly, finite element method requires solving of 2m equations (where m is the number of shaft

line elements). On the other hand, transfer matrix methods requires solving only of z+2 equations

(where z is the total number of stiff supports between the system ends). In addition to this, the

transfer matrix method is purely analytical, implementing the solutions to differential equations for

beams in bending and shear. So, the calculation model based on transfer matrices in a single (e.g.

vertical) plane is described further on.

The basic goal of the transfer matrix method is to determine the state vectors vi in each section

of the whole system. It is necessary to determine these vectors at each end section of each element:

vi = [− wi βi M i Qi 1]T

(1)

Considering the system element (i), the state vector (vi+1) at the right section of the element right

end is related with the state vector (vi) at the right section of the element left end as follows:

v(right ) i+1

=

Li,support

⋅

v

(left i+1

)

=

L i ,support

⋅ Li,elem

⋅

v

(left i

)

=

Li

⋅

v

(left i

)

(2)

In the equation (2) Li = Li,support ·Li,elem denotes the total transfer matrix of the element i (including the support at its right end). It may also be written in the expanded form (3):

⎡

λ2

⎢1 λi

i

λ3i − κ iλi

−

λ4i

⋅ ⎜⎜⎛ Ti

+ Fiλi

+

qi λ2i

⎟⎟⎞

+

κ iλi

⋅ ⎜⎛ F i

+

qiλi

⎞⎤ ⎟⎥

⎢

2EIi 6EIi GAi EIi ⎝ 2 6 24 ⎠ GAi ⎝

2 ⎠⎥

⎢

λ

λ2

⎢0 1 i

i

− λi ⋅ ⎜⎜⎛Ti + Fiλi + qiλ2i ⎟⎟⎞

⎥ ⎥ (3)

L =⎢

EIi

2EIi

EIi ⎝

2 6⎠

⎥

i⎢

⎛

q λ2 ⎞

⎥

⎢0 0 1

λi

⎢

− ⎜⎜Ti + Fi ⋅ λi + ⎝

ii

2

⎟⎟ ⎠

⎥ ⎥

⎢0 0 0

1

−

(F i

+

qiλi

+

) Ri+1

⎥

⎢⎣0 0 0

0

1

⎥ ⎦

3

The quantities in the equations (1) to (3) have the following meaning: λi – element length, [mm] EIi – element bending stiffness, [Nm2] GAi – element shear stiffness, [N] κi = f (du/dv) – shear form factor for the circular (solid or hollow) section, κi =1,11 … 1,45 qi – uniform distributed external loading along the element (see figure 2) [N/m], Fi – concentrated force at the element left end (see figure 2), [N] Ti – concentrated force at the element left end (see figure 2), [Nm] Ri+1 –reaction of the support (if any), at the element right end, positive downwards, [N]. w, β – displacement components (deflection, [m] and slope, [m/m]),

M, Q – internal forces (bending moment, [Nm] and shear force, [N]).

Note: In case there is no support at the element right end, the transfer matrix Li,elem = Li for the sole element is obtained from (3), taking Ri+1=0.

The initial parameters to be selected are the two additional unknowns at the whole system left end. They are finally determined from the two known parameters at the system right end, together with the reactions in all rigid supports. The system ends may either be free, propped, or fixed. Any case may be chosen, however, the most common situation is that both of the ends are free. In the case of free left end of the system the unknown initial parameters are:

w1 – deflection of the system left section; β1 – slope of the system left section.

These parameters, together with all the reaction forces in rigid supports (R1, R2, … , Rz) are determined from the known boundary conditions at the right end of the system, i.e.

Mm+1=0; Qm+1=0 – in case of free right end; wm+1=0; Mm+1=0 – in case of propped right end; wm+1=0; βm+1=0 – in case of fixed right end. The total number of equations to be solved is thus z+2 only.

2.4 Calculation of influence coefficients, initial reactions of supports and system response

The whole elastic system is described by means of the system matrix A, and the system vector

b. Both of them are assembled on the basis of the boundary conditions at each fixed support and at

the rightmost end of the system, by means of span transfer matrices. For each span these span

transfer matrices are simply matrix products of transfer matrices that relate the state vector in the

section of one stiff support (or system leftmost end) to the next one:

v (jR+1)

=

L span, j

⋅

v

(L j

)

=

Lk

⋅ L k−1 ⋅Κ

⋅

L1

⋅

v

(L) j

(4)

where:

k – number of elements in the present span.

In case of both the left and right ends free, the vector of unknowns k consists of the two initial

parameters (w0 and β0) and of the reaction forces in stiff supports (Rj), as follows:

k = [− w0 β 0 R1 R2 Κ Rz ]T

(5)

In this particular case, the boundary conditions used to assemble matrix A and vector b are: The zero displacements of the fixed supports (forming the first z equations). This condition may

also be expressed by the null-vector p0 of initial offsets of supports (p0=0). Mm+1=0 and Qm+1=0, used to form the remaining 2 equations (i.e. the 2 rows of A and the 2

components of b).

The best practice is to calculate the influence coefficients prior to the calculation of components

of k. That is the essential part of the complete analysis. The influence coefficient hij quantitatively

expresses the change of reaction force (in N) in the movement direction of the support i, when the support j moves for 1 mm in that direction.

4

The matrix of influence coefficients H, that is independent of the actual support offsets, is de-

termined as:

H = A −1

(6)

The vector of unknowns k0, containing the initial reactions in the stiff supports (i.e. those for

zero support offsets) then becomes:

k0 = H⋅b0

(7)

Once the components of the vector k0 are known, the state vectors in each section of the system

may be easily found by a simple matrix multiplication, beginning from the known state vector at

the leftmost end of the system. This is the system bending and shear response in terms of deflec-

tion, slope, bending moment and shear force at both ends of each element.

2.5 Calculation of bearing reactions and the system response for designed support offsets

Designed support offsets are to be determined in advance so that the system response satisfies

certain criteria for the final calculated case. This final case may be the static response of the assem-

bled shaft line during the ship outfitting, or even the pseudo-static response of the shaft line in op-

eration. If the external forces have not changed meanwhile, transferring from the system with zero

support offset to the present one, then the vector of unknowns k will be:

k = H ⋅ (p − p0 )+ k 0

(8)

Bending and shear response of the present system with the designed support offsets is deter-

mined according to the same procedure described in 2.4 for the system with zero support offsets.

2.6 Design acceptance criteria and their verification

Detailed description of the design acceptance criteria would be beyond the scope of this paper, so they will be briefly described here. These criteria are to be met for the pseudo-static response of the shaft line in operation, both for cold and hot working conditions, as follows [3]: The stresses in shafts are to be below the prescribed permissible limits. This criterion may be

applied either to the normal stresses or the equivalent stresses. Loading of the bearings is to be within prescribed limits. In case of vertical plane calculations,

bearing reactions are to be directed upwards (to avoid overloading of the neighbouring bearings) with the rule of thumb criterion for the minimal reaction value as 20% of the left and right total load of the span. Maximal reaction values shall not exceed the ones allowable by the specific pressure in the bearings, dependent upon the bearing material in question. Shaft line slope in the bearings are to be within allowable limits dependent upon the bearing pre-selected clearances, to avoid metal contact between the bearing and the shaft at the bearing ends. Otherwise, slope boring of the bearings will be unavoidable. The rule of thumb states that the slope may “spend” up to 50% of the bearing clearance. The shaft line is not to overload the gearbox itself, in case of propulsion systems with gearboxes. The gearbox manufacturers usually prescribe the maximal allowable load transmitted by the shaft line to the gearbox. In the absence of this data, the rule of thumb will be to limit the difference in reaction forces in the two bearings of the gearbox output shaft to maximum 20% of the weight of the bull gear. The shaft line is not to overload the main propulsion engine crankshaft or thrust shaft, in case of propulsion systems with directly coupled main engines. As a rule, the engine manufacturers usually prescribe the maximal allowable load transmitted by the shaft line to the engine flange in terms of shear forces and bending moments allowable range.

The stated design acceptance criteria shall be explicitly verified in the calculation phase, after all the results (system response values) have been obtained.

3. Shafting Alignment Validation On Board

The most common and widespread method to establish the elastic line and the designed position of the shaft line itself is to obtain, tune in and verify the measured differences on the flanges prior

5

to the joining of the shaft line parts during the assembly phase. After the shaft line is once assembled it is important to obtain, tune in and verify the bearing reactions onboard. Both these measurements (at the flanges and of the reactions) are always to be performed for the ship afloat.

The designer shall calculate the values of slope and deflection on each shaft line flange for the designed support offsets. This is to be done for each disassembled shaft and their combination during assembly phase (e.g. propeller shaft and aft intermediate shaft flanges joined together, prior to joining the fore intermediate shaft and the engine flanges). This will enable the calculation of the so-called SAG and GAP values (figure 3).

GAP1

SAG GAP2

Figure 3. Definition of SAG and GAP

The pre-calculated SAG and GAP values are to be tuned up within a certain tolerance (normally

0,05 mm) during the assembly phase onboard the ship afloat. They are based upon calculated de-

flections (w1, w2) and slopes (β1, β2) at the flanges of the shaft line parts (with diameter D) prior to

joining them together:

( ) SAG = w − w − D / 2 ⋅ (cos β − cos β ) ≈ w − w

(9)

GAP

=

2

GAP

1

− GAP

=

D ⋅ (sin

β

2

− sin

β

1

)≈

2) D⋅ β

1) −β

(10)

2

1

2

1

2

1

Figure 4. Measurement of bearing reactions Measurement of bearing reactions is shown in figure 4. As it is usually rather impractical or impossible to measure them onboard at the exact position (central section) of the bearing in question,

6

they are to be measured at a certain axial distance from the bearing. Consequently, the obtained measurement results are to be corrected by means of the bearing load correction factors. These correction factors are to be calculated from the system response at the position at which the measurements have been performed.

4. Calculation Example

The presented calculation procedure has been implemented in the computer program MarShAl (Marine Shafting Alignment), coded in MS Excel/VBA, specially developed for these purposes. For illustration a few characteristic diagrams obtained by this computer code, which have been implemented and verified on an inland navigation ship are presented hereafter. The propulsion system consists of a four-stroke engine (279 kW), connected to the shafting by a reduction gearbox.

System Elements Layout

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

-0,06 -0,04 -0,02

0 0,02 0,04 0,06 0,08

0,1 0

Deflection [mm]

1000

2000

3000

4000

5000

6000

7000

8000

9000

0,3 0,2 0,1

0 -0,1 -0,2 -0,3 -0,4

0

Bending Moment [kNm]

1000

2000

3000

4000

5000

6000

7000

8000

9000

7

1 0,5

0 -0,5

-1 -1,5

0

Shear Force [kN]

1000

2000

3000

4000

5000

6000

7000

8000

9000

Normal S tress [MPa]

12

10

8

6

4

2

0

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Figure 5. Example of calculated results

5. Conclusion

Shaft line is to be properly aligned in order to ensure its reliable functioning throughout the complete ship lifetime. Careful calculation, as well as setting up of its results onboard (for the ship afloat) and their validation during the assembly and the testing phase is essential. This paper describes details of the calculation procedure, with a brief description how to validate it onboard.

Details of the design acceptance criteria, real model of radial journal bearings and the review of classification society requirements are beyond the scope of this paper. However, from the authors’ long-term experience with this matter, it should be important to introduce the requirement for shafting alignment into classification societies technical rules even in the case of smaller ships. The proposal for this will be the matter of further work.

References

[1] Kozousek, W.M., Davies, P. G., “Analysis and Survey Procedures of Propulsion Systems: Shafting Alignment”, LR Technical Association, Paper No. 5, London 2000

[2] Vulić, N., “Advanced Shafting Alignment: Behaviour of Shafting in Operation”, Brodogradnja, Vol. 52, No. 3, 2004, pp 203-212.

[3] Vulić, N., “Shaft Alignment Computer Calculation (in Croatian)”, Bulletin Jugoregistar, No. 1, 1988, pp. 1-13.

Prof. dr. sc. Nenad Vulić, Croatian Register of Shipping, Marasovićeva 67, HR-21000 Split, Croatia, Telephone: +385 21 408 163, Telefax +385 21 358 159, e-mail: [email protected] Prof. dr. sc. Ante Šestan, University of Zagreb - FSB, Ivana Lučića 5, HR-10000 Zagreb, Telephone: +385 1 6168 378, Telefax +385 1 615 6940, e-mail: [email protected] Dr. sc. Vedrana Cvitanić, University of Split - FESB, Ruđera Boškovića bb, HR-21000 Split, Telephone: +385 21 305 865, Telefax +385 21 463 877, e-mail: [email protected]

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