Transient Response Analysis of Marine propulsion shafting systems by computer simulation + Appendix

DELFT UNIVERSITY OF TECHNOLOGY MECHANICAL ENGINEERING DEPARTMENT

OEMO SECTION

TRANSIENT RESPONSE ANALYSIS

OF

MARINE PROPULSION SHAFTING SYSTEMS

BY COMPUTER SIMULATION

By: G.A. Schouten March 1996

SUMMARY

The dynamic behaviour of marine propulsion shafting systems is transient in nature and can be described under either steady state or transient operating conditions; the

former being a special case of the latter. The differential equations describing the

dynamic behaviour of systems take into account the non-linearities associated with engine and propeller operation over the speed range. However, it is normal practice

to linearise such non-linearities about the steady operating conditions in the

frequency domain calculation to enable the formulation of the problem. This means

that a frequency domain calculation is unable to predict the response time of the

system in traversing the speed range of the system in response to changes in the

engine governor setting or variations in the propeller pitch settings.

Therefore, a time domain computer simulation program for the analysis of the

torsional dynamic behaviour of marine propulsion shafting systcms under transient

operating conditions has been developed. A modular approach is used for the

simulation, where the shafting system is considered to comprise a collection of

component models, each with its own set of dynamic characteristics. A fixed step,

fourth-order Runge-Kutta integration technique is used for time marching. The

correctness of the program has been demonstrated by comparing preliminary

torsional vibration results obtained from frequency domain vibration analysis and

time domain simulation of the system under steady operating conditions. Under

such conditions the frequency domain analysis and time domain simulation produce the same results. The initial conditions of shaft speeds and torques specified in the time domain simulation define the starting operating condition of a system and the simulation automatically converges to the steady operating condition, whereas the

frequency domain analysis implicitly assumes steady state operation at the given

conditions. The time domain simulation therefore presents a more realistic

representation of the system's dynamic energy balance by taking into account the

mean speed of the system and its velocity perturbations due to torsional

vibration

instead of "assuming" a steady system speed as in the frequency domain analysis

thus

eliminating erroneous matching of

the operation of

shafting system

components and the associated non-linearities. The full capabilities of the time

domain program are then demonstrated by the simulation of a marine propulsion

shafting system example under transient operating conditions.

The time domain simulation is a comprehensive method for analysing the torsional

vibration dynamic characteristics of marine propulsion shafting systems under

transient operating conditions. However, frequency domain analysis is

computationally much more efficient in determining the torsional vibration response under steady operating conditions.

Delft University of Technology 1996 acknowledgements

ACKNOWLEDGEMENTS

The author wishes to acknowledge Dr. Kian Banisoleiman for his invaluable help,

guidance and supervision throughout the duration of this project. Thanks are

extended to Prof. J. Klein Woud for his help and constructive criticism during his

visits to London. Special thanks to Mr J. Carlton for the opportunity to work in the

Technical Investigation,

Propulsion and Environmental Engineering (TIPEE)

Department of Lloyd's Register of Shipping. The author is also very grateful to the staff of the TIPEE and the Machinery Design and Dynamics (MDD) Department for their help and suggestions.

GENERAL NOMENCLATURE Ca; Ce kQ kr fill In (00

Q

No R, To old 1(4,, Vs (mm) (Nms/ rad) (Nms/ rad) (mm) (N)

(/)

(kg.m2), (NM/ rad)

(/)

(/)

(mm) (kg) (rpm) (rad/ s) (rad/ s) (Nm) (mm) (m)

(/)

(mm) (N) (Nm) (rad) (rad/ s) (rad/ s2)

(m/s)

(m)

(m/s)

(m/ s2), bore

absolute damping coefficient relative damping coefficient propeller diameter

force

advance coefficient mass moment of inertia torsional stiffness coefficient propeller torque coefficient propeller thrust coefficient conrod length

mass

rotational speed

fundamental frequency natural frequency

propeller torque (in open water) crankthrow radius

gear radius gear ratio stroke

propeller thrust (in open water) torque angular displacement angular speed angular acceleration ship speed displacement. velocity acceleration

LIST OF CONTENTS

Page

INTRODUCTION 1

MODERN MARINE PROPULSION SHAFTING SYSTEMS 3

7.1 Straight Shafting System 5

-).') _{Branched Shafting System 7}

FREQUENCY DOMAIN ANALYSIS OF PROPULSION SHAFTING SYSTEMS

TIME DOMAIN SIMULATION OF PROPULSION SHAFTING SYSTEMS 13

4.1 Modelling Technique 14

4.2 Sign Convention 14

4.3 Engine Crankshaft Model 17

4.4 Propeller Model 29

4.5 Shaft Model 33

4.6 Gearbox Model 35

4.7 Clutch Model 37

4.8 Epicyclic Gear Model 39

4.9 Ship Model.. 43

4.10 Program Structure 44

RESULTS OF TIME DOMAIN SIMULATIONS 46

5.1 Steady State Simulation and Correctness Testing 46

5.1.1 Simple Test System 46

5.1.2 Straight Marine Propulsion Shafting System 59

5.2 Transient Simulation of a Marine Propulsion Shafting System 69

CONCLUSIONS AND RECOMMENDATIONS 78

REFERENCES 80 APPENDIX 1 APPENDIX 2 APPENDIX 3 APPENDIX 4 APPENDIX 5

Lloyd's Register of Shipping Cylinder-line Model

Epicyclic Gear Simulation

LR 278 Vibration Analysis Results FORTRAN 77 Program Codes

1.

3.

1. INTRODUCTION

The dynamic behaviour of marine propulsion shafting systems can be considered

qualitatively as the temporal displacement, velocity, acceleration and jerk signatures of

the system along its geometry and relative to a fixed frame of reference. This

behaviour is primarily due to the elasticity of the shafting system, by twisting,

elongation and bending in response to external forces exerted on the system from the prime mover, the propeller and out of balance forces. The dynamic behaviour of the

shafting system is also a characteristic of the shafting system's geometry, the materials used and the supporting arrangement in the installation.

As such, the shafting system transmits power mechanically from the prime mover to

the propeller and exhibits vibration characteristics due to intermittent and cyclic

nature of the external forces acting on the system. These forces predominantly

comprise firing in multi-cylinder Diesel engines and propeller blade passing frequency

in a non-uniform wakefield distribution, which exists around the propeller. Such

vibration characteristics are predominantly exhibited in the torsional, axial and lateral

degrees of freedom. The torsional and the axial dynamic behaviour of propulsion

shafting systems are coupled through both the geometry of engine crankshaft and the

propeller hydrodynamics. The lateral

dynamic behaviour is due to whirling of

eccentric misaligned mass centres and is minimised by careful positioning of bearings and alignment of the shafting arrangement.

The dynamic behaviour of marine propulsion shafting systems is transient in nature and can be described under either steady state or transient modes of operation; the former being a special case of the latter. The steady state is typified when the engine

is running at a fixed operating point and a constant power is transmitted to the

propeller at a steady speed. The transient condition is when the operating point of the

engine and the propeller are changing, due to changes in demand and power

imbalance, engaging or disengaging of a clutch, impact loading of the propeller or a

combination of these.

The differential equations describing the dynamic behaviour of a shafting system and the associated non-linearities in the external forcing functions can be linearised about

the steady speed, for steady state operation. However, a description of the system

behaviour over a wide range of operating conditions requires the inherent

non-linearities in the total system over the entire speed and the torque range to be taken

into account.

Conventional techniques used for the analysis of shafting system dynamic behaviour

commonly adopt a frequency domain approach to

determine the

vibration

characteristics of the shafting system under steady operating conditions. This

approach is justified in the minimum energy state of the system at a steady speed

because the forcing functions required for operating conditions about a steady speed are cyclic in nature. Under transient operation however, when the speed is changing

significantly the non-linearities associated with speed and non cyclic forcing functions mean that the system dynamics can no longer be described in the frequency domain.

Delft University of Technology 1996

The objective of this thesis project is to develop a modular, interconnecting shafting system simulation program, which is suitable for the transient response analysis of

modern marine propulsion shafting systems directly in the time domain. Time domain

simulation of shafting system's dynamic behaviour allows a continuous transient

response analysis and forms a powerful tool in the range of marine

propulsion

shafting system analysis capabilities. It can be used specifically for; Analysis of transient response in propulsion shafting systems.

Assessment of stresses in the shafting system during ship emergency manoeuvring.

Analysis of shafting vibration peaks due to engine misfiring or propeller blade

impact loading.

Integration with time domain detailed engine and propeller simulation. Engine, shafting system and propeller matching.

A simulation environment for integrated ship propulsion system control for

traversing over the barred speed range.

The development of such la time domain computer simulation program is presented in this report.

The report is presented in 6 chapters. Chapter 1 is the introduction. Chapter 2 presents

examples of modern marine propulsion shafting systems from which two typical

examples are chosen for further detail analysis. Chapter 3 presents the conventional theory of the frequency domain vibration analysis, for comparison purposes. Chapter 4 presents the developed time domain simulation program and discusses the models

of shafting system components. Chapter 5 presents a correctness testing for the

developed time domain program by comparing the results of the time domain

simulation with the frequency domain analysis under steady operating conditions. The capabilities of the simulation program are then demonstrated using a selected example of the marine propulsion shafting systems from chapter 2. The conclusions

2. MODERN MARINE PROPULSION SHAFTING SYSTEMS

Lloyd's Register of Shipping (LR) is the world's premier ship classification society. The function of a ship classification society is to ensure that the construction of a ship is in

accordance with the standard of construction for safety of life at sea. A ship is

classified according to the standard of construction and equipment as stated in the

Rules of the classification society. The costs of the insurance of both ship and its cargo depend to a great extent upon this classification and it is therefore to the advantage of

the shipowner to have an internationally approved Classed ship. A brief overview of

LR's history, main activities, and organisation structure is presented in appendix 1. The ships which are built in accordance with the Rules of LR are assigned a class in the

Register book. For example, a ship is classed +100A1 when it is fully built to the

highest standard assigned by LR and surveyed during all stages of construction. The

additional +LMC notation is assigned when the propelling and essential auxiliary

machinery has been constructed, installed, and tested under the society's special

survey and in accordance with the society's Rules indicating that the ship has Lloyd's

Machinery Certificate.

Five typical examples of modern marine propulsion shafting systems are selected from actual ships. These ships are assigned the class +100A1 +LMC with LR. The schematic

representation of the five typical shafting system examples are presented in figure

[2.11. (Al BI IC)

>.<

2-STROKE ENGINE oKE ENGINE GEARBOX BEARING FLP-XIBLE coUPLING FLEXIBLE CLU1VH GENERATUR

MED PI 101PHI *FILER CONIBt)LIABLEBTTCH

PIP )PELLER

Figure 2.1: The schematic representation of _five typical marine propulsion shafting systems.

Delft University of Technology 1996 chapter?

The particulars of the systems shown are;

A two-stroke, low speed Diesel engine with a high number of cylinders coupled

directly to a fixed-pitch propeller through a straight shafting system.

A two-stroke, low speed Diesel engine with a low number of cylinders coupled

directly to a controllable-pitch propeller through a straight shafting system.

A two-stroke, low speed Diesel engine coupled directly to a controllable-pitch

propeller through a straight shafting system. A Power Take Off (PTO) branch is

coupled to the free end of the engine through a gearbox.

One of a pair of four-stroke medium speed Diesel engines distributing power by a

single reduction gearbox to a controllable-pitch propeller and a PTO-branch.

A four-stroke, medium speed Diesel engine distributing power by a single

reduction gearbox to a controllable-pitch propeller and to a PTO-branch which

comprises a shaft alternator and a flexible coupling. The PTO-branch is engaged or disengaged from the system by a flexible clutch.

The shafting systems examples shown in figure [2.1] are used to propel a container

ship, oil

tanker, bulkcarrier, passenger/freight

ferry,

and a chemical

tanker,

respectively.

Two modern marine propulsion shafting systems examples were selected from these for further detailed analysis. The first one is example (a), a straight shafting system,

and the second one is example (d), a branched shafting system. These examples were chosen because they included the most commonly used shafting system components.

2.1 STRAIGHT SHAFTING SYSTEM

The straight marine propulsion shafting system selected is a typical configuration used for the propulsion of single screw container ships and comprises a two-stroke, in-line,

nine cylinder Diesel engine driving a six-bladed, fixed-pitch propeller. The general

arrangement plan of a typical single screw container ship is presented in figure [2.2].

The particulars of the selected straight shafting system are as follows (N.B. the

operating condition of the engine is presented at the Maximum Continuous Rating

(MCR) speed of the engine);

MAIN ENGINE PARTICULARS

Engine type: 2-stroke, in-line, turbocharged, single acting piston with crosshead, low speed, reversible type Diesel engine. power output (MCR)

engine speed (MCR) mean effective pressure number of cylinders cylinder bore

stroke firing order

SHAFTING SYSTEM PARTICULARS

intermediate shaft 1 intermediate shaft 2 screw shaft PROPELLER PARTICULARS propeller diameter number of blades blade area ratio pitch at 0.7 radius propeller inertia (kW) 11415 (rpm) 120.7 (bar) 16.46

(/)

9 (mm) 500 (mm) 1950

(/)

1-9-2-7-3-6-54-8 (rrunxmm) 0 440 x 6540 L (mmxmm) 0 440 x 4000 L (mmxmm) 0 612 x 8317 L (mm) 5850

(/)

6

(/)

0.723 (m) 4.690 (kg.m2) 31700

/"7'1' r.. GO 01 7,---*7.-..-.."-.`... Er; -1,:r...71-41E111 a__ -,-..-f..1,-1.1-r---.41_,: _ _11.-__722,.7._.-7 I., --lilt 00 00700 , DO 00 I, ilL L..q. IF ! ' I ," 00 II' I1 8 II 1:L11 -11 $717 11

7I

I 1 ,I ; I I 10 01 nI I s 1 'r "Jal I I oI /i 1 .;1 ' .1. ''L zt,rii., : 1 .-. .41.. .14,.=.,1-,-,. .01:11313111, L.':". '''' ' L9 -0 ^02 WO oro3.1

tr..-

-F . rv t--- -r-I )""?':---V-. -., _{_F} 4, I.:.._,f ..._:

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I-tn., II' s !tn. na. -NI1101 W./ L._ ---4._ . .7: 7: r---F-- r---F-- _ _ _ -_.. IC -& f -42!:1. J1_la...132. p.m r4 _ .004 ,P,10 OM 01., IQ Fr I/ I -; 1- - I: ;,/Prr #4;;Vj A CORMASTLE MC. F. )01 Slt1.4-111 °N-41111.1.. 114_3111L,31. .:.-:1 ...:. :iv: : : ,.'1.' -2 ----, 0.---1 ....--.... ....,..e.:--1 - ---us :, 1":31.1 :L.:. 0.4104 ..0 e 10 P 77-0 .0.1 rOGO .70.0 1 . 1.0 .7 NMI, NO .00 NO 3 '00 .0.4 K. 3 1 .0.0 NI ;14.1odr...Nit....1 ...I ,.... f. ....1..;613,.4.(oj 1 ...L-....14 r"...t..-... 4....-...4... .I. ...-. --1 :a ... I. -. :-. - - ^1. 4,41 ....-t---i. 00 f M ".F...',

.

-...t. MO z.-= IS ILL --:-.:-,_-__- : ...-vb-.." ---:.,..c:: : .a._: ..;.::: 00 ".: 7. _. z....,----L .."..t.. e . --Laj-1-11 -1 II II I Tr- -t-L

\

-2.2 BRANCHED SHAFTING SYSTEM

The branched marine propulsion shafting system selected is used for the propulsion system of a twin screw, Roll On/ Roll Off (RO/ RO) passenger/ freight ferry. This

vessels is powered by four four-stroke, in-line, six cylinder Diesel engines operating at a constant engine speed. The two controllable-pitch propellers are each driven by one pair of engines through a single reduction gearbox. The electrical power is supplied by a shaft alternator driven from the gearbox.

As can be seen from example (d) in figure [2.1], the two engines and the generator on

the PTO-branch are separated from the gearbox by flexible clutches to prevent the torsional vibration being transmitted from the engines to the gearbox and from the

gearbox to the generator. Under normal operating conditions a constant engine

speed is maintained whilst the ship speed is governed by changing the pitch of the

controllable-pitch propeller..,

The particulars of the selected branched shafting system are as follows;

'ENGINE PARTICULARS

Engine type: 4-stroke, in-line, turbocharged, trunk piston, medium speed Diesel engine.

power output (MCR) engine speed (MCR) mean effective pressure number of cylinders cylinder bore,

stroke firing order

GEARBOX PARTICULARS

reduction ratio (engine-shaft) step-up ratio (engine-PTO),

3.2885 :1.0 0.2833 : 1.0

Not that the dimensions of the shaft are presented as the outer diameter, the inner

diameter, and the length of the shaft

(mmxmmxmrn) 0 350 x 0 150 x 72133 L (mmxmmxmm) 0 350 x 0 150 x 7134 L (minx mmx mm) 0 350 x0 150 x 4876 L (minx mrrixmm) 0 397 x 0 150 x 137783 L (kW) 4320 (rpm) 510 (bar) 24.07 (/)' 6 (mm) 400 (mm) 560 14-5-6-3-2

SHAFTING SYSTEM PARTICULARS

intermediate shaft 1 intermediate shaft 2 intermediate shaft 3 4 _{screw shaft}

Delft University of Technology

996--

- - _ chapter!.

PROPELLER PARTICULARS propeller diameter

number of blades

_(/)

4

blade area ratio propeller inertia

The general arrangement plan of a typical twin screw, RO/ RD passenger/ freight ferry Is presented in figure [2.3]. woriir

laM

-

ilUiroPPArJnas-1,411110

efilleiguscuom.drl=rta

"r=t1

1,1=c7=7

mai 0:L9AP 119011119 c te. OC19.0.1101111ML.

Imit! Pill

mirii

11/11/M2

Figure 2.3: General arrangement plan of a typical RO/RO passenger/freight ferry. (mm)

(/)

(kg.m2)

3. FREQUENCY DOMAIN ANALYSIS OF PROPULSION SHAFTING SYSTEMS

The dynamic behaviour of marine propulsion shafting systems under steady state operating conditions is normally analysed using frequency domain vibration analysis. The system's vibration behaviour at these conditions is described by the natural frequencies and forced damped response analysis of the shafting system. A standard matrix approach is presented

in this chapter to analyse the torsional vibration behaviour of a straight shafting system in

the frequency domain.

The equivalent mass-elastic torsional vibration model of this system is shown in figure [3.1].

This system comprises equivalent mass moments of inertia, flexibilities and damping,

representing the material properties and geometry of the engine, shafting arrangement, and

the propeller. 'where: (141 [C] 1[K] { x(kg.m2)

(Nms/rad)

(Nm/ rad) (Nm) (rad) (rad/ 0, (rad/ s2)

Figure 3.7: Torsional mass-elastic model of a straight marine propulsion shafting system.

The dynamic behaviour of this system under steady operating conditions can be described by a series of differential equations linearised about a predetermined steady speed. These

linearised differential equations may be expressed by the following equation in matrix form;

Pin {61 ± {C1 {OF} +[K]. {c4)

matrix of equivalent mass moments of inertia matrix of equivalent damping coefficients. matrix of equivalent stiffness coefficients vector of excitation moments

vector of displacement response vector of velocity response

vector of acceleration response

The dimensions of the matrices and vectors presented in this equation are (nxn) and (nxl), respectively, where n = number of inertia stations in the system. It should be noted that the

equivalent system for the axial vibration analysis cart also be described by the same form of matrix equation (3.1). However, the material properties of the system, the system geometry, and the excitations are then described in terms of mass, axial flexibility, axial damping, and forces.

10 II

Delft University of Technology 1996 _ chapter 3 A complex number description of the displacement, velocity and acceleration response IS

used to satisfy equation (3.1) using the standard solution of the second order

linear-differential equations. The displacement, velocity and acceleration response are described by

the following equations;

edmi = 0 ofele"" 041(cogp1 + j.siny

= 601 es'

...(3.3),

where:

=

(rad/ s)

angular speed

(rad) comp1,2x amplitude

00 j (rad) harmonic amplitude

(rad) phase angle

The complementary function on the Right Hand Side KRHS) of the equation is expressed as;

T, Tof To pp fio,

where: To., (Nm) complex amplitude

Tom (Nm) harmonic amplitude

Substituting for equations (3.2) through to (3.5) in equation (3.1), the matrix equation (3.1)

may be expressed by the following equation;

ci-(02.Em11,= Vol

w. (3.6)

The vibration behaviour of the equivalent mass-elastic system is described by both the

homogeneous and the particular solution of matrix equation (3.6). The homogeneous

solution of the equation, that is when the RHS of equation (3.6) is zero; i.e. a system without

excitation or external torques, gives the natural frequencies of the equivalent system. The

particular solution of equation (3.6)

that is the system with excitation, gives the forced

damped vibration response of the equivalent system.

(3.0

(3.5) 0,

The homogeneous solution of equation (16) may be expressed by the following Matrix

equation;

1{6,01[[K]-01.21A41.10,

It can be seen from equation (3.7) that the natural frequencies are determined without

considering damping in the system. Damping in the system will cause a slight shift in the natural frequencies, but analysis of an undamped system is considered sufficient to predict problem frequencies. Note that the natural frequencies depend on the system's geometry

and material properties only, i.e. the [KJ and [M] matrices..

The only non-trivial solution of equation (3.7) is obtained by setting the determinant of the

system matrix to zero, which gives the following equation [Newland, 1989];

detN

co 2 M =0

; (3.3)

There will be n values of corresponding to the eigenvalues 0)2 which satisfy equation (3.8),

for a n station system. These values are the natural frequencies of the system and can be expressed by the diagonal (nxn) matrix [ail. Letting [il] be the system matrix then, if and only if [Al is diagonal (N.B. a diagonal system matrix can be achieved by the Hessenberg

eduction technique [Newland, 1989]);

ifro2HAF[mr tic]

(3.9)

The homogeneous solution is normally expressed as two i(mm) matrices [w2] and [p]. The

former is the diagonal matrix containing all of the eigenvectors, while the latter contains the

corresponding mode shapes, arranged in columns known as

the eigenvectors. A

characteristic mode shape expressed generally by the vector {ca} exists for each natural

frequency and is found for the system by substituting cut into equation (3.7).

The obtained natural frequencies of the system indicate at what frequencies and their orders the system will freely vibrate or resonate. Resonance occurs when a natural frequency of the system coincides with an order of the fundamental forcing function frequency. This can be a dangerous condition that may cause very large response amplitudes and high stresses in the

structure. The shafting systems are designed to avoid possible resonance conditions within

the speed range by both active (e.g. damper) and passive (e.g. a barred speed range)

compensation.

11

Delft University of Technology 1996 chapter 3 The forced damped vibration response of the system under steady state operating condition

is given by the particular solution of matrix equation (3)6), and may be expressed by; {60 } 41-1.1to

(3.10)i

with

18] =E[K] + Jo).

(17141

Under steady state operating conditions the engine and propeller excitations are periodic

time functions and can therefore be represented by Fourier analysed series [Thomson, 1983].

These series give the magnitude and the phase angle of the forcing function for the

individual harmonic orders of the fundamental frequency with respect to the engine Mean

Indicated Pressure (MIP) at a steady speed. The Fourier analysed data provided by the

engine manufacturer is presented in the harmonic tables. For example, the magnitudes and

the phase angles of the first sixteen whole and the first thirty-two whole and halve orders of

the fundamental frequency are presented in such tables, for a two- and. four-stroke engine

respectively.

The torsional vibration response of the system due to excitation is determined by solving

matrix equation (3.10) for {00} . A particular solution is found for each order in the Fourier

analysis of the 'excitation. The magnitudes and the phase angles of the displacement

response for each order are calculated from the following equations;

dRe20-of +Im260j

true0, pc, arctan

Ree

(3.11)

The signs of the real and imaginary part of the response define the quadrant In which the

phase angle is placed.

The results of the frequency domain vibration analysis include, besides the amplitudes and phase angles of the displacement response of all the stations in the system, the additional stresses and torques in the shaft sections, and, as appropriate, the stresses in the flexible

couplings and dampers. The torsional vibration analysis must be carried out for all the

relevant, different shafting system configurations, for normal and cylinder misfiring

operating conditions.

00,1 =

4. TIME DOMAIN SIMULATION OF PROPULSION SHAFTING SYSTEMS A computer simulation program for the analysis of the transient dynamic behaviour of

marine propulsion shafting systems in the time domain has been developed. The

program is based on reformulation and solution of a number of simultaneous first and second order differential equations, representing the torsional dynamic behaviour of

the shafting system in the time domain. A fixed-step, fourth order Runge-Kutta

integration technique was used for time marching [Thomson, 1983]. This program also

caters for typical non-linearities in the operating conditions of the shafting system

during transient conditions, such as; Engine operation.

Propeller operation.

Changes in the shafting system configuration due to clutching operations. Torque transmission whilst slipping clutch plate.

A modular approach was used for the simulation of shafting systems. A system is considered to comprise a collection of component models, each with its own set of

dynamic characteristics. The component models are selected from a library. The

advantages gained from this modular approach are that;

The library of component models can be reused to simulate a variety of marine

propulsion shafting systems.

It easier to understand and test the functionality and limitations of the individual

components, than the complete system.

More sophisticated component models can be developed as may be required at a

latter stage.

Energy balance for individual components cater for the transmitted power and

automatically converge to steady operating conditions.

The development of the shafting system component library is presented in this

chapter. This library comprises tested transient response models of the following;

Engine crankshaft Propeller Shaft Layshaft gearbox Clutch Epicyclic gear Ship

Delft University of Technology

4.1 MODELLING TECHNIQUE

The shafting system components are represented in the simulation program by their

equivalent torsional mass-elastic models. These component models can then be linked together to represent an integral system.

The models are developed by considering a rigid body approach, using equivalent

mass moments of inertia, massless torsional springs, and viscous dampers to represent

the material properties (mass) and geometry (flexibility) of the component. A

mass-elastic free body diagram is presented to indicate the simplifications and the modelling

assumptions used, for each component. The torsional degrees of freedom, external torque boundary conditions and internal reaction torques are also presented for each

component.

4.2 SIGN CONVENTION

The sign convention adopted in the developed component models is illustrated by the

simple system of figure [4.2.1]. This shows a single cylinder engine coupled to a

propeller (load), by a shaft, along with its equivalent torsional mass-elastic model. In it the engine is represented by an equivalent mass moment of inertia, and an equivalent stiffness and a relative damping coefficient are used to represent the torsional elasticity of the shaft geometry and the vibration attenuation characteristics of the shaft material respectively. The inertia of the shaft is lumped with the engine and the load inertias at

each end of the shaft. An absolute damping coefficient is used to represent a speed

dependent load.

SHAFT

y Cal

Figure 4.2.1: A simple shafting system along with its torsional mass-elastic model.

.11 12

kl

ENGINE LOAD

The generalised free body diagram of the simple mass-elastic system is presented in

figure [4.22]. The external torque

act4

on the engine inertia, station (i), due to firing

force "Tex(i)" is assumed to be tir. in the same 'direction as 0. The internal reaction

torque acting on the RHS of

station (i), "Tr(i)", exerted by the shaft on the engine

inertia is considered to be -ye since it opposes the direction of motion. The torque,

exerted on the Left Hand Side (LHS) of the station (1+1), "T1(i+1)", due to the twist in

the shaft is considered to be +ve. Thus the following torque conservation equations

can be specified for a twist in the shaft, acted by displacement of 0, and 02+1 at stations (1) and (i+1), at either end of the shaft.

'Tx

-

-k1.(9

-

_{i+1) Cip(15/i -6}4+1) ...(4.21) and Tr(i) 71,.0

k1.0; 94J+ci.(ei

-(4.2.2) where: Ta,t (Nm) torque at the RHS of inertia station (i)

71,a+1 (Nm) torque at the LHS of inertia station (i+1) ki I(Nm/ rad) torsional stiffness coefficient between inertia

stations (i) and (i+/).

(Nms/rad), equivalent relative damping coefficient between

inertia stations (i) and (i+1). (rad) displacement of inertia station (i)

O (rad/ s); velocity of inertia station (i)i

NB. Thartd Tn4i, are zero in figure (4.2.2).

CO)

r3rn

Ca(i+L)

Figure 4.2.2: Generalised free body diagram. of the simple mass-elastic system.

DelajLiv

fl_i_Q_±c-si_s_Qgin ,} 1996

-

chapter 4,

Tex(i)

J(i)

K(i)

J(i+1) 0,

The viscous torque acting on the inertia station (HI) is determined from the absolute

damping coefficient with a negative sign indicating that it opposes the direction of motion;

Cai+, i±,

.. (4.2.3)

where: (Mm) absolute damping torque

(Nms/rad) equivalent absolute damping coefficient

The acceleration of the equivalent mass moments of inertia are then determined from the torques acting on each inertia, using the following equation:

6 (TLJ + TR., + TExi TA,i

J,

.. (4.2.4)

where. 6, (rad/ s2) torsional acceleration of inertia station (1)

TEX: (Nm) external torque acting on inertia station (i)

TA.! (Mm) absolute damping torque acting on station (i) (kg.m2) equivalent mass moment of inertia station (i) Equation (4.2.4) is presented in its generic form for torsional acceleration of the inertia

station (i) in terms of the external torque, the absolute damping torque, and the shaft

torques acting on each side of the inertia station. Depending on the connectivity one or

more of the torques above may be zero. Moreover,

it should be noted that

computation of the torques used in equation (4.2.4) are correctly signed as was shown in equations (4.2.1) through to (4.2.3).

Ca,+7,

4.3 ENGINE CRANKSHAFT MODEL

The engine crankshaft forms an integral part of the propulsion shafting system. Figure [4.3.1] shows a schematic representation of an in-line, nine cylinder engine crankshaft

along with its equivalent torsional mass-elastic model. The complex geometry of the

crankshaft structure

is modelled simply by a series of characteristic torsional

stiffnesses separated by representative mass moment of inertia at each crankthrow. The vibration attenuation characteristics of the crankshaft structure and the material

used are represented in the model by the relative and the absolute damping

coefficients.

Figure 4.3.1: A crankshaft along with its torsional mass-elastic model.

Each inertia station is assumed to comprise; the rotating inertia of the two crankwebs,

crankpin, and the two halves of the adjacent crank journals, referred to the axis of

rotation shown in figure [4.3.1]. The parallel axis theorem is used to refer the inertia of the crankpin to the axis of rotation. The inertia of the equivalent rotating mass part of the connecting rod (conrod) is also referred to the rotation axis of the crankthrow and

added to the total inertia. This lumped inertia at the crankthrow s then considered to

be constant throughout the engine cycle.

A procedure for calculating the torsional stiffness characteristics of a crankshaft in is

given by Homori [1991].

Analytical techniques for determining the relative and the absolute damping

coefficients are based on correlations to experimentally measured data [Jenzer, 1991]. One correlation suggested by Geislinger [LR, 1995] is as follows;

For an in-line engine: the absolute damping coefficient is given by;

chapter 4

and the relative damping coefficient is given by;

C1 =8000.

=12000.

( (b2

380

For a Vee type engine: the absolute damping coefficient is given by;

Ca, = 7.8 x 10-1°.(b x

and the relative damping coefficient is given by;

(0.5+.0625 x z)

(033+.0417 x

where: b (mm) bore

(mm) stroke

z

_(/)

number of cylinders.

The absolute damping coefficient can also be used to simulate the speed dependent component of the engine friction if the firing forces acting on the crankpin are not

corrected for engine friction.

Values of the crankshaft absolute and relative damping coefficients used in this report

were obtained from measurements performed by the engine manufacturer. The

method proposed by Geislinger was used to obtain these damping coefficients when

no data from the manufacturer was available.

The crankshaft excitation at the crankpin was determined by correcting the cylinder

gas-pressure excitation force for the cylinder-line

inertia

forces. Note that the

calculation of the crankshaft excitation was performed for an in-line, two-stroke,

crosshead Diesel engine. The in-line, four-stroke, trunk-piston engine configurations

can be treated in the same way as two-stroke, crosshead engines. However, a different

approach should be used for Vee type engine configurations because the angle

subtended between vertical and the cylinder banks [Hafner,1984].

4

(b 2

4.3.1 GAS PRESSURE EXCITATION FORCE

The cylinder gas-pressure excitation force was derived from knowledge of the cylinder pressure diagram versus crankangle throughout the engine cycle. A typical example of a pressure crankangle diagram of a two-stroke, slow speed Diesel engine is presented in figure [4.3.2]. The MERLIN engine simulation capability [Banisoleiman, 1993] was

used to determine this diagram. In it the gas-pressure crankangle diagram obtained

with retarded fuel injection timing.

where: 120 100 80 60 40 20 0

pressure crankangle diagram

---F' (NI) conrod force

Fp (N) piston gas-pressure force

p (N/ m2) cylinder gas-pressure

A (m2) piston area

4) (deg.) angle subtended between the conrod and vertical

0 50 100 150 200 250 300 350

crankangle [deg]

Figure 4.3.2: The indicator diagram oft? two-stroke, slow speed Diesel engine.

The gas-pressure force is transmitted through the cylinder-line geometry, comprising the piston, piston rod, crosshead and conrod, to the crankpin of the crankthrow. The

gas-pressure forces acting on the different parts of cylinder-line geometry of a

crosshead engine are presented in figure [4.3.3]. The gas-pressure force along the line of the conrod is determined using the following equation;

,=

p. A

F

cos

cos

1996

Xp Munn thsplaccmcnt L Comm] length Ro Crankihrow mhos

9.0)

Figure 4.3.3: Gas-pressure forces acting on the cylinder-lineofa crosshead engine.

The force transmitted through the conrod can be resolved into component in the

tangential and radial directions relative to the orbit of the crankpin, using the

following equations;

= F'. sin(0 +

.. (4.3.2)

= F'. cos(0 +

.. (4.3.3)

where: F"T (N) tangential force

F"R (N) radial force

0 (deg.) crankangle from Top Dead Centre (TDC)

The cylinder gas-pressure torque is obtained, from equations (4.3.1) & (4.3.2). The

radial component of the force is not used because it has a zero moment about the axis

of rotation. This force component is responsible for bending of the crank throw and

coupling with axial excitation. The gas-pressure torque may then be expressed as;

sin(0 +4))

7'pressure = Ro.F;' = R. p. A.

cosd)

)

.. (4.3.4)

where: Tpressurc (Nm) cylinder gas-pressure torque

Rearranging equation (4.3.4) and using the sine rules, the cylinder gas-pressure torque was expressed bythe following equation;

= R. p. A.(tan(0 cos() + sine)

.. (4.3.5)

The relationship between the angle between the conrod and the vertical, and the

cranIcangle may be expressed by the following equations using the cylinder-line

geometry; Ro

stilt]) = .sine

or coscl) = T pressure

R.

A. sine + )2 .

-L

.sin- 0 .. (4.3.6) where : L (m) conrod length between bearing centres

Substituting for tan 0 in equation (4.3.5) using equation (4.3.6), the gas-pressure torque is expressed as;

(

.. (4.3.7)

The gas-pressure torque acting on the crankshaft is determined from equation (4.3.7) using the cylinder-pressure diagram at the operating condition of the engine. In this

calculation, the components in the cylinder-line are considered to be rigid. Appendix 2 shows a model of a cylinder-line where the flexibility of the cylinder line components is taken into account.

An example of the calculated gas-pressure torque, for a typical, two-stroke Diesel

engine, is presented in figure [4.3.4].

It has been calculated from the pressure

crankangle diagram presented in figure [4.3.2]. The torque is plotted over one engine

cycle with TDC at U and Bottom Dead Centre (BDC) at 180 degrees crankangle 0, for each cylinder. The conrod length used is L = 1.638 (m) and the crankthrow radius is Ro = 0.680 (m). This torque was then corrected for the inertia forces and used to represent the external torque acting on a crankshaft inertia station.

-L.sin20

Delft University of Technology 1996 where: 500000 400000 300000 200000 (6' 100000 200000 0 50 100 150 200 250 300 350 crankangle [deg]

Figure 4.3.4: The gas-pressure torque acting on a crankshaft inertia station ofa two-stroke engine.

4.3.2 INERTIA FORCES

The cylinder gas-pressure force was corrected for the motion of the cylinder-line

inertias. The inertia forces can be thought as the forces required to accelerate and

decelerate the reciprocating and rotating mass components of the cylinder-line as a result of the rotation of the crankshaft. The rotating mass, for a crosshead engine, is

defined as the rotating mass part of the conrod whereas the reciprocating mass is

defined as the mass of the piston, piston rod, crosshead with guide shoes, and the

reciprocating mass part of the conrod.

The components of the inertia force acting on the cylinder-line of a crosshead engine and the sign convention used are presented in figure [4.3.5]. The inertia forces due to

the rotating and reciprocating mass parts are determined by using the following

equations;

= .R0.6

.. (4.3.8)

FI = -Mrecrp '17

.. (4.3.9)

F1 rotat (N) inertia force due to rotating mass

Fi,recip (N) inertia force due to reciprocating mass

in rota! (kg) rotating mass

m prop (kg) reciprocating mass

6 (rad/s2) crankshaft acceleration

3,, (m/ s2) piston acceleration pressure torque L. 0 100000

-F"T merrier

h.recip

= _F1,recip

FRumerna = FI"Tip '

The radial component of the inertia force is not used in the torsional vibration

treatment because it has no moment about the axis of rotation of the crankthrow. This

force component however is responsible for bending of the crankthrow and coupling

with axial excitation.

4))

Figure 4.3.5: The cylinder-line inertia forces of a crosshead engine.

The inertia force due to the rotating mass part is not used in the correction. Instead, the

equivalent inertia of the rotating mass part of the conrod is referred to the axis of

rotation of the crankthrow using the parallel axis theorem and is taken into account

through the dynamics of the shafting system.

The inertia force acting on the crankpin is resolved into components in the tangential

and radial directions relative to the orbit of the crankpin using the following

equations; sin(8 +4) COSS cos(0 +4)1 coscO ) .. (4.3.10) .. (4.3.11)

where: inertia (N) tangential inertia force

1996 chapter 4 The inertia torque is obtained by combining equations (4.3.9) & (4.3.10). It is given by;

where: T inertia

sin + 40)

= - Ro.Mre(11, XP '

(N) inertia torque

Rearranging and substituting for equation (4.3.6) in equation (4.3.12), the inertia torque may be expressed as;

R° .sin10

1

R2

0) .sir-0

\L}

.. (4.3.12) .. (4.3.11)

The piston displacement from TDC is obtained from the cylinder-line geometry and is expressed as;

xp = Ro.( I - cos° ) + L.( I -

cos)

.. (4.3.14)

where: (in) piston displacement from TDC

The piston displacement as a function of crankangle is obtained by substituting

equation (4.3.6) into equation (4.3.14), and it may be expressed by the following

equation;

.. (4.3.15)

Equation (4.3.15)

is then differentiated twice with respect to time to find the

acceleration of the piston from TDC, which is expressed as;

xp = Ro.( I

- cos()) + L.( I

-I - sin20

Ttnertio = R(, p sin() +

cos0

= Ro 62Xp 250 2001 150 100 50 -50 -100 -150 cos() + l cos 20 L r (RU)

.29

25 Kl.sin 70, 2L (RO) "_20 r ''" piston acceleration ...(4.3.16)

Equation (4.3.16) yields the plston acceleration at any cranlcangle position throughout the cycle of a typical two-stroke Diesel engine, as illustrated in figure [4.3.6]. On this

diagram a crankangle of 0 represents the position of the piston at TDC. The conrod length used is L = 1.638 (m), the crankthrow radius is Ro = 0.680 (m),, and the mean

engine speed n = 140.0 (rpm).

Figure 4.3.6: The piston acceleration against crank utgle position throughout one cycle of a typical two-stroke Diesel engine from TDC.

ote

a), 1~(1 where:

(rad/s)

angular speed of crankshaft

50r 100 150, 200 250 300, 350 crankangle Ide0

r

r 0

4

opt

1,44

1 1

a

V44--R, fl sThO + 0 0 . ( R 1 ₃ sin2 20 L

Del ft University of Technolouy chapter 4

The inertia torque is obtained by substituting equation (4.3.16) into equation (4.3.13).

Figure [4.3.7] shows the inertia torque, for the same two-stroke engine, plotted over

one engine cycle, with TDC at 0 and BDC at 180 degrees crankangle (L = 1.638 (m), Ro = 0.680 (m), mrp = 1872 (kg) and n = 140 (rpm)).

It should be noted that the acceleration, and both the gas-pressure and inertia torque

presented in the figures [4.3.6] & [4.3.7], respectively, were obtained using the

instantaneous engine speed, that is the engine speed with its velocity perturbations

due to torsional vibration. 500000 400000 300000 A

-

200000 100000 0 0 -100000 -200000 0

\

7 inertia torque pressure-torque 50 100 150 200 250 300 350 crankangle [deg]

Figure 4.3.7: The inertia torque of a typical two-stroke Diesel engine over one engine cycle from

TDC.

An approximate method of computing the inertia force correction has been proposed

by Fujii [1984], which disregards the terms proportional to the second or higher

powers of Ro/ L. The inertia torque according to Fujii is given by the following

equation;

3 R

Tinertia reciP .R(2, .6 2 R .sin 0 + sin 20 + sin 30

s L 2 L

R2 6

(

1 R0 cos() + cos20 + R0 .cos30

2

.. (4.3.17)

a)

Mrecip

The inertia torque calculated by the exact method of equations (4.3.16) & (4.3.13) is

compared to Fujii's approximation, for the same engine in figure [4.3.8]. It can be seen that there is a good agreement between the two methods.

200000 150000 100000 50000 0 -50000 100000 -150000 200000 200000 150000 100000 50000 a) 0 -50000 100000 150000 200000 FUJII approximation inertia torque _--50 100 150 200 250 300 350 crankangle [deg)

Figure 4.3.8: The Fujii 119841 approximation and the inertia torque.

The method proposed by the BICERA [Nestorides, 1958] only takes into account the

first term on the RHS of equation (4.3.17). This method is used to show the influence of

the crankthrow acceleration on the calculation of the inertia torque. The BICERA approximation of the inertia torque is shown in figure [4.3.9]. Superimposed on this

plot is the inertia torque presented in figure [4.3.7] It can be seen from the plot that the influence of the crankthrow acceleration of the two-stroke engine at these conditions is not significant.

BICERA --Inertia torque

0 50 100 150 200 250 300 350

crankangle [deg)

Delft University of Technology 1996

4.3.3 CRANKSHAFT EXCITATION

The crankshaft excitation of the typical, two-stroke, crosshead Diesel engine obtained from the gas-pressure torque and corrected for the inertia forces is presented in figure

[4.3.14 This plot also shows both the gas-pressure and inertia torque components for comparison purposes. 500000 400000 300000 200000 100000 4_) 19 10 0 10 29 crankshaft ecitatibn pressur t-orque inertia torque 50 100 150 200 250 300 350 crankangle [deg]

Figure 4.3.10: Crankshaft excitation including the inertia force correction.

As a measure of confirming the above computation, a plot of the tangential

gas-pressure, inertia, and crankshaft excitation force presented by Veritec [1985] is shown in figure [4.3.11]. As can be seen, similar trends to that predicted above were obtained

confirming the correctness of the sign convention and the algorithm used. Note that the gas-pressure, inertia, and crankshaft excitation torque can be derived multiplying

the tangential force by the crankthrow radius.

GAs FORCES + MASS FORCES (T)

Figure 4.3.11: Crankshaft excitation including the inertia force correction according to Veritec

11985]. GAS FOR

/

1

/

/

\

la. i / / /

//

N \ \ \ \

_\

/ 90 / /

/

/ ---...' CRANK ANGLE i I MASS FORCES -0 0 -100000 -200000

4.4 PROPELLER MODEL

The model of a fixed-pitch propeller has been developed using the equivalent mass

moment of inertia and the equivalent massless torsional spring stiffness. The latter was used to represent the flexibility of the propeller shaft. A schematic representation of a

typical 4-bladed fixed-pitch propeller is presented in figure [4.4.1], along with its

torsional mass-elastic model.

Figure 4.4.1: A _fixed-pitch propeller and its torsional mass-elastic model.

The equivalent inertia of the propeller comprises the dry propeller mass moment of inertia and half the mass moment of inertia of the propeller shaft. The other half is

added to the equivalent mass moment of inertia of the intermediate shaft connected to

the propeller shaft. The equivalent propeller inertia is corrected for the effects of

entrained water by adding 25 to 40 % of the dry propeller's mass moment of inertia. Methods were proposed by Parsons [1981] and Schwanecke [1963] to determine the

propeller added mass moment of inertia due to these effects.

The equivalent torsional stiffness coefficient of the propeller shaft was determined

using the method described by Nestorides [1958].

4.4.1 PROPELLER EXCITATION

The propeller excitation is obtained from the kT, kg

- J diagram of an equivalent

Wageningen B-screw series propeller, with the same power requirements as the

original propeller [Oosterveld, 1975]. This kT, kg- J presents the non-dimensionalised

propeller open water performance for the thrust coefficient kT and the torque

coefficient kg. These parameters are plotted against the advance coefficient J, which represents the non-dimensional form of the axial displacement per revolution of the

propeller.

A typical example of a kT, kg - J diagram from a Wageningen B-screw series propeller is presented in figure [4.4.2].

Jpropeller

410

1996

0.8

,(1.) oil

0

0.3

0.6

0.2 0.1 4 --4 01.1

0.2

0.3 0.4

0.5

0.6

0.7

Advance coefficient J

Figure 4.4.:A typical kr, kQ - J diagram of a Wageningen B-screw series propeller

The advance coefficient may be expressed as;

Vs w) n. D

where _{I (I)} advance coefficient

Vs

(m/s)

ship speed

(/)

wake fraction

D (m), propeller diameter

ri (rps) propeller rotational speed

The wake fraction is defined by Kuiper [199411 as the relation between the entrance.

velocity of the v. ater at the propeller tip, in the axial direction Viz ,and the speed of the

ship V9. where;

= Vs.(1 w)

The propeller efficiency rio is also presented on figure (4.4.21 and is defined by Kuiper 11994] as the ratio between the delivered and the effective power of the propeller. The propeller torque and thrust coefficient are obtained from the kr, k(2 - J diagram at the instantaneous value of the advance coefficient. The torque coefficient IQ is used to determined propeller torque from the following non-dimensionalised form;

Q = P

Ds le()

.. (4.4.2)

where: Q (Nm) propeller torque (in open water)

Pw (kg/ m3) sea water density

kg

_(/)

torque coefficient Ii -K4

eta

, L II ... ₎₁ III 7...: T -' -0.8 0.9 chapter 4 0 [I] -(4.4.1) 0.4 0 0.5 0 . 3

where: Cap (1) 11R (Nms/ rad) (Nm) (rad/ s)

(/)

Q (Nm) (rps)

absolute damping coefficient propeller torque

propeller angular speed relative rotative efficiency propeller torque (in open water) propeller rotational speed

The propeller thrust is calculated from the thrust coefficient using the following

equation;

= p

.n2 7'

.. (4.4.3)

where: T (N) propeller thrust (in open water)

kT

(/)

thrust coefficient

The acceleration of the propeller is then determined from the torques by using the

following equation, as was presented earlier in section 4.2; x Tps

0\

p

J,,

where: 0 (rad/ s2) acceleration of propeller

Tps (Nm) propeller shaft torque

Jp (kg.m2) equivalent mass moment of propeller

4.4.2 PROPELLER DAMPING COEFFICIENT

The absolute equivalent propeller damping coefficient is used to represent the

7ropeller loading in the frequency domain vibration analysis programs. Methods were

proposed by Parsons [1981] and by Schwanecke [1963] to determine this damping

coefficient.

The propeller equivalent damping coefficient can be also obtained from the propeller

KT, KQ - diagram. This damping coefficient is determined at the operating point of the

propeller using the following equation [Klein Woud, 1993];

am

52

car

= = 1 ao) 21t an .. (4.4.4) T

)

Delft University of Technology 1996

Substituting for equations (4.4.1) & (4.4.2), and rearranging equation (4.4.4), the

absolute propeller damping coefficient may be expressed as;

ak

Ca = w). + 2 n k n

27c1

-.. (4.4.5)

where: partial derivative of thek(2curve

4.5 SHAFT MODEL

The schematic representation of a typical straight shafting arrangement is shown in

figure [4.5.1], along with the torsional mass-elastic model of this arrangement. It can be

seen that this system comprises two intermediate shafts, a screw shaft, and a fixed-pitch propeller supported by three shaft bearings and a stern tube bearing (not shown in the figure). As can be seen from figure [4.5.1], the straight shaft arrangement is

modelled by representative inertias coupled by a series of equivalent torsional

massless spring stiffnesses.

where:

INTERMEDIATE I INTERMEDIATE 2

?.1

Figure 4.5.1: A typical straight shafting arrangement along with its torsional mass-elastic

model.

The inertias of the shafts are lumped at the connecting flanges. For example, half of the

inertia of shaft 1 is considered at the engine flywheel, and the other half at the

connecting flange to intermediate shaft 2 and so on. Thus, the following inertia stations are identified;

Psi (kg.m2) Inertia of the intermediate shaft 1

1152 (kg.m2) Inertia of the intermediate shaft 2

jps (kg.m2) Inertia of the propeller shaft

Jr (kg.m2) Inertia of the propeller

It should be noted that the inertia of the flanges are also included at each end of the

shaft. PROPELLER SHAFT Station Inertia 1 0.5 1N1 2 0.5. am + Its?) 0.5 .am + Ps) 4 0.5 p + f p 3 .

Delft University of Technology 1996

The torsional stiffnesses of intermediate shaft 1, intermediate shaft 2 and the propeller

shaft are modelled by stiffnesses 1051 , KIs2 and Kin , respectively. These equivalent

stiffness coefficients are determined using the method described by Nestorides [1958]j..

The accelerations of the 'equivalent inertias are determined using the equations from section 4.2. For example the acceleration of station 2 ( intermediate shaft 2) may be

expressed by;

° 1S2

TIM S2

IS2

where: _{0 .0} (rad/s2) torsional acceleration of station 2

T1.5/ (Nm) intermediate shaft 1 torque at LHS of station 2'

TIS2 (Nm) intermediate shaft 2 torque at RHS of station 2 1152 (kg.M2) inertia of station 2

Delft University of Technology 1996 chapter 4

4.6 GEARBOX MODEL

A layshaft gearbox model comprising two, three or four series meshing gears has been

developed for the simulation of branched marine propulsion shafting systems. These

models excluded any losses at the rolling surfaces of the gears. In reality there are

speed dependent losses due to oil churning and bearing housings, and torque

dependent losses due to friction at the rolling surfaces of involute cut gears. The gear

teeth were considered rigid in the gearbox model.

Figure 4.6.1: The torsional mass-elastic model and the free body diagram of a three gear layshaft

gearbox.

The modelling approach is presented in terms of a three gear layshaft gearbox. The

same technique was used for the two and the four gear gearboxes. The torsional mass-elastic model of the three gear geartrain is shown in figure [4.6.1]. As can be seen from

this diagram, the model comprises one speed and three torque degrees of freedom.

The sign convention on the free body diagram of this geartrain, shown on figure

[4.6.1], was used to formulate the equations of motion for individual gears. These

equations may be expressed as;

11.0i =

-

Ft .2 RI

2 F, .1.R,

.J I3. =

-where: J, (kg.m2) mass moment of inertia

(rad/ s2) acceleration of gear

.(Nm) external torque F,, 1+1 (N) internal force .. (4.6.1) .. (1.6.2) .. (4.6.3) F2, I F3,2

10

= /4

The following two equations were derived from the geartrain geometry; where: On (rad) Rn (m) °2 = Je R = .a

-angular displacement of gear gear radius

The acceleration of gear numbers 2 and a is determined by diffe:entiating equations,

(4.6.4) & (4.6.5) twice with respect to time. The acceleration of gear number 1 is then found by substituting for these equations, and rearranging equations (4.6.1) through to

(4.6.5). It should be noted that no losses were taken into account and therefore the, forces in between the gears were the same, that is F72 =F21 and F23=F32. Thus the

acceleration of gear numbers 1 through to 3 may then be expressed as:.

R, ..T R,,_T) R2 2 _R3 3 =

J +J

111112+.1 2 R23 3 14 ) .44.6.6) (4.6.5) .. (4.6:7) ... (4.6,8) Note that due to the single di.gree of speed freedom, the ratio of the,gear accelerations and speeds are in the same ratio as their displacement.

0 ₌

R,

4.7 CLUTCH MODEL

The operation of the clutch was modelled in each of its three working conditions:

engaged, slipping or disengaged. Figure [4.7.1] presents two typical dry friction clutch

examples used in marine propulsion shafting systems. They are the torsionally-stiff

Pneumastar and the torsionally-elastic Pneumaflex clutch, both manufactured by

Lohmann and Stolterfoht.

Ke I le2 INPUT AIR-PRESSURE RUBBER MEMBRAME OUTPUT

Figure 4.7.1: The Pneurnastar and Pneunzaflex dry friction clutches.

The mass-elastic model for an engaged, and a disengaged or a slipping clutch

are

shown in figure [4.7.2]. These models comprise two inertia stations connected by a

torsional spring, which represents the torsional stiffness characteristics of the rubber

membrane in the clutch. In a stiff friction clutch, the value of this stiffness is very high indicating almost a rigid connection.

1c12

Figure 4.7.2: The mass-elastic models of an engaged, disengaged or slipping clutch.

It should be noted that the equivalent mass moments of inertia of the disengaged

clutch halves are different from those specified for the engaged clutch. These inertias can be obtained from the clutch manufacturer.

When the clutch is disengaged, the input member rotates whilst the output member

is at rest. When the clutch is slipping, the torque in between the two halves is

a

function of the clearance between the friction plates between the input and output

member, irrespective of the relative speed between the two members. This clearance is

controlled by compressed air pressure. The slip time of the clutch is set by the

manufacturer for automatic sequencing of the clutch compressed air supply.

le I 1c1

CLUTCH ENGAGED CLUTCH DISENGAGED

Delft University of Technology 1996 chapter 4

The characteristics of the clutch in slipping conditions are accounted for in the

simulation model by specifying both the slip time and the slip torque against the

clearance between the friction plates in between the two clutch members. Therefore, many different types of clutches can be simulated by specifying the characteristics of

the slipping clutch in the simulation model.

The torque generated during the slipping operation will accelerate the stationary half and decelerate the rotating half of the clutch. The acceleration of the clutch, either in engaged, slipping or disengaged position is determined from equation (4.2.4). The acceleration of the input clutch in a Slipping condition is expressed by the following

equation;

(14C ="SHAFT ±

Ttitli

CL in

where:

_art

Jos (rad/ s2> acceleration of clutch input halve

TSHAFT (Nm) input shaft torque

TSLIP (Nm) clutch slipping torque

4.8 EPICYCLIC GEAR MODEL

A fully floating, transient model of an epicyclic gear was developed using Huckvale's [1978] model. Figure [4.8.1] shows a schematic diagram of an epicyclic gear with one

torque and two speed degrees of freedom. As can be seen from this diagram, the

epicyclic gear comprises a planet carrier, three planet gears, a ring and a sun gear. The planet gears are mounted on the planet carrier and mesh with the sun gear at one end, and with the ring member at the other end.

1=1 Ring Plana Sun Caner Fs I

Figure 4.8.1: A schematic representation of an epicyclic gear and its free body diagram.

The epicyclic gear used in a marine type application is illustrated by the constant

speed gear drive system of figure [4.8.2]. This drive consists of a three wheel geartrain,

epicyclic gear, hydrostatic speed control unit, and a constant frequency shaft

alternator. Both an elastic and a tooth coupling mounted on the free end of the

crankshaft isolate the generator and its drive from torsional and axial engine

vibrations.,

Figure 4.8.2: A schematic drawing of a constant speed gear drive system.

t iver it of chn lo 1

Such constant speed gear drive systems are used typically for electrical power

generation at sea from a low speed main Diesel engine directly coupled to a fixed-pitch propeller. The hydrostatic control unit continuously compensates the variable input speed of the main engine to keep the output speed of the epicylcic gear exactly

matched with the required constant electrical supply frequency. For example, a

variable input speed of 50 through to 70 rpm will be converted to a constant 1800 rpm for the 60 Hz shaft alternator.

The gear-ratio of the epicyclic gear presented in figure [4.8.1] may be expressed as;

where: (rad/ s) angular speed of sun 0R

(rad/s)

angular speed of ring

O.

(rad/s)

angular speed of carrier rip =

.. (4.8.1)

where: 11E.

(/)

epicyclic gear ratio

RR (m) ring radius

Rs (m) sun radius

The relationship of the carrier, ring, and sun gear speed was used to define the speed

ratio of the epicyclic. This relationship is expressed as;

.. (4.8.2)

The speed relationships of the epicyclic gear members are given by the following two equations; 1 0 = (ri,. +1) .. (4.8.3) where: 2rir 0 = +1).(rir

0,,

(rad/s)

velocity of a planet

.. (4.8.4)

The free body diagram of the epicyclic gear, presented in figure [4.8.1], is used to

calculate the resultant acceleration of the individual gears As can be seen, only 1

planet is shown, because the other planets are described by the same free body

diagram. The equations of motion for this system are;

-FRI .. (4.85) =Tr RE.FEI ..1(4.8.6) Rs. FR ... (4.8.7) Jo. R. FR, R. FR2 ..(4.8.8) where: Ocg.m2) (rad/ s2), T, (Nm) Fa, 12 (Nm) Rc (m) Rp (in)

mass moment of inertia acceleration of gear external torque

internal force in between gears carrier radius

planet radius

An extra equation is then derived to take into account the motion of the planet about

the rotation axis of the carrier, and it is given by the following equation;,

t

JE.1+2.(rlis '1 le

FR2+R .F +R F

riE 1

1

'' -

".

C C2' S

(4.8.9)

The acceleration of the carrier, planet, ring, and sun gear are obtained by rearranging

equations (4.8.3) through to (4.8.9) into the following form;

jy2

ix. z)

...(4.8.10)

J T

Ty Op Y

(lc

(4:8.11) R =

-,

4996 chapter 4_

Jz = J

Tr = 2ri ( -

ft, -1

+6,

2 rih. ri 11)2

+4

..0.8.16) (4.8.17)

The acceleration of the epicyclic gear members is determined assuming no losses, that

is Fo=Fo, FR7=FR2 and Fi57=F52 . The epicyclic gear model, excluded any losses at the

rolling surfaces. In reality there are speed dependent losses due to the bearing

housings, and torque dependent losses due to friction at the rolling surfaces of gears.

A time domain simulation program has been developed for the simulation of an

epicyclic gear under transient operating conditions. This program was used for a

simple transient simulation of an epicyclic gear, and the results of this simulation are

presented in appendix 3.

with

J

_{X -} R

+J

C

Jr =

+J +J

Si (rig I) 2) _0.8.12) .. (4.8.13) ...(4.8.14) ,..(4.8.15) 2 .) ' 1+ 2

Li' +1

[

P ri,,. - I

)

-( 2 2ri ri

J

T1 = T 2 (4.8.18)

4.9 SHIP MODEL

The propeller excitation the propeller torque and thrust is obtained from the KT, KQ

-J. diagram of an equivalent Wageningen B-screw series propeller. Both the propeller torque and thrust are obtained from this diagram for the advance coefficient J at the

propeller operating point. The advance coefficient at this point is also a function of the

ship speed. Therefore, a simple model of a ship has been developed to determine the

ship speed at the propeller operating point.

The mass of the ship can be determined from its dimensions using the following

equation;

ms/rio =pCLBT

.. (4.9.1) where: where: (kg) Ch

(/)

Lp (m) 13, (m) (rn) as (m/s) Vs (m/s) T (N) td

(/)

R (N) Ica

(/)

ship mass block coefficient

length between perpendiculars breadth moulded

design draught moulded

The acceleration of the ship is computed from the propeller thrust and the ship

resistance. A ship resistance curve is used based on a quadratic ship velocity, i.e. R a. V52. The acceleration of the ship is determined from the following equation;

dV,

T.(I - td) - R

a.,

-dt tn.( + ) .. (4.9.2) acceleration of ship ship speed propeller thrust

thrust deduction factor resistance of ship entrained mass factor

Mship

=