ittc symbols and terminology list -...

ITTC Symbols

Version 1993

International Towing Tank Conference

ITTC Symbols and Terminology List

Version 1993

VWS Mitteilungen Heft 57

_______________________________________

Versuchsanstalt für Wasserbau und Schiffbau, Berlin

ITTC Symbols

Version 1993

Versuchsanstalt für Wasserbau und SchiffbauVWS: the Berlin Model Basin

Mueller-Breslau-Strasse (Schleuseninsel)D-10623 Berlin, Germany

Phone +49-30-311 84-0, Fax +49-30-311 84-200

Bibliographical reference:

ITTC Symbols and Terminology List, Version 1993Prepared by ITTC Symbols and Terminology Group

Berlin: VWS, Mitteilungen, Heft 57 (1993)Edited by Michael Schmiechen

Short reference:

ITTC Symbols 1993, VWS Mitt. 57

ITTC Symbols Contents

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Contents

0 Introduction 10.1 Symbols and Terminology Group 1

0.1.1 Terms of Reference 10.1.2 Activities 20.1.3 Membership 4

0.2 List of Symbols 50.2.1 Classification 50.2.2 Structure 60.2.3 Organization 7

0.3 Principles of Notation 70.3.1 Objects: Quantities 70.3.2 Components: Subscripts 80.3.3 Operators: Superscripts 8

0.4 Details of Notation 100.4.1 Standard Symbols 100.4.2 Computer Symbols 100.4.3 Names, Definitions, SI-Units 11

0.5 References 110.5.1 ITTC Documents 110.5.2 Translations 120.5.3 Other References 12

1 General Mechanics 141.1 Fundamental Concepts 14

1.1.1 Concepts in General 141.1.1.1 Basic Concepts 141.1.1.2 Balances 161.1.1.3 Remarks 17

1.1.2 Space Related Concepts 201.1.2.1 Basic Concepts 201.1.2.2 Remarks 22

1.1.3 Time Related Concepts 251.1.3.1 Basic Concepts 251.1.3.2 Complex Transforms 261.1.3.3 Complex Quantities 271.1.3.4 Remarks 27

1.1.4 Random Quantities and Stochastic Processes 311.1.4.1 Random Quantities 311.1.4.2 Stochastic Processes 321.1.4.3 Probability Operators 331.1.4.4 Remarks 34

1.2 Solid Body Mechanics 351.2.1 Inertial properties 35

1.2.1.1 Basic Concepts 351.2.1.2 Remarks 36

1.2.2 Loads 37


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1.2.2.1 External Loads 371.2.2.2 Sectional Loads 381.2.2.3 Remarks 38

1.2.3 Rigid Body Motions 401.2.3.1 Motions 401.2.3.2 Attitudes 411.2.3.3 Remarks 42

1.3 Fluid Mechanics 431.3.1 Flow Parameters 43

1.3.1.1 Fluid Properties 431.3.1.2 Flow parameters 431.3.1.3 Boundary conditions 441.3.1.4 Remarks 44

1.3.2 Flow Fields 461.3.2.1 Velocities etc 461.3.2.2 Circulation etc 471.3.2.3 Remarks 47

1.3.3 Lifting Surfaces 491.3.3.1 Geometry 491.3.3.2 Sectional coefficients 49

1.3.4 Boundary Layers 511.3.4.1 Twodimensional Boundary Layers 511.3.4.2 Remarks 52

1.3.5 Cavitation 531.3.5.1 Flow parameters 531.3.5.2 Flow field 53

1.4 Environmental Mechanics 551.4.1 Waves 55

1.1.4.1 Periodic waves 551.4.1.2 Irregular waves 561.4.1.3 Time Domain Analysis 571.4.1.4 Frequency Domain Analysis 571.4.1.5 Directional Waves 581.4.1.6 Remarks 59

1.4.4 Ice Mechanics 61

2 Ships in General 622.1 Geometry and Hydrostatics 62

2.1.1 Hull Geometry 622.1.1.1 Basic Concepts 622.1.1.2 Derived concepts 642.1.1.3 Computer Symbols for Attributes 662.1.1.4 Remarks 67

2.1.2 Propulsor Geometry 682.1.2.1 Screw Propellers 682.1.2.2 Ducts 70

2.1.3 Appendage Geometry 722.1.3.1 Basic Concepts 72


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2.1.3.2 Identifiers for Appendages 732.1.3.3 Remarks 74

2.1.4 Hydrostatics 752.1.4.1 Stability levers 752.1.4.2 Various Concepts 772.1.4.3 Remarks 78

2.2 Resistance and Propulsion 792.2.1 Hull Resistance 79

2.2.1.1 Basic Concepts 792.2.1.2 Derived Concepts 802.2.1.3 Computer symbols for attributes 82

2.2.2 Ship Performance 832.2.2.1 Basic Concepts 832.2.2.2 Derived Concepts 842.2.2.3 Efficiencies etc 852.2.2.4 Remarks 86

2.2.3 Propulsor Performance 872.2.3.1 Basic Concepts 872.2.3.2 Derived Concepts 882.2.3.3 Induced Velocities etc 902.2.3.4 Remarks 91

2.2.4 Unsteady Propeller Forces 922.2.4.1 Concepts 922.2.4.2 Remarks 93

2.3 Manoeuvring and Seakeeping 942.3.1 Manoeuvring 94

2.3.1.1 Geometrical Concepts 942.3.1.2 Motions and Attitudes 942.3.1.3 Flow Angles etc 962.3.1.4 Forces and Derivatives 972.3.1.5 Linear Models 982.3.1.6 Turning Circles 992.3.1.7 Zig-Zag Manoeuvres 1002.3.1.8 Stopping Manoeuvres 1012.3.1.9 Remarks 101

2.3.2 Seakeeping 1022.3.2.1Basic Concepts 1022.3.2.2 Remarks 104

3 Special Craft 1053.1 Ice Going Vessels 105

3.1.2 Resistance and Propulsion 1053.2 Planing Boats 107

3.2.1 Geometry and Hydrostatics 1073.2.2 Resistance and Propulsion 110

3.3 Semi-displacement Vessels 1113.3.1 Geometry and Hydrostatics 1113.3.2 Resistance and Propulsion 115


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3.4 Catamarans 1213.4.1 Geometry and Hydrostatics 1213.4.2 Resistance and Propulsion 123

3.4.2.1 Resistance Components 1233.4.2.2 Power and Resistance Ratios 1233.4.2.3 Remarks 124

3.5 SWATH 1253.5.1 Geometry and Hydrostatics 125

3.6 Hydrofoil Boats 1273.6.1 Geometry and Hydostatics 1273.6.2 Resistance and Propulsion 130

3.7 ACV and SES 1323.7.1 Geometry and Hydrostatics 1323.7.2 Resistance and Propulsion 134

3.8 Sailing Vessels 1363.8.1 Geometry and Hydrostatics 1363.8.2 Resistance and Propulsion 138

Computer Symbol Index 140

ITTC Symbols 0 Introduction

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0 Introduction

0.1 Symbols and Terminology Group

0.1.1 Terms of Reference

In May 1985 the Executive Committee of the 18th International Towing Tank Conference (ITTC)reorganised the former Information Committee, earlier Presentation Committee, to form a Symbolsand Terminology (SaT) Group in the newly established ITTC Secretariat.

The task of the SaT Group for the 18th ITTC was to carry out Recommendations 1 through 5,related to the ITTC Standard Symbols, of the Information Committee of the 17th ITTC, whichwere:

1. The Information Committee should continue to monitor and co-ordinate the developmentof new symbols by the Technical Committees.

2. The Conference should adopt the new symbols for hydrostatics included in Appendix 4and the Information Committee should then include these in the ITTC Standard Symbols.

3. The Information Committee should restructure the ITTC Standard Symbols according tothe outline Proposal in Appendix 6 and include new symbols agreed by the TechnicalCommittees.

4. The Information Committee should continue to revise the Dictionary of ShipHydrodynamics as required.

5. The Information Committee shoud continue cooperation with other organizations toachieve a common agreement on symbols and terminology.

The 18th ITTC at Kobe adopted the following Recommendations to the Conference and for thefuture work of the SaT Group, respectively, related to Symbols:

Recommendations to the Conference:

1. The Conference should adopt the structure of the ITTC standard Symbols and Termino-logy List outlined by the Symbols and Terminology Group and used as the basis for the1987 Draft List distributed at the 18th ITTC in Kobe.

2. The Conference should urge the Technical Committees and individuals to contribute tothe completion of the List of Standard Symbols and should encourage the use of thesymbols and their further development in cooperation with the Symbols and TerminologyGroup.

3. The Conference should decide to delay the review and update of the ITTC Dictionaryof Ship Hydodynamics and the official translations of this into principal languages until thefinal Symbols and Terminology List is published in 1990.


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Recommendations for the future work of the Group:

1. The Symbols and Terminology Group should continue cooperation with otherorganizations to achieve a common agreement on symbols and terminology.

2. The Symbols and Terminology Group should continue to monitor and coordinate thedevelopment of new symbols and terminology by the Technical Committees of the ITTC.

3. The Symbols and Terminology Group should complete the ITTC Standard Symbols andTerminology List based on the 1987 Draft distributed at the 18th ITTC and distribute thefinal version with Volume 1 at the Proceedings of the 19th ITTC.

Finally the 19th ITTC at Madrid adopted the following recommendations related to symbols:

Recommendations to the Conferences:

The 1990 version of the List of Standard symbols should be used as a working documentwithout the formal approval of the Conference.

Recommendations for the future work of the Group:

The SaT Group to put the computer compatible symbols on a more rational basis in orderto make them useful for data exchange puposes.

0.1.2 Activities

The SaT Group took up its work immediately after it was established having its first meeting atWageningen in October 1985, and coming up with the plan to produce the present draft of arestructured and enlarged list of the ITTC Standard Symbols 1987. The first raw draft wasdiscussed at Berkeley in July 1986, the Draft 1987 published at the 18th ITTC in October 1987at Kobe by the Society of Naval Architects of Japan, having been finalized at Trondheim in June1987.

Work on various chapters has been continued by the 18th ITTC SaT Group and the results havebeen distributed to the Technical Committees at the Kobe Conference together with the printedDraft 1987.

The SaT Group of the 19th ITTC continued work on the Standard Symbols during meetings atGenova in March 1988, at the Hague and Berlin in September 1988 and in August 1989 atTrondheim, the 1990 version being completed at Genova in March 1990.

During this work, new and more rigorous requirements resulting from the proposed use of thesymbols in validation work and in data bases caused a reconsideration of the fundamental aspects.Duplication of computer symbols had to be carefully traced and avoided, in order to permitautomatic handling of symbols in data bases.

In order to facilitate the handling of the List of Symbols the earlier version was retyped as a series


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of WordPerfect files, which were available much too late for updating and were printed withouteven having been proof read! Consequently, the goal of finalizing the symbols list before the 19thITTC at Madrid could not be reached. From the document itself it is evident that was less thana draft.

The SaT Group of the 20th ITTC met at Madrid in September 1990, at Berlin in June 1991, atNewcastle in May 1992 and at Genova in January 1993. The primary task after many years offrustrations with the computerized list of symbols was to finally establish a computer implement-ation permitting direct expert corrections on a PC.

After the previous transscription into the WordPerfect files using the tabulator function the solutionwas achieved by transformation to the table format. With the appropriate tools being available afterall the next task tackled was to correct all the misprints and to implement all the improvementssuggested by colleagues of member organisations and members of the SaT Group. The List ofSymbols as printed is now available on a floppy discs using the format of a WordPerfect 5.2master document with the appropriate subdocuments, being themselves master documents accordingto the structure of the list.

The main concern after this still rather traditional approach was to achieve the goal set out in theRecommendations for the future work of the SaT Group, to put the computer symbols on a morerational basis. And it soon became evident that the accomplishment of this task could only beachieved by rigorously following the object oriented paradigma applied earlier in restructuring theList of Symbols. The basic principles and rules behind this work are outlined in the followingchapters of this preface.

Two problems had to be solved: to maintain the traditional, in many ways inconsistent "Standard"Symbols as an accepted interim and suggest new consistent symbols as alternatives. Some of theseare already used in computer work and SaT Group feels that due to their efficiency they willsooner or later completely replace the traditional symbols as has the system of SI-Units thetraditional systems.

In view of the increasing demands concerning quality assurance systems the SaT Group felt thatthe ITTC Symbols should no longer be called Standard Symbols as this name implies legalobligations, which are not existent. The International Standard Organisation and correspondingnational organisations may at a later stage take measures to adopt the ITTC Symbols as a Standardas was already intended with the earlier version; s. 0.5.3.

During the work to rationalize the computer compatible symbols for use in databases etc the SaTGroup became aware of a number of related efforts on an even more general level, which need tobe taken into account in the further development of the ITTC Symbols. As documented in theGroup Report to the 20th ITTC the development and and application of terminological databasesis dramatically increasing and has lead to a number of specialized workshops and symposia.

In the broadest sense terminological databases are basic for computer aided knowledge and scienceengineering, which are developing at a breath taking pace. In order to meet the forthcomingrequirements the ITTC Symbols will have to be further rigorously rationalized. Compared to thisformidable task, which has only been started with the new object oriented structure of the SymbolsList, the transformation from the present table format into one of the rapidly developing


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terminological database formats appears to be a minor task.

The software systems presently available do still not meet very basic requirements, as did the wordprocessors up to now, absorbing too much of the energy of the SaT Group which should have beendevoted to the symbols proper. While the problem of producing customized lists of symbols canbe solved rather easily, the much more interesting problem of deriving consistent submodels fromthe general models of the complete list needs still much more development work.

At this stage, it is appropriate to acknowledge with thanks the tremendous work done by the formerPresentation and Information Committees and the Technical Committees in their respective fields.It is only on the basis of their work that the task of the SaT Group could have been undertakenand can be carried on. Last but not least a word of thanks is due to the great number of typistswho have at all stages contributed to the actual production of the document.

All the ITTC Communnity, the Technical Committees in particular are invited to contribute to thecontinuing task of updating and further improvement.

0.1.3 Membership

The membership of the SaT Group as appointed by the 18th ITTC Executive Committee in May1985 and re-appointed by the 19th ITTC Executive Committee in October 1987 is as follows:

Prof. Bruce Johnson (Chairman)Naval Architecture, Ocean and Marine Engineering Department U.S. Naval AcademyAnnapolis, MD 21402USAPhone +1 410 267 3871Fax +1 410 267 2591

Dr. David ClarkeDepartment of Marine TechnologyUniversity of Newcastle upon TyneNewcastle upon Tyne, NE1 7RUUnited KingdomPhone +44 91 222 6721Fax +44 91 261 1182

Dr. Norihiro MatsumotoElectronics Research CenterNippon Kokan K.K.1-1 Minamiwatarida-cho, Kawasaki-kuKawasaki 210JapanPhone +81 44 322 6276Fax: +81 44 322 6523


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Prof. Carlo Podenzana-Bonvino (Secretary)Dipartimento di Ingegneria Navale e TecnologieMarine (DINAV), Universita di GenovaVia Montallegro 116145 GenovaItalyPhone +39 10 353 2426/2430Fax +39 10 353 2127

Prof. Michael SchmiechenVersuchsanstalt für Wasserbau und SchiffbauMüller-Breslau-Strasse (Schleuseninsel)10623 BerlinGermanyPhone +49 30 311 84 270Fax +49 30 311 84 200

Most of the members were re-appointed again by the Executive Committee in September 1990, Dr.Matsumoto being followed by:

Prof. Michio Nakato4-28-7 Kameyama-MinamiAsa-KitaHiroshima 731-02JapanPhone +81 82 814 5857Fax +81 82 814 7285

and as a new member was appointed:

Dr. Kostadin YossifovBulgarian Ship Hydrodynamics Centre9000 VarnaBulgariaPhone +359 52 776390Fax +359 52 772267

0.2 List of Symbols

0.2.1 Classification

The prime concern in setting up a revised and enlarged list of ITTC Standard Symbols was todesign an adequate system for the classification of concepts. As soon as the work started it becameclear that the outline proposed by the Information Committee of the 17th ITTC (Proc. 17th ITTC(1984) Vol.1, p.56) had to be reconsidered in view of the problems encountered.

Subsequently the following design requirements and goals have been established:


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1. produce a coherent document, meeting the present and possibly the future requirementsof the ITTC community in general and particular user groups

2. establish an open ended matrix structure that can be easily expanded as requirementsarise, without the need of restructuring and repetition or too many explicit cross-references

3. minimize departures from the well established and widely accepted previous list ofsymbols

After a series of attempts to meet these requirements the structure as listed in the table of contentsevolved very much in line with the past development of the symbols, for instance by the HighSpeed Craft Committee and others. The essential features are the subject areas of rather limitedscope, organized in an hierarchical order. Ideally each subject area represents a complete andcoherent model of that area under consideration, for example rigid body motion, hull geometry,propulsor performance.

0.2.2 Structure

The concepts related to a given subject area or model are designated by the ITTC Symbol andcalled by their Name. Their meaning can in principle only be concluded from the context of themodel, that is by coherent, so called 'implicit' definitions, to be derived from an explicit statementof the model, ideally an axiomatic system or any equivalent, for example a drawing.

The problem is that traditionally in lists of symbols as in dictionaries these explicit models aremissing for various reasons. One reason is that many subject areas under discussion are far frombeing developed and understood to the extent necessary. A consequence of this situation is that thesymbols proposed are not always as coherent as is necessary for advanced and systematic work,where the explicit models and adequate notations area are a prerequisite.

The problem under discussion is of course the same in national and international standards. Inorder to avoid the dilemma indicated, the ITTC Symbols should not only perpetuate past practiceand jargon but try to take the lead and step forward. This is particularly important in view of thedevelopment trends in marine technology. In a rapidly changing world adequate tools areprerequisite for efficient problem solving.

As expert system and knowledge engineering technologies evolve the importance of adequatesymbols and terminology is more widely acknowledged. The training of scientists working in theterminology field is being offered by the standards organizations. Some of these activities havebeen monitored but are felt to be lacking in clear-cut rules which may be readily understood andapplied in practice.

The original idea to add indices of symbols and names to the document had to be delayed as longas adequate tools were missing. Now such an undertaking is felt to be still premature at the presentstage, as it requires the resolution of a number of additional problems, such as standardisation ofnames.


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0.2.3 Organization

As has been emphasized the development of symbols is a continuing process and as the subjectdevelops, further amendments and additions, as approved by the Conference, will be included infuture editions of the list.

In order to avoid any extra problems the symbols are arranged in alphabetical order in each subjectarea as in previous lists. Continuous page numbering was discarded in earlier versions. The ideawas to establish a loose leaf organization as the most appropriate, in view of new drafts to beincorporated.

In view of the extremely powerful modern word processing systems the whole idea was discardedand advantage was taken of the indexing cababilities etc. permitting efficient production of realupdates including in future additional explanations and sketches or drawings related to particularsections where necessary, and as found in national and international standards.

But in view of the tremendous effort which explicit mathematical models, explanations, andsketches take for their preparation, the present SaT Group has only started to consider guidelinesfor these additions and has added only few examples of explanations to the present list. TheTechnical Committees and other interested parties are urged to provide further material for reviewby the SaT Group and future inclusion into the list.

It has been noted by the SaT Group that some users dislike the disruption of the list of symbolsby lengthly explanations. But the Group feels that the complexity of the subject and the sensibleuse of the symbols require such explanations, the more so as the fundamentals of the theory ofscience and terminology are not taught to students of naval architecture and marine engineering.

0.3 Principles of Notation

0.3.1 Objects: Quantities

Standard notations have to be adequate for the problems to be dealt with and preferably have tobe operational.

In general there is a body b, e. g. ship S or model M, in space s, referred to coordinates c withorigin o, and time t of which the values q of quantities of certain physical qualities Q are ofinterest, i. e.

q = Q (b, s, c, o, t) ,

q is a variable for numerical values of quantities, while Q is a variable for functors constants,quantities of qualities, e. g. of inertia, momentum, or energy.

In many cases the quantities in question are components of vectorial or tensorial quantities; andshould be denoted accordingly, s. 0.3.2.

Further, quite often various aspects of the same quantity are of interest, for example their spectra


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or aspects of those, in simpler cases just their expectation or estimates of these, e. g. time averages,all of them to be carefully distinguished; s. 0.3.3.

It should be evident, that the requirements concerning an adequate, operational notation are quitedemanding. At the same time it should be understood that it is worthwhile to create such anotation, as waste of effort due to confusion of concepts may be reduced drastically.

The question is of course how far one wants to depart from current practice in order to cope withthis situation. The example of the standard notation used in chemistry may serve as a guideline.

In the present context, the typical objects or "elements" referred to are the values of quantities intime or "signals". Consequently the symbols for the signals should be the primary symbol andcomponents and transforms should be denoted by sub- and superscripts, respectively.

0.3.2 Components: Subscripts

In view of vector and tensor components, it is felt that it is appropriate to introduce a simple tensornotation at least for orthogonal coordinates. This helps to limit the number of symbols as it requiresonly one symbol for the particular set of components in question. For example the various, say atleast two times thirty six "stability derivatives", i. e. generalized mass and damping, need not andcannot be introduced individually.

If vector or tensor components, in general matrix components are conveniently denoted bysubscripts, the above situation thus becomes in more general terms

qij = Qij (b, s, c, o, t).

Numerical subscripts are truly operational in most algorithmic languages, which can handlematrices, usually called one-, two-, or three-dimensional arrays.

0.3.3 Operators: Superscripts

Superscripts are traditionally used for exponentiation but can be generally used to denote operators;the most satisfactory approach being the inverse Polish notation.

The advantage of this notation is that no brackets are required and operators are listed exactly inthe sequence in which they are applied to the signal. As has been done with the matrix notationearlier this notation may in future be readily rendered operational in advanced softwareenvironments, object oriented languages in particular.

For convenience the computer symbols and symbols used in data bases should exactly reflect thisnotation in order to avoid any extra problems of translation. Consequently the earlier proposedprefixes in the computer symbols have been changed to suffixes. As an example the real part ofthe heave spectrum may be denoted as follows:


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standard computer data base XSR

3 XSR(3) or X_SR(3) or XxSpRe(3) XSR3 or X_SR3 or XxSpRe3

The main problem in any case is to define symbols for operations and not for the results of theoperations. In order to have the most compact notation agreement should be reached concerninga one character notation, and a corresponding two character notation for the computer symbols, forwell defined operations.

Due to the fact that it has not been possible to define symbols for concepts, qualifiers, operatorsetc uniformly in terms of two characters the above example show the presently used techniquesto introduce separators. X and Xx denote symbol variables, to be replaced by symbols proper inany particular application.

If necessary the meaning of a operator symbol may depend on the context, i.e. its position withrespect to others and the object it operates upon. This generic use of symbols is of course veryefficient, but needs special care not to confuse concepts.

It is most important to note that in any case definitions of concepts or operations should not beconfused with operational definitions, i.e. methods for determination or measurement of values.Separate identifiers have to be introduced in order to avoid confusion. A whole hierarchy of suchoperators and qualifiers is necessary.

Some 'operator' symbols are proposed in the following chapter on fundamental concepts. Theyconcern

1. identifiers of the object being tested, e. g. ship S or model M, or the various bodiesin a multi-body problem,

2. identifiers of coordinate systems and of the reference points, not only forward and aftperpendicular,

3. the various aspects of complex quantities,

4. the various aspects of spectra and

5. the various aspects of random quantities and stochastic processes.

So far no particular identifiers have been introduced for various estimators. As an example thepower spectra of stationary random processes may be estimated using Fourier techniques, as agreedupon by the oceanographic institutes world wide, or by autoregressive model techniques, avoidingsystematic i. e. bias errors inherent in the first technique. Another example is the interpretation ofthe conceptual frame-work of hull-propeller interaction based on propulsion, hull resistance, andpropeller open water tests or from the results of propulsion tests alone.


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0.4 Details of Notation

0.4.1 Standard Symbols

The symbols in the first column of the tables are primarily intended for use in technical writingand mathematical expressions. The following notes are relevant:

1. All symbols, their subscripts, and superscripts should be written as shown.

2. In a number of instances alternative symbols are given.

3. In many cases the symbols, their sub- and superscripts denote variables to be replacedby symbols for any object, component and qualifier or operator, respectively.

4. Where for one reason or another departures from the standard symbols are made, thesedepartures should be clearly indicated and stated.

0.4.2 Computer Symbols

Wherever possible the symbols in the second column of the tables have been chosen so that theirmeaning is readily apparent. They have been constructed from the CCITT International TelegraphAlphabet, restricted character set. They are therefore suitable for use in a wide range of situationse. g.: Telex messages, letters, computer printouts etc.

To ensure that the symbols can be used in a wide range of programming languages they currentlyhave been kept to less than six characters long. The symbols should be used as defined, and, inaccordance with modern programming practice, should have their type explicitly declared beforeuse. The following rules were applied in the derivation of the symbols:

1. Only upper case letter A - Z and digits 0 - 9 have been used.

2. Formerly Greek letters have been spelled out, if necessary in abbreviated form or withchanged spelling. This practice is considered obsolete.

3. The Froude 'circular' symbols are defined by the prefix CIRC.

4. All symbols start with a letter.

5. Qualifiers and operators, preferably two characters, are currently suffixed to the mainsymbol line, without spacing.

6. No one computer compatible symbol should be used for different concepts in a givencontext. This goal has not been completely achieved for the whole list. Ad hoc solutionshave been attempted but discarded as unsatisfactory.

7. Since the computer compatible symbols have been proposed as the basis of attributenames for data exchanges, the above rules will probably be further developed in the near


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future.

A final remark on the Computer Symbols: in the computer, the letter O and figure 0 (zero) havefundamentally different meanings, but owing to their resemblance they can be easily confused.Thus it is necessary to distinguish rigorously between them. As a matter of fact there arecontradictory conventions being widely used.

0.4.3 Names, Definitions, SI-Units

The third column in the tables contains the names of the concepts denoted by the symbols in thefirst and the second columns, while the fourth column usually contains a definition, or a shortexplanation where necessary. The last column gives the SI-Units for the concepts.

The dimensions of dimensionless quantities as well as their units are 1. They are measured incounts or "absolute units", which sometimes are given names, e.g. rad, rev, but this practice, usualin natural languages, is found to be not very useful in formal systems.

A number of concepts and their symbols are customarily defined and/or standardized differentlyin different fields of application. The SaT Group cannot resolve all of these discrepancies, buturges that in such cases the definitions and the units used are stated. Only a few examples havingbeen discussed may be mentioned.

While the SI-Units of angle and velocity are rad and meter/second, respectively, the traditionalunits degree and knot are still widely used and clearly this situation will not change in the nearfuture. In the spectral description of real deterministic or stochastic processes spectra and powerspectra, respectively may be defined as double- or single-sided as functions of frequency or circularfrequency. Any of these difinitions has its particular advantages, but has to be clearly distin-guished from the others.

A major step towards an unambiguous definition of the phase angle has been taken by explicitelydistinguishing phase lead and lag of complex quantities. Despite the fact that both have oppositesigns they are confused even in mathematically oriented standard textbooks!

0.5 References

0.5.1 ITTC Documents

1. International Towing Tank Conference, Standard Symbols 1971.BSRA Technical Memorandum No.400, August 1971.

2. International Towing Tank Conference, Standard Symbols 1976.BSRA T.M. No.500, 1976.

3. ITTC Dictionary of Ship Hydrodynamics.RINA Maritime Technology Monograph No.6, 1978.


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4. Translation of Overall Index of Titles of Dictionary of Ship Hydrodynamics.Vol. 1: CETENA, Genova, 1984,

Vol. 2: University of Tokyo, 1984.

5. Bibliography and Proposed Symbols on Hydrodynamic Technology as Related Model Tests of High Speed Marine Vehicles.

Prep. by 17th ITTC High-Speed Marine Vehicle Committee. SPPA Maritime Research and Consulting. Rep. No.101, 1984.

0.5.2 Translations

A number of translations of the List of ITTC Standard Symbols into languages other than Englishhas been made including French, German, Italian, Japanese, Russian, Spanish and Chinese. Forobvious reasons these translations are no longer up-to-date as the present accepted list in English.

1. French Translation of ITTC Standard Symbols 1971.Association Francaise de Normalisation (AFNOR).

2. International vereinbarte Buchstabensymbole und Bezeichnungenauf dem Gebiet der Schiffshydrodynamik. Collatz, G.Schiff und Hafen 27 (1975) No.10.

3. Italian Translation of ITTC Standard Symbols 1971. Luise E.Appendix II, Report of Presentation Committee.Proceedings 14th ITTC, Vol. 4, Ottawa 1975.

4. Japanese Translation of ITTC Standard Symbols.Transactions of the Society of Naval Architects of Japan, No.538, April 1974.

5. Russian Translation of ITTC Standard Symbols 1971.Brodarski Institute Publication No.28, Zagreb 1974.

6. Simbolos Internacionales en Arquitectura Naval.Asociacion de Investigacion de la Construccion Naval, Publication 7/75, Juli 1975, Madrid.

7. Report of Information Committee, Proc. 17th ITTC, Göteborg 1984.

8. Chinese Translation of ITTC Standard Symbols.China Ship Scientific Research Centre, Wuxi.

0.5.3 Other References

Apart form the organizations represented on the ITTC these symbols have been recommended foruse in technical writing on naval architecture by a number of organizations concerned with marinematters including The Royal Institution of Naval Architects, the American Society of Naval


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Architects and Marine Engineers and the American, British, Canadian, Australian, and ItalianNavies.

In 1985 the Draft International Standard ISO/DIS 7463 Shipbuilding - Symbols for ComputerApplications - has been published. The symbols are based on the list approved by the ITTC inOttawa 1975 and a related list produced by the ISSC in 1974, inconsistencies having beenremoved. The ISO/TC8/SC15 has been notified that major changes of the ITTC Symbols are underdiscussion. Subsequently processing of ISO/DIS 7463 has not been postponed, but the standardhas been published as ISO 7463 in 1990.

ITTC Symbols 1 General Mechanics1.1 Fundamental Concepts

Version 1993 1.1.1 Concepts in General 14

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Name Definition orExplanation

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1 General Mechanics

1.1 Fundamental Concepts

1.1.1 Concepts in General

1.1.1.1 Basic Concepts

s. Remark .1

a, a1 AC, A1 Linear or translatoryacceleration

dv / dt m/s2

A A, AR,AREA

Area in general m2

B B, BR Breadth m

C, FF2 FF(2) Cross force Force normal to lift and

drag (forces)N

D, FF1 FF(1) Drag (force) Force opposing translatory

velocity, generally for acompletely immersedbody

N

d, D D, DI Diameter m

E E, EN Energy J

f FC Friction coefficient Ratio of tangential forceto normal force betweentwo sliding bodies

1

f FR Frequency 1 / T Hz

F, F0 F, F0 Force N

g G, GR Specific gravity force W / m, strength of the earthgravity field, alias accel-eration of gravity

m/s2

h DE Depth m

H H, HT Height m

I I, IN Moment of inertia Second order moment of amass distribution

kg m2

L L, LE Length m



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L, FF3 FF(3) Lift (force) Force perpendicular to

translatory velocityN

m M, MA,MASS

Mass kg

M, F1 M1, F1 Moment of forces First order moment of aforce distribution

Nm

M MO Momentum Ns

n, N FR, N Frequency of revolution Alias RPM! Hz

P P, PO Power W

Q VF Volume flow rate m3/s

r, R RD Radius m

R, FF1 R, RE, FF(1) Resistance (force) Force opposing translatory

velocityN

s SP Length along path m

t TI Time s

t TE Temperature K

T TC Period Duration of a cycle of arepeating or periodic, notnecessarily harmonicprocess

s

U U, UN Undisturbed velocity of afluid

m/s

v, V1 V, V1 Linear or translatoryvelocity of a body

ds / dt m/s

V VO Volume m3

w WD Weight density, formerlyspecific weight

dW / dV N/m3

W WT Weight (force), gravityforce acting on a body

N

γ MR Relative mass or weight Mass density of a sub-stance divided by massdensity of distilled waterat 4°C

1



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η EF, ETA Efficiency Ratio of powers,s. Remark .2

ρ DN, RHO Mass density dm / dV kg/m3

τ ST, TAU Tangential stress Pa

λ SC Scale ratio Ship dimension divided bycorresponding modeldimension

1

σ SN, SIGS Normal stress Pa

ω FC, OMF Circular frequency 2 π f 1/s

ω, V0 V0, OMN Angular velocity 2 π n rad/s

1.1.1.2 Balances

s. Remark .3

q QQ Quantity of the qualityunder consideration storedin a control volume

QU

Q Quality under consideration QU/s

QC QCF Convective flux QU/s

QD QDF Diffusive flux QU/s

QF QFL Total flux across thesurface of the controlvolume

Inward positive! QU/s

QM QDM Molocular diffusion QU/s

QP QPN Production of sources in thecontrol volume

QU/s

QS QRT Storage in the controlvolume, rate of change ofthe quantity stored

dq / dt QU/s

QT QDT Turbulent diffusion QU/s



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1.1.1.3 Remarks

.1 Greek Symbols

For traditional reasons the computer symbols of the concepts denoted by Greek ITTC Symbols doin general not refer to the concepts, but rather to the Greek symbol. This state of affairs is morethan unsatisfactory. The SaT Group feels that at the present stage it may be time for a radicalchange.

An example is the efficiency, the universally accepted symbol being the Greek η. The computersymbol should of course be EF, instead of ETA.

Another example is the traditional symbol ω for circular frequency and angular velocity. Clearlythe computer symbols FC and V0, respectively, or similar would be much more reasonable than thetraditional symbols listed.

.2 Velocities, Forces

In the following sections more general concepts are proposed, which permit an even more rationalapproach. Appropriate symbols for the linear and the angular velocity would be v1 and v0 ,respectively, in precisely that order! In terms of the generalized velocity v, the complete motionwith six degrees of freedom, the components of the angular velocity are then uniquely denoted byv0

i = v3+i with i = 1, 2, 3 and 'resulting' in the the computer symbols V0(I) = V(3+I), again withI = 1, 2, 3; s. the following section on space related concepts and the section on rigid body motions.

Concerning the hydrodynamic forces acting on a body due to translatory motion only the rationalcomputer symbols are given. As a matter of fact this type of notation is used more and more invarious applications. The advantages need not to be elaborated upon.

.3 Efficiencies

The concept of efficiency or factor of merit is that of a ratio of powers, preferably powers proper,but sometimes virtual powers are considered as well. The most appropriate notation for efficiencieswould therefore be the following with two indices, namely the identifiers of the two powers put intoproportion, i. e.

ηXY = PX / PY .

This notation together with the computer notation ETXY would of course greatly improve the datahandling as it is truly operational.



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.4 Balances

Traditionally balances of various extensive or socalled "conservative" qualities or properties aredescribed by ad hoc symbols, disguising the similarities and essentials. For any quality Q enclosedin a control volume the balance may be written in the format

QS = QF + QP ,

implying, that the net storage of the quality in a given boundary equals the net flux of the qualityacross the boundary into the control volume and the net production of sources within the boundary.

The symbol Q is the variable for the symbol of the particular extensive qualitiy under investigation,e. g. mass, momentum, and energy. QS, QF, and QP are variables for values of the storage, flux, andproduction, respectively.

The net storage is nothing else but the net rate of change of the quantity q of the quality Q storedin the control volume:

QS = dq / dt .

q is the variable for values of the quantity of the quality Q stored in the control volume.

Concerning the flux there are two types to be clearly distinguished according to their mechanisms,the convective and the diffusive fluxes, i. e.

QF = QC + QD .

The diffusive flux itself may be due to two types of diffusion, the molecular diffusion and theturbulent diffusion, i. e.

QD = QT + QM .

Traditionally the time rate of change is denoted by a dot, i. e.

dq / dt = q

According to some standards, e. g. the German DIN, fluxes and the productions may be denoted bysymbols with a dot as well, apparently due to the fact, that they have the same dimension as timerates of change. This usage is misleading and confusing and therefore totally unacceptable.

The concepts of flux and source are fundamental concepts and essentially different, due to the totallydifferent nature of the mechanisms, from the concept of rate of change of the quantity they cause tochange, although they may each, in the absence of the other, be equal in value and balancing the rateof change.



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Much more reasonable is to denote rate of change by an operator symbol as well, e. g. by R, as willbe done in this version of the symbols, and write any balance in the format

qR = QS = QC + QT + QM + QP ,

clearly indicating the four totally different physical mechanisms taking part in the change of anyquantities of extensive qualities.

If instead of the object oriented notation the function oriented notation is being used the balancewould e. g. look like

qR = SQ = CQ + TQ + MQ + PQ .

This is not very practical if the quality under consideration is of tensorial character or of even morecomplex matrix nature. QU is the variable for the SI unit of the quality Q under consideration.

It will become evident from this very elementary exposition that precisely the most fundamentalconcepts are mostly used extremely carelessly. The concepts "variable", "quantity", and "quality" arerarely clearly distinguished as they ought to be.

E. g.: momentum is a quality and a body may have stored a certain quantity of it at a given time.M and MO are variables for vectors of numerical values of the quantity measured in Ns. t and TIare variables for values of the quantity of the quality time measured in s .


Version 1993 1.1.2 Space related Concepts 20

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1.1.2 Space Related Concepts


s. Remarks .1and .2

s S Any scalar quantitydistributed, maybesingularly, in space

∫ds

S0ij SM0(I,J) Zeroth order moment of a

scalar quantity∫δijds =δijS

S1ij SM1(I,J) First order moment of a

scalar quantity,formerly static moments ofa scalar distribution

∫εikjxkds

S2ij SM2(I,J) Second moment of a scalar

quantity, formerly momentsof inertia of a scalardistribution

∫εklixlεjkm xmds

Suv S(U,V) Generalized moment of ascalar quantity distributedin space

s. Remark .3

Sij = S0ij

Si, 3+j = S1ij

T

S3+i, j = S1ij

S3+i, 3+j = S2ij

Tij T(I,J) Tensor in space referred toan orthogonal system ofCartesian coordinatesfixed in the body

Tijs + Tij

a

TijA TAS(I,J) Anti-symmetric part of a

tensor( Tij - Tji ) / 2

TijS TSY(I,J) Symmetric part of a tensor ( Tij + Tji ) / 2

TijT TTR(I,J) Transposed tensor Tji

Tij vj Tensor product ∑ Tij vj



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ui, vi U(I), V(I) Any vector quantities

ui vi UVPS Scalar product uivi

ui vj UVPD(I,J) Diadic product uivj

u×v UVPV(I) Vector product εijkujvk

V0i ,V i V0(I),V(I) Zeroth order moments of a

vector quantity distributedin space, referred to anorthogonal system ofCartesian coordinates fixedin the body

∫dvi

V1i V1(I) First order moments of a

vector distribution∫εijkxjdvk

Vu V(U) Generalized vector V i = V0i

V3+i = V1i

x, x1 y, x2 z, x3

X, X(1)Y, X(2)Z, X(3)

Body axes andcorresponding Cartesiancoordinates

Right-hand orthogonalsystem of coordinates fixedin the body, s. Remark .2

m

x0, x01

y0, x02

z0, x03

X0, X0(1)Y0, X0(2)Z0, X0(3)

Space axes andcorresponding Cartesiancoordinates

Right-hand orthogonalsystem of coordinates fixedin relation to the space, s.Remark .2

m

xF, xF1

yF, xF2

zF, xF3

XF, XF(1)YF, XF(2)ZF, XF(3)

Flow axes andcorresponding Cartesiancoordinates

Right-hand orthogonalsystem of coordinates fixedin relation to the flow, s.Remark .2

m

εijk EPS(I,J,K) Epsilon operator +1 : ijk = 123, 231, 312- 1 : ijk = 321, 213, 132 0 : if otherwise

δij DEL(I,J) Delta operator +1 : ij = 11, 22, 33 0 : if otherwise



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1.1.2.2 Remarks

.1 Notation

The symbols s, S, T, u, v, V denote variables to be replaced by the symbols of the specificquantities under consideration in any particular application. The range of the operational indices i, j, k is from 1 to 3, while for the generalized concepts theoperational indices u, v, w range from 1 to 6.

.2 Coordinate systems

Orientation of cordinates

A problem of general interest, the orientation of the axes of coordinate systems, has been treatedextensively in the Report of the 17th ITTC Information Committee. For ready reference therecommendation is quoted in the following.

"In order to adapt ITTC nomenclature to common practice a proposal for a standard coordinatesystem was published in the newsletter No 7, March 1983, to generate discussion. The response wasquite diverse. On the one hand it was suggested that instead of the two orthogonal right handedsystems with the positive x-axis forward and the positive z-axis either up- or downward as proposedonly one system should be selected, in particular the one with the positve z-axis upwards. On theother hand the attention of the Information Committee was drawn to the fact that in ship flowcalculations neither of the two systems proposed is customary. Normally the x-axis is directed inthe main flow direction, i.e. backwards, the y-axis is taken positive to starboard and the z-axis ispositive upwards. The origin of the co-ordinates in this case is usually in the undisturbed freesurface half way between fore and aft perpendicular.

In view of this state of affairs the Information Committee may offer the following recommendation,if any:

Axes, coordinates

Preferably, orthogonal right handed systems of Cartesian co-ordinates should be used, orientationand origin in any particular case should be chosen for convenience.

Body axes (x,y,z)

Coordinate systems fixed in bodies or ships.

For the definition of hull forms, for structural deflections, and exciting forces usually the x-axispositive forward and parallel to the reference or base line used to describe the body's shape, the y-



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axis positve to port, and the z-axis positive upwards.

For seakeeping and manoeuvring problems usually the x-axis as before the y-axis positive tostarboard, and the z-axis positive downwards, the origin customarily at the centre of mass of thevehicle or at a geometrically defined position.

For ship flow calculations usually the x-axis positive in the main flow direction, i.e. backwards, they-axis positive to starboard,and the z-axis positive upwards, the origin customarily at theintersection of the plane of the undisturbed free-surface, the centre plane, and the midship section.

Fixed or space axes (x0,y0,z0)

Coordinate systems fixed in relation to the earth or the water. For further references see ISOStandard 1151/1 ...6: Terms and symbols for flight dynamics.

The Information Committee is aware that there may be other coordinate systems in use and seesno possibility for the adoption of a single system for all purposes. Any problem requires anadequate coordinate system and transformations between systems are simple, provided thatorientations and origins are completely and correctly documented for any particular case."

Origins of coordinates

In seakeeping and manoeuvring problems customarily the centre of mass of the vehicle is chosenas the origin of the coordinates. This is in most cases not necessarily advantageous, as all thehydrodynamic properties entering the problems are related rather to the geometries of the bodiesunder investigation. So any geometrically defined point may be more adequate for the purposes athand.

.3 Generalized vector or 6-D notation

Most mechanical problems related to bodies moving in three dimensional space are six dimensionaldue to the six degrees of freedom involved. Consequently it is extremely convenient to have anappropriate notation available. Historically a symbolic 'motor' notation has been proposed andsuccessfully used by Richard von Mises (1924). Much later the operational notation ready forcomputer applications adopted here has been independently developed (Schmiechen, 1962) and usedfor the efficient solution of complex problems, including the motions of robots in flows(Schmiechen, 1989) .

The basic idea is to combine the two vectorial balances for the translational momentum and therotational momentum, respectively, into only one 6-D balance of the generalized momentum, andconsequently to deal with generalized forces, i. e. loads, generalized velocities, i. e. motions,generalized masses, i. e. inertia, etc. The generalized vectors, i. e. von Mises' motors, and the



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generalized tensors are simple matrices of vectors and tensors, respectively. As ordinary vectors andtensors their genralized counterparts obey certain transformation rules related to changes in theorientations and the origins of the coordinate systems.

The introduction of this notation at this very early stage is of course in line with the object orientedapproach adopted and permitting an extremely efficient notation not only for the motions of bodiesin general but the seakeeping and manouvring of ships, the notation for which was so far in a quiteunacceptable state.


Version 1993 1.1.3 Time related Concepts 25

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1.1.3 Time Related Concepts


a ADMP Damping sr, in Laplace variable 1/s

f FR Frequency Hz

fC FC Basic frequency inrepeating functions

1 / TC Hz

fS FS Frequency of sampling 1 / TS

period in repeating spectraHz

i I Imaginary unit sqrt(-1) 1

I IM Imaginary variable i

j J Integer values -∞ ...+∞ 1

R R Complex variable exp(s TS)Laurent transform

s S Complex variable a + 2πifLaplace transform

1/s

t TI Time -∞ ... +∞ s

tj TI(J) Sample time instances j TS

TC TC Period of cycle 1 / fC

duration of cycles inperiodic, repeatingprocesses

s

TS TS Period of sampling Duration between samples s

x x Values of real quantities x(t)

X Real "valued" function

xj X(J) Variables for samples values of real quantities

x(tj) = ∫x(t)δ(t - tj)dt

z Z Complex variable



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1.1.3.2 Complex Transforms

xA XA Analytic function XA(t) = X(t) + iXH(t)

xDF XDF Fourier transform ofsampled function

XDF(f) = ∑xjexp(-i2πfjT S)i.e. periodically repeating

= X(0)/2 + fS∑XF(f + jfS)sample theorem: aliasing!

xDL XDL Laurent transformSampled function

XDL(s) = ∑xjexp(-sjTS)

xF XFT Fourier transform XF(f) = ∫X(t)exp(-i2πft)dtinverse form:= ∫XF(f)exp(-i2πft)dtif X(t) = 0 and a = 0 thenXF(f)=XL(f)

xFj XFT(J) Fourier transform of

periodic function 1/TC∫X(t)exp(-i2πjt/TC)dt t = 0 . . TC XF = ∑xF

jδ(f - j/TC)inverse form:X(t) = ∑xF

jexp(-i2πfjT C)

xH XHT Hilbert transform XH(t) = 1/π ∫X(τ)/(t - τ)dτ

xHF XHF Fourier transform ofHilbert transform

XHF(f) = XF(f)(-i sgn f)(1/t)F = -i sgn f

xL XLT Laplace transform XL(s) = ∫X(t)exp(-st)dtif X(t<0) = 0 then= (X(t)exp(-at))F

xR XRT Laurent transform XR(r) = ∑xjr-j=XDL

xS XS Single-sided complexspectra

XS(f) = XF(f)(1 + sgn f)= XAF

i.e. = 0 for f < 0

xSj XS(J) Single-sided complex

Fourier seriesXF

j(1 + sgn j)line spectra



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1.1.3.3 Complex Quantities

za ZAM Amplitude mod(z) = sqrt(zr2+zi2)

zc ZRE Real or cosine componentzc = real(z) = zacos(zp)

zi ZIM Imaginary or sinecomponent

imag(z) = zasin(zp) = zs

zj ZCJ Conjugate zr - izi

zl ZLG (Phase) Lag - zp

zp ZPH Phase arc(z) = arctg(zi / zr)

zr ZRE Real or cosine component real(z) = zacos(zp) = zc

zs ZIM Imaginary or sinecomponent

zs = imag(z) = zasin(zp)

1.1.3.4 Remarks

.1 Fourier transforms and spectra

The notation proposed has proved to be adequate for "real" problems at hand, these notes givingsome useful background information in the most concise form.

The complex "values" may be quantities of any "complexity", e.g. tensors, matrices, and tensors ofmatrices as e.g. encountered in 6-D parameter identification.

The uniform use of the "natural" frequency instead of artifical circular frequency has the advantagethat no factors are occuring in the Fourier transform pair.

.2 Group properties

The Fourier and Hilbert transforms are the unit elements of cyclic groups with the followingproperties:

X(t)F = XF(f), XF(f)F = X(-t), X(-t)F = XF(-f), XF(-f)F = X(t)

X(t)H = XH(t), XH(t)H = -X(t), -X(t)H = -XH(t), -XH(t)H = X(t) .

Consequently among others the following fundamental relations hold:



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F4 = H4 = 1.

.3 Fourier series

Due to the fact that in most cases only real functions and single-sided spectra are used the usualformat of the Fourier series is

X(t) = real(∑xSj exp(i2πjt/TC) = ∑ xSc

j cos(2πjt/TC) + ∑ xSsj sin(2πjt/TC)

The reason for this step is that the spectra are in fact Fourier transforms not of the real functionbeing studied but of the corresponding analytic function.

For ready reference the following formulae are given

xS j = xF

j (1 + sgn j)

xFc = 1/TC ∫X(t) cos(2πjt/TC)dt

xFs = 1/TC ∫X(t) sin(2πjt/TC)dt

where the integration has to be extended over the cycle TC .

.4 Causal functions

Causal functions, defined by

X(t<0) = 0,

are conveniently expressed as

X(t) = Xe(t)(1 + sgn t)

with the even function

Xe(t) = (X(t) + X(-T))/2.

Noting the property

XeF = XFr

the Fourier transform

XF = XeF - iXeFH



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leads to the relations

XFi = -XFrH, i.e. XFiF(t) = -XFrF(t)(-i sgn t)

and, taking advantage of the group properties,

XFr = +XFiH, i.e. XFrF(t) = +XFiF(t)(-i sgn t).

These relationships are known under various names and guises, the derivations sometimes obscuredby irrelevant or misleading arguments., the worst being hydrodynamic.

.5 Minimal phase functions

From the format

XF = XFaexp(iXFp)

the logarithm

ln(XF) = ln(XFa) + iXFp

is derived and it can be proved that the relations

XFp = - (ln(XFa))H, i.e. XFpF(t) = - (ln(XFa))F(t)(-i sgn t)

and

ln(XFa) = +XFpH, i.e. (ln(XFa))F(t) = +XFpF(t)(-i sgn t)

hold for phase minimal functions; s.e.g. Papoulis, A.: The Fourier Integral and Its Applications.New York: McGraw-Hill, 1964.

.6 Spectral estimates

While for periodic functions the estimation of Fourier transforms, spectra, etc. can be efficientlyperformed by fast Fourier algorithms (FFA) the same is not true in general. Due to neccessarytruncation FFT will in general produce results with systematic errors. These are a consequence ofthe implied periodic repetition, which in most cases is simply inadequate.

In these cases only autoregressive model techniques lead to unbiased estimates of the transforms.The reason is that these models provide proper harmonic descriptions of the truncated record; s.e.g.Childers, D.G.: Modern spectrum analysis. New York: IEEE Press, 1978.



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In any case the algorithm used has to be clearly identified, possibly by reference to a fulldescription or, ideally and unambiguously, a subroutine. At this stage it appears premature to tryand introduce standard symbols for various standard procedures.

So far standard procedures not been agreed upon by the ITTC community, but in the near futureit will be necessary to do so in order to arrive at comparable results. Agreement should not bereached by "vote", as has been tried by Ocean Engineeering Committee. The standard adopted bythe hydrographic institutes for the estimation of power spectra is in general quite disputable as well.


Version 1993 1.1.4 Stochastic Processes 31

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1.1.4 Random Quantities and Stochastic Processes

s. Remark .1 and .2

1.1.4.1 Random Quantities

gE, gM, gMR GMR Expected value of afunction of a randomquantity

E(g) = ∫g(x)fx(x)dx x = -∞ ... ∞

x, y X, Y Random quantities x(ζ), y(ζ)

xi, yi X(I), Y(I) Samples of randomquantities

i = 1... n n : sample size

xmE XmMR m-th moment of a randomquantity

xmE

xD, xDR, σx XDR Standard deviation of arandom quantity

xVR 1/2

xDS, sx XDS Sample deviation of arandom quantity

xVS 1/2,unbiased random estimateof the standard deviation

xxR, xxMR,Rxx

XXMR Auto-correlation of arandom quantity

x xE

xyR, xyMR,Rxy

XYMR Cross-correlation of tworandom quantities

x yE

xE, xM , xMR,µx

XMR Expectation or populationmean of a random quantity

E(x)

xA, xMS, mx XMS Average or sample mean of a random quantity

1/n ∑ xi , i = 1...nunbiased random estimateof the expectation with xAE = xE

xVSE = xV / n

xPD, fx XPD Probability density of arandom quantity

d Fx / dx

xyPD, fxy XYPD Joint probability density oftwo random quantities

∂2 Fxy / (∂x ∂y)



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xPF, Fx XPF Probability function(distribution) of a randomquantity

1

xyPF, Fxy XYPF Joint probability function(distribution) function oftwo random quantities

1

xV, xVR, xxVR XVR, XXVR Variance of a randomquantity

x2 E - xE 2

xVS, xxVS XVS, XXVS Sample variance of arandom quantity

1/ (n - 1) ∑ (xi - xA)2

i = 1...nunbiased random estimateof the variance xVSE = xV

xyV, xyVR XYVR Variance of two randomquantities

x yE - xE yE

ζ Outcome of a random "experiment"

1.1.4.2 Stochastic Processes

gMR GMR Mean of a function of arandom quantity

M(g(t)) = lim(1/T ∫g(t)dt) t =-T/2 ... +T/2 T =-∞ ... +∞

gMS GMS Average or sample mean ofa function of a randomquantity

A(g(t)) = 1/T ∫g(t)dt t =0 ... +T

x, y X, Y Stationary stochasticprocess

x(ζ,t), y(ζ,t)

xxC, xxCR, Cxx XXCR Auto-covariance of astationary stochastic process

(x(t) - xE)(x(t + τ) - xE)E

xyC, xyCR, Cxy XYCR Cross-covariance of twostationary stochasticprocesses

(x(t) - xE)(y(t + τ) - yE)E



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xxR, xxRR, Rxx XXRR Auto-correlation of astationary stochastic process

x(t)x(t + τ)E = Rxx(τ)

Rxx(τ) = Rxx(-τ) if x is ergodic: Rxx(τ) = x(t)x(t + τ)MR

Rxx(τ) = ∫ Sxx(ω)cos(ωτ)dτ τ = 0 ... ∞

xyR, Rxy XYRR Cross-correlation of twostationary stochasticprocesses

x(t)y(t + τ)E = Rxy(τ) Ryx(τ) = Rxy(-τ)

if x, y are ergodic:Rxy(τ) = x(t)y(t + τ)MR

xxS, Sxx XXSR Power spectrum orautospectral power densityof a stochastic process

xxRRSR

xyS, Sxy XYSR Cross-power spectrum oftwo stationary stochasticprocesses

xyRRSR

τ TICV Covariance or correlationtime

ζ Outcome of a random "experiment"

1.1.4.3 Probability Operators

A, MS MS Average, sample mean

C, CR CR Population covariance

CS CS Sample covariance

D, DR DR Population deviation

DS DS Sample deviation

E, M, MR MR Expectation, populationmean

PD PD Probability density



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PF PF Probability function

S SR (Power) Spectrum

SS SS Sample spectrum

R, RR RR Population correlation

RS RS Sample correlation

V, VR VR Population variance

VS VS Sample variance

1.1.4.4 Remarks

.1 Concepts

An adequate introduction into the conceptual world of "Probability, Random Variables (Quantities!),and Stochastic Processes" is provided by A. Papoulis in his book with that same title.

.2 Estimates

Apart of the fundamental theory of probability with its concepts outlined here, in practice thetheory of statistics is necessary, providing for the estimation of probabilities and or their parameters,e.g. expected values. In any case these estimates are at best free of bias, but they are randomvariables themselves and as such clearly distinct from the quantities for which they are estimates.

In the solution of real problems it is absolutely mandatory to account for this distinction. As themost important quantities of this type the sample mean and the sample variance have beenintroduced. It is important to note that as a matter of fact the terminology is still not standardized.The foregoing symbols and terminology are proposed in an attempt to provide tools for the tasksat hand in sytems identification and in quality assurance.

.3 Sample Variance

It should be noted that in contrast to the practice elsewhere the sample variance is not defined asaverage of the squared sample deviations from the sample average. This provides for an unbiasedestimate of the variance and the standard deviation right away. In some text books and somesoftware packages the definition of the sample variance is different from the one proposed here. Socare is necessary if unbiased estimates for small samples are being determined.

ITTC Symbols 1 General Mechanics1.2 Solid Body Mechanics

Version 1993 1.2.1 Properties 35

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1.2 Solid Body Mechanics

1.2.1 Inertial properties


s. Remarks

Ix , Ixx ,m2

11 ,m44

IX, IXX,M2(1,1),MA(4,4)

Roll moment of inertiaaround the principal axis x

kg m2

Iy , Iyy ,m2

22 ,m55

IY, IYY,M2(2,2),MA(5,5)

Pitch moment of inertiaaround the principal axis y

kg m2

Iz , Izz ,m2

33 ,m66

IZ, IZZ,M2(3,3),MA(6,6)

Yaw moment of inertiaaround the principal axis z

kg m2

Ixy , I12

Iyz , I23

Izx , I31

IXY, I2(1,2)IYZ, I2(2,3)IZX, I2(3,1)

Real products of inertia incase of non-principal axes

kg m2

kx, kxx RDGX Roll radius of gyrationaround the principal axis x

(Ixx/m)1/2 m

ky, kyy RDGY Pitch radius of gyrationaround the principal axis y

(Iyy/m)1/2 m

kz, kzz RDGZ Yaw radius of gyrationaround the principal axis z

(Izz/m)1/2 m

m MA mass kg

m0ij ,

mij M0(I,J),MA(I,J)

Zeroth moments of mass,i.e. inertia distribution, masstensor

mij = m δij kg

m1ij M1(I,J) First moments of mass, i.e.

inertia distribution Alias static moments ofmass

kg m


Version 1993 1.2.1 Properties 36

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m2ij ,

Iij

M2(I,J),IN(I,J)

Second moments of mass,i.e. inertia distribution

Alias mass moments ofinertia

kg m2

Muv MA(U,V) Generalized mass, i. e. generalized inertia tensorof a (rigid) body referredto a body fixed coordinatesystem

Mij = M0ij

Mi, 3+j = M1Tij

M3+i, j = M1ij

M3+i , 3+j = M2ij

1.2.1.2 Remarks

.1 Notation

The operational indices i, j, k range from 1 to 3, the indices u, v ,w of the generalized tensors from1 to 6.

Refer to 1.1.2 Space Related Concepts for definition of generalized concepts.

.2 Reference Points

In any particular case the orientation and the origin of the coordinate system have to be specified andindicated, if necessary. If the coordinate system coincides with the principal axes system thegeneralized tensor has only components in the main diagonal, the first order moments as well as thereal moments of inertia are vanishing.

While this aspect may be of interest in cases, where the translational and rotational motions may beconsidered as uncoupled, as in the case of gravitational forces acting alone acting on a solid body,or for qualitative considerations, where this condition holds at least approximately, it is not at allimportant for computational purposes. Quite to the contrary it requires the extra, in generalunnecessary operation of transformation to the principal axes of the inertia tensor. Due to thehydrodynamic forces the translational and the rotational motions can in general not be consideredfrom each other in the ordinary way just by construction of a special reference point.


Version 1993 1.2.2 Forces 37

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1.2.2 Loads

s. Remark .1

1.2.2.1 External Loads

s. Remark .2

Fu F(U) Force, generalized, load,in body coordinates

MFu = MM

u

Fi = F0i

F3+i = F1i

gu G(U) Gravity field strength,generalized, in bodycoordinates

gi = g1i

g3+i = 0

gi G1(I) Gravity field strength,in body coordinates!

m/s2

K, Mx ,F1

1 , F4

K, M(1),F1(1), F(4)

Moment around body axis x Nm

M, My ,F1

2 , F5

M, M(2), F1(2), F(5)

Moment around body axis y Nm

N, Mz ,F1

3 , F6

N, M(3), F1(3), F(6)

Moment around body axis z Nm

X, Fx ,F0

1 , F1

X, FX,F0(1), F(1)

Force in direction of bodyaxis x

Nm

Y, Fy ,F0

2 , F2

Y, FY,F0(2), F(2)

Force in direction of bodyaxis y

Nm

Z, Fz ,F0

3 , F3

Z, FZ,F0(3), F(3)

Force in direction of bodyaxis z

Nm

Gu G(U) Gravity or weight force,generalized, in body coor-dinates!

Gu = muv gv

G0i , Gi G0(I) Gravity or weight forcein body coordinates!

Gi = G0i = m0

ij gj

= mgiN

G1i G1(I) Gravity or weight moment

in body coordinates!G3+i = G1

i = εikj xk G0j

= m1ij gj

Nm



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q UNQ Load per unit length N/m

w WPUL Weight per unit length dW / dx1 N/m

1.2.2.2 Sectional Loads

s. Remark .3

FSu FS(U) Force or load acting at a

given planar cross-sectionof the body, generalized, insection coordinates!

FSi = FS0

i

FS3+i = FS1

i = MBi

NNm

FSi FS(I) Shearing force FS0

2 , FS0

3 N

FT FT,FS(1)

Tensioning or normal force FS01 N

MBi MB(I) Bending moment FS1

2 , FS1

3 Nm

MT MT,MB(1)

Twisting or torsionalmoment

FS11 Nm

1.2.2.3 Remarks

.1 Operational Indices

The operational vector and tensor indices i, j, k range from 1 to 3, the corresponding indices u, v, w for their generalized counterparts range from 1 to 6.

.2 Momentum Balance

For the fundamental balance of quantities of extensive qualities see Section 1.1.1 on FundamentalConcepts and the Remarks on Balances. For definition of the generalized concepts see Section1.1.2 on Space Related Concepts.

According to the fundamental balance of extensive quantities applied to momentum two differenttypes of 'external' forces have to be distinguished, namely the momentum flux across theboundaries, in the case of solid bodies by molocular diffusion only, i. e. stresses, the socalledsurface forces, and the momentum sources in the volumes of the bodies, the socalled volume forces.In the usual applications the weight is the only momentum source, while all other forces acting ona body, distributed over the surface or concentrated, may be considered as surface forces.



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.3 Sectional Loads

Sectional loads are surface loads, i. e. moments of stresses due to molecular momentum fluxesacross the section. Sectional loads are only meaningful relative to the coordinates of the section,on which they act. If the components are referred to body coordinates as usual, this implies sectionsnormal to the longitudinal axis. The former terminology referring to horizontal and vertical shearforces and bending moments is to be considered obsolete even in this context. Lateral and normalare the appropriate names in the context of body coordinates.


Version 1993 1.2.3 Rigid Body Motions 40

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1.2.3 Rigid Body Motions

1.2.3.1 Motions

p , ωx ,v0

1 , v4

P, OMX,V0(1), V(4)

Angular velocity aroundbody axis x

rad/s

q , ωy ,v0

2 , v5

Q, OMY,V0(2), V(5)

Angular velocity aroundbody axis y

rad/s

r , ωz ,v0

3 , v6

R, OMZ,V0(3), V(6)

Angular velocity aroundbody axis z

rad/s

u , vx ,v1

1 , v1

U, VX,V1(1), V(1)

Translatory velocity in thedirection of body axis x

m/s

v , vy ,v1

2 , v2

V, VY,V1(2), V(2)

Translatory velocity in thedirection of body axis y

m/s

w , vz ,v1

3 , v3

W, VZ,V1(3), V(3)

Translatory velocity in thedirection of body axis z

m/s

vu V(U) Components of generalizedvelocity or motion relativeto body axes

vi = v1i

v3+i = v0i

s.Remark .2

m/s

rad/s

p q r

PRQRRR

Rates of change ofcomponents of angularvelocity relative to bodyaxes

s.Remark .3 rad/s2

u v w

URVRWR

Rates of change ofcomponents of linearvelocity relative to bodyaxes

s. Remark .3 m/s2

α AA Angular acceleration dω/dt rad/s2



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1.2.3.2 Attitudes

s.Remark .4

α ATALFA

Angle of attack The angle of thelongitudinal body axis fromthe projection into theprincipal plane of symmetryof the velocity of the originof the body axes relative tothe fluid, positive in thepositive sense of rotationabout the y-axis

rad

β DRBET

Angle of drift or side-slip The angle to the principalplane of symmetry from thevelocity vector of the originof the body axes relative tothe fluid, positive in thepositive sense of rotationabout the z-axis

rad

γ ROGAMR

Projected angle of roll orheel

The angular displacementabout the xo axis of theprincipal plane of symmetryfrom the vertical, positivein the sense of rotationabout the xo axis

rad

φ X(4), RO,PHIR

Angle of roll, heel or list Positive starboard sidedown

rad

θ X(5), TR,TETP

Angle of pitch or trim Positive bow up rad

ψ X(6), YA,PSIY

Angle of yaw, heading orcourse

Positive bow to starboard rad



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1.2.3.3 Remarks

.1 Operational Indices

The operational vector and tensor indices i, j, k range from 1 to 3, the corresponding indices u, v, w for their generalized counterparts range from 1 to 6.

.2 Angular Velocities

The operational ("exponential") notation for the linear and angular velocities reflects the fact thatthe angular velocity of a rigid body is independent of the reference point, while the linear velocitychanges with the change of reference point.

.3 Time Rates of Change

The computer symbols for the time derivatives have been either DXDT or XDOT, both being veryunsatisfactory. The notation proprosed is XRT etc for "x rate", in full "x time rate of change". See1.1.1.3 on Balances.

.4 Angles

The proposed computer symbols for the various angles are an attempt to get away from the oldcryptic notation. The Euler angles roll, pitch, and yaw are evidently to be considered as the naturalextension of the position vector to the generalized position vector. It has of course to noted thatcontrary to the translatory motion the rotational motion can not directly integrated to obtain theattitudes in question.

Further, if extreme motions are to be considered the Euler angles may be not adequate forcomputational purposes, e. g. in numerical simulations, as the corresponding matrix of the directionscosines can become singular. This problem can be avoided if Euler parameters (quaternions) areemployed.

ITTC Symbols 1 General Mechanics1.3 Fluid Mechanics

Version 1993 1.3.1 Flow Parameters 43

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1.3 Fluid Mechanics

1.3.1 Flow Parameters

1.3.1.1 Fluid Properties

c CS Velocity of sound (E / ρ)1/2 m/s

E EL Modulus of elasticity Pa

κ CK Kinematic capillarity σ / ρ m3/s2

µ VI Viscosity kg/ms

ν VK Kinematic viscosity µ / ρ m2/s

ρ DN, RHO Density kg/m3

σ CA Capillarity Surface tension per unitlength

kg/s2

1.3.1.2 Flow parameters

s. Remark .1

Bn BN Boussinesq number V / (g RH)1/2 1

Cn CN Cauchy number V / (E / ρ)1/2 1

Fn FN Froude number V / (g L)1/2 1

Fnh FH Froude depth number V / (g h)1/2 1

Fn FV Froude displacementnumber

V / (g 1/3)1/2 1

Mn MN Mach number V / c 1

Rn RN Reynolds number V L / ν 1

Sn SN Strouhal number f L / V 1

Tn TN Thoma number 1

Wn WN Weber number V2 L / κ 1



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1.3.1.3 Boundary conditions

k HK Roughness height ormagnitude

Roughness height, usuallyin terms of some average

m

ks SK Sand roughness Mean diameter of theequivalent sand grainscovering a surface,s. Remark .2

RH RH Hydraulic radius Area of section divided bywetted perimeter

m

1.3.1.4 Remarks

.1 Flow parameters

The ITTC notation for the flow parameters is not in accordance with that of Physics in general andsomewhat redundant, but the SaT Group feels that the usage is so established now that there is nochance for a change.

The flow parameters are the normalised fluid properties, although mostly not written in that way.E. g. the inverse of the Reynolds number is the normalized viscosity

µn = µ / (ρ U L ) = 1 / Rn ,

with the reference quantities ρ, U and L for steady motion problems. For other problems otherreference quantities may be more appropriate.

The Cauchy number is not identical with the Mach number. The modulus of elasticity entering is notthat of the fluid but that of an elastic structure in the flow.

The search for "characteristic" reference quantities is a matter of physical argument or the evaluationof experiments, i. e. is a matter either of previous knowledge or a cura posterior. Dimensionalanalysis does not provide any apriory arguments!

The usage of scale factor in model testing relates full scale and model scale. A scale factor inabsolute physical terms would be the normalized length

Ln = (Rn / Fn)2/3 = L g1/3 / ν2/3 .

.2 Sand roughness

Although still widely used to characterize the roughness of a surface it is now well understood that



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sand roughness and the resulting roughness resistance are not typical for technical surfaces, ships'surfaces in particular.

So far no sound correlation between the surface description and the additional resistance has beenestablished.


Version 1993 1.3.2 Flow Fields 46

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1.3.2 Flow Fields

1.3.2.1 Velocities etc

s. Remark .1

e ED Density of total flowenergy

ρ V2 / 2 + p + ρ g h Pa

fi FS(I) Mass specific force Strength of force fields,usually only gravity field gi

m/s2

h HS Static pressure head ∆z0 ,z0-axis positive vertical up!

m

H HT Total head e / w = h +p/w +q/w m

p PR, ES Pressure, density of staticflow energy

Pa

p0 P0 Ambient pressure inundisturbed flow

Pa

q PD, EK Density of kinetic flowenergy, dynamic pressure

ρ V2 / 2 Pa

Q QF,QFLOW

Rate of flow Volume passing across acontrol surface in time unit

m3/s

sRij SR(I,J) Turbulent or Reynolds

stressρ vivj

CR Pa

sij ST(I,J) Total stress tensor Density of total diffusivemomentum flux due tomolecular and turbulentexchange

Pa

sVij SV(I,J) Viscous stress Pa

u, vx ,v1

v,vy ,v2

w,vz ,v3

VX, V1VY, V2VZ, V3

Velocity component indirection of x, y, z axes

m/s

vi V(I) Velocity m/s

V VA Velocity V = v ivi1/2 m/s

V0 V0 Velocity of undisturbedflow

m/s



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1.3.2.2 Circulation etc

Γn CN Nomalized circulation Γ / (π D V)π is frequently omitted

1

I ID Induction factor Ratio between velocitiesinduced by helicoidal andby straight line vortices

1

γ VD Vortex density Strength per length or perarea of vortex distribution

m/s

Γ CC Circulation V ds along a closed line

m2/s

φ PO Potential function m2/s

ψ SF Stream function ψ = constis the equation of a streamsurface

m3/s

1.3.2.3 Remarks

.1 Equation of Motion

The universal equation of motion for any continuum in space is the balance of mass specificmomentum vi , the Cauchy equation, in Cartesian coordinates,

ρ dt vi =ρ (∂t +vj ∂j) vi =ρ (∂t vi +vj ∂j vi) = ∂j sji + ρ fi ,

which can be derived if the balance of mass density ρ , the equation of continuity

dt ρ = (∂t + vj ∂j) ρ = ∂t ρ + vj ∂j ρ = - ρ ∂j vj

is taken into account.

The notation used for differentiation is evidently

dt = d / dt ,∂t = ∂ / ∂t ,∂i = ∂ / ∂xi .

Further Einstein's summing convention is conveniently implied:



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xi yi = Σi xi yi , i = 1, 2, 3 .

In hydrodynamics incompressibility is a further adequate idealisation and consequently the universalequations reduce to the two equations

ρ dt vi = ρ (∂t vi + vj ∂j vi) = ∂j sji + ρ fi ,

∂j vj = 0 .

In addition the balance of moments requires that the stress tensor is symmetric

sji = sij ,

(Boltzmann's axiom). The stress consists of three constituents: the pressure term, the stress proper,and the Reynolds stress:

sji = - p δji +sVij +ρ vjvi

CR .

The first two terms represent the molecular diffusion of momentum, the last term the turbulentdiffusion.

.2 Constitutive Laws

Only at this stage the individual properties of fluids have to be introduced through constitutive laws,i. e. the laws for the stress tensor s. Newtonian fluids, i. e. incompressible linear viscous fluids, aredefined by the law

sVij = µ ∂i vj

S = µ (∂i vj + ∂j vi) / 2 .

Introducing the stress terms with the constitutive law into the universal Cauchy's equation results inthe "Reynolds averaged" Navier-Stokes equation (RANSE) in its kinematic form

dt vi = ∂tvi + vj ∂j vi = - ∂i p/ρ + ν ∂j ∂j vi + ∂j vjviCR + gi .

Apart of the equation of continuity the closure of the problem requires further "constitutive"equations for the turbulent Reynolds stresses, the socalled turbulence models and, even worse,boundary conditions including details of the surface structure, i. e. roughness.

A very popular turbulence model is the k-ε model, with two balances for the density of the turbulentenergy k and its dissipation ε , respectively. There are fundamental investigations under way toconstruct more advanced models in accordance with the rational theory of constitutive laws.


Version 1993 1.3.3 Lifting Surfaces 49

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1.3.3 Lifting Surfaces

1.3.3.1 Geometry

A AP Planform area b cm m2

b SP Wing span m

cm CHME Mean chord length A / b m

ct CHTP Tip chord length m

cr CHRT Root chord length m

δf ANFL Flap deflection angle rad

δs ANSL Slat deflection angle rad

γ ANSW Sweep angle rad

λ TA Taper ratio ct / cr 1

Λ AS Aspect ratio b2 / A 1

1.3.3.2 Sectional coefficients

CD CDSE Section drag coefficient 1

CDI CDSE Section induced dragcoefficient

1

CL CLSE Section lift coefficient 1

CM CMSE Section moment coefficient 1

1.3.3.3 Flow angles etc

vI VI Induced velocity m/s

VT VT Resultant velocity offlow approaching ahydrofoil

Taking vortex inducedvelocities into account

m/s

α AA,ALFA

Angle of attack or inci-dence

Angle between the directionof undisturbed relative flowand the chord line

rad


Version 1993 1.3.3 Lifting Surfaces 50

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αE AAEF,ALFE

Effective angle of attack or incidence

The angle of attack relativeto the chord line includingthe effect of inducedvelocities

rad

αG AAGE,ALFG

Geometric angle ofattack or incidence

The angle of attack relativeto the chord line neglectingthe effect of inducedvelocities

rad

αH AAHY,ALFI

Hydrodynamic angle of attack

In relation to the position atzero lift

rad

αI AAID,ALFS

Ideal angle of attack For thin airfoil or hydrofoil,angle of attack for whichthe streamlines are tangentto the mean line at theleading edge. This conditionis usually referred to as"shock-free" entry or"smooth"

rad

α0 AAZLALF0

Angle of zero lift Angle of attack or incidenceat zero lift

rad


Version 1993 1.3.4 Boundary Layers 51

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1.3.4 Boundary Layers

s. Remark .1

1.3.4.1 Twodimensional Boundary Layers

Cf CFL Skin friction coefficient τ / (ρ Ue2 / 2) 1

F CQF Entrainment factor 1 / (Ue dQ / dx) 1

H HBL Boundary layer shapeparameter

δ* / Θ 1

HE HQF Entrainment shapeparameter

(δ - δ*) / Θ 1

p PR Static pressure Pa

P PT Total pressure Pa

Q QF Entrainment b ∫U dy a

m2/s

Rδ* RDELS Reynolds number based on displacement thickness

U∞ δ* / ν or Ue δ* / ν 1

Rθ RTHETA Reynolds number based on momentum thickness

U∞ Θ / ν or Ue Θ / ν 1

u UFL Velocity fluctuations inboundary layer

m/s

us UFLS Root mean square value of velocity fluctuations

m/s

u+ UPLUS U / uτ 1

uτ UTAU Shear (friction) velocity (τ / ρ)1/2 m/s

Um UMR Time mean of velocity inboundary layer

m/s

Ui UIN Instantaneous velocity m/s

U∞ UFS Free-stream velocity farfrom the model

m/s

Ue UE Velocity at the edge of theboundary layer at y=δ995

m/s


Version 1993 1.3.4 Boundary Layers 52

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∆U UDEF Velocity defect in boundarylayer

(Ue- U) / uτ 1

y+ YPLUS y uτ / ν 1

β BETE Equilibrium parameter δ* / (τw dp / dx) 1

δ995 DEL Thickness of a boundarylayer at U=0.995Ue

m

δ*, δ1 DELS Displacement thickness ofboundary layer

∫(Ue- U) / Ue dy m

∆ CD Clauser thickness ∫(Ue- U) / uτ dy m

K K Von Karman constant 0.41 1

Λ PRGR Pressure gradient parameterδ99 / (ν dUe / dx) 1

θ*, δ** ENTH Energy thickness ∫(U / Ue) (1 - U2 / Ue2)dy m

Θ THETA Momentum thickness ∫(U / Ue) (1 - U / Ue)dy m

τw TAUW Shear stress at a wall µ (∂U / ∂y)y=0 Pa

1.3.4.2 Remarks

.1 Future work

In future the section should have an additional subsection on threedimensional boundary layers. Andboth subsections should be structured as follows:

Basic Concepts,Differential Formulation,Integral Formulation.

The Restistance and Flow Committee is strongly urged to provide a complete revision of the wholechapter along this line and accordance with the general concepts put forward.


Version 1993 1.3.5 Cavitation 53

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1.3.5 Cavitation

1.3.5.1 Flow parameters

as GR Gas content ratio α / αS 1

α GC Gas content Actual amount of solvedand undissolved gas in aliquid

ppm

αS GS Gas content of saturatedliquid

Maximum amount of gassolved in a liquid at a giventemperature

ppm

σ CNPC Cavitation number (pA - pC) / q 1

σV CNPV Vapor cavitation number (pA - pV) / q 1

1.3.5.2 Flow field

DC DC Cavity drag N

lC LC Cavity length Streamwise dimension of afully-developed cavitatingregion

m

pA PA Ambient pressure Pa

pAC PACO Collapse pressure Absolute ambient pressureat which cavitaties collapse

Pa

pAI PAIC Critical pressure Absolute ambient pressureat which cavitationinception takes place

Pa

pC PC Cavity pressure Pressure within a steady orquasi-steady cavity

Pa

pCI PCIN Initial cavity pressure Pressure, maybe negative,i. e. tensile strength,necessary to create a cavity

Pa

pV PV Vapor pressure of water At a given temperature! Pa

UI UNIN Critical velocity Free stream velocity atwhich cavitation inceptiontakes place

m/s


Version 1993 1.3.5 Cavitation 54

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VL VOLS Volume loss WL / w m3

WL WTLS Weight loss Weight of material erodedfrom a specimen during aspecified time

N/s

δC HC Cavity height or thickness Maximum height of a fully-developed cavity, normal tothe surface and the stream-wise direction of the cavity

m

1.3.5.3 Pumps

HN HTNT Net useful head of turbo-engines

m

HU HTUS Total head upstream ofturbo-engines

m

Tn TN Thoma number (HU - pV / w) / HN 1

ITTC Symbols 1 General Mechanics1.4 Environmental Mechanics

Version 1993 1.4.1 Waves 55

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1.4 Environmental Mechanics

1.4.1 Waves

s. Remark .1

1.1.4.1 Periodic waves

s. Remark .2

cW VP Wave phase velocity orcelerity

LW / TW m/s

cWi VP(I) Wave phase velocity ofharmonic components of aperiodic wave

const = cWfor periodic waves

m/s

cG VG Wave group velocity orcelerity

m/s

fW FW Basic wave frequency 1 / TW Hz

fWi FW(I) Frequencies of harmoniccomponents of a periodicwave

i fW Hz

HW HW Wave height ηC - ηT m

LW , λW LW Wave length Measured in the direction ofwave propagation

m

TW TW Basic wave period 1 / fW s

α WD Wave direction rad

η EW Instantaneous waveelevation at a givenlocation

z-axis positive vertical up,zero at mean water level;s. Remark .3

m

ηai EWAM(I) Amplitudes of harmonic

components of a periodicwave

ηFSa m

ηpi EWPH(I) Phases of harmonic

components of a periodicwave

ηFSp rad

ηC EC Wave crest elevation m



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ηT ET Wave trough elevation Negative values! m

κ WN Wave number 2 π / LW 1/m

λW , LW LW Wave length Measured in the direction ofwave propagation

m

ζ DW Instantaneous wavedepression

z-axis positive verticaldown, zero at mean waterlevel

m

ωW FC Circular wave frequency 2 π fW = 2 π / TW 1/s

1.4.1.2 Irregular waves

s. Remark .3

Hd HD Wave height by zero down-crossing

m

Hu HU Wave height by zero up-crossing

m

Td TD Wave periods by zerodowncrossing

s

Tu TU Wave periods by zero up-crossing

s

ηC EC Maximum of elevations ofwave crests in a record

m

ηT ET Elevations of wave troughsin a record

Negative values! m

λd LD Wave length by zero down-crossing

m

λu LU Wave length by zero up-crossing

m



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1.4.1.3 Time Domain Analysis

HV HV Wave height estimated fromvisual observation

m

TR TR Duration of record 1 / fR s

TS TS Sample interval 1 / fS ,time between twosuccessive samples

s

TV TV Wave period estimated fromvisual observation

s

1.4.1.4 Frequency Domain Analysis

b B Bandwidth of spectralrevolution

Sampling frequency dividedby the number of transformpoints

Hz

Cr CRA Average reflectioncoefficient

1

Cr(f) CRF Reflection coefficientfunction

1

fP FRPK Spectral peak in frequency Frequency at which thespetrum has its maximum

Hz

fR FRRC Frequency resolution 1 / TR Hz

fS FRSA Sample frequency 1 / TS Hz

Hmo HMO Significant wave heightbased on zeroth moment

4 (m0)1/4 m

H1/3d H13D Zero downcrossingsignificant wave height

Average of the highest onethird zero downcrossingwave heights

m

H1/3u H13U Zero upcrossing significantwave height

Average of the highest onethird zero upcrossing waveheights

m



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Hσ HWDS Estimate of significant waveheight from sampledeviation of wave elevationrecord

m

mn Mn n-th moment of wave power spectral density

∫fn S(f)df m2/ sn

Si(f), Si(ω)

EISF, EISC

Incident wave powerspectral density

m2/Hz

Sr(f), Sr(ω)

ERSF, ERSC

Reflected wave powerspectral density

m2/Hz

Sη(f), Sη(ω)

EWSF,EWSC

Wave power spectraldensity

m2/Hz

T0,1 T1 Average period from zerothand first moment

m0/m1 s

T0,2 T2 Average period from zerothand second moment

(m0/m2)1/2 s

α,θ WD Wave direction rad

1.4.1.5 Directional Waves

D(f,θ),D(ω,µ)

DIRSF Directional spreadingfunction

S(f,Θ)=S(f)D(f,Θ)

2π ∫D(f,Θ)dΘ=1 0

rad

f FR Frequency Hz

Sζ(ω,µ)Sθ(ω,µ)etc.

S2ZETS2TETetc.

Two dimensional spectraldensity

1

Sρ(f,θ)Sζ(ω,µ)

STHETA Directional spectral density m2/Hz/rad

θ, µ Component wave direction rad



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SI-Unit

1.4.1.6 Remarks

.1 General

This section is of course in many ways related to the Sections 1.1.3 Time Related Concepts and1.1.4 Random Quantities and Stochastic Processes. In terms of the object oriented paradigma onlythe time function, the wave elevation at a given location, denoted by η and EW, respectively, hasto be introduced and the operations defined earlier along with the corresponding notation may beapplied without modification and repetition.

.2 Periodic waves

The basic concepts on waves are derived from the model of periodic, not necessarily harmonicwaves, but which may be considered as composed of harmonic components. Even periodic wavesmay be considered as samples of stochastic processes. In this case the wave parameters arerandom quantities with given joint probability functions. In practice only samples of suchprocesses will be available and consequently only random sample estimates of the parameters canbe obtained.

.3 Irregular waves

In the section on non-priodic waves only random quantities have been introduced as e. g. thecrest height, to which all the probabilty concepts and parameters can be applied as defined earlierin Section 1.1.4., e. g. the population mean and variance of the crest height. If waves are not periodic any individual infinite record may be considered as a random sample ofstationary stochastic process, which is usually assumed to be ergodic, thus permitting to replacepopulation means by appropriate time means. In future ergodicity may be required to be checkedat least for research and quality assurance purposes.

.4 Finite records

In practice only records of finite duration are available of the hypothetical stochastic processes forthe estimation of the population parameters. This should be reflected in the symbols andterminology, e. g. in the case of the wave crest only the random sample mean ηC

A (ECMS) maybe determined. And as long as in most cases no agreement has been reached on the optimumestimators to be used the symbols and terminology should even indicate the special estimatorsused in order to avoid confusion.

.5 Sampled values

Usually not even finite records are available for the estimation of spectra etc, but only finite setsof sampled values, namely ηi or EW(I).



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.6 Research Parameters

Currently discussed research parameters may be found in the IAHR/PIANC List of Sea StateParameters, Supplement to Bulletin No 52, January 1986.


Version 1993 1.4.4 Ice Mechanics 61

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SI-Unit

1.4.4 Ice Mechanics

SI SAIC Salinity of ice Weight of salt per unitweight of ice

1

SW SAWA Salinity of water Weight of dissolved salt perunit weight of saline water

1

t°A TEAI Temperature of air °C

t°I TEIC Local temperature of ice °C

t°W TEWA Temperature of water °C

δI ELIC Deflection of ice sheet Vertical elevation of icesurface

m

εI STIC Ice strain Elongation per unit length 1

I ε STRTIC Ice strain rate ∂ε / ∂τ 1/s

µI POIIC Poisson's ratio of ice 1

νA POAI Relative volume of air Volume of gas pores perunit volume of ice

1

νB POBR Relative volume of brine Volume of liquid phase perunit volume of ice

1

νO POIC Total porosity of ice νO = νA + νB 1

ρI DNIC Mass density of ice Mass of ice per unit volume kg/m3

ρSN DNSN Mass density of snow Mass of snow per unitvolume

kg/m3

ρW DNWA Mass density of water kg/m3

ρ∆ DNWI Density difference ρ∆ = ρW - ρI kg/m3

σCI SCIC Compressive strength of ice Pa

σFI SFIC Flexural strength of ice Pa

σTI SNIC Tensile strength of ice Pa

τSI STIC Shear strength of ice Pa

ITTC Symbols 2 Ships in General2.1 Geometry and Hydrostatics

Version 1993 2.1.1 Hull Geometry 62

ITTCSymbol

ComputerSymbol


SI-Unit

2 Ships in General

2.1 Geometry and Hydrostatics

2.1.1 Hull Geometry


ABL ABL Area of bulbous bow inlongitudinal plane

The area of the ramprojected on the middle lineplane forward of the foreperpendicular; s. Remark .1

m2

ABT ABT Area of transverse cross-section of a bulbous bow(full area port and star-board)

The cross sectional area atthe fore perpendicular. Where the water lines arerounded so as to terminateon the forward perpendic-ular ABT is measured bycontinuing the area curveforward to the perpendic-ular, ignoring the finalrounding; s. Remark .1

m2

AM AM Area of midship section Midway between fore andaft perpendiculars

m2

AT ATR Area of transom (full areaport and starboard)

Cross-sectional area oftransom stern below theload waterline

m2

AV AV Area exposed to wind Area of portion of shipabove waterline projectednormally to the direction ofrelative wind

m2

AW AW Area of water-plane m2

AWA AWA Area of water-plane aft ofmidship

m2

AWF AWF area of water-plane forwardof midship

m2

AX AX Area of maximumtransverse section

m2



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SI-Unit

B B Beam or breadth, moulded,of ships hull

m

D DEP Depth, moulded, of a shiphull

m

iE ANEN Angle of entrance, half Angle of waterline at thebow with reference tocenterplane, neglecting localshape at stem

rad

iR ANRU Angle of run, half Angle of waterline at thestern with reference to thecenter-plane, neglectinglocal shape of stern frame

rad

IL IL Longitudinal moment ofinertia of water-plane

About transverse axisthrough center of floatation

m4

IT IT Transverse moment ofinertia of water-plane

About longitudinal axisthrough center of floatation

m4

L L Length of ship Reference length of ship(generally length betweenthe perpendiculars)

m

LE LEN Length of entrance From the forwardperpendicular to the forwardend of parallel middle body,or maximum section

m

LOA LOA Length, overall m

LOS LOS Length, overall submerged m

LP LP Length of parallel middlebody

Length of constanttransverse section

m

LPP LPP Length betweenperpendiculars

m

LR LRU Length of run From section of maximumarea or after end of parallelmiddle body to waterlinetermination or otherdesignated point of the stern

m

LWL LWL Length of waterline m



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SI-Unit

LFS LFS Frame spacing m

LSS LSS Station spacing m

S S, AWS Area of wetted surface m2

t TT Taylor tangent of the areacurve

The intercept of the tangentto the sectional area curveat the bow on the midshipordinate

1

T, d T Draft, moulded, of ship hull m

TA, da TA, TAP Draft at aft perpendicular m

TF, df TF, TFP Draft at forwardperpendicular

m

TM, dm TM, TMS Draft at midship m

∇, V DISV Displacement volume ∆ / (ρ g) m3

λ SC Linear scale of ship modelλ = LS / LM = BS / BM

= TS / TM

1

µ PMVO Volumetric permeability The ratio of the volume ofwater entering acompartment to the volumeof the compartment

1

2.1.1.2 Derived concepts

B CIRCB R.E. Froude's breadthcoefficient

B / ∇1/3 1

CB CB Block coefficient ∇ / (L B T) 1

CIL CWIL Coefficient of inertia ofwaterplane, longitudinal

12 IL / ( B L3) 1

CIT CWIT Coefficient of inertia ofwaterplane, transverse

12 IT / (B3 L) 1

CM CMS Midship section coefficient(midway between forwardand aft perpendiculars)

AM / (B T) 1



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SI-Unit

CP CPL Longitudinal prismaticcoefficient

∇ / (AX L) or ∇ / (AM L) 1

CPA CPA Prismatic coefficient,afterbody

∇A / (AX L / 2) or∇A / (AM L / 2)

1

CPE CPE Prismatic coefficient,entrance

∇E / (AX LE) or∇E / (AM LE)

1

CPF CPF Prismatic coefficient forebody

∇F / (AX L / 2) or∇F / (AM L / 2)

1

CPR CPR Prismatic coefficient, runs. Remark .2

∇R / (AX LR) or∇R / (AM LR)

1

CS CS Wetted surface coefficient S / (∇ L)1/2 1

CVP CVP Prismatic coefficientvertical

∇ / (AW T) 1

CWA CWA Aft water plane areacoefficient

AWA / (B L / 2) 1

CWF CWF Forward water plane areacoefficient

AWF /(B L / 2) 1

CWP CW Water-plane areacoefficient

AW /(L B) 1

CX CX Maximum transversesection coefficient

AX / (B T)where B and T aremeasured at the position ofmaximum area

1

C∇ CVOL Volumetric coefficient ∇ / L3 1

fBL CABL Area coefficient for bul-bous bow

ABL / (L T) 1

fBT CABT Taylor sectional areacoefficient for bulbous bow

ABT / AX 1

fT CATR Sectional area coefficientfor transom stern

AT / AX 1



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SI-Unit

M CIRCM R.E. Froude's lengthcoefficient, or length-displacement ratio

L / ∇1/3 1

S CIRCS R.E. Froude's wettedsurface area coefficient

S / ∇2/3 1

T CIRCT R.E. Froude's draftcoefficient

T / ∇1/3 1

2.1.1.3 Computer Symbols for Attributes

AB After body

AP After perpendicular

BH Bare hull

DW Design waterline

EN Entry

FB Fore body

FP Fore perpendicular

FS Frame spacing

LP Based on LPP

LW Based on LWL

MS Midships

PB Parallel body

RU Run

SS Station spacing

WP Water plane

WS Wetted surface



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2.1.1.4 Remarks

.1 Bulbous Bows

Below the load water line the stem contour sometimes recedes aft of the fore perpendicular beforeprojecting forward to define the outline of the ram or the fore end of the bulb. In such instances thisarea should be calculated using as datum the aftermost vertical tangent to the contour instead of thefore perpendicular.

.2 Reference Quantities

The prismatic coefficient should generally be based upon maximum section area rather than onmidsection area, as in the 1960 Committee Report, but it should be clearly stated which area has beenused. Whatever ship length considered appropriate may be used for this end and another coefficient,but this length should be clearly indicated and stated.


Version 1993 2.1.2 Propulsor Geometry 68

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SI-Unit

2.1.2 Propulsor Geometry

2.1.2.1 Screw Propellers

AD AD Developed blade area Developed blade area of ascrew propeller outside theboss or hub

m2

AE AE Expanded blade area Expanded blade area of ascrew propeller outside theboss or hub

m2

AO AO Disc Area π D2 / 4 m2

Ap AP Projected blade area Projected blade area of ascrew propeller outside theboss or hub

m2

c LCH Chord length m

cm CHME Mean chord length The expanded or developedarea of a propeller bladedivided by the span fromthe hub to the tip

m

cS CS Skew displacement The displacement betweenmiddle of chord and theblade reference line.Positive when middle chordis at the trailing sideregarding the bladereference line

m

d DH Boss or hub diameter m

D DP Propeller diameter m

f FBP Camber of blade profile m

GZ GAP Gap between the propellerblades

2 π r sin (φ / z) m

h0 HO Immersion The depth of submergenceof the propeller measuredvertically from the shaftaxis to the free surface

m



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SI-Unit

iG RAKG Rake The displacement from thepropeller plane to thegenerator line in thedirection of the shaft axis. Aft displacement isconsiderate positive rake

m

iS RAKS Axial displacement, skew-induced

The axial displacement of ablade section which occurswhen the propeller isskewed. Aft displacementis considered positive rake

m

iT RAKT Axial displacement, total The axial displacement ofthe blade reference linefrom the propeller planeiG + iS . Positive direction isaft.

m

Ρ PITCH Propeller pitch in general m

r RL Blade section radius m

rn RR Relative radius ratio r / R 1

rh RH Hub radius m

R RDP Propeller radius m

t TM Blade section thickness m

t0 TO Thickness on axis ofpropeller blade

Thickness of propeller bladeas extended down topropeller axis

m

xp XP Longitudinal propellerposition

Distance of propeller centerforward of the afterperpendicular

m

yp YP Lateral propeller position Transverse distance of wingpropeller center frommiddle line

m

z NPB Number of propeller blades 1

zp ZP Vertical propeller position Height of propeller centerabove base line

m



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SI-Unit

θs TETS Skew angle The angular displacementabout the shaft axis of thereference point of any bladesection relative to thegenerator line measured inthe plane of rotation. It ispositive when opposite tothe direction of aheadrotation

rad

θ RAKA Angle of rake rad

θEXT TEMX Skew angle extent The difference betweenmaximum and minimumlocal skew angle

rad

φ PHIP Pitch angle of screwpropeller

arctg (P / (2 π R)) 1

φF PHIF Pitch angle of screwpropeller measured to theface line

1

ψ PSI Propeller axis angle Angle between horizontalplane and propeller shaftaxis

rad

2.1.2.2 Ducts

ADEN ADEN Duct entry area m2

ADEX ADEX Duct exit area m2

dD CLEARD Propeller tip clearance Clearance between propellertip and inner surface of duct

m

fD FD Camber of duct profile m

LD LD Duct length m

LDEN LDEN Duct entry part length Axial distance betweenleading edge of duct andpropeller plane

m



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SI-Unit

LDEX LDEX Duct exit length Axial distance betweenleading edge of duct andpropeller plane

m

tD TD Thickness of duct profile m

αD AD Duct profile-shaft axis angle Angle between nose-tailline of duct profile andpropeller shaft

rad

βD BD Diffuser angle of duct Angle between inner ducttail line and propeller shaft

rad


Version 1993 2.1.3 Appendage Geometry 72

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SI-Unit

2.1.3 Appendage Geometry

s. Remark .1


AFB AFB Area of bow fin m2

AC AC Area under cut-up m2

ARF AF Flap area m2

AR ARU Rudder area Area of the rudder,including flap

m2

ARX ARX Area of the fixed part ofrudder

m2

ARP ARP Area of rudder in thepropeller race

m2

ART ART Total rudder area ARX + ARF m2

AFS AFS Area of stern fin m2

ASK ASK Skeg area m2

AWBK AWBK Wetted surface area of bilgekeels

m2

c CH Chord length of an aerofoilor a hydrofoil

m

cm CHME Mean chord length ART / S m

cr CHRT Chord length at the root m

ct CHTP Chord length at the tip m

f FM Camber of an aerofoil or ahydrofoil

Maximum separation ofmedian and nose-tail line

m

t TMX Maximum thickness of anaerofoil or a hydrofoil

Measured normal to meanline

m

δFB ANFB Bow fin angle s. Remark .2 rad

δFS ANFS Stern fin angle s. Remark .2 rad

δFR ANFR Flanking rudder angle s. Remark .2 rad



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SI-Unit

δFRin ANFRIN Assembly angle of flankingrudders

Initial angle set up duringthe assembly as zero angleof flanking rudders

rad

δR ANRU Rudder angle s. Remark .2 rad

δRF ANRF Rudder-flap angle s. Remark .2 rad

λR TARU Rudder taper ct / cr 1

λFR TAFR Flanking rudder taper 1

ΛR ASRU Rudder aspect ratio S2 / ART 1

ΛFR ASRF Flanking rudder aspect ratio 1

2.1.3.2 Identifiers for Appendages

BK Bilge keel

BS Bossing

FB Bow foil

FR Flanking rudder

FS Stern foil

KL Sail yacht keel

RU Rudder

RF Rudder flap

SA Stabilizer

SH Shafting

SK Skeg

ST Strut

TH Thruster

WG Wedge



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2.1.3.3 Remarks

.1 Related Information

Related information may be found in Section 1.3.3 on Lifting Surfaces.

.2 Sign Convention

Positive angles are defined as clockwise when viewed from the center of axes along the appropriatebody axis, i. e. nose-up fin angles and port rudder angles are positive.


Version 1993 2.1.4 Hydrostatics 75

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SI-Unit

2.1.4 Hydrostatics

2.1.4.1 Stability levers

AB XAB Longitudinal center of

buoyancy from aftperpendicular

Distance of center ofbuoyancy from aftperpendicular

m

AF XAF Distance of center of

flotation from afterperpendicular

m

AG XAG Longitudinal center of

gravity from aftperpendicular

Distance of center ofgravity from aftperpendicular

m

BM ZBM Transverse metacenter

above center of buoyancyDistance from the center ofbuoyancy B to thetransverse metacenter M

- KM

KB

m

L

BM ZBML Longitudinal metacenter

above center of buoyancy L - KM

KB

FB XFB Longitudinal center of

buoyancy from forwardperpendicular

Distance of center ofbuoyancy from forwardperpendicular

m

FG XFG Longitudinal center of

gravity from forwardperpendicular

Distance of center ofgravity from forwardperpendicular

m

GM GM Transverse metacentric

heightDistance of center ofgravity to the metacenter

- KM

KG

m

L

GM GML Longitudinal center of

metacentric heightDistance from the center ofgravity G to thelongitudinal metacenter ML

L - KM

KG

m

KA ZKA Assumed center of gravity

above moulded base or keel

Distance from the assumedcenter of gravity A to themoulded base or keel K



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SI-Unit

KB ZKB Center of buoyancy above

moulded base or keel Distance from the center ofbuoyancy B to the mouldedbase or keel K

m

KM ZKM Transverse metacenter

above moulded base or keelDistance from thetransverse metacenter M tothe moulded base or keel K

m

L

KM ZKML Longitudinal metacenter

above moulded base or keelLongitudinal ML m

T

AG YAG Transverse distance from

assumed center of gravityA, to actual centre ofgravity G

m

v

AG ZAG Vertical distance from

assumed center of gravityA, to actual center ofgravity G

m

AZ YAZ Righting arm based on

horizontal distance fromassumed center of gravityA, to Z

m

b Mean center of floatation ofadded buoyant layer

FF XFF Longitudinal center of

floatation from forwardperpendicular

Distance of center offloatation from forwardperpendicular

m

g Center of gravity of anadded or removed weight(mass)

GZ GZ Righting arm or lever - ( V + T) sinφ

AZ

AG

AG m

KG ZKG Center of gravity above

moulded base or keelDistance from center ofgravity G to the mouldedbase or keel K

m

Kg ZKAG Vertical center of gravity of

added or removed weightabove moulded base orkeel

Distance from center ofgravity, g, to the mouldedbase or keel K

m



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SI-Unit

2.1.4.2 Various Concepts

CMTL CMTL Longitudinal trimmingcoefficient

B ML / L or trimming moment divided by trim

1

f FREB Freeboard From the freeboardmarkings to the freeboarddeck, according to officialrules

m

l XTA Longitudinal trimming arm xcb - xcg m

MS MS Moment of ship stability in general

Other moments such asthose of capsizing, heeling,etc. will be represented byMS with additionalsubscripts as appropriate

NM

m MA Ship mass w / g kg

MTC MTC Moment to change trim onecentimeter

Nm/cm

MTM MTM Moment to trim one meter ∆CMTL Nm/m

t YHA Transverse heeling arm m

xcb XACB Longitudinal mean center of floatation of addedbuoyant layer

Longitudinal distance froma midship to the center ofthe added buoyant layer

m

xCB XCB Longitudinal center ofbuoyancy

Longitudinal distance froma midship to the center ofbuoyancy, B

m

xCF XCF Longitudinal center offlotation

Longitudinal distance froma midship to the center offlotation, F

m

xcg XACG Longitudinal center ofgravity of added weight(mass)

Longitudinal distance froma midship to the center ofgravity,g, of an added orremoved weight (mass)

m



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SI-Unit

xCG XCG longitudinal center ofgravity

Longitudinal distance froma midship to the center ofgravity, G

m

T TR Trim TA - TF m

W WT Displacement weight, shipweight

m g N

δ D Finite increment in... Prefix to other symbol

δ T DTR Change in trim m

δTPS DTPS Parallel sinkage Added weight in tonsdivided by tons per unitimmersion

m

∆ DISF Displacement force(buoyancy)

g ρ ∇ N

∇ DISV Displacement volume ∆ / (g ρ) m3

2.1.4.3 Remarks

.1 Other Notation

Alternatively, the position of the center of buoyancy B may be expressed in terms of thecoordinate axes with the appropriate suffix e.g. XB,YB,ZB the position of other items such as thecenter of gravity, G, metacenter M and center of floatation F could also be treated in the sameway.

ITTC Symbols 2 Ships in General2.2 Resistance and Propulsion

Version 1993 2.2.1 Hull Resistance 79

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SI-Unit

2.2 Resistance and Propulsion

2.2.1 Hull Resistance


RA RA Model-ship correlationallowance

Incremental resistance to beadded to the smooth shipresistance to complete themodel-ship prediction

N

RAA RAA Air or wind resistance N

RAP RAP Appendage resistance N

RAR RAR Roughness resistance N

m BLCK Blockage parameter Maximum transverse areaof model ship divided bytank cross section area

1

RC RC Resistance corrected fordifference in temperaturebetween resistance and selfpropulsion tests

((1 + k) (CFMC) + CR) / /((1 + k) (CFM) + CR)) * *RTM

where CFMC is the frictionalcoefficient at thetemperature of the selfpropulsion test

N

RF RF Frictional resistance of abody

Due to fluid friction on thesurface of the body

N

RFO RFO Frictional resistance of aplate

N

RP RP Pressure resistance Due to the normal stressesover the surface of a body

N

RPV RPV Viscous pressure resistance Due to normal stress relatedto viscosity and turbulence

N

RR RR Residuary resistance RT - RF or RT - RFO N

RS RS Spray resistance Due to generation of spray N

RT RT Total resistance Total towed resistance N

RV RV Total viscous resistance RF + RPV N



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SI-Unit

RW RW Wavemaking resistance Due to formation of surfacewaves

N

RWB RWB Wavebreaking resistance Associated with the breakdown of the bow wave

N

SH THL Total head loss m

∆CF DELCF Roughness allowance (obsolete, see CA) 1

V V Speed of the model or theship

m/s

VR VR Wind velocity, relative m/s

τW LSF Local skin friction N/ m2

2.2.1.2 Derived Concepts

CA CA Incremental resistancecoefficient for model shipcorrelation

RA / (S q) 1

CAA CAA Air or wind resistancecoefficient

RAA / (AV qR) 1

CD CD Drag coefficient D / (S q) 1

CF CF Frictional resistancecoefficient of a body

RF / (S q) 1

CFO CFO Frictional resistancecoefficient of acorresponding plate

RFO / (S q) 1

Cp CP Local pressure coefficient 1

CPR CPR Pressure resistancecoefficient, including waveeffect

1

CPV CPV Viscous pressure resistancecoefficient

RPV / (S q) 1

CR CR Residuary resistancecoefficient

RR / (S q) 1

CS CSR Spray resistance coefficient RS / (S q) 1



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SI-Unit

CT CT Total resistance coefficient RT / (S q) 1

CTL CTLT Telfer's resistancecoefficient

g R L / (∆V2) 1

CTQ CTQ Qualified resistancecoefficient

CT∇ / (ηH ηR)

CT CTVOL Resistance displacement RT / (∇2/3 q) 1

CV CV Total viscous resistancecoefficient

RV / (S q) 1

CW CW Wavemaking resistancecoefficient

RW / (S q) 1

CWP CWP Wave pattern resistancecoefficient, by waveanalysis

1

C CIRCC R.E. Froude's resistancecoefficient

1000 R / ∇ K3 1

Cτ CFUL Local friction coefficientbased on velocity at theedge of the boundary layerat y=δ

Cτ = τW / (ρ Uδ2 / 2) 1

d DJWS Jones wake strength 1

F CIRCF R.E. Froude's frictionalresistance coefficient

1000RF / (∆ K2) 1

K K Three dimensional formfactor on flat plate friction

(CV - CFO) / CFO 1

K CIRCK R.E. Froude's speeddisplacement coefficient

(4 π Fn∆ )1/2

KR KR Resistance coefficientcorresponding to KQ,KT

R / (ρ D4 n2) 1

cSf CSF Sinkage coefficient at FP ∆TF / L 1

cSa CSA Sinkage coefficient at AP ∆TA / L 1

cS CSNK Sinkage coefficient (∆TF + ∆TA) / L 1

cτ CTRIM Trim coefficient (∆TA - ∆TF) / L 1



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SI-Unit

2.2.1.3 Computer symbols for attributes

FW Fresh water

MF Faired model data

MR Raw model data

OW Open water

SF Faired full scale data

SR Raw full scale data

SW Salt water


Version 1993 2.2.2 Ship Performance 83

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SI-Unit

2.2.2 Ship Performance


FD FDT Drag reduction Towing force applied to themodel in a self propulsiontest carried out at the shipself-propulsion point

N

FP FP Force pulling or towinga ship

N

FPO FPO Pull during bollard test N

n N Frequency, commonly rateof revolution

Hz

PB PB Brake power W

PD , PP PD,PP Delivered power,propeller power

Q ω W

PE , PR PE,PR Effective power,resistance power

R V W

Pi PI Indicated power Determined from pressuremeasured by indicator

W

PS PS Shaft power Power measured on theshaft

W

PT PTH Thrust power T VA W

Q Q Torque PD / ω Nm

V V Ship speed m/s

VA VA Propeller advance speed Equivalent propeller openwater speed based on thrustor torque identity

m/s

zV ZV Sinkage of model or shipdue to forward speed

m

ω V0,OMN Angular shaft velocity 2 π n rad/s



ITTCSymbol

ComputerSymbol


SI-Unit


a RAUG Resistance augment fraction (T + Fp) / RT - 1 1

CD∇ CDVOL Power-displacementcoefficient

PD / (ρ V3 ∇2/3 / 2) 1

CN CN Trial correction forpropeller rate of revolutionat speed identity

nT / nS 1

CNP CNP Trial correction forpropeller rate of revolutionat power identity

PDT / PDS 1

CP CP Trial correction fordelivered power

1

K1 C1 Ship model correlationfactor for propulsiveefficiency

ηDS / ηDM 1

K2 C2 Ship model correlationfactor for propeller raterevolution

nS / nM 1

CADM CADM Admiralty coefficient ∆2/3 V3 / PS 1

s SINK Sinkage, dynamic Change of draft, fore andaft, divided by length

1

t TRIM Trim, dynamic Change of the trim due todynamic condition, dividedby length

1

t THDF Thrust deduction fraction 1 - (RT - FP) / T 1

w WFT Taylor wake fraction ingeneral

(V - VA) / V 1

wF WFF Froude wake fraction (V - VA) / VA 1

wQ WFTQ Taylor torque wake fraction Propeller speed VA

determined from torqueidentity

1



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SI-Unit

wT WFTT Taylor thrust wake fraction Propeller speed VA

determined from thrustidentity

1

∆w DELWC Ship-model correlationfactor for wake fraction

WT,M - WT,S 1

∆Wc DELWC Ship-model correlationfactor with respect to WT,s

method formula of ITTC1978 method

1

x XLO Load fraction in powerprediction

ηD PD / PE - 1 1

β APSF Appendage scale effectfactor

Ship appendage resistancedivided by modelappendage resistance

1

2.2.2.3 Efficiencies etc

ηB ETAB,EFTP

Propeller efficiency behindship

PT / PD = T VA / (Q ω) 1

ηD ETAD,EFRP

Propulsive efficiency orquasi-propulsive coefficient

PE / PD = PR / PP 1

ηG ETAG,EFGP

Gearing efficiency 1

ηH ETAH,EFRT

Hull efficiency PE / PT = PR / PT

= (1 - t) / (1 - w)1

ηM ETAM,EFSI

Mechanical efficiency PS / P1 or PB / P1 1

ηO ETAO Propeller open waterefficiency

1

ηR ETAR,EFRO

Relative rotative efficiency ηB / ηO 1

ηS ETAS,EFPS

Shafting efficiency PD / PS = PP / PS 1



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SI-Unit

2.2.2.4 Remarks

.1 Basic Concepts

Traditionally the basic concepts resistance and propeller advance speed are implicitely understoodto have certain traditional operational, i. e. experimental interpretations, namely in terms of hulltowing and propeller open water tests, respectively. Very clearly these are not the only possibleinterpretations. In many cases, where the traditional interpretations are not possible, as in the caseof full scale ships under service conditions, or where they are not meaningful, as e. g. in the case ofwake adapted propellers, more adequate conventional interpretations have to be agreed upon.

The traditional set of basic concepts for the ship performance analysis is incomplete. It does e. g. notallow for the separation of displacement and energy wakes, fundamental for the analysis of hull-propeller interaction.


Version 1993 2.2.3 Propulsor Performance 87

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2.2.3 Propulsor Performance


AO AO Propeller disc area π D2 / 4 m2

D DP Propeller diameter m

n FR Propeller frequency ofrevolution

Hz

kS KS Roughness height ofpropeller blade surface

m

qA QA Dynamic pressure based onadvance speed

ρ VA2 / 2

s. Remark .1Pa

qS QS Dynamic pressure based onadvance speed

ρ VS2 / 2 Pa

QS QSP Spindle torque About spindle axis ofcontrollable pitch propeller QS=QSC+ QSH positive if it increases pitch

Nm

QSC QSPC Centrifugal spindle torque Nm

QSH QSPH Hydrodynamic spindletorque

Nm

T TH Propeller thrust N

TD THDU Duct thrust N

TP THP Ducted propeller thrust N

TT THT Total thrust of a ductedpropeller unit

N

VA VA Advance speed of propeller m/s

VP VP Mean axial velocity atpropeller plane of ductedpropeller

m/s

VS VS Section advance speedat 0.7 R

(VA2+ (0.7 R ω)2)1/2

s. Remark .2m/s

ρP DNP Propeller mass density kg/m3

ω V0P Propeller angular velocity 2 π n 1/s



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BP BP Taylor's propellercoefficient base ondelivered horse power

n PD½ / VA

2.5

with n is revs/min, PD in horsepower,and VA in knots (obsolete)

1

BU BU Taylor's propellercoefficient based on thrusthorsepower

n PT½ / VA

2.5

with n is revs/min, PT in horsepower,and VA in knots (obsolete)

1

CP CPD Power loading coefficient PD / (AP qA VA) 1

CQ* CQS Torque index Q / (AP qS) 1

CTh CTH Thrust loading coefficient,energy loading coefficient

T / (AP qA) = (TP / AP) / qA

1

CT* CTHS Thrust index T / (AP qS) 1

J , JP JEI, JP Propeller advance ratio VA / (D n) = VP / (D n) 1

JA , JH JA, JH Apparent or hull advanceratio

V / (D n) = VH / (D n) 1

JT , JPT JT, JPT Advance ratio of propellerdetermined from thrustidentity

1

JQ , JPQ JQ, JPQ Advance ratio of propellerdetermined from torqueidentity

1

KP KP Delivered power coefficient PD / (ρ n3 D5) = 2 π KQ 1

KQ KQ Torque coefficient Q / (ρ n2 D5) 1

KSC KSC Centrifugal spindle torquecoefficient

QSC / (ρP n2 D5) 1

KSH KSH Hydrodynamic spindletorque coefficient

QSH / (ρ n2 D5) 1



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KT KT Thrust coefficient T / (ρ n2 D4) 1

KTD KTD Duct thrust coefficient TD / (ρ n2 D4) 1

KTP KTP Ducted propeller thrustcoefficient

TP / (ρ n2 D4) 1

KTT KTT Total thrust coefficient for a ducted propeller unit

KTP+ KTD 1

KQO KQO Torque coefficient ofpropeller converted frombehind to open watercondition

KQ.ηR 1

KQT KQ Torque coefficient ofpropeller determined fromthrust coefficient identity

1

PJ PJ Propeller jet power ηTJ T VA

SA SRA Apparent slip ratio 1 - V / (n P) 1

SR SRR Real slip ratio 1 - VA / (n P) 1

δ ADCT Taylors's advancecoefficient

n D / VA with n in revs/min, D in feet, VA in knots (obsolete)

1

ηJP EFJP Propeller pump or hydraulicefficiency

PJ / PD = PJ / PP 1

ηJP0 ZETO,EFJPO

Propeller pump efficiency atzero advance speed,alias static thrust coefficient

T / (ρ π / 2)1/3 / (PD D)2/3 1

ηI EFID Ideal propeller efficiency Efficiency in non-viscousfluid

1

ηTJ EFTJ Propeller jet efficiency 2 / (1 + (1 + CTh)1/2) 1

ηO , ηTPO ETAO,EFTPO

Propeller efficiency in openwater

PT / PD = T VA / (Q ω) allquantities measured in openwater tests

1

λ ADR Advance ratio of a propeller VA / (n D) / π = J / π 1



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τ TMR Ratio between propellerthrust and total thrust ofducted propeller

TP / TT 1

2.2.3.3 Induced Velocities etc

UA UA Axial velocity induced bypropeller

m/s

UAD UADU Axial velocity induced byduct of ducted propeller

m/s

URP URP Radial velocity induced bypropeller of ductedpropeller

m/s

URD URDU Radial velocity induced byduct of ducted propeller

m/s

UAP UAP Axial velocity induced bypropeller of ductedpropeller

m/s

UR UR Radial velocity induced bypropeller

m/s

UTD UTDU Tangential velocity inducedby duct of ducted propeller

m/s

UTP UTP Tangential velocity inducedby propeller of ductedpropeller

m/s

UT UT Tangential velocity inducedby propeller

m/s

β BETB Advance angle of apropeller blade section

arctg (VA / (R ω)) rad

β1 BET1 Hydrodynamic flow angleof a propeller blade section

Flow angle taking intoaccount induced velocity

rad

β* BETS Effective advance angle arctg (VA/ (0.7 R ω)) rad



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2.2.3.4 Remarks

.1 Dynamic Pressure

It has become bad practice to write

q = ρ/2 V2

for the dynamic pressure. This is not meaningful and should be avoided.

.2 Section Advance Speed

In the earlier versions of this list the notation for the concept of section advance speed deterioratedto the completely meaningless form

VS = (VA2+ (0.7 π n D)2)1/2 ,

hiding the very simple meaning of the concept.


Version 1993 2.2.4 Unsteady Propeller Forces 92

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2.2.4 Unsteady Propeller Forces

2.2.4.1 Concepts

Cuv SI(U,V) Generalized stiffness s. Remark .1

Duv DA(U,V) Generalized damping s. Remark .1

Fu FG(I) Generalized vibratoryforce

u = 1,.., 6u = 1, 2, 3: forceu = 4, 5, 6: moment

NN

Nm

Fi F(I) Vibratory force i = 1, 2, 3 N

KFu KF(U) Generalized vibratoryforce coefficients

According to definitions ofKFi and KMi

1

KFi KF(I) Vibratory forcecoefficients

Fi / (ρ n2 D4) 1

KMi KM(I) Vibratory momentcoefficients

Mi / (ρ n2 D5) 1

Kp KPR Pressure coefficient p / (ρ n2 D2) 1

Mi M(I) Vibratory moment i = 1, 2, 3 Nm

Muv MA(U,V) Generalized mass s. Remark .1

p PR Pressure Pa

Ru R(U) Generalized vibratorybearing reaction

u = 1,.., 6u = 1, 2, 3: forceu = 4, 5, 6: moment

NN

Nm

Vi V(I) Velocity field of the wake i = 1, 2, 3 m/s

xyz

XYZ

Cartesian coordinates Origin O coinciding withthe centre of the propeller.The longitudinal x-axiscoin-cides with the shaftaxis, positive forward; thetrans-verse y-axis, positiveto port; the third, z-axis,positive upward

mmm


Version 1993 2.2.4 Unsteady Propeller Forces 93

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xar

XATTR

Cylindrical coordinates Cylindrical system withorigin O and longitudinal x-axis as defined before;angular a-(attitude)-coordinate , zero at 12o'clock position, positiveclockwise looking forward,r distance measured fromthe x-axis

m1

m

u DP(U) Generalized vibratorydisplacement

u = 1,.., 6u = 1, 2, 3: linearu = 4, 5, 6: angular

mm

rad

u DPVL(U) Generalized vibratoryvelocity


m/sm/s

rad/s

u DPAC(U) Generalized vibratoryacceleration


m/s2

m/s2

rad/s2

2.2.4.2 Remarks

.1 General concepts

The generalized concepts have been introduced in Section 1. General Mechanics. .2 Equation of motion

In terms of the notation introduced the linear equation of motions may be rendered in the conciseform

Muv v +Duv v + Cuv v = Fu .

In spectral terms it is just as simple

(Muv (iω)2 +Duv iω + Cuv) Sv = FS

u .

ITTC Symbols 2 Ships in General2.3 Manoeuvring and Seakeeping

Version 1993 2.3.1 Manoeuvring 94

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2.3 Manoeuvring and Seakeeping

s. Remark .1

2.3.1 Manoeuvring

2.3.1.1 Geometrical Concepts

AFB AFBO Area of bow fins m2

AFS AFST Area of stern fins m2

AHL AHLT Lateral area of the hull The area of the profile ofthe underwater hull of aship when projectednormally upon the longi-tudinal centre plane

m2

ALV AHLV Lateral area of hull abovewater

m2

AR ARU Total lateral area of rudder m2

ARmov ARMV Movable area of rudder m2

ARN ARNO Nominal area of rudder (AR + ARmov) / 2 m2

bR SPRU Rudder span m

bRM SPRUME Mean span of rudder m

CAL CAHL Coefficient of lateral area ofship

AHL / (L T) 1

h DE Water depth m

hM DEME Mean water depth m

xR XRU Longitudinal position ofrudder axis

m

λR ASRU Aspect ratio of rudder bR2 / AR 1

2.3.1.2 Motions and Attitudes

p OX, P Roll velocity, angularvelocity about body x-axis

1/s



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q OY, Q Pitch velocity, angularvelocity about body y-axis

1/s

r OZ, R Yaw velocity, angularvelocity about body z-axis

1/s

p OXRT, PR Roll acceleration, angularacceleration about body x-axis

dp / dt 1/s2

q OYRT, QR Pitch acceleration, angularacceleration about body y-axis

dq / dt 1/s2

r OZRT, RR Yaw acceleration, angularacceleration about body z-axis

dr / dt 1/s2

u VX, U Surge velocity, linearvelocity along body x-axis

m/s

v VY, V Sway velocity, linearvelocity along body y-axis

m/s

w VZ, W Heave velocity, linearvelocity along body z-axis

m/s

u VXRT, UR Surge acceleration, linearacceleration along body x-axis

du / dt m/s2

v VYRT, VR Sway acceleration, linearacceleration along body y-axis

dv / dt m/s2

w VZRT, WR Heave acceleration, linearacceleration along body z-axis

dw / dt m/s2

V V Linear velocity of origin inbody axes

m/s

VA,VO VA, VO Approach speed m/s

Vu V(U) Generalized velocity

u V V(U) Generalized acceleration

VF VF Flow or current velocity m/s



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VWR VWRL Relative wind velocity m/s

VWT VWAB True wind velocity m/s

ψ YA Yaw or course angle rad

ψ YART Rate of change of course rad/s

ψ0 YAOR Original course rad

θ PI Pitch angle rad

φ RO Roll angle rad

2.3.1.3 Flow Angles etc

α AAPI Pitch angle Angle of attack in pitch onthe hull

rad

β AADR Drift angle Angle of attack in yaw onthe hull

rad

βWR ANWIRL Angle of attack of relativewind

1

δeff ANRUEF Effective rudder inflowangle

rad

δ0 ANRU0 Neutral rudder angle 1

δB ANFB Bow fin angle rad

δS ANFS Stern fin angle rad

δR ANRU Rudder angle 1

δR0 ANRUOR Rudder angle, ordered 1

ψC COCU Course of current velocity 1

ψWA COWIAB Absolute wind direction rad

ψWR COWIRL Relative wind direction rad



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2.3.1.4 Forces and Derivatives

s. Remark .2

Dhuv DH(U,V) Generalized hydrodynamic

damping∂Fh

u / ∂Vv

Fhu FH(U) Generalized hydrodynamic

force

Ihuv IH(U,V) Genralized hydrodynamic

inertia∂Fh

u / ∂ v V

K MX Roll moment on body,moment about body x-axis

Nm

M MY Pitch moment on body,moment about body y-axis

Nm

N MZ Yaw moment on body,moment about body z-axis

Nm

Nr NR Derivative of yaw momentwith respect to yaw velocity

∂N / ∂r Nms

N r NRRT Derivative of yaw momentwith respect to yawacceleration

∂N / ∂ r Nms2

Nv NV Derivative of yaw momentwith respect to swayvelocity

∂N / ∂v Ns

N v NVRT Derivative of yaw momentwith respect to swayacceleration

∂N / ∂ v Nms2

Nδ ND Derivative of yaw momentwith respect to rudder angle

∂N / ∂δ Nm

QFB QFB Torque of bow fin Nm

QR QRU Torque about rudder stock Nm

QFS QFS Torque of stern fin Nm

X FX Surge force on body, forcealong body x-axis

N

XR XRU Longitudinal rudder force N



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Xu XU Derivative of surge forcewith respect to surgevelocity

∂X / ∂u Ns/m

X u

XURT Derivative of surge forcewith respect to surge acceleration

∂X / ∂ u Ns2/m

Y FY Sway force on body, forcealong body y-axis

N

Yr YR Derivative of sway forcewith respect to yaw velocity

∂Y / ∂r Ns

YR YRU Transverse rudder force N

Y r

YRRT Derivative of sway forcewith respect to yawacceleration

∂Y / ∂ r Ns2

Yv YV Derivative of sway forcewith respect to swayvelocity

∂Y / ∂v Ns/m

Y v YVRT Derivative of sway forcewith respect to swayacceleration

∂Y / ∂ v Ns2/m

Yδ YD Derivative of sway forcewith respect to rudder angle

∂Y / ∂δ N

Z FZ Heave force on body, forcealong body z-axis

N

2.3.1.5 Linear Models

Cr CRDS Directional stabilitycriterion

Yv (Nr - muxG) - - Nv (Yr - mu)

N2s2

Lb LSB Static stability lever Nv / Yv m

Ld LSR Damping stability lever (Nr - muxG) / (Yr - mu) m

T TIC Time constant of the 1storder manoeuvring equation

s



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T1 TIC1 First time constant ofmanoeuvring equation

s

T2 TIC2 Second time constant ofmanoeuvring equation

s

T3 TIC3 Third time constant ofmanoeuvring equation

s

K KS Gain factor in linearmanoeuvring equation

1/s

Pn PN P-number, heading changeper unit rudder angle in oneship length

1

2.3.1.6 Turning Circles

DC DC Steady turning diameter m

DC DCNO Non-dimensional steadyturning diameter

DC / LPP 1

D0 DC0 Inherent steady turningdiameter δR = δ0

m

D0 DC0N Non-dimensional inherentsteady turning diameter

D0 / LPP 1

lr LHRD Loop height of r-δ curvefor unstable ship

1/s

lδ LWRD Loop width of r-δ curve forunstable ship

1

rC OZCI Steady turning rate 1/s

rC OZCINO Non-dimensional steadyturning rate

rC LPP / UC or 2 LPP / DC m

RC RC Steady turning radius m

t90 TI90 Time to reach 90 degreechange of heading

s

t180 TI180 Time to reach 180 degreechange of heading

s



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UC UC Speed in steady turn m/s

x090 X090 Advance at 90° change ofheading

m

x0180 X0180 Advance at 180° change ofheading

m

x0max XMX Maximum advance m

y090 Y090 Transfer at 90° change ofheading

m

y0180 Y0180 Transfer at 180° change ofheading, tactical diameter

m

y0max Y0MX Maximum transfer m

βC DRCI Drift angle at steady turning rad

2.3.1.7 Zig-Zag Manoeuvres

ta TIA Initial turning time s

tc1 TIC1 First time to check yaw(starboard)

s

tc2 TIC2 Second time to check yaw(port)

s

thc TCHC Period of changes inheading

s

tr TIR Reach time s

y0max Y0MX Maximum transversedeviation

m

δmax ANRUMX Maximum value of rudderangle

rad

ψS PSIS Switching value of courseangle

rad

ψ01 PSI01 First overshoot angle rad

ψ02 PSI02 Second overshoot angle rad



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2.3.1.8 Stopping Manoeuvres

sF SPF Distance along track,track reach

m

x0F X0F Head reach m

y0F Y0F Lateral deviation m

tF TIF Stopping time s

2.3.1.9 Remarks

.1 Solid Body Motions

The whole Chapter 2.3 on Manoeuvring and Seakeeping relies heavily on the Section 1 on GeneralMechanics, Chapter 1.2 on Solid Body Mechanics in particular. Members of the ManoeuvringCommittee are strongly urged to try and finalize the work that has been started for them.

.2 Derivatives

The traditional notation for the "stability" derivatives is not very efficient and not in accordance withthe notation outlined in Section 1 on General Mechanics. Instead of completely denoting the conceptsof generalized hydrodynamic damping and inertia, respectively, by adequate symbols, the traditionalsymbols indicate some measuring procedures for the components.


Version 1993 2.3.2 Seakeeping 102

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2.3.2 Seakeeping

2.3.2.1Basic Concepts

s. Remark .1

ai AT(I) Attitudes of the floatingsystem

i = 1, 2, 3, e. g. Eulerangles roll, pich, and yaw,respectively

rad

f FR Frequency 1 / T Hz

fe FE Frequency of waveencounter

1 / Te Hz

fz Natural frequency of heave 1 / Tz Hz

fθ Natural frequency of pitch 1 / Tθ Hz

fφ Natural frequency of roll 1 / Tφ Hz

FL FS(2) Wave excited lateral shearforce

Alias horizontal!s. Remark .2

N

FN FS(3) Wave excited normal shearforce

Alias vertical!s. Remark .2

N

ML MB(3),FS(6)

Wave excited lateralbending moment

Alias horizontal!s. Remark .2

Nm

MN MB(2),FS(5)

Wave excited normalbending moment

Alias vertical!s. Remark .2

Nm

MT MT(1),FS(4)

Wave excited torsionalmoment

Nm

nAW NAW Mean increase of rate ofrevolution in waves

1/s

PAW PAW Mean power increase inwaves

W

QAW QQAW Mean torque increase inwaves

Nm

RAW RAW Mean resistance increase inwaves

N



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Sη(f),Sηη(f),Sη(ω),Sηη(ω)

EWSF,

EWSC

Wave elevation autospectral density

m2s

xi X(I) Absolute displacement ofthe ship at the referencepoint

u = 1, 2, 3 :surge, sway, and heave respectively

m

xu X(U) Generalized absolutedisplacement of the shipat the reference point

u = 1,.., 6u = 1, 2, 3 : diplacementsu = 4, 5, 6 : attitudes,e. g. Euler angles

mrad

TAW TAW Mean thrust increase inwaves

N

T TC Wave period s

Te TE Wave encounter period s

Tz TNHE Natural period of heave s

Tθ TNPI Natural period of pitch s

Tφ TNRO Natural period of roll s

Yz(ω),Azζ(ω)

Amplitude of frequencyresponse function fortranslatory motions

za(ω) / ζa(ω) 1

Yθζ(ω),Aθζ(ω)

Amplitude of frequencyresponse function for rotarymotions

Θa(ω) / ζa(ω) or Θa(ω) / (ω2/ (gζa(ω)))

1

µ Wave encounter angle Angle between ship positivex axis and positive directionof waves (long crested) ordominant wave direction(short crested)

rad



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2.3.2.2 Remarks

.1 Related Information

Related information is to be found in Chapter 1 on General Mechanics, in Sections 1.1.3 on TimeRelated Concepts, 1.1.4 on Stochastic Processes, 1.2 on Solid Body Mechanics, and 1.4.1 onWaves.

Evidently this whole chapter has not reached a final state in any sense. Therefore the Seakeepingand Ocean Engineering Committees are strongly urged to come up with a detailed draft forconsideration by the SaT Group and inclusion into a future version of the SaT List.

.2 Sectional Loads

Sectional loads are meaningful only referred to body fixed coordinates. The traditionalterminology speaking of horizontal and vertical forces and moments, referring to space fixedcoordinates, is adequate only for very special conditions of little interest for the sectional loadsand should consequently be avoided as obsolete.

ITTC Symbols 3 Special Craft3.1 Ice going Vessels

Version 1993 3.1.2 Resistance and Propulsion 105

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3 Special Craft

3.1 Ice Going Vessels

3.1.2 Resistance and Propulsion

CI CI Coefficient of net iceresistance

RI / (ρI g h2 B) 1

CIW CIW Coefficient of waterresistance in the presence of ice

RIW / (S qIW) 1

FIN FNIC Normal ice force on a body Projection of hull-iceinteraction force on theexternal normal

N

FIT FTIC Tangential ice force on abody

Projection of the hull iceinteraction force on thedirection of motion

N

FnI FNIC Froude number based on icethickness

V / (g hI)1/2 1

FXI

FYI

FZI

FXICFYICFZIC

Components of the local ice force

NNN

fID CFRD Coefficient of frictionbetween surface of bodyand ice (dynamic)

Ratio of tangential force tonormal force between twobodies (dynamic condition)

1

fIS CFRS Coefficient of frictionbetween surface of bodyand ice (static)

The same as above (staticcondition)

1

hI HTIC Thickness of ice m

hSN HTSN Thickness of snow cover m

KQIA KQICMS Average coefficient oftorque in ice

QIA / (ρW nIA2 D5) 1

KTIA KTICMS Average coefficient ofthrust in ice

TIA / (ρW nIA2 D4) 1

nIA FRICMS Average rate of propellerrevolution in ice

Hz

ITTC Symbols 3 Special Craft3.1 Ice going Vessels


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PDI PDI Delivered power atpropeller in ice

2 π QIA nIA W

QIA QIMS Average torque in ice Nm

RI RI Net ice resistance RIT - RIW N

RIT RIT Total resistance in ice Ship towing resistance inice

N

RIW RIW Hydrodynamic resistance in presence of ice

Total water resistance ofship in ice

N

TIA TIMS Average total thrust in ice N

ηICE ERIC Relative propulsiveefficiency in ice

ηID / ηD 1

ηID EFDIC Propulsive efficiency in ice RIT V / (2 π nIA QIA) 1

ITTC Symbols 3 Special Craft3.2 Planing Boats

Version 1993 3.2.1 Geometry and Hydrostatics 107

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3.2 Planing Boats

3.2.1 Geometry and Hydrostatics

APB APB Planing bottom area Horizontally projectedplaning bottom area (atrest), excluding area ofexternal spray strips

m2

ΒLCG BLCG Beam at LCG Breadth over spray strips atLCG

m

ΒC BC Beam over chines Beam over chines,excluding external spraystrips

m

ΒPA BPA Mean breadth over chines APB / LPR m

ΒPTR BPTR Transom breadth Breadth over chines attransom, excluding externalspray strips

m

ΒPX BPX Maximum breadth overchines

Maximum breadth overchines, excluding externalspray strips

m

dTR DTRA Immersion of transom Vertical depth of trailingedge of boat at keel belowwater surface level

m

e EN Lever of bottom normalforce N

Distance between N andcenter of gravity (measurednormally to N)

m

eAP ENA Lever of appendage liftforce NAP

Distance between NAP andcenter of gravity (measurednormally to NAP)

m

eF ENF Lever of propeller normalforce FZ

Distance between propellercenterline and center ofgravity (measured alongshaft line)

m

eP ENP Lever of resultant ofpropeller pressure forcesNP

Distance between NP andcenter of gravity (measurednormally to NP)

m



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eR ENPR Lever of resultant of rudderpressure forces NPR

Distance between NPR andcenter of gravity (measurednormal to NPR)

m

eS ENS Lever of resultant propellersuction forces NS

Distance between NS andcenter of gravity (measurednormal to NS)

m

fAA FRAA Lever of wind force RAA Distance between RAA andcenter of gravity (measurednormal to RAA)

m

fAP FRAP Lever of appendage dragRAP

Distance between RAP andcenter of gravity (measurednormal to RAP)

m

fF FRF Lever of frictionalresistance RF

Distance between RF andcenter of gravity (measurednormal to RF)

m

fK FRK Lever of skeg or keelresistance RK

Distance between RK andcenter of gravity (measurednormal to RK)

m

fR FDRR Lever of augmented rudderdrag ∆RRP

Distance between ∆RRP andcenter of gravity (measurednormal to ∆RRP)

m

fS FSL Lever of axial propellerthrust

Distance between axialthrust and center of gravity(measured normal to shaftline)

m

hP HSP Wetted height of strutpalms

m

hR HRU Wetted height of rudders m

LC LC Wetted chine length m

LCP LCP Lever of resultant ofpressure forces

Distance between center ofpressure and aft end ofplaning surface

m

LK LK Wetted keel length m

Lm LM Mean wetted length (LK + LC) / 2 m



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LSH LSHB Total length of shafts andbossings

m

LPR LPRC Projected chine length Length of chine projected ina plane parallel to keel

m

SWHP SWHP Wetted area underway ofplaning hull

Principal wetted areabounded by trailing edge,chines and spray root line

m2

α ALFB Angle of attack of the afterportion of the bottom

1

β BETD Deadrise angle of planingbottom

Angle between a straightline approximating bodysection and the intersectionof the basis plane with thesection plane

1

γ GAMSP Spray angle Angle between stagnationline and keel line (measuredin plane parallel to keel)

1

λW LAMS Mean wetted length-beamratio

(LK + LC) / (2 BLCG) 1

τ TAUB Angle of attack of bottom Angle between planingbottom or mean buttock lineand horizontal

1

φ PHISP Spray angle Angle between stagnationline and keel (measured inplane of bottom)

1

∆λ DLAM Dimension increase in totalfriction area

Effective increase in frictionarea length-beam ratio dueto spray contribution todrag

1



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LO CLOD Lift coefficient for zerodeadrise

∆ / (BLCG2 q) 1

CL CLBET Lift coefficient for deadrisesurface

∆ / (BLCG2 q) 1

CV CSP Speed coefficient V / (BLCG g)1/2 1

C∆ CDL Load coefficient ∆ / (BLCG3 ρ g ) 1

NAP NAP Appendage lift forces Lift forces arising fromappendages inclined toflow, assumed to actnormally to keel line

N

∆RP DELRP Resistance component dueto pressure force

RP sinτ N

∆DR DELDR Augmented rudder drag Augmented drag from therudder due to the propellerslip-stream

N

ITTC Symbols 3 Special Craft3.3 Semi-displacement Vessels


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3.3 Semi-displacement Vessels


AAP AAP Appendage area Reference area of anappendage element

m2

AFR AFR Frontal area Projected frontal area of aprotruding body

m2

BTR BTR Breadth moulded oftransom at design water line

m

BX BX Maximum molded breadthof design water line

m

BM BM Breadth moulded ofmidship section at designwater line

m

FWL FWL Wetted length factor LWE / LWL 1

FWS FWS Wetted area factor SWHE / SW(v=0) 1

HTC HTC Hull tip clearance Distance between thepropeller sweep circle andthe hull

m

LF LF Length of flap or wedge Measured in directionparallel to keel

m

LK LK Wetted length of keel m

LW LWT Wetted length (general) m

NPR NPR Number of propellers 1

SAP SAP Wetted surface ofappendages

m2

SW SW Wetted surface (general) m2

SWAPE SWAPE Total wetted surfaceincluding the wetted surface of the appendages

Total wetted surface of thehull underway, includingspray area, wetted side areaand wetted surface of theappendages, without wettedtransom area

m2



ITTCSymbol

ComputerSymbol


SI-Unit

SWB SWB Wetted bottom area Area bounded by stagnationline, chines or water surfaceunderway and transom

m2

SWH SWH Wetted surface of the hullwithout wetted surface ofappendages (general)

m2

SWHE SWHE Wetted hull area Total wetted surface of thehull underway, includingspray area and wetted sidearea, without wettedtransom area

m2

SWHS SWSH Area of wetted sides Wetted area of the hull sideabove the chine or thedesign water line

m2

SWS SWS Area wetted by spray Wetted area between designline or stagnation line andspray edge

m2

SWAP SWAP Area of hull andappendages at rest

Total wetted area of hulland appendages at rest

m2

SW(v=0) SWV0 Wetted surface of the hullat rest

Immersed area of the hullwithout the transom area

m2

TH THUL Draft of the hull Maximum draft of the hullwithout keel or skeg

m

TTR TTR Draft at transom Maximum immersion oftransom

m

TTRM TTRM Mean immersion of thetransom

ATR / BT m

XBL XBL Longitudinal position ofcalculated boundary layerthickness

Distance from leading edgeto the point of calculatedboundary layer thickness

m

αB ALFSL Angle of stagnation line Angle between keel andstagnation line in plane ofkeel

rad

αBABR ALFBAR Barrel flow angle Angle between barrel axisand assumed flow lines

rad



ITTCSymbol

ComputerSymbol


SI-Unit

αSH ALFSH Shaft flow angle Angle between shaft axisand assumed flow lines

rad

αO ALFO Buttock inclination Angle between tangent toafterbody buttock lines andhorizontal plane at rest

rad

β BETAD Angle of deadrise (general) Angle between a straightline approximating thesection in a specific regionand the intersection of thebasis plane with the sectionplane

rad

δF DELFS Flap angle Angle between the planingsurface of a flap and thebottom before the leadingedge

rad

δW DELWG Wedge angle Angle between the planingsurface of a wedge and thebottom before the leadingedge

rad

∆AP MAAP Displacement force of theappendages

N

∆WAP MAWAP Displacement force withappendages

N

δTA DTA Change of draft at AP Sinkage or rise of aftperpendicular

m

δTF DTF Change of draft at FP Sinkage or rise of foreperpendicular

m

δTLCG DTLCG Change of draft at LCG Sinkage or rise of center ofgravity

m

εO EPSO Shaft inclination Angle between shaft axisand flow direction

rad

εS EPSS Relative shaft inclination Angle between shaft axisand mean buttock line

rad

εSO EPSSO Shaft angle Angle between shaft axisand horizontal plane at rest

rad



ITTCSymbol

ComputerSymbol


SI-Unit

θ TRIM Running trim angle Angle between ship's designwater line and theundisturbed water surfaceunderway

rad

θ0 TRIM0 Trim angle at rest Angle between ship's designwater line and theundisturbed water surface atrest

rad

τ TAUBUT Angle of attack of thespecific buttock

Angle between flow and aspecific buttock

rad

ψS PSISI Shaft inclination Angle between shaft axisand horizontal planeunderwayεSO + θ

rad

∇AP DISVAP Displacement volume ofthe appendages

∆ / (ρ g) m3

∇WAP DISWAP Displacement volumeincluding the volume of theappendages

m3



ITTCSymbol

ComputerSymbol


SI-Unit


CDAP CDAP Specific pressure dragcoefficient of theappendages

CTAP - CF 1

CRH CRH Specific residuary resistancecoefficient of theappendaged hull, excludingthe appendages drag

(RTwAP - RF - RAP) / / (Sw q)

1

CRwAP CRWAP Specific residuary resistancecoefficient of the hull withappendages

(RTwAP - RF) / (Sw q) 1

CRwoAP CRWOAP Specific residuarycoefficient of the hullwithout appendages

(RwoAP - RF) / (Sw q) 1

CDP CDP Specific pressure dragcoefficient (general)

RP / (A q) 1

CDINT CDINT Specific interference dragcoefficient (general)

DINT / (A q) 1

DINT DINT Interference drag (a force) N

FDAP FDAP Towing force in a selfpropulsion test including the thrust reduction due to the scale effects of themodel appendage drag

(CFM - CFS - CA)(SWH q)M +KAP RAP,M

N

FN FN Propeller normal force Propeller force normal toshaft and in vertical centerplane

N

FTR FTR Trial power factor (PEwAP + ∆PETR) / PEwAP 1

FX FX Horizontal component ofpropeller transverse force

FN sinψ N

FZ FZ Vertical component ofpropeller transverse force

FN cosψ N

Jψ ADVCPS Advance coefficient atoblique inflow

J cosψ 1



ITTCSymbol

ComputerSymbol


SI-Unit

KAP KAP Appendage correction factor

Scale effect correctionfactor for model appendagedrag applied at the towingforce in a self-propulsiontest

1

KN KN Normal force coefficient FN / (ρ n2 D4) 1

KQO KQO Torque coefficient, openwater condition

Torque coefficient forhomogeneous axial inflowcondition

KQB KQB Torque coefficient in behindcondition

1

KQψ KQPSI Torque coefficient underoblique inflow condition

Qψ / (ρ n2 D5) 1

KTψ KTPSI Thrust coefficient underoblique inflow condition

Tψ / (ρ n2 D4) 1

KTO KTO Thrust coefficient, openwater condition

Thrust coefficient forhomogeneous axial inflowcondition

1

KTB KTB Thrust coefficient in behindcondition

1

LA LAP Appendage lift force Resultant lift force arisingfrom appendages inclined toflow (assumed normal tokeel)

N

LD LIDY Vertical component ofhydrodynamic lift

N

LS LIBU Hydrostatic lift Due to buoyancy N

N NF Bottom force Resultant of pressure andbuoyant forces assumedacting normally to thebottom

N

NPP NPP Induced propeller force Resultant of propellerpressure forces actingnormally to the bottom

N



ITTCSymbol

ComputerSymbol


SI-Unit

NS NPS Induced propeller force Resultant of propellersuction forces actingnormally to the bottom

N

NYO NYO Horizontal bottom force Horizontal component ofbottom force. Hull withdeadrise N sinβ

N

NZ0 NZO Vertical bottom force Vertical component ofbottom force. Hull withdeadrise N cosβ

N

PDTRIAL PDTR Delivered power under trial conditions

(2 π Q n)TRIAL W

PEAA PEAA Effective power windresistance

RAA V W

PEAP PEAP Effective power appendagesdrag

RAP V W

PEPAR PEPAR Effective power parasiticdrag

RPAR V W

PETRIAL PETR Effective power under trialcondition

(RTwAP + ∑ ∆RTRIAL) V W

PEwAP PEWAP Effective power withappendages

RTwAP V W

PEwoAP PEWOAP Effective power withoutappendages

RTwoAP V W

QO QHO Torque, open watercondition

Torque at axial andhomogeneous inflowcondition

Nm

QB QB Torque in behind condition Nm

Qψ QPSI Torque at oblique inflowcondition

Nm

RF RF Frictional resistance Force assumed as actingparallel to keel or baseline

N

RFXO RFXO Horizontal component of RF Force opposite to motion N

RFZO RZFO Vertical component of RF Force normal to motion N

Rk RKEEL Keel drag N



ITTCSymbol

ComputerSymbol


SI-Unit

Rπ RPI Induced drag g ρ ∇ tg τ N

RPAR RPAR Parasitic drag Drag due to inlet and outletopenings

N

RRH RRH Residuary resistance of thenaked hull

N

RRwAP RRWAP Residuary resistanceincluding appendages drag

N

RSP RSP Pressure component ofspray drag

N

RSV RSV Viscous component of spray drag

CF SWS qS N

RTwoAP RWOAP Resistance of naked hull N

RTwAP RWAP Resistance of hull withappendages

N

TAψ THPSI Axial thrust under obliqueinflow condition

N

TB THB Thrust in behind condition N

TO THO Thrust, open watercondition

Thrust at axial andhomogeneous inflowcondition

N

TX THX Horizontal thrust component TA cosψ - FN sinψ N

TZ THZ Vertical thrust component TA sinψ + FN cosψ N

tψ THDFPS Thrust deduction fraction at oblique inflow

1 - RT / (TAcosψ - FN sinψ) 1

VABS VSABS Resultant velocity in sprayarea

m/s

VBm VBM Bottom velocity Mean velocity over bottomof the hull

m/s

VREL VSREL Spray velocity Velocity in direction of thespray

m/s



ITTCSymbol

ComputerSymbol


SI-Unit

WTψ WFTPSI Taylor wake fraction foroblique inflow

Taylor wake fractiondetermined from thrustidentity for oblique inflowcondition 1 - Jψ / V / (n D)

1

ZR ZR Resultant towing force atself-propulsion test undertrial condition

FDAP - RAA,M - RPAR,M - ∆RST,M - ∆RAW,M

N

∆PEAW PEAW Effective power due torippling sea

∆RAW V W

∆PEST PEST Effective power due tocourse keeping

∆RST V W

∆PETRIAL SDPE Effective power incrementdue to trial condition

PEAA + PEPAR + ∆PEAW +∆PEST

W

∆RAW DELAW Resistance increment due torippling seas

N

∆RRS DELTR Rudder drag increment Change of rudder drag dueto propeller slip-stream

N

∆RST DELST Resistance increment due tocourse keeping

N

ε EPSG Drag-weight ratio ingeneral

R / (g ρ ∇) 1

εH EPSH Residuary drag-weight ratio of the appendagedhull, excluding theappendage drag

(RTwAP - RF - RAP) / / (g ρ ∇)

1

εM EPSM Model drag-weight ratio RT,M / (g ρM ∇M) 1

εR EPSR Residuary drag-weight ratio of the naked hull

(RTwoAP - RF) / (g ρ ∇) 1

εRAP EPSRAP Residuary drag-weight ratioincluding the appendagedrag

(RTwAP - RF - KAP RAP) // (ρ g ∇)

1

εS EPSFS Full scale drag-weight ratioof the appendaged hull

RT,S / (g ρS ∇S ) 1



ITTCSymbol

ComputerSymbol


SI-Unit

ηAP ETAAP Appendage efficiency PEwoAP / PEwAP 1

ηOψ ETAPSI Propeller efficiency in openwater at non-axial inflowconditions

(KT cosψ - KN sinψ) J // (2 π KQ)

1

ITTC Symbols 3 Special Craft3.4 Catamarans


ITTCSymbol

ComputerSymbol


SI-Unit

3.4 Catamarans


AMX AMX Area of midship section ofdemi-hull without keel orskeg

m2

AXH AXH Maximum transverse sect-ion of demihull withoutkeel or skeg

m2

BAXDH BAXDH Breadth at maximumsection of demihull

At design water line m

BOADH BOADH Overall breadth of demihull m

BMDH BMDH Breadth of midship sectionat midship section anddesign water line

m

BPXDH BPXDH Maximun breadth overchines

Exclusive external sprayrails

m

BT BTWL Tunnel width Minimal distance of thedemihulls at the waterline

BXDH BXDH Maximum breadth ofdemihull

At design water line

CDK CLDK Deck clearance Minimum clearance of wetdeck from water surface atrest

m

iEI ANENIN Half angle of entrance attunnel (inner) side

Angle of inner water linewith reference to centre lineof demihull

rad

iEO ANENOU Half angle of entrance atouter side

Angle of outer water linewith reference to centre lineof demihull

rad

αCH ALFCH Angle of attack of the chineat 0.25 LPP

rad

αMB ALFMB Angle of attack of thebottom at 0.25 LPP

In the plane with maximumlongitudinal area

rad



ITTCSymbol

ComputerSymbol


SI-Unit

∆α DALF Twist of the bottom of theafterbody

Difference between theangleof attackof the centre linebuttock and the chine

rad

βFB BETFB Angle of deadrise at 0.9 LPP rad

βM BETM Angle of deadrise midships rad

βTR BETTR Angle of deadrise at tran-som

rad



ITTCSymbol

ComputerSymbol


SI-Unit


3.4.2.1 Resistance Components

RFCA RFCA Frictional resistance ofcatamarane

2 RFDH N

RFDH RFDH Frictional resistance ofdemihull

N

RFINT RFINT Frictional interferenceresistance

RFCA - 2 RFDH N

RRCA RRDH Residual resistance ofcatamarane

RTCA - RFCA N

RRDH RRDH Residual resistance ofdemihull

RTDH - RFDH N

RRINT RRINT Residual interferenceresistance

RRCA - 2 RRDH N

RTCA RFCA Total resistance of catama-rane

2 RFDH N

RTDH RFDH Total resistance of demihull N

RTINT RTINT Total interference resistance RTCA - 2 RTDH N

3.4.2.2 Power and Resistance Ratios

s. Remark .1

εP EP Power ratio in general P / (∆ V) 1

εPD EPD Delivered power ratio PD / (∆ V) 1

εR ER Drag weight ratio in general R / ∆ 1

εRTCA ERTCA Full scale total drag weightratio

RT,S / ∆S 1



ITTCSymbol

ComputerSymbol


SI-Unit

3.4.2.3 Remarks

.1 Ratios

The power and resistance ratios listed are examples only. According to the definition of thevarious power and resistance components other symbols may be created.

ITTC Symbols 3 Special Craft3.5 SWATH


ITTCSymbol

ComputerSymbol


SI-Unit

3.5 SWATH


AI AIA Strut-hull intersection area m2

BB BB Box beam Beam of main deck m

BS BS Hull spacing Distance between hullcenter lines

m

DH DHUL Hull diameter Diameter of axis symmetricsubmerged hulls

m

DX DX Hull diameter at thelongitudinal position "X"

m

hc HC Box clearance Clearance of main deckrelative to the water surfaceat rest

m

LCH LCH Length of center section ofhull

Length of prismatic part ofhull

m

LCS LCS Length of center section ofstrut

Length of prismatic part ofstrut

m

LH LH Box length Length of main deck m

LNH LNH Length of nose section ofhull

Length of nose section ofhull with variable diameter

m

LNS LNS Length of nose section ofstrut

Length of nose section ofstrut with variable thickness

m

LS LS Strut length Length of strut from leadingto trailing edge

m

LSH LSH Length of submerged hull m

LSS LSS Strut submerged depth Depth of strut from stillwater line to strut-hullintersection

m

LTS LTS Length of foil section ofstrut

Length of foil section ofstrut with variable thickness

m

ts TSTR Maximum thickness of strut m

ITTC Symbols 3 Special Craft3.5 SWATH


ITTCSymbol

ComputerSymbol


SI-Unit

tx TX Thickness of strut at thelongitudinal position "X"

m

ITTC Symbols 3 Special Craft3.6 Hydrofoil Boats


ITTCSymbol

ComputerSymbol


SI-Unit

3.6 Hydrofoil Boats

3.6.1 Geometry and Hydostatics

AF AFO Foil area (general) Foil area in horizontal plane m2

AFS AFS Submerged foil area m2

AFT AFT Total foil plan area c b m2

AFTO AFTO Total immersed foil planarea at take-off speed

c b m2

AR AR Aspect ratio b2 / A 1

AFR AFRE Reserve area of foil m2

ASS ASS Submerged strut area m2

BFOA BFOA Maximum breadth of foils m

b BSP Foil span (general) m

bF BSPF Flap span m

bST BSTR Transverse horizontaldistance of struts

m

bw BSPW Foil span wetted m

CC CHC Chord length at center plane(general)

m

CF CFL Chord length of flap m

CLm CLM Mean lift coefficient forfoils with twist

1

Cm CHM Mean chord length m

CPF CPFL Center of pressure on a foilor flap from leading edge

m

CS CSTR Chord length of a strut m

CSF CHSF Chord length of strut atintersection with foil

m

CT CHTI Chord length at foil tips m



ITTCSymbol

ComputerSymbol


SI-Unit

fL FML Camber of lower side(general)

m

fU FMU Camber of upper side m

FnL FNF Froude Number based onfoil distance

V / (g LF)1/2 1

FnC FNC Froude Number based onchord length

V / (g Cm)1/2 1

hC HVCG Height of center of gravityfoilborne

Distance of center ofgravity above mean watersurface foilborne

m

hF HFL Flight height Height of foil chord atfoilborne mode aboveposition at rest

m

hK HKE Keel clearance Distance between keel andmean water surfacefoilborne

m

LF LEFF Horizontal distance ofcenter of pressure of frontfoil to center of gravity

m

LR LERF Horizontal distance ofcenter of pressure of rearfoil to center of gravity

m

LS LSTR Length or span of struts m

M ML Load factor based ondisplacement weight ofvessel (general)

1

MF MLF Load factor front foil LF / (∇ ρ g) 1

TF TFO Foil immersion orsubmergence (general)

Distance between foil chordand mean water surface

m

TFD TFD Depth of submergence ofapex of a dihedral foil

Distance between foil apexand mean water surface

m

TFM TFOM Mean depth of foilsubmergence

m

WF WTF Weight of foil N



ITTCSymbol

ComputerSymbol


SI-Unit

xR XFRG Distance of real foil tocenter of gravity

m

αC ALFTW Geometric angle of twist 1

αm ALFM Angle of attack of mean lift coefficient for foils with twist

1

αs AFS Angle of attack for whichflow separation (stall)occurs

1

αTO ATO Incidence angle at take-offspeed

1

δ DELTT Thickness ratio of section(general)

t / C 1

δB DELTB Thickness ratio of trailingedge of struts

tB / CS 1

δF DELTF Camber ratio of mean line(general)

f / C 1

δFL DLTFL Angle of flap deflection 1

δL DELTL Camber ratio of lower sideof foil

fL / C 1

δS DELTS Thickness ratio of strut tS / CS 1

δSTH DELTT Theoretical thickness ratioof section

tS / CSTH 1

δU DELTU Camber ratio of upper side fu / C 1

ε EPSF Downwash angle 1

θ TETAD Dihedral angle Angle between foil paneland horizontal

1

τ TAUT Taper ratio CT / CC 1

Λ LAMBS Sweep angle Angle between a normal tofoil motion and the ¼ chordline

1

∇F DISVF Displacement volume of a foil

m3



ITTCSymbol

ComputerSymbol


SI-Unit


CDF CDF Drag coefficient of foil DF / (A q) 1

CDI CDI Induced drag coefficient DI / (b c q) 1

CDINT CDINT Interference drag coefficient DINT / (t2 q) 1

CDO CDO Section drag coefficient forangle of attack equal tozero

DP / (b c q) 1

CDS CDSP Spray drag coefficient DS / (t2 q) 1

CDVENT CDVENT Ventilation drag coefficient DVENT / (b t q) 1

CDW CDW Wave drag coefficient DW / (b TF q) 1

CLF CLF Flap lift coefficient LFL / (AFL q) 1

CLO CLO Profile lift coefficient forangle of attack equal tozero

L / (A q) 1

CLTO CLTO Lift coefficient at take-offcondition

L / (AF qTO) 1

CLX CLA Slope of lift curve dCL / dα 1

CM CM Pitching moment coefficient M / (AF (LF - LR) q) 1

DF DRF Foil drag Force in the direction ofmotion of an immersed foil

N

DFA DFA Drag force on aft foil CDF AFR q N

DFF DFF Drag force on front foil CDF AFF q N

DI DRIND Induced drag Due to lift N

DINT DRINT Interference drag Due to mutual interaction ofthe boundary layers ofintersecting foil

N

DP DRSE Section or profile drag Streamline drag N



ITTCSymbol

ComputerSymbol


SI-Unit

DS DRSP Spray drag Due to spray generation N

DST DRST Strut drag N

DW DRWA Wave drag Due to propagation ofsurface waves

N

DVENT DRVNT Ventilation drag Due to reduced pressure atthe rear side of the strutbase

N

LFA LFA Lift force on aft foil CL AFR q N

LFF LFF Lift force on front foil CL AFF q N

M MSP Section pitching moment CM AF (LF - LR) q Nm

VC VC Additional velocity due tocamber

m/s

VT VT Additional velocity due tothickness

m/s

ε EPS Drag-lift ratio (general) D / L 1

ITTC Symbols 3 Special Craft3.7 ACV and SES


ITTCSymbol

ComputerSymbol


SI-Unit

3.7 ACV and SES


AC CUA Cushion area Projected area of ACV orSES cushion on watersurface

m2

BC BCU Cushion beam SES cushion beammeasured between the sidewalls

m

BOAWL BOAWL Maximum breadth of SES At the water line m

BXSH BXSH Maximum breadth of sidehulls

At the water line m

FWS FWS Wetted surface factor SWHES / SWH 1

GH GH Height of center of gravityabove mean water planebeneath craft

m

hBS HBS Bow seal height Distance from side wallkeel to lower edge of bowseal

m

HSK HSK Skirt depth m

hSS HSS Stern seal height Distance from side wallkeel to lower edge of sternseal

m

LB LB Deformed bag contactlength

m

LC LAC Cushion length m

LE LACE Effective length of cushion AC / BC m

SWH SWH Wetted area of side hullsat rest off cushion

Total wetted area of sidewalls under way on cushion

m2

SWHES SWHES Wetted area of side hullsunder way on cushion

Total wetted area of sidewalls under way on cushion

m2

SWSW SWSW Wetted area of side hullsunder way off cushion

Total wetted area of sidewalls under way off cushion

m2

W WT Craft weight ρ g ∇ N



ITTCSymbol

ComputerSymbol


SI-Unit

XH XH Horizontal spacing betweeninner and outer side skirthinges or attachment pointsto structure

m

XS XS Distance of leading skirtcontact point out-board orouter hinge of attachmentpoint to structure

m

ZH ZH Vertical spacing betweeninner and outer side skirthinges or attachment pointsto structure

m

θBC TETBC Increase in cushion beamdue to water contact

m

θB TETB Bag contact deformationangle

1

θF TETF Finger outer face angle 1

θW TETW Slope of mean water planefor surface level beneathcushion periphery

1

ζC ZETAC Height of cushion generatedwave above mean waterplane at leading edge sideof the skirt

m



ITTCSymbol

ComputerSymbol


SI-Unit


C∆ CLOAD Cushion loading coefficient∆ / (g ρA AC3/2) 1

CR CR Drag coefficient 1

CPR CPR Aerodynamic profile drag R0 / (ρA VR2 AC / 2) 1

CWC CWC Cushion wavemakingcoefficient

1

pB PBM Mean bag pressure Pa

pBS PBS Bow seal pressure Pressure in the bow sealbag

Pa

pCE PCE Mean effective skirtpressure

Pa

pCU PCU Cushion pressure Mean pressure in thecushion area

Pa

pFT PFT Fan total pressure Pa

pLR PLR Cushion pressure to lengthratio

PCU / LC Pa/m

pSK PSS Skirt pressure in general Pa

pSS PSS Stern seal pressure Pressure in the stern sealbag

Pa

PFCU PFCU Power of lift fan kW

PFSK PFSK Power of skirt fan kW

QBS QBS Bow seal air flow rate Air flow rate to the bowseal

m3/s

QCU QCU Cushion air flow rate Air flow rate to cushion m3/s

QSS QSS Stern seal air flow rate Air flow rate to the sternseal

m3/s

QT QT Total air volume flow m3/s

RAT RAT Total aerodynamicresistance

RM + R0 N

RH RH Hydrodynamic resistance RW + RWET N



ITTCSymbol

ComputerSymbol


SI-Unit

RM RM Intake momentum resistancein general

ρA QT VA N

RMCU RMCU Intake momentum resistanceof cushion

ρA QTCU VA N

RMSK RMSK Intake momentum resistanceof skirt

ρA QTSK VA N

RO RO Profile resistance CDP ρA VR2 AV / 2 N

RWET RWET Resistance due to wetting N

TC TC0 Cushion thrust N

ITTC Symbols 3 Special Craft3.8 Sailing Vessels


ITTCSymbol

ComputerSymbol


SI-Unit

3.8 Sailing Vessels


A A Projected area of a body or lifting surface (one sideonly)

Reference area of anappendage element

m2

AH AH Area of hull m2

AS AS Sail area (P E + I J) / 2 m2

AW AW Area of waterplane m2

AX AX Area of maximumtransverse section

m2

b BSP Span of a lifting surface m

B B Beam m

BOA BOA Beam, overall m

BWL BWL Beam, waterline m

C CH Mean chord of a liftingsurface

m

CB CB Block coefficient ∇ / (L B T) 1

CP CP Prismatic coefficient 1

CVP CVP Vertical prismaticcoefficient

∇ / (AW TC) 1

CWP CWP Waterplane coefficient AW / (L B) 1

CX CX Maximum sectioncoefficient

AX / (B TC) 1

E EM Mainsail base m

I Fore triangle height m

J Fore triangle base m

P Mainsail height m

L Lenghts m

LOA LOA Lenghts, overall m

LWL LWL Lenghts, waterline m



ITTCSymbol

ComputerSymbol


SI-Unit

Le LEFF Effective lenghts forcomputing ReynoldsNumber

m

S S Wetted surface area m2

SH SH Hull wetted surface m2

T T Draft m

Tc TCAN Draft of canoe body m

∇ DISV Displaced volume m3

∆ DISF Displacement force(weight)

N



ITTCSymbol

ComputerSymbol


SI-Unit


CD CD Drag coefficient D / (S q) 1

CF CF Friction resistancecoefficient

RF / (S q) 1

CR CR Residuary resistancecoefficient (upright)

RR / (S q) 1

CT CT Total resistancecoefficient

RT / (S q) 1

CA CA Incremental resistancecoefficient

N

CW CW Wave resistance coefficient 1

Cφ CPHI Heel resistance coefficient 1

CI Induced resistancecoefficient

1

Cx, Cy, Cz Force coefficients 1

CL Lift coefficient L / (S q) 1

D Drag N

F Force N

F, FH Heeling force of sails N

FR Driving force of sails N

FV Vertical force of sails N

H Side force N

L Lift force N

R Resistance (drag) N

RF Friction resistance N

RR Residuary resistance(upright)

N

Rφ Resistance increase due toheel (with zero side force)

N

RI Resistance increase due toside (induced resistance)

N

RT Total resistance N



ITTCSymbol

ComputerSymbol


SI-Unit

X,Y,Z Components of resultantforce along designated axis

N

U Boat velocity m/s

Uaw Apparent wind velocity m/s

Vtw True wind velocity m/s

Vmg Velocity made goodtowards a given location (contrary to wind direction)

m/s

ITTC Symbols Computer Symbol Index

Version 1993 140

Computer Symbol Index

A . . . . . . . . 136A . . . . . . . . 14A1 . . . . . . . 14AA . . . . . . 40AA . . . . . . 49AADR . . . . 96AAEF . . . . . 50AAGE . . . . 50AAHY . . . . 50AAID . . . . . 50AAP . . . . . . 111AAPI . . . . . 96AAZL . . . . 50ABL . . . . . . 62ABT . . . . . . 62AC . . . . . . . 72AC . . . . . . . 14AD . . . . . . 71AD . . . . . . 68ADCT . . . . 89ADEN . . . . 70ADEX . . . . 70ADMP . . . . 25ADR . . . . . 89ADVCPS . . 115AE . . . . . . . 68AF . . . . . . . 72AFB . . . . . . 72AFBO . . . . 94AFO . . . . . . 127AFR . . . . . . 111AFRE . . . . . 127AFS . . . . . . 127AFS . . . . . . 129AFS . . . . . . 72AFST . . . . . 94AFT . . . . . . 127AFTO . . . . . 127AH . . . . . . 136AHLT . . . . 94AHLV . . . . 94AIA . . . . . . 125ALF0 . . . . . 50ALFA . . . . . 49ALFA . . . . . 41ALFB . . . . . 109ALFBAR . . 112

ALFCH . . . 121ALFE . . . . . 50ALFG . . . . . 50ALFI . . . . . 50ALFM . . . . 129ALFMB . . . 121ALFO . . . . . 113ALFS . . . . . 112ALFS . . . . . 50ALFSH . . . . 113ALFTW . . . 129AM . . . . . . 62AMX . . . . . 121ANEN . . . . 63ANENIN . . 121ANENOU . . 121ANFB . . . . 96ANFB . . . . 72ANFL . . . . . 49ANFR . . . . 72ANFRIN . . . 73ANFS . . . . . 96ANFS . . . . . 72ANRF . . . . 73ANRU . . . . 96ANRU . . . . 63ANRU . . . . 73ANRU0 . . . 96ANRUE . . . 96ANRUM . . . 100ANRUOR . . 96ANSL . . . . . 49ANWIRL . . 96AO . . . . . . 68AP . . . . . . . 49AP . . . . . . . 68AP . . . . . . . 66APB . . . . . . 107APSF . . . . . 85AR . . . . . . . 127AR . . . . . . . 14AREA . . . . 14ARMV . . . . . . .ARNO . . . . 94ARP . . . . . . 72ART . . . . . . 72ARU . . . . . 72

ARU . . . . . 94ARX . . . . . 72AS . . . . . . . 49AS . . . . . . . 136ASK . . . . . . 72ASRF . . . . . 73ASRU . . . . 73ASRU . . . . 94ASS . . . . . . 127AT . . . . . . . 41AT(I) . . . . . 102ATO . . . . . 129ATR . . . . . . 62ATT . . . . . . 93AV . . . . . . 62AW . . . . . . 136AW . . . . . . 62AWA . . . . . 62AWBK . . . . 72AWF . . . . . 62AWS . . . . . 64AX . . . . . . 62AX . . . . . . 136AXH . . . . . 121

B . . . . . . . . 14B . . . . . . . . 57B . . . . . . . . 136B . . . . . . . . 63BAXDH . . . 121BB . . . . . . . 125BC . . . . . . . 107BCU . . . . . 132BD . . . . . . . 71BET . . . . . . 41BET1 . . . . . 90BETAD . . . 113BETB . . . . . 90BETD . . . . . 109BETE . . . . . 52BETFB . . . . 122BETM . . . . 122BETS . . . . . 90BETTR . . . . 122BFOA . . . . 127BH . . . . . . . 66BK . . . . . . . 73

BLCG . . . . 107BLCK . . . . 79BM . . . . . . 111BMDH . . . . 121BN . . . . . . . 43BOA . . . . . 136BOADH . . . 121BOAWL . . . 132BP . . . . . . . 88BPA . . . . . . 107BPTR . . . . . 107BPX . . . . . . 107BPXDH . . . 121BR . . . . . . . 14BS . . . . . . . 125BS . . . . . . . 73BSP . . . . . . 127BSP . . . . . . 136BSPF . . . . . 127BSPW . . . . 127BSTR . . . . . 127BTR . . . . . . 111BTWL . . . . 121BU . . . . . . . 88BWL . . . . . 136BX . . . . . . . 111BXDH . . . . 121BXSH . . . . 132

C . . . . . . . . 96C1 . . . . . . . 84C2 . . . . . . . 84CA . . . . . . . 80CA . . . . . . . 138CA . . . . . . . 43CAA . . . . . 80CABL . . . . 65CABT . . . . 65CADM . . . . 84CAHL . . . . 94CATR . . . . 65CB . . . . . . . 64CB . . . . . . . 136CC . . . . . . . 47CD . . . . . . . 138CD . . . . . . . 80CD . . . . . . . 52


Version 1993 141

CDAP . . . . 115CDF . . . . . . 130CDI . . . . . . 130CDINT . . . . 115CDINT . . . . 130CDL . . . . . . 110CDO . . . . . . . .CDP . . . . . . 115CDSE . . . . . 49CDSE . . . . . 49CDSP . . . . . 130CDVENT . . 130CDVOL . . . 84CDW . . . . . 130CF . . . . . . . 80CF . . . . . . . 138CFL . . . . . . 51CFL . . . . . . 127CFO . . . . . . 80CFRD . . . . 105CFRS . . . . . 105CFUL . . . . . 81CH . . . . . . . 49CH . . . . . . . 72CH . . . . . . . 136CHC . . . . . 127CHM . . . . . 127CHME . . . . 68CHME . . . . 72CHRT . . . . 72CHRT . . . . 49CHSF . . . . . 127CHT . . . . . . 72CHTI . . . . . 127CHTP . . . . . 49CI . . . . . . . 105CIRCB . . . . 64CIRCC . . . . 81CIRCF . . . . 81CIRCK . . . . 81CIRCM . . . 66CIRCS . . . . 66CIRCT . . . . 66CIW . . . . . . 105CK . . . . . . . 43CLA . . . . . . 130CLBET . . . . 110CLDK . . . . 121CLEARD . . 70

CLF . . . . . . 130CLM . . . . . 127CLO . . . . . . 130CLOAD . . . 134CLOD . . . . 110CLSE . . . . . 49CLTO . . . . . 130CM . . . . . . 130CMS . . . . . 64CMSE . . . . 49CMTL . . . . 77CN . . . . . . . 47CN . . . . . . . 84CNP . . . . . . 84CNPC . . . . 53CNPV . . . . 53COWIAB . . 96COWIRL . . 96CP . . . . . . . 84CP . . . . . . . 136CP . . . . . . . 80CPA . . . . . . 65CPD . . . . . . 88CPE . . . . . . 65CPF . . . . . . 65CPFL . . . . . 127CPHI . . . . . 138CPL . . . . . . 65CPR . . . . . . 65CPR . . . . . . 134CPR . . . . . . 80CPV . . . . . . 80CQF . . . . . . 51CQS . . . . . . 88CR . . . . . . . 134CR . . . . . . . 138CR . . . . . . . 33CR . . . . . . . 80CRA . . . . . 57CRDS . . . . 98CRF . . . . . . 57CRH . . . . . 115CRWAP . . . 115CRWOAP . . 115CS . . . . . . . 68CS . . . . . . . 43CS . . . . . . . 33CS . . . . . . . 65CSA . . . . . . 81

CSF . . . . . . 81CSNK . . . . 81CSP . . . . . . 110CSR . . . . . . 80CSTR . . . . . 127CT . . . . . . . 138CT . . . . . . . 81CTH . . . . . . 88CTHS . . . . . 88CTLT . . . . . 81CTQ . . . . . . 81CTRIM . . . . 81CTVOL . . . 81CUA . . . . . 132CV . . . . . . . 81CVOL . . . . 65CVP . . . . . . 136CVP . . . . . . 65CW . . . . . . 65CW . . . . . . 138CW . . . . . . 81CWA . . . . . 65CWC . . . . . 134CWF . . . . . 65CWIL . . . . . 64CWIT . . . . . 64CWP . . . . . 81CWP . . . . . 136CX . . . . . . . 65CX . . . . . . . 136

D . . . . . . . . 14D . . . . . . . . 78D . . . . . . . . 33D . . . . . . . . 33DA(U,V) . . . 92DALF . . . . . 122DC . . . . . . . 53DC0 . . . . . . 99DC0N . . . . . 99DCNO . . . . 99DE . . . . . . . 14DE . . . . . . . 94DEL . . . . . . 52DEL(I,J) . . . 21DELAW . . . 119DELCF . . . . 80DELDR . . . 110DELFS . . . . 113

DELRP . . . . 110DELS . . . . . 52DELST . . . . 119DELTB . . . 129DELTF . . . . 129DELTL . . . . 129DELTR . . . 119DELTS . . . . 129DELTT . . . . 129DELTT . . . . 129DELTU . . . 129DELWC . . 85DELWC . . . 85DELWG . . . 113DEME . . . . 94DEP . . . . . . 63DFA . . . . . . 130DFF . . . . . . 130DH . . . . . . 125DH . . . . . . 68DI . . . . . . . 14DINT . . . . . 115DIRSF . . . . 58DISF . . . . . 137DISF . . . . . 78DISV . . . . . 78DISV . . . . . 137DISV . . . . . 64DISVAP . . . 114DISVF . . . . 129DISWAP . . 114DJWS. . . . . 81DLAM . . . . 109DLTFL . . . . 129DN . . . . . . 16DN . . . . . . 61DN . . . . . . 43DNIC . . . . . 61DNSN . . . . 61DNWA . . . . 61DP . . . . . . . 87DP . . . . . . . 68DP(U) . . . . 93DPAC(U) . . 93DPVL(U) . . 93DR . . . . . . . 41DRCI . . . . . 100DRF . . . . . . 130DRIND . . . . 130


Version 1993 142

DRINT . . . . 130DRSE . . . . . 130DRSP . . . . . 131DRST . . . . . 131DRVNT . . . 131DRWA . . . . 131DTA . . . . . 113DTF . . . . . . 113DTLCG . . . 113DTPS . . . . . 78DTR . . . . . . 78DTRA . . . . 107DW . . . . . . 66DW . . . . . . 56DX . . . . . . 125

E . . . . . . . . 14E . . . . . . . . 123E . . . . . . . . 123E . . . . . . . . 123EC . . . . . . . 56EC . . . . . . . 55ED . . . . . . . 46EF . . . . . . . 16EFDIC . . . . 106EFGP . . . . . 85EFID . . . . . 89EFJP . . . . . 89EFJPO . . . . 89EFPS . . . . . 85EFRO . . . . . 85EFRP . . . . . 85EFRT . . . . . 85EFSI . . . . . 85EFTJ . . . . . 89EFTP . . . . . 85EFTPO . . . . 89EISC . . . . . 58EISF . . . . . 58EK . . . . . . . 46EL . . . . . . . 43ELIC . . . . . 61EM . . . . . . 136EN . . . . . . . 107EN . . . . . . . 14EN . . . . . . . 66ENA . . . . . 107ENF . . . . . . 107ENP . . . . . . 107

ENPR . . . . . 108ENS . . . . . . 108ENTH . . . . 52EPS . . . . . . 131EPS(I,J,K) . . 21EPSF . . . . . 129EPSFS . . . . 119EPSG . . . . . 119EPSH . . . . . 119EPSM . . . . 119EPSO . . . . . 113EPSR . . . . . 119EPSRAP . . . 119EPSS . . . . . 113EPSSO. . . . 113ERIC . . . . . 106ERSC . . . . . 58ERSF . . . . . 58ERTCA . . . 123ES . . . . . . . 46ET . . . . . . . 56ET . . . . . . . 56ETA . . . . . . 16ETAAP . . . 120ETAB . . . . . 85ETAD . . . . 85ETAG . . . . 85ETAH . . . . 85ETAM . . . . 85ETAO . . . . 89ETAO . . . . 85ETAPSI . . . 120ETAR . . . . . 85ETAS . . . . . 85EW . . . . . . 55EWAM(I) . . 55EWPH(I) . . 55EWSC . . . . 103EWSC . . . . 58EWSF . . . . 103EWSF . . . . 58

F . . . . . . . . 38F . . . . . . . . 37F . . . . . . . . 14F(1) . . . . . . 37F(2) . . . . . . 37F(3) . . . . . . 37F(4) . . . . . . 37

F(5) . . . . . . 37F(6) . . . . . . 37F(I) . . . . . . 92F0 . . . . . . . 14F0(1) . . . . . 37F0(2) . . . . . 37F0(3) . . . . . 37F1 . . . . . . . 15F1(1) . . . . . 37F1(2) . . . . . 37F1(3) . . . . . 37FBP . . . . . . 68FC . . . . . . . 25FC . . . . . . . 56FC . . . . . . . 14FC . . . . . . . 16FD . . . . . . . 70FDAP . . . . . 115FDRR . . . . 108FDT . . . . . . 83FE . . . . . . . 102FF(1) . . . . . 15FF(1) . . . . . 14FF(2) . . . . . 14FF(3) . . . . . 15FG(I) . . . . . 92FH . . . . . . . 43FM . . . . . . 72FML . . . . . 128FMU . . . . . 128FN . . . . . . . 115FN . . . . . . . 43FNC . . . . . . 128FNF . . . . . . 128FNI . . . . . . 105FNIC . . . . . 105FP . . . . . . . 66FP . . . . . . . 83FPO . . . . . . 83FR . . . . . . . 58FR . . . . . . . 102FR . . . . . . . 15FR . . . . . . . 14FR . . . . . . . 25FRAA . . . . 108FRAP . . . . . 108FREB . . . . . 77FRF . . . . . . 108FRICMS . . . 105

FRK . . . . . . 108FRPK . . . . . 57FRRC . . . . . 57FRSA . . . . . 57FS . . . . . . . 25FS . . . . . . . 66FS(1) . . . . . 38FS(2) . . . . . 102FS(3) . . . . . 102FS(4) . . . . . 102FS(5) . . . . . 102FS(6) . . . . . 102FS(I) . . . . . 46FS(I) . . . . . 38FSL . . . . . . 108FT . . . . . . . 38FTIC . . . . . 105FTR . . . . . . 115FV . . . . . . . 43FW . . . . . . 82FW . . . . . . 55FW(I) . . . . . 55FWL . . . . . 111FWS . . . . . 111FWS . . . . . 132FX . . . . . . . 37FX . . . . . . . 97FX . . . . . . . 115FXIC . . . . . 105FY . . . . . . . 98FY . . . . . . . 37FYIC . . . . . 105FZ . . . . . . . 37FZ . . . . . . . 115FZ . . . . . . . 98FZIC . . . . . 105

G . . . . . . . . 14G(U) . . . . . 37G(U) . . . . . 37G0(I) . . . . . 37G1(I) . . . . . 37G1(I) . . . . . 37GAMR . . . . 41GAMSP . . . 109GAP . . . . . . 68GC . . . . . . . 53GH . . . . . . 132GM . . . . . . 75


Version 1993 143

GML . . . . . 75GMR . . . . . 31GMR . . . . . 32GMS . . . . . 32GR . . . . . . . 53GR . . . . . . . 14GS . . . . . . . 53GZ . . . . . . . 76

H . . . . . . . . 14H13D . . . . . 57H13U . . . . . 57HBL . . . . . . 51HBS . . . . . . 132HC . . . . . . . 125HC . . . . . . . 54HD . . . . . . 56HFL . . . . . . 128HK . . . . . . 44HKE . . . . . 128HMO . . . . . 57HO . . . . . . 68HQF . . . . . . 51HRU . . . . . 108HS . . . . . . . 46HSK . . . . . . 132HSP . . . . . . 108HSS . . . . . . 132HT . . . . . . . 46HT . . . . . . . 14HTC . . . . . . 111HTIC . . . . . 105HTNT . . . . 54HTSN . . . . . 105HTUS . . . . . 54HU . . . . . . 56HV . . . . . . 57HVCG . . . . 128HW . . . . . . 55HWDS . . . . 58

I . . . . . . . . 25I . . . . . . . . 14I2(1,2) . . . . 35I2(2,3) . . . . 35I2(3,1) . . . . 35ID . . . . . . . 47IL . . . . . . . 63IM . . . . . . . 25

IN . . . . . . . 14IN(I,J) . . . . 36IT . . . . . . . 63IX . . . . . . . 35IXX . . . . . . 35IXY . . . . . . 35IY . . . . . . . 35IYY . . . . . . 35IYZ . . . . . . 35IZ . . . . . . . 35IZX . . . . . . 35IZZ . . . . . . 35

J . . . . . . . . 25JA . . . . . . . 88JEI . . . . . . . 88JH . . . . . . . 88JP . . . . . . . 88JPT . . . . . . 88JQ, JPQ . . . 88JT . . . . . . . 88

K . . . . . . . . 81K . . . . . . . . 52K . . . . . . . . 37KAP . . . . . . 116KF(I) . . . . . 92KL . . . . . . . 73KM(I) . . . . . 92KN . . . . . . 116KP . . . . . . . 88KPR . . . . . . 92KQ . . . . . . 89KQ . . . . . . 88KQB . . . . . 116KQICMS . . 105KQO . . . . . 116KQO . . . . . 89KQPSI . . . . 116KR . . . . . . . 81KS . . . . . . . 99KS . . . . . . . 87KSC . . . . . . 88KSH . . . . . . 88KT . . . . . . . 89KTB . . . . . . 116KTD . . . . . 89KTICMS . . . 105KTO . . . . . 116

KTP . . . . . . 89KTPSI . . . . 116KTT . . . . . . 89

L . . . . . . . . 63L . . . . . . . . 14LAC . . . . . . 132LACE . . . . . 132LAMBS . . . 129LAMS . . . . 109LAP . . . . . . 116LB . . . . . . . 132LC . . . . 53LC . . . . . . . 108LCH . . . . . . 68LCH . . . . . . 125LCP . . . . . . 108LCS . . . . . . 125LD . . . . . . . 56LD . . . . . . . 70LDEN . . . . 70LDEX . . . . 71LE . . . . . . . 14LEFF . . . . . 128LEFF . . . . . 137LEN . . . . . . 63LERF . . . . . 128LF . . . . . . . 111LFA . . . . . . 131LFF . . . . . . 131LFS . . . . . . 64LH . . . . . . . 125LHRD . . . . 99LIBU . . . . . 116LIDY . . . . . 116LK . . . . . . . 108LK . . . . . . . 111LM . . . . . . 108LNH . . . . . 125LNS . . . . . . 125LOA . . . . . 136LOA . . . . . 63LOS . . . . . . 63LP . . . . . . . 63LP . . . . . . . 66LPP . . . . . . 63LPRC . . . . . 109LRU . . . . . . 63LS . . . . . . . 125

LSB . . . . . . 98LSF . . . . . . 80LSH . . . . . . 125LSHB . . . . . 109LSR . . . . . . 98LSS . . . . . . 125LSS . . . . . . 64LSTR . . . . . 128LTS . . . . . . 125LU . . . . . . . 56LW . . . . . . 55LW . . . . . . 56LW . . . . . . 66LWL . . . . . 63LWL . . . . . 136LWRD . . . . 99LWT . . . . . 111

M . . . . . . . 37M . . . . . . . 15M(1) . . . . . 37M(2) . . . . . 37M(3) . . . . . 37M(I) . . . . . . 92M0(I,J) . . . . 35M1 . . . . . . . 15M1(I,J) . . . . 35M2(1,1) . . . 35M2(2,2) . . . 35M2(3,3) . . . 35M2(I,J) . . . . 36MA . . . . . . 35MA . . . . . . 77MA . . . . . . 15MA(4,4) . . . 35MA(5,5) . . . 35MA(6,6) . . . 35MA(I,J) . . . 35MA(U,V) . . 92MA(U,V) . . 36MAAP . . . . 113MASS . . . . 15MAWAP . . 113MB . . . . . . 38MB(1) . . . . 38MB(2) . . . . 102MB(3) . . . . 102MF . . . . . . 82ML . . . . . . 128


Version 1993 144

MLF . . . . . 128Mn . . . . . . . 58MN . . . . . . 43MO . . . . . . 15MR . . . . . . 33MR . . . . . . 15MR . . . . . . 82MS . . . . . . 33MS . . . . . . 77MSP . . . . . . 131MT . . . . . . 38MT(1), . . . . 102MTC . . . . . 77MTM . . . . . 77MX . . . . . . 97MY . . . . . . 97MZ . . . . . . 97

N . . . . . . . . 15N . . . . . . . . 37N . . . . . . . . 83NAP . . . . . . 110NAW . . . . . 102ND . . . . . . 97NF . . . . . . . 116NPB . . . . . . 69NPP . . . . . . 116NPR . . . . . . 111NPS . . . . . . 117NR . . . . . . . 97NRRT . . . . 97NV . . . . . . 97NVRT . . . . 97NYO . . . . . 117NZO . . . . . 117

OMF . . . . . 16OMN . . . . . 83OMN . . . . . 16OMX . . . . . 40OMY . . . . . 40OMZ . . . . . 40OW . . . . . . 82OX . . . . . . 94OXRT . . . . 95OY . . . . . . 95OYRT . . . . 95OZ . . . . . . . 95OZCINO . . . 99

OZRT . . . . . 95

P . . . . . . . . 94P . . . . . . . . 40P . . . . . . . . 15P0 . . . . . . . 46PA . . . . . . . 53PACO . . . . 53PAIC . . . . . 53PAW . . . . . 102PB . . . . . . . 134PB . . . . . . . 83PBS . . . . . . 134PC . . . . . . . 53PCE . . . . . . 134PCIN . . . . . 53PCU . . . . . . 134PD . . . . . . . 46PD . . . . . . . 34PD . . . . . . . 83PDI . . . . . . 106PDTR . . . . . 117PE . . . . . . . 83PEAA . . . . . 117PEAP . . . . . 117PEAW . . . . 119PEPAR . . . . 117PEST . . . . . 119PETR . . . . . 117PEWAP . . . 117PEWOAP . . 117PF . . . . . . . 34PFCU . . . . . 134PFSK . . . . . 134PFT . . . . . . 134PHIF . . . . . 70PHIP . . . . . 70PHIR . . . . . 41PHISP . . . . 109PI . . . . . . . 83PITCH . . . . 69PLR . . . . . . 134PMVO . . . . 64PN . . . . . . . 99PO . . . . . . . 15PO . . . . . . . 61PO . . . . . . . 47POAI . . . . . 61POBR . . . . 61

POIIC . . . . . 61PP . . . . . . . 83PR . . . . . . . 95PR . . . . . . . 40PR . . . . . . . 51PR . . . . . . . 83PR . . . . . . . 92PR . . . . . . . 46PRGR . . . . 52PS . . . . . . . 83PSI . . . . . . 70PSI01 . . . . . 100PSI02 . . . . . 100PSIS . . . . . . 100PSIS . . . . . . 114PSIY . . . . . 41PSS . . . . . . 134PSS . . . . . . 134PT . . . . . . . 83PT . . . . . . . 51PV . . . . . . . 53

Q . . . . . . . . 40Q . . . . . . . . 95QQAW . . . . . 102Q . . . . . . . . 83QB . . . . . . . 117QBS . . . . . . 134QCF . . . . . . 16QCU . . . . . 134QDF . . . . . . 16QDM . . . . . 16QDT . . . . . 16QF . . . . . . . 51QF . . . . . . . 46QFB . . . . . . 97QFL . . . . . . 16QFLOW . . . 46QFS . . . . . . 97QHO . . . . . 117QPN . . . . . . 16QPSI . . . . . 117QQ . . . . . . 16QR . . . . . . . 40QR . . . . . . . 95QRT . . . . . . 16QRU . . . . . 97QSP . . . . . . 87QSPC . . . . . 87

QSPH . . . . . 87QSS . . . . . . 134QT . . . . . . . 134

R . . . . . . . . 25R . . . . . . . . 40R . . . . . . . . 95R . . . . . . . . 15R(U) . . . . . 92RA . . . . . . . 79RAA . . . . . 79RAKA . . . . 70RAKG . . . . 69RAKS . . . . 69RAKT . . . . 69RAP . . . . . . 79RAR . . . . . 79RAT . . . . . . 134RAUG . . . . 84RAW . . . . . 102RC . . . . . . . 79RD . . . . . . . 69RD . . . . . . . 15RDELS . . . . 51RDGX . . . . 35RDGY . . . . 35RDGZ . . . . 35RE . . . . . . . 15RF . . . . . . . 117RF . . . . . . . 79RFCA . . . . 123RFCA . . . . 123RFDH . . . . 123RFDH . . . . 123RFINT . . . . 123RFO . . . . . . 79RFXO . . . . 117RH . . . . . . . 44RH . . . . . . . 134RH . . . . . . . 69RHO . . . . . 16RHO . . . . . 43RI . . . . . . . 106RIT . . . . . . 106RIW . . . . . . 106RKEEL . . . 117RL . . . . . . . 69RM . . . . . . 135RMCU . . . . 135


Version 1993 145

RMSK . . . . 135RN . . . . . . . 43RO . . . . . . . 41RO . . . . . . . 135RO, . . . . . . 41RP . . . . . . . 79RPAR . . . . 118RPI . . . . . . 118RPV . . . . . . 79RR . . . . . . . 40RR . . . . . . . 34RR . . . . . . . 79RR . . . . . . . 69RR . . . . . . . 95RRDH . . . . 123RRDH . . . . 123RRH . . . . . 118RRINT . . . . 123RRWAP . . . 118RS . . . . . . . 34RS . . . . . . . 79RSP . . . . . . 118RSV . . . . . . 118RT . . . . . . . 79RTHETA . . 51RTINT . . . . 123RU . . . . . . . 73RV . . . . . . . 79RW . . . . . . 80RWAP . . . . 118RWB . . . . . 80RWET . . . . 135RWOAP . . . 118RZFO . . . . . 117

S . . . . . . . . 25S . . . . . . . . 20S . . . . . . . . 64S . . . . . . . . 137S(U,V) . . . . 20S2TET . . . . 58S2ZET . . . . 58SA . . . . . . . 73SAIC . . . . . 61SAP . . . . . . 111SAWA . . . . 61SC . . . . . . . 16SC . . . . . . . 64SCIC . . . . . 61

SDPE . . . . . 119SF . . . . . . . 82SF . . . . . . . 47SFIC . . . . . 61SH . . . . . . . 73SH . . . . . . . 137SI(U,V) . . . 92SIGS . . . . . 16SIN . . . . . . 84SK . . . . . . . 73SM0(I,J) . . . 20SM1(I,J) . . . 20SM2(I,J) . . . 20SN . . . . . . . 43SN . . . . . . . 16SNIC . . . . . 61SP . . . . . . . 15SP . . . . . . . 49SPF . . . . . . 101SPRU . . . . . 94SPRUME . . 94SR . . . . . . . 82SR . . . . . . . 34SR(I,J) . . . . 46SRA . . . . . . 89SRR . . . . . . 89ST . . . . . . . 16ST . . . . . . . 73ST(I,J) . . . . 46STHETA . . 58STIC . . . . . 61STIC . . . . . 61STRTIC . . . 61SV(I,J) . . . . 46SW . . . . . . 82SW . . . . . . 111SWAP . . . . 112SWAPE . . . 111SWB . . . . . 112SWH . . . . . 132SWH . . . . . 112SWHE . . . . 112SWHES . . . 132SWHP . . . . 109SWS . . . . . 112SWSH . . . . 112SWSW . . . . 132SWV0 . . . . 112

T . . . . . . . . 137T . . . . . . . . 64T . . . . . . . . 25T(I,J) . . . . . 20T1 . . . . . . . 58T2 . . . . . . . 58TA . . . . . . . 64TA . . . . . . . 73TA . . . . . . . 49TAFR . . . . . 73TAP . . . . . . 64TAS(I,J) . . . 20TAU . . . . . 16TAUB . . . . 114TAUB . . . . 109TAUT . . . . 129TAUW . . . . 52TAW . . . . . 103TC . . . . . . . 103TC . . . . . . . 15TC . . . . . . . 25TC . . . . . . . 135TCAN . . . . 137TCHC . . . . 100TD . . . . . . . 71TD . . . . . . . 56TE . . . . . . . 15TE . . . . . . . 103TEA . . . . . . 61TEIC . . . . . 61TEMX . . . . 70TETAD . . . 129TETB . . . . . 133TETBC . . . . 133TETF . . . . . 133TETP . . . . . 41TETS . . . . . 70TETW . . . . 133TEWA . . . . 61TF . . . . . . . 64TFD . . . . . . 128TFO . . . . . . 128TFOM . . . . 128TFP . . . . . . 64TH . . . . . . . 73TH . . . . . . . 118TH . . . . . . . 87THB . . . . . . 118THDF . . . . . 84

THDFPS . . . 118THDU . . . . 87THETA . . . 52THL . . . . . . 80THP . . . . . . 87THPSI . . . . 118THT . . . . . . 87THUL . . . . 112THX . . . . . 118THZ . . . . . . 118TI . . . . . . . 15TI(J) . . . . . . 25TI180 . . . . . 99TI90 . . . . . . 99TIA . . . . . . 100TIC . . . . . . 98TIC1 . . . . . 100TIC1 . . . . . 99TIC2 . . . . . 99TIC2 . . . . . 100TIC3 . . . . . 99TICV . . . . . 33TIMS . . . . . 106TIR . . . . . . 100TM . . . . . . 69TM . . . . . . 64TMR . . . . . 90TMS . . . . . 64TMX . . . . . 72TN . . . . . . . 54TNHE . . . . 103TNPI . . . . . 103TNRO . . . . 103TO . . . . . . . 69TR . . . . . . . 41TR . . . . . . . 57TR . . . . . . . 78TRIM . . . . . 84TRIM . . . . . 114TS . . . . . . . 125TS . . . . . . . 25TS . . . . . . . 57TSY(I,J) . . . 20TT . . . . . . . 64TTR . . . . . . 112TTR(I,J) . . . 20TTRM . . . . 112TU . . . . . . . 56TV . . . . . . . 57


Version 1993 146

TW . . . . . . 55TX . . . . . . . 126

U . . . . . . . . 95U . . . . . . . . 15U . . . . . . . . 40U(I) . . . . . . 21UA . . . . . . 90UADU . . . . 90UAP . . . . . . 90UDEF . . . . . 52UE . . . . . . . 51UFL . . . . . . 51UFLS . . . . . 51UFS . . . . . . 51UIN . . . . . . 51UM . . . . . . 51UN . . . . . . 15UNIN . . . . . 53UNQ . . . . . 38UPLUS . . . . 51UR . . . . . . . 40UR . . . . . . . 90UR . . . . . . . 95URDU . . . . 90URP . . . . . . 90UT . . . . . . . 90UTAU . . . . 51UTDU . . . . 90UTP . . . . . . 90

V . . . . . . . . 80V . . . . . . . . 95V . . . . . . . . 34V . . . . . . . . 83V . . . . . . . . 15V . . . . . . . . 40V(1) . . . . . . 40V(2) . . . . . . 40V(3) . . . . . . 40V(4) . . . . . . 40V(5) . . . . . . 40V(6) . . . . . . 40V(I) . . . . . . 92V(I) . . . . . . 21V(I) . . . . . . 46V(I) . . . . . . 21V(U) . . . . . 40V(U) . . . . . 21

V0 . . . . . . . 46V0 . . . . . . . 16V0 . . . . . . . 83V0(1) . . . . . 40V0(2) . . . . . 40V0(3) . . . . . 40V0(I) . . . . . 21V0P . . . . . . 87V1 . . . . . . . 46V1 . . . . . . . 15V1(1) . . . . . 40V1(2) . . . . . 40V1(3) . . . . . 40V1(I) . . . . . 21V2 . . . . . . . 46V3 . . . . . . . 46VA . . . . . . 95VA . . . . . . 87VA . . . . . . 83VA . . . . . . 46VBM . . . . . 118VC . . . . . . . 131VD . . . . . . 47VF . . . . . . . 95VF . . . . . . . 15VG . . . . . . 55VI . . . . . . . 49VI . . . . . . . 43VK . . . . . . 43VO . . . . . . 95VO . . . . . . 15VOLS . . . . . 54VP . . . . . . . 55VP . . . . . . . 87VP(I) . . . . . 55VR . . . . . . . 40VR . . . . . . . 95VR . . . . . . . 80VR . . . . . . . 34VSABS . . . 118VSREL . . . . 118VT . . . . . . . 131VT . . . . . . . 49VWAB . . . . 96VWRL . . . . 96VX . . . . . . 46VX . . . . . . 95VX . . . . . . 40VXRT . . . . 95

VY . . . . . . 46VY . . . . . . 95VY . . . . . . 40VYR . . . . . 95VZ . . . . . . . 40VZ . . . . . . . 95VZ . . . . . . . 46VZRT . . . . . 95

W . . . . . . . 95W . . . . . . . 40WD . . . . . . 15WD . . . . . . 58WFF . . . . . 84WFT . . . . . 84WFTPSI . . . 119WFTQ . . . . 84WFTT . . . . 85WG . . . . . . 73WN . . . . . . 56WN . . . . . . 43WPUL . . . . 38WR . . . . . . 40WR . . . . . . 95WT . . . . . . 78WT . . . . . . 132WT . . . . . . 15WTF . . . . . 128WTLS . . . . 54

x . . . . . . . . 25X . . . . . . . . 37X . . . . . . . . 21X . . . . . . . . 93X . . . . . . . . 92X(1) . . . . . . 21X(2) . . . . . . 21X(3) . . . . . . 21X(4) . . . . . . 41X(5) . . . . . . 41X(6) . . . . . . 41X(I) . . . . . . 103X(J) . . . . . . 25X(U) . . . . . 103X0 . . . . . . . 21X0(1) . . . . . 21X0(2) . . . . . 21X0(3) . . . . . 21X0180 . . . . 100

X090 . . . . . 100X0F . . . . . . 101XA . . . . . . 26XAB . . . . . 75XACB . . . . 77XACG . . . . 77XAF . . . . . . 75XAG . . . . . 75XBL . . . . . . 112XCB . . . . . 77XCF . . . . . . 77XCG . . . . . 78XD . . . . . . 31XDF . . . . . . 26XDL . . . . . 26XDS . . . . . . 31XF . . . . . . . 26XF . . . . . . . 21XF(1) . . . . . 21XF(2) . . . . . 21XF(3) . . . . . 21XFB . . . . . . 75XFF . . . . . . 76XFG . . . . . . 75XFRG . . . . 129XFT(J) . . . . 26XH . . . . . . 133XHF . . . . . . 26XHT . . . . . 26XLO . . . . . 85XLT . . . . . . 26XmMR . . . . 31XMR . . . . . 31XMS . . . . . 31XMX . . . . . 100XP . . . . . . . 69XPD . . . . . . 31XPF . . . . . . 32XRT . . . . . . 26XRU . . . . . 97XS . . . . . . . 133XS . . . . . . . 26XS(J) . . . . . 26XTA . . . . . 77XU . . . . . . 98XUR . . . . . 98XVR . . . . . 32XVS . . . . . . 32XXCR . . . . 32


Version 1993 147

XXM . . . . . 31XXRR . . . . 33XXSR . . . . 33XXVR . . . . 32XXVS . . . . 32XYCR . . . . 32XYM . . . . . 31XYPD . . . . 31XYPF . . . . . 32XYRR . . . . 33XYSR . . . . 33XYVR . . . . 32

Y . . . . . . . . 37Y . . . . . . . . 21Y . . . . . . . . 92Y0 . . . . . . . 21Y0180 . . . . 100Y090 . . . . . 100Y0F . . . . . . 101Y0MX . . . . 100Y0MX . . . . 100YA . . . . . . 96YA . . . . . . 41YAG . . . . . 76YAOR . . . . 96YART . . . . 96YAZ . . . . . 76YD . . . . . . 98YF . . . . . . . 21YHA . . . . . 77YP . . . . . . . 69YPLUS . . . . 52YR . . . . . . . 98YRR . . . . . 98YRU . . . . . 98YV . . . . . . 98YVRT . . . . 98

Z . . . . . . . . 25Z . . . . . . . . 37Z . . . . . . . . 21Z . . . . . . . . 92Z0 . . . . . . . 21ZAG . . . . . 76ZAM . . . . . 27ZBM . . . . . 75ZBML . . . . 75ZCJ . . . . . . 27

ZETAC . . . 133ZETO,ZF . . . . . . . 21ZH . . . . . . . 133ZIM . . . . . . 27ZIM . . . . . . 27ZKA . . . . . 75ZKAG . . . . 76ZKB . . . . . . 76ZKG . . . . . 76ZKM . . . . . 76ZKML . . . . 76ZLG . . . . . . 27ZP . . . . . . . 69ZPH . . . . . . 27ZR . . . . . . . 119ZRE . . . . . . 27ZRE . . . . . . 27ZV . . . . . . . 83