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Constanta Maritime Univesity

Electromechanical Faculty

Introduction into Naval-Mechanical Engineering

 

Student: Efteme Ionut Alexand

rou!: MA

Su!ervisor: drd$ asist$univ$ in

%UMI&'ESCU (AU'ENȚ

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  )*ME+*', N*$"

A$ &)E*'E&ICA( ASEC&S

1. THE BASICS.

"$" ive and ex!lain the definition of :

%is!lacement.of gravity/ of mass and of volume01

  2 2 3uoyancy$

  E4ually/ 5rite the measure unit of them$

It 5as Archimedes 5ho first realised/ in his 6eure7a8 moment/ that a 9ody that is immersed

com!letely in 5ater 5ill dis!lace a volume of 5ater e4ual to the volume of the 9ody and that

the a!!arent 5eight of the 9ody/ 5hen immersed/5ill 9e reduced 9y the 5eight of 5ater e4ua

to its o5n volume/ then it 5ill not 9ecome immersed com!letely$ It 5ill float 5ith !art if its

o5n volume a9ove the 5ater surface such that the 5eight of the dis!laced 5ater e4uals the

5eight of the 9ody$ &hus a 9ody/ floating freely in 5ater 5ill 9e acted u!on 9y forces due to

the 5ater !ressure at each !oint of its 5etted surface$ &he resultant of these forces 5ill 9e an

u!5ard force e4ual to the 5eight of the 5ater dis!laced 9y the 9ody$ &his is the 9odys

dis!lacement/ often denoted 9y the sym9ol ;$ If the 5ater density is < then the !ressure actinu!on a small area =A 5ill 9e

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characteristics of the hull sha!e/ e$g$ ho5 6full8 it is$ &here are some 6coefficients8 5hich can

 9e o9tained for the under5ater hull 5hich !rovide clues as to its general nature and its li7ely

 9ehavior at sea$ &hey are derived 9y relating certain areas and volumes to their circumscri9in

rectangles or !risms$ &hese coefficients are 7no5n as the coefficients of fineness if:

@ is the volume of dis!lacement1A  ᴡ   is the 5ater!lane area1

AM is the under5ater area of the midshi! section1and

(/3 and & are the length/9eam and draught$

&hen the coefficients of fineness are defined as follo5s:

3loc7 coefficient/ C3 ? @D(3&

+ater!lane coefficient/ C+  ? AD(3

Midshi! area coefficient/ CM ? AMD3&

)oriontal !rismatic coefficient/ C ? @DAM(

@ertical !rismatic coefficient/ C@ ?@D A+&

In com!aring values of these coefficients 9et5een shi!s it is im!ortant to ensure that the samdefinitions of (/ 3 and & are used$ Usually ( is the length 9et5een !er!endiculars exce!t for

the 5ater!lane coefficient 5here the length on 5aterline is ta7en$ 3 is usually the mean

draught 9et5een !er!endiculars$ Also/ usually (/3 and & are ta7en as defining the external hu

 9ut sometimes moulded dimensions are used/ that is to the inside of the !lating$ &he external

hull dimensions hel! to determine the 9ehavior of a shi! in res!onding to the thrust of the

 !ro!ellers/ to 5aves and to the movements of control surfaces/ such as rudders and sta9ilisers

&he moulded dimensions assist in finding the internal volume availa9le for e4ui!ment/

accommodation and cargo$

Exam!le

A shi! of length " m and 9eam "G m floats at a mean draught of H/ m 5hen in 5ater of

density "/B tonnesDm$ Assuming her 9loc7 coefficient is / and her 5ater!lane coefficien

of fineness is /J / calculate the shi!s dis!lacement and her a!!roximate draught 5hen it

enters 5ater of "/" tonnesDm$

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Solution:

@olume of circumscri9ing rectangular 

Solid ? 150 X 18 X 7.5 = 20,250 m3

@olume of dis!lacement ? 0.5 X 20250 = 10,125 m3

)ence dis!lacement ? 10,125 X 1,025 = 10,378 tonnes&he 5ater!lane area ? 0,6 X 150 X 18 = 1620 m2

In the less dense 5ater the shi! must dis!lace more 5ater to !roduce a 9uoyancy force e4ual

its 5eight$

  Ne5 volume in the less dense 5ater ? 10,378/1,015 = 10,225 m3

  &he shi! 5ill sin7 in the 5ater to com!ensate for the added volume

  Added volume ? 10,225 – 10,125 = 100 m3

  Sin7age ? 100/1620 = 6,17 cm

  Ne5 draught is a!!roximately 7, 56 m

  "$ +rite and ex!lain the Archimedes (a5 for a freely floating 9ody . e4ual

in scalar and

vectorial form 0$ +rite the e4uili9rium e4uations

ractically/ the Archimedes !rinci!le allo5s the 9uoyancy of an o9Kect !artially or

5holly immersed in a li4uid to 9e calculated$ &he do5n5ard force on the o9Kect is sim!ly its

5eight$ &he u!5ard/ or 9uoyant/ force on the o9Kect is that stated 9y ArchimedesL !rinci!le/

a9ove$ &hus the net u!5ard force on the o9Kect is the difference 9et5een the 9uoyant force an

its 5eight$ If this net force is !ositive/ the o9Kect rises1 if negative/ the o9Kect sin7s1 and if er

the o9Kect is neutrally 9uoyant - that is/ it remains in !lace 5ithout either rising or sin7ing$ In

sim!le 5ords/ ArchimedesL !rinci!le states that 5hen a 9ody is !artially or com!letely

immersed in a fluid/ it ex!eriences an a!!arent loss in 5eight 5hich is e4ual to the 5eight of

the fluid dis!laced 9y the immersed !art of the 9ody$Consider a cu9e immersed in a fluid/ 5ith its sides !arallel to the direction of gravity$

&he fluid 5ill exert a normal force on each face/ and therefore only the forces on the to! and

 9ottom faces 5ill contri9ute to 9uoyancy$ &he !ressure difference 9et5een the 9ottom and th

to! face is directly !ro!ortional to the height .difference in de!th0$ Multi!lying the !ressure

difference 9y the area of a face gives the net force on the cu9e the 9uoyancy/ or the 5eight

the fluid dis!laced$ 3y extending this reasoning to irregular sha!es/ 5e can see that/ 5hateve

https://en.wikipedia.org/wiki/Normal_forcehttps://en.wikipedia.org/wiki/Pressurehttps://en.wikipedia.org/wiki/Pressurehttps://en.wikipedia.org/wiki/Normal_force

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the sha!e of the su9merged 9ody/ the 9uoyant force is e4ual to the 5eight of the fluid

dis!laced$ a!!arent loss in 5t of 5ater? 5t of o9Kect in air-5t of o9Kect in 5ater 

&he 5eight of the dis!laced fluid is directly !ro!ortional to the volume of the dis!laced fluid

.if the surrounding fluid is of uniform density0$ &he 5eight of the o9Kect in the fluid is reduce

 9ecause of the force acting on it/ 5hich is called u!thrust$ In sim!le terms/ the !rinci!le state

that the 9uoyant force on an o9Kect is e4ual to the 5eight of the fluid dis!laced 9y the o9Kect/the density of the fluid multi!lied 9y the su9merged volume times the gravitational constant/

&hus/ among com!letely su9merged o9Kects 5ith e4ual masses/ o9Kects 5ith greater volume

have greater 9uoyancy$

Su!!ose a roc7Ls 5eight is measured as " ne5tons 5hen sus!ended 9y a string in

a vacuum 5ith gravity acting on it$ Su!!ose that 5hen the roc7 is lo5ered into 5ater/ it

dis!laces 5ater of 5eight ne5tons$ &he force it then exerts on the string from 5hich it hang

5ould 9e " ne5tons minus the ne5tons of 9uoyant force: " ? H ne5tons$ 3uoyancy

reduces the a!!arent 5eight of o9Kects that have sun7 com!letely to the sea floor$ It is

generally easier to lift an o9Kect u! through the 5ater than it is to !ull it out of the 5ater$

For a fully su9merged o9Kect/ ArchimedesL !rinci!le can 9e reformulated as follo5s:

then inserted into the 4uotient of 5eights/ 5hich has 9een ex!anded 9y the mutual volum

yields the formula 9elo5$ &he density of the immersed o9Kect relative to the density of

the fluid can easily 9e calculated 5ithout measuring any volumes:

Exam!le: If you dro! 5ood into 5ater/ 9uoyancy 5ill 7ee! it afloat$

Exam!le: A helium 9alloon in a moving car$ +hen increasing s!eed or driving in a

curve/ the air moves in the o!!osite direction to the carLs acceleration$ )o5ever/ due to

 9uoyancy/ the 9alloon is !ushed Oout of the 5ayO 9y the air/ and 5ill actually drift in the sam

direction as the carLs acceleration$

+hen an o9Kect is immersed in a li4uid/ the li4uid exerts an u!5ard force/ 5hich is 7no5nas the 9uoyant force/ that is !ro!ortional to the 5eight of the dis!laced li4uid$ &he sum force

acting on the o9Kect/ then/ is e4ual to the difference 9et5een the 5eight of the o9Kect .Ldo5nL

force0 and the 5eight of dis!laced li4uid .Lu!L force0$ E4uili9rium/ or neutral 9uoyancy/ is

achieved 5hen these t5o 5eights .and thus forces0 are e4ual$

https://en.wikipedia.org/wiki/Weighthttps://en.wikipedia.org/wiki/Densityhttps://en.wikipedia.org/wiki/Gravity_of_Earthhttps://en.wikipedia.org/wiki/Newton_(unit)https://en.wikipedia.org/wiki/Vacuumhttps://en.wikipedia.org/wiki/Weighthttps://en.wikipedia.org/wiki/Densityhttps://en.wikipedia.org/wiki/Gravity_of_Earthhttps://en.wikipedia.org/wiki/Newton_(unit)https://en.wikipedia.org/wiki/Vacuum

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2. DA!"HTS

B$" +hat means 6draughts8P Unit measureP

&he #$%&t .American0 or #$%'()t .3ritish0 of a shi!Ls hull is the vertical distance 9et5een

the 5aterline and the 9ottom of the hull .7eel0/ 5ith the thic7ness of the hull included1 in the

case of not 9eing included the draft outline 5ould 9e o9tained$ %raft determines the minimum

de!th of 5ater a shi! or 9oat can safely navigate$ &he draft can also 9e used to determine the

5eight of the cargo on 9oard 9y calculating the total dis!lacement of 5ater and then

using ArchimedesL !rinci!le$ A ta9le made 9y the shi!yard sho5s the 5ater dis!lacement for

each draft$ &he density of the 5ater .salt or fresh0 and the content of the shi!Ls 9un7ers has to

 9e ta7en into account$ &he closely related term OtrimO is defined as the difference 9et5een th

for5ard and aft drafts$

• &he draft aft .stern0 is measured in the !er!endicular of the stern$

• &he draft for5ard . 9o50 is measured in the !er!endicular of the 9o5$

• &he mean draft is o9tained 9y calculating from the averaging of the stern and 9o5

drafts/ 5ith correction for 5ater level variation and value of the !osition of F 5ith res!ect

to the average !er!endicular$

 

B$B Ex!lain the conce!t of 6metacentre8.e4ually/ transversal and longitudinal0$

So far the shi! has 9een considered as floating in a state of e4uili9rium/ u!right and at

the design draught$It is no5 necessary to consider de!artures from this state$First ta7ethe case 5hen a shi! is heeled through a small angle$ &he centre of 9uoyancy moves to

ne5 !osition 3" and the 9uoyancy force/5hich acts vertically/ that is/ normal to the ne5

5aterline/acts through a !oint M on the centerline$M is 7no5n as the t$%ns*e$se

met%cente$$For angles of u! to a9out " degrees/M can 9e regarded as a fixed !oint$

&he !ositions of 3 and M de!end only u!on the geometry of the shi! and are fixed for

the draught at 5hich it is floating$ &he designer can !rovide information on 3 and M fo

https://en.wikipedia.org/wiki/Sternhttps://en.wikipedia.org/wiki/Length_between_perpendicularshttps://en.wikipedia.org/wiki/Bow_(ship)https://en.wikipedia.org/wiki/Sternhttps://en.wikipedia.org/wiki/Length_between_perpendicularshttps://en.wikipedia.org/wiki/Bow_(ship)

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each mean draught/ assuming the 5aterline is !arallel to the design 5aterline$ Small

de!artures from this state are unli7ely to 9e significant$ +ith such a !lot/7no5n as a

met%cent$+c #+%($%m/ a master can find M and 3 for the condition the shi! is in$

+e have seen ho5 the centre of 9uoyancy/ the centroid of the volume of the dis!laced

5ater/ can 9e found$3M is given 9y:

3M?ID@/ 5here I is the transverse second moment/ or inertia/ of the 5ater!lane and @

the volume of dis!lacement$

 A mathematical !roof of this relationshi! can 9e found in standard text9oo7s on naval

architecture$&he student is as7ed to acce!t it and to note that 3M 5ill vary directly 5it

the s4uare of the 9eam and inversely 5ith the draught$ A large 9eam and shallo5

draught 5ould lead to a very sta9le shi! one difficult to roll over$ )o5ever/ there

5ould 9e disadvantages in terms of ra!id rolling in 5aves and slamming .see later0$

Also/ the high angle sta9ility 5ould 9e !oor$ &he %esigner has to 9alance u! a num9erfactors in deciding u!on the degree of sta9ility to 9uild into a shi!$ &he conce!t of

sta9ility is addressed in more detail in the next cha!ter$

&here is a corres!onding on(+t'#+n% met%cente$ for small changes in angle a9out a

transverse axis$ It 5ill 9e sho5n later that the t5o metacentres are critical to a study of

shi!s sta9ility$ &he !osition of the longitudinal metacenter is defined 9y:

3)(?I(D@/ 5here I( is the longitudinal inertian of the 5ater!lane a9out a transverse axi

through the centroid of the area of the 5ater!lane

Exam!le:

Consider a uniform rectangular solid/ length (/ 9eam 3 and de!th %/ floating in 5ater

5ith its long dimension horiontal$ Assuming the solids density is 7 times tha of

5ater/discuss the form of the metacentric diagram for a range of 7 values$ &he second

moment .or inertia0 of a rectangle a9out a longitudinal axis is 3(D"B$

Solution:

&he solid 5ill float at a draught & 5here: & ? 7%

For a rectangular cross-section:,3?draughtDB ? 7%DB

&hus for this case ,3 increases linearly 5ith % and its !lot 5ill 9e a straight line !assi

through the origin and 5ith slo!e tan-" Q

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For a rectangular 5ater!lane: I?3(D"B .&he student is as7ed to acce!t this0

 No5 3M ? ID@ ? R3(D"B(3& ? 3BD"B& ? 3BD"B7%

3M 5ill 9e infinite 5hen 7 is ero and reduce as 7 increases$

  B$ +rite and ex!lain the formula for 6metacentric height8.relation 9et5een ,

, and ,M0$

&he metacentric height .M0 is a measurement of the initial static sta9ility of a floatin 9ody$ It is calculated as the distance 9et5een the centre of gravity of a shi! and its

metacentre$ A larger metacentric height im!lies greater initial sta9ility against

overturning$ Metacentric height also influences the natural !eriod of rolling of a hull/

5ith very large metacentric heights 9eing associated 5ith shorter !eriods of roll 5hich

are uncomforta9le for !assengers$ )ence/ a sufficiently high 9ut not excessively high

metacentric height is considered ideal for !assenger shi!s

+hen a shi! heeled/ the centre of 9uoyancy of the shi! moves laterally$ It may alsomove u! or do5n 5ith res!ect to the 5ater line$ &he !oint at 5hich a vertical line

through the heeled centre of 9uoyancy crosses the line through the original/ vertical

centre of 9uoyancy is the metacentre$ &he metacentre remains directly a9ove the centre

of 9uoyancy 9y definition$

In the diagram to the right the t5o 3s sho5 the centres of 9uoyancy of a shi! in the

u!right and heeled condition/ and M is the metacentre$ &he metacentre is considered to

 9e fixed for small angles of heel1 ho5ever/ at larger angles of heel the metacentre can n

longer 9e considered fixed/ and its actual location must 9e found to calculate the shi!Ls

sta9ility$&he metacentre can 9e calculated using the formulae:

,M ? ,3 T 3M

3M ?fracVIWV@W

+here ,3 is the centre of 9uoyancy .height a9ove the 7eel0/ I is the Second moment o

area of the 5ater!lane in metres# and @ is the volume of dis!lacement in metres$ ,M

is the distance from the 7eel to the metacentre$ R"

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Sta9le floating o9Kects have a natural rolling fre4uency li7e a 5eight on a s!ring/ 5her

the fre4uency is increased as the s!ring gets stiffer$ In a 9oat/ the e4uivalent of the s!ri

stiffness is the distance called OMO or Ometacentric heightO/ 9eing the distance 9et5ee

t5o !oints: OO the centre of gravity of the 9oat and OMO/ 5hich is a !oint called the

metacentre$

Metacentre is determined 9y the ratio 9et5een the inertia resistance of the 9oat and the

volume of the 9oat$ .&he inertia resistance is a 4uantified descri!tion of ho5 the

5aterline 5idth of the 9oat resists overturning$0 +ide and shallo5 or narro5 and dee!

hulls have high transverse metacenters .relative to the 7eel0/ and the o!!osite have lo5

metacenters1 the extreme o!!osite is sha!ed li7e a log or round 9ottomed 9oat$

Ignoring the 9allast/ 5ide and shallo5 or narro5 and dee! means the shi! is very 4uic7

to roll and very hard to overturn and is stiff$ A log sha!ed round 9ottomed means slo5

rolls and easy to overturn and tender$

OO/ is the center of gravity$ OMO/ the stiffness !arameter of a 9oat/ can 9e lengthened

 9y lo5ering the center of gravity or changing the hull form .and thus changing the

volume dis!laced and second moment of area of the 5ater!lane0 or 9oth$

An ideal 9oat stri7es a 9alance$ @ery tender 9oats 5ith very slo5 roll !eriods are at ris7

of overturning 9ut are comforta9le for !assengers$ )o5ever/ vessels 5ith a higher

metacentric height are Oexcessively sta9leO 5ith a short roll !eriod resulting in high

accelerations at the dec7 level$

Sailing yachts/ es!ecially racing yachts/ are designed to 9e stiff/ meaning the distance

 9et5een the centre of mass and the metacentre is very large in order to resist the heelin

effect of the 5ind on the sails$ In such vessels the rolling motion is not uncomforta9le

 9ecause of the moment of inertia of the tall mast and the aerodynamic dam!ing of the

sails$

+hen setting a common reference for the centres/ the molded .5ithin the !late or

 !lan7ing0 line of the 7eel .,0 is generally chosen1 thus/ the reference heights are:

,3 - to Centre of 3uoyancy

, - to Centre of ravity

,M& - to &ransverse Metacentre

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B$ +hat means : &C and MC&P

Tonnes -e$ Cent+met$e Imme$s+on T-C

For each 5ater!lane the area defines/ for a given density of 5ater/ the cange in 9uoyancy in

tonnes is ex!erienced due to unit increase or decrease in draught$ &he unit of draught change

considere is one centimeter/ giving the tonnes e$ cm +mme$s+on T-C. &he other from tha

is sometimes used is &I/ standing for tonnes !er nit immersion$ If the area of a 5ater!lane is

A mB and the 5ater densiy is ! tonnes !er m/ then the &C 5ill 9e A !D"$

oment to C)%n(e T$+m ne et$e CT

 No5 consider de!artures from the steady state 9y rotation a9out a transverse axis/ that is 5he

the shi! is trimmed$ (et it trim through a small angle/  ,ϕ a9out an axis through the CF/ extra 9uoyancy 5ill 9e created at the end of the shi! at 5hich the draught increases and a loss of

 9uoyancy at the end 5hich emerges from the 5ater$&he forces due to these 9uoyancy forces

5ill !roduce a moment 5hich 5ill act so as to o!!ose the change of trim$ &he total 9uoyancy

5ill remain constant and e4ual to the shi!s 5eight/ + / other5ise the shi! 5ould move u! or

do5n$ If M( is the longitudinal metacenter/ the righting movement 5ill 9e:

+$M(X

&his 5ill 9e the moment re4uired to hold the shi! at the ne5 trim angle$ If the trim change isone metre 9et5een !er!endiculars/ the moment is reterred to as the moment to c)%n(e t$+m

one met$e CT$

It 5ill 9e noted that the MC& does de!end u!on the !osition of / so it 5ill vary 5ith the

loading of the shi!$ )o5ever/ 3M(.?I(D@0 is large/ ty!ically of the order of the length of the

shi!$ It 5ill 9e a!!reciated that the variation in the height of a9ove the 7ell 5ll 9e small

com!ared 5ith M($ )ence / M( 5ill not vary much 5ith the loading of the shi!$ &he stude

5ill no5 a!!reciate 5hy/ in the !revious cha!ter/ the transverse and longitudinal inertias of th

5ater!lane 5ere mentioned$ &hey are im!ortant in defining the heights of the transverse and

longitudinal metacentres a9ove the centre of 9uoyancy$

+e have seen that for small changes in trim a9out the CF/ there is no change in overall

dis!lacement$ If the trim/ ϕ/ is a9out amidshi!s / there 5ill 9e a change of draught at the CF

given 9y:

X 5here x is the distance of the CF from amidshi!

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&here 5ill 9e a corres!onding change in dis!lacement given 9y:

X .&C0

&his change 5ill 9e an increase 5hen the trim is such that the draught at the CF !osition

increase$

A !lot of the a9ove data against draught gives rise to a set of curves 7no5n as the H4#$ost%t

c'$*es or sim!ly )4#$ost%t+cs$

3. STABIIT

  $" Ex!lain the sta9ility conce!t.transversal and longitudinal0$

&'ANS@E'SE S&A3I(I&Y/SMA(( AN(ES

So far/ consideration has 9een given only to a shi! 5hen in e4uili9rium or moving slo5ly fro

one !osition of e4uili9rium to another$ No5 consider 5hat ha!!ens 5hen a shi! is su9Kect to

small heeling moment$

For small distur9ances/ it is reasona9le to assume that there is no change in trim and that the

u!right and heeled 5aterlines 5ill intersect on the centerline of the shi!$ &he shi! 5ill ta7e u!

a small angle of heel$&here 5ill 9e a ne5ly su9merged 5edge of hull on one side of thecenterline and a corres!onding emerged 5edge on the other$&he centre of 9uoyancy 5ill mov

from its initial !osition in the centerline !lane/ 3 / to a ne5 !osition 3" such that the distance

 9et5een the t5o 5edges$ &he line 33" 5ill 9e !arallel to the line Koining the centres of volum

of the t5o 5edges$

&he total 9uoyancy force remains constant 9ut it no5 acts in a vertical line through 3"/ that is

normal to the inclined 5aterline$ &his line intersects the shi!s centerline !lane in M/ the

metacenter$ As !reviously noted for most common shi! forms/ M can 9e ta7en as a fixed !oin

for small angles/ ty!ically u! to a9out " degrees$

&he 5eight + .e4ual to the 9uoyancy0 5ill act through the centre of gravity of the shi! 5ich

remains fixed$ &he resultant momen acting on the shi! 5ill 9e given 9y:

+$M$X 5here X is the angle of heel/ assumed small

• From a sta9ility !oint of vie5/ there are three !ossi9ilities:

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• M is a9ove $ In this case / the moment acting/ due to the shift in 9uoyancy/ is tending

restore the shi! to the u!right !osition$ &he shi! is said to 9e st%e1 it has! os+t+*e

st%++t4$

• M is 9elo5 $ In this case/ the moment is trying to !ush the shi! further from the

u!right$ &he shi! is said to 9e 'nst%e1 it has ne(%t+*e st%++t4$

• M and coincide$ In this case/ there is no moment acting on the shi! 5hich 5ill rema

in the heeled !osition$&he shi! is said to have ne't$% st%++t4$

As stated !reviously/ the vertical distance 9et5een the centre of 9uoyancy and the metacenter

is given 9y:

3M ? ID@/ 5here I is the second moment of area/ or the inertia/ of the 5ater!lane a9out a

longitudinal axis and @ is the total volume of dis!lacement$

For a rectangular 5ater!lane of length/ ( / and 9readth/ 3 / the inertias a9out its centerline ar

given 9y:

I ?3(D"B

In general terms/ @ 5ill 9e !ro!ortional to (3& 5here & is the draught of the vessel concerne

&hus/ in this case/ 3M ? Const$3BD& RNote 3 changed to 3B

Although a normal shi!s 5ater!lane is not rectangular/ this general relationshi! 5ill 9e

a!!roximately true for shi!s$ &hus/ the 9eam has a very great influence on the initial sta9ility

of a shi!$ It might a!!ear/ at first/ that good sta9ility could 9e o9tained 9y ma7ing the draugh

small$ +hilst true for small angles/ the sta9ility at large angles 5ould suffer$

In the a9ove 4uestion/ the cylinder floated 5ith its axis vertical$ &he case of a circular cylind

floating 5ith its axis horiontal is interesting$ As the cylinder turns a9out its axis/ that is as it

heels/ the vertical through the centre of 9uoyancy 5il al5ays act through the centre of the

circular cross-section$ &hus/ M is at the centre of the circle$ &his a!!lies at 5hatever draught

the cylinder is floating at !rovided its axis is horiontal$

(ongitudinal Sta9ility

 

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arameters such as centre of gravity and centre of 9uoyancy have 9een used in descri9ing

transverse sta9ility/ so far$ &hey can also 9e used to descri9e longitudinal sta9ility$

 

(C3 and (C

 

+('$e 3.16 CB o$9%$# o& C" : ;esse t$+ms 4 t)e ste$n 

In Fig $"J (C3 is the longitudinal centre of 9uoyancy$ &his is the longitudinal centre of the

under5ater volume/ and is the !oint through 5hich all the 9uoyancy can 9e said to act

vertically u!5ards$

 

(C is the longitudinal centre of gravity$ &his is the !oint through 5hich all of the 5eight of

the vessel can 9e said to act vertically do5n5ards$

 

If the !osition of (C and (C3 are as sho5n in Fig $"J then the actions of 9uoyancy and

5eight 5ill cause the vessel to rotate as sho5n 9y the arro5$ &he stern 5ill sin7 dee!er/ the

 9o5 5ill rise higher$ (C3 is the longitudinal centre of all under5ater volume$ As the vessel

rotates/ the sha!e of the under5ater volume 5ill change and (C3 5ill move to the ne5 centr

 

+hen (C and (C3 are in the same vertical line/ the rotation 5ill sto!/ the vessel 5ill 9e

trimmed 9y the sternas sho5n in Fig $"H$

 

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 +('$e 3.17 C" %n# CB +n s%me *e$t+c% +ne : no t$+mm+n( moment

If the vessel had started 5ith (C3 aft of (C as sho5n in Fig $"G then the rotation 5ould

cause a trim 9y the 9o5$

+('$e 3.18 CB %&t o& C" : *esse t$+ms 4 t)e )e%#

  $B Ex!lain 5hat means 6&he Metacentric %iagram8 and 5rite the formula

for sta9ility at

6small angle8

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T)e et%cent$+c D+%($%m

It has 9een seen that the !ositions of !oints 3 and M de!end only u!on the geometry of the

shi! and are fixed for the draught at 5hich it is floating$ de!ends u!on the loading conditio

of the shi! and 5ill vary 5ith time$ +ith the metacentric diagram/ a master can find M for thdraught at 5hich the shi! is floating$ From the loading of the shi!/ the master 5ill 9e a9le to

assess the !osition of $ &he value of M/ and hence the initial sta9ility follo5s$

TA

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Z M and coincide$ In this case/ there is no moment acting on the shi! 5hich 5ill rema

in the heeled !osition$&he shi! is said to have neutral sta9ility$

As stated !reviously/ the vertical distance 9et5een the centre of 9uoyancy and the metacenter

is given 9y:

3M ? ID@/ 5here I is the second moment of area/ or the inertia/ of the 5ater!lane a9out a

longitudinal axis and @ is the total volume of dis!lacement$

For a rectangular 5ater!lane of length/ ( / and 9readth/ 3 / the inertias a9out its centerline ar

given 9y:

I ?3(D"B

In general terms/ @ 5ill 9e !ro!ortional to (3& 5here & is the draught of the vessel concerne

&hus/ in this case/ 3M ? Const$3BD& RNote 3 changed to 3B

Although a normal shi!s 5ater!lane is not rectangular/ this general relationshi! 5ill 9e

a!!roximately true for shi!s$ &hus/ the 9eam has a very great influence on the initial sta9ility

of a shi!$ It might a!!ear/ at first/ that good sta9ility could 9e o9tained 9y ma7ing the draugh

small$ +hilst true for small angles/ the sta9ility at large angles 5ould suffer$

In the a9ove 4uestion/ the cylinder floated 5ith its axis vertical$ &he case of a circular cylind

floating 5ith its axis horiontal is interesting$ As the cylinder turns a9out its axis/ that is as itheels/ the vertical through the centre of 9uoyancy 5il al5ays act through the centre of the

circular cross-section$ &hus/ M is at the centre of the circle$ &his a!!lies at 5hatever draught

the cylinder is floating at !rovided its axis is horiontal$

 

$ Ex!lain in details / the ty!ical 6 [ curve8 or 6 curve of statical sta9ility8 of

figure $$

T)e " C'$*e o$ C'$*e o& St%t+c% St%++t4

A ty!ical !lot of [ against angle of heel is sho5n in the next image$

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Some interesting information can 9e deduced from the statical sta9ility curve:

• &he slo!e of the curve at the origin re!resents the value of M for small angles$

• &he maximum value of [ multi!lied 9y the dis!lacement re!resents the greatest stea

heeling moment the shi! can sustain 5ithout turning right over$

•

&he angle at 5hich [ 9ecomes ero is 7no5n as the !oint of *%n+s)+n( st%++t4 anddefines the $%n(e o& st%++t4$

• &he !oint of inflexion on the 5ay to the maximum is an indication of the angle at 5hic

a reasona9le length of dec7 edge 9ecomes immersed$

• &he area under the curve u! to any given angle/ multi!lied 9y the dis!lacement/

re!resents the energy needed to heel the shi! to that angle$ It is referred to as the shi!s

#4n%m+c% st%++t4 and is a measure of the a9ility of the shi! to a9sor9 the energy

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im!arted 9y 5aves and gusting 5inds$ &his should not 9e confused 5ith a shi!s

dynamic sta9ility 5hich is a measure of its a9ility to maintain a constant heading$

  $# +hat means 6 angle of loll8P

In calm conditions/ the shi! 5ill float at an angle e4ual to that at 5hich [ 9ecomes !ositive

&his is the %n(e o& o. &he shi! 5ill loll to !ort or star9oard de!ending on ho5 she arrived

the initial condition$ A small moment a!!lied to 9ring it u!right 5ill cause her to 6flo!8

suddenly to a similar angle on the other side$ An ex!erienced master 5ill recognie from themotion of the shi! 5hen she exhi9its this characteristic and can decide 5hat action needs to 9

ta7en$

>. STE

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 !ure moments .or tor4ue0 can 9e measured RJ$ 'ecently/ an a!!lication consisting in six-

com!onent

measurements of the actions torsor during cutting !rocess 5as carried out for the case of high

s!eed milling

RH/ drilling RG/ \/ etc$ Cahuc et al$/ in R"/ !resent another use of this six-com!onent

dynamometer in anex!erimental study: ta7ing into account of the cut moments allo5s a 9etter machine tool !o5

consum!tion

evaluation$ It allo5s a 9etter a!!roach of the cut RG/ ""/ "B and should thus allo5 in the

dynamic case to

reach ne5 !ro!erties of the vi9rations of the system !iece-tool-matter$

Moreover/ the tool torsor has the advantage of 9eing trans!orta9le in any !oint of s!ace and i

es!ecially/ at the tool ti! in * !oint$ &he follo5ing study is carried out in several stages/

including t5o maKor 

stages1 the first of is related to the analysis of forces$ &he second of is dedicated to the

determining of thecentral axis and a first moments analysis to the central axis during the cut$

In !aragra!h B 5e !resent first the ex!erimental device used and the associated elements of 

measurement$ aragra!h is devoted to the measurement of the tor4ue of the cutting actions

An analysis of 

the forces exerted during the cut action is carried out$ It allo5s to esta9lish in ex!eriments

certain !ro!erties

of the resultant of the cutting actions$ &he case of the moments at the tool ti! !oint is also

examined 5ith

 !recision$ &he central axis of the tor4ue is re4uired .!aragra!h #0$ &he 9eams of central axesdeduced from

the multi!le tests confirm es!ecially the !resence of moments at the tool ti! !oint$ In !aragra

/ 5e more

 !articularly carry out the analysis of the moments at the central axis 9y loo7ing at the case th

most sensitive

to vi9rations .a! ? mm/ f ? $" mmDrev0$ 3efore concluding/ this study gives a certain

num9er of !ro!erties

and drive to some innovative reflexions$

 #$B +rite and ex!lain the 6&*'S*' *F E]&E'NA( F*'CES8

 

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&he torsor of the external forces exerted on a solid .S0/ denoted RF/ is determined 9y its

reduction elements com!rising the resultant A of these forces and 9y their moment mA at a

 !oint A$3y com!arison 5ith the reduction elements at another !oint B/ 5e deduce:

  A ? 3 ? / mA ? m3 T AB ]

+e 5rite sym9olically" RF?R/ mA$

&he tensor of forces is RT ?RT.s0/ .s0 and in M 5e have M ? .s0T- ] T. 3y

derivation/ 5e o9tain the reduction elements of the torsor Rd T / d s/ so:

Rd t/ ds?Rd T. s0Dd s, d  . s0D d s+t ] T. s0

 

#$ %ra5 and ex!lain the 6SI] F'EE%*M %E'EE8 of the general

shi! motions in +aves. three translations and three rotation0$

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&he term is im!ortant in mechanical systems/ es!ecially 9iomechanical systems for analying

and measuring !ro!erties of these ty!es of systems that need to account for all six

degrees of freedom$ Measurement of the six degrees of freedom is accom!lished today

through 9oth AC and %C magnetic or electromagnetic fields in sensors that transmit !ositional and angular data to a !rocessing unit$ &he data are made relevant through

soft5are that integrate the data 9ased on the needs and !rogramming of the users$

Ascension &echnology Cor!oration has recently created a J%oF device small enough to fit in

 9io!sy needle/ allo5ing !hysicians to 9etter research at minute levels$ &he ne5 sensor

 !assively senses !ulsed %C magnetic fields generated 9y either a cu9ic transmitter or a

flat transmitter and is availa9le for integration and manufactura9ility 9y medical *EMs

An exam!le of six degree of freedom movement is the motion of a shi! at sea$ It is descri9edas

&ranslation:

Moving u! and do5n .heaving01

Moving left and right .s5aying01

Moving for5ard and 9ac75ard .surging01

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'otation

&ilting for5ard and 9ac75ard .!itching01

&urning left and right .ya5ing01

&ilting side to side .rolling0$

  #$# +rite the 6%A(EM3E'& 'INCI(E8 and ex!lain the com!onent

of the 6&*'S*' *F IN&E'NA( F*'CES8.inside the shi! 9ody0$

%LAlem9ertLs !rinci!le/ also 7no5n as the (agrangedLAlem9ert !rinci!le/ is a statement of th

fundamental classical la5s of motion$ It is named after its discoverer/ the French

 !hysicist and mathematician ^ean le 'ond dLAlem9ert$ It is the dynamic analogue to th !rinci!le of virtual 5or7 for a!!lied forces in a static system and in fact is more genera

than )amiltonLs !rinci!le/ avoiding restriction to holonomic systems$ A holonomic

constraint de!ends only on the coordinates and time$ It does not de!end on the velociti

If the negative terms in accelerations are recognied as inertial forces/ the statement of

dLAlem9ertLs !rinci!le 9ecomes &he total virtual 5or7 of the im!ressed forces !lus the

inertial forces vanishes for reversi9le dis!lacements$ &he !rinci!le does not a!!ly for

irreversi9le dis!lacements/ such as sliding friction/ and more general s!ecification of th

irreversi9ility is re4uired$

&he !rinci!le states that the sum of the differences 9et5een the forces acting on a system of

mass !articles and the time derivatives of the momenta of the system itself along any

virtual dis!lacement consistent 5ith the constraints of the system/ is ero$ &hus/ in

sym9ols dLAlem9ertLs !rinci!le is 5ritten as follo5ing/

+here

  is an integer used to indicate .via su9scri!t0 a varia9le corres!onding to a !articular

 !article in the system/

  is the total a!!lied force .excluding constraint forces0 on the -th !article/

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  is the mass of the i-th !article/

  is the acceleration of the i-th !article/

  together as !roduct re!resents the time derivative of the momentum of the i-th !article

and

  is the virtual dis!lacement of the i-th !article/ consistent 5ith the constraints$

  %LAlem9ertLs !rinci!le of inertial forces

%LAlem9ert sho5ed that one can transform an accelerating rigid 9ody into an e4uivalent stati

system 9y adding the so-called Oinertial forceO and Oinertial tor4ueO or moment$ &he

inertial force must act through the center of mass and the inertial tor4ue can act

any5here$ &he system can then 9e analyed exactly as a static system su9Kected to this

Oinertial force and momentO and the external forces$ &he advantage is that/ in thee4uivalent static system one can ta7e moments a9out any !oint .not Kust the center of

mass0$ &his often leads to sim!ler calculations 9ecause any force .in turn0 can 9e

eliminated from the moment e4uations 9y choosing the a!!ro!riate !oint a9out 5hich t

a!!ly the moment e4uation .sum of moments ? ero0$ Even in the course of

Fundamentals of %ynamics and ,inematics of machines/ this !rinci!le hel!s in

analying the forces that act on a lin7 of a mechanism 5hen it is in motion$ In text9oo7

of engineering dynamics this is sometimes referred to as dLAlem9ertLs !rinci!le$

  #$ Ex!lain the 6FA&IUE8 conce!t$ +hat means 6F'AC&U'E8P

ATI"!E

%uring a shi!s life/ her structure is su9Kect to fluctuating strains due to o!erations such as loading/a

very im!ortantly in a sea5ay as the hull flexes in the 5aves$

f cyclic stresses are very high/ then the num9er of reversals a steel structure can 5ithstand 5ithoutailre is limited$ Consider a stri! of steel 9eing 9ent to and froin a vice$If 9ent through a small angle

can 5ithstand many reversals$ If 9ent through large angles/ it 5ill soon fracture$ Further/ if a notch

made in the stri! at the !oint of 9ending it 9rea7s much more 4uic7ly due to the stress concentration

roduced 9y the notch$ &he surface of the fracture 5ill 9e fi9rous in character$

or many materials/ including mild steel/ 5hen su9Kect to cyclic stressing in the la9oratory there

!!ears to 9e a level of stress 5hich can 9e re!eated indefinitely 5ithout failure$ &his is 7no5n as th

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atigue limit and results are usually !lotted as stress level .S0 against num9er of reversais to failure

N0/ the S-N curve for the material$ &he curve is asym!totic to the fatigue limit$ Unfortunately/ there

o5 evidence that steels have no such limit 5hen used in a corrosive environment such as that at sea

Avoiding/ or reducing/ stress concetrations and 9uild-in stresses 5ill increase a structures fatigue lif

ut it must al5ays 9e considered in design$ Ex!erience sho5s that most !ro9lems arise from theetailed design of 5elded connections$ (a9oratory test on ty!ical connectons are carried out to

rovide design guildance$At various mean stress levels/ the s!ecimens are su9Kect to stress reversals

arious ranges of stress$ lots are usually in logDlog from 5ith the stress range !lotted against the

um9er of reversals to failure$ &y!ical !lots for mild steel are straight lines 5ith the num9er of

eversals increasing as the stress range reduces$ At the fatigue limit/ a second line !arallel to the N ax

omes into !lay$

are is needed in using la9oratory results to !redict !erformance at sea$ A!art from the corrosive

tmos!here/ already referred to/ the range of stress and the mean stress to 5hich the shi! is su9Kect a

onstantly varying$ Safety factors are a!!lied to the la9oratory results to allo5 for these influences$

ACT!E

he characteristics of a fracture in steel de!ends u!on its toughness/ and tem!erature$ At reasona9le

em!eratures fracture initiation is 9y the coalescence of voids in the material$ &he crac7 then gro5

nly if/ and 5hen/ it is su9Kected to increased strain or load$ Such crac7s 5ill extend at moderate rate

measured in mmDsecond and are li7ely to 9e detected 9y regular ins!ections 9efore they can cause

erious trou9le for the shi!$ At lo5er tem!eratures/ fracture is 9y cleavage$ &he fracture is then said e 9rittle and rates of !ro!agation can 9e a9out mDsecond/ causing catastro!hic failure$ &he

em!erature mar7ing the transition from one mode of fracture to the other is 7no5n as the transition

em!erature$ It de!ends u!on the microstructure of the material/ the loading/ rate/ structural geometr

nd the nature of the notch$ In thic7 materials tri-axial stresses can 9e set u! 5hich favour 9rittle

acture$ )igh loading rates are 9ad$

he Char!y. for details of this test the student can refere to a standard text on strength of materials0

est is often used in 4uality control to indicate 5hether a material is li7ely to exhi9it 9rittle fracture

ver a range of tem!eratures$ It is a sim!le standard test that is easy to carry out 9ut it does note!roduce the geometry and method of loading$ )o5ever / 5ithin the general !arameters of shi!

ructures and their loading/ the Char!y test can indicate 5hich steels are li7ely to 9e effective$

ecommendations are 9ased on the !ercentage of the Char!y fracture that is crystalline$ In some cas

!ecially tough steel stra7es are !rovided to act as crac7 arresters and for these the !ercentage shoul

e ero$ &hat is/ the fracture must 9e totally fi9rous$ A figure of _ !rovides good crac7 arrest

a!a9ility and should avoid fracture initiate H_ gives a god !ro9a9ility that 9rittle fracture 5ill not

nitate$

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is clear that crac7s cannot 9e !revented com!letely although their accourrence can 9e reduced 9y

ood design$ *nce initated/ they can extend due to fatigue or 9rittle fracture$ Steels having high notc

oughness should 9e selected to reduce the rsi7 of catastro!hic 9rittle fracture occurring$ As steel is

more 9rittle at lo5 tem!eratures/ shi!s intended to o!erate in ice 5ill re4uire steels 5ith higher

oughness$ So 5ill shi!s 5hich must 9e ca!a9le of sustaining high loading rates/ as steels react inmore 9rittle fashion 5hen the rate of stress 9uild-u! is high$ Such conditions may a!!ly in collisions

r ex!losions$

3$ 'AC&ICA( '*3(EMS

 

"$ THE BASICS.

  -1. T)e so*e# $oem %s %n e?eme %&te$ &+('$e 1.6.

Exam!le

A shi! of length " m and 9eam "G m floats at a mean draught of H$m 5hen in 5ater of

density "$B tonnesDm$Assuming her 9loc7 coefficient is $ and her 5ater!lane coefficient

of fineness is $J/ calculate the shi!s dis!lacement and her a!!roximate draught 5hen it ente

5ater of "$" tonnesDm

Solution

@olume of circumscri9ing rectangular 

Solid ? " ] "G ] H$ ? B/B m

@olume of dis!lacement ? $ ] BB ? "/"B m

)ence dis!lacement ? "/"B ] "/B ? "/HG tonnes

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&he 5ater!lane area ? /J ] " ] "G ? "JB mB

In the less dense 5ater the shi! must dis!lace more 5ater to !roduce a 9uoyancy force e4ual

its 5eight$

  Ne5 volume in the less dense 5ater ? "/HGD"/" ? "/BB m  &he shi! 5ill sin7 in the 5ater to com!ensate for the added volume

  Added volume ? "/BB "/"B ? " m

  Sin7age ? "D"JB ? J/"H cm

  Ne5 draught is a!!roximately H/ J m

  -2. Se&:%ssessment @'est+on 1.

A shi! has a length of "G m/ a 9eam of "Hm and a draught of J/H m 5ith a dis!lacement of

"/ tonnes and an immersed midshi! section area of "# mB$ Assuming a 5ater density of

"$B tonnes !er cu9ic metre/ find the 9loc7/ longitudinal !rismatic and midshi!-section

coefficinets$

&he 9loc7 coefficient .C30 is the ratio of the under5ater hull volume of a shi! at a !articulardraft to the volume of a rectangular !rism .the circumscri9ing !rism0 of the same length/

 9readth/ and draught as the shi!$

C3 ? @D.( x 3 x &0

C3? "#D."G]"H]J/H0

C3?"H#$"mB

&he longitudinal !rismatic coefficient .C0 is the ratio of the under5ater hull volume of a shi

to the volume of a !rism 5ith length e4ual to the shi!Ls and cross-section area identical to the

midshi! section$ In other 5ords/ the !rismatic coefficient is e4ual to the 9loc7 coefficient .C3divided 9y the midshi! section coefficient .CM0$

C ? @D.Am x (0 ? C3DCM

C?"#D."$B ] "G0 ? "H#$"D"""$\"

C?"$HBG"?"$HBG"

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&he midshi! section coefficient .CM0 is the ratio of the area of the under5ater midshi! sectio

.Am0 of a shi! at a !articular draft to the area of a rectangle .the circumscri9ing rectangle0 of

the same 9readth and draught as the shi!$

CM ? AmD.3 x &0

CM?"$BD."H]J/H0

CM?"""$\"

  -3. Se&: %ssessment @'est+on >.

A 9arge m long 9y G m 9eam floats in 5ater of density "$B tonnesDm$ &he centre of

gravity of the 9arge itself is on the centerline amidshi!s$ +eights are added at the !ositions

indicated in the ta9le$

+here must an additional 5eight of " tonnes 9e added so that the 9arge 5ill sin7 9odily

5ithout heel or trimP

+hat 5ill 9e the increase in draughtP

2. DA!"HTS.

  ->. T)e so*e# $oem %s %n e?eme %&te$ &+('$e 2.7.

A vessel dis!lacement B/ tonnes/ has a length 9et5een !er!endiculars of B m/ and has

level draught of J m$ It has a 5ater!lane area of $ mB/ 5ith its centre of flotation m aft

amidshi!s$Calculate the ne5 draughts for5ard and aft if a 5eight of B tonnes is added m

for5ard of the centre of flotation$ &he longitudinal metacenter is " m a9ove the shi!s

centre of gravity$ Assume the 5ater density is "$B tonnesDm$

Solution

&he &C of the 5ater!lane is $$ ] "$BD" ? "$B tonnes !er cm$

If the 5eight 5ere added at the CF the !arallel sin7age 5ould 9e BD"$B ? $\ cms ? $m

Moving the 5eight for5ard m causes a moment of B ] ? J$ tonnes m trimming th

shi! 9y 9o5

&he resulting trim is J$DB$ ] " ? "D radians

&he shi! trims a9out the CF so the increase in draught f5d ? "D m ? B" cm

Aft ? Jm T .$\ "\0 cms ? $G#\ m

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  -5. Se& –%ssessment @'est+on 3.

For the shi! in the a9ove 5or7ed exam!le/ in her initial condition/ calculate the ne5 draughtsand if a 5eight of " tonnes is removed from a !oint B m for5ard of amidshi!s$

Find the !osition at 5hich a " tonne 5eight must 9e added in order that the draught for5ar

does not change$

  -6.Se&:%ssessment @'est+on >.

+hat condition must 9e met in order that a shi! 5ill not change trim in moving 9et5een 5atof different densitiesP

If a shi! is of " m length 5ith its centre of flotation m aft of amidshi!s/ 5hat 5eight !lac

" m for5ard of amidshi!s 5ill cause the same trim as a 5eight of J tonnes !laced " m from

the 9o5P

3EINNE%: "H$""$B"

%EA%(INE: F'I%AY/ "$"B$B"$ Su!ervisor: drd$ asist$univ$ in

  %UMI&'ESCU (AU'EN IȚ 

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