small angle stability longitudinal

SMALL ANGLE STABILITY:SMALL ANGLE STABILITY:L n itudin l St bilitL n itudin l St bilitLongitudinal StabilityLongitudinal Stability

INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION•• In the previous section, we have calculated vertical and In the previous section, we have calculated vertical and

transverse weight shifts, weight additions, and weight transverse weight shifts, weight additions, and weight removals In this section we will look at removals In this section we will look at longitudinallongitudinal weight weight removals. In this section we will look at removals. In this section we will look at longitudinallongitudinal weight weight shifts, weight additions, and weight removals. Longitudinal shifts, weight additions, and weight removals. Longitudinal problems are done in a different manner because we are problems are done in a different manner because we are usually not concerned with the final position of G, but the usually not concerned with the final position of G, but the newnew trim conditiontrim condition of the ship of the ship newnew trim conditiontrim condition of the ship. of the ship.

•• The consequence of longitudinal shifts, additions, and The consequence of longitudinal shifts, additions, and removals of weight is that the ship undergoes a change in removals of weight is that the ship undergoes a change in th f d d ft d ft Wh th f d d ft th f d d ft d ft Wh th f d d ft the forward and after drafts. When the forward and after the forward and after drafts. When the forward and after drafts have drafts have different magnitudesdifferent magnitudes the ship is said to have the ship is said to have trim. trim.

•• Trim is defined by the difference between the forward and Trim is defined by the difference between the forward and after drafts. after drafts.

T i TT i T TTTrim = TTrim = Taftaft -- TTfwdfwd

INTRODUCTION Cont.INTRODUCTION Cont.INTRODUCTION Cont.INTRODUCTION Cont.

•• If a ship is If a ship is "trimmed by the bow,""trimmed by the bow," then the forward draft is then the forward draft is bigger than the after draft A ship bigger than the after draft A ship "trimmed by the stern""trimmed by the stern"bigger than the after draft. A ship bigger than the after draft. A ship "trimmed by the stern""trimmed by the stern"has an after draft bigger than the forward draft. has an after draft bigger than the forward draft.

•• Recall that the ship rotates about the Recall that the ship rotates about the center of flotation (F)center of flotation (F)eca a e s p o a es abou eeca a e s p o a es abou e ce e o o a o ( )ce e o o a o ( )which is the centroid of the waterplane area. (It does not which is the centroid of the waterplane area. (It does not rotate about midships!) When the centroid of the rotate about midships!) When the centroid of the waterplane area is aft of midships the forward draft will waterplane area is aft of midships the forward draft will change by a larger amount than the after draft. This is change by a larger amount than the after draft. This is g y gg y gusually the case since a typical ship is wider aft of midships usually the case since a typical ship is wider aft of midships than forward of midships. than forward of midships.

LONGITUDINAL STABILITY AND TRIMLONGITUDINAL STABILITY AND TRIMLONG UD N L S L Y ND MLONG UD N L S L Y ND M

•• A ship floating at equal draught all along is said to be on an A ship floating at equal draught all along is said to be on an even keeleven keel, or to have , or to have zero trimzero trim. If the . If the ddraughts are not the raughts are not the same from bow to stern, the ship is floting with a trimsame from bow to stern, the ship is floting with a trim..

LONGITUDINAL STABILITY AND TRIM LONGITUDINAL STABILITY AND TRIM C tC tCont.Cont.

•• Two waterlines (or waterplanes) are shown on the ship in Two waterlines (or waterplanes) are shown on the ship in Figure Figure 1 01 0; a trimmed waterline (W1L1) and the even keel ; a trimmed waterline (W1L1) and the even keel Figure Figure 1.01.0; a trimmed waterline (W1L1) and the even keel ; a trimmed waterline (W1L1) and the even keel waterline (WL) corresponding to the same displacement. waterline (WL) corresponding to the same displacement. They are shown intersecting at the centre of flotation of the They are shown intersecting at the centre of flotation of the even keel waterplane. The quantities shown in the figure even keel waterplane. The quantities shown in the figure are defined as followsare defined as follows::are defined as followsare defined as follows::

•• Centre of Flotation (F):Centre of Flotation (F): Geometric center of the ship's Geometric center of the ship's waterline plane. The ship trims about this point. May be waterline plane. The ship trims about this point. May be p p p yp p p yforward or aft of the midships depending on the ship's hull forward or aft of the midships depending on the ship's hull shape at the waterline.shape at the waterline.

•• Longitudinal Centre of Flotation (LCF) :Longitudinal Centre of Flotation (LCF) : Distance from Distance from •• Longitudinal Centre of Flotation (LCF) :Longitudinal Centre of Flotation (LCF) : Distance from Distance from the centre of flotation (F) to the midships. Used to the centre of flotation (F) to the midships. Used to distribute changes of trim between the fwd and aft distribute changes of trim between the fwd and aft draughts.draughts.


•• Trim (t) :Trim (t) : The difference between the forward and after The difference between the forward and after draughtsdraughtsdraughtsdraughts

•• Parallel Rise/Sinkage (PR/PS):Parallel Rise/Sinkage (PR/PS): When weight is When weight is removed/added from/to a ship at LCF, the forward and aft removed/added from/to a ship at LCF, the forward and aft e o ed/added o / o a s p a C , e o a d a d ae o ed/added o / o a s p a C , e o a d a d adrafts will change by the same amount.drafts will change by the same amount. Means, no trim Means, no trim occur.occur.

•• Change in Trim (CT):Change in Trim (CT): The sum total of the absolute The sum total of the absolute •• Change in Trim (CT):Change in Trim (CT): The sum total of the absolute The sum total of the absolute values of the change in forward and after drafts.values of the change in forward and after drafts.

•• Trimming Arm (Trimming Arm (dd):): The distance from the center of The distance from the center of g (g ( ))gravity of the weight to the LCF. If the weight is shifted, gravity of the weight to the LCF. If the weight is shifted, ((d)d)is the distance shifted.is the distance shifted.


T i i M (TM)T i i M (TM) M b h LCF M b h LCF •• Trimming Moment (TM):Trimming Moment (TM): Moment about the LCF Moment about the LCF produced by weight additions, removals, or shifts (wd), produced by weight additions, removals, or shifts (wd), where w is the amount of weight added, removed, or where w is the amount of weight added, removed, or shifted.shifted.

•• Moment to Moment to change change Trim One cm (MTrim One cm (MCTCCTC):): The moment The moment necessary to produce a change in trim (CT) of one cm. necessary to produce a change in trim (CT) of one cm. Found using the hydrostatic curvesFound using the hydrostatic curvesFound using the hydrostatic curves.Found using the hydrostatic curves.

•• Tons Per cm Immersion (TPTons Per cm Immersion (TPCC):): The number of tons The number of tons added or removed necessary to produce a change in mean added or removed necessary to produce a change in mean d ftd ft ( ll l i k )( ll l i k ) f f P ll l i k i h draftdraft (parallel sinkage)(parallel sinkage) of one cm. of one cm. Parallel sinkage is when the ship changes it’s forward and after drafts by the same amount so that no change in trim occurs.


TT D h f d D h f d ••TTFF :: Draught forward Draught forward

••TTAA :: Draught aftDraught aft

••TTMM :: Mean draught at amidships. It is the average of TMean draught at amidships. It is the average of TFF and Tand TAA..

••TTOO :: Draught at centre of floDraught at centre of floaation also called the corresponding tion also called the corresponding ••TTOO :: Draught at centre of floDraught at centre of floaation, also called the corresponding tion, also called the corresponding even keel draught.even keel draught.

••δδTTFF :: Change in draughts forwardChange in draughts forward

••δδTTAA :: Change in draughts afterChange in draughts after

θθ t i lt i l•• θθ:: trim angletrim angle

Figure 2.0Figure 2.0 shows more detail on hull geometry and sign conventions adopted in this notesshows more detail on hull geometry and sign conventions adopted in this notes

Figure 2.0Figure 2.0

Trim due to Movement of WeightsTrim due to Movement of Weightsr m u to Mo m nt of W ghtsr m u to Mo m nt of W ghts

ML

GG1FW1 L

L

d

w

BB1

1

WL1

Figure 1.0Figure 1.0

Trim due to Movement of Weights Trim due to Movement of Weights C tC tCont…Cont…

•• Consider the ship as in Consider the ship as in Figure 1.0Figure 1.0 above, if the weight above, if the weight ww is is moved a distance moved a distance dd meter G will move to G’ parallel to the meter G will move to G’ parallel to the moved a distance moved a distance dd meter, G will move to G parallel to the meter, G will move to G parallel to the direction of movement of w.direction of movement of w.

×dwΔ×

=dwGG '

•• The shift in weight results in a trimming moment wd and the The shift in weight results in a trimming moment wd and the ship will trim until G and B are in line. LCF, the centre of ship will trim until G and B are in line. LCF, the centre of floatation is the centre of area of the water plane. For small trim, floatation is the centre of area of the water plane. For small trim, the ship is assumed to be trimming about LCF. The trimming the ship is assumed to be trimming about LCF. The trimming p g gp g gmoment causes change in trim and hence change in draughts at moment causes change in trim and hence change in draughts at AP and FP.AP and FP.

Trim due to Movement of WeightsTrim due to Movement of WeightsC tC tCont…Cont…

•• Change in trim (CT), Change in trim (CT),

MCTCmoment trimming)( =CTtriminChange

•• Changes in draught forward, Changes in draught forward, δδTTFF and aft, and aft, δδTTAA can be obtained can be obtained by dividing trim in proportion to the distance from LCF to the by dividing trim in proportion to the distance from LCF to the

MCTC

y g p py g p ppositions where the draughts are measured, normally AP and FP.positions where the draughts are measured, normally AP and FP.


AmidshipAmidship

xδTA

FδTF

T

Trim

T

TA

TF

LBP

Baseline


• Trim is defined as the difference in the draughts aft and forward.

TATFAF TTt δδ +=−=

• The angle of trim may be expressed as follows

LBPLBPLBPt TATF ===tan δδθ

LCFLBPLCFLBPLBP −+22


• Change in draughts forward

⎟⎟⎟⎞

⎜⎜⎜⎛ +

×=LBP

LCFLBP

tTF2δ

Ch i d ht ft

⎟⎟⎠

⎜⎜⎝

LBPTF

• Change in draughts after

⎟⎟⎞

⎜⎜⎛ − LCFLBP

2

⎟⎟⎟

⎠⎜⎜⎜

⎝

×=LBP

tTA2δ

Small Weight ChangesSmall Weight ChangesSmall Weight ChangesSmall Weight Changes

• If a small weight w is added or removed from a ship, the draught of the ship will change as follows:draught of the ship will change as follows:

wTPC

w=rise/ sinkageParallel

MCTC todistance

MCTC(TM)moment trimming(CT) in trim Change LCFw×

==

• Once the trim is obtained, the changes δTF and δTA can be calculated and the final draughts will include the parallel rise/sinkage and δTF and δTA rise/sinkage and δTF and δTA.

Exercise 1Exercise 1• A ship LBP 100m has MCTC 125 tonne.m while its LCF p

is 2.0 m aft of amidships. Its original draughts are 4.5 m at AP and 4.45 m at FP. Find new draughts when a 100 tonne weight already on board is moved 50 m afttonne weight already on board is moved 50 m aft.

Exercise 2Exercise 2• A ship LBP 100 m has LCF 3 m aft of amidships and p p

floats at 3.2 m and 4.4 m at FP and AP respectively. Its TPC is 10 tonne while MCTC 100 tonne.m. 50 tonne cargo is removed from 20 m forward of amidships whilecargo is removed from 20 m forward of amidships while 30 tonne is unloaded from cargo hold 15 m aft of amidships. Find the final draughts at the perpendiculars.

small angle stability longitudinal

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