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Section 3.1
Archimedes Principle:The Buoyant Force on an object is equal to the weight of the volume of the water displaced by the object
FB=rgVForces on a body in water
Distributed forces:Gravity: Distributed throughout volume of body based on mass
density.Buoyancy: Distributed over wetted surface of body based on
hydrostatic pressureDrag/Lift: Distributed over surface of body based on flow field
when moving relative to medium
Box Shaped Barge:
FB=PA;P=rgz;FB=rgzA;
V=zA;FB=rgV
Az
BG
Weight
FB
Horizontalcomponents ofpressure forceare negated byequal force onopposite side ofbarge.
Buoyant Forces
F=0Weight, Buoyancy, Drag and Lift forces all sum tozero in each dimension
M=0All forces in each dimension are colinear and cancel;i.e. there are no separation of the action points offorces such that couples or moments are generated.
Static Equilibrium
Example ProblemA boxed shaped barge 50ft wide, 100ft long, and 15ft deep has 10ft of
freeboard in sea water:
– What is its draft?
– What is the hydrostatic pressure (psi) acting on the barge’s keel?
– What is the magnitude (LT) of the total hydrostatic force acting on the barge’s keel?
– What is the weight (LT) of the water displaced by the barge?
– Assuming that the buoyant force acts through a single point, what is the location of that point in 3 dimensions?
– Assuming minimum freeboard is 5ft, how many ft³ of coal at 50lb/ft³ can we load on the barge in seawater?
– If we then take the barge into freshwater, what will the new draft be?
Example Answer
Draft=Depth-Freeboard=15ft-10ft=5ft
Phyd=rgz=64lb/ft³×5ft×[1ft²/144in²]=2.22psi
Fhyd=Phyd×A=2.22lb/in²×50ft×100ft×[144in²/ft²]×[1LT/2240lb]=714LT
w=rgV=64lb/ft³×50ft×100ft×5ft×[1LT/2240lb]=714LT
Center of Buoyancy=at amidships, on centerline, 2.5ft above keel
Example Answer
TPI=AWP[ft²]{LT/in}/420=50ft×100ft/420{LT/in}=11.9LT/in
Change in draft=10ft-5ft=5ft×[12in/ft]=60in
Change in weight=60in×11.9LT/in=714LT
V=w/rg=714LT/(50lb/ft³)×2240lb/LT=32,000ft³
Example Answer
Current draft=TSW=10ft
w=rgSWVSW=64lb/ft³×50ft×100ft×10ft=3,200,000lb
VFW=w/rgFW=3,200,000lb/62.4lb/ft³=51,280ft³
TFW=VFW/AWP=51,280ft³/(50ft×100ft)=10.26ft
Increased draft means reduced freeboard below minimum spec
Section 3.2Center of Mass/Gravity
The weighted average over area or volume based on given distribution summed such that result is equivalent to the total force applied through a single point.
What can change the Center of Gravity?– Add/subtract weight– Move weight/change distribution
Notation:G=Location of Center of Gravity for shipg=Location of Center of Gravity for objectDs= Displacement of ship (LT)W = Magnitude of Gravitational Force/Weight of object
(LT)
CL
BL
Go
Bo
K
So far we’ve looked at ships that are in STATIC EQUILIBRIUM: • SFx = 0• SFy = 0• SFz = 0• SMp = 0
Now let’s take a look at what happens when a weight is added to disturb this equilibrium
CL
BLK
Go
• A change in VCG (or KG)
KGnew
A change in weight (either adding or removing it) will cause a change inthe location of G, the center of gravity of the ship
G1
• A change in the TCG
TCG
It also causes a change in thelongitudinal CG (LCG), butwe’ll discuss that later...
BL
When a weight is ADDED, the CG shifts TOWARD the added weight in line with the CG of the ship and the cg of the weight
CL
BLK
G1
When a weight is REMOVED, the CG shifts AWAY from the added weight in line with the CG of the ship and the cg of the weight
•g
G0
CL
BLK
G0•g
In the case of a weight SHIFT, the CG first shifts AWAY from theremoved weight….
G1
…and the TOWARDS the relocated weight
•g
G2
CL
BLK
•g
Let’s first consider a weight added directly over the centerline
G0
… Causing a change in the VERTICAL distance, or KG
KGold
G1
This will cause the location of the CG to move TOWARD the weight ...
KGnew
CL
BLK
G0
G1
•g
KGnew = + wadd x Kg
Dsold + wadd
Dsold x KGold
Use the concept of weighted averages to determine the new CG:
CL
BLK
G1
G0
•g
KGnew = + (-w) x Kg
Dsold + (-w)
Dsold x KGold
KGoldKGnew
Kg
It’s the same deal for removing a weight, only this time the weight is negative (i.e. removed):
CL
BLK
•g
•g
KGold
KGnew
G0
G1
Kg1
Kg2
In a relocation of a weight, look at it as SUBTRACTING one weight, and ADDING another weight.
In this unique case, Dsnew and Dsold and are the SAME THING! • w1 and w2 are also the same thing!• The weight has only moved, not been removed• So we can rearrange the formula:
...This is ONLY for a single vertical weight shift!!
Ds GnewGold = w g2g1
Ds GnewGold = w g2g1
Where:• GnewGold is the distance between the old and new CG’s
• g2g1 is the distance between the old and new Cg locations of the relocated weight
...This relation will become important in the Inclining Experiment
We can generalize the formula for vertical changes in CG by the following:
KGnew = Dsold x KGold + Swi x Kgi
Dsold +Swi
Example:
Given• USS CURTS (FFG-38) floats on an even keel at a draft of 17ft• KG = 19.5ft• Lpp = 408ft• It takes on 150LT of fresh water in a tank 6ft above the keel on the CL
Find• New vertical center of gravity (KG) after taking on water
Step 1: Draw picture!
CL
BLK
G1
G0
150LT
6’19.5’
?
Ds
Step 2: Find Ds when floating at 17ft draft
• Go to curves of form for FFG in appendix• Using curve 1, find the intersection w/ 17ft
Ds = 147 x 30LT
Ds = 4410LT
Step 3: Write the GENERAL Equation
KGnew = + Swi x Kgi
Dsnew
Dsold x KGold
Step 4: Substitute in values into the general equation
KGnew = 4410LT x 19.5ft + 150LT x 6ft
4410LT + 150LT
KGnew = 86215.5 LT-ft + 900LT-ft
4560 LT
KGnew =87115.5 LT-ft
4560 LT
KGnew = 19.10 ft
CHECK: Does this answer make sense?
YES! The CG shifts toward the added weight, lower than the original CG
Example Problem
A 688 Class Submarine is in port, pier side undergoing a maintenance period. The tender will be pulling periscopes tomorrow which requires the ship to maintain zero list, i.e. TCG=0ft.
The Engineering Dept needs to pump #2RFT dry to perform a tank inspection. What impact will this have on the sub’s TCG?
The sub has 10 MK 48 ADCAP torpedoes at 2LT each. How far and in which direction should these torpedoes be shifted to restore the sub’s TCG to zero?
Data: Do=6900LT Tcg#2RFT= -12ft (i.e. port of centerline); TCGo=0ft Capacity#2RFT=5000galfw
Example Answerw#2RFT=rgV
=5000gal×[1ft³/7.4805gal]×rgfw=668.4ft³×62.4lb/ft³×[1LT/2240lb]=18.62LT
Df= D0+Swa-Swr=6900LT-19LT=6881LT
TCGf =(TCG0D0+STcgawa-STcgrwr)/Df=(0ft×6900LT-[-12ft]×19LT)/6881LT=0.033ft (stbd of centerline)(Removed weight from port side)
Example Answer
TCGf=(TCG0D0+STcgawa-STcgrwr)/Df
0ft=(0.033ft×6881LT+dTcg×10torps×2LT/ torp)/6881LT
dTcg = -(0.033ft×6881LT)/20LT= -11.4ft (to port)
Shift 10 torpedoes each 11.4ft to port to compensate for the loss of weight on the port side of the sub.
Section 3.3:What happens when “G” leaves the Centerline?
WL
CL
KBL
G0
Initial Condition:
B0
WL
CL
K
G1
G shifts:
B0
WLG1
Ship responds:
B1
As the ship lists/trims, the shape of thesubmerged volume changes movingB outboard until it slides under G.*Since the total weight of the ship hasnot changed, the total submerged volumeremains constant, but its shape changes.
FB
D D
FB FB
D
F1
WL
Ship respondsto opposite weightshift:
B2
G2
WL
CL
K
Where the lines of action of the variouscenters of buoyancy cross* is the Metacenter
B0BL
B1B2
M
G0
*Lines of action cross at a single point only for“small” angles of inclination (<10º).
FBFB
F2
F1(+)F2(-)
Shapes which impact KM:
WL
CL
KB0BL
B1B2
M
WL
CL
KBL
B0B1B2
Highly curved hull cross-section:Little buoyant volume at largelever arm: M is at/near center ofcurvature
Very flat hull cross-section:Large buoyant volume at largelever arm: M is high
M
WL
CLK
BL
G
B
MT
KB
BMTKMT
KG
GMT
ML
Locations and Line Segments for Hydrostatic Calculations
Distance from G to MT = Metacentric Height =
Major player in stability calculations (+ keeps
ship upright)
Distance from B to MT = Transverse Metacentric Radius
TCG/TCB (+)TCG/TCB (-)
Example Problem
A rocking chair’s “skids” have a radiusof curvature of 3ft. The chair’s initial center of gravity is 2.5ft above the skids. A box is put on the seat which raises the combined center of gravity to 3ft above the skids. Another box is put on top of the first which raises the combined center of gravity to 3.5ft above the skids.
For each of these conditions, when the chair is tipped 45°, show how the forces of gravity and support are spatially related and predict how the chair will react when released.
What point in this scenario is analogous to a ship’s metacenter?
Radius=3ft
G1
G2
G3
Example AnswerG1: Support is outboard center of gravity
creating a couple which returns the chair upright.
G2: Support is aligned with center of gravity eliminating any couple. The chair maintains position.
G3: Support is inboard center of gravity creating a couple which tips the chair over.
The center of curvature of the rocking chair’s “skids” correspond to a ship’s metacenter.
G1
G2
G3
Support
Support
Support
Section 3.4:Angle of List for Small Angles after Transverse Weight Shift
For a given transverse weight shift, what is the corresponding change in list angle?
WL
BL
CL
MT
G0 Gf
Bf
B0
g0
gf
FB
DF
t
CL
BL
Go
Bo
K
Up to now we’ve considered ship’s floating on an even keel …(no list or trim).
The following points are noted:• K, keel• B, center of buoyancy• G, center of gravity
One point of particular note remains….
…MT, or the Transverse Metacenter
CL
BL
Go
Bo
K
The Transverse Metacenter (MT) represents a convenient point of reference for small changes in the angle of inclination, F, (less than 10o)
MT
For small changes in inclination, the point MT is where the ship is assumed to rotate.
MT
...The MT is generally about 10-30ft above the keel
F
B1
There is also a Longitudinal Metacenter, or ML...
ML
…usually in the magnitude of 100 to 1000ft above the keel
ϑ
MT
F
When the ship reacts to an off-center load (whichwill change the ship’s CG),...
FB
B1
...the center of buoyancy will shiftuntil it is vertically aligned
with the new CG...
Remember, this is only for listingof 10o or less
G1
…G1 can be assumed to movePERPENDICULAR from the CL
MT
F
FB
B1
G1
Look at the right triangle formed by this shifting…
The long leg is G0MT
The hypotenuse is G1MT
tan F = G0G1G0MT
SO….
The short leg is G0G1
(tan F = opp/adj… remember?)
With that fact understood, we can now determine the ANGLE OF LISTof a vessel due to a change in loading.
tan F = G0G1G0MT
• G0G1 is the change in the transverse Center of Gravity
CL
Go
Bo
K
MT
• From the Curves of Form you can get KMT
• The Vertical Center of Gravity is KG0
• G0MT = KMT - KG0
KM
T
KG
How?
Example:
The USS Simpson (FFG-56) floats on an even keel at a 16ft draft. The KG is 20ft above the keel. After 1 week, 50LT of fuel has been used from a tank 11ft to port and 15ft above the keel.
Find the angle of list after the fuel has been used.
Step 1: Find the ship’s displacement
From the curves of form, curve #1, 16ft draft crosses at 132
132 x 30LT = 3960LT
KG1 = Ds0 x KG0 - (w x Kg)
Ds1
KG1 = 3960LT x 20ft - (50LT x 15ft)(3960 - 50)LT
KG1 = 78,450LT-ft3910LT
KG1 = 20.06ft
Step 2: Find the new vertical CG (KG)
TCG1 = Ds0 x TCG0 - (w x Tcg)
Ds1
TCG1 = 3960LT x 0ft - (50LT x -11ft)(3960 - 50)LT
TCG1 = 0 - (-550LT-ft)3910LT
TCG1 = 0.141ft
(minus because it’s to port)
(shifts to starboard, away from removed weight)
Step 3: Find the Transverse CG (TCG)
Step 4: Define lengths of G0G1 and G0MT
G0G1 is the change in the Transverse CG:
• G0 = 0 (on the centerline)• G1 = .141ft
G0G1 = .141ft
G0MT = KMT - KG0
• KMT from curves is 113 x .2ft = 22.6ft• KG0 = 20ft
G0MT = 2.6ft
Step 5: (Almost there!) Find tan f:
tan f = opposite adjacent
tan f = G0G1G0MT
tan f = 0.0541
atan 0.0541 = f
3.10o = f
MT
F
G1
tan f = .141ft2.6ft
Section 3.5The Inclining Experiment
In the previous section, we derived the relationship between a shift in weight and the resultant list/trim angle:
tan(F) = wt/(DG0MT)
w,t are the weight and distance moved – usually known
The location of MT and the magnitude of D are properties of the hull shape read from the Curves of Form for the appropriate draft (T).
How do we find the location of G0?
How do we find the location of G0?We determine it experimentally after new construction
for a class or any major permanent complex weight redistributions for a given ship (alteration/conversion).
Inclining Experiment Procedure:
1. Configure the ship in a “light” condition
2. Bring on large weights (~2% of Dship), move to known distances port and starboard of centerline and measure tan(F) using “plum bob”. Measure & record Dincl using draft and Curves of Form.
3. Plot wt vs. tan(F); divide slope by Dincl to get GinclMT
4. Calculate KGincl = KMT(from Curve of Form)–GinclMT
5. KG0=KGlight=(KGinclDincl–Kginclwtswinclwts)/(Dincl–winclwts)
Inclining Experiment Tools-Plot: -Plumb Bob:
Tangent of Inclining Angle (Tan[F])
Incl
inin
g M
omen
t, w
t (LT
-ft)
Fdadj
doppScale
Mast
tan(F)=dopp/dadj
MT
F
FB
B1
G1
The short leg is G0G1
The long leg is G0MT
The hypotenuse is G1MT
tan F = G0G1G0MT
SO….
So far we’ve established that the angle of list can be found usingthe right triangle identified here:
...And so we can find the angle of list
MT
F
FB
B1
G1
Up to now, however, G0MT hasbeen given based upon a KG thathas been provided.
We’ll now see how KG can befound by determining G0MT
This is done by the Inclining Experiment
By using a known weight and placing it at a known distancean angle of list can be measured
By repeating this process - port and starboard- we can graph the relationship between the moment created by the weight and the angle of inclination
This will allow an average inclined KG to be determined,and from that a KG for the ship in an condition of no list or trim can be established
Ds G0G1 = w g0g1
In earlier discussions an equation was derived for a shift in of a single weight:
…where g0g1 was the distance that the weight was
shifted. Let’s call that distance “t”. Sooo,...
Ds G0G1 = wt
And re-look at the equation for the angle of list:
tan F = G0G1G0MT
Note that the common term in both equations is G0G1. Solet’s isolate it in each equation:
tan F = G0G1G0MT
Ds G0G1 = wt
G0MT tan F = G0G1 G0G1 = wtDs
G0MT tan F = wtDs
G0MT tan F = wtDs
That’s nice,… but not nice enough... One more rearrangementand we’ll have what we really want, G0MT:
G0MT = wttan F Ds
G0MT = wttan FDs
Let’s review what we know:
• “w” is a known weight that is relocated• “t” is the distance the weight is moved• “tan f” is the angle created by the weight shift
• “Ds” is the displacement of the ship
This will be the formula that governs the Inclining Experiment
G0MT = wttan FDs
In the Inclining Experiment:
The distance “t” is varied, changing the angle of list, tan f
“w” and Ds will remain constant
By varying t, thus varying the created moment of wt,the angle of inclination will change
By plotting wt versus tan f, you can determine the average G0MT
G0MT = wttan FDs
Remember, slope is Dy/Dx:
Or...
=DyDx
DWtDtan f
So...
Average G0MT = (slope of wt vs tan F curve)
Ds
When you vary the distance t, and thus the moment, you’ll vary the inclination angle. The result is plotted in an example here:
The slope of the “best fit” line, Dy/Dx, when divided by the displacement,will give the average G0MT distance:
Average G0MT = (slope of wt vs tan F curve)
Ds
Moment v. tan f
-1000-800-600-400-200
0200400600800
1000
-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06
tan f
mom
ent (
LT-ft
)
Having found the Average G0MT, you can find the KG when theship is loaded with the inclining weight:
KG = KMT - G0MT
The problem now degenerates to a simple “’change in verticalcenter of gravity, KG, equation:
KG light = KG inclined x Ds old - Kg x wDs new
KG light, the KG of the ship with considering the ship’s weight only-no crew, stores, fuel, etc.- is what we wanted!!
In Summary:
Using a known weight and a measured distance, a moment is created
The moment creates a list that can be measured
By repeating the process with the same weight over different distances and plotting the results, the average G0MT can be found
Once G0MT is found, you can find KG of the light ship
Example Problem• The USS OHIO has just completed her Overhaul and Conversion
from an SSBN to an SSGN and Special Operations Forces platform. She is pierside performing a required Inclining Experiment. Dlightship=18700LT; KMT=21ft. The inclining gear weighs 400LTs and is centered 47ft above the keel. 375LTs is moved to the following transverse distances resulting in the corresponding list angles.
Distance to Starboard(ft) List Angle(°)-50 -12.8-25 -6.40 025 6.550 12.7
What is KGlight?
Example AnswerMultiply transverse distances by 375LT to get inclining moment. Take tangent of list angle and plot the two derived sets of data against one another:
Data From Inclining Experiment
-25000
-20000
-15000
-10000
-5000
0
5000
10000
15000
20000
25000
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25
Tangent of the Inclining Angle
Incl
inin
g M
omen
t (LT
-ft)
Example AnswerSlope=(18750-[-18750]LT-ft)/(.225-[-.227]) =83000LT-ft
GMTincl=slope/Dincl=83000LT-ft/19100LT =4.35ft
KGinc l=KMT-GMTincl=21ft-4.35ft=16.65ft
KGlight =(KGinclDincl-Kgwtswwts)/Dlight=(16.65ft×19100LT-47ft×400LT)/18700LT=16ft
Section 3.6 Longitudinal Changes
Tm=(Taft+Tfwd)/2
Trim=Taft - Tfwd
– If ship is “trimmed by the stern”,– Bow is up– Taft> Tfwd
– Trim is (+)
WLTfwdTaft
DWL
FFpAp daft dfwd
_.
Consider a ship floating on an even keel, that is, no list or trim...
_.
When a weight, w, is added, it causes a change draft.
_.
_.
The ship will pivot about the center of flotation, F.
F
dTfwd
The change in draft will be evident in a change of draft forward...
dTaft
…and aft.
dTrim
The difference between the fore and aft drafts is the change in trim:
Trim = dTaft - dTfwd
Graphically, it looks like this. First, the ship is represented with a line representing its initial state:
O)( _.F
FPAP
You can simulate this on your paper by turning the sheet in the direction that thebow or stern would sink because of the added weight, then drawing a line to represent the new position.
As weight is added, the the ship rotates about F:
O)(F
_.AP FP
w _.
Now, rotate the sheet so that the line drawn becomes level and acts as the new waterline:
w _.dTfwd
dTaft
dTrim
The changes in draft can now be read directly…
dTaft is below the WL, so it’s subtracted. dTfwd is above the waterline, so it’s added to the draft.
FPAP
There are two aspects of draft to consider when finding the change in draft:
1. Change in draft due to the parallel sinkage of the vessel due to the added weight, “w”:
dTPS = wTPI
2. Change in draft due to the moment created by the added weight at a distance from F, or “wl”:
dTrim = wlMT1”
Tfwd new = Tfwd old +/- dTPS +/- dTrim
AND
Taft new = Taft old +/- dTPS +/- dTrim
These two measurements- change due to parallel sinkage andchange in trim due to moment- when added with the initial draft will give you the TOTAL draft, forward and aft:
dTrim = wl MT1”
dTPS = w TPI
Let’s consider change due to the parallel sinkage of the vessel first:
dTPS = wTPI
TPI, Tons Per Inch Immersion is a geometric function of the vessel at a given draft and is taken from the Curves of Form
• The added weight, w, will cause the vessel to “sink” a small distance for the length of the entire vessel
• We assume that the weight is applied at F! This assures that the sinkageis uniform over the length of the ship
Now consider the change in trim due to the created moment of the added weight:
dTrim = wlMT1”
MT1”, or the Moment to Trim 1”, is also from the Curves of Form
The weight, w, at a distance, l, from the center of flotation, F, creates a moment that causes the ship to rotate about F
This rotation causes one end to sink and the other end to rise
The degree of rise or fall depends on the location of F with regard to the entire length of the ship as given by Lpp
The value for dTrim will be for the entire length of the ship:
dTrim = wlMT1”
...Now we need to find how much of the trim is aft and how much is forward!
w _.dTfwd
dTaft
dTrim
l
Lpp
To find the trim distribution, consider the similar triangles formed below:
The largest triangle shows the TOTAL change in trim, dTrim
The hatched green triangle shows the forward trim dTfwd
The hatched yellow area triangle shows the aft trim, dTaft
w _.dTfwd
dTaft
dTrim
l
Lppdaft dfw
d
For these similar triangles there is a ratio aspect that relates to each:
dTrimLpp
dTaft
daft
dTfwd
dfwd= =
(The short leg divided by the long leg of the triangle!)
w _.dTfwd
dTaft
dTrim
l
Lpp
daft dfwd
Tfwd new = Tfwd old +/- dTPS +/- dT
AND
Taft new = Taft old +/- dTPS +/- dT
Knowing how to find the change in draft from both parallel sinkage and from the induced moment, you can now find the total draft change, fore and aft:
Calculating Draft Changes
Procedure:Calculate impact of weight addition/removal to mean
draft using TPI.
Calculate impact of weight addition/removal to trim at given distance from center of floatation.
Calculate trim effect on fwd and aft drafts separately.
Separately add mean draft impact to trim effects to determine final drafts fwd and aft.
GIVEN:Lpp = 101.7 ft Draft = (10.5 + 10.1)/2 = 10.3ftamidships = 50.85 ft Ds = 2LT x 205 = 410LTLCF = 55.8 ft from FP, or 4.95 ft aft of amidshipsDRAW A PICTURE!
Example:
The YP floats at a draft 10.5 ft aft and 10.1ft forward. A load of 10LT is placed15ft forward of amidships. Find the final forward and aft drafts.
101.7
Daft = 45.9 Dfwd = 55.8
)(_. dTaft
dTfwd dTrimF
O 10LT
19.95
Step 1: Find change due to parallel sinkage
dTPS = w TPI
dTPS = 10LT 235 x .02LT/in
dTPS = 2.13in
101.7
Daft = 45.9 Dfwd = 55.8
)(_. dTaft
dTfwd dTrimF
O 10LT
19.95
Step 2: Find change due to moment
dTrim = wl MT1”
dTrim = 10LT x 19.95ft 252.5 x .141 LT-ft/in
dTrim = 5.60in
101.7
Daft = 45.9 Dfwd = 55.8
)(_. dTaft
dTfwd dTrimF
O 10LT
19.95
Step 3: Divide the dTrim based on similar triangles
dTrimLpp
dTaftdaft
dTfwd
dfwd= =
4.21in101.7ft
dTaft45.9ft
dTfwd
55.8ft= =
5.60in101.7ft
dTaft 45.9ft x= 2.53 in=
5.60in101.7ft
dTfwd 55.8ft x= 3.07 in=
101.7
Daft = 45.9 Dfwd = 55.8
)(_. dTaft
dTfwd dTrimF
O 10LT
19.95
Step 4: Sum the changes in draft fore and aft
Forward:Tfwd new = Tfwd old +/- dTPS +/- dTmoment
Tfwd new = 10.1ft + (2.13in + 3.07in) x (1ft/12in)Tfwd new = 10.1ft + .43ft
Tfwd new = 10.53ft
101.7
Daft = 45.9 Dfwd = 55.8
)(_. dTaft
dTfwd dTrimF
O 10LT
19.95
Step 4: Sum the changes in draft fore and aft
Aft:Taft new = Taft old +/- dTPS +/- dTmoment
Taft new = 10.5ft + (2.13in - 2.53in) x (1ft/12in)Taft new = 10.5ft - .033ft
Taft new = 10.467ft
101.7
Daft = 45.9 Dfwd = 55.8
)(_. dTaft
dTfwd dTrimF
O 10LT
19.95
Background Lab 2
Lab ObjectivesReinforce students’ understanding of
Archimedes PrincipleReinforce student’s concept of static
equilibriumReinforce student’s concept of the center of
floatation
Concepts/Principles:Archimedes PrincipleStatic EquilibriumCenter of FloatationSimpson’s First RuleInterpolationHydrostatic ForceTPIMT1”
Background Lab 2
Terminology– Displacement– Buoyant Force
Equations– D=rg=FB
General Safety– Immediately clean up any water spilled to avoid fall
hazard
Background Lab 2
ApparatusEquipment
– Floating bodies– Tanks with weirs and spillways– Buckets– Scale– Rulers– 5 lb weights
Procedures for taking measurements– Record results measurements of models and weighing
of buckets
Data Collection/ReductionData to be collected & Expected results
– These should be equalWeight of modelWeight of waterCalculated water volume displacedHydrostatic Force
– Longitudinal Center of Floatation (LCF)
Sources of error– Measurements– Insufficient drip time
Data Collection/ReductionCalculationsFB
TPIMT1”
Plots/sketches– None
Section 3.8: Dry DockingHow is the ship’s weight shared between docking blocks and buoyant force?
Requirements for Static Equilibrium still apply: SF=0; SM=0
SFV=(-)D+FB+Fblocks=0
FB=rgS
D =rgS+ Fblocks
Since ship’s weight remains constant, as hull comes out of water, submerged volume decreases, hence buoyant force decreases, and force from the blocks increases.
(P= Fblocks)
Dry-DockingIf a list develops during docking, the increasing force
from the blocks can work to capsize the ship
Solutions:– Use side blocks to force a zero list– Stop docking evolution and correct problem, if ship
develops an increasing list
WLG
M
BFBFblock=P=D-FB
D
WLG
M
B
FB
P
D
Impact on StabilityConsider force of blocks to be the same as a weight removal from the keel:
– What is the impact on KG and GMT?
– Df= D0-wr= D0-P• Ship’s weight/displacement is decreased
– KGfDf= KG0D0-Kgrwr, but Kgr=0;
– KGfDf= KG0D0;
– KGf= KG0D0/Df= KG0D0/(D0-P);• Center of Gravity moves up due to keel weight removal
– GMT= KMT – KGf• Shorter distance between Center of Gravity and Metacenter
gives less distance to develop a righting moment
WLG0
M
BFB
P=weightremoved
D0Gf
DfDisturbancetrying to rollthe ship
Comparison to Grounding:Same stability concerns for both evolutions although grounding is obviously not planned or controlled.
Since re-floating after grounding is generally not on level sea bed with a zero list, it should only be done at highest available tide to maximize buoyant force and righting moment and avoid capsizing.
WLG
M
BFBFground=P=D-FB
D
Pulling the ship directly off the shoal.
Floating the ShipUndocking has the same concerns as docking plus:
– The Center of Gravity may have been shifted by the work done in dock.
– All holes in the ship below the waterline need to be confirmed properly closed.
Recovery from grounding concerns:– The Center of Gravity may have been changed by
flooded or damaged compartments.
– When ship floats again, damage previously held above the water could be submerged resulting in further damage.
Example ProblemDD963 is preparing to enter drydock. It is currently moored pier side on an even keel and a draft of 18.5 feet. To ensure that the sonar dome rests properly on the blocks, the forward draft of the ship must be Tf=17.5 feet. How much ballast must be removed from a tank located 100 feet forward of amidship? Give the answer in gallons of saltwater.
Lpp=465 feet TPI=50LT/in MT1”=1400ft-LT/in LCF=25 feet aft of amidships
Example AnswerTfinal fwd=Tinitial fwd±dTps±dTfwd
dTps=w/TPIdTfwd=dTrim×Dfwd/LppdTrim=wl/MT1”
Tfinal fwd= Tinitial fwd± w/TPI ± wl/MT1”×Dfwd/Lpp= w/TPI ± wl/MT1”×Dfwd/Lpp =(17.5ft-18.5ft)×12in/ft= -12 in=-w/(50LT/in) – w(125ft)/(1400ft-LT/in)×257.5ft/465ft= (-)12in
-12 in = -w/(50LT/in) – w/(20.23LT/in)= -w/(14.4LT/in)
w= -12in×(-14.4LT/in)=172.8LT
w
AP
100ft25ft
232.5ftl=125ftF amidship FP
Lpp=465ft
Dfwd=257.5ftDaft=207.5ft
V=w/(rg)=172.8LT/[(64lb/ft³)×2240lb/LT×7.4805gal/ft³]=45,243gal
This is just another application of moments!
Example Problem
An FFG-7 is in the process of undocking when the evolution is halted at 10ft of water on the hull.
– If D=3600LT, how much weight is being supported by the blocks?
– If the water level is raised 1in, how much additional weight is removed from the blocks?
Example Answer
At T=10ft, FB= 62×30LT = 1860LT;
P=D-FB=3600LT-1860LT = 1740LT
At T=10ft, TPI=128×0.2LT/in = 25.6LT/in;
Raising water level 1in removes an additional 25.6LT from the blocks
Background Lab 3Lab Objectives
– Reinforce students’ understanding of the theory behind inclining experiments
– Provide students with practical experience in conducting an inclining experiment
– Determine the KG of the 27-B-1 model for future laboratories
Concepts/Principles– KG– TCG– MT
– Inclining Experiment
Background Lab 3
Terminology– Light-ship condition– Inclined ship condition– Plum bob
Equations– GinclMT= wt/tan(F)×1/D– KGincl = KMT(from Curve of Form)–GinclMT
– KG0=KGlight=(KGinclDincl–Kginclwtswinclwts)/(Dincl–winclwts)
Background Lab 3
Apparatus
General Safety– Minimize water on the floor
Equipment– 27-B-1 Models– Weights– Plum bobs
Procedures for taking measurements– Record measurements
Data Collection/ReductionData to be collected & Expected results
– 27-B-1 Model Numbers– Weight of Models– Drafts– Model dimensions– Water temperature– tan(F)– Where do you expect KG to be?
Sources of error– Measurement error– Round off
Data Collection/Reduction
Calculations– Use equations
Plots/sketches– w×t vs. tan(F)
Review of Chapters 1-3for
Six Week Exam
• Chapter 1: Engineering Fundamentals• Chapter 2: Hull Form and Geometry• Chapter 3: Hydrostatics• Review Equation & Conversion Sheet
Chapter 1: Engineering Fundamentals
• Drawings, sketches, graphs• Dependent/independent variables• Region under and slope of a curve• Unit analysis• Significant figures• Linear interpolation• Forces, moments, couples, static equilibrium,
hydrostatic pressure, mathematical moments• Six degrees of freedom• Bernoulli’s Equation
Force × distanceEqual and opposite forces applied with an offset
distance to produce a rotation F=0; M=0 P= rgzMx= ydATranslational: heave, surge, swayRotational: roll, pitch, yawList, trim, heel p/r+V²/2+gz=constant
Chapter 1: Engineering Fundamentals
Chapter 2: Hull Form and Geometry
Categorizing shipsWays to represent the hull formTable of OffsetsHull form characteristicsCentroidsCenter of Flotation, Center of BuoyancySimpson’s RuleCurves of Form
PlansBody: Section LinesSheer: Buttock LinesHalf-Breadth: Waterlines
Depth(D), draft(T), beam(B), freeboard
Centroid (location): LCF=(2/AWP)* xdACenter of waterplane areaCenter of submerged volume ydx=Dx/3*[1y0+4y1+2y2+4y3+…+2yn-2+4yn-1+1yn]
D, LCB, KB, TPI, AWP, LCF, MT1”, KML, KMT
Draft->proper curve, proper axis, proper multiple/units
Chapter 2: Hull Form and Geometry
Simpson IntegralsSee your “Equations and Conversions” Sheet
Waterplane AreaAWP=2 ydx; where integral is half breadths by station
Sectional AreaAsect=2 ydz; where integral is half breadths by waterline
X
Y
Half-Breadths(feet)
Stations
y(x)
dx=Station Spacing
0
Z
YHalf-Breadths (feet)0
Waterlines
y(z)
dz=Waterline Spacing(Body Plan)
(Half-Breadth Plan)
Simpson IntegralsSee your “Equations and Conversions” Sheet
Submerged VolumeS= Asectdx; where integral is sectional areas
by station
Longitudinal Center of FloatationLCF=(2/AWP)* xydx; where integral is product of distance
from FP & half breadths by station
X
Asect
SectionalAreas(feet²)
Stations
A(x)
0
dx=Station Spacing
X
Y
Half-Breadths(feet)
Stations
y(x)
dx=Station Spacing
0
(Half-Breadth Plan)
x
Chapter 3: HydrostaticsArchimedes Principle/Static Equilibrium
Impact to G of weight addition, removal, movement
Metacenter
Angle of list
Inclining Experiment
Trim calculations
Drydocking
The Buoyant Force on an object is equal to the weight of the volume of the water displaced by the object: FB=rgV
For box shaped barge, FB= rgV = P×Awp= rgzAwp
F=0; M=0
Center of Gravity (G)
Df= D0+Swa-Swr
KGfDf= KG0D0+SKgawa-SKgrwr
TCGfDf= TCG0D0+STcgawa-STcgrwr
WL
CL
KBL
G0g0
Gi
gf Gf
G moves parallel toweight shift
Chapter 3: Hydrostatics
tan(F) = wt/(DG0MT)
To find KG:– Plot wt vs. tan(F); divide slope by Dincl to get GinclMT– KGincl = KMT(from Curve of Form)–GinclMT– KG0=KGlight=(KGinclDincl–Kginclwtswinclwts)/(Dincl–winclwts)
M
WL
CL
KB0BL
B1B2G0
FB
F1(+)F2(-)
WL
CL
KBL
G
B
MT
KB
BMTKMT
KG
GMT
ML
TCG/TCB (+)TCG/TCB (-)
Chapter 3: Hydrostatics
Trim Equations:– dTPS=w/TPI– dTrim=wl/MT1”– dTfwd/aft/dfwd/aft =dTrim/Lpp
– Tfinal fwd/aft=Tinitial fwd/aft±dTPS±dTfwd/aft
w
dTPSFl
Ap Fpdaft dfwd
WeightAdded
dTfwddTaftdTrim
Lpp
q
Tfinal fwd
Tfinal aft
Chapter 3: Hydrostatics
General Problem Solving TechniqueWrite down applicable reference equation which contains the desired
“answer variable”.Solve the reference equation for the “answer variable”.
Write down additional reference equations and solve for unknown variables in the “answer variable” equation, if needed.
Draw a quick sketch to show what information is given and needed and identify variables, if applicable.
Rewrite “answer variable” equation, substituting numeric values with units for variables.
Simplify this expanded equation, including units, to arrive at the final answer.
Check the answer:Do units match answer?Is the answer on the right order of magnitude?
Summary
Equation SheetAssigned homework problemsAdditional homework problemsExample problems worked in classExample Problems worked in text