Hawaii Marine Company
Naval Architectural, Marine Engineering & Computer Aided Drafting & Engineering

E-mail - hmt@hawaii-marine.com    Web Site - http://hawaii-marine.com
45-302 Makalani St., Kaneohe, HI, USA   96744-2819
Phone/Fax (808) 247-6443

Some important characteristics of this picture are:

	B is the Buoyancy Force provided by the vessel with level trim.  Usually obtained from a "Curves of Form" plot.  In this procedure B is numerically equal to the vessel's displacement (i. e. B = D).
	W is the total weight applied to the vessel.  Usually obtained from a "Weights and Moments" analysis.
	L is the length between the forward and after draft marks
	LCG is the Longitudinal Center of Gravity location, normally obtained from a "Weights and Moments" analysis.
	LCB is the Longitudinal Center of Buoyancy location, usually obtained from a "Curves of Form" plot.
	LCF is the Longitudinal Center of Floatation, normally obtained from a "Curves of Form" plot.
	TA is the draft and the Aft Draft marks
	TF is the draft at the Forward Draft marks
	TM is the Draft Amidships, located midway between forward and aft draft marks
	TLCF is the draft located at the Longitudinal Center of Floatation.

The sign conventions adopted for this presentation are:

	Distances Aft of Amidships are Positive, applies to LCG and LCB
	Trim by the stern is defined as Positive
	Trim Moment causing trim by the stern is Positive
	Trim Lever that causes trim by the stern is Positive

Now that the geometry is defined and the sign conventions are stated, we can proceed with the two basic trimming cases.
 

Case I - Displacement & CG Location Known Find Forward & Aft Drafts

Step 1 - Obtain Equilibrium

	For equilibrium the vessel weight must equal the vessel's displacement, W = D.
	With this displacement enter the "Curves of Form" and obtain a draft.  This draft (TLCF) obtained is the draft present at the LCF location.

Step 2 - In the "Curves of Form" at this LCF draft obtain the following:

	Moment to Trim MTI Moment to Trim one Inch, for English Units of long tons·feet/inch or MTC Moment to change Trim one Centimeter, for Metric Units of metric tons·meters/cm.
	Longitudinal Center of Buoyancy, LCB, feet or meters, with aft of amidships defined as positive
	Longitudinal Center of Floatation, LCF, feet or meters, with aft of amidships defined as positive

Step 4 - Find the Trim. 

	First the trim lever is defined mathematically as Trim Lever = LCG - LCB, in either feet or meters.  If this value is positive trim by the stern should be produced.  If it is negative the vessel should trim by the bow.  Sign convention consistency is extremely important.  For instance if the LCG is 5 feet aft of amidships and the LCB is 2 foot forward, the trimming lever would be equal to 5 - (-2) = positive 7 feet.  Since the numerical value is positive this scenario will cause trim by the stern.
	The applied trimming moment is defined mathematically as TM = D·(LCG - LCG)
	The hydrostatic response trimming moment is defined mathematically as: TM = TRIM·MTI for English units TM=TRIM·MTC for metric units.
	For equilibrium to occur, the applied trimming moment must equal the response trimming moment.  The previous defined equations are combined, algebraically rearranged with the following expressions for trim obtained: TRIM = D·(LCG-LCB)/MTI for English units of inches, the value obtained must be converted to feet, by dividing by 12,  prior to applying it in the formulas which follow. TRIM = D·(LCG-LCB)/MTC for metric units of centimeters, the value obtained must be converted to meters, by dividing by 100, prior to applying it in the formulas which follow. When the above expressions are satisfied, there is corresponding subtle hydrostatic physical reality for the trimmed vessel condition.  This reality is that the LCB has moved to a new location that is either directly above or below the LCG location.  However, the initial LCB that must be applied in these trim calculations correspond to the vessel in a level condition (i. e. obtained from "Curves of Form" values).

Step 5 - Find the Forward and After Draft Via Geometry

	This method involves the use of similar triangles and the position of the LCF.
	For the forward draft the similar triangles present yield the following expression TRIM/L = dTF/(LCF+L/2), solve this for dTF to obtain dTF = (TRIM/L)·(L/2+LCF) = TRIM·(1/2+LCF/L), then apply the following formula from geometry to obtain the forward draft TF = TLCF - dTF = TLCF - TRIM·(1/2+LCF/L).
	For the aft draft the similar triangles present yield the following expression TRIM/L = dTA/(L/2-LCF), solve this for dTA to obtain dTA = (TRIM/L)·(L/2-LCF) = TRIM·(1/2-LCF/L), then apply the following formula from geometry to obtain the forward draft TA = TLCF + dTA = TLCF + TRIM·(1/2-LCF/L).
	Alternatively, based on geometry, the after draft may be more simply computed as follows: TA = TF + TRIM.
	With the forward and aft drafts known the mean draft can be quickly computed as follows: TM = (TF + TA)/2.

Step 6 - Important Points to Remember

	If the LCB is aft of the LCG the vessel will trim by the bow.  If the LCB is forward of the LCG then the vessel will trim by the stern.  These principles apply regardless of the position of the LCF.
	Sign convention consistency is extremely important.  If they are not followed the formulas presented here will not work properly.

Step 7 - Improvements Made to this site's Trim and Stability Sheets

The MTI or MTC  values that are presented in the "Curves of Form" are based on the assumption that metacentric radius in the longitudinal direction is equal to the metacentric height in the longitudinal direction (i. e. BML = GML).  This assumption yields approximations for moment to trim values.  These approximations are normally adequate since in most cases there is not much difference between the BML and GML values.  Furthermore the approximations must be made because the VCG values are not known at the time that the "Curves of Form" are made.

However in the "Trim and Stability Sheets," that are available on this website, the VCG values are known for the conditions at hand, so the moment to trim values are computed accurately.  Three basic formulas are applied.  First, by definition, the restoring moment = GMLD·Tanq.  Second geometry present requires that Tanq = Opposite/Adjacent = TRIM/L.  Three, by definition GML = KML - VCG, where KML is obtained from the "Curves of Form" instead of MTI or MTC.  All three of these equations are combined and rearranged yielding: MTF = (KML - VCG)·D/L.  MTF in this case is moment to trim one foot, where TRIM equals one foot.  Note that ML can be obtained from the following formula: KML = BML + VCB.  The "Curves of Form" may just give BML and VCB, but this is alright since these can be summed to obtain the KML value.  Another article in this website, "Understanding Stability" explains the theory discussed in this paragraph.  However, a little adaptation is required by the reader because that article applies to stability in the transverse direction and this article applies to stability in the longitudinal direction.

Case II - Forward & Aft Drafts Known, Find Displacement & LCG Location

Step 1 - Calculate the Mean Draft & Trim Present

	Compute the mean draft present, where TM = 1/2·(TF + TA).  Remember the "Curves of Form" are not based on this mean draft but on the LCF draft.  However this mean draft serves its purpose as a close estimate for the LCF draft and is initially used to retrieve preliminary data from the "Curves of Form."
	Compute the trim present, with this formula  TRIM  = TA - TF.
	These values of draft and trim are now used to help determine the LCF draft (TLCF).

Step 2 - Obtain the LCF Draft Through Iteration

	At TM go into the Curves of Form and obtain a initial value for LCF.
	An expression for the LCF draft needs to be derived.  Fortuitously the waterline slope (or Tanq = TRIM / L) and the ship length (L between forward and aft draft marks) are known.  From similar triangles we have  dTLCF / TRIM = LCF / L.  From geometry we have  TLCF = TM + dTLCF.  Combining the preceding two equations we have:  TLCF = TM + (TRIM)·LCF / L
	Compute the initial guess for LCF draft through application of TLCF = TM + (TRIM)·LCF / L
	Go back to the "Curves of Form" with initial TLCF just computed and obtain a new value for LCF.
	Recompute the LCF draft, by using the LCF value just obtained into the following formula: TLCF = TM + (TRIM)·LCF / L.
	The LCF just obtained should be close to the one previously calculated.  If not, repeat this process using the most recent LCF draft value to enter the "Curves of Form" to get a new LCF value.  Recompute another LCF draft using the formula TLCF = TM + (TRIM)·LCF / L and compare it with the preceding LCF draft computed, they now should be very close.  Usually only need to iterate once.  The last value for LCF draft is the considered the actual LCF draft and it is applied in the rest of this analysis.

Step 3 - Obtain "Curves of Form" Data Based on the LCF Draft

	With the last TLCF value enter the "Curves of Form" and obtain the following:
	Displacement, D
	Longitudinal Center of Buoyancy, LCB
	Moment to Trim, MTI or MTC.  Which term depends on applicable units system. MTI for English units of inches, the value obtained from "Curves of Form" must be converted to feet, by dividing by 12,  prior to applying it in the formulas which follow. MTC for metric units of centimeters, the value obtained from "Curves of Form" must be converted to meters, by dividing by 100, prior to applying it in the formulas which follow.

Step 4 - Derive Relationships Between Trim and LCG

	Two relationships for trimming are presented and then equated to each other, the combined result is then manipulated to give an expression for computing LCG.
	First the applied trimming moment is defined as TM = D·(LCG - LCB).
	Second the hydrostatic response moment is defined as TM = MTI·TRIM.
	These equations are equated to each other and solved for LCG to obtain the following result: LCG = LCB + MTI · TRIM / D

Step 5 - Calculate the LCG Value

With the LCB, TRIM, MTI (converted to per foot or meter) and displacement compute the LCG using the formula just derived in Step 4.

Step 6 - Important Points to Remember

	It  trim value is positive, the vessel is has trim by the stern (the stern is submerged deeper than the bow) then the LCG must be located aft of the LCB.
	If trim value is negative, the vessel has trim by the bow (the bow is submerged deeper than the stern), then the LCG must be located forward of the LCB.
	Sign convention consistency remains extremely important!  If they are not followed exactly the formulas presented here will not work properly.