How to make a GZ curve if you know GM?

Discussion in 'Stability' started by HaveANiceDay, Feb 25, 2012.

  1. HaveANiceDay
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    HaveANiceDay Junior Member

    I have calculated GM for my ship and now Im trying to make a GZ curve. Is it possible to make just if you know GM or do I need some additional information?
    Can anyone explain me how to do it please.
     
  2. daiquiri
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    daiquiri Engineering and Design

    Consider this drawing:
    [​IMG]
    If you have some knowledge of trigonometry, you will notice that, for small angles of heel theta:
    GZ = GM * sin(theta)
    For small angles (theta < 5°) the following approximation is also valid:
    sin(theta) = theta
    And hence the first equation becomes:
    GZ = GM * theta (for theta <5°).​

    Please note that metacenter M as a quasi-fixed point generally exists only for small angles. At larger angles it becomes a locus of false metacenters N, so the above linear geometric equations with fixed M are generally not valid. For larger angles the more general relationship between GZ and ZN is used:
    GZ = ZN * tan(theta)​

    Hope it helps.
     
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  3. DCockey
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    DCockey Senior Member

    The first and last equations are equivalent.

    The small heel angle metacenter, M, does not usually stay above the center of buoyancy, B1 in the diagram, as the heel angle increases. To obtain an accurate GZ curve the location of the B1 needs to be determined as heel angle changes, and GZ calculated as the horizontal (relative to the water surface) offset between G and B1.
     
  4. daiquiri
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    daiquiri Engineering and Design

    Thanks for noting that. I see that you have quoted the version of my reply before I had corrected it, probably we were both writing in the same time.
    It does if the ship is stable, which is a mandatory requirement.


    @HaveANiceDay:

    I wholeheartedly suggest you to get a copy of the book "Ship Hydrostatics and Stability" by A. B. Biran. It contains all you need to know about this subject, but contains a lots of math.
    If you don't need the in-depth mathematical analysis and demonstrations of ship stability formulae, then I could suggest you "Merchant ship Stability" by H. J. Pursey. A crystal-clear textbook on ship stability, with a very simple and immediately usable math - it contains all the practical info mariners and most of boat designers need to know in order to keep the ship operation safe.
    Incidentally, I have a 1971 metric version of the book given to me by my father, a former Master Mariner, when he retired. It was such a great gift! It displays a price tag of £ 2.80 :D Gosh, good old times!

    If you need to perform your calculations at large angles of heel, you will have to use the so-called "Attwood's Formula". You will find it in any book of Ship Stability. If the hull of your boat can be approximated with the wall-sided model, then a special case of Attwood's formula can be derived:
    GZ = [GM + 0.5 * BM * tan(theta)^2] * sin (theta)​

    where symbols have the same meaning as in the picture from my first reply.
    The quantity BM can be calculated with the following formula:
    BM = I / V​

    where I is the moment of inertia of the waterplane area around ships' longitudinal axis (which varies with the heel angle theta), and V is the volume displacement (which is held constant).
    Hope this helps too. :)

    Cheers
     
  5. HaveANiceDay
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    HaveANiceDay Junior Member

    Thank you very much for your help. I guess I will use for small angles this formula: "GZ = GM * sin(theta)"
    and for larger angles I will use: GZ = [GM + 0.5 * BM * tan(theta)^2] * sin (theta)
     
  6. daiquiri
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    daiquiri Engineering and Design

    Then please note that the latter formula can and should be used throughout the whole range of heel angles, in order to avoid discontinuities in the stability curves when shifting from one formula to other. The simplified formula (GZ = GM sin(theta) ) is justified only if you want to do a quick calculation and are interested in theta angles up to max. 7-8 °.

    Cheers
     
  7. DCockey
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    DCockey Senior Member

    I'll try again. The small angle metacenter, M, does not stay above the center of buoyancy, B, as the heel angle increase beyond a small angle.

    There may be confusion about terminilogy. When I say "small angle metacenter" I mean the metacenter for very small angle of heel. The location of it is fixed relative to the hull and does not change with angle. However, as the heel angle increases it generally does not stay vertical (perpendicular to the surface of the water) above the center of buoyancy.

    The term "metacenter" is sometimes considered to be a function of heel angle and used for the intersection of a vertical (perpendicular to the surface of the water) line from the center of buoyancy and the center plane of the hull.
     
  8. daiquiri
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    daiquiri Engineering and Design

    Terminology again, yes... The same old problem. :)

    First it is necessary to define what is a metacenter. Consider a ship floating listed at an angle Theta1, and then imagine it heeled to a new angle Theta2, where the difference (Theta2-Theta1) is very small (mathematically speaking - infinitesimal). A metacenter relative to this differential heel angle is a point of intersection of lines of action of buoyancy forces in the two angular positions. In other words, a metacenter is a center, or pivot, around which points B (points of application of the buoyancy force) move as the heel is increased.

    If Theta1=0 (upright position), then the corresponding metacenter is called initial metacenter and is placed at the centreline.

    At any other angle of heel, metacenters are defined exactly in the same way, by applying a differential heel (at constant displacement) and finding the intersection of the new line of action of the buoyancy force and it's line of action in the previous position. In this way a locus of metacenters is constructed and is called M-curve or metacentric evolute. Points B also describe a curve during this process, which is called B-curve or metacentric involute.

    From the way these two curves are defined and constructed, it can be demonstrated that, at any heel angle Theta, a tangent to the B-curve is always parallel to the waterline at that heel angle, and the corresponding metacenter (being the radius of curvature of the B-curve) is always perpendicular to the waterline.

    So the statement "However, as the heel angle increases it generally does not stay vertical (perpendicular to the surface of the water) above the center of buoyancy." is not correct. The tangents to the M-line are always perpendicular to the waterline, at any given angle of heel, and hence the relative metacenter is always vertical above the waterline.

    Not "sometimes" - it is always a function of heel angle, as explained above.

    This is true. The intersection of the line of action of the buoyancy force with the section centreline at some larger angle of heel is more correctly called pro-metacenter or false metacenter, and is sometimes indicated with N.

    Hope this explanation, together with the attached picture, will help clarifying things and terminology.

    Cheers
     

    Attached Files:

  9. Ad Hoc
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    Ad Hoc Naval Architect

    Indeed, and a nice job of explaining for others the inaccuracies too.

    As also noted below from any text book:

    metacentric curve def.jpg metacentric curves.jpg
     
  10. DCockey
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    DCockey Senior Member

    From the ITTC Dictionary of Hyrdromechanics, Version 2011
    Metacentre, transverse (M) and longitudinal (ML) The intersection of the vertical through the centre of buoyancy of an inclined body or ship with the upright vertical when the angle of inclination approaches to zero as limit, for transverse or longitudinal inclinations respectively. (underline added for emphasis.)

    Let's look at what I actually said rather than taking it out of context.

    When I say "small angle metacenter" I mean the metacenter for very small angle of heel. The location of it is fixed relative to the hull and does not change with angle. However, as the heel angle increases it generally does not stay vertical (perpendicular to the surface of the water) above the center of buoyancy. (bold added for emphasis.)

    "It" in the sentence you quoted refers to the subject of the previous sentence which you did not include, "small angle metacenter". What I called the "small angle metacenter" and defined as the metacenter for a very small angle of heel is identical to what is labeled on your chart as "initial metacenter". "Small angle metacenter" which I used and "inital metercenter" on your chart are exactly the same thing.

    The small angle metacenter / initial metacenter generally does not stay vertical (perpendicular to the surface of the water) above the center of buoyancy. Do you disagree?

    See definition from ITTC dictionary at the top of this post. Perhaps you should let ITTC know that their definition is wrong. :)

    I'm aware of the use of "metacenter" as varying with heel angle. That is why I took care to use the qualifier "small angle metacenter" to make clear I was talking about what the ITTC dictionary defines at metacenter.

     
  11. DCockey
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    DCockey Senior Member

    So will you also be notifying ITTC that their definition is incorrect?
     
  12. Ad Hoc
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    Ad Hoc Naval Architect

    This is the point. You are using a definition that is not used in naval architecture. No such thing as small angle metacentre. You have come up with a definition, for whatever reason, and decided that it is the only accept method of definition the metacentric height. You have now tried to clarify by attempting to link the accepted definitions to your own as justification..why I have no idea.

    As noted on the other thread, it is your understanding that is incorrect.

    This just highlights your flawed understanding. Perhaps you should design a boat and then see how it relates to its stability and meeting codes of acceptance rather than trying to find references to support your incorrect definitions. But if you wish to continue coming up with your own definitions, that is your own prerogative.
     
  13. daiquiri
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    daiquiri Engineering and Design

    Please let me note that the initial metacenter is defined and used only at small angles of heel. So whether or not it is normal to the water surface at larger angles of heel is a non-isssue and has no practical importance.
     
  14. DCockey
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    DCockey Senior Member

    Agree. Someone who read your earlier posts they could be confused if they understood "metacenter" to mean the intial metacenter rather than the meaning you and many other use then
     

  15. DCockey
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    DCockey Senior Member

    I was just attempting to make clear that the comments were for what daiquiri called "initial metacenter" and I called "small angle metacenter". What name would you suggest for the metacenter as the heel angle approaches zero degrees?

    To show that metacenter is sometimes defined in naval architecture as what daiquiri defined as "initial metacenter". And other times, perhaps most of the time, it is defined as dependent on heel angle. There are two similar but different definitions of "metacenter" used in naval architecture.

    My understanding is that there are two definitions used by others for "metacenter". One is what daiquiri described above, the other is as described by ITTC.

    Basic Ship Theory, Volume 1, Fourth Edition by Rawson & Tupper describes "metacenter" consistent with the ITTC definition:

    The term metacentre is reserved for small inclinations from an upright condition. p 20

    Another way of defining the line of action of the buoyancy force is to use its point of intersection, M, with the z-axis. As the angle (beta) is indefinately diminished M tends to a limiting position termed the metacentre pp92-93

    The second is interesting because it implies that "M" which Rawson and Tupper have as varying with heel angle is not synonymous with "metacentre" except at vanishingly small heel angles.

    Amusing but .......

    Perhaps you could just note where and why you disagree with my use of terminology.
     
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