proof of d(GZ)/d(teta)= ZM

Discussion in 'Stability' started by nettersheim, Sep 13, 2011.

  1. nettersheim
    Joined: Sep 2010
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    nettersheim Consultant

    Dear Colleagues,

    I can't find anywhere the proof of following formula (found in "Ship Hydrostatics and Stability" by A.B Biran, page 116, but without any proof):

    d(GZ)
    ------ = ZM (derivative of GZ)
    d(teta)

    teta being the angle of inclination, G center of gravity, Z foot of perpendicular from G to the line of action of the buoyancy force and M metacenter.

    Could you provide me with suitable reference or whatever ?
    Many thanks in advance,

    François-Xavier Nettersheim
     
  2. daiquiri
    Joined: May 2004
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    daiquiri Engineering and Design

    The proof is rather immediate. You have to consider that quasi-fixed point "M" (metacenter) exists only for small angles of heel (otherwise it becomes "N" - false metacenter), for which the following simplifications are valid:
    sin(theta) = theta
    cos(theta) = 1.

    The assumption of small value of heel angle theta, together with the above approximations and with the following explanatory drawing:

    [​IMG]

    will lead you to the following equations:
    GZ = GM * sin(theta) = GM * theta
    ZM = GM * cos(theta) = GM

    hence, by substituting GM with ZM in the first equation, you get:
    GZ = ZM * theta

    Now, take a derivative with respect to theta, and that's it.

    Cheers!
     
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  3. nettersheim
    Joined: Sep 2010
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    nettersheim Consultant

    Thanks for your reply, Daiquiri.

    Alas, I am looking for the rigourous math demo, for pedagogic needs.

    The formula is quite straightforward to "feel" at small inclination angle as you mention it, you are right.

    My concern is to get the rigourous general proof of :

    d(GZ)
    [-------] = ZM for any angle of inclination and therefore up to large angles !
    d(theta)

    Cheers,

    François-Xavier Nettersheim
     
  4. daiquiri
    Joined: May 2004
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    daiquiri Engineering and Design

    The above demonstration is rigorous from the mathematical point of view, as long as the angle of heel is sufficiently small (less than around 5°).
    The above linear equation is not valid for large angles, because the metacenter M exists as a quasi-fixed point only for small angles. At larger angles it becomes a locus of metacenters (called N), and the above linear geometric equations are no more valid.
    The relationship between GZ and ZN for larger angles become:
    GZ = ZN * tan(theta)

    and it's derivative is:
    dGZ/d(theta) = ZN / cos^2(theta) + dZN/d(theta) * tan(theta).

    Nothing linear there, as you can see.

    Cheers!
     
  5. nettersheim
    Joined: Sep 2010
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    nettersheim Consultant

    Yes, you are absolutely right.

    I missed to indicate that in the above mentioned formulae the ZM is associated to the move of metacenter. I missed it because I was not able to display a subscript (theta) on the ZM, shame on me !

    According to Biran (Ship Hydrostatics and Stability , page 116) the relation is valid for any angle and therefore to the "real" moving metacenters situated on the locus of metacenters.

    d(GZ)
    -------- = ZM , M being considered as real metacenters not only initial one
    d(theta)

    Therefore above relation is not linear, as you mention it.

    May be Biran is wrong ... but I don't think so, because he quotes that a proof of above formulae could be found in e.g Birbanescu-Biran (1979).

    Can't find this text.

    Cheers,
    François-Xavier
     
  6. daiquiri
    Joined: May 2004
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    Location: Italy (Garda Lake) and Croatia (Istria)

    daiquiri Engineering and Design

    Ok, now it's more clear what you meant. I have Biran's book and have opened the page 116.
    What he basically says is that the above differential equation is valid for large angles of heel angle, but... for small increases of theta. :)

    In other words, the way I understand it, for a given large angle of heel he fixes this angle (making it become the angle of list), then applies a small incremental value dTheta and measures the (small) resultant variations of GZ and ZM. Since all incremental values involved all small, the equations again become linear and can be analized with simplifications seen in my post #2.

    In more other words, he basically performs a linearization of the curve which describes locus of metacenters (N), in proximity of each single fixed angle of list (theta).

    The main source of incomprehension (for me) here was the fact that the author has used the term "M" to indicate the metacenter at large values of heel, when "N" would have been more appropriate, imho. That's what has confused me.

    Cheers
     
  7. nettersheim
    Joined: Sep 2010
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    nettersheim Consultant

    Well, Daiquiri, this is definitely interesting discussion. Thank you !

    Don't be too rude with Biran because he said bottom of page 116 : "Mtheta is the metacentre corresponding to heel angle theta" therefore that means he was perfectly aware of the different metacenters involved...

    I see your idea on this bloody formulae. You helped me doing half the way to get the final proof. The french way is traditionnally quite "academic" and in need for formal things, then I still have some home work !

    Cheers,
    François-Xavier
     
  8. daiquiri
    Joined: May 2004
    Posts: 5,373
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    Location: Italy (Garda Lake) and Croatia (Istria)

    daiquiri Engineering and Design

    The last thing I'd want to be is - rude. So, I'm sorry if my words sounded like that.
    Actually, I realize that many authors do use KM even for high heel angles (like Biran does), while at the same time many others will use KN notation for the same condition. So I might have been too rigid in my initial assumptions here. :)

    Cheers
     

  9. nettersheim
    Joined: Sep 2010
    Posts: 47
    Likes: 7, Points: 8, Legacy Rep: 152
    Location: France

    nettersheim Consultant

    Coming back to my original question, and after some head scratching, I am still looking after the proof of the general formulae !

    You said :
    """"In other words, the way I understand it, for a given large angle of heel he fixes this angle (making it become the angle of list), then applies a small incremental value dTheta and measures the (small) resultant variations of GZ and ZM. Since all incremental values involved all small, the equations again become linear and can be analized with simplifications seen in my post #2""""

    This does not provide us with the final demo, even if I appreciate very much the attempt made...

    François-Xavier Nettersheim
     
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