I need to know the exact equation of a ship's natural rolling period. Wiki and most of the sources say it is: T = 2*pi*k/(g*GM)^.5 (k = radius of gyration), which is logical. k is statistically .35-.45 B. but then, I got something like T= .44B/(GM)^.5 I can't understand how it came from k = .44B, recently in "Sea loads on Ships and offshore structure" I have found a completely different equation. Does anyone know whats going on? and another thing- is there any easy, standard procedure to measure the k of a ship about a certain axis?
Dear shadman,
If you consider roll equation to be linear you will have:
(I_xx + a_44) d^2(phi)/dt^2 + ( b_44) d(phi)/dt + c_44 phi=Exciting Moment
The natural frequency for this differential eq. is:
w_n = sqrt(c_44/I_xx + a_44) ---> T=2*pi * w_n
You can find c_44 using the plot of Gz versus. phi. Write a linear relation for this plot, the slope pf this plot may be c_44. For finding a_44 use the empirical equations. Finally, I_xx of a ship of marine vehicle may be determined by you. However, you can use:
I_xx = m * k^2, where k =~ 0.36 B to 0.42 B.
P.S. Maybe in the relations you are talking about, both a_44 and I_xx are considered together.
Good luck
yes, a_44 and I_xx were considered together :)
thanks a lot.. I think I have already found the answer last night from a faculty member of my university..
thanks again :)
Hello,
I would not talk about an exact solution'. For a practical one see pages 139-142 in my book (with Ruben Lopez Pulido as second coauthor)
Ship Hydrostatics and Stability, 2nd edition, Butterworth-Heinemann (Elsevier), 2014.
Adrian Biran
Dr Biran,
The book is really amazing. It is a new book for hydrostatic which can be considered as the text book of the course. The text is simple and there are also good exercises.
Thank you.
Dr. Biran,
I am a graduate student of Naval architecture and in my university we read your book as one of the main textbooks. Its really good :)
thanks for your comment...
Dear Mr. Sakib,
I thank you very much, indeed, for your appreciation. I wish you success in your studies.
Adrian Biran
Reading the answers here is so amusing. If you ask anyone, say Sasan Tavakoli , Adrian Biran , Jan P. Michalski how they define the moment of inertia I_xx they will claim that is moment of inertia around the mass centroid which is wrong. The reason that the answer is wrong because the ship is not rotating around the center of mass. It just cannot, and only a special case when the draft is half. The same can be said on the added mass it is not defined and value of what denoted as a_44, Sasan Tavakoli symbol used, has errors of hundreds percents in many cases. The governing equation is wrong as well. And they suppose to be the experts.
Now you can get the derivations with equation in
https://www.researchgate.net/publication/374898374_Simplified_Rolling_Calculations_Part_1_The_Basics
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