Ships and Science : The Birth of Naval Architecture in the Scientific Revolution, 1600-1800 by Ferreiro Larrie D

Author:Ferreiro, Larrie D. [Ferreiro, Larrie D.]
Language: eng
Format: mobi, pdf
Published: 2006-08-24T14:27:58+00:00

230

Chapter 4

Figure 4.21

Bouguer’s diagram of the metacenter (1746). (Bouguer, Traité du navire [1746], plate 6) Where: p

= submerged volume, Γ = upright CB, g = inclined CB, AB = upright waterline, ab = inclined waterline, I = CG of an unstable ship, G = CG of a stable ship, g = metacenter, 1 = centroid of immersed wedge, 2 = centroid of emerged wedge, 3 = centroid of body without wedges.

Step 6: Results

In the next section, Bouguer used calculus to determine the three

unknowns: the distance from point 1 to point 2; the volume of the wedges; and the volume of the hull. He imagined that figure 4.21 represents the largest section of the ship, which actually extends through the plane of the page in the longitudinal direction x, with the immersed and emerged wedges actually an infinite sequence of triangles of width y (the largest being b = F − B, or the half-breadth of the hull at the waterline) and height e (H − B) going through the length of the hull at a distance dx from each other; integrating, the volume of the wedge is

e

V

=

2

wedge

y dx.

2 b ∫

The second unknown is the distance between the centers of the wedges, point 1 to 2; but since the center of a triangle is two-thirds the height from apex to base, it is straightforward to obtain this distance as

4 y d

3 x

∫

Dis

ce

tan

1−2 =

.

3 y d

2

x

∫

Inventing the Metacenter

231

The third unknown, the volume of the hull p, is derived using the method of trapezoids. Putting the three together, the distance is

2 ∫ 3

e y dx

Γg =

.

3bp

Finally, observing that the triangle Γ g g (i.e., between the centers of buoyancy and the intersection of their vertical lines of force) is similar to the triangles formed by the immersed and emerged wedges, the height of the metacenter g above the center of buoyancy Γ is found, via Euclid, to be the now well-known equation for the height of the metacenter, also known as the metacentric radius, since it is applicable across the full rotation of the body:

2∫ 3

y dx

Γ g =

.

3p

In the modern notation (GΓ = GB, Γ g = BM ), the height of the metacenter above the center of gravity is

BM − GB = GM.

Step 7: Implications of the metacenter

Bouguer developed in thorough detail both the

theoretical and the practical aspects of the metacenter. He derived the metacenter for various solids (ellipsoid, parallelepiped, prismatic body) and presented the procedures for its practical, numerical calculation for ships. These explanations were detailed enough for practical applications and became the foundation for later textbooks.

But Bouguer also charged ahead beyond the initial metacenter for infinitesimal

angles of heel when he introduced the concept of the métacentrique (i.e., the metacentric curve for finite angles of heel). First, he clearly recognized that the same physical principles and stability criteria apply to an inclined position of the ship as to the upright case. Second, he recognized that the metacentric curve for finite angles is in fact the locus of the centers of curvature of the curve of the centers of buoyancy, which form a class of curves known as evolutes.

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