Ship Computer Aided Design Displacement and Weight

Outline Hydrostatic Forces and Moments; Archimedes’ Principle. Numerical Integration. Planimeters and Mechanical Integration. Areas, Volumes, Moments, Centroids, and Moments of Inertia. Weight Estimates, Weight Schedule. Hydrostatic Stability.

1. Hydrostatic Forces and Moments; Archimedes’ Principle

In a stationary fluid of uniform density, and in a uniform vertical gravitational field of magnitude g, the static pressure increases linearly with depth (-z) below the free surface : P= P o – ρgz Where: P o is the atmospheric pressure acting on the Surface. The atmospheric pressure P o acts not only on the wetted surface of the body but also on all non wetted surfaces, producing zero resultant force and moment. Consequently, it is normally omitted from hydrostatic calculations.

A solid boundary in the fluid is subject to a force on any differential area element ds equal to the static pressure p times the element of area The contribution to force is: where n is the unit normal vector. The contribution to moment about the origin is: where r is the radius vector from the origin to the surface element.

By application of Gauss’ theorem the surface integrals are converted to volume integrals, so Where: kˆ is the unit vector in the vertical upward direction, and V is the displaced volume. Because this force is vertically upward, it is called the “buoyant force” Its moment about the origin is; This Equations are the twin statements of “Archimedes’ principle”

The net buoyant force: is vertically upward and is equal to the weight of fluid displaced by the body (the displacement) The buoyant force effectively acts through the centroid of the immersed volume..

2. Numerical Integration Many of the formulas involved in calculation of hydrostatic and mass properties are expressed in terms of single or multiple integrals. The integral expression (representing the area between the curve y vs. x and the x- axis) is only meaningful if y is defined at all values of x in the range of integration, a to b.

Sum of Trapezoids The simplest interpolate is a piecewise linear function joining the tabulated points (xi, yi) with straight lines. so the integral is approximated by Trapezoidal Rule When the tabulation is at uniformly spaced abscissae (including the endpoints of the interval), then the intervals are constant, and the sum of trapezoids takes the simpler form (the “trapezoidal rule”) Note: The trapezoidal rule can be seriously in error if the function has discontinuities; in such cases, the sum of trapezoids will usually give a much more accurate result

Simpson’s First Rule When : 1.the tabulation is at uniformly spaced abscissae 2.the number of intervals is even (number of abscissae is odd) then a piecewise parabolic function can be a more accurate interpolant. This leads to “Simpson’s first rule”

3. Planimeters and Mechanical Integration A planimeter : an area-measuring mechanical instrument. This is a clever device with a stylus and indicator wheel; when the user traces one full circuit of a plane figure with the stylus, returning to the starting point, the indicator wheel rotates through an angle proportional to the area enclosed by the figure. More complex versions of this instrument, known as integrators, are able to additionally accumulate readouts proportional to the moments of area and moments of inertia of the figure.

4. Areas, Volumes, Moments, Centroids, and Moments of Inertia. Volume is usually calculated as an integral of areas. In the general volume integral The area of a plane section normal to the x-axis at location x, the so called section area curve or section area distribution of the ship.

5. Weight Estimates, Weight Schedule Archimedes’ principle states the conditions for a body to float in equilibrium : 1.its weight must be equal to that of the displaced fluid; 2.its center of mass must be on the same vertical line as the center of buoyancy The intended equilibrium will only be obtained if the vessel is actually built, and loaded, with the correct weight and weight distribution.

Weight is the product of mass times acceleration due to gravity, g. The total mass will be the sum of all component masses, and the center of mass (or center of gravity) can be figured by accumulating x, y, z moments : where m i is a component mass and {x i, y i, z i } is the location of its center of mass. The resultant center of mass (center of gravity) has coordinates

The weight schedule is a table of weights, centroids, and moments arranged to facilitate the above calculations. Often it is useful to categorize weight components into groups, e.g., hull, propulsion, tanks, and cargo. Some component weights can be treated as points, e.g., an engine or an item of hardware. Weight analysis and flotation calculations are an ongoing concern during operation of the vessel, too, as cargo and stores are loaded and unloaded. Often this is performed by on- board computer programs which contain a geometric description of the ship and its partitioning into cargo spaces and tanks.

6. Hydrostatic Stability Stability can depend on the nature of the disturbance. Illustration of various types of equilibrium. (a) Unconditionally stable. (b) Unconditionally unstable. (c) Neutral. (d) Conditionally stable, globally unstable

A ship can be stable with respect to a change of pitch and unstable with respect to a change in roll, or (less likely) vice versa. In order to be globally stable, the system must be stable with respect to all possible “directions” of disturbance, or degrees of freedom. A 3-D rigid body has in general six degrees of freedom : linear displacement along three axes and rotations with respect to three axes.

For a fully submerged rigid body, the stability of these degrees of freedom depends entirely on the vertical position of the center of gravity (CG) with respect to the center of buoyancy (CB).

There will be exactly one attitude of stable equilibrium, with the CG below the CB, and exactly one attitude of unstable equilibrium with the CG above the CB.