Unstable Equilibrium

For a floating body in static equilibrium there may be multiple equilibrium conditions; some stable and some unstable. A stable equilibrium condition is one in which a small perturbation leads to restoring forces/moment which tend to cause the body to return to the initial condition. Conversely, an unstable equilibrium condition is one in which a small perturbation results in upsetting forces/moments which tend to cause the body to move further from the initial condition and assume a new (generally stable) equilibrium condition.

Consider the example of a floating cube with a density half that of the fluid in which it is floating and with a center of gravity located at the geometric center of the cube. One equilibrium solution for this problem has the cube floating at half its depth with top and bottom parallel to the water surface. This, however, represents an unstable equilibrium. If you were to perturb the cube in any way it would immediately assume an orientation with one of the corners moving downward some amount. This new condition represents a stable equilibrium. Exactly which one of the corners moves down depends on which way the cube was perturbed, but it should be clear that there are multiple potential stable and unstable equilibrium solutions for the cube.

When Orca3D solves the static equilibrium condition, the equation solver generally finds the solution closest to the initial condition. The solution might represent an unstable equilibrium (as indicated in the output by a negative transverse or longitudinal metacentric height). In the case of the cube, if the initial condition has top and bottom faces parallel to the free surface, the Orca3D solution will likely be an unstable one. In reality it is highly unlikely that the cube will remain in such an unstable equilibrium since any small perturbation (such as a wave) will cause the cube to seek its stable equilibrium (which for certain geometries and mass properties might be upside down).

One might ask why finding an unstable equilibrium is important if it is unlikely to ever occur in the real world. In the context of a design tool for ships and boats, it is often useful to know not only when an unstable equilibrium condition exists but also how unstable it is. By presenting the unstable result, the designer can determine how much the KG (vertical center of gravity) needs to shift to make the vessel stable.

Finding an unstable equilibrium does not necessarily mean that a ship or boat will capsize. There may be a stable equilibrium condition that results before capsize, such as in the case of cube. So there are also times where it is desired to find the nearest stable equilibrium. If you are getting an unstable equilibrium result from Orca3D and want to obtain the stable equilibrium, one approach is to compute a righting arm curve for the model at the specific weight and center of gravity over a range of heel angles. By looking at the righting arm curve you can identify heel angles corresponding to stable equilibrium conditions. These are conditions where the righting arm is zero and the righting arm curve has a positive slope. You can then rotate the model to a “stable” heel angle so that its initial condition will be closer to the final desired condition. Remember of course that if you rotate your Rhino model, this implies a change to the location of the center of gravity that is input to the hydrostatics calculation. One method for accounting for this is to define an Orca3D weight/cost point at the initial CG location (prior to heeling the model) and with a weight equal to the total vessel weight. Then select this point along with the rest of the model geometry when you rotate it. When computing the hydrostatics, select the button to get the weight/cg from Orca3D weight/cost items. Hopefully the resulting equilibrium condition should be the desired stable condition.

In a future version of Orca3D we plan to provide the user with greater control over which equilibrium condition to find.