浮動のコートの安定性

When evaluating floating vessels and structures, the most important performance metric, apart from staying afloat, is arguably that it can stay upright. In fact, for many vessels, the ability to stay float depends heavily on the ability to maintain a particular orientation. A capsized vessel is likely to flood and subsequently lose positive buoyancy. Even in less extreme scenarios, the safety and comfort of crew and cargo are at stake. This tendency of a vessel to right itself or to capsize when disturbed is characterized by its stability. Unfortunately, changes that improve stability often negatively impact other important performance metrics such as fuel efficiency and maneuverability. Because of this tradeoff, optimizing a design for safety and performance generally demands ensuring sufficient but not maximum stability. In the remainder of this video, we will illustrate how the shape and weight distribution of a floating structure impact its stability. We will then test these principles experimentally on a model boat and compare the results with theoretical predictions made by computer-aided design software.

In a previous video, we covered the basics of buoyancy and gravity. Now we will examine how these two forces can affect the orientation of an object. Recall that for an extended object, the cumulative effect of gravity is a force passing through the center of mass equivalent to the total weight of the object. Similarly, the net buoyant force passes through the center of buoyancy at the centroid of the submerged portion of the object. Therefore, if the object is only partially submerged or the mass is not evenly distributed, a torque can develop. If the center of mass is below the center of buoyancy, any sideways rolling or heeling motion will impart a restoring moment to right the structure. This configuration is always stable, but generally requires a larger volume to be submerged. Now if the center of mass is raised above the center of buoyancy, the structure might become unstable and any heeling motion will be accelerated by the imparted moment, causing it to capsize. Note though that a higher center of mass does not guarantee that the structure will be completely unstable. A carefully-designed hull can make the structure metastable, that is stable up to a critical angle. This happens because in general, the shape of the submerged portion changes with heeling angle so the center of buoyancy shifts as the structure tilts. If it shifts laterally outside the center of mass, then that moment will act to right the structure. Equivalently, the vessel will be stable as long as the center of mass is below the metacenter, which is the point of intersection between the center line of the hull and the line of action of buoyancy. The dynamic behavior of a floating structure is also important since strong impulses from the environment could drive it past its metastable limit. The frequency and amplitude of oscillation also impact the safety and comfort of passengers and cargo. The rotational motion of a vessel can be predicted with a moment balance around its center of mass, which results in a second order differential equation for the heeling angle, that depends on the moment of inertia about the vessel’s center of mass, the total mass, the acceleration due to gravity, and the distance L along the vessel’s center line from the center of mass to the metacenter. Solutions to this equation for small angles are sines and cosines fluctuating at the natural oscillation frequency of the vessel denoted by omega. Now that we’ve seen how to determine stability in theory, let’s use this knowledge to analyze a hull design experimentally.

Set up a water bath in an area shielded from air currents and place a solid white background behind it. Now procure a small, preferably white boat with a simple hull design. Attach a lightweight brightly-colored mast at the center of the boat and float it on the water so that it points toward the camera. Mount a camera in front of the bath so that the boat is centered on the screen and adjust the camera height so that the field of view captures the portion of the mast above the boat. Ensure that the area is well illuminated and record a reference video of the boat at rest. We’ll use some custom code to track the angle of the mast by isolating the mast color in recordings from the camera. Refer to the text for details and example code. Analyze the reference video to verify that the tracking is working correctly and adjust the code as necessary to isolate the mast. Finally, level the camera until the code reports no tilt angle with the boat at rest. Once the code and camera are adjusted, remove the boat from the water and dry the hull. Snugly affix a cable tie about one centimeter from the bottom of the mast so that it can support a weight. Now slide a weight down onto the mast and weigh the total mast of the boat when dry. Next, record the height of the weight on the mast and then use a straight edge to balance the boat on its side. This balance point identifies the center of mass of the boat. Record the distance from the bottom of the hull to the center of mass. Place the boat back in the water and record a video while gradually tipping the boat, pressing sideways on the top of the mast until it capsizes. Now capture a second video with the boat initially tipped approximately 10 degrees and then suddenly released. Record the oscillations for 10 to 15 seconds. Repeat the capsizing procedure three or four more times for increasing heights of the weight. At the final height, record another video of the oscillations as before. Analyze each of the capsizing videos using the analysis script. The maximum stable angle can be determined by inspection of the chart, looking for the point beyond which the boat rapidly rolls over. In this case, this occurs around minus 26 degrees. Complete a table with the heights of the weight and center of mass and capsize angle. Next, analyze the two oscillation videos. Determine the dominant oscillation frequency by inspection of the animation of the mast motion or graph of the mast angle with time or by using a power spectral density estimate function. This experimental procedure is useful for small-scale testing and simple designs, but it is not always practical in real-world scenarios or for rapidly optimizing a design. In the next section, we’ll demonstrate a numerical approach to analyzing the boat and compare the results with these experimental findings.

We’ll use a Computer-Aided Design or CAD package to analyze the stability of the model boat. First, let’s see how to determine the center of buoyancy. Use the CAD software to create a solid to scale model of the boat hull. Position the model so that the center line of the keel is coincident with the origin in the CAD environment and the mast is parallel with the vertical axis. Recall that the center of buoyancy is at the centroid of the submerged portion of the hull. So to find the center of buoyancy, we must first isolate the submerged portion of the vessel. Create a horizontal plane intersecting the hull to represent the fluid surface and then remove everything above the plane. If the plane was at the correct height, the remaining volume will be equal to the total mass of the boat divided by the fluid density. Undo the cut and adjust the height of the plane as necessary until the remaining volume is correct. When the correct submerged portion of the hull has been found, use the mass properties function of the CAD software to evaluate the lateral offset of the centroid of this volume. In this case, since the hull is symmetric and level, you should find no lateral offset. In other words, the centroid will be on the center line of the hull. Repeat this process for increasing heeling angles of the boat to build up a table of the centroid offset as a function of heeling angle. When you are finished, plot the results and fit a cubic polynomial for the center of buoyancy. Now plot the lateral offset of the center of mass, which is its height times the sine of the heeling angle. At the critical angle, the center of mass will be at the metacenter and the lateral offsets will be equal. You should find that the predicted critical angle matches the experimental value within a reasonable uncertainty. Now let’s numerically predict the natural oscillation frequency of the model boat. Refine the CAD model to match the actual thickness of the hull and add the mast and weight. Adjust the weight height to match the position in the first oscillation test. Match the density of materials in the model to actual values and then use the mass properties function to evaluate the moment of inertia around the center of mass along the heeling axis. Repeat this process for the second position of the weight at which you measured the oscillation frequency. Calculate the height of the metacenter during small oscillations by assuming a small heeling angle such as five degrees. Subtract the height of the center of mass that you measured earlier to determine the length of the moment arm L. Now use the solution we found earlier to calculate the natural frequency of the rolling motion. Compare these calculated frequencies to the measured frequencies you observed before. You should find a close match. Notice that in the more stable case shown on the top row, which has a lower center of mass hCM, the restoring moment arm length L is larger. This results in a higher frequency of rolling than in the less stable case on the bottom row.

Now that we’ve seen a few methods for analyzing a hull design, let’s see how these are applied in real scenarios. Stability is an extremely important consideration in the design of all floating structures and vessels. Ships operating with shallow drafts, that is with most of the vessel above water level, have reduced drag and better maneuverability. In large cargo vessels, shipping containers can be stacked high above the top deck, increasing cargo capacity and facilitating loading and unloading operations. Both of these improvements require a higher center of mass and are made practical by careful design of the hull to ensure that the vessels are metastable. In cruise ships, shallow drafts permit more windows and decks for the passengers. These ships are designed not just to be metastable, but also to have a comfortable, natural oscillation frequency. Higher stability yields higher rocking frequency which may be uncomfortably snappy for those onboard.

You’ve just watched Jove’s introduction to the stability of floating vessels. You should now understand how the relative positions of the center of mass and center of buoyancy of a floating structure impact the structure’s stability and natural oscillation frequency. You’ve also seen how to analyze a hull design both experimentally and with computer-aided design tools. Thanks for watching.