### Overview

Source: Alexander S Rattner and Kevin Rao Li Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park, PA

The objective of this experiment is to demonstrate the phenomenon of stability of floating vessels - the ability to self-right when rolled over to the side by some external force. Careful design of hull shapes and internal mass distribution enables seagoing vessels to be stable with low drafts (submerged depth of hull), improving vessel maneuverability and reducing drag.

In this experiment, a model boat will first be modified to enable adjustment of its center of mass (representing different cargo loadings) and automated tracking of its roll angle. The boat will be placed in a container of water, and tipped to different angles with varying heights of its center of mass. Once released, the capsizing (tipping over) or oscillating motion of the boat will be tracked with a digital camera and video analysis software. Results for the maximum stable roll angle and frequency of oscillation will be compared with theoretical values. Stability calculations will be performed using geometric and structural properties of the boat determined in a computer aided design environment.

### Principles

The buoyant force, which supports floating vessels, is equal to the weight of fluid displaced by the submerged portion of such vessels. The buoyant force acts upward, along the vertical line passing through the centroid (center of volume) of this submerged volume. This point is called the center of buoyancy. If the center of mass of a floating structure is below its center of buoyancy, any sideways rolling (*heeling* motion) will impart a moment to *right* the structure, returning it to the upright orientation (Fig. 1a). If the center of mass is above the center of buoyancy, the structure may be unstable, causing it *capsize* if disturbed (Fig. 1b). However, if the hull of a floating vessel is designed carefully, it can be stable, even if its center of mass is above its center of buoyancy. Here, tipping the vessel slightly causes the shape of its submerged volume to change, shifting its center of buoyancy outward in the direction of tipping. This results in a net *righting* moment as long as the line of action of buoyancy is outside of the center of mass of the structure (Fig. 1c). Equivalently, a vessel will be stable if the point of intersection of the line of action of buoyancy and the centerline of the hull (metacenter) is above its center of mass. Some vessels are metastable - only self-righting up to some critical angle.

It is also important to consider the dynamic behavior of a floating vessel. Strong impulses from waves may cause a boat to rotate past its metastable limit, even if the initial tipping angle is small (*i.e.*, large for small ). The frequency and amplitude of oscillation may also affect passenger comfort. The rotational motion of a vessel can be predicted with a moment balance about its center of mass. Here, *I*_{zz} is the moment of inertia about the center of mass, *θ* is the roll angle, *m* is the vessel mass, and *L*_{cm,mc} is the distance along the boat centerline from its center of mass to its metacenter.

(1)

**Figure 1: a. Stable vessel with center of mass below center of buoyancy, ensuring righting moment. b. Unstable vessel with center of mass above center of buoyancy. c. Hull shape that causes the center of buoyancy to acts outside the center of mass (metacenter above center of mass). This yields stability even with the center of mass above the center of buoyancy.**

### Procedure

1. Measuring maximum angle of stability

- Select a small model boat. A relatively simple hull design is recommended to reduce analysis complexity in Sections 3 and 4.
- Connect a lightweight brightly-colored vertical mast to the boat (blue recommended). The provided MATLAB code tracks the position of the mast in the video by looking for bright blue pixels in the image. If a different color mast is used, the image analysis code will have to be adjusted accordingly.
- Snugly affix a cable tie to the mast to act as a stop for a weight. Slide a weight (
*e.g.*, coupling nut) onto the mast so that it rests on the stop. - Place the boat in a larger container of water, and allow it to settle (Fig. 2a). Position the setup so that airflow in the room does not disturb the boat. Mount a video camera facing the mast along the length of the boat. A white backdrop is recommended.
- Collect a reference video of the boat at rest, and analyze it using the provided MATLAB function (
*TrackMast.m*). Adjust the orientation of the camera until it correctly reads 0-tilt when the boat is at rest. You may need to adjust the masking parameters to isolate the mast on line 17 of the code. - Collect videos of very gradually tipping the boat by pressing sideways on the top of the mast until it falls over on its own (capsizes). Keep the mast in the video frame as long as possible during each test. Perform this procedure for different heights of the weight. Record the height of the weight on the mast for each case.
- Analyze these videos using the provided MATLAB script. For each case, the maximum stable angle can be determined by inspection of the output angle and time arrays. Complete a table of capsize angle vs. weight height.

**Figure 2****: a. Model boat with adjustable weight on mast, b. Roll angle variation with when released from slight angle (Step 2.1), c. Power spectrum density plot of (b) showing peak oscillation frequency of 1.4 Hz** Please click here to view a larger version of this figure.

2. Measuring the oscillation frequency

- Perform a second set of tipping experiments with two different mast-weight heights. This time, only tip the boat slightly (~10°), and collect videos of the rocking boat for 10 - 15 s.
- Rerun the mast tracking function on the video. After calling the function, evaluate the following MATLAB expression on the output: pwelch( theta,[],[], [],1/(t(2)-t(1)) ); . This will plot the power spectrum density for the rocking boat. The primary rolling frequency is the peak value on this plot (Fig. 2b-c).

3. Prediction of the tipping angle

- Using a scale, measure the mass of the model boat, including the mast and weight.
- For each position of the mast weight evaluated in Step 1.5, balance the boat on its side with the mast on a straight edge. Record the height of the balance point from the bottom of the hull as the center of mass (
*H*_{cm}). - Using a CAD software package, create a to-scale model of the boat and mast with weight. Ensure that the hull is filled-in (solid) in this model (Fig. 3a).
- Position the model so that the centreline of the bottom hull (keel) is coincident with the origin in the CAD environment and the mast is (initially) parallel with the vertical (y) axis.
- In the CAD environment, rotate the boat about the z-axis, which is along the length of the hull, in small increments (e.g., 5°, 10°, 15°…).
- After each rotation, cut away all of the boat above a vertical level such that the volume of the remaining lower portion equals the total boat mass divided by the density of water (
*m*/*ρ*_{w},*ρ*_{w}= 1000 kg m^{-3}). This represents the portion of the boat below the water line when it is floating at that angle (Fig. 3b). - Using the "Mass Properties" feature in the CAD software, evaluate the x-position of the centroid of the remaining hull. Here, the origin should be along the lowest edge of the boal (the keel), and the x-axis should point in the horizontal direction. This represents the center of buoyancy (xb); the buoyant force acts through this point. Prepare a table of
*x*_{cm}vs.*θ*. - For each maximum stable angle (
*θ*) identified in Step 1.6, compare the moment arm of the boat weight () and the moment arm of the restoring buoyant force (). You may need to interpolate between the values obtained in Step 3.7. Do these balance approximately?

**Figure 3****: a. Filled in model of the boat hull, b. Vertical cutaway of the hull, revealing the submerged volume of the vessel, c. Physically accurate model of the vessel.**

4. Predicting the period of oscillation

- Produce a second CAD model of the boat with the position of the weight corresponding to the cases in Step 2.1. This time model the actual thickness of the hull (
*i.e.*, not filled-in, Fig. 3c). Match the density of the materials with actual values. - Using the CAD software "Mass Properties" feature, evaluate the moment of inertia of the boat about its center of mass along the roll axis (
*I*_{zz}) for the weight heights. - Using results from preceding steps, and the
*x*-position of the center of buoyancy measured when (Step 3.7), evaluate the theoretical oscillation frequencies:

(2) - Compare the theoretical result from Step 4.3 with the measured oscillation frequencies. Do these values agree reasonably well?

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.

### Results

Total mass( m, kg) |
Center of mass( H_{cm}, m) |
Center of buoyancy( , m) |
Moment of Inertia(I_{zz}, kg m^{2}) |

0.088 (Step 3.1) |
0.053 (Step 3.2) |
0.0078 (Step 3.7) |
0.00052 (Step 4.2) |

Table 1. Properties of model boat with 24 g weight positioned 13 cm above keel.

Procedure Step |
Experimental Value |
Predicted Value |

Maximum stable roll angle (1.6, 3.8) | ~25° |
28.5° |

Natural roll frequency (2.2, 4.3) | 1.4 Hz | 1.24 Hz |

Table 2. Maximum stable roll angle and rolling frequency of boat with 24 g weight 13 cm above keel.

### Applications and Summary

This experiment demonstrated the phenomena of stability of floating vessels and how ships can stay upright even with relatively high centers of mass. For example, in the representative results, a small model boat with a center of mass (*H*_{cm} = 5.3 cm) well above the water line (*H*_{water line} ~ 1 - 2 cm) could return to its upright position after being tipped to a ~25° angle. In the experiments, the maximum stable angle was measured for a model boat with different vertical centers of mass. The effect of center of mass height on oscillation (rolling) frequency was also evaluated. Both of these measurements were compared with theoretical values obtained using geometric parameters in CAD packages. These results and procedures can serve as a starting point for students seeking to design and analyze floating structures.

The property of stability is crucial for the design and operation of seagoing vessels. Ships operating with shallow drafts (most of the vessel above water) have reduced drag and increased maneuverability. In large cargo vessels, shipping containers can be stacked high above the top deck, increasing cargo capacity and facilitating loading and unloading operations. In cruise ships, shallow drafts permit many windows and decks for passenger. While stability is critical for safety, very stable hull shapes (high ) yield fast rocking frequencies (Eqn. 2), which may be uncomfortably *snappy* for passengers. Hydrostatic stability analyses, as demonstrated in this experiment, are thus crucial tools to guide marine engineering.