The baby-boot system was modeled as a thin rod and solid plate, and was decomposed into a free-body diagram (Image 1). The thin rod was represented as rigid body A, whereas the latter represented as rigid body B. Simplifying assumptions were made:

• The rod and plate are rigid bodies.

• The revolute joints are massless, frictionless, and inflexible.

• Uniform gravity forces.

• No air resistance.

In dynamics, the governing equations of a system could be modeled by using Newton's, Lagrange's method, or Kane's method. Lagrange's method of the second kind was chosen as it was considered to be the most straightforward approach.

Images 2 - 4 shows the process to obtain the governing equations of the system. In essence, the Lagrangian (L = T - V) is solved for by finding the kinetic energy and potential energy of each rigid body. The generalized coordinates were defined as the shoelace's angle with respect to the vertical axis, and the angle between the baby-boot and second revolute joint. The governing equations were then obtained by substituting all relevant terms into Lagrange's equation.

Lagrange method yielded two second order differential equations to represent the dynamics of the system. However, these equations were reduced into four first order differential equations for the Runge-Kutta 4 numerical solver to function. 

The  RK-4 solver was created in MATLAB. The four states in the system-- qA, qȦ, qB, and qḂ--were placed in a row vector called y.  By setting the appropriate initial conditions, the behavior of the system was modeled.

Various parameters were altered to examine their effect on the system's dynamics. In each case only one or two parameters were altered at a time to maintain consistency. Image 2 shows the starting initial value conditions where no parameter was modified.

The Runge-Kutta 4 solver was compared against stiff and non-stiff ODEs to verify whether the solver was created correctly. Image 3 shows that the RK-4 with a time step of 1/10^3 does not follow the general trend of the other ODEs. However, Image 4 shows that the RK-4 more closely follows the other ODEs when the time step is reduced. This trend is more evident in Image 5 where the RK-4 is undistinguishable from the other ODEs unless the plot is zoomed in. Note that the RK-4 shares the same time step with the other ODEs in each case. This verification shows that the RK-4 was accurately created.

The baby-boot or chaotic pendulum system exhibits dynamic regions of instability and stability. For unstable behavior, there is no clear trend between any of the ODEs regardless of the time-step's size (Image 1). For stable regions, all ODEs model the behavior of the system in a similar fashion (Image 2).  ΔT could be further decreased to potentially resolve the chaotic behavior, but this is not practical due to computational time.

Stable regions interval (qA):  [0°, 71.3°] U [111.78°, 159.9°].

Unstable regions interval (qA): [71.4° , 111.77°] U [160°, 180°].

The angle of the baby-boot (qB) was selected as the primary comparison metric because this parameter provides the most information on how the baby-boot rotates. The other parameters either show the behavior of a simple pendulum or information that is not as insightful.

The first parameters that were altered were the plate's center of mass position and its mass (Image 1). By doubling these parameters the frequency of qB increased. Therefore, as the length of the shoelace and the mass of the baby-boot increases, the baby-boot will rotate more slowly. This is because the heavier mass of the baby-boot would cause there to be a greater energy requirement for the baby-boot to swing back up. Conversely, by reducing the length of the shoelace and mass of the baby-boot, the baby-boot should spin faster. This expectation was fulfilled as shown by Image 2.

The next set of parameters that were altered were the masses of the plate and rod. From previous conclusions, it is expected that doubling the mass would reduce the angular velocity of the baby-boot due to inertia. However, Image 3 shows that this configuration had a negligible impact on the system. A minor phase shift occurred around 25 s for the RK-4 and ODE23 plots; decreasing the time step or increasing the simulation time could potentially show if this impact is more substantial.

The vertical starting angle (qA) of the rod was then altered. Increasing the starting angle causes the frequency of qB to increase. Furthermore, the ODE solutions start to separate. Based on MATLAB documentation ODE89 and ODE113 are the most accurate due to their small error tolerances. qA was increased to 75° and 179° (Image 5 - 6). The previous expectation could not be verified because the initial starting conditions were within the unstable region.

qB was altered directly to examine the plate's behavior. The angle of qA and qB were both set to 45° to serve as the new baseline plot. Image 7 shows that decreasing qB increases its frequency; the ODEs separate as well. In both plots it is evident that sustained oscillation occurs at the starting angle's amplitude. Although the unstable regions of qA were specified earlier, it appears that qB has an unstable region at 90°. At this point, all ODE solvers fail; an incredibly small time step is likely required to see the full behavior of the system. At qB = 179°, the plot exhibits high-frequency behavior (Image 9). Both the qB = 1° and qB = 179 plots share similar system behavior. The difference is that one starts off as a cosine wave (Image 7) and the other as a sine wave (Image 9). Both start at their respective amplitudes. Thus, it can be concluded that the frequency of qB increases as qB's angle gets further away from 90°; the same can be said for 270°.

The governing equations of each point particle of the plate.

Initially, the centroid's position for each rigid body was used to create a 3D plot. However, since this plot only modeled simple pendulum motion, point particles were attach to each corner of the plate. The resulting governing equations were used to animate the system in MATLAB; ODE89 was used to animate the system due to its high accuracy.

Animations were created to model the dynamics of the system for when the starting angles of qA and qB were varied. As previously concluded, the baby-boot swings at a greater frequency the further it is away from 90°. By comparing Figure 1 and Figure 2, this conclusion is reinforced. This is because the plate in Figure 1 swings more frequently than the plate in Figure 2. Another conclusion that is reinforced is that how far the plate could twist is based on its starting angle. For example, the plate in Figure 2 twists more than the plate in Figure 1. This is because their maximum respective amplitudes are 45° and 1°.

A case where the system experienced unstable behavior was plotted (Figure 3) and animated (Figure 4). It is not directly clear on what is happening to the system by examining by the plot. But with the animation, the whole picture is obtained.  Considering the ODE89 plot with the animation, qB initially starts at 179°. Past the first time instant, the pendulum swings down causing the plate to rapidly rotate in one direction; since the plate rotates so fast, multiple full rotations occur. Once the pendulum fully swings to the other side--it's maximum amplitude of -179° is reached--, the plate rotates in the other direction. The system's behavior repeats itself, which is why there's a zig-zag pattern in the plot.