After plotting the motion profile of the penguin's body, I wanted to plot the position of the penguin's fin tip and its motion profile. First, I needed to determine how the angle φ changed as the penguin's body moved upwards using the following triangle and equations.
Figure 13. Ready To Fly Penguin Displacement And Trend Line.
In this triangle, d is the horizontal distance between the fin slit and the fin "hinge" (this distance is assumed to stay constant), g is the length of the fin inside the penguin's body, and fo is the initial vertical distance between the bottom of the slit and the fin hinge. As the penguin's body moves upward, the distance between the bottom of the slit and the hinge is reduced (if the fin is below the horizontal) or increased (if the fin is above the horizontal) by the amount the penguin is displaced. However, the penguin body does not contact the fin at zero degrees. Contact between the fin and body does not occur until about 110 degrees. Therefore, the displacement function in this case, s, is adjusted by taking the displacement trend line from the previous section and subtracting the displacement of the body at 110 degrees. Now that I have an equation for angle φ, I can solve for the position of the fin tip using the following triangle and equations.
I then took these x and y values and plotted them as seen in Figure
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