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 20. Inside-Body Triangle.
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.
Figure 21. Entire Fin Triangle.
I then took these x and y values and plotted them as shown in Figure 22. I checked this plot by measuring the fin tip at its maximum vertical position. Both of my measurements for the vertical position and horizontal position of the fin tip were about equal to the values calculated from the above equations (1.6491, 2.6309).
Figure 22. Fin Tip Position. Figure 23. Measurement of Fin Tip Horizontal Position. Figure 24. Measurement of Fin Tip Vertical Position.
I then moved on to creating the motion profile of the fin tip. I took the first derivative of the position vector with respect to time to calculate the velocity, and I took the second derivative of the position vector with respect to time to calculate the acceleration. These derivatives are shown in the equations below.
These equations require that I know how the angle φ is changing over time. From Equation 18, I have values for angle φ as a function of angle θ (because the displacement function is in terms of θ). I plotted angle φ vs. angle θ and converted θ to time using the equation θ = ωt, shown in Figure 25. I gave angular velocity a value of ω = 2 rad/s.
Figure 25. Angle φ over time for ω = 2 rad/s.
I plotted the trend line of φ against time in Excel, also shown in Figure 25, and used the trend line to calculate the first and second derivatives of φ with respect to time. I could then calculate all components of Equations 21-23 and the results are plotted in Figures 26-28 below.
Figure 26. Fin Tip Position, Velocity, and Acceleration Magnitudes.
Figure 27. Fin Tip Motion Profile (Real Component). Figure 28. Fin Tip Motion Profile (Imaginary Component).