Position Analysis of the Body and Wings

The vertical position of the body of the penguin is measured as the distance between the bottom of the penguin and the base. Thus, it is the instantaneous radius of the cam (the vertical distance from the center of rotation to the top of the cam, see previous page) minus the minimum radius of the cam. This plot is shown below. A constant input angular velocity of -1 rad/s (clockwise) is assumed. 

Theta=0 was defined as when the penguin was at its highest point and about to drop down. We can see that at this point, the position of the penguin is at 2.3 cm, or 3.9-1.6 cm, matching the maximum radius of the cam (radius of the larger semicircle) minus the minimum radius of the cam (radius of the smaller semicircle). During the drop, the point of the cam which is in contact with the base of the follower is the sharp corner of the snail cam. Thus, the vertical distance is the maximum radius of the cam, multiplied by the cosine of theta. A rest period of no motion follows, as the radius is at a constant minimum when the small semicircle of the cam points upwards. At theta=180, the cam radius begins to increase, following the simple harmonic motion mentioned in the previous section.

The graph of the position of the wings is shown above. The wing length from the grounded connection point to the wing tip was measured to be about 7.6 cm and the positions in the graph are the position of the wingtip with respect to the grounded connection point of the wings. The angle of the wings was measured to go from about -45° when the penguin is at its lowest to 45° when it is at its highest. Since the penguin body pushes up on the wing, the part of the wing that is just at the slot has the same vertical position as the body. Thus, the vertical position of the wing follows the same general curve as the position of the body. It should be noted that this assumes the distance from the grounded point of the wing to the slot is constant, which is not exactly true, as it decreases as the wing becomes more horizontal, and increases as it becomes more vertical. By Pythagorean theorem, the horizontal position is then the square root of the difference of the wing length squared and the vertical position squared.