In order to determine if the slider mechanisms on the carousel would be able to capture the desired motion we performed a few analysis to determine the following values:
theta A: the angle that the sliding link makes with the x-axis
b: the length of the slider between the pole and point A, where the slider is fixed to the stationary pole
b dot (or b'): the speed at which the length value "b" is changing at any given time
omega A: the rotational speed of the sliding link
b dot dot (or b''): the acceleration of the sliding link along itself, the rate at which the linear speed of the link is changing
alpha A: the rotational acceleration of the sliding link
To identify these values we solved for each of them symbolically by hand and then used MATLAB to calculate the numerical values for one full revolution of the servo, the graph of each value can be seen below.
Figure 7. Position, Velocity, and Acceleration for Slider Mechanism of Carousel
To further verify that our design will work we used PMKS to model one of the pole and slider pairs as it rotates around the ring. We utilized the data from the MATLAB and PMKS simulation to verify the length of our slider and the radius of the sun gear we used for our design, (6" and 9.5" respectively) as we could determine there would be room for the slider bar to fully extend without interfering with the rotation of the pole around the carousel.
Figure 8. Animation for Slider using PMKS