3) Kinematic Analysis and Synthesis [ATD]

Mobility Calculation

In order to calculate the mobility (degrees-of-freedom) of the system, Gruebler's equation was employed:

Figure 3.1: Links, lower pairs, and higher pairs

Figure 3.1 displays the location of the links, lower pairs, and higher pairs on the mechanism. Dotted lines are used to denote that two components are fixed to the same axis. Note that gears (and other components) fixed to the same axis are considered one link.

The diagram shows:

• L = 8

• J_L = 8
• J_H = 4

Thus,

The mechanism has a mobility of 1.

Position Analysis

The most complex part of the position analysis is determining the location of the slider with respect to the crank angle.

Figure 3.2: Crank slider mechanism (and vector representation)

Figure 3.2 shows physical and vector representations of the crank slider mechanism. The length c denotes the distance of the slider from the crank shaft. A vector loop can be set up to determine this distance:

This relationship is plotted below:

Figure 3.3: Graph of distance of container from center of turntable

Figure 3.4: Crank slider and turntable (to scale)

Figure 3.3 shows a to-scale representation of the crank slider mechanism in combination with the turntable. The end of the connecting link (position of the "slider") is offset from the actual topping container by a fixed amount. Adding this quantity onto c will provide the distance of the edge of the topping container from the crank axis. This distance should range between a maximum and a minimum value, determined by the lengths of the crank and connecting links. The minimum distance should correspond to the near (w.r.t. the crank axis) edge of the topping container reaching the center of the turntable. This position occurs when the crank is at 270° (when measured from the positive x-axis). The maximum distance of the container from the crank axis should correspond to the far (w.r.t. the crank axis) edge of the topping container reaching the edge of the turntable. This position occurs when the crank is at 90° (when measured from the positive x-axis). The lengths of the linkages were chosen such that the desired behavior was produced, as shown in the following animation:

The animation also displays the turntable rotating at 6x the speed of the crank, as a result of the 6:1 gear ratio between the two components. This can be represented mathematically:

Below is an animation showing the topping container's path as the turntable moves below it:

The saw-wave profile of the the cam also causes 6 "drops" of the container for every one rotation of the turntable. Combining all of the positional information synthesized above, the specific locations of each topping "drop" can be overlayed onto the visualization:

We also know the size of the open end of the topping container and thus we can visualize the actual coverage of the topping distribution for one cycle of the crank:

Velocity Analysis

The motor, turntable, and crank are all spinning at constant angular velocities, and related to one another through a fixed gear ratio. The container, however, has a non-constant velocity. The calculations for determining this velocity with respect to the crank position over the full range of motion are as follows:

Figure 3.5: Graph of angular velocity of connecting link

Figure 3.6: Graph of velocity of container