Kinematic Analysis & Synthesis- JH

Mobility Calculations:

One degree of freedom is desirable for this mechanism since only the output should be free to move linearly and not rotationally. Mobility calculations were performed as follows:

M=3L-3-2J1-J2

where L=5, J1=5, and J2=1 (pin in slider). This results in: 

M=3(5)-3-2(5)-1

M=1

Grashoff conditions were also calculated, with

S+L=3.5+11.24 (ground from 02 to slider)=14.74

P+Q=6+7=13

and S+L>P+Q, resulting in Class II Grashof meaning no link can fully rotate.

Kinematic Analysis:

For each of the analyses explained below, the code as well as the hand calculations are included and labeled in the appendix.

Position Analysis:

Position Analysis was first performed on the linkage to show the positions of points 1,2, and 3 as theta 2 rotated. Point 1 is defined as the joint between the input and the sliding linkage, point 2 is the median point on the sliding linkage, and point 3 is the joint between the coupler and the sliding linkage. A graph of each position vs. theta 2 is shown below: 

Velocity Analysis:

Next, velocity analysis was performed to understand how the two rotating links and slider were affected by the input velocity. A constant velocity of 5 rad/s was assumed for the downstroke motion. The graphs below represent the velocity of the two rotating links vs theta 2 as well as the sliding velocity vs theta 2. 

These make sense, as the input begins to rotate CCW, link three rotates CW around the pin it slides on and link 4 doesn't rotate very much. However, as link three begins to rotate less as it approaches the end of the sliding portion of the link, link 4 beings to rotate CCW to adapt to this change in velocity from link 3. The slider velocity also matches the motion- as it returns quickly downwards after link 4 begins to rotate, it settles and then moves up slightly once link reaches maximum velocity and then settles at a resting position slightly below the upwards movement. 

Acceleration Analysis:

Finally, acceleration analysis was performed. The angular input acceleration was assumed to be zero for link 2 since it was rotated at constant velocity. The graphs for acceleration are shown below: 

PMKS Simulation:

The results of the simulation are again shown below. Using measured lengths, this model was constructed and shows the position paths of each link. Compared to the actual motion of the mechanism, it matches up very closely and is an accurate representation of the paths that each link and point will take during a complete cycle of the mechanism.