First manual calculations
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For our kinematic analysis, we modeled the linkage mechanism as a crank connected to two four-bar offset slider-crank mechanisms, as shown in the image below. The slider-crank kinematic analysis was the simpler portion of the problem because the mechanism operates in a 2D plane and the rocker arm undergoes pure planar translational motion. This allowed us to simplify the mathematics and avoid accounting for rotational effects of the rocker arm and how those effects would influence the oar output.
For the oar-output kinematics, we then used the 2D translational motion of the rocker arm as the input to a 3D geometric model. Specifically, we defined a varying unit vector between the paddle-handle base position and the fixed output/rest location. By scaling this unit vector by the paddle length, we were able to determine the oar-end position. Velocity and acceleration of the oar end were then obtained numerically in Python through differentiation of the position results. The plots below show the important points of the mechanism described above.
Midpoint of Link 3 Kinematics
Paddle Output Kinematics
For the force analysis, rather than performing a full dynamic force analysis, we treated the mechanism as quasi-static. This eliminated the inertial effects of the linkages and allowed us to neglect dynamic force terms. Under this assumption, we were also able to model link 3 as a two-force member, which greatly reduced the complexity of the analysis. In addition, we assumed ideal pinned joints, neglected frictional losses between links, and treated all links as rigid in order to avoid deflection and compliance calculations.
Using a constant input torque and the assumptions stated above, we developed a force-output function at the midpoint of the coupler link and evaluated it over the full range of mechanism motion. Under the quasi-static assumption, the rocker arm was also treated using pinned-joint force transmission, with reactions occurring at the base where the paddles connected to the rocker arm. This allowed us to determine the force output at the paddle base, but not directly at the paddle end.
To determine the force output at the oar end, we used the unit vector previously developed in the kinematic analysis together with the paddle-base position array. By defining an initial offset from the paddle base to the oar slot, we generated an additional array describing the x-, y-, and z-position of the paddle base relative to the oar slot as the crank angle varied from 0 to 360. We then used this geometry in a quasi-static three-dimensional moment balance between the input force at the paddle base and the output force at the paddle end, taken about the oar-slot location. This provided the final three-dimensional force results and corresponding plots for the oar output.
