Before considering the kinematic design of the cutting arm linkage mechanism, we had to learn how professional chefs cut vegetables. When using a large chef knife for relatively small vegetables, the knife blade is typically left in contact with the cutting surface. This blog was helpful in deciding a knife path (and is the source of figure 1).
Initially a fourbar mechanism was considered to drive more complex motion. However, this would leave the knife end unsupported, which could result in poor cuts or possible safety issues. To better constrain the system, a slider crank was chosen. This is also a good choice because the knife needs to stay in constant contact with the cutting surface, so the sliding axis will be parallel to the cutting board. Lastly, the knife itself can be used as the slider crank output coupler, further simplifying the design.
In order to choose the input coupler length, the effective cutting length of the knife was found. This is the length of the knife available for cutting, which is equal to the overall blade length minus the end clamp length, maximum vegetable diameter (chosen to be 1.5"), and tolerance for minor misalignment. This came out to about 5", which was set equal to the diameter of the input coupler's circlular path. This leads to the input coupler having a length of 2.5".
For safety reasons and practicality, machining modifications to the knife were as limited as possible. The handle connection was initially intended to be a drill and tap into the dowel pin; however, the knife was cheap and the "dowel pins" ended up being thin aesthetic disks. Because of this, a portion of the rubber handle was removed and a custom part was made for the connection. At the knife end, a simple clamp was made to safely hold the blade. This allowed for small adjustments in case the knife CAD wasn't fully accurate. After these attachments were modeled, the knife linkage length from the handle pin to the clamp shaft was found to be 11.75".
The last step in completing knife kinematics was finding the position of the sliding axis relative to the driven axis. Since the knife has complex geometry, it is difficult to do a standard simulation to choose the best linkage ratios. Given this, it was important to create a precise 3D model of the knife. To do this, the knife was photographed next to a ruler. Using the ruler as a scale, the outline of the knife was traced in SolidWorks using splines, and extruded to measured specifications. After this step, the vertical distance between the driven axis and the slider rail was found by incrementally adjusting the value in CAD, eventually approaching a value of 2.275". A small notch was machining in the cutting board to account for slight errors, assist with alignment, and ensure the vegetable was fully cut.
Once the vertical distance was found, the slider crank was examined at its toggle positions to find the rail length. Figure 2 shows the fully defined mechanism. At this point, every dimension was fully defined, so kinematic analysis could take place. To do this step, a tool called PMKS (Planar Mechanism Kinematic Simulator) was used. This software calculates position, velocity, and acceleration of linkages in most standard mechanisms. (Unfortunately, it only runs in Internet Explorer.) The slider crank used for this project can be found here, as seen in figure 3.
A force analysis was conducted to find the pressure applied on the vegetable by the knife. The potentially troublesome points in the cutting cycle are located at the toggle positions and where the knife meets the vegetable, due to the vegetable's skin being the hardest surface. Using the free body diagram in figure 4, the force applied to the vegetable could be found at the motor's stall torque. This analysis does not include the full cutting effect of the knife due to its material properties. Chef knives have microscopic serrations that assist in cutting when the knife edge moves relative to the cutting object. By not including this effect, the force analysis is more conservative. Overall, the force analysis yielded a minimum force of 7.4N when analyzed at the vegetable surface. Since the vegetable resistance was unknown, the project was continued with the given motor. Future iterations should use a stronger motor.