Kinematic Modeling

System Mechanics

This device is a combination of a four bar linkage as well as a geared platter which runs off the same input as the crank for the linkage. To analyze this system these features will be first described separately before synthesizing their relative motions to describe the full output. Because they work simultaneously however the full position of the pen relative to the paper cannot be sufficiently described with just one or the other. The goal was to first break them into manageable sub systems based on our knowledge of both, and then utilize out understanding of the physics that connect the two to develop the math that can coherently describe the full motion taking place. 

Pertinent measurements are the turntable pivot relative to the mechanism as it will determine the translation and rotation transforms necessary on the fundamental linkage locus. For the purposes of this analysis we are not going to move the rocker pivot though it is technically also a variable. The same goes for the crank arm as well as the gearing, these are all variable but not without cumbersome modification, so we are limiting our dependent variables to just coupler length, rocker length, and pen holder length and angle. The pen holder is what defines the coupler curve in this device and the trace of point P is the roulette of interest for this analysis. 

Figure 6. Annotated Machine with Link Labels and Dimensions

Four-Bar

This simple mechanism has the following attributes, some of which are functionally fixed while others take on a range. This allows different designs to be traced by the machine. The dimensions shown are approximate to within 1/8". 

• Link d (ground) - 20.5"
• Link a (crank arm) - 1.5"
• Link b (coupler) - 14", 15", 16", 17", 18"
• Link c (rocker) - 6.25", 7.25", 8.25"
• Pen Holder - 4" over about 150 deg range below coupler depending on link assignments and paper size

Gearing 

The two gear centers include the crank pivot and the turntable pivot. They are 11.5" from each other and connected via a belt and pulley. 

• Crank pulley diameter - 3.5"
• Turntable pulley diameter - 4"
• Gear ratio 1.14 

In practice the gearing seems to follow the relation on 5:5 for some as yet to be explained reason. I believe there is enough slop and slip in the system to make idealized ratios be unreliable but from experimentation the platter lags the crank by about 

Figure 7. Drawing Machine Demonstration

Figure 8. Matlab Partial Simulation Still Frame

Figure 9. Movie of Matlab Simulation in Action

In the clip above you can see the functionality of the Matlab script which does a partial simulation of the trajectory. The plan was to manipulate the point plot data by using a transformation matrix that could account for both the linear offset in X and Y of the platter from the crank pivot, which was my origin. And then it would also apply a rotational adjustment with a angle indexed to the crank by via the gear ratio. However those plans did not materialize and I was only able to plot the four bar curve by itself. In theory this is acheivable by simply multiplying the 2x1 position vector by the following for the appropriate translation and angle values. 

Where 11.5" and 2.75" are the translations in x and y respectively. And in this case phi would be a function of the crank angle and would therefore need to be updated for each point. 

Matlab code for simulation presented. Completed in R2018a