Kinematic Analysis

he mechanism was analyzed using a 5 bar vector loop as shown in Figure 1. In order to get a desired output at the pen (point P), the input angles (angles of R2 and R5) were calculated. This was accomplished using the Law of Cosines as well as Freudenstein's Equations. Once the angles of links 2 and 5 were obtained, the y positions of points E and F were found and these values were converted into radii of the cam to create an entire cam profile. More detailed information will be provided next to each corresponding figure. (For this analysis, all angles are measured counterclockwise from the positive x-axis. The global coordinate frame XY uses θ for angles and the local coordinate frame xy uses Φ for angles. All equations can be found in the attached MATLAB code.)

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The 5 bar mechanism can be viewed as two symmetric arms working in parallel. First, the top arm was analyzed by creating a fictitious link R7P to form a 4 bar linkage. Another fictitious link R3P was created to replace R3. (Note that the angles of links R3 and R3P have a constant relation which will be discussed in the next paragraph.) Since we know where we want the pen to draw, the position of point P is known. The distance between P and the origin can easily be found to get |R7P|. The angle of R7P (Φ7P) can be found using the inverse tangent. These values can be inserted into Freudenstein's equations to solve for all the angles of the vector loop in Figure 2.

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The previous step used vector R3P . The distance between BP or |R3P| can be found using Law of Cosines, since the distance between BC and CP are known, . The results of the above 4 bar analysis provide the angle of R3P, but we are interested in the angle of R3. Again using Law of Cosines, the angle of R3 can be found. Finding the angle of R3 allows us to add vectors R2 and R3 together to find the coordinates of point C which will be necessary to find the angle of link 5.

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Since the coordinates of point C are known, the magnitude and direction of R7C can be found the same way as R7P. Using Law of Cosines, angle COD can be found. Adding angle COD to the angle of R7C yields the angle of R5.

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Now that the angles of both links 2 and 5 are known, the displacement of points E and F must be determined. The mechanism will read the outside of the cam at point E and the inside of the cam at point F. The displacements caused by the cam will act as the input to the mechanism and will be discussed in the next section.

FIGURE 1: 5 Bar Vector Loop for Writing Mechanism

FIGURE 2: Four Bar Vector Loop Used to Obtain Angle of R2

FIGURE 3: Close-up of Link 3

FIGURE 4: Triangle Used to Obtain Angle of R5

FIGURE 5: Using the angle of R2 to Determine E_y