3. Results and Conclusions

The position analysis script generated the following dynamic plot:

Figure 3. Position Analysis Results.

This figure visualizes the motion of the system very well, especially the motion of Link B.  It is clear that Link B must travel with non-planar motion for the mechanism to function, which means that the spherical joints at its ends are indeed essential to proper performance.  However it is not obvious from this plot that Link C is traveling with the exact circular arc that is expected.  The perspective of this figure does not display this adequately, so the following supplemental plot was also constructed:

Figure 4. Output Link Trajectory.

After isolating the vector solutions generated for Link C, a sum of squares of the x and z components was plotted against the array of y components, resulting in a plot that clearly demonstrated the circular path of the mechanism output.  Further, the information calculated for this plot was used to determine the output angle, Θo, to be 108.3109º.  This was interesting because the simplest version of a rotary valve would rotate 90º so that the internal tubing (see Figure 1) could be symmetrical.  The trombone's rotary valve therefore has a more complex internal mechanism, probably so that the extra tubing accessed by the valve requires less complicated machining to reach the valve.

Finally, the ratio of output to input ratios can be calculated as r = 7.5636.  This is very impressive: an input from a user can be amplified nearly 7.5x to achieve a precise result each and every time.  From this result and from the motion plots above, it can safely be said that the trombone's F-Attachment trigger mechanism achieves its design goals very well.