1. Background

The subject of this mechanism analysis will be the trigger linkage on a trombone's F-Attachment.  The modern trombone is commonly equipped with a single valve, which is useful because it lengthens the horn as much as a full arm extension of the slide (about 3 feet).  This reduces the number of large, fast arm movements musicians must make.  The mechanism is called an F-Attachment because it enabled tenor trombonists to play an F natural in an octave normally only achievable on the bass trombone (a completely separate instrument with larger proportions).  The valve works by rotating a very specific amount so as to orient sections of tubing seamlessly together at either end of its path of rotation.  This rotary valve is very similar to those of other brass instruments like the French horn, of which a simplified version is displayed in the diagram below. 

Figure 1. Diagram of French Horn Rotary Valves. Image retrieved from https://www.normans.co.uk/blog/2017/07/piston-valves-vs-rotary-valves-whats-the-difference/

The rotation of the valve must be very precise to ensure proper contact between the sections of tubing, meaning that the mechanism must produce the exact same rotation at its output each use.  Furthermore, the valve must be simple to use and require a small input from the user.  To achieve simplicity of use, the trigger is positioned very deliberately: the input link is located near where the left thumb would normally rest while holding the trombone.   In addition, it is positioned so that the input motion is produced with a natural and comfortable motion of the user's thumb.

Figure 2. Trombone F-Attachment.

Designing this mechanism required solving the difficult problem of translating a small rotation about one axis to a large and precise rotation in another non-parallel axis.  This design problem is solved by shortening the output link (the small arm that rotates the valve) and implementing spherical joints at each end of the middle link to allow for the non-planar motion it must undergo to follow the arc-shaped paths of both the input and output links.  This report will seek to characterize the mechanism's ability to perform its function by computing the ratio between input and output angles as well as performing a comprehensive position analysis in MATLAB to verify its output.