Understanding the mechanical advantage of a Pneumatic linkage system

[DBTStud]

TL;DR Summary

I need to derive an equation for the mechanical advantage between the piston force and the force at the wall. I need help to get the right answer

Hi all,

I need to derive an equation for the mechanical advantage between the force applied at the wall and the force at the piston for a linked system. I have attempted this but my equation suggests that the wall force increases expenantionally as the connecting rod approaches the horrizontal.

I've attached my derrivation. He fixed pivot of the bellcrank is ridgedly attached to the vertical piston cylinder. The module works to push the complete unit from the wall.

Please can you suggest any modifications where I might have went wrong

 

[Lnewqban]

... my equation suggests that the wall force increases exponentially as the connecting rod approaches the horizontal.

That is correct, reason for which the horizontal position of the connecting rod should be avoided.
Consider that the lateral force, and subsequent friction, on the piston-cylinder and stem-bushing would be also huge.
In some cases, a "jammed" situation takes place, making the reverse movement very difficult.

The mechanical advantage is proportional to the relative velocities of piston (point A) and point B (which reaches zero velocity when the connecting rod and piston stem are 90° apart).

Please, see:
https://www.engineeringtoolbox.com/toggle-joint-d_2077.html