Another linear transmission mechanism

Thanks go to reader Richard, who commented on a previous article, suggesting that the levers could be removed. After some doodling, I realized that he was right, and here is the modification of the linear transmission mechanism, also showing the driving mechanism:

The little ovals are holes or eyelets that the string passes through, so that straight connections to the rods are maintained. They are effectively pulleys. Spring 3 is a compression string, and acts to push Rod 2 away from the (fixed) hole for the string.

The maximum distance between Rod 1 and the plunger is fixed by means of a stop (not shown), so that the plunger can push against Spring 2 which pushes against Rod 1, but Spring 2 cannot push Rod 1 unless the plunger is pushed. In other words, the distance between Rod 1 and the plunger can be no more than one unit.

When the plunger on the left is pushed by the linear cam (i.e. the ramp), Spring 1 is compressed, and as long as Rod 1 is not inhibited from movement, Spring 2 is pushed so that Rod 1 is also pushed. This creates slack in the string, which allows Spring 3 to expand, pushing Rod 2 the same distance that Rod 1 is pushed.

When the plunger is released, Spring 1 expands, which pulls Rod 1 back (remember that the distance can be no more than one unit). This pulls on the string, which compresses Spring 3 and pulls Rod 2 back.

The expansion of Spring 1 must be strong enough to compress Spring 3. The force that a compressed Spring 3 exerts when it is fully compressed by one unit is F3 = k3x (x = one unit). This is counterbalanced by the force that Spring 1 exerts when it is expanded, F1 = k1(N-x). Note that "expanded" does not mean that Spring 1 is now loose. It must maintain tension against Spring 3. Fully expanded would mean x = N, but N > 1, which means that Spring 1 will never be fully expanded, and therefore will always maintain tension. So clearly F1 must be greater than F3 at x=1, and so (N-1)k1 must be greater than k3.

Richard's other idea was to eliminate Spring 1 (and all its concerns) by replacing the linear cam track with a guided cam track. That is, the roller on the plunger slides in a groove and so does not require anything to push it against the track. I'll probably adopt this idea as well after some more doodling.