# How to calculate the coupler length in a crank rocker mechanism

I'm trying to learn a bit about building kinematic mechanisms by building a small project. The goal is to be able to open and close a box programmatically via a servo motor.

I have a small box that I've reproduced in fusion 360 and I'm trying to design a crank rocker mechanism to open and close the box. After reading up on the crank rocker concept (I have no background in engineering so I'm brand new to all of these concepts) and building a couple of free standing proof of concepts (I put them up on thingiverse) I tried to start fitting the linkages into the box model.

The issue I'm running into is that, given the size constraints of the box and the length of the servo's horn, I'm not sure how to calculate the correct coupler link length.

I know that according to grashof's law I need to make sure that the sum of the shortest and longest links are less than the sum of the other two links; $s + l < q + p$.

I started by modeling the rocker arm ($q$) first so that I could figure out the length of the sliding slot (not sure what to call this) I would have to add to the lid so that the arm could actually open the lid. I placed the points for the rocker arm on the box and the lid so that there was room for the rocker arm to move without hitting the back of the box when the lid was fully opened, so it was kind of an arbitrary placement choice.

I already know the length of my default servo horn so I was going to stick with that (though I can 3d print a new horn if necessary).

I figured that between:

• The positioning of the servo and the bottom of the rocker arm (making the grounded link $p = 55.228mm$
• The length of the rocker arm $q = 60mm$
• The length of the crank arm $s = 25.53 mm$

I could just calculate the length of the coupler arm ($l$).

$$25.53 + x < 55.228 + 60$$

The part that I'm not sure how to figure out the correct X value. It seems like, if I understand all of this correctly, I have a range of values I can pick from and the top of that range is $89.698 mm$. I know that the bottom of somewhere around $68.081 mm$ (I don't actually know the proper way to calculate this, I just closed the lid on my model and measured the distance between the end of the servo horn and the top of the rocker arm :| ).

I know I can figure it out by noodling around with the coupler arm length until I find one that will let me fully extend the lid, but I'm trying to use this all as a learning exercise/introduction into these concepts.

• Is there a way of determining the ideal coupler length?
• Am I even right on the way that I've approached it so far?
• Am I missing anything super obvious?

Thanks in advance for the help!

• As a design engineer, and not a mathematician, I'd use software to work this out for me... You may find this app useful (best on a tablet, or an android emulator on a computer) stonybrook.edu/commcms/motiongen There are good tutorial videos on the site showing how it can be used for synthesis. Let me know if you have any trouble! Hopefully another user can provide the 'pure' answer... Aug 28 '18 at 16:21
• Oh fantastic! I'm going to dig into this tonight! Ultimately I'm def ok with letting the software tell me if that's the best way of doing it, I just want to try to understand the underlying concepts too. Thanks for the link! Aug 28 '18 at 19:24
• There are also graphical (pen/paper) ways to work out linkages from fixed poses, which I remember learning at university, but in my line of work, time is the most expensive resource so computers take all of the work like this... Aug 28 '18 at 20:01