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I want to build a table which lays flat, or can be raised to a specific height.

I found a mechanism that does this:

enter image description here

I've tried to build a cardboard model, but it was very difficult to find the correct radius, as well as the positioning of the fixation bolts which would be in equilibrium at two different heights.

I've tried to sketch this: enter image description here

But what are the inherent equations? I've tried to build a cardboard model to simulate the above, in order to customize height and so on.

The model depends on:

  • the desired equilibrium heights, e.g. y=H1 and y=H2
  • the radius of the two beams
  • the positioning of the bottom beam fixation (Xa1, Ya1) and (Xb1, Yb1)
  • the positioning of the upper beam fixation (Xa2, Ya2) and (Xb2, Yb2)

The constraints are:

  • for the table to be flat, the difference in height of the upper fixation should be: Yb2-Ya2 = Yb3-Ya3 = k at the two equilibrium heights

Is there an engineering/mechanism software for quickly doing these kinds of simulations?

Or did the engineers behind this mechanism use math?

Any simple/quick ideas on how I might set up the equations to calculate this with constraints? Perhaps an intuitive approach that can be done with cardboard and lines, while adhering to some ratio or likewise.

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    Could you please clarify exactly what variables you are interested in, preferably with them drawn into your figure? – 2017-02-10
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    I've tried to add something. I'm not quite sure how to properly describe the problem. Is there something I can read about this type of mechanisms and the geometry involved? – 2017-02-10
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    Much better! As a last thing to improve readability: Use MathJax (see here http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference) – 2017-02-10
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    Computer-aided mechanical engineering came into vogue decades ago, so I think it very likely the engineers used software that was able to simulate the raising and lowering of the table. Depending on other design criteria, they might or might not also have done some additional math outside the simulation. But the basic math to get the dimensions is not necessarily so laborious that it would require a computer. – 2017-02-10
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    Question: Does the table surface (A) stay level as it moves between the two "equilibrium" positions, or (B) tilt while moving between those positions? Answer A would indicate the two beams are the same length (or at least the parts between pivot points have the same length), while answer B would indicate they have different lengths. – 2017-02-10
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    @DavidK, I tested with the cardboard model, and by having the beams at same length and same placement on the y-axis causes the table to stay level and greatly simplifies the problem, to the point where height is entirely determined by beam length. I think the reason for dissimilarity is that the inherent circle differences would prevent movement beyond a certain point, thus causing the beams to be structural. Do you have any insights regarding this? What are the benefits of dissimilar beam size/position? – 2017-02-10
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    I found a video of the mechanism: https://www.youtube.com/watch?v=wCpm5zBwYAc – 2017-02-10
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    In some applications it might be necessary to tilt a flat surface while it was moving in order to allow it to pass between obstructions. In the video there is no need for that, but you would want the table to stay as level as possible so that your computer, coffee cup, and lunch don't slide off. So I would bet on equal-length beams. – 2017-02-10

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Maybe something like this:

enter image description here

On this picture $|A_1A_2| = |B_1B_2| = r$ and the whole time $A_1A_2$ is parallel to $B_1B_2$. You fix the points $A_1 = (x_1,0)$ and $B_1 = (0,y_2)$. To make sure that the corner $K$ does not bump into corner $N$ you have to look at the fictitious triangles $K'A_1A_2$ and $KA_2B_2$ which are congruent and in fact translates of each other. Then $|K'N| \leq |K'K|$. All points from the moving part of the mechanism follow circular arcs of radius $r$ as trajectories, each arc centered at a corresponding point: $K$ follows an arc of radius $r$ centered at $K'$, point $A_2$ follows an arc of radius $r$ centered at $A_1$, point $B_2$ follows an arc of radius $r$ centered at $B_1$, point $Q$ follows an arc of radius $r$ centered at the origin $O$ of the coordinate system.

Does it make sense?