Hello there,

Here is a math problem I thought would be really easy. It turns out it's not that easy, and even though I have not solved it yet I believe there is a solution.
You'll have to refer the diagram to understand it most likely.

The brace is a brace for a table leg. It connects to the top of the table (horizontal line is the top of the table) and to the leg on the left (vertical line). This of course creates a right triangle that is easy to calculate.
Because the top side is 4.5 and the left side is 6 units, the hypotenuse C1 is 7.5 units exactly (stroke of pure luck it came out like that).

The question is, what is the point of intersection (circled in green) when we only reduce the brace by 1 percent?
Note that I had only shown the left side (leg) rotate (red line) about the original 90 degree angle pivot point (upper left corner) but it would seem that the hypotenuse also rotates (not sure about that yet though). As the hypotenuse C1 gets shorter by an amount we can call dC1, it becomes C2 (shorter now) and the 6 unit side rotates and becomes shorter also.

I'm really after the angle that the 90 degree angle changes to, but the other information would allow us to double check any results. This would be the length of the 6 unit side after rotation. The length of the hypotenuse is known that will be 7.5 times 0.99 for a 1 percent decrease in length. If preferred, change the length of the hypotenuse to a more convenient amount like 10 percent shorter or something else.

What I am after here is a fix for a table that wobbles back and forth as the brace becomes shorter. It becomes shorter because of the somewhat poor design. It's a folding table. I'd like to solve this so I can calculate how much I need to decrease the length from changing in order to reduce the angle of wobble to some specified value (like from now 90 to maybe 85 degrees with the wobble, to maybe 89.9 degrees or even 89.5 maybe. There will always be some wobble but I'd like to reduce it.

Note: It could be that we have to prevent C2 from rotating from it's original angle to the horizontal top. I know for a fact that the angle of the 6 unit side does change though. With the rotation C2 there might be a lot of solutions perhaps.

 

As the hypotenuse C1 gets shorter by an amount we can call dC1, it becomes C2 (shorter now) and the 6 unit side rotates and becomes shorter also.

Hi, why would the 6 unit side become shorter also? It doesn't need to. With a variable length leg the problem becomes underconstrained.

A beer coaster might fix the wobble.

 

It becomes shorter because of the somewhat poor design.

Can't you modify the design, e.g. by adding a screw or clamp or whatever, to stop the effective brace length from changing?

 

The angle between the brace and the short side is 53.13°. Does this angle change when the brace shortens? - your diagram implies that it doesn't. If so, you can split the triangle into to back-to-back right angle triangles. The larger has a short side of (7.5*0.99*cos(53.13)) which is 4.455 and a long side of 5.94 (actually the triangle is just scaled by 0.99).
So now you have a very thin triangle with short side of 4.5-4.455 = 0.045 and long side of 5.94.
Your answer is arctan(5.94/0.045) which is 89.57°.

If the angle DOES change as the brace shortens, which parts remain the same?

 

Hi, why would the 6 unit side become shorter also? It doesn't need to. With a variable length leg the problem becomes underconstrained.

A beer coaster might fix the wobble.

Hi,

Well if you look at the circled dot, it moves up the angled brace and that means that the red line is shorter than the vertical black line. That's if it was the simplest case.

 

Can't you modify the design, e.g. by adding a screw or clamp or whatever, to stop the effective brace length from changing?

Hi,

Well thanks for the idea, but the goal was not only to fix it but to calculate the way it changes as it wobbles.
I think it is possible that if we let the upper right corner angle vary, then there may be a LOT of solutions. What I will have to do is look at it more carefully in order to try to determine if that angle really changes or not. That would make a big difference.

 

The angle between the brace and the short side is 53.13°. Does this angle change when the brace shortens? - your diagram implies that it doesn't. If so, you can split the triangle into to back-to-back right angle triangles. The larger has a short side of (7.5*0.99*cos(53.13)) which is 4.455 and a long side of 5.94 (actually the triangle is just scaled by 0.99).
So now you have a very thin triangle with short side of 4.5-4.455 = 0.045 and long side of 5.94.
Your answer is arctan(5.94/0.045) which is 89.57°.

If the angle DOES change as the brace shortens, which parts remain the same?

Hi there,

Oh yes ok that's very good. That's if the upper right corner angle does not change. And you're right, it's just something like two congruent triangles so not too hard to figure out.
What has me wondering is if the upper right hand corner angle also changes. I may have to inspect it more carefully to determine that. That seems to require more variables though so it might not have a single solution. Then the thing that bothers me is if it does not have a single solution, how can it have a single solution in real life.

There is another way of looking at it too that might help. The view would be where the starting angle was like a text forward slash, and as the table top moves to the right, the angle gets closer to 90 degees.

What I might do is take a picture of it and post it here. That might be the only way to figure out if we can assume the other angle does not change. The strange thing is, if it does change, then that is allowing the end with the dot to slide UP the vertical line (leg) as the upper left hand corner angle changes (in the diagram as is), or could it slide down along that side.

Next I'll take a real-life photo(s) of it and see what comes of it. I didn't think of doing that before this.

 

Photos would certainly help, if you're trying to solve a real-world problem rather than an academic one.

 

Photos would certainly help, if you're trying to solve a real-world problem rather than an academic one.

Hi,

Both really.

Here is a picture of the brace and how it connects to the table and the leg on the left. The brace forms a triangle.

I was able to determine that the upper right-hand corner of that triangle does in fact rotate. The lower left-hand corner of that triangle just follows that groove up.
I just checked this out again a little while ago and took the pic too so this will have to be thought about for a while.

A funny but not too practical solution (not mathematical though) is to place a folding chair up against the left side so that the back of the chair touches the table top. That makes it wobble a lot less, but of course that's not a math solution and not really a real-life solution that I would want to implement every time I use the table. I know that welding would solve this, but I REALLY don't want to do that and I won't do that.

 

don't see any picture of the brace

 

don't see any picture of the brace

Geeze did I forget to upload it?

Here it is...

 

If that were my table I'd extend the slot at the lower end of the brace by a few mm. There's plenty of meat left in the brace. That would enble the leg to splay out slightly and resist wobbling.

 

If that were my table I'd extend the slot at the lower end of the brace by a few mm. There's plenty of meat left in the brace. That would enble the leg to splay out slightly and resist wobbling.

Hi,

Well that would mean replacing four of those braces. Not sure I want to do that, although that's an interesting idea.

Here is a drawing showing it the opposite way which might make everything simpler.

[getting 6 tubes of super glue out of cabinet] haha
I would glue it if I never wanted to fold it up ever again. Epoxy.

 

Well that would mean replacing four of those braces.

Modify, not replace.

 

Modify, not replace.

How do you modify a steel brace to make it longer when it has a groove along the entire length? It needs that for folding up. It's thin steel but decently heavy gauge.
I'm open to any ideas though.

 

Like this:

 

it has a groove along the entire length?

Not quite. See post 16 for what I had in mind. I did say a few mm, not 1m !

 

Like this:View attachment 353829

Not sure what you are trying to say. You show a red dot, that's good, but no explanation of what that represents.

 

Not quite. See post 16 for what I had in mind. I did say a few mm, not 1m !

Yes but how do you extend a grooved piece of metal without cutting and welding.

 

Give it a few light passes with a rat tail file. If the front and back leg from your photo move together you need to be precise about it such that the pins on the legs reach the end of both slots at the same angle.