Trig Problem: Please Help!

[John_Lascelles]

Hello,

This is a real world problem that I am struggling to solve. As you can see from the image below I am trying to find the distance shown as a "?". I have spent hours trying to use the sine, cosine and Pythagoras but I just can't see it!

The far right point is actually a pivot and the two known lines on the left are the change in length of a gas strut.

Any help would be much appreciated at this point as it is giving me a headache.

Many thanks,

 

[hoosie]

 

[John_Lascelles]

Great work! However the angle you have as 108.37 deg is incorrect. I have attached an image to explain.

 

[Cubist]

I added some labels to your image...



I don't think this can be solved algebraically. However numerically I found the solution q=365.43631

My strategy...
1. Find "h" from constants c,a,b
2. Find "f" in terms of "q", and constant a
3. Find the gradient of line KO in terms of "q", and constants a,e
4. Find the x coord of point K - there could be two options (intersection of a circle and line KO. Circle centre J radius d)
5. Find "g", in terms of results from steps 3 & 4

THEN there is the requirement that "g" should equal "f". This can be solved numerically by finding a value of "q" that gives "g-f" close to zero.

My equations...
1. h = sqrt(c^2 - (a+b)^2)
2. f = sqrt(q^2 + a^2)
3. m = -tan(e + atan(a/q))
4. Two possible solutions x = (-(q+h) - m*b ± (sqrt((-m*b - (q+h))^2 - (1 + m^2)*((q+h)^2 + b^2 - d^2))) )/(1 + m^2)
5. g = -x*sqrt(1 + m^2)


Why can't "q" be found algebraically? I tried combining the above together and the result I obtained was:-
g-f = -abs(cos(e + atan(a/q)))*(-q - sqrt(c^2 - (a + b)^2) + tan(e + atan(a/q))*b ±(sqrt((q + sqrt(c^2 - (a + b)^2) - tan(e + atan(a/q))*b)^2 - ((q + sqrt(c^2 - (a + b)^2))^2 + b^2 - d^2)/cos(e + atan(a/q))^2))) - sqrt(q^2 + a^2)

I don't think this can be rearranged to "q=<xxx>" form! (after letting "g-f=0"). In fact I'd recommend NOT combining the steps together, it is far clearer left as a set of separate equations.


FYI: For the solution q=365.43631 then f=370.4105; gradient of KO =-1.040620; x coord of point K= -256.6552; g=370.4105. Only the +ve option of the ± at step 4 gives a solution with your set of constants.

 

[hoosie]

 

[hoosie]

 

[hoosie]

 

[hoosie]

 

[Cubist]

View attachment 14932View attachment 14933View attachment 14934

I feel bad to point this out, because you did a lot of work, but I did an approximate to-scale drawing of your solution and it doesn't join up. See below:-


The 510 arrow should reach the end of the black line that it points towards. There is quite a large gap, but obviously this could be caused by just a single mistake somewhere.

 

[hoosie]