Solving 5 Unknowns with 6 Equations 1

[newreynolds]

Probably a stupid question but I am doing a static force analysis using a Method of Joints. There are 3 joints of interest, so 6 equations but I only need to solve for 5 unknowns.

The last unknown is in the last 2 equations so I solve one and then do a check with the final equation but it's never zero. My question is then, should that last equation be satisfied since I only really needed 5 equations?

I believe it should but after looking over my algebra and not finding an error, I'm questioning things.

 

[electricpete]

somehow I missed that ph1 and ph2 were not given so you must assume some values to complete the geometry

In effect, they are specified. We have theta1 and thet2 specified as well as L1, L2, L3, L4. This means phi1 and phi2 can be solved.

Starting at the bottom, we can traverse L1 and L2 or L4 and L3 to arrive at same point:
L1*exp(i*theta6)+L2*exp(i*<theta6+phi1>) = L4*exp(i*theta7)+L3*exp(i*<theta7+phi2>)

Taking the real and imaginary parts yields two scalar equations in 2 scalar unknowns phi1 and phi2

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(2B)+(2B)' ?

 

[electricpete]

If we could vary any of the lamda's or theta's, there might be a solution but..

.... but it would seem a contradition to the stated problem which was supposed to achieve target values of theta6 and theta7 (and by extension the associated phi1, phi2, psiL).

The problem must be malformed, moment equilibrium of the whole structure can't be satisfied with the spring fixed to the pivot. [./quote]
.... and if you changed the pivot type, you would have bending moments which completely change the approach.

And we don’t know for sure if the angle psiL happens to be the single exact one required one to satisfy our force balance (until we do the calc), but it would seem a very artificial situation where we analyse a geometry that happens to have exact right value of psiL (along with the fact that we are ingoring lack of moment balance which also would make it artificial).

I agree, the problem seems malformed to me.


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(2B)+(2B)' ?

 

[zekeman]

You are right,but there are 2 solution sets for ph1 and ph2. Using geometry:

The angles given together with the link lengths determine the location of pivot points at intersections of L1- L2 and L3-l4.
Now it is easy to see that the point L2-L3 is determined by swinging two arcs (with radii L2 and L3) from the pivot points noted above; so you get TWO intersections or two different L2-L3 points; you can pick one to complete the geometry and determine ph1 and ph2. And treat the spring as a solid link, since its stretched length is determined and the problem leaves you with no other choice.

If you want to speculate further on this, it could be that the author was mistaken about theta6 and theta 7 and all of the lambdas are correct.So instead of fixing lambda3, fix these thetas for rotational equilibrium.But why ..?

 

[electricpete]

Yes, I did realize there were 2 solutions. It didn’t seem relevant to the point I was driving to which was that the angle psiL is not a continuous variables that can be freely chosen to suit force balance, rather it is pre-defined by the problem statement. But good to clarify.

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(2B)+(2B)' ?

 

[FeX32]

Good point pete. I didn't realize at first you could solve for phi1 and phi2. This means what I said about the 2nd method is not the way. But, the first one is valid.




Fe

 

[Drexl]

Hi,

I did not look at the equations but generally when you have an overdetermined linear equation system you can use a least-squares method. Just write it in matrix form and solve it in matlab, or look up the matlab c=A\y and see what it does for overdetermined systems.

Without further thought what I would do with an non-linear system say Fn(x) = 0 where Fn represents a set of functions F1-FN and x is a vector of variables is to solve

min Y, where Y = F1(x)^2 + F2(x)^2 + .. + FN(x)^2

You can put in weighting factors above for specific use...

br
Drex

 

[rb1957]

in this particular problem i think equations 5 and 6 are inter-dependent ... i think they both solve for Ps given the loads in the other links.

however, the question in my mind is can you apply static equilibrium to points on the structure when it is not in moment equilibrium ? you can apply static equilibrium to the whole structure (i think) and determine the ground reactions; but i doubt the results of applying static equilibrium to say the RH load point (can you determine the loads in the two links given the applied loads ?)

 

[Cockroach]

Oh man, this is all wrong.

NewReynolds, you need to be factual and accurate in the forum and not post interpretations of problems as you have done in the start of this thread. Like some sort of exam problem, write the problem out accurately and give us the input information. You don't start by saying something like 5 equations in 6 unknowns, this presumes a ton of work which may be in error.

Okay, you got said linkage, four "To" means ropes applying tensile load to the system at 200 N each, lamba two thru five exclusively are 50, 55, 50 and -20 degrees. The linkage one thru four are 9, 5, 7 and 9 meters long. The question is to compute spring force required keeping theta 6 at 100 and theta 7 at 20 degrees.

Is this correct? This is the original problem?

Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada

 

[FeX32]

This is great. My last input is all "dynamic" .
I'd be happy to solve the model as given (in a dynamic fashion). But, I think this could expand this thread to oblivion.




Fe

 

[77JQX]

Since there's a moment about the fixed pin, no spring force applied to the fixed pin will be able to keep Theta6 and Theta7 to 100 and 20 degrees. If we were mathematicians instead of engineers we might have realized that when we reduce our equations down to something like Fs != Fs, then we have proven that the original premise (Theta6=100 and Theta7=20 in a static condition) is impossible. Thanks, newreynolds, for taking me back to school. Sometimes old dogs can relearn old tricks.

 

[FeX32]

Hence dynamic


Fe

 

[zekeman]

"This is great. My last input is all "dynamic" smile.
I'd be happy to solve the model as given (in a dynamic fashion). But, I think this could expand this thread to oblivion"

Fex32,

Oh, really. If you could do that accurately I would eat all these posts and post a picture of it.

I don't think that even Matlab could help you.
--------------------------------------

Drexl,

Get real, how do you solve a 5X6 linear set with Matlab?

 

[zekeman]

I have a proposal to handle the dilemma posed by the 5X6 linear set

Use the first 4 equations previously presented.

Instead of using eq 5 and 6, consider the following

Point L3-L4 (intersection of L3 and l4) is in equilibrium, so instead of using the T0 inputs at that point, use -F3 and -F4 as the external vector forces that put that point in equilibrium and now write the moment equation in vector form

eq5 T01xL1=-F3xL4-F3x4=-F3xL3

x is the cross product symbol (see Pete's explanation)

which is now eq 5

where T01 is the vector force of the rope at point L1-L2 and note that F3xL3=0Now you have the 5x5linear set which has a solution.
And if you have a valid problem , -F3 -F4 must be equal to the rope vector force at the point L3-L4 .

 

[FeX32]

Relax Zeke.
I specialize in dynamic analysis. This would not be a problem. Even not assuming the spring is a rod it is numerically possible. It also may be analytically possible using a powerful Udwadia analytic method....

"I don't think that even Matlab could help you."....
This honestly makes me laugh.

Also,
"Get real, how do you solve a 5X6 linear set with Matlab?"
This is possible within numerical tolerances....


Fe

 

[zekeman]

Fex32,

"specialize in dynamic analysis. This would not be a problem. Even not assuming the spring is a rod it is numerically possible. It also may be analytically possible using a powerful Udwadia analytic method...."

Powerful methods only depend on accurately modeling the components. That was my point; the dangling spring would be a huge problem and also where would you start the problem at t=0 in this case? he only gave you what he thought was the equilibrium position.

 

[FeX32]

Fair enough Zekeman. There would be some assumptions in the analysis. (IC's ect.)




Fe

 

[rb1957]

maybe now the OP understands why we need to see the problem, rather than be given a peice of it to solve.

the engineering solution to th eproblem is quite different to the mathematical approach, it is easy to solve the equations given; possibly there isn't a unique solution, but i'm sure there is a solution (mathematically). but from the eningeering viewpoint, i think we've seen that the basic formulation of the equations is in error.

on a different tack, what was the answer (ie the approach to derive Ps)

 

[electricpete]

Dynamic solution would require knowledge of mass per length of each bar and knowledge of the spring constant of the spring (and by the way, do we assume the bars are rigid = infinite spring constant) And I'm pretty sure it would never come to equilibrium without damping. If anyone can do it, I'm sure FeX32 can. I'm not sure exactly what this author had in mind, but I think we'd all agree it wasn't dynamic analysis.

A correction to my own comments - I stated the value of PsiL would not be correct for static equilibrium of the middle joint (balancing the forces F2 and F3 computed from left and right joints), but I never verified that

As a purely academic/programming excercize, attached I have now verified that the solution of PsiL will not be correct for static equilibrium of the middle joint..... i.e. the required line of action of spring force is not in the direction along the spring (all of this based on static analysis which ignores the lack of moment balance).

The result is shown in graphical form on page 6 of 6.
The bars are in blue.
Tension vectors are in green.
Applied force vectors are in red.
Spring force vector is in purple.

From the graphical solution, you can re-create the calculation for yourself:
Tension forces at left and right pin have to balance the force vectors.
Then spring force vector in the center pin has to balance the tension vectors.
The direction of the purple spring force vector is nowhere close to being toward the origin.

I apologize for prolonging a thread which some of you seeem impatient with (will it hit 100 posts?). As for me, it doesn't bother me how long it goes. Everyone has their own slant, and that's what makes it interesting.








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(2B)+(2B)' ?

 

[electricpete]

Forgot to hit insert - here's the attachment
(that brings us one lcoser to 100)

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(2B)+(2B)' ?

 

[electricpete]

Honestly, I'm not creating extra posts on purpose. I even used preview last time. Let me try to post the attachment one more time.

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(2B)+(2B)' ?