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]

I agree you can solve the left and right joints to find a tension in bars 1, 2, 3, 4.
Then when you try to find the required force of the spring to balance the force applied by bars 2 and 3 at the center joint, you will find you need a vector force. But the directio of that vector will not align with the pre-defined PsiL (it is pre-defined by virtue of specified values for theta6, theta7, L1, L2, L3, L4 are all specified).


=====================================
(2B)+(2B)' ?

 

[rb1957]

not all the links will be in tension; i think link4 is in compression.

the line of action of the spring is defined by the geometry. if and that's If) the problem is static then it Should work, the spring force should balance the upper links.

If it doesn't balance (ie one variable will satisfy only one equation) then it isn't in static equilibrium and the angles will change untill it does (as the geometry changes, the length of the spring changes, and so the load.; but note we not working with the spring stiffness).

 

[electricpete]

if and that's If) the problem is static then it Should work,

That's the point. The applied moments do not sum to zero. There is no restraining moment. Net moment is not zero. Static equilibrium is not satisfied.

Attached computes the moment associated with applied forces assuming T0=1. It is 1.97-54.6 which does not come out to zero.

=====================================
(2B)+(2B)' ?

 

[electricpete]

1.97-54.6 Should'be been 1.97-5.46

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(2B)+(2B)' ?

 

[FeX32]

pete beat me to it

...from your solution...then 1.97-5.46=I*alpha, where I is the mass moment of inertia of the entire mechanism about the pivot point.

Dynamic force analysis was one of my favourite topics in uni. .. and still is


Fe

 

[rb1957]

agreed, but the force in the spring is calculatable given the geometry, and i think the result of plotting the two load vectors and the ground reaction vector will show that the three forces don't intersect and so it isn't in balance ... it is in force equilibrium ('cause we've used sum Forces to determine the results) but fails moment equilibrium (as you've shown).

 

[electricpete]

You're right: force equilibrium and moment equilibrium are two different topics that perhaps I mixed up. Neither will be satisfied in the problem as specified.

=====================================
(2B)+(2B)' ?

 

[rb1957]

no, i believe you can satisfy force equilirbium (in that the ground is going to prevent the structure from translating), but as you've pointed out there is no moment equilibrium (so it'll want to rotate). still the problem can be answered, the load in the spring can be determined from the geometry.

 

[electricpete]

The problem with the force equilibrium results from specifying theta6, theta7, L1, L2, L3, L4, which in effect specifies psiL. The angle of psiL does not correspond to the angle of spring force required for static force equilibrium.

=====================================
(2B)+(2B)' ?

 

[rb1957]

that's a problem ! maybe that's a symptom of the moment imbalance, in that this joint (and the others neither ?) isn't in force balance ... 'cause if the links are rotating, the loading points will be translating.

i wonder if the problem also told you the unstrained length of the spring and it's stiffness, so you'd determine the spring load from F = k*x ?

 

[FeX32]

The problem asks to determine the spring force that keeps the structure at Theta6 = 100deg and Theta7 = 20deg

Jeeze. I neglected to read that. Haha.

I think given one thing, both of you are correct.
With giving the values for phi1 and phi2 (along with all the other angles you can get 4 equations and 4 unknowns which solve for F1,F2,F3,F4. With these you can solve for Fs.
(this is neglecting the non-zero moment, that whoever came up with the problem didn't really notice/ care about...)
For example, not in vectoral form the 2 equations for the left side can be simply: -To*cos(lam5)+To*cos(lam4)=F1*cos(90-theta6)+F2*cos(-theta6-phi1) and
-To*sin(lam5)+To*sin(lam4)=-F1*sin(90-theta6)-F2*sin(-theta6-phi1)
bascially we get 4 equations like the OP's equations 1,2,5,6, which can be solved for F1,F2,F3,F4. You then use F2 and F4 to easily solve for Fs.
But, if you don't have phi1 and phi2 then you need 2 more equations. agree?

So, if that's the case then we can do the following. Determine the 6 equations similar to the OP's (if not the same as I haven't checked all of them). Assume that phi2=180-phi1 and we have gotten rid of one of the 7 variables (F1 to F4, Fs, phi1, phi2), and we now have 6 equations and 6 unknowns with 1 unique solution.
Done deal.




Fe

 

[FeX32]

In real life I would do a dynamic analysis however....


Fe

 

[electricpete]

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 (I was assuming based on my previous calcs using ficticious angles... never tried the real PsiL).

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(2B)+(2B)' ?

 

[newreynolds]

The problem must be malformed, moment equilibrium of the whole structure can't be satisfied with the spring fixed to the pivot.

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

All of the theta's and the phi's are related by the linkage closure equations. I guess I was the one that over-thought the problem.

 

[zekeman]

I think somebody ought to take the bold step of MAKING lambda3=29.665 deg ( correct me if I'm wrong) forcing the contraption into equilibrium so one of us can mercifully put this thing to bed with a single unique solution.

 

[FeX32]

I concur


Fe

 

[zekeman]

"The problem must be malformed, moment equilibrium of the whole structure can't be satisfied with the spring fixed to the pivot.'

The spring is no problem; you make it a solid link and fix lamda 3 as I suggested. Now you have a system in equilibrium and each of its parts are in equilibrium.

 

[newreynolds]

Well, I assumed that the loads wouldn't change and that it was the linkage that needed to be adjusted to go into equilibrium.

 

[rb1957]

@ the OP, since you wanted to solve this problem,

you have a solution for the link loads, based on force equilibirum at the loaded points. you could solve these "old school" using a force polygon (= triangle) diagram.

you said you couldn't find the spring force; but i think you can from force balance of the upper spring attmt point.

is this the complete problem ? were you told anything else about the spring ?

 

[zekeman]

Oh, somehow I missed that ph1 and ph2 were not given so you must assume some values to complete the geometry and use a solid link for the spring.