Linkage analysis of forces and stresses?

[jai_helsing]

The attached image shows a linkage mechanism with 2 parallel bars pivoted to a fixed base. The lengths of the the bars are equal. A connecting linkage and gears are used to connect it to a 3rd bar. I would like to know how to analyze this linkage system to find the forces, moments and loads for particular lengths.I thought approach this problem by an assumption of 50lb load on one end and going from there. I am not sure if that's the right approach. Could anyone please guide me to understand how to go about analyzing the dynamics of this linkage system.
Thanks a lot for your time and support.

 

[jrmichler]

Start by writing down on your diagram all known information. Show the extremes of motion, any external loads or forces, desired speeds and accelerations. If you are at the conceptual stage, you can estimate the weight of the parts.

...analyzing the dynamics of this linkage system.

After you have defined the motion (displacement, velocity, acceleration) of anyone part, you can calculate the motion of the remaining parts. Total forces are the sum of acceleration forces, gravity, and external forces.

When you have all of the forces, you can calculate stresses and bearing loads. Iterate as necessary until you have a complete, fully analyzed design.

 

[Baluncore]

Could anyone please guide me to understand how to go about analyzing the dynamics of this linkage system.

The term “dynamics” suggests you are concerned with mass and velocity of the load and links. But it seems you need to first analyse the “static” forces on the links in different positions resulting from load and gravity. Are you assuming the links have no mass and the pivots are perfect?

I would define pivot #1 to be at the origin of a coordinate system. I would define a variable, angle theta, that sweeps the primary red link through up to 180°. For any theta, I could compute the position of all linkages using vector geometry. Then working backwards from the load to the origin, the horizontal and vertical components of the force in each link can be resolved.

 

[Dr.D]

It looks like the red bar and the grey bar are fixed at right angles; is this true? It is not clear in the diagram, and it makes a lot of difference how/where things are connected.

 

[Baluncore]

I think pivot #1 is shown incorrectly in the diagrams.

This appears to be an attempt to design a linkage that can remain balanced in any static position.
Similar to an Anglepoise lamp. https://en.wikipedia.org/wiki/Anglepoise_lamp

For conservation of energy reasons, it will be unable to pull itself up by it's bootstraps.

 

[jai_helsing]

Thank you all for your replies. Apologies if the diagram wasn't clear to understand what I am looking for.
@jrmichler Yes the whole system is at a conceptual stage. The only know quantities are the weights of all the parts. You suggested me to define the motion of one part and hence calculate the motion of remaining parts. Can you please help me understand that better. How to go about it. I want this linkage arms to move to from lower position to open position (around 90 degree) in 20 seconds. That is a considerably slow opening. The mass of the whole system is around 70lbs.
@Baluncore Pivot point 1 is not the hole seen on the base. Its a actually not clearly seen because it a pin on the inside of the red bar. You have suggested me to do a static analysis of the system resulting from load and gravity. Can you please guide me how to go about it?

 

[jai_helsing]

@Dr.D No they are not fixed at the right angle. They bars are parallel and are intended to be moved from the fully extended position to the lower position as shown in the photos. That is the only range of motion I am looking to achieve with this mechanism.This whole thing can be assumed as an arm with an shoulder(pivot 1 and 2) with a forearm (the red and green bars), the gears (act as elbow) with a forearm (light green bar). Hope this gives you a better clarity of the mechanism. Could you please guide me to understand forces in the system.

 

[jrmichler]

If your linkage will take 20 seconds to move 90 degrees, then you can ignore dynamic effects. It is then a static problem.

The analysis procedure is as follows:
1) Start with the linkage in one of the extreme positions. Flip a coin to decide which extreme.
2) Create a free body diagram of the outermost piece (the piece with the 50 lb force on it). If the gear is rigidly attached to the outer piece, then it is part of the outer piece. Remember that an FBD includes the weight of the part.
3) Use the forces from Step 2 for free body diagrams of the parts that control the outer piece (the other gear and the link). This will be two FBD's.
4) Use the forces from Step 3 for free body diagrams ...
5) Repeat until you have a free body diagram for every part in your system.
6) Now go back, put the linkage in the other extreme position, and repeat Steps 2 through 5.
7) Decide if you need to analyze intermediate positions.
8) You now have multiple FBD's for each part. For each part, select the FBD with the highest forces. Use that FBD to analyze that parts. Do this for all parts.

Note that your analysis is only as good as your assumed external force(s).

 

[Dr.D]

I would suggest a procedure similar to what JRMICHNER has outlined, but I'd analysis the system once at an intermediate position with the shoulder angle as a parameter. Then I'd write a computer code to sweep through the shoulder angle values, computing forces at each position.

 

[Baluncore]

Then I'd write a computer code to sweep through the shoulder angle values, computing forces at each position.

I have a generalised linkage analyser that I wrote for solving problems with the geometry and hydraulic pressures in earthmoving equipment.
It would simplify the analysis here if we knew which links remained parallel and which did not.
I can't really help here without the coordinates of the invisible base pivot points and the initial lengths of all the linkages.

 

[jai_helsing]

Arm1 and arm 2 are parallel. the lengths are specified in the image. This is all the information I knew put in. Could you please guide me from here.
@Dr.D @jrmichler

 

[Baluncore]

@ jai_helsing.
Do you understand complex numbers and how they can represent vectors?
Do you understand vector addition and subtraction?
Can you say what happens when you multiply two vectors?

 

[jai_helsing]

I do have a basic understanding. But not a very in depth knowledge of it.

 

[Baluncore]

We now know the distance between the base pivot points for the parallel arms is the same as link1 length = 62mm.
But we do not know their relative orientation, horizontal or vertical offset on the base.

 

[Dr.D]

@ jai_helsing.
Do you understand complex numbers and how they can represent vectors?
Do you understand vector addition and subtraction?
Can you say what happens when you multiply two vectors?

I would never advise bringing complex numbers into a simple kinematics problem; there is simply no need. It makes part of the problem simply unreal, when the actual mechanism is totally real.

 

[Baluncore]

It makes part of the problem simply unreal, when the actual mechanism is totally real.

I'm sorry. I found that vectors eliminated the errors made by engineers in their spreadsheets, where the profusion of incorrect signs of trig functions make a mockery of the result.

 

[Dr.D]

I'm sorry. I found that vectors eliminated the errors made by engineers in their spreadsheets, where the profusion of incorrect signs of trig functions make a mockery of the result.

If I believed that I'd be very worried.

 

[Tom.G]

If I believed that I'd be very worried.

Even the astronauts on the Space Shuttle had one or two failed experiments due to the wrong sign in a calculation. The one I participated in was a plasma experiment to generate and investigate the propagation of radio waves in the ionosphere as influenced by the Earths magnetic field. A sign was wrong in the experimental protocol. This caused the plasma jet that was to be aimed toward the Earth being sent into space instead. There were an awful lot of disappointed experimenters, both paid and volunteers, scattered around the world.

IIRC, there was also a planetary probe that was lost during its landing sequence from the same problem.

Oh well, Sh_t happens.

 

[Tom.G]

Back to the subject on hand.
@jai_helsing It looks like the gears are an extra 'feature.' They are not needed to rotate the 50lb. load, they can be replaced with only a lever.

Also If there is room, the Green bar could be moved to be above the Red bar. This would put the Green bar in tension, thereby reducing the cross section area needed for the Green bar.

Cheers,
Tom

 

[Baluncore]

If I believed that I'd be very worried.

I am concerned that a 2D vector technique used for the majority of accelerated PC graphics computations, might somehow be real in the X dimension, but unreal in the Y dimension.

I am going to watch while Dr.D guides the OP through this problem.

 

[Dr.D]

Even the astronauts on the Space Shuttle had one or two failed experiments due to the wrong sign in a calculation. The one I participated in was a plasma experiment to generate and investigate the propagation of radio waves in the ionosphere as influenced by the Earths magnetic field. A sign was wrong in the experimental protocol. This caused the plasma jet that was to be aimed toward the Earth being sent into space instead. There were an awful lot of disappointed experimenters, both paid and volunteers, scattered around the world.

IIRC, there was also a planetary probe that was lost during its landing sequence from the same problem.

Oh well, Sh_t happens.

@Tom.G So, what is your point? Are you suggesting that this one experience proves that spread sheets used by engineers suffer from a profusion of sign errors, as alleged by Baluncore? Or, perhaps you are suggesting that the use of imaginary numbers might have prevented the error? Its really not clear how this relates, and your vulgarity does not improve clarity.

 

[Dr.D]

I am concerned that a 2D vector technique used for the majority of accelerated PC graphics computations, might somehow be real in the X dimension, but unreal in the Y dimension.

It may (or may not) come as a surprise to you to learn that the computationally fastest technique is not necessary the means to the greatest human understanding. I have taught kinematics and dynamics of machines many times at several different universities, and I found rather early on that most students understand operations with real numbers better than those that involve complex or imaginary numbers.

You speak of "vectors" as helping to eliminate errors, but are your referring to "complex number vectors" or to vectors that transform tensorially?