Extending arm with counterweight balance question

[Whymars]

Hi guys, I'm working on a prototype for a lamp and have quickly overreached my meagre school boy mechanics skills, almost at the first hurdle, in fact. Embarrassing, I know, but hopefully some kind soul here will be able to point me in the right direction.

A picture is worth a thousand words, so:

825mm long arm - horizontal

700mm long arm (125mm less) - tilting slightly up

635mm long arm (65mm less) - very up

590mm long arm (45mm less) - almost vertical

The structure is locked at each of these lengths. The fulcrum is always 4/5ths of the way along the arm, the long side is always four times longer than the short side. The total weight of each side is constant

Now I have already exhibited my ignorance, but I will compound it thus: What gives? I thought if this thing balances horizontally it's because the forces pulling down on either side are equal, and since extending or retracting the arm affects both sides in equal proportions, then doesn't that it should stay balanced regardless of the lengths of the arms?

Ok, I've a few ideas, and would love some guidance on which is wrong(est).
1. The counterweight hangs from a pin through the final link - it isn't inline with the arm and that makes that lever bent (does it?) and that makes for a less simple problem.
2. The weight of the long side is evenly distributed over the length of that side, but the weight of the short side is right at the end. Does this make any difference?
3. The amount of "play" that is in the joints of this structure means that the short side tends to be "stretched" by the weight and ends up slightly longer than exactly one fifth of the total length. This becomes more pronounced when the arm is shorter, and that unbalances it. This doesn't make sense for when it is already horizontal though, because the weight is not yet stretching the short side.

My working conclusion is that to stay balanced, the short side needs to retract towards the fulcrum slightly more rapidly than the long side. This could be achieved by making the last struts of the structure slightly shorter. However, if this is a solution, I don't want to just rely on trial and error.

Any insight into this would be very very welcome, and any ideas or pointers towards a working out a mechanical solution would be brilliant!

sn

 

[Lok]

This is a hopeless trial. There is no ultimate solution as gravity will unbalance equal weights just by one being more closely to the Earth than the other.

Clearly not the case here, as manufacturing tolerances are at play. Best if you use friction to hold things in their place, it's not a lot needed just a bit.

 

[Whymars]

Thanks Lok, but if the beam is horizontal, isn't gravity acting equally on both sides? I don't want to be able to pose it at particular angles, just to try and understand what makes it tip from horizontal when it gets shorter.

I've resisted using friction in the design so far because [it seems like cheating and] I really wanted moving the lamp to be extremely light - floaty - in motion. But you're right in that only a tiny amount of friction would do the trick, probably a spring washer.

sn

 

[Q_Goest]

Hi Whymars, welcome to the board. To get the arms to balance, they need to be in "static equilibrium". For a relatively simple system like this, that means the moments on the arm must add up to zero. The term "moments" means torque in this case. There's a torque produced by the long arm around the pivot acting in a counterclockwise direction (from the photo) and there's a torque produced by the short arm that has a weight attached that acts in the opposite direction, or clockwise direction as seen in the photo.

If these two moments (torques) are equal, the system will balance.

To determine the moment of the long arm producing a counterclockwise moment, simply multiply the weight of the long arm times the horizontal distance from the pivot to the center of gravity of the arm. For example, assuming the four sets of links are identical, the center of gravity is between the second and third links. Multiply the distance from the pivot to this point on the links by the weight of these four links. Note that the distance is the horizontal distance, not the actual distance from the pivot.

Do the same thing for the opposite side. In this case, you can break up the short arm into two components, the arms and the weight. Do each separately. You should have an equation that looks something like this:
w_L*r_L = w_S*r_S + w_E*r_E
Where
w_L is the weight of the arm on the Long side
r_L is the horizontal distance from the pivot to the CG of the Long arm
w_S is the weight of the arm on the Short side
r_S is the horizontal distance from the pivot to the CG of the Short arm
w_E is the weight of the Extra balance weight
r_E is the horizontal distance from the pivot to the CG of the extra balance weight

Now you say that the short arm stretches as it tilts up. This means that the distance r_E doesn't decrease as fast (in proportion) as the Long arm. From the equation above, you can then see why the system becomes unbalanced as this happens. If r_E doesn't decrease at the same rate as the long arm, the right side of the equation gets bigger than the left side, and the additional moment tends to make it rotate in that direction.

So if you wanted to make sure it stays balanced, one thing you might consider is moving the location of the weight. Imagine the arms in the horizontal position. Now imagine moving the pivot where the weight is attached - down along the Short arm, towards the point where there's another pivot that looks like the bottom of a V. I suspect you'll find that somewhere along that arm is a spot that helps to balance things better even though the Short side starts to stretch. You can do this by trial and error, or if you can determine how much the arms 'stretch' you could do it mathematically using the above equation. The reason this might help is because the horizontal distance r_E gets shorter faster as the whole arm is rotated. The horizontal distance is a function of the geometry, so if you know a bit of trig, you could put in the correct cosine function for r_E and determine how it shrinks as a function of arm rotation. If you can't figure that out, I'll show you how.

I suspect moving the location of the weight like this will help some, but it won't do everything. It won't get rid of the problem when it goes straight up, and you may still need some additional friction. To do it right, the arms must be very rigid, so you'll need to get the slop out of it. Then you can keep the weights where they are. Making the short arms shrink slightly as they rotate as you've suggested might also help.

 

[Whymars]

Thanks very much Q_Goest, that's outstandingly helpful - I've already trialled the counterweight in a different position (where there happened to already be some holes, fortunately), with enlightening results!

New placement (was through the metal pivot hole previously)

Fully extended and horizontal

(almost) fully retracted and still horizontal!

But with the short arm "stretched", up it goes!

I had to add a few more nails to the counterweight gondola to make it balance, but otherwise: Ace. The arm stays balanced as long as the short side is truly in proportion to the long side. As soon as the short side gets stretched because of the slop in the joints, it quickly tilts the whole arm.

Which made me think that the slop is the real villain here, and sure enough, with the counterweight in the old position (right at the end), and the framework screwed tight so that no stretch can occur, then it balances regardless of extension. This means it works upright too. Because the counterweight hangs, it's centre of mass will always be the same horizontal distance from the fulcrum as the actual thing it hangs from (the end joint), so no adjustment in it's position should be necessary.

I think Lok was probably right in suggesting this was a hopeless trial because of this sloppy prototype. Practically, I don't think it's feasible to make this arm stay balanced if I am relying on the scissor mechanism to keep both sides in proportion: If the joints are tight enough to eliminate slop, they'll be too tight to have the "free and easy" motion I'm looking for, not without investing in proper bearings or something like that. Judicial use of spring washers is perhaps the key. And metal bushings in the holes to prevent too much wear.

There are a few more problems to overcome for extra things that will be involved in this design, and I suspect the guidance you've given me (equations et al) will serve me very well to take that forward.

Thanks again for such a helpful answer Q_Goest, very much appreciated indeed!

sn

 

[nvn]

Whymars: Are you sure all of your linkages have identical dimensions? It almost sounds like your five mechanism segments are extending different amounts even in the horizontal position. Does the imbalanced behavior, in horizontal positions, depend on whether you pushed the mechanism to shorten it to a certain length, or pulled the mechanism to lengthen it to that length?

 

[Phrak]

The obtain balance throughout the extension, the counter weight should hang from the center of mass of the short arm, rather than the end. The weight of the counterbalance would have to double.

To obtain equilibrium you'd want to place the balancing pivot higher than the center of mass. Of course, you'll get this by moving the counterbalance to the bottom hing of the short arm, anyway.

 

[Whymars]

Whymars: Are you sure all of your linkages have identical dimensions? It almost sounds like your five mechanism segments are extending different amounts even in the horizontal position. Does the imbalanced behavior, in horizontal positions, depend on whether you pushed the mechanism to shorten it to a certain length, or pulled the mechanism to lengthen it to that length?

Hi nvn - Each segment is cut from identical plans, but only by hand so there is some inaccuracy. The main problems with this model is that the holes have been filed out slightly too large for the bolts, and the hardboard is fairly thin so the bolts aren't inclined to stay very perpendicular unless done up tight.

In it's horizontal position, it does matter whether it's pushed or pulled, particularly if one end is screwed up tighter than the other. If you look on the last picture in my last post, you can see that the top-most fifth is much more compressed than the bottom one.

cheers!
sn

 

[Whymars]

The obtain balance throughout the extension, the counter weight should hang from the center of mass of the short arm, rather than the end. The weight of the counterbalance would have to double.

I did think that would be ideal, but then couldn't decide if there was any geometric difference in having the cw at the end, and it's certainly preferable to have a lighter weight if possible.

To obtain equilibrium you'd want to place the balancing pivot higher than the center of mass. Of course, you'll get this by moving the counterbalance to the bottom hing of the short arm, anyway.

I guess this is why it balances a little more stably with the cw lower down, I hadn't really thought about that. I'm resigning myself to the idea that absolutely stable equilibrium isn't practical with this design, and given that in it's final incarnation I want it to be movable, from horizontal to vertical, if there's any bias in it, I'd maybe prefer it to be stable upright rather than horizontal. But.. not quite sure about that stuff yet!

thanks!
sn

 

[Phrak]

OK. Alternatively you could add an extra diamond on the short end.

 

[nvn]

Sorry, but both statements in post 7 are incorrect. See the equation by Q_Goest in post 4 to understand what causes a moment balance in this problem.

Whymars: I see your point regarding the last two pictures of post 5. The hole tolerance stack-up problem is quite evident there, and appears to be the main contributing factor for the random behavior.

It is interesting to note that, for zero hole tolerance and zero friction, the mechanism (a) is in unstable (neutral) equilibrium at any position if the moments are perfectly balanced, and (b) tilts directly to 90 deg if the moments are imbalanced (if the counterweight pin is on the truss centre line).

 

[Phrak]

Sorry, but both statements in post 7 are incorrect. See the equation by Q_Goest in post 4 to understand what causes a moment balance in this problem.

Absolutely not. You do understand that the moments of the two balance arms remain proportional where their center of mass is at their center of extension, don't you?

Why not draw a picture and use a couple test values, instead.

I didn't read Q_Goest; too many words, where invoking a few simple symmetries would suffice. Explain it in your own word if you still have doubts.

 

[nvn]

Hi, Phrak. When summation of moment is zero, the moments are balanced. Rearranging, you get the equation posted by Q_Goest. There is no requirement that r_E = r_S.

 

[Phrak]

Hi, Phrak. When summation of moment is zero, the moments are balanced. Rearranging, you get the equation posted by Q_Goest. There is no requirement that r_E = r_S.

There is. The requirement that the arms balance at all extensions requires that the counterweight be placed at the center of gravity of the short arm. This isn't rocket surgery, just a simple balance.

You will notice that Q_Goest, didn't actually determine the placement of the counterweight. I did. QG and I are not in disagreement.

 

[Whymars]

There is. The requirement that the arms balance at all extensions requires that the counterweight be placed at the center of gravity of the short arm. This isn't rocket surgery, just a simple balance.

Intuitively, I did think that the counterweight should be in the centre of the mass. But if it's right at the end instead, that'll move the centre of mass for the whole arm towards the end - but won't that centre of mass will still move in the same proportion as if it were in the centre?

I am curious to know if there is any significance of having the counterweight hanging, as opposed to having it in-line with the arm. The horizontal distance from the fulcrum will always be the same as the point it hangs from, but I am aware that the overall centre of mass will not lie on the centre line of the arm, and as the arm tilts the centre of mass will end up lying beyond the end of the arm. I don't know if this makes any difference!

It is interesting to note that, for zero hole tolerance and zero friction, the mechanism (a) is in unstable (neutral) equilibrium at any position if the moments are perfectly balanced, and (b) tilts directly to 90 deg if the moments are imbalanced (if the counterweight pin is on the truss centre line).

That is interesting nvn, I have gone through dozens of ideas for ways to deal with the tilting but always assumed that getting it to stay at any particular angle would be accomplished by actively introducing some imbalance.

Thanks for you discussion here guys, it's really helpful!


whymars

 

[Q_Goest]

Hi Phrak,

The obtain balance throughout the extension, the counter weight should hang from the center of mass of the short arm, rather than the end. The weight of the counterbalance would have to double.

Not sure what you're getting at here. There's no reason the counter weight has to act at the CG of the short arm. I agree the weight of the counterbalance would have to double if it was moved to that location (because it would be half the distance), but there's no reason to do so. All one needs to do is sum moments around a single point.

 

[Q_Goest]

Hi Whymars,

I am curious to know if there is any significance of having the counterweight hanging, as opposed to having it in-line with the arm. The horizontal distance from the fulcrum will always be the same as the point it hangs from, but I am aware that the overall centre of mass will not lie on the centre line of the arm, and as the arm tilts the centre of mass will end up lying beyond the end of the arm. I don't know if this makes any difference!

From the pictures you've provided, I'm assuming the weight is hanging from a pivot point, so the weight can be assumed to act vertically downward at that location. The weight could even be hung from a long string of any length instead of a pivot, it wouldn't make any difference. The moment the weight produces around the central pivot point is simply the verticle force times the horizontal distance from pivot to that force.

 

[nvn]

Intuitively, I did think the counterweight should be in the centre of the mass.

That's incorrect; this is a misconception.

I am curious to know if there is any significance of having the counterweight hanging, as opposed to having it in-line with the arm.

None whatsoever, if the counterweight pin is anywhere on the arm centre line. There is, however, an advantage to having the counterweight pin as low as possible, which is probably what Phrak meant to say in the second statement in post 7. The actual centre of mass location of the counterweight doesn't matter, but the counterweight pin location does matter. Therefore, yes, the lower the counterweight pin location, the greater the arm stability in the horizontal position.

Therefore, my statement about neutral equilibrium in post 11 applies only if the counterweight pin location is on the arm centre line. If the counterweight pin location is below the arm axial centre line, you will be in stable equilibrium when the arm is horizontal, but the arm will be unstable when tilted.

 

[Phrak]

nvn. You're right. Why didn't you simply say w_L (e r_L) = w_S (e r_S) + w_E (e r_E)?

 

[Whymars]

Therefore, my statement about neutral equilibrium in post 11 applies only if the counterweight pin location is on the arm centre line. If the counterweight pin location is below the arm axial centre line, you will be in stable equilibrium when the arm is horizontal, but the arm will be unstable when tilted.

nvn, phrak and QGoest, thanks for all your help here, very very helpful indeed. My aim with the structure was always to go for a finger-tip sensitive neutral equilibrium - able to be posed upright or horizontal and any point inbetween (and at any extension). On reflection, I suspect this ideal is unrealistic, but I think i can probably temper any kind of inherent stable equilibrium with a rubber washer or two, since the forces involved should be fairly small, and make it appear neutral to the user. I guess you wouldn't want it being wafted away from you by the breeze of turning your page anyway...

At the moment, this prototype is sitting on my desk, screwed up tight, and in stable equilibrium horizontally. If I tilt it up vertically, then it falls back down, albeit gently. This must be because even though all the available nuts are tightened up, there is still the loose pin the the cw hangs from, and the pin upon which the whole structure bears that pass through holes that are not locked - and even this small amount of off-axis weight is enough to imbalance the arm. And that that would be true even if the rest of the structure was a "perfect" one, straight as a die, and without any slack at all. Is that right?

cheers everyone... I'm getting there slowly!
sn

 

[nvn]

Whymars: I currently assume the arm pivot hole and counterweight pin hole would not be causing this behavior. Maybe (?) the arm falls back down because the counterweight pin location is currently below the arm centre line, as mentioned in the last sentence of post 18. What do you think? Can you try it with the counterweight pin location returned to the arm centre line, in its original location?

 

[Whymars]

Can you try it with the counterweight pin location returned to the arm centre line, in its original location?

Hi nvn, I'd moved the cw back to it's old position already - it still exhibits the horizontal stability, albeit more gently. Looking at the arm along it's length, even with it screwed up tightly, there is still a visible slight bend in it though, partly a property of the material maybe, but more significant around the couple of pivots that couldn't be locked down. I guess this puts the overall centre of mass below the pivot, and that'd make it stay down. Does that sound realistic?

Cheers!
whymars

 

[nvn]

Try to see if the arm centre of mass (CM) is below the arm centre line (perhaps due to nonuniform density of materials) by rotating the arm to the other side and seeing if it exhibits a different behavior. If it exhibits the same behavior, and the arm CM is below the arm centre line due to beam bending (i.e., if the deflected arm forms an arc when loaded, thereby always placing the arm CM slightly below the arm centre line), then you could perhaps compensate for the arc (i.e., you could bring the arm CM back to the arm centre line) by placing weights (washers) above the arm centre line.

 

[Whymars]

perhaps compensate for the arc (i.e., you could bring the arm CM back to the arm centre line) by placing weights (washers) above the arm centre line.

Hi nvn, that might be a good plan - although it will mean that when the arm is upright, and any bend disappears then the arm will then be biased to roll right over. In practice this won't be a problem as there'll be a mechanical stop on the pivot so it can't, and in any case, I'd like it to have a nice positive vertical behaviour, and a little extra weight might just help with that.

I did try flipping it right over: the arm didn't appear to exhibit the same stability when horizontal in this orientation, and it was a lot more "touchy" - took much longer and did more pendulum-ing before it settled down - I assume this is because it was straighter, and therefore less stable. But after a bit of time and wiggling in this orientations, I guess the joints adjusted, and the bend re-formed in the opposite direction to before and it eventually settled down to the same stability.

Interesting experiment, and I am very very glad I asked here for help. I have a much better picture about what the properties of the theoretical version of this contraption might be and because of that, I can see more clearly where my prototypes are diverging from that theory! I thank all who contributed, wholeheartedly!

If anyone's still interested, the development of the lamp is covered on www.euphy.co.uk.

Thanks everyone!

whymars

 

[nvn]

Whymars: You could perhaps use a plastic sleeve (a very short piece of plastic or brass tube) to act as a bushing to fill the clearance in your bolt holes.

 

[Q_Goest]

Hi Whymars,
I think nvn really hits the nail on the head. You're trying to make up for a faulty design by moving the weight around, which actually isn't such a bad idea, but it's just making up for a fault in the original design. I understand you're trying to minimize cost, but having a hole drilled and reamed to +/- 0.0005", and the shaft turned to an equal tolerance is cheap and easy. If the issue is metal to metal rubbing (because you want to make both the pin and dog bones both out of metal) then there are cheap solutions to that issue. If the dogbones are actually bending that much, there are alternatives there as well.

Try coming up with 2 or 3 different approaches to the problem you're solving with this design. I think most people tend to think very linearly, focusing on one idea they feel will work and not seeing alternatives that may be even more appealing than the original.

 

[nvn]

Whymars: What is the diameter of your current bolt holes? I don't know the thread size of your current bolts, but you could perhaps go to a hobby shop (or look on-line) to see if they have plastic sleeve bushings already the exact size you need. If not, you could perhaps buy plastic or bronze tubes in a hobby (or hardware) shop, and a very small pipe cutter, and cut your own bushings. You could buy, e.g., 10-mm-long shoulder bolts having a 3 or 4 mm shoulder diameter, and an M2.5 x 0.45 or M3 x 0.5 thread (such as part number https://www.misumi-ec.com/uk/PDFViewer.aspx?Metric=true&Page=1374" (although these are too long, and would instead need to be 9.8 mm long, and would use 5.3 mm ID washers). Now when you torque the bolt, you have a near-frictionless joint with almost no clearance. The shoulder bolt is free to rotate, but the nut will not loosen.

 

[Whymars]

Hi folks, thanks for thinking about this so carefully!

Whymars: What is the diameter of your current bolt holes? I don't know the thread size of your current bolts

The bolts in this prototype are M4, and the holes are... a bit bigger... I believe I would have used a 4mm drillbit in the first instance, but I also remember playing fast and loose with the needle files at some point. This one is a fairly mushy model, made with hand tools without a good workbench - I'm using this as my testbed for this tilting version because it is easy to tighten and loosen, and to a certain extent (and this may be revisionist) because it exaggerates the problems of wear and looseness and things that the final version will have only after lots and lots of use. The material is 3mm hardboard which is really soft.

Looking at the way this model flexes, I can see that the necessary looseness in the joints makes the bolts able to move away from perpendicular (a shearing kind of action), and that's helped by the small difference between the bolt and the hole size. I have thought about using bushings, but not necessarily in the way you described. I couldn't see a way of having the joint tight enough to hold the bolt (bushing or not) perpendicular and still be loose enough to move. My best solution working along those lines was to tighten a bolt and a nut with washers in one of the dogbone holes (so it stays perpendicular), and then use another nut to keep it's other dogbone on. I actually don't see too many problems with that either, except the two dogbones could not be very close and so the second one would end up cantilevered out somewhat, with the nut inbetween it and the first, which would encourage the first dogbone to twist.

Having the bolts being continuous axles going right through both sides of links would remove this cantilevering/perpendicularity problems entirely, but... I'm not sure I like the look of that and I don't know if solid through-axles would allow me to dispense with the little struts holding the dogbones together entirely.

I understand you're trying to minimize cost, but having a hole drilled and reamed to +/- 0.0005", and the shaft turned to an equal tolerance is cheap and easy. If the issue is metal to metal rubbing (because you want to make both the pin and dog bones both out of metal) then there are cheap solutions to that issue. If the dogbones are actually bending that much, there are alternatives there as well.

The parts will be laser cut from 4mm plywood eventually (as in the most recent prototype on www.euphy.co.uk) so the size of the holes can be very accurate, and I'll be using standard off-the-shelf connector parts, so they can be very closely matched. But there's a lot of flex in the ply and it's quite springy, and along with the soft (wearing) bearing surfaces that's just the nature of this particular beast. Maybe metal is in the future, but there's no need for such heavyweight stuff right now.

My most "advanced" iteration of this uses thin-wall 1/4" brass tubing for the pivots, fastened with starlock washers on the ends, or with wire clips. This has the benefit of allowing me to pass the cabling through the centres of the pivots so there is no slackening of it when the lamp is extended, and that's quite important. The larger diameter of the tubes, and the extra thickness of the ply improves perpendicularity too. The new model that uses these parts are much more rigid, but harder to take to pieces to experiment with, which is why I haven't been. I think I will bite the bullet and disassemble it next week and try the same experiments with the new model as I have with this old one, and see how much better, or how much more neutral it is.

It might sound counter-productive, but I don't want to have to rely on a lot of precision to make this work. I was kind of working on the tacit assumption that if I throw lots of money and quality materials and engineering at it then I can get something pretty close to perfect, but the law of diminishing returns applies. I'm currently more interesting in exploring the lower range of the quality spectrum - how cheap/with fewer parts/less assembly/lower precision can it go and still behave nicely.

Cheers folks!
whymars

 

[nvn]

I would currently say glue the brass tubing into the external struts using epoxy. If epoxy is not strong enough, perhaps try cyanoacrylate. Use a very small diameter, very thin washer, of any material (steel, teflon, etc.), between the two struts, and use a very small diameter washer at the cotter pin. If the friction is still too high with no bushing, consider inserting a bushing into the internal strut. The bushing could be made from the next larger size of brass tubing, if these are typical, very thin-walled brass tubes, which are perfectly telescoping. Lubricate any metal-to-metal surface with a very thin film of molybdenum disulfide (typical wheel bearing) grease.

 

[Whymars]

Actually gluing them in is a great idea. I hadn't really thought about that before because I've been consciously avoiding a messy sort of build process, but the right amount of the right glue would be ideal, and would look great too - could dispense with fasteners on the outside entirely! You're right that the tube sizing is perfect for telescoping... Some good ideas there. In fact, if I made the holes in the outside set of dogbones (I can't help but call them dogbones now, thanks QGoest) a good tight fit for the tubes, I could maybe dispense with the adhesive too, or at least use an easier-handling one.