# Extending arm with counterweight balance question

1. Aug 8, 2009

### 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

2. Aug 8, 2009

### 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.

3. Aug 8, 2009

### 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

4. Aug 8, 2009

### 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:
wL*rL = wS*rS + wE*rE
Where
wL is the weight of the arm on the Long side
rL is the horizontal distance from the pivot to the CG of the Long arm
wS is the weight of the arm on the Short side
rS is the horizontal distance from the pivot to the CG of the Short arm
wE is the weight of the Extra balance weight
rE 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 rE 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 rE 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 rE 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 rE 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.

Last edited: Aug 8, 2009
5. Aug 9, 2009

### 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

6. Aug 9, 2009

### 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?

7. Aug 9, 2009

### 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.

8. Aug 9, 2009

### Whymars

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

9. Aug 9, 2009

### Whymars

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.

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

10. Aug 9, 2009

### Phrak

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

11. Aug 9, 2009

### 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).

12. Aug 9, 2009

### Phrak

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.

Last edited: Aug 9, 2009
13. Aug 9, 2009

### 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 rE = rS.

14. Aug 10, 2009

### Phrak

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.

Last edited: Aug 10, 2009
15. Aug 10, 2009

### Whymars

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!

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

16. Aug 10, 2009

### Q_Goest

Hi Phrak,
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.

17. Aug 10, 2009

### Q_Goest

Hi Whymars,
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.

18. Aug 10, 2009

### nvn

That's incorrect; this is a misconception.
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.

19. Aug 10, 2009

### Phrak

nvn. You're right. Why didn't you simply say wL (e rL) = wS (e rS) + wE (e rE)?

20. Aug 11, 2009

### Whymars

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

21. Aug 11, 2009

### 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?

22. Aug 12, 2009

### Whymars

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

23. Aug 12, 2009

### 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.

24. Aug 15, 2009

### Whymars

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

25. Aug 15, 2009

### 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.