How to Calculate Angular Movement Amplification in a Mechanical System

[polyperson]

Homework Statement

http://www.geocities.com/zakcapriturbo/bourdon_diag02.jpg

A pointer is connected to an arm that moves along a single axis via a pinion gear wheel and gear quadrant. A displacement of the arm (x) results in an angular movement of the pointer (X3).

Knowns:

Effective radius of the pinion wheel (R3) and gear quadrant (R2)
Distance between the quadrant and arm pivot point (R1)
Displacement of arm (x)

Question:

How can I work out angular movement X3 using only the above information?

Homework Equations

None but I know it's just an amplification factor...see below

The Attempt at a Solution

Well, this has confused me because some movements are angular and some linear. I have worked on similar problems before but the movements were all linear, making it simple to derive an amplification factor. I've scoured the web for tips and hints on this and cannot find anything.

Really, I just need some help to get going!

 

[Astronuc]

It's a matter of successive levers to go from x to x3 or vice versa.

The \Delta{s} of the surfaces must be equal, assuming no slipping, so R_3\Delta{\theta_3}=R_2\Delta{\theta_2}, and its much the same between R1 and R2.

 

[polyperson]

Hi Astronuc,

Grateful for your reply.

Could you explain that a bit more? I don't quite understand your notation.

What do you mean by "the https://www.physicsforums.com/latex_images/12/1201015-0.png of the surfaces" ?

And [PLAIN]https://www.physicsforums.com/latex_images/12/1201015-1.png ? Delta means change or difference, so [PLAIN]https://www.physicsforums.com/latex_images/12/1201015-1.png means what exactly?

I know that le/lr=le/lr (Where le = length to effort, lr = length to resistance).

 

[polyperson]

More thoughts

From what you said, I realize now that this diagram is a better representation of what I have. It certainly makes things easier for me to understand:

http://www.geocities.com/zakcapriturbo/bourdon_diag03.jpg

So, I've omitted that X3 is an angle and just called it linear displacement X and this is what I'm thinking:

X = (x/R1) x (R2/R3)

Does this seem OK?

 

[polyperson]

Even more thoughts

Looking at the above again, I see that it is wrong since it doesn't include the length of the pointer (which we can call R4).

But I think I'm on the right track here but both radius R4 and angle X3 are unknown, so...more work needed !

 

[polyperson]

Solution?

OK, I think I have it. This is entirely thanks to Astronuc getting me to treat this as successive levers:

http://www.geocities.com/zakcapriturbo/bourdon_diag04.jpg

And:

http://www.geocities.com/zakcapriturbo/bourdon_formulas01.jpg

Anybody agree?