How to Calculate Angular Movement Amplification in a Mechanical System

In summary, Astronuc helped me to understand how to solve a problem involving angular movement using only knowns. The equation for linear displacement is X = (x/R1) x (R2/R3), which can be simplified to X = R1*R2. To solve for X3, I need to find R4 and then use the equation for angular movement to find X3.
  • #1
polyperson
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0

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!
 
Last edited:
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  • #2
It's a matter of successive levers to go from x to x3 or vice versa.

The [tex]\Delta{s}[/tex] of the surfaces must be equal, assuming no slipping, so [tex]R_3\Delta{\theta_3}[/tex]=[tex]R_2\Delta{\theta_2}[/tex], and its much the same between R1 and R2.
 
  • #3
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 [Broken] of the surfaces" ?

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

I know that le/lr=le/lr (Where le = length to effort, lr = length to resistance).
 
Last edited by a moderator:
  • #4
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?
 
  • #5
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 !
 
  • #6
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?
 

1. What is mechanical amplification?

Mechanical amplification is the process of increasing the amplitude or power of a mechanical signal. This can be achieved through various mechanisms such as levers, pulleys, gears, and springs.

2. How does mechanical amplification work?

Mechanical amplification works by using a mechanical system to increase the force, displacement, or velocity of a signal. This can be achieved through the principles of mechanical advantage and energy conservation.

3. What are the applications of mechanical amplification?

Mechanical amplification has a wide range of applications in engineering and technology. It is commonly used in machinery, vehicles, and tools to increase the power and efficiency of mechanical systems. It is also used in musical instruments, such as guitars and pianos, to produce louder and more resonant sounds.

4. What are the advantages of mechanical amplification?

The primary advantage of mechanical amplification is its ability to increase the output of a mechanical system without requiring additional power sources. This makes it a cost-effective and efficient method for boosting the performance of various devices and machines.

5. Are there any drawbacks to mechanical amplification?

While mechanical amplification can be beneficial, it also has some limitations. It can introduce noise and vibrations into a system, which can affect its accuracy and precision. It can also add complexity and weight to a device, making it less portable and more prone to mechanical failures.

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