Spring Theory 8

[treddie]

Howdie.
Would anyone happen to know of a good book that explains the dynamics of springs? I have searched and searched on the web forever for the equations governing the dynamics of all types of springs and come up zero, and can't seem to find anything on something that MUST be covered in detail SOMEWHERE. You know it's getting bad when you reach somebody's New Age page about "Spring" theory and how it will change your life....Jeez!

Thanks,
treddie

 

[prex]

Everything is OK for me, of course replacing the ? with the corresponding symbols (look in the Process TGML link at bottom of the Reply window, then 'Special characters', for how to use symbols on this site).
Look here to see what I got with those formulae using Excel: in this exercise the factor Pr³/EJ was 0.04 (at the inner end of the spiral) and the radius was increasing by 20% at each half turn (we are dealing with archimedean spirals, aren't we?).
To obtain these result you only need to be careful about signs and origin of displacements: the above formulae are valid only between 0 and [π].
You'll see that, for a spiral with many turns, the turns will come into contact for a very small deformation.
Is this what you were looking for?

prex

http://www.xcalcs.com

Online tools for structural design

 

[treddie]

Howdie prex.
Finally got a chance to get back to this thing.
Have been running some test values, and one thing has me stumped. The 2 equations for, [ξ] and [η], have only 2 variables, (P) and ([θ]); (E and J are constants for a given spring). How are [ξ] and [η] applied to the x and y outputs of the circle? My tests simply scale the size but not the shape of the half-segments, because simply changing for instance, load (P), does nothing more than scale both outputs, which is obvious from the equations. Therefore, doing any of the following really doesn't make sense (where xc and yc are outputs from a simple circular arc from 0 to [π]):
As offsets:
x = xc + [ξ]
y = yc + [η]

As coefficients:
x = xc *[ξ]
y = yc * [η]

As "self-contained" coords:
x = [ξ]
y = [η]

(E), modulus of elasticity and (J), moment of inertia are constants, for a given spring. This leaves (P) to do nothing more than scale the results. What's going on here that I'm missing?

It's difficult without seeing the actual discussion by Belluzzi.

treddie
p.s. Thanks for the tip on special characters.

 

[prex]

[ξ] and [η] are displacements, so that your first equation (as offsets) is the correct one.
And OK the independent variable of the equations for [ξ] and [η] is [θ]: can't see what worries you. P is more a parameter than a variable (just like EJ).

prex

http://www.xcalcs.com

Online tools for structural design

 

[treddie]

I agree (P) being more a parameter.
Let me go back to my code and see if I can find anything stupid. Right now, I get a sort of elliptical shape for one 1/2 turn (to be expected), but changing the load does not change the proportions of the curve, just its size. Unless my understanding of the behaviour of the equations is wrong to begin with; that the curve proportions WOULD stay the same regardless of load, only the DISPLACEMENT AND SCALE of the curves should change. This is easy to test...I've only been looking at 1 half-segment up to now to test the basic code. Time to connect them all and look at the overall shape.

treddie

 

[treddie]

My last post doesn't make sense either. In looking at your image again, it's clear that each of your segments is definitely changing its proportions. This leaves only one other possibility I can think of:
x = [Σ](x_c + [ξ]_c)
y = [Σ](y_c + [η]_c)

where:
x_c = rcos[θ]
y_c = rsin[θ]

accumulating the displacements after each step [θ].

treddie

 

[treddie]

God! I’m losing it!
What I MEANT to say, was…
x = x_c + [Σ]( [ξ]_T)
y = y_c + [Σ]( [η]_T)

where:
x_c = rcos[θ]
y_c = rsin[θ]

[ξ]_T = Total accumulated [ξ] up to the present [θ]. ( [ξ]_T not
reset to zero after each half-segment)

[η]_T = Total accumulated [η] up to the present [θ]. ( [η]_T not
reset to zero after each half-segment)

treddie

 

[prex]

Yes, you are lost!
Try to follow the following steps (or ask for help to a mathematically minded colleague):
1- Rewrite my equation for [ξ] so that [ξ]=0 for [θ]=0, you should obtain:
[ξ]=(Pr³/4EJ)(2[θ]+2sin[θ]cos[θ]+8([θ]sin[θ]+cos[θ]-1)/[π]-4sin[θ])
2-For the first semicircle
x=rcos[θ]+[ξ]([θ])
y=rsin[θ]+[η]([θ])
3-Now call x_e,y_e the endpoint of the first semicircle, namely
x_e=-r+[ξ]([π])
y_e=0
4-The next semicircle must start from this endpoint end has a new radius R=r+[Δ]r
Using again [θ] going from 0 to [π] its undeformed equation is
x=x_e+R(1-cos[θ])
y=-Rsin[θ]
(hope with no error on my part)
5-Add to the above [ξ] and [η] calculated with the new R
6-Loop from point 3 for subsequent semicircles.


prex

http://www.xcalcs.com

Online tools for structural design

 

[prex]

Follow up: step 5 above should read:
5-Add to the above [ξ] and [η] calculated with the new R and with the sign of P reversed

prex

http://www.xcalcs.com

Online tools for structural design

 

[treddie]

Well, it likes like I DID understand you correctly in the beginning, because that is exactly the approach I took, except that I let the centerpoint of all the semicircles line up; I was going to deal with that problem after all of the semicircles were completed and in an array.
I also think that one of the problems I had was that I was putting in unrealistic values for the load (P), resulting in wacked out results. But to know for sure could you check these three equations for correctness, if you can one last time, because I STILL am getting “puckered” like deformations (I would upload the image to you, but didn’t want you to have to go through the web server hassle to retrieve it):

For distribution relative to spiral center:

Pr³ 8([θ]sin[θ] + cos[θ])
[ξ] = ______ * ( 2[θ] - [π] + 2sin[θ]cos[θ] + ______________ - 4sin[θ] )

4EJ [π]



For distribution relative to spiral end:

Pr³ 8([θ]sin[θ] + cos[θ] - 1)
[ξ] = ______ * ( 2[θ] + 2sin[θ]cos[θ] + ________________ - 4sin[θ] )

4EJ [π]


“Vertical” displacement:

Pr³ 8(sin[θ] - [θ] cos[θ])
[η] = ______ * ( 4cos[θ] - 2 - 2cos ² [θ] + ______________ )

4EJ [π]

 

[prex]

OK

prex

http://www.xcalcs.com

Online tools for structural design

 

[treddie]

Got it!
As I thought, a stupid high-school error in my code. Shapes are correct, now I'll link up all the half-segments.
treddie

 

[treddie]

It's working. FINALLY!
Pretty much gives me what I need. I think I'll toy with it and get it to do applied loads at other than along the x-axis, so I can make it more versatile.

Anyway, prex, thank you very much for all of your help and patience. I really appreciate your time.

treddie