Auger flight sections into flat pattern?
Hey harry.doldersum,
Fusion doesn't have this capability at this time. Exactflat would make short work of this though: https://apps.autodesk.com/FUSION/en/Detail/Index?id=3499335255055878377
I am not advocating for ExatFlat, but if working with surfaces is not part of your workflow, then it's time to reconsider, and upgrade your workflow.
@harry.doldersum - I am not an expert in ExactFlat. However, if it is true that it only works with surfaces (which actually makes sense, because the goal is a flat pattern, which is inherently 2D), why not just create a zero-offset surface from the auger geometry, and send that to ExactFlat? You don't need to change your design at all, just have an extra surface body in it for unwrapping.
Jeff Strater
Engineering Director
Hi Mr. Harry Doldersum,
AUGER tool is based on simple topology objects: cylinder and helical surface.
Let us skip the cylinder and concentrate on the helical part. I assume that we are talking here about a classical auger (not, e.g., one designed for removing vax from ear cavity … images of which pop-up profusely everywhere on media in my jurisdiction). Considering the diversity of the F360 Forum, I am sure that there will be time for this one also.😏
The equation of helical surface can be drawn as follows:
x = <r>*sin(φ) r ∈ <R1,R2> Eq.1
y = <r>*cos(φ) r ∈ <R1,R2> Eq.2
z = <φ>*h φ ∈ < Φ0=0,Φ> Eq.3
Compel ourselves to do a simple mental exercise. What will happen if we squeeze (stretch) out a helical auger.
A carpenter's intuition will deduce that inner, outer arc perimeter lengths and the surface area will be preserved (sort off). At the same time, R1, R2 and Φ will change … slightly.
The helix inner & outer arc lengths can be derived from the equations:
L1 = Φ*sqrt(R1^2 + H^2) Eq.4
L2 = Φ*sqrt(R2^2 + H^2) Eq.5
and after the squeeze:
L1_ = Φ_*sqrt(R1_^2 + H_^2) Eq.6
L2_ = Φ_*sqrt(R2_^2 + H_^2) Eq.7
… but why stop … just flatten-out this thing!!!
_L1 = _Φ*sqrt(_R1^2) = _Φ * _R1 Eq.8
_L2 = _Φ*sqrt(_R2^2) = _Φ * _R2 Eq.9
If our carpenter's intuition is at least close to the target, then:
L1 = L1_ = _L1 Eq.10
L2 = L2_ = _L2 Eq.11
What about the surface area? The simple integration yields:
S = ∫ (L, r, dr) r ∈ <R1,R2> Eq.12
S = ∫Φ* sqrt (r^2 + H^2)dr r ∈ <R1,R2> Eq.13
Which gives:
S = Φ*(r*sqrt(H^2 + r^2))/2 + (H^2*Log(r + sqrt(H^2 + r^2)))/2 |R1, |R2 Eq.14a
S = Φ*(R2*sqrt(H^2+R2^2))/2 + (H^2*Log(R2 + sqrt(H^2+R2^2)))/2 –
Φ*(R1*sqrt(H^2+R1^2))/2 + (H^2*Log(R1 + sqrt(H^2+R1^2)))/2 Eq.14b
and for the flattened helical surface where lim H ⇒ 0
_S = Φ*(R2*sqrt(R2^2))/2 + (0*Log(R2 + sqrt(R2^2)))/2 –
Φ*(R1*sqrt(R1^2))/2 + (0*Log(R1 + sqrt(R1^2)))/2 Eq.15
_S = _Φ*(_R2^2 – _R1^2)/2 Eq.16
… so finally … surprise … surprise … we have got three equations with three variables.
_L1 = _Φ * _R1 Eq.17
_L2 = _Φ * _R2 Eq.18
_S = _Φ*(_R2^2 – _R1^2)/2 Eq.19
It's time to solve them!
Observe that:
_L1/_L2 = _R1/_R2 = L1/L2 = R1/R2 Eq.20
So,
_R1 = _R2*R1/R2 Eq.21
_S = _Φ*_R2^2*(1 – (R1/R2)^2)/2 Eq.22
_S = _L2 *_R2*(1 – (R1/R2)^2)/2 Eq.23
S = L2 *_R2*(1 – (R1/R2)^2)/2 Eq.24
Thus,
_R2 = 2*S/(L2*(1 – (R1/R2)^2)) …. Bingo Eq.25
_R1 = _R2*R1/R2 …. Bingo Eq.26
_Φ = 2*S/*(_R2^2 – _R1^2) …. Bingo Eq.27
The solution of R2_, R2_,Φ_ for arbitrary ‘squeeze phase’ as per Eq.1,2,3
would be challenging as the nasty Eq.14 is on the way. However, considering dependence of R2_, R2_, Φ_ on the squeeze deformation, by knowing the boundary condition one can resort to the simple linear approximation approach where:
R1(h) = R1 + (_R1 – R1)*_H/H_0 Eq.28
R2(h) = R2 + (_R2 – R2)*_H/H_0 Eq.29
Φ (h) = Φ + (_Φ – Φ)*_H/H_0 Eq.30
The simulation presented in the attached videos has adopted such a path.
The accuracy seems to be better than half a finger carpenter's gauge, and as measured by the standard deviation of the calculated area of helical surface, it is about -0.012.
The purely theoretical approach will not address many physical phenomena when shaping metal plates to helical forms… although in some instances … the method might be good enough.
One significant benefit is … it is inexpensive (not even the cost of a second-hand diesel car). So don't worry, mate. I will charge a small interest rate and afterlife only, … but forever.😐
Attached files:
FlattenTheAuger_mono.png 4K_mono (0.2MB) https://a360.co/3ubewZO
FlattenTheAuger _mono.mp4 4K_mono ( 6MB) https://a360.co/3IgiO7b
FlattenTheAuger _arcd.png 4K_stereo ( 4MB) https://a360.co/3JtXay0
FlattenTheAuger _arcd.mp4 4K_stereo ( 20MB) https://a360.co/3CTKrC1
FlattenTheAuger _al.mp4 4K_stereo ( 12MB) https://a360.co/34QRr64
To be viewed on 4K media devices (monitors, UHD TVs, projectors...) of reasonable performance. For the best experience, use stand-alone media applications (WMP, VLC) and the native resolution 3840x2160 - full screen. The '_arcd' files require an anaglyph red/cyan glasses, while '_al' is for an active shutter glasses 3D hardware (~30 deg viewing angle is recommended). Download the files over a network, where the cost of doing so is not a concern. The files are to be used for private, non-commercial purposes only.
Regards
MichaelT
NOTE To TF360
There is still a deficiency (and bipolar craziness) in managing user parameter units. Also, as many probably agree, the whole interface/input method is very inefficient and cumbersome. In this example, I have had to resort to implementing a surface area parameter as unitless. The other only choices were acres and circular_mil (whatever this means). Here is my earlier … well … I know … the sarcastic post.
Units in 21st century - Autodesk Community - Fusion 360
Hi Fellows,
So, … what is the problem …?
We have arrived at the set of equations:
_L1 = _Φ * _R1 Eq.17
_L2 = _Φ * _R2 Eq.18
_S = _Φ*(_R2^2 – _R1^2)/2 Eq.19
… and at this point, the issue emerged!
The following equation implemented the constrain condition on R1 & R2
In the form:
_L1/_L2 = _R1/_R2 = L1/L2 = R1/R2 Eq.20
It was a weak condition!
When we look at Eq.17 and Eq.18 in the context of it and perform based on this constraint, the substitution:
_R1/_R2 = _L1/_L2
_R2 = _R1*_L2/_L1
Then:
_L2 = _Φ * R1*_L2/_L1 Eq.18 .. sub
After div by _L2 and mul by _L1 becomes Eq.17.
It means that we have got not three but only two independent equations with three variables.
They still can be solved in funiverse, giving a legit infinite number of solution pairs ( Φ, (R1, R2)) we are after.
The problem is that we live in the universe (aka. reality, …. really?), so we need to find out only one definite and deterministic, unavoidable guidance imposed on us by the law of physics.
To discover it, we need to reject the weak condition and replace it with the strong conviction, which is …
W = R2 – R1 = const
Where W is a radial width of a helical surface (auger spiral). It is perhaps very obvious to any rational carpenter. Therefore, I must repent … I was fallacious … believing in … R1/R2=const although … well it is still could be true, … but only in the weak sense.
After improving on truthiness of the truth … let's correct the equation's flow to get the only proper answer. Replace Eq.20 with the new one.
W = _R2 - _R1 = R2 - R1 = const Eq.20_n
Thus,
_L1 = _Φ * _R1 = L1 = Φ*sqrt(R1^2 + H^2) Eq.18,4 recall
_L2 = _Φ * (R2 - R1 + _R1) = L2 = Φ*sqrt(R2^2 + H^2) Eq.21_n,5 recall
_L1/_L2 = _R1/(R2 - R1 + _R1) Eq.22_n
L1/L2 = sqrt((R1^2 + H^2)/ (R2^2 + H^2)) Eq.23_n
_L1/_L2 = L1/L2
_R1/(R2 - R1 + _R1) = sqrt((R1^2 + H^2)/ (R2^2 + H^2)) Eq.24a_n
_R1 = (R2 - R1 + _R1) *sqrt((R1^2 + H^2)/(R2^2 + H^2) Eq.24b_n
_R1*(1-_sqrt((R1^2+H^2)/(R2^2+H^2))=(R2-R1)*sqrt((R1^2+H^2)/(R2^2+H^2) Eq.24c_n
_R1=(R2-R1)*sqrt((R1^2+H^2)/(R2^2+H^2))/(1-sqrt((R1^2+H^2)/(R2^2+H^2)) Eq.24d_n
Finally ( I hope so)…
_R1 = (R2-R1)*sqrt((R1^2+H^2)/(R2^2+H^2))/(1-sqrt((R1^2+H^2)/(R2^2+H^2))) Eq.25_n
_R2 = (R2-R1)*sqrt((R2^2+H^2)/(R1^2+H^2))/(1-sqrt((R2^2+H^2)/(R1^2+H^2))) Eq.26_n
_Φ = _L1/_R1 = _L2/_R2 Eq.27a_n
_Φ = Φ *sqrt(R1^2 + H^2)/_R1 = Φ *sqrt(R2^2 + H^2)/_R2 Eq.27b_n
If we look at the previous answer, it is not difficult to discover that the new one is much neater, and thanks … there is no reference to the helical surface area in them, … hence no need for the strenuous gymnastic in dealing with its userParameter representations.
The last step is to rebuild the auger's F360 model file. For the sake of clarity, rehearse:
Input data:
_F – ‘squeeze’ phase factor, <0,1>=<flattened, at rest>
_H – helical surface vertical span (auger flight height)
_T – auger flight plate thickness
R1 – inner helical surface radius (auger pole radius)
R2 – outer helical surface radius (auger external radius)
Φ – helical surface angular span (auger flight span angle)
Intermediary:
H_ – HR/(2*π), helical surface vertical span 2π normalized, Eq.3 parameter
Output data:
R1F – inner radius of the flattened helical surface
R2F – outer radius of the flattened helical surface
Φ.F – flattened helical surface angular span (flattened ager’s plate angular span)
L1 – theoretical length of inner helical arc
L2 – theoretical length of outer helical arc
ST – theoretical helical surface area (auger flight plate area)
Φx – maximum helical surface angular span (auger flight span angle) ) when ΦF=360 °
Hx – maximum helical surface vertical span (auger flight height) when ΦF=360 °
Sx – maximum theoretical helical surface area (auger flight plate area)
Add one important note here. In order to avoid falling into F360 userParameters units’ grinding machinery … both input and output values will be represented as unitless. Angles will be assumed to be in radians and lengths/distances in millimeters (I have allergy to inches, … hope that this long-lasting viral imposition will be cured … mercilessly & soon. Don’t wary … if you have one, after three i-shots you will remember nothing, you shouldn’t). Units will be applied post-factum directly on the modelled objects (aka. model’s units’ tattooed skin) giving it … customized identity.
Attached files:
FlattenTheAuger_mono.png 4K_mono (0.2MB) https://a360.co/3qtaD1f
FlattenTheAuger _mono.mp4 4K_mono ( 6MB) https://a360.co/37LQBsl
FlattenTheAuger _arcd.png 4K_stereo ( 4MB) https://a360.co/3IrQ8rX
FlattenTheAuger _arcd.mp4 4K_stereo ( 20MB) https://a360.co/3tutUS4
FlattenTheAuger _al.mp4 4K_stereo ( 12MB) https://a360.co/3ipGxrg
To be viewed on 4K media devices (monitors, UHD TVs, projectors...) of reasonable performance. For the best experience, use stand-alone media applications (WMP, VLC) and the native resolution 3840x2160 - full screen. The '_arcd' files require an anaglyph red/cyan glasses, while '_al' is for an active shutter glasses 3D hardware (~30 deg viewing angle is recommended). Download the files over a network, where the cost of doing so is not a concern. The files are to be used for private, non-commercial purposes only.
Regards
MichaelT
