Model a Conical Logarithmic Spiral 3

[pillao75]

Hello everyone and thanks for let me be part of this forum.

I have tried to model something like the attached sketch, sorry for the sketch quality... I think is a conical logarithmic spiral.

I have tried a lot of ways to do it, projecting spirals and helix in different directions, creating logarithmic helix manually, using the wrap tool but I don´t find the solution.
The problem for me is define the wrap angle.
If somebody can help me I will be very grateful.

Regards.

 

[ferdo]

Hi,

I suggest you to check few posts here

In my opinion, you can learn a lot of things from Kapitan posts.

Regards
Fernando

https://picasaweb.google.com/102257836106335725208

- Romania

https://picasaweb.google.com/103462806772634246699/

- EU

 

[jackk]

one way to do it:

1. draw a cone by revolving a line about the axis
2. draw a Spiral curve at the base of the cone
3. project the Spiral curve onto the cone, in the direction of the cone's axis

(now to figure out how to get the 60° angle????)

 

[jackk]

another way with a Helix:

1. determine the angle (slope) of the cone
2. draw a point representing the large radius of the cone
3. add a helix:
-- starting point: the point on the base
-- axis: axis of the cone
-- pitch: approx distance between coils
-- height: height of the cone
-- taper angle: angle (slope) of the cone
-- way: inward
4. adjust the pitch to get the angle you want

 

[itsmyjob]

the problem with spiral or helix is that we do not control the angle but the pitch.

Helix pitch could be defined by the angle : pitch = Radius *2*PI / tan (angle) that will do for a cylinder where the radius is fixed.

As the result curve should be on a cone, the pitch does have to change in order to keep the angle value.

The problem I have is that I could not use a defined Law for the pitch.

May be with the equation...

Eric N.
indocti discant et ament meminisse periti ​

 

[pillao75]

First of all thanks for your reply.

Ferdo, I have looked a lot before ask on the forum and effectively Kapitan have the closer solution I have found, this link but is not the final solution because this is for an standard logarithmic spiral.
Jackk, your idea is ok for a archimedes spiral, is the only spiral Catia create by one operation. But in this case I need a conical logarithmic spiral with an specific angle (this spiral have the same angle regard to the cone surface in all its points) in this specific cone.
The more visible difference is the pitch and the angle, in your case is a constant pich and variable angle, in the log spiral the pitch varies because the radius is reduced. I think the only way to obtain this spiral is with some kind of function for obtain a 2D spiral and projected it into the cone or create the 3D spiral.
I have been investigating about this kind of spirals and my conclusions are that depends of two factors: The start radius (Is a data that I know) and other factor that varies between 0 and 1, so, there are infinite logarithmic spirals with the same start radius. I suppose that in this case, the factor is related with the slope of the cone and the angle of the spiral but I don't know how obtain the correct factor.
Itsmyjob explained the problem better than me.
I have found this imagen that could help to understand what I am looking.

Regards.

 

[pillao75]

I GET THE SOLUTION!
I have done the model with NX but can be solved with any program. I'll explain without images because right now I don't have any PC with CAD.
The idea is to get the 2D spiral and then project it onto the cone. To the spiral I used its parametric equations, which one time developed are:

x (t) = a * exp (b * t) * cos (t)
y (t) = a * exp (b * t) * sin (t)

Where:

a = start radius of the spiral
b = 1 / tan (α)

α is the angle between between the tangent and radial line. One of the main properties of these spirals is that any straight line from the origin cut the logarithmic spiral at the same angle, so also called equiangular.


To kwon α, create a cone with the wanted angle. Creates a tangent plane to the surface of the cone. In this plane draw a line that cut the axis of the cone with the desired slope angle, in this case 60 degrees. Project this line in the plane of the base of the cone. In a top view we can see the angle of the radial line of the cone to the tangent of the spiral, measuring the angle and we have α and therefore b.

In NX, create the necessary expressions and then a curve defined by these expressions, in other programs is a similar process. We must not forget the expression to define how many revolutions we want, this will simply make the spiral bigger or small initial radius is defined by "a" and the end radius by the number of revolutions. Also take special care in all the places where is the variable "t" it is multiplied by the number of revolutions (at least in NX "t" only varies between 0 and 1 and we have no control over it) because otherwise the spiral could looks right but is not the solution and something is wrong in this equation. For example for 4 revolutions and in radians, the equation for "x" would be:

x (t) = a * exp (b * 4 * 2 * pi * t) * cos (4 * 2 * pi * t)

Once we have the 2D spiral is projected onto the cone and that is all.

I hope this will be helpful if someone has the same problem in the future .

Regards.

 

[itsmyjob]

in CATIA, create 3 parameters :

1 real , name 'nb of rotation', value = 2
1 real, name 'radius', value = 0.01 I wish we can have 0 but I have error.
1 angle, name 'tgt angle', value = 78.13268deg

get Curve from Equation function

for X define first law:

create 2 real variables x and t
x = radius * exp ( `nb of rotation` * 2 * PI * t / tan('tgt angle') ) * cos ( `nb of rotation` * 2 * PI * t * 1rad)​

for Y define second law:

create 2 real variables y and t
y = radius * exp ( `nb of rotation` * 2 * PI * t / tan('tgt angle') ) * sin ( `nb of rotation` * 2 * PI * t * 1rad)​

for Z define last law:

create 2 real variables z and t
z = 0 * t​

Eric N.
indocti discant et ament meminisse periti ​

 

[jackk]

Pillao75: Sorry my solutions create the wrong type of spiral, but thanks for explaining the difference! Glad you were able to get a solution.