Equation of rotated cylinder in 3-D 1

[flamby]

I need to find out intersection of 2 cylinders crossing themselves. Actually this is a manhole (600 mm dia) on a 5500 mm dia pipe, but with an angle like

______//_______


_______________

If one pipe is y^2+z^2=2750
what the equation of other pipe is, considering an angle of rotation of "theta" and 300 mm radius?

Hope my question is clear.

Ciao.

 

[quark]

If z is perpendicular to the screen and y along the height and x along the width and [Θ] is angle of the branch connection with the horizontal pipe then

x²/(rcosec[Θ])²+z²/r² = 1

 

[prex]

quark, your equation can't be correct as it doesn't contain the y coordinate ...
This is my trial.
In the system given by quark with the origin at the intersection of the centerlines, the run pipe has the equation
y²+z²=R²
R being run's radius.
Branch pipe equation may be obtained by observing that its section in the xz plane is an ellipse with axes 2r in the z direction and 2r/cos[θ] in the x direction, r being branch's radius.
Moreover its center lies on the line of equation y=x tan[θ]; by combining one should obtain:
(x cos[θ])²-2xy cos³[θ]/sin[θ]+(y cos²[θ])²+z²=r²
(hope with no residual mistake after the fourth correction ...)

prex

http://www.xcalcs.com

Online tools for structural design

 

[gwolf]

I solved the equations for the general cylinder-cylinder intersection in the late 1980's, it's very complicated. If you limit it to one case like yours it's still very complicated but doable by hand - quite a few pages of maths.

The easiest way to do this is to model the two cylinders in a 3D CAD package and then let it perform the intersection. The result will be a spline line.

The next easiest way to calculate this is to solve using MathCad or similar software. Set up the 3D equation for each cylinder in parametric vector notation and press the button. Remember to put the origin at the intersection of the two centre lines and align one cylinder along a primary axis.

 

[GregLocock]

I don't see how it can be several pages of maths, since a skilled draughty could solve it /for each point/ by drawing four to six lines (from memory).



Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[flamby]

Fairly complicated stuff this! Now let me think what I can make out of this!

Thanks all!

Ciao.

 

[gwolf]

In response to GregLocock - yes you can do it on a piece of paper with construction lines but is the paper result useable - the real intersection is a 3D saddle shape. If you don't think it's a few pages of maths to get the true equation of the intersection curve (which is what Flamby asked for) - go try it.

Flamby - if this is for a fabrication that you are doing yourself you could simplify things. I notice that the manhole cover is quite a lot smaller than the main pipe and the saddle shape will not be very deep. Treat the intersection as a cylinder with a plane - this is a simple elipse equation. Cut out a piece of paper with the elipse on it and draw around it on the main pipe. Cut just inside the line and fettle the hole until the fit is good.

 

[flamby]

prex,

I think there was a mixup in your post. The axis of the elllipse should be 2r/sin?, not 2r/cos?, if you maintain the equation of branch centre-line as y=tan(?).

Only if I was as smart as you in maths after this point...

Ciao.

 

[prex]

Oh yes flamby, you caught me...
Now the equation of the branch takes a more elegant form:
(x sin[θ]-y cos[θ])²+z²=r²
To get at this simply observe that the equation of an ellipse in the xz plane is
(x-x_c)²/a²+(z-z_c)²/c²=1
with x_c=y/tan[θ],z_c=0,a=r/sin[θ],c=r.
Now by substituting z²=R²-y² a quite simple equation for the intersection is obtained:
(x sin[θ]-y cos[θ])²-y²=r²-R²
where y can be obtained as a function of x, x being in the range R/tan[θ][±]r/sin[θ].
That's not really a very difficult math, just a second order algebraic equation...
For x=R/tan[θ][±]r/sin[θ] as expected one has y=R: hope this indicates that I'm correct now...


prex

http://www.xcalcs.com

Online tools for structural design

 

[zekeman]

Not exactly,Prex. Since the intersection involves x, y and z, your equation is only the projection of that curve in the x-y plane. You must add the eq of the main cylinder,
y^2 +z^2= R^2
to complete the solution.

 

[prex]

Well of course zekeman,
one may take a value for x, determine y from my last equation, then z is calculated from the equation of main cylinder.

prex

http://www.xcalcs.com

Online tools for structural design

 

[JStephen]

If you want to complicate it further, include wall thickness and bevel details for both pipe and manway.

If the angle is not too severe, a good welder can set that neck in on the pipe, cut the hole, and trim the neck inside quicker than you can calculate all the curves and plot them out.