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aplesch

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Joined: 05/10/2015

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X3D: behind-the-scenes

23/12/2015 - 14:56:38
more gears, fast

11/11/2015 - 01:28:48
general smoothness

02/11/2015 - 08:18:08
smoother gears

30/10/2015 - 07:33:05
helical gears

26/10/2015 - 04:09:02
3d involute gears

21/10/2015 - 07:30:11
involute gear

16/10/2015 - 22:49:13
circle involute

09/10/2015 - 06:50:27
Flying Boxes

07/10/2015 - 00:07:10
A paper dragon

05/10/2015 - 15:41:53

helical gears

Submitted by aplesch on Mon, 26/10/2015 - 04:09

The real advantage of gear modelling in 3d is that gears in a mesh do not have to parallel each other. It is posible to demonstrate how gears can transfer rotation (and forces) to orientations other than the driving gear. This kind of transfer is possible by use of helical gears.

Helical gears have their teeth at an angle to the axis of the gear but are otherwise very similar to spur gears. In fact, profiles through a helical gear are all identical and rotated by an amount proportional to their distance from the front.

Based on the 3d involute gears program, helical gears therefore did not require too many changes. The main idea is that instead of just extruding the gear profile in one step, now several quads in the direction of the helical angle are generated, with the correct amount on distorttion (rotation) based on the width of each segment.

Here is the logo which is quite similar to the 3d involute gear logo but has the rotated extrusion and some smaller changes including the addition of a third gear. It takes quite a long time to run.

The triangles across the teeth were qute noticable. Changing the crease angle by editing the x3d helped.

;

; involute: draw involute
; rb: base radius
; limit: max. generating angle
; steps: number of segments on involute to draw
; returns list of positions: top is first
 
TO involute :rb :limit :steps
 
  LOCALMAKE "start POS
  LOCALMAKE "i 0
  LOCALMAKE "step 0.5 * :limit / :steps
  LOCALMAKE "coords []
  PUSH "coords POS
  PU
 
  REPEAT ABS(:steps)
  [
    ;push "coords :start
    MAKE "i :i + :step
    ;move along circle
    LT :step
    FD 2 * :rb * SIN :step
    ;find tangent
    RT 180 - :step
    ;unroll string
    FD 2 * :i * :deg2rad * :rb 
    ;make "start pos
    ;record position
    PUSH "coords POS
    BACK 2 * :i * :deg2rad * :rb
    ;turn forward
    LT 180
  ]
  SETPOS :start
  OUTPUT THING "coords
 
END
 
; gearinvolute: draw involute side of gear tooth
; rb: base radius
; PA: pressure angle
; P: diametral pitch
; steps: number of subdivisions
; returns list of position from landing to base
 
TO gearinvolute :rb :PA :P :steps
 
  LOCALMAKE "a 1 / :P
  LOCALMAKE "ro ( :rb / COS :PA ) + :a
  ;calculate generating angle for top of tooth
  ;(needs drawing)
  LOCALMAKE "l2 :ro ^ 2 - :rb ^ 2
  LOCALMAKE "limit ( SQRT :l2 ) / :rb
  LOCALMAKE "coords involute :rb (:limit / :deg2rad) :steps
  OUTPUT THING "coords
 
END
 
; geartooth: draw one full tooth
; rp: radius of pitch circle
; PA: pressure angle
; P: diametral pitch
; hdelta: half angular base width (expected)
; segments: number of subdivisions on each side
; returns list of positions from root to landing to root
TO geartooth :rp :PA :P :hdelta :segments
 
  LOCALMAKE "rb :rp * COS :PA
  LOCALMAKE "clr (2.157 - 2) / :P ;coarser 20P
  LOCALMAKE "ddif (1/:P + :clr) - (:rp - :rb)
  LOCALMAKE "coords []
  PUSH "coords POS
  ; record straight edge
  RT 90
  FD :ddif PUSH "coords POS
  LT 90
  ; save position
  LOCALMAKE "base POS
  LOCALMAKE "head HEADING
  ; record side
  MAKE "coords SENTENCE ( gearinvolute :rb :PA :P :segments ) :coords
  ;record with negative step
  LOCALMAKE "top POS
  SETPOS :base
  SETHEADING :head
  ;turn and move to other side of tooth
  ;half angular base width
  LT :hdelta
  FD 2 * :rb * ( SIN ( :hdelta ) )
  LT :delta / 2
  ;save pos
  MAKE "base POS
  MAKE "head HEADING
  ;draw other side
  ;prepend reverse
  MAKE "coords SENTENCE REVERSE ( gearinvolute :rb :PA :P :segments * -1 ) :coords
  ;return to base
  SETPOS :base
  SETHEADING :head
  ;record straight edge
  LT 90
  FD :ddif PUSH "coords POS
  RT 90
  OUTPUT THING "coords
 
END
 
; gear: draw involute gear
; rp: radius at pitch circle
; PA: pressure angle
; P: number of teeth normalized by pitch diameter
; segments: subdivions on each involute side of each tooth
; returns list of positions
TO gear :rp :PA :P :segments
 
  LOCALMAKE "cpitch :PI / :P 
  LOCALMAKE "apitch (:cpitch / :rp) / :deg2rad
  LOCALMAKE "N :P * :rp * 2
  LOCALMAKE "rb :rp * COS :PA
  LOCALMAKE "clr (2.157 - 2) / :P ;coarser 20P
  LOCALMAKE "ded (1/:P + :clr)
  LOCALMAKE "rr :rp - :ded
  ;this is the main calculation
  ;for angular base width delta
  ;(needs drawing)
  LOCALMAKE "alpha ( TAN :prA ) / :deg2rad
  LOCALMAKE "beta :alpha - :prA
  LOCALMAKE "delta ( 0.5 * :apitch ) + ( 2 * :beta )
  LOCALMAKE "coords []
  ; move from center to root
  PU
  RT 90 FD :rr LT 90
  REPEAT :N
  [
    ;tooth from root to root
    MAKE "coords FPUT ( geartooth :rp :PA :P :delta / 2 :segments ) :coords
    ;advance to root of next tooth
    LT (:apitch - :delta) / 2
    FD 2 * :rr * SIN ((:apitch - :delta) / 2)
    LT (:apitch - :delta) / 2
  ]
  ;close up
  ;push "coords pos
  LT 90 FD :rr RT 90
  OUTPUT THING "coords
 
END
 
TO polygon :positions
 
  JUMPPOS FIRST :positions
  FOREACH "setpos BF :positions
 
END
 
TO firstlast :positions
 
  OUTPUT LIST FIRST :positions LAST :positions
 
END
 
TO plate :positions
 
  PU SETPOS FIRST FIRST :positions
  FACE PCOFF
  PD
  FOREACH "polygon :positions
  ;fill circle
  polygon REDUCE "sentence MAP "firstlast :positions
  ;show reduce "skinny :flatpositions
  PU
  LINE
  OUTPUT OBJECT
 
END
 
TO skinny :p1 :p2
  JUMPPOS :p1
  SETPOS :p2
  SETPOS :pcenter
  OUTPUT :p2
END
 
TO gear3d :positions :width :helix :radius :fsegments
 
  LOCALMAKE "posflat FPUT (LAST LAST :positions) (REDUCE "sentence :positions)
  fence :posflat :width
  ; does not seem to work
  ;set object "creaseAngle 1.57
  COPY plate :positions
  SET OBJECT "rotation [0 0 1 0]
  PASTE
  ;object not updated after paste
  LOCALMAKE "pasted WORD "obj_ (REDUCE "word BF BF BF BF OBJECT) + 1
  SET :pasted "rotation (LIST 0 0 1 (TAN :helix) * :width / :radius)
  SET :pasted "translation ( LIST 0 0 :width ) 
 
END
 
TO fence :positions :width
 
  PU SETPOS FIRST :positions
  FACE
  PCOFF
  PD
  SHOW REDUCE "quads :positions
  PU
  LINE
 
END
 
TO quads :p1 :p2
 
  LOCALMAKE "p22 :p2
  LOCAL "p3
  LOCAL "p4
  ;fsegments inherited from gear3d
  REPEAT :fsegments
  [
    JUMPPOS :p1
    SETPOS :p2
    MAKE "p3 offsetz :p2 :width / :fsegments
    MAKE "p4 offsetz :p1 :width / :fsegments
    SETPOS :p3
    SETPOS :p4
    MAKE "p1 :p4
    MAKE "p2 :p3    
  ]    
  OUTPUT :p22
 
END
 
TO offsetz :point :offset
 
  LOCALMAKE "x FIRST :point
  LOCALMAKE "y FIRST BF :point
  LOCALMAKE "z LAST :point
  LOCALMAKE "rad SQRT (:x ^ 2) + (:y ^ 2)
  ;helix and radius inherited from gear3d
  LOCALMAKE "rot (ARCTAN :x :y) + (TAN :helix) * :offset / :radius / :deg2rad
 
  OUTPUT (LIST (:rad * COS :rot) (:rad * SIN :rot) :z + :offset)
 
END
 
;involute (for gears)
;comprehensive guide
;http://bostongear.com/pdf/gear_theory.pdf
 
RESET
CLEARSCREEN
SETBACKGROUND 34
POINTLIGHT
 
RU 90
MAKE "PI 4 * RADARCTAN(1)
MAKE "deg2rad :PI / 180
MAKE "prA 14.5
MAKE "piR 0.5
MAKE "piR2 0.25
MAKE "pitch 24
MAKE "facesegments 4
 
MAKE "gear1center POS
MAKE "gear1line gear :piR :prA :pitch 10
MAKE "gear2line gear :piR2 :prA :pitch 10
 
RT 90
FD :piR + :piR2 + :piR2/20 ;some slack
LT 90
MAKE "gear2center POS
 
SETPOS :gear1center
 
TRANSFORM
MAKE "gear1spin OBJECT
SET :gear1spin "rotation [0 0 1 0]
SET :gear1spin "center :gear1center
SETPARENT :gear1spin
TRANSFORM
MAKE "gear1 OBJECT
SET :gear1 "rotation [0 0 1 0]
SET :gear1 "center :gear1center
SETPARENT :gear1
SETMATERIAL 3 33
gear3d :gear1line -0.2 45 :piR :facesegments
SETPARENT "root
 
TRANSFORM
MAKE "gear2spin OBJECT
SET :gear2spin "rotation [0 0 1 0]
SET :gear2spin "center :gear1center
SET :gear2spin "translation :gear2center
SETPARENT :gear2spin
TRANSFORM
MAKE "gear2 OBJECT
SET :gear2 "rotation [0 0 1 0]
SET :gear2 "center :gear1center
;set :gear2 "translation :gear1center
SETPARENT :gear2
SETMATERIAL 3 28
gear3d :gear2line -0.2 -45 :piR2 :facesegments
SETPARENT "root
 
TRANSFORM
MAKE "gear3rot OBJECT
SET :gear3rot "rotation [0 1 0 1.57]
SET :gear3rot "center :gear1center
SET :gear3rot "translation (LIST 0.1 (:piR + :piR2) -0.1)
SETPARENT :gear3rot
TRANSFORM
MAKE "gear3spin OBJECT
SET :gear3spin "rotation [0 0 1 0]
SET :gear3spin "center :gear1center
SET :gear3spin "translation :gear1center
SETPARENT :gear3spin
TRANSFORM
MAKE "gear3 OBJECT
SET :gear3 "rotation [0 1 0 0]
SET :gear3 "center :gear1center
;set :gear2 "translation :gear1center
SETPARENT :gear3
SETMATERIAL 3 28
gear3d :gear2line -0.2 45 :piR2 :facesegments
SETPARENT "root
 
MAKE "beta (TAN :prA) - (:prA * :deg2rad)
;inital rotations
SET :gear1 "rotation (LIST 0 0 1 -:beta)
SET :gear2 "rotation (LIST 0 0 1 :PI/:pitch/:piR2 - :beta)
SET :gear3 "rotation (LIST 0 0 1 :PI/:pitch/:piR2/2 - :beta)
 
MAKE "period1 60
SPIN :gear1spin "tl :period1
SPIN :gear2spin "tr :period1 * :piR2/:piR
SPIN :gear3spin "tl :period1 * :piR2/:piR
 
PU
SHOWALL

This update take advantage of how scope works in logo, eg. that variables are available to subprocedures.

aplesch's blog
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Complication

Submitted by Andy on Tue, 27/10/2015 - 23:15.

Using Logo to write one is possble, and the SET command should work.

I was actually thinking about in javascript to handle this so users do not need to worry. The complication is that the Color also uses the same index, and if coordinate index is going to be changed, I will need to keep a copy to the Color index, or even do the same to remove duplicated colors...(in fact, colors are mostly the same or repeating the 16 built-in colors. Removing duplicates can reduce file size).

At the moment, colors and coordinates are using the same index. If changed the coordinate index, x3dom should report errors. But if PCOFF (PenColorOff) for FACE, then no Color node will be created but only Material node. This way changing the coordinate index should be fine.

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