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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

involute gear

Submitted by aplesch on Fri, 16/10/2015 - 22:49

It turned out that producing a set of involute gears requires quite a bit of geometry in addition to involutes.

It is useful to get familiar with gear terminology and with the special properties of an involute of a circle. In particular, it is necessary to figure the angular width of a tooth at the base circle which is where the involute, curved section of a tooth begins.

So there is a lot to explain in this logo program.

Next steps include to figure out why there is a slight overlap between teeth from both gears, and to make the gears 3d with some depth. Is there a way to use the extrusion functionality of x3d ?

[Update: the initial configuration was slightly off. Scene and logo updated.]

Well, here is the logo:

; involute: draw involute
; rb: base radius
; limit: max. generating angle
; steps: number of segments on involute to draw
 
TO involute :rb :limit :steps
 
  LOCALMAKE "start POS
  LOCAL "end
  LOCALMAKE "i 0
  LOCALMAKE "step 0.5 * :limit / :steps
  ;(print :step :steps)
  PU
 
  REPEAT ABS(:steps)
  [
    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 "end POS
    ;connect end points
    PD
    SETPOS :start
    ;move back
    PU
    SETPOS :end
    MAKE "start :end
    BACK 2 * :i * :deg2rad * :rb
    ;turn forward
    LT 180
    ;nextcolor
  ]
  SETPOS :start
 
END
 
; gearinvolute: draw involute side of gear tooth
; rb: base radius
; PA: pressure angle
; P: diametral pitch
; steps: number of subdivisions
 
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
  involute :rb (:limit / :deg2rad) :steps
 
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
 
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)
  RT 90
  FD :ddif
  LT 90
 
  ; save position
  LOCALMAKE "base POS
  LOCALMAKE "head HEADING
 
  gearinvolute :rb :PA :P :segments
 
  ;mirror
  ;or draw 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
  gearinvolute :rb :PA :P :segments * -1
 
  ;connect sides at top
  PD SETPOS :top PU
 
  SETPOS :base
  SETHEADING :head
 
  PD
  LT 90
  FD :ddif
  RT 90
 
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
 
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 )
 
  PU
  RT 90 FD :rr LT 90
  PD
 
  REPEAT :N
  [
    geartooth :rp :PA :P :delta / 2 :segments
    PD
    LT (:apitch - :delta) / 2
    FD 2 * :rr * SIN ((:apitch - :delta) / 2)
    LT (:apitch - :delta) / 2
    NEXTCOLOR
  ]
 
  PU
  LT 90 FD :rr RT 90
 
END
 
; mycircle: draw circle
; crd: circle radius
 
TO mycircle :crd
 
  PU
  RT 90 FD :crd LT 90
  PD
 
  REPEAT 180
  [
  LT 2
  FD 2 * :crd * SIN ( 2 )
  LT 2
  ]
 
  PU
  LT 90 FD :crd RT 90
  PD
 
END
 
;involute (for gears)
;comprehensive guide
;http://bostongear.com/pdf/gear_theory.pdf
 
RESET
CLEARSCREEN
SETBACKGROUND 33
RU 90
 
PD
MAKE "PI 4 * RADARCTAN(1)
MAKE "deg2rad :PI / 180
MAKE "prA 14.5
MAKE "piR 0.5
MAKE "pitch 16
 
;mycircle :baR
;mycircle :piR
;mycircle :ouR
 
TRANSFORM
SET OBJECT "rotation [0 0 1 0]
MAKE "gear1 OBJECT
SET :gear1 "center POS
 
SETPARENT OBJECT
gear :piR :prA :pitch 10
SETPARENT "root
 
MAKE "piR2 0.25
PU
RT 90
FD :piR + :piR2
LT 90
 
TRANSFORM
SET OBJECT "rotation [0 0 1 0]
MAKE "gear2 OBJECT
SET :gear2 "center POS
 
SETPARENT OBJECT
gear :piR2 :prA :pitch 10
SETPARENT "root
 
;adjust by beta, angular distance between point at base and point at pitch circle
MAKE "beta (TAN :prA) - (:prA * :deg2rad)
SET :gear1 "rotation (LIST 0 0 1 -:beta)
SET :gear2 "rotation (LIST 0 0 1 :PI/:pitch/:piR2 - :beta)
 
MAKE "period1 60
SPIN :gear1 "tl :period1
SPIN :gear2 "tr :period1 * :piR2/:piR
 
PU
SHOWALL

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Comments

gear ratio

Submitted by aplesch on Sat, 17/10/2015 - 06:17.

Yes, it should be possible to produce any standard involute gear. Gears fit together if they same pitch.

The velocity of the gear rotations needs to be inversely proportional the pitch radius. It is easiest to think about wheels instead. For a unit period of time both wheels need to cover the same distance on their circumference.

Good point about the possible lack of synchronization for the spinning of the gears. I will try to use just one time sensor, so both orientation interpolator should get the same input.

TimeSensor

Submitted by Andy on Sat, 17/10/2015 - 17:11.

Another way to do can be changing the startTime of the TimeSensors (if two). TimeSensor is fully implemented in X3DOM. The fields startTime and stopTime etc. will not show in DOM but I remembered the SET command can still get it to work.

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