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Andy

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Spur gears

Submitted by Andy on Tue, 20/10/2015 - 21:51

Finally, not perfect at all but working gear set in 3D. Thanks to asplech's blog about involute gear and this Spur Gear document, I finally get to know better about these cogged wheels.

The 3D world below shows 4 gears meshing together. Because they have different number of teeth, they would have created some mechanical advantages/disadvantages in changing directions, speeds, and forces. Click on the cylinder can start or stop the animation. You can also navigate and zoom in to see the gears are actually meshing well.

The Logo program is commented below. I thought that I can make it short but still end up with almost 200 lines. The good thing is that you can change the pressure angle, diametral pitch, teeth number, and thickness. The gear generated should have same tooth length and as long as they have the same pressure angle and diametral pitch, they should mesh well. However, it will be a little bit tricky to spin at the right speed and direction, particularly when the number of teeth is an odd number.

; Spur Gear http://www.gearseds.com/files/6.3.1_Gear_Terms_Lesson_rev3.pdf
RESET
; Global parameters for meshing gears
MAKE "pa 20 ; pressure angle, common angles are 14.5, 20
MAKE "dp 24 / 2.54 ; diametral pitch = tn / pd (in inch) 
 
; below are derived from global parameters above
MAKE "add 1 / :dp ; addendum
MAKE "ddd 1.157 / :dp ; dedendum
 
; tn: number of teeth
; thickness: thickness of gear
; clearance: not implemented in this prototype
; gear3d will output the parent transform id for the gear.
TO gear3d :tn :thickness
  ; find four key diamater from outside to inside
  MAKE "od (:tn + 2) / :dp ; overall diameter of the gear = (tn+2)/dp
  MAKE "or :od / 2 ; overall radius
  MAKE "pd :tn / :dp ; pitch diameter
  MAKE "pr :pd / 2 ; pitch radius
  MAKE "bd :pd * COS :pa ; base diameter
  MAKE "br :bd / 2 ; base radius
  MAKE "rd (:tn - 2) / :dp ; root diameter
  MAKE "rr :rd / 2 ; root radius
  ; other derivitives
  MAKE "ta 360 / :tn ; angle between two teeth
  ; oc, oc1: outside circle coordinates array
  ; pc, pc1: pitch circle coordinates array
  ; bc, bc1: base circle coordinates array
  ; rc, rc1: root circle coordinate array
  MAKE "oc []
  MAKE "oc1 []
  MAKE "pc []
  MAKE "pc1 []
  MAKE "bc []
  MAKE "bc1 []
  MAKE "rc []
  MAKE "rc1 []
  ; record initial pos and ori
  MAKE "pos POS
  MAKE "ori ORI
  RU 90
  ; collect coordinates
  REPEAT :tn [
    ; collect oc, oc1
    LT :ta/14 FD :or QUEUE "oc POS RD 90 FD :thickness QUEUE "oc1 POS
    BK :thickness RU 90 BK :or
    RT :ta/14*2 FD :or QUEUE "oc POS RD 90 FD :thickness QUEUE "oc1 POS
    BK :thickness RU 90 BK :or
    LT :ta/14
    ; collect pc, pc1
    LT :ta/4 FD :pr QUEUE "pc POS RD 90 FD :thickness QUEUE "pc1 POS
    BK :thickness RU 90 BK :pr
    RT :ta/4*2 FD :pr QUEUE "pc POS RD 90 FD :thickness QUEUE "pc1 POS
    BK :thickness RU 90 BK :pr
    LT :ta/4
    ; collect bc, bc1
    LT :ta/3.6 FD :br QUEUE "bc POS RD 90 FD :thickness QUEUE "bc1 POS
    BK :thickness RU 90 BK :br
    RT :ta/3.6*2 FD :br QUEUE "bc POS RD 90 FD :thickness QUEUE "bc1 POS
    BK :thickness RU 90 BK :br
    LT :ta/3.6
    ; collect rc, rc1
    LT :ta/3.4 FD :rr QUEUE "rc POS RD 90 FD :thickness QUEUE "rc1 POS
    BK :thickness RU 90 BK :rr
    RT :ta/3.4*2 FD :rr QUEUE "rc POS RD 90 FD :thickness QUEUE "rc1 POS
    BK :thickness RU 90 BK :rr
    LT :ta/3.4
    RT :ta
  ]
  SETPOS :pos
  SETHEADING :ori
  FD :thickness/2 MAKE "posCenter POS
  FD :thickness/2 MAKE "pos1 POS
  ; draw gear
  SETPOS :pos
  SETHEADING :ori
  TRANSFORM
  MAKE "obj OBJECT
  SETPARENT :obj
  SET OBJECT "center :posCenter
  FACE PCOFF
  ; draw first cross section
  SETMAT 0 33
  SETTR 200 ; make it a little transparent
  PD
  SETPOS LAST :rc
  REPEAT :tn [
    MAKE "n 2 * REPCOUNT
    SETPOS ITEM :n-1 :rc
    SETPOS ITEM :n-1 :bc
    SETPOS ITEM :n-1 :pc
    SETPOS ITEM :n-1 :oc
    SETPOS ITEM :n :oc
    SETPOS ITEM :n :pc
    SETPOS ITEM :n :bc
    SETPOS ITEM :n :rc
  ]
  PU
  SET OBJECT "rotation :ori
  SET OBJECT "center :pos
  ; draw second cross section
  SETPOS :pos1
  PD
  SETPOS LAST :rc1
  REPEAT :tn [
    MAKE "n 2 * REPCOUNT
    SETPOS ITEM :n-1 :rc1
    SETPOS ITEM :n-1 :bc1
    SETPOS ITEM :n-1 :pc1
    SETPOS ITEM :n-1 :oc1
    SETPOS ITEM :n :oc1
    SETPOS ITEM :n :pc1
    SETPOS ITEM :n :bc1
    SETPOS ITEM :n :rc1
  ]
  PU
  SET OBJECT "rotation :ori
  SET OBJECT "center :pos
  ; draw side faces
  SETPOS FIRST :rc
  SETMAT 0 25
  PD
  REPEAT :tn [
    MAKE "n 2 * REPCOUNT
    SETPOS ITEM :n-1 :rc1 SETPOS ITEM :n-1 :bc1 SETPOS ITEM :n-1 :bc JUMP
    SETPOS ITEM :n-1 :bc1 SETPOS ITEM :n-1 :pc1 SETPOS ITEM :n-1 :pc JUMP
    SETPOS ITEM :n-1 :pc1 SETPOS ITEM :n-1 :oc1 SETPOS ITEM :n-1 :oc JUMP
    SETPOS ITEM :n-1 :oc1 SETPOS ITEM :n :oc1 SETPOS ITEM :n :oc JUMP
    SETPOS ITEM :n :oc1 SETPOS ITEM :n :pc1 SETPOS ITEM :n :pc JUMP
    SETPOS ITEM :n :pc1 SETPOS ITEM :n :bc1 SETPOS ITEM :n :bc JUMP
    SETPOS ITEM :n :bc1 SETPOS ITEM :n :rc1 SETPOS ITEM :n :rc JUMP
    IFELSE :n+1 > COUNT :rc
      [ SETPOS ITEM :n :rc1 SETPOS FIRST :rc1 SETPOS FIRST :rc ]
      [ SETPOS ITEM :n :rc1 SETPOS ITEM :n+1 :rc1 SETPOS ITEM :n+1 :rc JUMP ]
  ]
  PU
  SET OBJECT "rotation :ori
  SET OBJECT "center :posCenter
  ; draw pitch circle
  LINE
  SETPOS LAST :pc
  PD
  REPEAT :tn [
    MAKE "n 2 * REPCOUNT
    SETPOS ITEM :n-1 :pc
    SETPOS ITEM :n :pc
  ]
  PU
  ; return to starting pos and ori
  SETPOS :pos
  SETHEADING :ori
  SETPARENT "root
  OUTPUT :obj
END
 
TO getPR :tn
  OUTPUT :tn / :dp / 2 ; output pitch radius
END
 
CLEAN HOME
RU 90 SETSCALE .2 4 .2
SETMAT 0 30 CYLINDER
RESET HOME
; gear3d :tn :thickness
MAKE "g1n 20
MAKE "g2n 11
MAKE "g3n 8
MAKE "g4n 14
MAKE "gear1 gear3d :g1n .2
RT 90 FD getPR :g1n FD getPR :g2n LT 90
TR 90 - 180/:g2n
MAKE "gear2 gear3d :g2n .2
; collect rotation value
MAKE "g2r [] QUEUE "g2r ORI
REPEAT 4 [ TL 90 QUEUE "g2r ORI ]
HOME
BK 1
MAKE "gear3 gear3d :g3n .2
LT 90 FD getPR :g3n FD getPR :g4n RT 90
; tr 90 - 180/:g2n
MAKE "gear4 gear3d :g4n .2
SPIN :gear1 "tr 60
SPIN :gear2 "tl 60 / (:g1n/:g2n)
SET WORD "spin_oi_ :gear2 "keyvalue :g2r
 
SPIN :gear3 "tr 60
SPIN :gear4 "tl 60 / (:g3n/:g4n)
 
TOGGLE "act1 WORD "spin_ts_ :gear1 "enabled
TOGGLE "act2 WORD "spin_ts_ :gear2 "enabled
TOGGLE "act3 WORD "spin_ts_ :gear3 "enabled
TOGGLE "act4 WORD "spin_ts_ :gear4 "enabled
 
CLICK "obj_0 "act1
CLICK "obj_0 "act2
CLICK "obj_0 "act3
CLICK "obj_0 "act4

The gear tooth is not as smooth as the involute gear. But it demonstrates a Logo way to collect circle coordinates and geometric shapes created later by connecting these coordinates. Same principle can be applied when I have implemented EXTRUSION, 2D coordinates can be collected as the cross section and be extruded. The rotation (SPIN) of the 11 teeth gear is a bit tricky. Line 175-176 collects the orientation and apply to the OrientationInterpolator's keyvalue in order to syn witht the gear ratio (applied to TIMESENSOR's cycleInterval).

Files: gear3d.x3d gear3d_proto1.logo

Andy's blog
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Comments

Mesh

Submitted by aplesch on Wed, 21/10/2015 - 09:53.

The combined movement of the gears looks really good. Is it the example from the gear document for teaching ?

Good idea to customize the keyvalues. I ended up using two transforms.

Teaching gears

Submitted by Andy on Wed, 21/10/2015 - 13:45.

The gear document for teaching is a surprise I found online. It even maps to the English, Mathematics and Science and Technology standards for Year 9-12 in the US. One of my major goals of VRMath2 is to support the teaching and learning in STEM areas. That Spur Gear teaching document is a good example for this website.

I just noticed that it has many examples of meshing gears but I did not follow them when I did. I just want to try how to mesh gears with different numbers of teeth. A 3D world like that can teach/learn about gear ratio, rotation speeds and directions, which are topics in STEM. Of course, the best way of learning is for students to go though the construction as we did, not just see what we have created.

Yes, two transforms will work.. but I wonder why not one parent transform in my case... My gear coordinates are already rotated. This is something I need to look into...

You have spent much time on VRMath2 sites, which I really appreciate. I hope you found VRMath2 interesting and useful for some of your project.

Converted

Submitted by Andy on Tue, 20/10/2015 - 22:44.

I tried to import this gear3d.x3d into Blender and MeshLab and they both did, including the cylinder. This means that I can export as STL or PLY for 3D printing of gears.

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