BBC Mi­cro Bit projects

IN THIS IS­SUE, WE EN­JOY THE MATH­E­MAT­ICS OF CYCLOID CURVES TO MAKE A DRAW­ING MA­CHINE — AN EASY-TO-BUILD PRO­JECT TO DRAW IN­CRED­I­BLE GRAPHIC TEX­TURES AND PAT­TERNS

The Shed - - Contents - By En­rico Miglino Pho­to­graphs: En­rico Miglino

A draw­ing ma­chine in the spirit of the 1960s Spiro­graph

Acy­cloid ge­o­met­ri­cal move­ment has a num­ber of ap­pli­ca­tions in the study of equa­tions, as well as in me­chan­ics, science, and other fields.

‘Cy­cloids’ — curves traced from a point on a cir­cle ro­tat­ing along a lin­ear path — are in­volved in a num­ber of equa­tions in trigonom­e­try and ge­om­e­try. For ex­am­ple, cy­cloids are used in me­chan­ics to cal­cu­late the move­ment and size of the gears to de­sign gear­boxes for speed re­duc­tion.

As shown in the cycloid scheme

(page 73), if we ide­ally put a pen where the ro­tat­ing point lays, it draws a se­ries of chained arcs. Things be­come more in­ter­est­ing if we ap­ply some con­straints to the cir­cle with the draw­ing point.

Sup­pose, for ex­am­ple, you add sup­port to the ro­tat­ing cir­cle by plac­ing the point ex­ter­nal to its cir­cum­fer­ence — the de­sign is more in­ter­est­ing than the pre­vi­ous scheme but is it al­ways a se­ries of re­peated curves along a lin­ear path.

Toys from the ’60s

Do you re­mem­ber the Spiro­graph from the 1960s? When I was a child I spent hours mak­ing de­signs with this sim­ple but fas­ci­nat­ing toy. It is based on cycloid maths, where the ideal point on the cir­cum­fer­ence ro­tates inside a larger cir­cle in­stead of a straight line; ev­ery full ro­ta­tion the curve is repli­cated, shifted by some cir­cu­lar de­grees, cre­at­ing the draw­ing tex­ture. Spiro­graph draw­ing tools are lim­ited to cre­at­ing cir­cu­lar draw­ings.

Fol­low­ing the same math­e­mat­i­cal prin­ci­ple of the cycloid are the more com­plex curves of the same fam­ily, named ‘hypocy­cloids’.

How can we make me­chan­i­cal draw­ings over rec­tan­gu­lar pa­per sheets? We should con­nect two cycloid move­ments in­ter­act­ing to­gether. Us­ing a BBC Mi­cro Bit and a few more com­po­nents we can eas­ily create a ver­sa­tile draw­ing ma­chine.

Me­chan­i­cal prin­ci­ples

Be­fore de­sign­ing the parts in prac­tice, let me ex­plain the me­chan­i­cal prin­ci­ple.

Do you re­mem­ber the Spiro­graph from the ’60s?

Based on the math­e­mat­i­cal as­sump­tions of cycloid move­ment, the draw­ing scheme (top right page 73) shows the min­i­mal me­chan­i­cal com­po­nents to gen­er­ate the move­ment. D1 and D2 are the two wheels gen­er­at­ing the move­ment. D1 al­ters the sim­ple ro­ta­tion of the wheel D2 that drives the pen in S. The length of the two bars, L1 and L2, change the pat­tern of the draw­ing, as well as the dis­tance of the cen­tre of the two wheels and the at­tach point of the bars to the ro­tat­ing wheels.

Start­ing from this es­sen­tial de­sign we can add other ro­tat­ing wheels — and up to four mo­tors — and bars, whose di­am­e­ter and length can create new pat­terns of different sizes and shapes.

I de­signed the plat­form as a mod­u­lar and gen­eral-pur­pose tool, easy to make and as­sem­ble, as well as mod­ify or change some of its fea­tures. To move the wheels with a good speed con­trol I have used mi­cro-geared mo­tors from Kitronik, which I have used for other projects and found they per­form very well.

Start­ing from the few geo­met­ric and math­e­mat­i­cal con­cepts de­scribed I have de­signed the me­chan­i­cal com­po­nents to gen­er­ate the re­cur­sive move­ment. I sug­gest ex­plor­ing the pos­si­bil­i­ties for us­ing this de­sign with other wheels and bars to create different curves that can be cus­tom­ized to change the speed and ro­ta­tion di­rec­tion of the wheels.

Cre­at­ing the com­po­nents

De­ter­mined by the shaft di­am­e­ter and the size of the mo­tors, I de­signed a

first draft of the idea, then cre­ated the com­po­nents with Fu­sion 360 for the 3D-printed parts, and Inkscape for the 2D vec­to­rial de­signs for the parts that I have laser-cut us­ing 3mm Per­spex sheet.

The Mi­cro Bit soft­ware has been de­signed us­ing the Mi­crosoft Make­code web plat­form, ac­ces­si­ble from the Mi­cro Bit web­site. The same soft­ware, based on graphic blocks, has been con­verted to JavaScript (one of the fea­tures avail­able from the web ap­pli­ca­tion). It has been saved on the GitHub repos­i­tory, where you can also find the STL files to build the com­po­nents (al­icemir­ror.github.io/ draw­ing­ma­chine).

Pro­gramme us­age notes

The draw­ing process can be con­trolled by the Mi­cro Bit with the two but­tons, A and B, and shak­ing the Mi­cro Bit board:

• Press but­ton A + B: Switch be­tween con­trol­ling the big wheel or the small wheel.

• Press But­ton A: De­crease the mo­tor speed.

• Press But­ton B: In­crease the mo­tor speed.

• Shake the board: Start/Stop the draw­ing process.

The Java source and the cor­re­spond­ing blocks pro­gramme from the Mi­crosoft Make­code site. The full source is avail­able on the GitHub repos­i­tory

Left: Draft sketches of the me­chan­i­cal parts and sizes for 3D de­sign of the 3D-printed parts and the laser-cut flat parts (wheels and draw­ing bars)

Right: Schematic draw­ing of the me­chan­i­cal parts. D1 and D2 are the two wheels gen­er­at­ing the move­ment. D1 al­ters the sim­ple ro­ta­tion of the wheel D2 that drives the pen in S. The length of the two bars L1 and L2 change the pat­tern of the draw­ing, as well as the dis­tance of the cen­tre of the two wheels and the at­tach point of the bars to the ro­tat­ing wheels

Right: Cy­clodic move­ment

Be­low left: One of the wheels with the mo­tor shaft, top, and bot­tom side. The mo­tor shaft hole has been 3D printed about 0.25mm smaller, then the shaft has been in­serted by heat­ing the metal

Be­low right: The mo­tors con­nected for test­ing, and the mo­tor sup­ports with the mo­tors as­sem­bled inside

Above and left: De­tails of the 3D-printed pen holder. A screw is used to lock the pen at the right height. Note the nut in­serted in the plas­tic part to drive the lock screw

Above: The fin­ished kit. The laser-cut pieces are red while the 3D printed are black. The holes have a di­am­e­ter to hold an M2 Allen screw

Left and right: 3D print­ing the mo­tor sup­ports

Be­low, left to right: As­sem­bling the me­chan­i­cal parts of the draw­ing ma­chine

Above and right: The draw­ing ma­chine fin­ished with the set-up con­nected to the BBC Mi­cro Bit. Ev­ery­thing is ready for de­sign­ing

Be­low: The draw­ing ma­chine at work

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