Type "Eppendahl"

Dr. Adam Eppendahl devised this configuration from the constraints given by LEGO blocks. Make sure you understand the green and red symbols in the schematic to the right (otherwise see description syntax first).
Note: Don't be confused by the obvious lack of gears in the LEGO realization. As can be seen easily, the bevel gear (5) in my "packaged differential" schematic and the spur gear (4/10) in the schematic to the right can be omitted in pairs, as the gear integral to the differential can be meshed from the "wrong" side with LEGO blocks.

Equations:

(a:)	rot₂ + rot₅ + k * rot₃ = 0
(b:)	rot₆ + rot₈ - k * rot₇ = 0
(c:)	n₄ * rot₄ + n₁₁ * rot₁₁ = 0
(d:)	n₁₀ * rot₁₀ + n₁₁ * rot₁₁ = 0
(e:)	n₁₂ * rot₁₂ + n₁₁ * rot₁₁ = 0

The remaining constraints:

(f:)	rot₁ = rot₂
(g:)	rot₃ = rot₄
(h:)	rot₅ = rot₆
(i:)	rot₈ = rot₉
(j:)	rot₇ = rot₁₀

Composition:

(a,f=>k)	rot₅ = - rot₁- k * rot₃
(b,h,i=>l)	rot₅ = - rot₉ + k * rot₇
(k,l=>m)	rot₉ - rot₁ = k * (rot₃ + rot₇)
(c,g=>n)	rot₃ = - rot₁₁ * (n₁₁ / n₄)
(d,j=>o)	rot₇ = - rot₁₁ * (n₁₁ / n₁₀)
(m,n,o=>p)	rot₉ - rot₁ = - rot₁₁ * k * n₁₁ * (n₄ + n₁₀) / (n₄ * n₁₀)
(e,p=>q)	rot₁₂ = (rot₉ - rot₁) * n₄ * n₁₀ / ( k * n₁₂ * (n₄ + n₁₀))

To make it "south-pointing" requires, that if rot₁ is held 0, rot₁₂ has to be exactly one turn when wheel 9 of diameter d₉ covered one full circle (centre at wheel 1's contact to the ground, radius equal to track width t):

(r:)	d₉ * rot₉ * PI = 2 * t * PI	d₁ * rot₁ * PI = 2 * t * PI
(s:)	rot₁ = 0	rot₉ = 0
(t:)	rot₁₂ = 1	rot₁₂ = -1
(q=>)	rot₉ = k * n₁₂ * (n₄ + n₁₀) / (n₄ * n₁₀)	rot₁ = k * n₁₂ * (n₄ + n₁₀) / (n₄ * n₁₀)
(r=>)	d₉ = 2 * t / rot₉	d₁ = 2 * t / rot₁
(n₄ = n₁₀) => d₁ = d₉ = t * n₄ / ( k * n₁₂)

Reference	Size	Description
4/10	n₁₀ = [teeth]	Spur gear, fixed to drive shaft 3/7 of differential
12	n₁₂ = [teeth]	Spur gear, fixed to pointer
k	k =	Differential reduction factor
T	t = [length units]	Track width

Reference	Size	Description
1/9	d₁ = [units of t]	Road wheel, free running on axle

Type "Eppendahl"

Given Parameters (Your choice !)

Derived Parameters