Locked State

Box lid prototype in the locked state

This is part of a project I have been playing with on and off for the last 9 months. It is similar in concept to a reverse geocaching box but with cooler locking mechanism. I have always appreciated the giant vault door bolt mechanisms and figured designing a laser cut version controlled by a servo would be an interesting learning exercise.

To accomplish the desired effect I had to solve a fairly simple geometry problem for the linkage mechanism. I figured this might be of use to someone else or my future self so I will walk through the derivation of the basic equations for finding functional linkage lengths.

The Setup

Linear actuation

Rotary to linear actuation geometry

This problem has 5 variables:

  • L_1 – Length of the linkage when fully extended
  • L_2 – Length of the linkage when fully retracted
  • A – Length of the linkage attached to the rotary actuator
  • B – Length of the linkage attached to the bolt
  • \Theta – The number of degrees of rotation required to fully retract the bolt

To the Mathing!

I am going to start by assuming that A and B are the two unknowns as it simplifies the process dramatically. This is followed by a fairly straight forward application of the Law of Cosines which solves the problem nicely with a bit of manipulation:

    \[B^2 = A^2 + L_2^2 - 2 A L_1 cos\Theta\]

Solving for two variables is hard so since L_1 = A + B we can substitute in so we only have to solve one variable:

    \[(L_1 - A)^2 = A^2 + L_2^2 - 2 A L_2 cos\Theta\]

Expand the polynomial:

    \[L_1^2 - 2 A L_1 + A^2 = A^2 + L_2^2 - 2 A L_2 cos\Theta\]

Cancel out the A^2 on both sides:

    \[L_1^2 - 2 A L_1 = L_2^2 - 2 A L_2 cos\Theta\]

Move the constants to the left and all of the A terms to the right:

    \[L_1^2 - L_2^2 = 2 A L_1 - 2 A L_2 cos\Theta\]

Factor:

    \[L_1^2 - L_2^2 = 2 A (L_1 - L_2 cos\Theta)\]

Dive through by 2 (L_1 - L_2 cos\Theta) so we are left with A:

    \[\frac{1}{2} \frac{L_1^2 - L_2^2}{(L_1 - L_2 cos\Theta)} = A\]

End Result

How theta effects linkage lengths A (Blue), and B (Red) for L1=100, and L2=75

How Theta effects linkage lengths A (Blue) and B (Red) for L_1 = 100, and L_2 = 75

Tidying it up and bringing all of the variables over to the left hand side we are left with:

    \[L_1 > L_2\]

    \[A = \frac{1}{2} \frac{L_1^2 - L_2^2}{(L_1 - L_2 cos\Theta)}\]

    \[B = L_1 - A\]

Unlocked State

Box lid prototype in the unlocked state

I am hoping to have the full build for the box done by September and will be posting full build details and schematics.