Wunderlich Directory Reference

Detailed Description

[Introduction] [Geometry] [Process] [Statistics] [Results][References]

Introduction

(taken from Karl Wohlhart)

In 1954 Walter Wunderlich published a paper [Wunderlich, 1954] in which he describes a planar twelve-bar mechanism with six parallelogram or antiparallelogram loops which can be arranged in four different closure modes, all of them movable with either one or two degrees of freedom. What makes this mechanism especially remarkable is the fact that by passing a singularity position the mechanism might change its movability. In that case the singularity position is, therefore, not a bifurcation position, but a sort of mobility turning position. The figure above shows Wunderlich's mechanism in two different positions. In the position in which it has two parallelogram and four antiparallelogram loops, the mechanism is movable with mobility 1 because the two angles $ \Psi $ and $ \Phi $ are related by the equation:

\[ \tan^2 \frac{\Psi}{2} \: k_2 \: [1 - k_1\: \tan^2 \frac{\Phi}{2}]+ \tan \frac{\Psi}{2} \tan \frac{\Phi}{2} \: (1+k_1+k_2+k_1 k_2) + k_1 \:\tan^2\frac{\Phi}{2} -1 =0. \]

wherein

\[k_1 = \frac{a + b}{a - b},\]

\[ k_2=\frac{c + b}{c - b} \]

are the system parameters. In the position, however, in which Wunderlich's mechanism has four parallelogram and two antiparallelogram loops, its mobility is 2, as $Psi$ and $Phi$ can be chosen arbitrarily. The passage from one position to the other goes smoothly through a singularity position ( $\Phi = 0$). The Wunderlich Mechanism, therefore, belongs to the category of kinematotropic linkages.

Geometry

The Wunderlich we formulate has

  • a =9
  • b =6
  • c =3

Process

This example is treated following these steps (from the main CuikSuite folder):

  • Generate the equations: Execute
  • Solve the positional analysis problem: Execute the parallel version of cuik (the execution time takes long)
    • bin/cuik examples/Wunderlich/Wunderlich
  • Examine the isolated configuration space:
    • bin/cuikplot3d examples/Wunderlich/Wunderlich 3 21 37 0 wunderlich.gcl
    • geomview wunderlich.cgl
  • Examine the configuratoins:

Statistics

Characteristics of the problem:

Nr. of indep. loops 5
Nr. of links 12
Nr. of joints 8
Nr. of equations (in the simplified system) 21
Nr. of variables (in the simplified system) 22

Here you have the statistics about the execution (on an Intel Core i7 at 2.9 Ghz).

Nr. of Empty boxes 3616
Nr. of Solution boxes 82483
Execution time (s) 4000

Results

This is a snapshot of the projection that is obtained with the procedure described above:

This is a couple of the computed configurations (out of 82483 :)

References

  • W. Wunderlich, Ein merkwürdiges Zwölfstabgetriebe, Österreichisches Ingenieurarchiv, Band 8, Heft 2/3, pp. 224-228, 1954.
  • K. Wohlhart, "Kinematotropic Linkages", Recent Advances in Robot Kinematics, J. Lenarcic; and V. Parenti-Castelli (eds.), Kluwer Academic Publishers, 1996.

Files

file  Wunderlich.world [code]
 The Wunderlich mechanism.