Mambo Sample Projects - Bead on Sphere

Reference: Chapter 2 in Multibody Mechanics and Visualization by Harry Dankowicz, published by Springer Verlag UK, 2004.

Objectives

To illustrate the use of sequences of pure rotations to effect changes in position.

Procedure

This Mambo project contains a non-stationary bead, a non-stationary block and one stationary sphere. The non-stationary bead is fixed with respect to the non-stationary block. The non-stationary block may be reoriented relative to the world observer in two independent steps, namely

Impose a rotation about the 3-direction of the world reference triad.
Impose a subsequent rotation about the 1-direction of the world reference triad.

Parameters

l1, l2, and l3: lengths of the sides of the block.
R: distance of the bead from the center of the block.
marker: radius of the bead.

State variables

theta: angle of rotation of the block about the 3-direction of the world reference triad.
phi: angle of the subsequent rotation of the block about the 1-direction of the world reference triad.

Tasks

Animate the predefined motion and make use of the wiremesh view.

Download

bead.zip (zip file, 7 kb, December 30, 2012)
This zip archive contains MAMBO and the MAMBO toolbox source files to visualize the MAMBO project and regenerate visualization files using maple 16, mupad 5.6.0 (matlab R2011a), and mathematica 8.0 versions of the code found below.

Initialize

>Restart():

Establish a geometry

>DeclareObservers(W,B,A):
>DeclarePoints(W,seq(seq(cat(E,i,j),i=1..3),j=1..4)):
>DeclareTriads(w,b,a):
>DefineObservers([W,W,w],[B,W,b],[A,W,a]):
>DefinePoints([W,E11,a,0,l2/2,l3/2],[W,E12,a,0,l2/2,-l3/2],[W,E13,a,0,-l2/2,-l3/2],[W,E14,a,0,-l2/2,l3/2],[W,E21,a,l1/2,0,l3/2],[W,E22,a,l1/2,0,-l3/2],[W,E23,a,-l1/2,0,-l3/2],[W,E24,a,-l1/2,0,l3/2],[W,E31,a,l1/2,l2/2,0],[W,E32,a,l1/2,-l2/2,0],[W,E33,a,-l1/2,-l2/2,0],[W,E34,a,-l1/2,l2/2,0]):
>DefineTriads(w,b,theta,3):
>DefineTriads(b,a,phi,seq(MakeTranslations(w,1) &oo MakeTranslations(b,i),i=1..3)):
>DefineNeighbors([W,B],[B,A]):

Associate MAMBO objects

>DefineObjects([W,'Sphere',radius=R*(1-exp(-(t-2))),color=yellow]):
>DefineObjects(
  seq([A,'Block',point=cat(E,1,j),xlength=l1,ylength=(l1+l2+l3)/30,zlength=(l1+l2+l3)/30],j=1..4),
  seq([A,'Block',point=cat(E,2,j),xlength=(l1+l2+l3)/30,ylength=l2,zlength=(l1+l2+l3)/30],j=1..4),
  seq([A,'Block',point=cat(E,3,j),xlength=(l1+l2+l3)/30,ylength=(l1+l2+l3)/30,zlength=l3],j=1..4)):
>DefineObjects([A,'Sphere',point=MakeTranslations(a,R,0,0),radius=marker,color=red]):

Export geometry and motion description

>GeometryOutput(main=W,states=[theta,phi],parameters=[R,l1,l2,l3,marker],checktree,checkargs,filename="bead.geo");
>MotionOutput(ode={thetat=u1,phit=u2},states=[theta,phi],parameters=[R=1,l1=1,l2=.5,l3=.25,marker=.2],insignals=[u1=(t&>=0)*(t&<1)+cos(sin(t))*(t&>=5),u2=(t&>=1)+sin(cos(t))*(t&>=5)],checkargs,checksings,filename="bead.dyn");

Sample projects

Bead on Sphere Bead on Block Bicycle Block Cone Crane Flopping Tetrahedron Hinge Lamp Motion Along Line Motion Along Plane No Translation Point Contact Puck Pyramid Radial Arm Saw Rigid-body Dynamics Rolling Contact Switching Constraints Tangential Contact Tetrahedron


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