Mambo Sample Projects - Rolling Contact

Reference: Chapter 9 in Multibody Mechanics and Visualization by Harry Dankowicz, published by Springer Verlag UK, 2004.

Objectives

To illustrate the significance of using initial conditions that satisfy the configuration constraints.

Procedure

This Mambo project contains two non-stationary spheres and a stationary plane. The configuration of the non-stationary spheres relative to the world observer may be specified in three independent steps, namely

Specify the position of the center of the first sphere.
Specify the orientation of the first sphere.
Specify the position of the second sphere on the surface of the first sphere.

The motion of the first sphere is constrained in such a way that it makes rolling contact with the plane at the location of the second sphere provided that tangential contact is established by the initial configuration.

Parameters

rho: scaling factor corresponding to the radius of the first sphere.

State variables

q1, q2, and q3: coordinates of the center of the sphere in a coordinate system with origin at the reference point of the world observer and axes parallel to the basis vectors of the reference triad of the world observer.

q4, q5, and q6: 3-1-3 sequence of Euler angles representing the orientation of the first sphere relative to the reference triad of the world observer.

q7 and q8: spherical angle coordinates of the center of the second sphere in a coordinate system with origin at the center of the first sphere and axes parallel to the basis vectors of the reference triad of the first sphere.

Tasks

Use initial conditions that do not satisfy the configuration constraint and explain the resulting motion.

Download

rollingcontact.zip (zip file, 17 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,A):
>DeclarePoints(W,A,P):
>DeclareTriads(w,a):
>DefineObservers([W,W,w],[A,A,a]):
>DefinePoints([W,A,w,q1,q2,q3],[A,P,a,rho*sin(q7)*cos(q8),rho*sin(q7)*sin(q8),rho*cos(q7)]):
>DefineTriads(w,a,[q4,3],[q5,1],[q6,3]):
>DefineNeighbors(W,A):

Associate MAMBO objects

>DefineObjects(
   [W,'Block',xlength=rho*10,ylength=rho*10,zlength=rho/20,color=green],
   [A,'Sphere',radius=rho,color=red],
   [A,'Sphere',point=P,radius=rho/7,color=blue]):

Constrain geometry and motion

>DeclareStates(q1,q2,q3,q4,q5,q6,q7,q8):
>confconst1:=FindTranslation(W,P) &oo MakeTranslations(w,3)=0:
>confconst2:=MakeTranslations(a,rho*cos(q7)*cos(q8),rho*cos(q7)*sin(q8),-rho*sin(q7)) &oo MakeTranslations(w,3)=0:
>confconst3:=MakeTranslations(a,-rho*sin(q7)*sin(q8),rho*sin(q7)*cos(q8),0) &oo MakeTranslations(w,3)=0:
>motionconst1:=DiffTime(confconst1):
>motionconst2:=DiffTime(confconst2):
>motionconst3:=DiffTime(confconst3):
>linvel:=LinearVelocity(W,A):
>angvel:=AngularVelocity(w,a):
>motionconst4:=(linvel &++ (angvel &xx FindTranslation(A,P))) &oo MakeTranslations(w,1)=0:
>motionconst5:=(linvel &++ (angvel &xx FindTranslation(A,P))) &oo MakeTranslations(w,2)=0:
>kde:={seq(angvel &oo MakeTranslations(a,i)=cat(u,i+3),i=1..3),motionconst1,motionconst2,motionconst3,motionconst4,motionconst5}:

Export geometry and motion description

>GeometryOutput(main=W,parameters=[rho],states=[q1,q2,q3,q4,q5,q6,q7,q8],checktree,checkargs,filename="rollingcontact.geo"):
>MotionOutput(ode=kde,states=[q1=0,q2=0,q3=1,q4=.6,q5=1.8,q6=-0.1,q7=7.624777961,q8=-7.753989756],parameters=[rho=1],insignals=[u4=cos(2t),u5=sin(t),u6=.4*cos(t)],checkargs,checksings,filename="rollingcontact.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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