UniversalSpherical

Description

This component consists of a universal joint at frame_a anda spherical joint at frame_b that are connected together witha rigid rod, see default animation figure (the arrows are notpart of the default animation):

This joint aggregation has no mass and no inertia and introduces the constraintthat the distance between the origin of frame_a and the origin of frame_b is constant(= Frames.length(rRod_ia)). The universal joint is defined in the following way:

• The rotation axis of revolute joint 1 is along parameter vector n1_a which is fixed in frame_a.
• The rotation axis of revolute joint 2 is perpendicular to axis 1 and to the line connecting the universal and the spherical joint.

The definition of axis 2 of the universal joint is performed accordingto the most often occurring case. In a future release, axis 2 mightbe explicitly definable via a parameter. However, the treatment is much morecomplicated and the number of operations is considerably higher,if axis 2 is not orthogonal to axis 1 and to the connecting rod.

Note, there is a singularity when axis 1 and the connecting rod are parallelto other. Therefore, if possible n1_a should be selected in such a way that itis perpendicular to rRod_ia in the initial configuration (i.e., thedistance to the singularity is as large as possible).

An additional frame_ia is present. It is fixed in the connectingrod at the origin of frame_a. The placement of frame_ia on the rodis implicitly defined by the universal joint (frame_a and frame_ia coincidewhen the angles of the two revolute joints of the universal joint are zero)and by parameter vector rRod_ia, the position vectorfrom the origin of frame_a to the origin of frame_b, resolved in frame_ia.

The easiest way to define the parameters of this joint is by moving theMultiBody system in a reference configuration where all framesof all components are parallel to other (alternatively,at least frame_a and frame_ia of the UniversalSpherical jointshould be parallel to other when defining an instance of thiscomponent). Since frame_a and frame_ia are parallel to other,vector rRod_ia from frame_a to frame_b resolved in frame_ia can be resolvedin frame_a (or the world frame, if all frames are parallel to other).

This joint aggregation can be used in cases wherein reality a rod with spherical joints at end are present.Such a system has an additional degree of freedom to rotatethe rod along its axis. In practice this rotation is usuallyof no interest and is mathematically removed by replacing oneof the spherical joints by a universal joint. Still, in mostcases the Joints.SphericalSpherical joint aggregation can be used insteadof the UniversalSpherical jointsince the rod is animated and its mass properties are approximated bya point mass in the middle of the rod. The SphericalSpherical jointhas the advantage that it does not have a singular configuration.

In the public interface of the UniversalSpherical joint, the following(final) parameters are provided:

  parameter Real rodLength(unit="m")  "Length of rod";  parameter Real eRod_ia[3] "Unit vector along rod, resolved in frame_ia";  parameter Real e2_ia  [3] "Unit vector along axis 2, resolved in frame_ia";

This allows a more convenient definition of data which is related to the rod.For example, if a box shall be connected at frame_ia directing fromthe origin of frame_a to the middle of the rod, this might be defined as:

    Modelica.Mechanics.MultiBody.Joints.UniversalSpherical jointUS(rRod_ia={1.2, 1, 0.2});    Modelica.Mechanics.MultiBody.Visualizers.FixedShape    shape(shapeType       = "box",                                              lengthDirection = jointUS.eRod_ia,                                              widthDirection  = jointUS.e2_ia,                                              length          = jointUS.rodLength/2,                                              width           = jointUS.rodLength/10);  equation    connect(jointUS.frame_ia, shape.frame_a);