.Modelica.Mechanics.MultiBody.Joints.Assemblies.JointUSR

Information

This component consists of a universal joint at frame_a, a revolute joint at frame_b and a spherical joint which is connected via rod1 to the universal and via rod2 to the revolute joint, see the default animation in the following figure (the axes vectors are not part of the default animation):

model Joints.Assemblies.JointUSR

This joint aggregation has no mass and no inertia and introduces neither constraints nor potential state variables. It should be used in kinematic loops whenever possible since the non-linear system of equations introduced by this joint aggregation is solved analytically (i.e., a solution is always computed, if a unique solution exists).

The universal joint is defined in the following way:

The definition of axis 2 of the universal joint is performed according to the most often occurring case for the sake of simplicity. Otherwise, the treatment is much more complicated 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 parallel to each other. Therefore, if possible n1_a should be selected in such a way that it is perpendicular to rRod1_ia in the initial configuration (i.e., the distance to the singularity is as large as possible).

The rest of this joint aggregation is defined by the following parameters:

An additional frame_ia is present. It is fixed in the rod connecting the universal and the spherical joint at the origin of frame_a. The placement of frame_ia on the rod is implicitly defined by the universal joint (frame_a and frame_ia coincide when the angles of the two revolute joints of the universal joint are zero) and by parameter vector rRod1_ia, the position vector from the origin of frame_a to the spherical joint, resolved in frame_ia.

An additional frame_ib is present. It is fixed in the rod connecting the revolute and the spherical joint at the side of the revolute joint that is connected to this rod (= rod2.frame_a = revolute.frame_a).

An additional frame_im is present. It is fixed in the rod connecting the revolute and the spherical joint at the side of the spherical joint that is connected to this rod (= rod2.frame_b). It is always parallel to frame_ib.

The easiest way to define the parameters of this joint is by moving the MultiBody system in a reference configuration where all frames of all components are parallel to each other (alternatively, at least frame_a and frame_ia of the JointUSR joint should be parallel to each other when defining an instance of this component).

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

parameter Real rod1Length(unit="m")  "Length of rod 1";
parameter Real eRod1_ia[3] "Unit vector along rod 1, 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 rod 1. For example, if a box shall be connected at frame_ia directing from the origin of frame_a to the middle of rod 1, this might be defined as:

  Modelica.Mechanics.MultiBody.Joints.Assemblies.JointUSP jointUSR(rRod1_ia={1.2, 1, 0.2});
  Modelica.Mechanics.MultiBody.Visualizers.FixedShape     shape(shapeType       = "box",
                                             lengthDirection = jointUSR.eRod1_ia,
                                             widthDirection  = jointUSR.e2_ia,
                                             length          = jointUSR.rod1Length/2,
                                             width           = jointUSR.rod1Length/10);
equation
  connect(jointUSP.frame_ia, shape.frame_a);

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