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Subsubsection 3.11.1.5: Initial Rigid Angular Velocity Up Section 3.11: Rigid Section Subsubsection 3.11.2.1: Rigid Spherical Joint

### 3.11.2 Rigid Connectors

The rigid connector section allows users to define different type of connections between two rigid bodies. The specific connector is defined via the type attribute.
The rigid connectors fall into two categories. The first category define different joints that contstrain the relative motion of one rigid body with respect to the second one.
Rigid joints produce nonlinear constraints between rigid bodies, which prevent relative motion except along the degrees of freedom of the joint. The term rigid' refers to the bodies, not to the joints. Each rigid joint needs to define two rigid bodies ( and , a joint origin common to both bodies, and a set of axes that determine the relative orientation of the joint degrees of freedom. These axes define orthonormal basis vectors and on each rigid body, with both bases being coincident, , at the start of the analysis, and given in world coordinates.
The rigid joint nonlinear constraints produce reaction forces and moments that are enforced with penalty parameters and Lagrange multipliers. The penalty parameters, force_penalty and moment_penalty, may be conceptualized as stiffnesses of linear/torsional springs that prevent relative translations/rotations of the rigid bodies along degrees of freedom that must remain constrained for that joint. The penalty values should be selected based on a rough estimation of the maximum reactions forces and moments acting at these joints, divided by the maximum amount of linear/angular separation that your analysis can tolerate for that joint. Alternatively, set the force_penalty and moment_penalty parameters to 1 and turn on the auto_penalty; this setting will automatically adjust the force_penalty and moment_penalty to an appropriate value.
The augmented Lagrangian method is used by default, where the joint reaction force and moment are treated as Lagrange multipliers, augmented at each time step by the product of the force/moment penalty parameter and the linear/angular gap. Use the parameter maxaug to control the maximum allowable number of augmentations at each time step (maxaug=0 produces the penalty method); use a non-zero value for the parameter minaug to ensure a minimum number of augmentations. Augmentations will proceed until the relative change in the reaction force/moment magnitude is less than tolerance, and/or the linear gap is less than gaptol, and/or the angular gap is less than angtol. Setting any of these parameters to zero disables that check.
The nonlinear constraints that enforce these joints produce a non-symmetric stiffness matrix. Therefore, when using rigid joints, use the analysis setting for a non-symmetric formulation, see Section 3.3.4↑.
The following rigid joints can be defined.
 type Description rigid spherical joint Connect rigid bodies at point. rigid revolute joint Rotation about a single prescribed axis rigid prismatic joint Translation along a single prescribed axis. rigid cylindrical joint Rotation and translation about prescribed axis. rigid planar joint 2D plane translation and rotation about normal. rigid lock Prevent any relative motion between rigid bodies.
The second category of connectors apply a force on the rigid bodies that is proportional to some relative motion. The term rigid' refers to the bodies, not to the connectors.
The following rigid connectors can be defined.
 type Description rigid spring Applies a spring that connects the two rigid bodies. rigid damper Applies a damper based on relative linear velocity. rigid angular damper Applies a damper based on relative angular velocity. rigid contractile force Constant contractile force between rigid bodies. rigid planar joint 2D plane translation and rotation about normal. rigid lock Prevent any relative motion between rigid bodies.
All of these joints and connectors define two parameters that reference the two rigid bodies.
 parameter Description body_a The first rigid body in the connector body_b The second rigid body in the connector
In addition, the joints definde above are all based on the Augmented Lagrangian method, and consequently, also share the following parameters.
 parameter Description laugon Augmentation flag tolerance Specifies the augmentation tolerance minaug The minimum number of augmentations. maxaug The maximum number of augmentations. gaptol Gap tolerance angtol Angular separation tolerance force_penalty Penalty factor applied to force augmentations. moment_penalty Penalty factor applied to moment augmentations. auto_penalty Flag to calculate penalty factor automatically.
Additional parameters, unique to a specific connector, are defined in the sections below.
Subsubsection 3.11.1.5: Initial Rigid Angular Velocity Up Section 3.11: Rigid Section Subsubsection 3.11.2.1: Rigid Spherical Joint