The ALIGN connection imposes kinematic constraints only. The local directions at node are set equal to those at node . If the local directions do not align initially, the ALIGN connection holds fixed the Cardan angles between the local orientation directions at node , , and those at node , . These fixed angular positions are the connector position output quantities. See connection type CARDAN for a definition of Cardan angles.

The constraint moment enforcing the alignment of the local directions is

The AXIAL connection does not constrain any component of relative motion. The distance between nodes and is

The kinetic force is

An optional orientation can be provided at one of the nodes in an AXIAL connection to provide direction for the force if the nodes are coincident or when one of the nodes is a “ground node.” If the orientation is provided at both of the coincident nodes, the orientation at the first node in the connectivity will be used. The orientation definitions will be ignored when the two nodes separate. Rotational degrees of freedom are not activated for connection type AXIAL.

Connection type BEAM imposes kinematic constraints and uses local orientation definitions equivalent to combining connection types JOIN and ALIGN.

Connection type BUSHING does not constrain any components of relative motion and uses local orientation definitions equivalent to combining connection types PROJECTION CARTESIAN and PROJECTION FLEXION-TORSION.

The CARDAN connection does not impose kinematic constraints. A CARDAN connection is a finite rotation connection where the local directions at node are parameterized in terms of Cardan (or Bryant) angles relative to the local directions at node . Local directions are positioned relative to by three successive finite rotations , , and as follows:

The three available components of relative motion in the CARDAN connection are the changes in the Cardan angles positioning the local directions at node relative to the local directions at node . Therefore,

The kinetic moment in a CARDAN connection is determined from the three component relationships:

The CARTESIAN connection does not impose kinematic constraints. It defines three local directions at node and measures the change in position of node along these local coordinate directions. The local directions at node follow the rotation of node .

The position of node relative to node is

The kinetic force is

In two-dimensional analysis , , , and .

The shaft direction at node is , and the shaft direction at node is . The constant velocity constraint is stated as follows. In any configuration there are two unit length orthogonal vectors and in the plane perpendicular to the shaft at node . These vectors can be written

The name “constant velocity” for this connection type derives from the following property. If the angular velocities of the two shafts, and , have components only along each shaft, respectively, and in the direction of the normal to the plane containing the two shafts (that is, along the direction), the components of angular velocity along the respective shaft directions are equal:

The constraint moment imposing the constant velocity constraint has a single component about the average shaft direction and is written

Connection type CVJOINT imposes kinematic constraints and uses local orientation definitions equivalent to combining connection types JOIN and CONSTANT VELOCITY.

Connection type CYLINDRICAL imposes kinematic constraints and uses local orientation definitions equivalent to combining connection types SLOT and REVOLUTE.

The connector constraint forces and moments reported as connector output depend strongly on the order and the location of the nodes in the connector (see “Connector behavior,” Section 17.2.1). Since the kinematic constraints are enforced at node (the second node of the connector element), the reported forces and moments are the constraint forces and moments applied at node to enforce the CYLINDRICAL constraint. Thus, in most cases the connector output associated with a CYLINDRICAL connection is best interpreted when node is located at the center of the device enforcing the constraint. This choice is essential when moment-based friction is modeled in the connector since the contact forces are derived on the connector forces and moments, as illustrated below. Proper enforcement of the kinematic constraints is independent of the order or location of the nodes.

Predefined Coulomb-like friction in the CYLINDRICAL connection defines the friction force (CSFC) along the instantaneous slip direction on the two contacting cylindrical surfaces (the pin and the sleeve) illustrated above. The table below summarizes the parameters that are used to specify predefined friction in this connection type as discussed in detail next.

The frictional effect is formally written as

The normal force is the sum of a magnitude measure of contact friction-producing connector forces, , and a self-equilibrated internal contact force (such as from a press-fit assembly), :

The contact force magnitude is defined by summing the following two contributions:

The magnitude of the frictional tangential tractions, is computed using

The EULER connection does not impose kinematic constraints. An EULER connection is a finite rotation connection where the local directions at node are parameterized in terms of Euler angles relative to the local directions at node . Local directions are positioned relative to by three successive finite rotations , , and as follows:

If the intermediate rotation is an even multiple of , , where , the other two Euler angles become non-unique. In this case

The available components of relative motion in the EULER connection are the changes in the Euler angles that position the local directions at node relative to the local directions at node . Therefore,

The kinetic moment in a EULER connection is determined from the three component relationships:

The FLEXION-TORSION connection does not impose kinematic constraints. The FLEXION-TORSION connection describes a finite rotation by three angles: flexion, torsion, and sweep (, , and ). However, the flexion, torsion, and sweep angles do not represent three successive rotations. The flexion angle between two shafts measures the angle of misalignment of the two shafts and is always reported as a positive angle. The torsion angle measures the twist of one shaft relative to the other.

The sweep angle orients the rotation vector, in the – plane, for the flexion motion. See Figure 17.1.5–12. Since the flexion angle is never negative, the sweep angle may undergo discontinuous jumps by up to radians when the flexion angle passes through zero. An analysis may give inaccurate results or may not converge if any jump occurs in the sweep angle. In general, the sweep angle is not used as an available component of relative motion for which connector behavior is defined. Rather, it is used to define angular dependence for the elastic constitutive response in flexion deformations (as an independent component in the connector elastic behavior definition). Since the sweep angle is restricted to the interval to radians, any dependence on the sweep angle should be periodic, such that the behavior for is the same as . Since is a singular point for which the sweep angle is not uniquely defined, it is strongly recommended that any connector behavior that defines flexural moment versus flexion angle gives zero moment at zero flexion angle. If connector behavior is defined in the sweep available component, the sweep moment must be zero at flexion angles and .

The FLEXION-TORSION connection is similar to a finite successive rotation parameterization 3–2–3. However, in terms of the 3–2–3 parameterization, the sweep angle is the first rotation angle, the flexion angle is the second rotation angle, and the torsion angle is the sum of the first and third rotation angles.

The first shaft direction at node is , and the second shaft direction at node is . Let the two shafts form an angle , called the flexion angle. Then,

The sweep angle measures the angle from to the projection of onto the – plane. With this definition

A singularity in the definition of the sweep angles occurs when the flexion angle vanishes. In this case ; that is, the torsion and sweep angle axes are coincident, and the two angles are no longer independent. When , the sweep angle is assumed zero, .

The available components of relative motion , , and are the changes in the flexion, torsion, and sweep angles and are defined as

The kinetic moment in a FLEXION-TORSION connection is determined from the three component relationships:

Connection type HINGE imposes kinematic constraints and uses local orientation definitions equivalent to combining connection types JOIN and REVOLUTE.

The connector constraint forces and moments reported as connector output depend strongly on the order and the location of the nodes in the connector element (see “Connector behavior,” Section 17.2.1). Since the kinematic constraints are enforced at node (the second node of the connector element), the reported forces and moments are the constraint forces and moments applied at node to enforce the HINGE constraint. Thus, in most cases the connector output associated with a HINGE connection is best interpreted when node is located at the center of the device enforcing the constraint. This choice is essential when moment-based friction is modeled in the connector since the contact forces are derived from the connector forces and moments, as illustrated below. Proper enforcement of the kinematic constraints is independent of the order or location of the nodes.

Predefined Coulomb-like friction in the HINGE connection relates the kinematic constraint forces and moments in the connector to a friction moment (CSM1) in the rotation about the hinge axis. The table below summarizes the parameters that are used to specify predefined friction in this connection type as discussed in detail next. A typical interpretation of the geometric scaling constants is illustrated in Figure 17.1.5–14.

Since the rotation about the 1-direction is the only possible relative motion in the connection, the frictional effect is formally written in terms of moments generated by tangential tractions and moments generated by contact forces, as follows:

The normal moment is the sum of a magnitude measure of contact friction-producing connector moments, , and a self-equilibrated internal contact moment (such as from a press-fit assembly), :

The contact moment magnitude is defined by summing the following contributions:

The moment magnitude of the frictional tangential tractions, .

The JOIN connection makes the position of node equal to that of node . If the two nodes are not coincident initially, the Cartesian coordinates of node relative to node are fixed. See connection type CARTESIAN for a definition of the Cartesian coordinates of node relative to node . If rotational degrees of freedom exist at node , the local directions co-rotate with the node.

The constraint force in the JOIN connection acts in the three local directions at node and is

When used by itself, there is no predefined Coulomb-like friction in the JOIN connection, since there are no available components of relative motion for which friction can be defined. However, when the JOIN and REVOLUTE connection types are used together, the predefined friction is the same as the HINGE connection. When the JOIN and UNIVERSAL connection types are used together, the predefined friction is the same as the UJOINT connection.

The LINK connection constrains the position of node , , to a constant distance from node . The distance between the two nodes is

Connection type PLANAR imposes kinematic constraints and uses local orientation definitions equivalent to combining connection types SLIDE-PLANE and REVOLUTE.

Predefined Coulomb-like friction in the PLANAR connection relates the kinematic constraint forces and moments in the connector to the friction forces in the translations in the local 2–3 plane and the rotations about the local 1-direction. These two frictional effects are discussed separately below.

The PROJECTION CARTESIAN connection does not impose kinematic constraints. It defines three local directions as a function of the directions at both nodes and . These directions are the projection directions defined by the PROJECTION FLEXION-TORSION connection. The PROJECTION CARTESIAN connection measures the change in position of node relative to node along the (projection) coordinate directions .

The position of node relative to node is

The local directions in a PROJECTION CARTESIAN connection are “centered” between the systems at the two connector nodes. PROJECTION CARTESIAN connections are appropriate where isotropic or anisotropic material response is modeled and the local material directions evolve as a function of the rotations at both ends of the connection. The kinetic force is

In two-dimensional analysis , , , and .

The PROJECTION FLEXION-TORSION connection does not impose kinematic constraints. The PROJECTION FLEXION-TORSION connection describes a finite rotation by three angles: flexion 1, flexion 2, and torsion (, , and ). However, the flexion 1, flexion 2, and torsion angles do not represent three successive rotations. The two component flexion angles ( and ) make up the total flexion angle between two shafts and measure the angle of misalignment of the two shafts. The torsion angle measures the twist of one shaft relative to the other.

The first shaft direction at node is , and the second shaft direction at node is . Let the two shafts form an angle , called the total flexion angle. Then,

The PROJECTION FLEXION-TORSION connection is formulated in terms of the unit vector normal to a plane, , and two unit vectors spanning this plane, and . See Figure 17.1.5–20. The plane with normal vector is referred to as the flexion-torsion plane. The component flexion angles and are determined from and by projection onto the two in-plane directions:

The torsion angle in a PROJECTION FLEXION-TORSION connection can be understood from a finite successive rotation parameterization 3–2–3. In terms of the 3–2–3 parameterization the total flexion angle is the second successive rotation angle, and the torsion angle is the sum of the first and third successive rotation angles. The torsion angle between the two shafts is defined as

The PROJECTION FLEXION-TORSION connection avoids the singularity that occurs in the sweep angle of the FLEXION-TORSION connection when the total flexion angle vanishes. As a result, the PROJECTION FLEXION-TORSION connection is better suited for defining bushing-like behavior for flexion response that varies with the direction of in the equivalent plane.

The available components of relative motion , , and are the changes in the two flexion angles and the torsion angle and are defined as

The kinetic moment in a PROJECTION FLEXION-TORSION connection is

The RADIAL-THRUST connection does not impose kinematic constraints. An orientation at node is required to define the axis of the rectangular coordinate system, . The position of node relative to node is given by the radial and axial-direction distances

The kinetic force is

A REVOLUTE connection constrains two rotational degrees of freedom between two nodes and allows one free rotation. The two kinematic constraints imposed by the REVOLUTE connection are

Node can rotate about the shared local direction . The relative angular position of the local directions at node relative to is

The available component of relative motion, , measures the change in angular position and is defined as

The kinetic moment in the REVOLUTE connection is

When used by itself, there is no predefined Coulomb-like friction in the REVOLUTE connection. However, when the REVOLUTE connection is used in combination with a JOIN, SLIDE-PLANE, or SLOT connection, the predefined friction is the same as the HINGE, PLANAR, and CYLINDRICAL connections, respectively.

The rotation connection does not impose kinematic constraints. The rotation connection is a finite rotation connection where the local directions at node are parameterized relative to the local directions at node by the rotation vector. Let be the rotation vector that positions local directions relative to ; that is,

The available components of relative motion in the rotation connection are the change in the rotation vector components positioning the local directions at node relative to the local directions at node . Therefore,

The kinetic moment in a rotation connection is

In two-dimensional and axisymmetric analyses and .

The SLIDE-PLANE connection constrains the position of node , , to remain on a plane defined by the local normal direction . The normal direction distance from node to the plane is constant:

Node can move in the plane defined at node . The position of node in the plane relative to node is

The kinetic force in the plane is

Predefined Coulomb-like friction in the SLIDE-PLANE connection relates the kinematic constraint forces in the connector to the friction forces (CSFC) in the translations along the two local directions in the 2–3 plane.

The frictional effect is formally written as

The normal force is the sum of a magnitude measure of contact friction-producing connector forces, , and a self-equilibrated internal contact force, :

The contact force magnitude .

The magnitude of the frictional tangential tractions, is computed using

The predefined Coulomb-like friction is computed differently when the SLIDE-PLANE connection is used in combination with a REVOLUTE connection. See the description of the PLANAR connection for the predefined friction definition in this case.

The line of the slot is defined by the first local direction at node , , and the initial position of node . The SLOT connection constrains the position of node , , to remain on the line of the slot. Therefore, the relative position of node is fixed in the directions perpendicular to the slot:

Node can move along the line of the slot. The relative position in the slot is the distance between node and node along the -direction and is defined as

The kinetic force in the slot is

Predefined Coulomb-like friction in the SLOT connection relates the kinematic constraint forces in the connector to the friction force (CSF1) in the translation along the slot.

The frictional effect is formally written as

The normal force is the sum of a magnitude measure of contact friction-producing connector forces, , and a self-equilibrated internal contact force, :

The contact force magnitude is computed using

The magnitude of the frictional tangential tractions .

The predefined Coulomb-like friction is computed differently when the SLOT connection is used in combination with a REVOLUTE or an ALIGN connection. See CYLINDRICAL and TRANSLATOR, respectively, for the predefined friction definition in these cases.

Connection type TRANSLATOR imposes kinematic constraints and uses local orientation definitions equivalent to combining connection types SLOT and ALIGN.

The connector constraint forces and moments reported as connector output depend strongly on the order and location of the nodes in the connector (see “Connector behavior,” Section 17.2.1). Since the kinematic constraints are enforced at node (the second node of the connector element), the reported forces and moments are the constraint forces and moments applied at node to enforce the TRANSLATOR constraint. Thus, in most cases the connector output associated with a TRANSLATOR connection is best interpreted when node is located at the center of the device enforcing the constraint. This choice is essential when moment-based friction is modeled in the connector since the contact forces are derived from the connector forces and moments, as illustrated below. Proper enforcement of the kinematic constraints is independent of the order or location of the nodes.

Predefined Coulomb-like friction in the TRANSLATOR connection relates the kinematic constraint forces and moments in the connector to the friction forces (CSF1) in the translation along the slot.

The frictional effect is formally written as

The normal force is the sum of a magnitude measure of contact friction-producing connector forces, , and a self-equilibrated internal contact force, :

The contact force magnitude is defined by summing the following three contributions:

The magnitude of the frictional tangential tractions, is .

Connection type UJOINT imposes kinematic constraints and uses local orientation definitions equivalent to combining connection types JOIN and UNIVERSAL.

The connector constraint forces and moments reported as connector output depend strongly on the order of the nodes and location of the nodes in the connector (see “Connector behavior,” Section 17.2.1). Since the kinematic constraints are enforced at node (the second node of the connector element), the reported forces and moments are the constraint forces and moments applied at node to enforce the UJOINT constraint. Thus, in most cases the connector output associated with a UJOINT connection is best interpreted when node is located at the center of the device enforcing the constraint. This choice is essential when moment-based friction is modeled in the connector since the contact forces are derived from the connector forces and moments, as illustrated below. Proper enforcement of the kinematic constraints is independent of the order or location of the nodes.

Predefined Coulomb-like friction in the UJOINT connection relates the kinematic constraint forces and moments in the connector to friction moments about the unconstrained rotations (about the two directions of the connection cross). The UJOINT connection type consists of four hinge-like connections placed at the four ends of the connection cross (see Figure 17.1.5–27) that generate frictional moments about the cross axes. The frictional moments in each of these hinges are computed in a fashion similar to the HINGE connection.

The constraint forces and moments are used first to compute a reaction force, (the magnitude of the constraint forces enforcing the JOIN constraint), and a “twisting” constraint moment, (the magnitude of the constraint moment enforcing the UNIVERSAL connection), as follows:

Friction in the UJOINT connection is the superposition of four HINGE-like frictional effects due to rotations about the two cross axes. Since the rotations about the local 1- and 3-directions are the only possible relative motions in the connection, the frictional effects (CSM1 and CSM3) are formally written in terms of moments generated by tangential tractions and moments generated by contact forces. In the following equations subscript 1 refers to frictional effects about the local 1-direction, and subscript 3 refers to frictional effects about the local 3-direction. The frictional effects are written as follows:

The normal moments and are the sums of magnitude measures of contact force-producing connector moments, and , and self-equilibrated internal contact moments (such as from a press-fit assembly), and , respectively:

The contact moment magnitudes and are defined by summing the following contributions:

The moment magnitudes of the frictional tangential tractions are and .

A UNIVERSAL connection constrains the rotation about the shaft directions at two nodes. The shaft directions at nodes and are and , respectively. A UNIVERSAL connection requires that local direction be perpendicular to . This single constraint is written

The constraint moment imposed by the UNIVERSAL connection is

A UNIVERSAL connection allows two free rotational degrees of freedom between two nodes. The first and third Cardan angles that position local directions at node relative to those at node are

The kinetic moment in the UNIVERSAL connection is

When used by itself, there is no predefined Coulomb-like friction in the UNIVERSAL connection. However, when the UNIVERSAL connection is used in combination with the JOIN connection type, the predefined friction is the same as the UJOINT connection.

Connection type WELD imposes kinematic constraints and uses local orientation definitions equivalent to combining connection types JOIN and ALIGN.