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and will be ignored when the two nodes separate. Rotational degrees of freedom are not activated for connection type AXIAL.

Summary 

<table><tr><td colspan="2">AXIAL</td></tr><tr><td>Basic, assembled, or complex:</td><td>Basic</td></tr><tr><td>Kinematic constraints:</td><td>None</td></tr><tr><td>Constraint force output:</td><td>None</td></tr><tr><td>Available components:</td><td> $u_1$ </td></tr><tr><td>Kinetic force output:</td><td> $f_1$ </td></tr><tr><td>Orientation at a:</td><td>Optional</td></tr><tr><td>Orientation at b:</td><td>Optional</td></tr><tr><td>Connector stops:</td><td> $l_1^{min} \leq l \leq l_1^{max}$ </td></tr><tr><td>Constitutive reference lengths:</td><td> $l_1^{ref}$ </td></tr><tr><td>Predefined friction parameters:</td><td>None</td></tr><tr><td>Contact force for predefined friction:</td><td>None</td></tr></table>

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# BEAM

Connection type BEAM provides a rigid beam connection between two nodes.

![](images/page-602_2945fb55e86d2878cbcfde24c512c88b71f273b6a9b95033401be0f8674e2ddf.jpg)

<details>
<summary>text_image</summary>

e₂ᵃ
e₁ᵃ
e₃ᵃ
e₂ᵇ
e₁ᵇ
e₃ᵇ
</details>

Figure 31.1.5–5 Connection type BEAM.

# Description

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

# Summary

<table><tr><td colspan="2">BEAM</td></tr><tr><td>Basic, assembled, or complex:</td><td>Assembled</td></tr><tr><td>Kinematic constraints:</td><td>JOIN + ALIGN</td></tr><tr><td>Constraint force and moment output:</td><td> $f_1, f_2, f_3, m_1, m_2, m_3$ </td></tr><tr><td>Available components:</td><td>None</td></tr><tr><td>Kinetic force and moment output:</td><td>None</td></tr><tr><td>Orientation at a:</td><td>Optional</td></tr><tr><td>Orientation at b:</td><td>Optional</td></tr><tr><td>Connector stops:</td><td>None</td></tr><tr><td>Constitutive reference lengths and angles:</td><td>None</td></tr><tr><td>Predefined friction parameters:</td><td>None</td></tr><tr><td>Contact force for predefined friction:</td><td>None</td></tr></table>

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# BUSHING

Connection type BUSHING provides a bushing-like connection between two nodes. It cannot be used in two-dimensional or axisymmetric analyses.

![](images/page-603_ce9afdf5456e5c6c96ae960b4db40621e55f1660e6bb65f8ce7b5778d758e4d8.jpg)

<details>
<summary>text_image</summary>

attached to Part A
attached to Part B
deformable material
(e.g. rubber)
or
attached to Part A
deformable material
attached to Part B
e₁ᵇ
e₃ᵇ
e₁ᵃ
e₂ᵇ
e₂ᵃ
e₃ᵃ
e₃ᵃ
e₂ᵇ
e₂ᵃ
e₁ᵃ
e₁ᵇ
e₁ᵃ
e₂ᵇ
e₂ᵃ
e₁ᵃ
e₁ᵇ
</details>

Figure 31.1.5–6 Connection type BUSHING.

# Description

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.

# Summary

<table><tr><td colspan="2">BUSHING</td></tr><tr><td>Basic, assembled, or complex:</td><td>Assembled</td></tr><tr><td>Kinematic constraints:</td><td>None</td></tr><tr><td>Constraint force and moment output:</td><td>None</td></tr><tr><td>Available components:</td><td> $u_{1}, u_{2}, u_{3}, ur_{1}, ur_{2}, ur_{3}$ </td></tr></table>

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BUSHING 

<table><tr><td>Kinetic force and moment output:</td><td> $f_{1}, f_{2}, f_{3}, m_{1}, m_{2}, m_{3}$ </td></tr><tr><td>Orientation at a:</td><td>Required</td></tr><tr><td>Orientation at b:</td><td>Optional</td></tr><tr><td>Connector stops:</td><td> $l_{1}^{min} \leq x \leq l_{1}^{max}$ , $l_{2}^{min} \leq y \leq l_{2}^{max}$ , $l_{3}^{min} \leq z \leq l_{3}^{max}$  $\theta_{1}^{min} \leq \alpha_{1} \leq \theta_{1}^{max}$ , $\theta_{2}^{min} \leq \alpha_{2} \leq \theta_{2}^{max}$ , $\theta_{3}^{min} \leq \beta \leq \theta_{3}^{max}$ </td></tr><tr><td>Constitutive reference lengths and angles:</td><td> $l_{1}^{ref}, l_{2}^{ref}, l_{3}^{ref}$  $\theta_{1}^{ref}, \theta_{2}^{ref}, \theta_{3}^{ref}$ </td></tr><tr><td>Predefined friction parameters:</td><td>None</td></tr><tr><td>Contact force for predefined friction:</td><td>None</td></tr></table>

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# CARDAN

Connection type CARDAN provides a rotational connection between two nodes where the relative rotation between the nodes is parameterized by Cardan (or Bryant) angles. A Cardan-angle parameterization of finite rotations is also called a 1–2–3 or yaw-pitch-roll parameterization. Connection type CARDAN cannot be used in two-dimensional or axisymmetric analysis.

When connection type CARDAN is used with connector behavior, the relative rotation axis with the highest resistance to rotational motion should be assigned to the second component of relative rotation (component number 5) to avoid “gimbal lock,” a singularity in the rotation parameterization for relative rotation angles $\beta = \pm \pi / 2$ .

![](images/page-605_29f50fe9721fc35eb331ddbd2d6a749bc67b56f4b8a931beaba5965503769bd9.jpg)  
Figure 31.1.5–7 Connection type CARDAN.

# Description

The CARDAN connection does not impose kinematic constraints. A CARDAN connection is a finite rotation connection where the local directions at node b are parameterized in terms of Cardan (or Bryant) angles relative to the local directions at node a. Local directions $\{ \mathbf { e } _ { 1 } ^ { b } , \mathbf { e } _ { 2 } ^ { b } , \mathbf { e } _ { 3 } ^ { b } \}$ are positioned relative to $\{ { \bf e } _ { 1 } ^ { a } , { \bf e } _ { 2 } ^ { a } , { \bf e } _ { 3 } ^ { a } \}$ by three successive finite rotations $\alpha , \beta _ { ; }$ and $\gamma$ as follows:

1. Rotate by radians about axis ${ \bf e } _ { 1 } ^ { a }$ ;   
2. Rotate by $\beta$ radians about the intermediate 2-axis, $\mathbf { e } _ { 2 } = \mathrm { c o s } \alpha \mathbf { e } _ { 2 } ^ { a } + \mathrm { s i n } \alpha \mathbf { e } _ { 3 } ^ { a }$ ; and   
3. Rotate by $\gamma$ radians about axis $\mathbf { e } _ { 3 } ^ { b }$ .

Rotation angle $\beta$ should be moderate (magnitude less than $\pi / 2 )$ , whereas and may be arbitrarily large (i.e., magnitude greater than $2 \pi )$ . The Cardan angles are determined by the local directions as

$$
\alpha = - \tan^ {- 1} \left(\frac {\mathbf {e} _ {2} ^ {a} \cdot \mathbf {e} _ {3} ^ {b}}{\mathbf {e} _ {3} ^ {a} \cdot \mathbf {e} _ {3} ^ {b}}\right) + m \pi ;
$$

$$
\beta = \sin^ {- 1} \left(\mathbf {e} _ {1} ^ {a} \cdot \mathbf {e} _ {3} ^ {b}\right), - \frac {\pi}{2} <   \beta <   \frac {\pi}{2};
$$

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$$
\gamma = - \tan^ {- 1} \left(\frac {\mathbf {e} _ {1} ^ {a} \cdot \mathbf {e} _ {2} ^ {b}}{\mathbf {e} _ {1} ^ {a} \cdot \mathbf {e} _ {1} ^ {b}}\right) + n \pi .
$$

Here, m and n are integers that account for rotations with a magnitude greater than .

The three available components of relative motion in the CARDAN connection are the changes in the Cardan angles positioning the local directions at node $\pmb { b }$ relative to the local directions at node a. Therefore,

$$
u r _ {1} = \alpha - \alpha_ {0}; \quad u r _ {2} = \beta - \beta_ {0}; \quad \text {and} \quad u r _ {3} = \gamma - \gamma_ {0};
$$

where $\alpha _ { 0 } , \beta _ { 0 }$ , and $\gamma _ { 0 }$ are the initial Cardan angles. The connector constitutive rotations are

$$
u r _ {1} ^ {m a t} = \alpha - \theta_ {1} ^ {r e f}; \quad u r _ {2} ^ {m a t} = \beta - \theta_ {2} ^ {r e f}; \quad \mathrm{and} \quad u r _ {3} ^ {m a t} = \gamma - \theta_ {3} ^ {r e f}.
$$

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

$$
\mathbf {m} _ {C a r d a n} = m _ {1} \mathbf {e} _ {1} ^ {a} + m _ {2} \bigl (\cos \alpha \mathbf {e} _ {2} ^ {a} + \sin \alpha \mathbf {e} _ {3} ^ {a} \bigr) + m _ {3} \mathbf {e} _ {3} ^ {b}.
$$

Summary

CARDAN 
<table><tr><td>Basic, assembled, or complex:</td><td>Basic</td></tr><tr><td>Kinematic constraints:</td><td>None</td></tr><tr><td>Constraint moment output:</td><td>None</td></tr><tr><td>Available components:</td><td> $ur_1, ur_2, ur_3$ </td></tr><tr><td>Kinetic moment output:</td><td> $m_1, m_2, m_3$ </td></tr><tr><td>Orientation at a:</td><td>Required</td></tr><tr><td>Orientation at b:</td><td>Optional</td></tr><tr><td>Connector stops:</td><td> $\theta_1^{min} \leq \alpha \leq \theta_1^{max}$ , $\theta_2^{min} \leq \beta \leq \theta_2^{max}$ , $\theta_3^{min} \leq \gamma \leq \theta_3^{max}$ </td></tr><tr><td>Constitutive reference angles:</td><td> $\theta_1^{ref}, \theta_2^{ref}, \theta_3^{ref}$ </td></tr><tr><td>Predefined friction parameters:</td><td>None</td></tr><tr><td>Contact force for predefined friction:</td><td>None</td></tr></table>

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# CARTESIAN

Connection type CARTESIAN provides a connection between two nodes where the change in position is measured in three local connection directions for node a, shown in Figure 31.1.5–8.

![](images/page-607_9b4c4f1de67e70460b9e272ea6e4355fddef91a09629a5a74ddac8fb1213802f.jpg)

<details>
<summary>text_image</summary>

e₃ᵃ
b
a
e₂ᵃ
e₁ᵃ
</details>

Figure 31.1.5–8 Connection type CARTESIAN.

# Description

The CARTESIAN connection does not impose kinematic constraints. It defines three local directions $\{ \mathbf { e } _ { 1 } ^ { a } , \mathbf { e } _ { 2 } ^ { a } , \mathbf { e } _ { 3 } ^ { a } \}$ at node a and measures the change in position of node b along these local coordinate directions. The local directions at node a follow the rotation of node a.

The position of node b relative to node a is

$$
x = \mathbf {e} _ {1} ^ {a} \cdot (\mathbf {x} _ {b} - \mathbf {x} _ {a}); \quad y = \mathbf {e} _ {2} ^ {a} \cdot (\mathbf {x} _ {b} - \mathbf {x} _ {a}); \quad \text {and} \quad z = \mathbf {e} _ {3} ^ {a} \cdot (\mathbf {x} _ {b} - \mathbf {x} _ {a}).
$$

The available components of relative motion are

$$
u _ {1} = x - x _ {0}; \quad u _ {2} = y - y _ {0}; \quad \mathrm{and} \quad u _ {3} = z - z _ {0};
$$

where $x _ { 0 } , y _ { 0 }$ , and $z _ { 0 }$ are the initial coordinates of node b relative to the local coordinate system at node a. The connector constitutive displacements are

$$
u _ {1} ^ {m a t} = x - l _ {1} ^ {r e f}; \quad u _ {2} ^ {m a t} = y - l _ {2} ^ {r e f}; \quad \mathrm{and} \quad u _ {3} ^ {m a t} = z - l _ {3} ^ {r e f}.
$$

The kinetic force is

$$
\mathbf {f} _ {C a r t} = f _ {1} \mathbf {e} _ {1} ^ {a} + f _ {2} \mathbf {e} _ {2} ^ {a} + f _ {3} \mathbf {e} _ {3} ^ {a}.
$$

In two-dimensional analysis , , $u _ { 3 } ^ { m a t } = 0$ , and $f _ { 3 } = 0$

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Summary 

<table><tr><td colspan="2">CARTESIAN</td></tr><tr><td>Basic, assembled, or complex:</td><td>Basic</td></tr><tr><td>Kinematic constraints:</td><td>None</td></tr><tr><td>Constraint force output:</td><td>None</td></tr><tr><td>Available components:</td><td> $u_1, u_2, u_3$ </td></tr><tr><td>Kinetic force output:</td><td> $f_1, f_2, f_3$ </td></tr><tr><td>Orientation at a:</td><td>Optional</td></tr><tr><td>Orientation at b:</td><td>Ignored</td></tr><tr><td>Connector stops:</td><td> $l_1^{min} \leq x \leq l_1^{max}$ , $l_2^{min} \leq y \leq l_2^{max}$ , $l_3^{min} \leq z \leq l_3^{max}$ </td></tr><tr><td>Constitutive reference lengths:</td><td> $l_1^{ref}, l_2^{ref}, l_3^{ref}$ </td></tr><tr><td>Predefined friction parameters:</td><td>None</td></tr><tr><td>Contact force for predefined friction:</td><td>None</td></tr></table>

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# CONSTANT VELOCITY

Connection type CONSTANT VELOCITY provides the rotational part of connection type CVJOINT. It cannot be used in two-dimensional or axisymmetric analysis. Furthermore, the connection type does not have available components of relative motion. To include connector behavior in flexural motion, use connection type FLEXION-TORSION with the torsion angle set to zero.

This connection type models physical connectors that under certain conditions transmit a constant spinning velocity about misaligned shafts.

![](images/page-609_edf89b9a418fe8fe47d93a0f4df1eab5eca1383e55356a47fd558f945bde4722.jpg)

<details>
<summary>text_image</summary>

Diagram illustrating vector relationships with labeled points and directional arrows, including e1, e2, e3 and their primed counterparts.
</details>

Figure 31.1.5–9 Connection type CONSTANT VELOCITY.

# Description

The shaft direction at node a is ${ \bf e } _ { 3 } ^ { a }$ , and the shaft direction at node b is $\mathbf { e } _ { 3 } ^ { b }$ . The constant velocity constraint is stated as follows. In any configuration there are two unit length orthogonal vectors $\mathbf { b } _ { 1 }$ and $\mathbf { b } _ { 2 }$ in the plane perpendicular to the shaft at node b. These vectors can be written

$$
\mathbf {b} _ {1} = \cos \beta \mathbf {e} _ {1} ^ {b} + \sin \beta \mathbf {e} _ {2} ^ {b} \quad \mathrm{and} \quad \mathbf {b} _ {2} = - \sin \beta \mathbf {e} _ {1} ^ {b} + \cos \beta \mathbf {e} _ {2} ^ {b}.
$$

The angle $\beta$ is chosen such that

$$
\mathbf {e} _ {1} ^ {a} \cdot \mathbf {b} _ {2} = \mathbf {e} _ {2} ^ {a} \cdot \mathbf {b} _ {1}.
$$

The constant velocity constraint requires that the angle $\beta$ is constant at all times. The constant velocity constraint is equivalent to constraining the torsion angle to be constant in a FLEXION-TORSION connection.

The name “constant velocity” for this connection type derives from the following property. If the angular velocities of the two shafts, ${ \bf w } _ { a }$ and $\mathbf { w } _ { b }$ , 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 $\mathbf { e } _ { 3 } ^ { b } \times \mathbf { e } _ { 3 } ^ { a }$ direction), the components of angular velocity along the respective shaft directions are equal:

$$
\mathbf {w} _ {a} \cdot \mathbf {e} _ {3} ^ {a} = \mathbf {w} _ {b} \cdot \mathbf {e} _ {3} ^ {b}.
$$

Hence, the “spinning” angular velocity component is the same about each shaft.

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The constraint moment imposing the constant velocity constraint has a single component about the average shaft direction ${ \bf e } _ { 3 } ^ { a } + { \bf e } _ { 3 } ^ { b }$ and is written

$$
\bar {\bf m} = m _ {2} \frac {({\bf e} _ {3} ^ {a} + {\bf e} _ {3} ^ {b})}{\| {\bf e} _ {3} ^ {a} + {\bf e} _ {3} ^ {b} \|}.
$$

Summary

CONSTANT VELOCITY 
<table><tr><td>Basic, assembled, or complex:</td><td>Basic</td></tr><tr><td>Kinematic constraints:</td><td> $\mathbf{e}_{1}^{a} \cdot \mathbf{b}_{2} = \mathbf{e}_{2}^{a} \cdot \mathbf{b}_{1}$ </td></tr><tr><td>Constraint moment output:</td><td> $m_{2}$ </td></tr><tr><td>Available components:</td><td>None</td></tr><tr><td>Kinetic moment output:</td><td>None</td></tr><tr><td>Orientation at a:</td><td>Required</td></tr><tr><td>Orientation at b:</td><td>Optional</td></tr><tr><td>Connector stops:</td><td>None</td></tr><tr><td>Constitutive reference angles:</td><td>None</td></tr><tr><td>Predefined friction parameters:</td><td>None</td></tr><tr><td>Contact force for predefined friction:</td><td>None</td></tr></table>
