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$$
\mathrm{F} _ {\mathrm{N} _ {\mathrm{C}}} = | \mathrm{F} _ {\mathrm{C}} + \mathrm{F} _ {\mathrm{C}} ^ {\mathrm{int}} | = | g (\mathbf {f}) + \mathrm{F} _ {\mathrm{C}} ^ {\mathrm{int}} |.
$$

The contact force magnitude $\mathrm { F _ { C } }$ is defined by summing the following two contributions:

• a force contribution, $F _ { 1 } = | f _ { 1 } |$ (the constraint force enforcing the SLIDE-PLANE constraint); and   
• a force contribution from “bending,” $F _ { b e n d }$ , obtained by scaling the bending moment, $M _ { b e n d }$ (the magnitude of the constraint moments enforcing the REVOLUTE constraint), by a length factor, as follows:

$$
M _ {b e n d} = \sqrt {m _ {2} ^ {2} + m _ {3} ^ {2}},
$$

$$
F _ {b e n d} = \frac {M _ {b e n d}}{R},
$$

where R represents a characteristic radius of the “puck” (as illustrated in Figure 31.1.5–20) in the local 2–3 plane. If R is $0 . 0 , M _ { b e n d }$ is ignored.

![](images/page-631_94b77f805098b601826768ecd7c951582d7c29019cc1d8b871827dc8fa5c4559.jpg)

<details>
<summary>text_image</summary>

M_bend
F_1
R
M_bend
2R
M_bend
2R
</details>

Figure 31.1.5–20 Illustration of the effective internal friction contact forces.

Thus,

$$
\mathrm{F} _ {\mathrm{C}} = g (\mathbf {f}) = F _ {1} + F _ {b e n d} = | f _ {1} | + \sqrt {(\beta m _ {2}) ^ {2} + (\beta m _ {3}) ^ {2}},
$$

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where $\textstyle { \beta = { \frac { 1 } { R } } }$

The magnitude of the frictional tangential moment, $\mathrm { P _ { C } } ( \mathbf { f } )$ is computed using

$$
\mathrm{P} _ {\mathrm{C}} (\mathbf {f}) = \sqrt {f _ {2} ^ {2} + f _ {3} ^ {2}}.
$$

B. Since the frictional effects due to rotation about the 1-direction are quantified, the frictional effect is formally written in terms of moments generated by tangential tractions and moments generated by contact forces as

$$
\Phi_ {\mathrm{R1}} = \mathrm{P} _ {\mathrm{R1}} (\mathbf {f}) - \mu \mathrm{M} _ {\mathrm{N} _ {\mathrm{R1}}} \leq 0,
$$

where the potential $\mathrm { P _ { R 1 } } ( \mathbf { f } )$ represents the magnitude of the frictional tangential moment in the connector about the 1-direction, $\mathrm { M } _ { \mathrm { N } _ { \mathrm { R 1 } } }$ is the friction-producing normal moment about the same axis, and $\mu$ is the friction coefficient. Frictional stick in rotation occurs if $\Phi _ { \mathrm { R 1 } } < 0 ;$ and sliding occurs if $\Phi _ { \mathrm { R 1 } } = 0$ , in which case the friction moment (CSM1) is $\mu \mathrm { M _ { N _ { R 1 } } }$ .

The normal moment $\mathrm { M } _ { \mathrm { N } _ { \mathrm { R 1 } } }$ is the sum of a magnitude measure of friction-producing connector moments, $\mathrm { M } _ { \mathrm { R 1 } } = g ( \mathbf { f } )$ , and a self-equilibrated internal contact moment, $\mathrm { M } _ { \mathrm { R 1 } } ^ { \mathrm { i n t } }$ :

$$
\mathrm{M} _ {\mathrm{N} _ {\mathrm{R} 1}} = | \mathrm{M} _ {\mathrm{R} 1} + \mathrm{M} _ {\mathrm{R} 1} ^ {\mathrm{int}} | = | g (\mathbf {f}) + \mathrm{M} _ {\mathrm{R} 1} ^ {\mathrm{int}} |.
$$

The contact moment magnitude $\mathrm { { M } _ { R 1 } }$ is defined by summing the following two contributions:

• a moment from a contact force in the 2–3 plane, $M _ { 1 }$ (the constraint moment enforcing the SLIDE-PLANE constraint):

$$
M _ {1} = \frac {2}{3} F _ {1} R,
$$

where $\begin{array} { r c l } { F _ { 1 } } & { = } & { \left| f _ { 1 } \right| } \end{array}$ , R represents a characteristic radius of the “puck” (as illustrated in Figure 31.1.5–20) in the local 2–3 plane (if R is $0 . 0 , M _ { 1 }$ is ignored), and the 2/3 factor comes from integrating moment contributions from a uniform pressure $\scriptstyle \left( { \frac { F _ { 1 } } { \pi R ^ { 2 } } } \right)$ over the circular contact patch; and

• a moment contribution from “bending,” $M _ { b e n d }$ (the magnitude of the constraint moments enforcing the REVOLUTE constraint):

$$
M _ {b e n d} = \sqrt {m _ {2} ^ {2} + m _ {3} ^ {2}}.
$$

Thus,

$$
\mathrm{M} _ {\mathrm{R1}} = g (\mathbf {f}) = M _ {1} + M _ {b e n d} = \frac {2}{3} R | f _ {1} | + \sqrt {m _ {2} ^ {2} + m _ {3} ^ {2}}.
$$

The magnitude of the frictional tangential tractions, $\mathrm { P _ { R 1 } } ( \mathbf { f } )$ is computed using

$$
\mathrm{P} _ {\mathrm{R} 1} (\mathbf {f}) = | m _ {1} |.
$$

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Summary 

<table><tr><td colspan="2">PLANAR</td></tr><tr><td>Basic, assembled, or complex:</td><td>Assembled</td></tr><tr><td>Kinematic constraints:</td><td>SLIDE-PLANE + REVOLUTE</td></tr><tr><td>Constraint force and moment output:</td><td> $f_1, m_2, m_3$ </td></tr><tr><td>Available components:</td><td> $u_2, u_3, ur_1$ </td></tr><tr><td>Kinetic force and moment output:</td><td> $f_2, f_3, m_1$ </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> $l_2^{min} \leq y \leq l_2^{max}$ , $l_3^{min} \leq z \leq l_3^{max}$ , $\theta_1^{min} \leq \alpha \leq \theta_1^{max}$ </td></tr><tr><td>Constitutive reference lengths and angles:</td><td> $l_2^{ref}, l_3^{ref}, \theta_1^{ref}$ </td></tr><tr><td>Predefined friction parameters:</td><td>Optional:  $R, F_{C}^{int}, M_{R1}^{int}$ </td></tr><tr><td>Contact forces and moments for predefined friction:</td><td> $F_{C}, M_{R1}$ </td></tr></table>

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# PROJECTION CARTESIAN

Connection type PROJECTION CARTESIAN provides a connection between two nodes where the response in three local connection directions (that is, the axes of the local Cartesian coordinate system) is measured. Unlike the CARTESIAN connection, which uses an orthonormal coordinate system that follows node ${ \pmb a } ,$ the PROJECTION CARTESIAN connection uses an orthonormal system that follows the systems at both nodes a and b.

The connector local directions used in the PROJECTION CARTESIAN connection are identical to those used in the PROJECTION FLEXION-TORSION connection. Connection type PROJECTION CARTESIAN is compatible with connection type PROJECTION FLEXION-TORSION and is appropriate for modeling the displacement response of bushing-like or spot-weld-like components.

![](images/page-634_9dbea8c52522c0685002e5d2017aefb15818e4caed1af41a818a0e41ca1595b2.jpg)

<details>
<summary>text_image</summary>

α
e₃ᵃ
e₃
α
e₃ᵇ
e₁
a, b
e₂
</details>

Figure 31.1.5–21 Connection type PROJECTION CARTESIAN.

# Description

The PROJECTION CARTESIAN connection does not impose kinematic constraints. It defines three local directions $\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } , \mathbf { e } _ { 3 } \}$ as a function of the directions at both nodes a and b. These directions are the projection directions defined by the PROJECTION FLEXION-TORSION connection. The PROJECTION CARTESIAN connection measures the change in position of node b relative to node a along the (projection) coordinate directions $\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } , \mathbf { e } _ { 3 } \}$ .

The position of node b relative to node a is

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

The available components of relative motion are

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

where $x _ { 0 } , y _ { 0 }$ , and $z _ { 0 }$ are the initial coordinates of node b relative to node a along the initial $\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } , \mathbf { e } _ { 3 } \}$ directions. The connector constitutive displacements are

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

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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

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

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

Summary 
<table><tr><td colspan="2">PROJECTION 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>Optional</td></tr><tr><td>Connector stops:</td><td> $l_1^{min} \leq x \leq l_1^{max}$ , $l_2^{min} \leq y \leq l_2^{max}$ , $l_3^{min} \leq z \leq l_3^{max}$ </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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# PROJECTION FLEXION-TORSION

Connection type PROJECTION FLEXION-TORSION provides a rotational connection between two nodes. It models the bending and twisting of a cylindrical coupling between two shafts. In this case the response to twist rotations about the shafts may differ from the response to bending of the shafts. Connection type PROJECTION FLEXION-TORSION is similar to connection type FLEXION-TORSION. Whereas the FLEXION-TORSION connection has rotation parameterization angles consisting of total flexion, torsion, and sweep, the PROJECTION FLEXION-TORSION connection has rotation parameterization angles consisting of two component flexion angles and a torsion angle. The flexion angle of the FLEXION-TORSION connection is the resultant flexion angle resulting from the two component flexion angles of the PROJECTION FLEXION-TORSION connection. Connection type PROJECTION FLEXION-TORSION cannot be used in two-dimensional or axisymmetric analysis.

The flexural part of the connection resists angular misalignment of the two shafts, whereas the torsional part of the connection resists relative rotations about the shafts. Connection type PROJECTION FLEXION-TORSION can be used in conjunction with connection type PROJECTION CARTESIAN when modeling the response of bushing-like or spot-weld-like components.

![](images/page-636_a201e96b02be67db4856e44634878c775c14e7daf9986ef30a8425ec5bf89947.jpg)

<details>
<summary>text_image</summary>

α
e₃ᵃ
e₃
α
e₃ᵇ
e₁
a, b
e₂
</details>

Figure 31.1.5–22 Connection type PROJECTION FLEXION-TORSION.

# Description

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 $\beta )$ . However, the flexion 1, flexion 2, and torsion angles do not represent three successive rotations. The two component flexion angles $( \alpha _ { 1 }$ and $\alpha _ { 2 } )$ 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 a is ${ \bf e } _ { 3 } ^ { a }$ , and the second shaft direction at node b is $\mathbf { e } _ { 3 } ^ { b }$ . Let the two shafts form an angle $\alpha ,$ called the total flexion angle. Then,

$$
\alpha = \cos^ {- 1} \left(\mathbf {e} _ {3} ^ {a} \cdot \mathbf {e} _ {3} ^ {b}\right), \quad \text { where } \quad 0 \leq \alpha \leq \pi .
$$

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The flexion angle is a rotation by about the (unit) rotation vector,

$$
\mathbf {q} = \frac {1}{\sin \alpha} \mathbf {e} _ {3} ^ {a} \times \mathbf {e} _ {3} ^ {b}, \quad \mathrm{where} \quad \sin \alpha = \| \mathbf {e} _ {3} ^ {a} \times \mathbf {e} _ {3} ^ {b} \|.
$$

The PROJECTION FLEXION-TORSION connection is formulated in terms of the unit vector normal to a plane, $\mathbf { e } _ { 3 }$ , and two unit vectors spanning this plane, $\mathbf { e } _ { 1 }$ and $\mathbf { e } _ { 2 }$ . See Figure 31.1.5–22. The plane with normal vector ${ \bf e } _ { 3 }$ is referred to as the flexion-torsion plane. The component flexion angles $\alpha _ { 1 }$ and $\alpha _ { 2 }$ are determined from and $\mathbf { q }$ by projection onto the two in-plane directions:

$$
\alpha_ {1} = \alpha (\mathbf {e} _ {1} \cdot \mathbf {q}) \quad \text {and} \quad \alpha_ {2} = \alpha (\mathbf {e} _ {2} \cdot \mathbf {q}).
$$

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 $\beta$ between the two shafts is defined as

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

where positive torsion angles are rotations about the positive $\mathbf { e } _ { 3 }$ -direction and m is an integer.

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 flexion-torsion plane.

The available components of relative motion $u r _ { 1 } , u r _ { 2 }$ , and $u r _ { 3 }$ are the changes in the two flexion angles and the torsion angle and are defined as

$$
u r _ {1} = \alpha_ {1} - \alpha_ {1 0}, \quad u r _ {2} = \alpha_ {2} - \alpha_ {2 0}, \quad \mathrm{and} \quad u r _ {3} = \beta - \beta_ {0},
$$

where $\alpha _ { 1 0 } , \alpha _ { 2 0 }$ , and $\beta _ { 0 }$ are the initial flexion component angles and torsion angle, respectively. The connector constitutive rotations are

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

The kinetic moment in a PROJECTION FLEXION-TORSION connection is

$$
\mathbf {m} _ {p r o j f l e x - t o r} = m _ {1} \mathbf {e} _ {1} + m _ {2} \mathbf {e} _ {2} + m _ {3} \mathbf {e} _ {3}.
$$

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Summary 

<table><tr><td colspan="2">PROJECTION FLEXION-TORSION</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 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> $\theta_1^{min} \leq \alpha_1 \leq \theta_1^{max}$ , $\theta_2^{min} \leq \alpha_2 \leq \theta_2^{max}$ , $\theta_3^{min} \leq \beta \leq \theta_3^{max}$ </td></tr><tr><td>Constitutive reference 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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# RADIAL-THRUST

Connection type RADIAL-THRUST provides a connection between two nodes where the response differs in the radial and cylindrical axis directions. Connection type RADIAL-THRUST models situations such as a point inside a cylindrical bearing where the response to radial displacements differs from the response to thrusting motions. Connection type RADIAL-THRUST cannot be used in two-dimensional or axisymmetric analysis.

If the rotational degrees of freedom at the two nodes are connected through flexural and torsional resistance, connection type FLEXION-TORSION can be used in conjunction with connection type RADIAL-THRUST.

![](images/page-639_e011915c30eb7e9d7a4a8899770a8dd42297d130aa978fd3eae3af515cc36081.jpg)

<details>
<summary>text_image</summary>

r
b
e₃ᵃ
l
a
</details>

Figure 31.1.5–23 Connection type RADIAL-THRUST.

# Description

The RADIAL-THRUST connection does not impose kinematic constraints. An orientation at node a is required to define the axis of the rectangular coordinate system, ${ \bf e } _ { 3 } ^ { a } .$ . The position of node b relative to node a is given by the radial and axial-direction distances

$$
r = \sqrt {[ \mathbf {e} _ {1} ^ {a} \cdot (\mathbf {x} _ {b} - \mathbf {x} _ {a}) ] ^ {2} + [ \mathbf {e} _ {2} ^ {a} \cdot (\mathbf {x} _ {b} - \mathbf {x} _ {a}) ] ^ {2}} \quad \mathrm{and} \quad l = \mathbf {e} _ {3} ^ {a} \cdot (\mathbf {x} _ {b} - \mathbf {x} _ {a}).
$$

The RADIAL-THRUST connection has two available components of relative motion, $u _ { 1 }$ and $u _ { 3 }$ . The radial displacement $u _ { 1 }$ measures the change in distance from node b to the axis of the cylindrical coordinate system and is defined as

$$
u _ {1} = r - r _ {0},
$$

where $r _ { 0 }$ is the initial radial distance from node b to the axis. The thrust displacement $u _ { 3 }$ measures the change in distance from node a to node b along the cylindrical axis and is defined as

$$
u _ {3} = l - l _ {0},
$$

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where $l _ { 0 }$ is the initial distance along the axis from node b to node a. The connector constitutive displacements are

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

The kinetic force is

$$
\mathbf {f} _ {r a d - t h r} = f _ {1} \mathbf {e} _ {r} + f _ {3} \mathbf {e} _ {3} ^ {a},
$$

where the radial unit vector is

$$
\mathbf {e} _ {r} = \frac {1}{r} \left[ \mathbf {e} _ {1} ^ {a} \cdot (\mathbf {x} _ {b} - \mathbf {x} _ {a}) \mathbf {e} _ {1} ^ {a} + \mathbf {e} _ {2} ^ {a} \cdot (\mathbf {x} _ {b} - \mathbf {x} _ {a}) \mathbf {e} _ {2} ^ {a} \right].
$$

The radial resistance of the RADIAL-THRUST connector is analogous to a single spring in the ${ \bf e } _ { 1 } ^ { a } { - \bf e } _ { 2 } ^ { a }$ plane. Loads applied in this plane and perpendicular to the current radial unit vector will initially encounter no resistance and may lead to numerical singularity and/or zero pivot warnings from the solver during static analyses. If the numerical singularities cause convergence difficulties, one modeling option is to overlay the RADIAL-THRUST connector with a CARTESIAN connector with a very small elastic stiffness.

# Summary

RADIAL-THRUST 
<table><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> $u_1, u_3$ </td></tr><tr><td>Kinetic force output:</td><td> $f_1, f_3$ </td></tr><tr><td>Orientation at a:</td><td>Required</td></tr><tr><td>Orientation at b:</td><td>Ignored</td></tr><tr><td>Connector stops:</td><td> $l_1^{min} \leq r \leq l_1^{max}$ , $l_3^{min} \leq l \leq l_3^{max}$ </td></tr><tr><td>Constitutive reference lengths:</td><td> $l_1^{ref}, l_3^{ref}$ </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>
