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김경종
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<!-- source-page: 621 -->
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# FLOW-CONVERTER
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Connection type FLOW-CONVERTER converts the relative rotation about a user-specified axis between the two nodes of the connector into material flow degree of freedom (10) at the second node of a connector element. This connection type can be used to model retractor and pretensioner devices in automotive seat belts (see “Seat belt analysis of a simplified crash dummy,” Section 3.3.1 of the Abaqus Example Problems Guide) or cable drums in winch-like devices. Belt or cable material is considered to be wrapped around an axle or a drum, and material can be spooled either into or out of the connector element.
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In certain cases, material flow needs to be converted into a displacement rather than a rotation. Examples include pretensioner devices for which experimental force vs. displacement data need to be specified. Although this connection type always converts the material flow into a rotation, the two modeling cases are equivalent. The experimentally available force vs. displacement data can be input directly as moment vs. rotation data for the same end result.
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This connection type activates degree of freedom 10 at the second node of a connector. As with any other nodal degree of freedom, you must be careful in constraining it. This is typically done by attaching the connector to a SLIPRING connector that is part of the belt system or by applying a boundary condition. FLOW-CONVERTER connections cannot be used in two-dimensional and axisymmetric analyses in Abaqus/Explicit.
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<details>
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<summary>text_image</summary>
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axle or drum
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belt or cable
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wrapped belt
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or cable
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Lw
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e₃ᵃ
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a, b
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e₃ᵃ
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e₁ᵃ
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a, b
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e₂ᵃ
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</details>
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Figure 31.1.5–14 Connection type FLOW-CONVERTER.
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# Description
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The FLOW-CONVERTER connection constrains the relative rotation between the two nodes about the third local direction, ${ \bf e } _ { 3 } ^ { a }$ , to the material flow at node $b , \Psi _ { b }$ . The constraint can be written as
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$$
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u r _ {3} = \mathbf {e} _ {3} ^ {a} \cdot (\theta_ {a} - \theta_ {b}) - \beta_ {s} \Psi_ {b} = 0,
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$$
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where $\theta _ { a } - \theta _ { b }$ is the relative nodal rotation between node a andb and $\beta _ { s }$ is a scaling factor specified as part of the associated connector section definition. By default, $\beta _ { s } = 1 . 0$ . The local direction ${ \bf e } _ { 3 } ^ { a }$ rotates with the nodal rotation at node a.
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<!-- source-page: 622 -->
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There are no available components of relative motion for this connection type; hence, kinetic behavior cannot be specified. However, the following kinematic quantities are available for output:
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$$
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u r _ {1} = \mathbf {e} _ {3} ^ {a} \cdot (\theta_ {a} - \theta_ {b}) \quad \mathrm{and} \quad u r _ {2} = \Psi_ {b},
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$$
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which will be output as CPR1 and CPR2, respectively.
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The constraint moment is
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$$
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\bar {\bf {m}} = m _ {3} {\bf {e}} _ {3} ^ {a}.
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$$
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# Limitation
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At most two FLOW-CONVERTER connectors can share their second node where degree of freedom 10 is active.
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# Summary
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<table><tr><td colspan="2">FLOW-CONVERTER</td></tr><tr><td>Basic, assembled, or complex:</td><td>Specialized basic rotational</td></tr><tr><td>Kinematic constraints:</td><td> $ur_{3} = 0$ </td></tr><tr><td>Constraint moment output:</td><td> $m_{3}$ </td></tr><tr><td>Available components:</td><td>None</td></tr><tr><td>Kinetic force output:</td><td>None</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>None</td></tr><tr><td>Constitutive reference lengths:</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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<!-- source-page: 623 -->
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# HINGE
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Connection type HINGE joins the position of two nodes and provides a revolute constraint between their rotational degrees of freedom. Connection type HINGE cannot be used in two-dimensional or axisymmetric analysis.
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<details>
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<summary>text_image</summary>
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e₂ᵃ
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a, b
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e₃ᵃ
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e₃ᵇ
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e₂ᵇ
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e₁ᵃ, e₁ᵇ
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</details>
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Figure 31.1.5–15 Connection type HINGE.
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# Description
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Connection type HINGE imposes kinematic constraints and uses local orientation definitions equivalent to combining connection types JOIN and REVOLUTE.
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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 31.2.1). Since the kinematic constraints are enforced at node b (the second node of the connector element), the reported forces and moments are the constraint forces and moments applied at node b to enforce the HINGE constraint. Thus, in most cases the connector output associated with a HINGE connection is best interpreted when node b 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.
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# Friction
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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 31.1.5–16.
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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:
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$$
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\Phi = \mathrm{P} (\mathbf {f}) - \mu \mathrm{M} _ {\mathrm{N}} \leq 0,
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$$
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<!-- source-page: 624 -->
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<details>
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<summary>text_image</summary>
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Ls
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Part B
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Pin
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2Ra
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Part A
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2Rp
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Contact on this face
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between Part A and Part B
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</details>
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Figure 31.1.5–16 Illustration of the geometric scaling constants for a HINGE connection.
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where the potential $\mathrm { P } ( \mathbf { f } )$ represents the moment magnitude of the frictional tangential tractions in the connector in a direction tangent to the cylindrical surface on which contact occurs, $\mathrm { M _ { N } }$ is the frictionproducing normal moment on the same cylindrical surface, and $\mu$ is the friction coefficient. Frictional stick occurs if $\Phi < 0 ;$ and sliding occurs if $\Phi = 0$ , in which case the friction moment is $\mu \mathrm { M _ { N } }$ .
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The normal moment $\mathrm { M _ { N } }$ is the sum of a magnitude measure of friction-producing connector moments, $\mathrm { { M } } _ { \mathrm { { C } } } { \bf \Xi } = { \bf \Lambda } _ { g } ( { \bf f } )$ , and a self-equilibrated internal contact moment (such as from a press-fit assembly), $\mathrm { M } _ { \mathrm { C } } ^ { \mathrm { i n t } }$ :
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$$
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\mathrm{M} _ {\mathrm{N}} = | \mathrm{M} _ {\mathrm{C}} + \mathrm{M} _ {\mathrm{C}} ^ {\mathrm{int}} | = | g (\mathbf {f}) + \mathrm{M} _ {\mathrm{C}} ^ {\mathrm{int}} |.
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$$
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The magnitude measure of friction-producing connector contact moments, $\mathrm { M } _ { \mathrm { C } } ,$ is defined by summing the following contributions:
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<!-- source-page: 625 -->
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• a moment from an axial force, $F _ { a } R _ { a }$ , where $F _ { a } = | f _ { 1 } |$ and $R _ { a }$ is an effective friction arm associated with the constraint force in the axial direction (the $R _ { a }$ radius could be interpreted as an average radius of the outer sleeve cylindrical sections as found in a typical door hinge or as an effective radius associated with the hinge end caps, if they exist; if $R _ { a }$ is $0 . 0 , F _ { a }$ is ignored); and • a moment from normal forces to the cylindrical face, $F _ { n } R _ { p }$ , where $R _ { p }$ is the radius of the pin crosssection in the local 2–3 plane and $F _ { n }$ is itself a sum of the following two contributions:
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– a radial force contribution, $F _ { r }$ (the magnitude of the constraint forces enforcing the translation constraints in the local 2–3 plane):
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$$
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F _ {r} = \sqrt {f _ {2} ^ {2} + f _ {3} ^ {2}}, \mathrm{and}
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$$
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– 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:
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$$
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M _ {b e n d} = \sqrt {m _ {2} ^ {2} + m _ {3} ^ {2}},
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$$
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$$
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F _ {b e n d} = 2 \frac {M _ {b e n d}}{L _ {s}},
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$$
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where $L _ { s }$ represents a characteristic overlapping length between the pin and the sleeve. If $L _ { s }$ is $0 . 0 , M _ { b e n d }$ is ignored.
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Thus,
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$$
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\mathrm {M_ {C}} = g (\mathbf {f}) = F _ {a} R _ {a} + F _ {n} R _ {p} = | f _ {1} R _ {a} | + \sqrt {(R _ {p} f _ {2}) ^ {2} + (R _ {p} f _ {3}) ^ {2}} + \sqrt {(\beta m _ {2}) ^ {2} + (\beta m _ {3}) ^ {2}},
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$$
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where $\begin{array} { r } { \beta = \frac { 2 R _ { p } } { L _ { s } } } \end{array}$ Ls .
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The moment magnitude of the frictional tangential tractions, $\mathrm { P } ( \mathbf { f } ) = | m _ { 1 } |$ .
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# Summary
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<table><tr><td colspan="2">HINGE</td></tr><tr><td>Basic, assembled, or complex:</td><td>Assembled</td></tr><tr><td>Kinematic constraints:</td><td>JOIN + REVOLUTE</td></tr><tr><td>Constraint force and moment output:</td><td> $f_1, f_2, f_3, m_2, m_3$ </td></tr><tr><td>Available components:</td><td> $ur_1$ </td></tr><tr><td>Kinetic force and moment output:</td><td> $m_1$ </td></tr><tr><td>Orientation at $a$ :</td><td>Required</td></tr><tr><td>Orientation at $b$ :</td><td>Optional</td></tr></table>
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<!-- source-page: 626 -->
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<table><tr><td colspan="2">HINGE</td></tr><tr><td>Connector stops:</td><td> $\theta_{1}^{min} \leq \alpha \leq \theta_{1}^{max}$ </td></tr><tr><td>Constitutive reference lengths:</td><td> $\theta_{1}^{ref}$ </td></tr><tr><td>Predefined friction parameters:</td><td>Required: $R_{p}$ ; optional: $R_{a}$ , $L_{s}$ , $M_{C}^{int}$ </td></tr><tr><td>Contact moment for predefined friction:</td><td> $M_{C}$ </td></tr></table>
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<!-- source-page: 627 -->
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# JOIN
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Connection type JOIN makes the position of two nodes the same. If the two nodes are not colocated initially, the position of node b is fixed relative to that of node a in a Cartesian coordinate system attached to node a.
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Even though an orientation is optional at node a, connection type JOIN does not activate rotational degrees of freedom at node a.
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<details>
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<summary>natural_image</summary>
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Simple 3D illustration of a stylized object resembling a hammer or stylus, with blue and red sections (no text or symbols)
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</details>
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<details>
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<summary>text_image</summary>
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e₂ᵃ
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e₁ᵃ
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a, b
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e₃ᵃ
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</details>
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Figure 31.1.5–17 Connection type JOIN.
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# Description
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The JOIN connection makes the position of node b equal to that of node a. If the two nodes are not coincident initially, the Cartesian coordinates of node b relative to node a are fixed. See connection type CARTESIAN for a definition of the Cartesian coordinates of node b relative to node a. If rotational degrees of freedom exist at node a, the local directions corotate with the node.
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The constraint force in the JOIN connection acts in the three local directions at node a and is
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$$
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\bar {\mathbf {f}} = f _ {1} \mathbf {e} _ {1} ^ {a} + f _ {2} \mathbf {e} _ {2} ^ {a} + f _ {3} \mathbf {e} _ {3} ^ {a},
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$$
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where $f _ { 3 } = 0$ in two-dimensional analysis.
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# Friction
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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.
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# Summary
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# JOIN
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Basic, assembled, or complex:
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Basic
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Kinematic constraints:
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$$
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u _ {1} = 0, u _ {2} = 0, u _ {3} = 0
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$$
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<!-- source-page: 628 -->
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JOIN
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<table><tr><td>Constraint force output:</td><td> $f_{1}, f_{2}, f_{3}$ </td></tr><tr><td>Available components:</td><td>None</td></tr><tr><td>Kinetic force output:</td><td>None</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>None</td></tr><tr><td>Constitutive reference lengths:</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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<!-- source-page: 629 -->
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# LINK
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Connection type LINK maintains a constant distance between two nodes. Rotational degrees of freedom, if they exist, are not affected at either node.
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Figure 31.1.5–18 Connection type LINK.
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# Description
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The LINK connection constrains the position of node b, , to a constant distance from node a. The distance between the two nodes is
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$$
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l = | | \mathbf {x} _ {b} - \mathbf {x} _ {a} | |
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$$
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and is constant. The constraint force in the LINK connection acts along the line connecting the two nodes and is
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$$
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\bar {\mathbf {f}} = f _ {1} \mathbf {q}, \quad \text { where } \quad \mathbf {q} = \frac {1}{\| \mathbf {x} _ {b} - \mathbf {x} _ {a} \|} (\mathbf {x} _ {b} - \mathbf {x} _ {a}).
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$$
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# Summary
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<table><tr><td colspan="2">LINK</td></tr><tr><td>Basic, assembled, or complex:</td><td>Basic</td></tr><tr><td>Kinematic constraints:</td><td> $l = constant$ </td></tr><tr><td>Constraint force output:</td><td> $f_1$ </td></tr><tr><td>Available components:</td><td>None</td></tr><tr><td>Kinetic force output:</td><td>None</td></tr><tr><td>Orientation at a:</td><td>Ignored</td></tr><tr><td>Orientation at b:</td><td>Ignored</td></tr><tr><td>Connector stops:</td><td>None</td></tr><tr><td>Constitutive reference lengths:</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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<!-- source-page: 630 -->
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# PLANAR
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Connection type PLANAR provides a local two-dimensional system in a three-dimensional analysis. Connection type PLANAR cannot be used in two-dimensional or axisymmetric analysis.
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<details>
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<summary>text_image</summary>
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e₁ᵃ
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a
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e₃ᵃ
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e₂ᵃ
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e₁ᵇ
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ur₁
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b
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e₃ᵇ
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u₂
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u₃
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e₂ᵇ
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</details>
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Figure 31.1.5–19 Connection type PLANAR.
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# Description
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Connection type PLANAR imposes kinematic constraints and uses local orientation definitions equivalent to combining connection types SLIDE-PLANE and REVOLUTE.
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# Friction
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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 frictional moment in the rotation about the local 1-direction. These two frictional effects are discussed separately below.
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A. The frictional effect due to sliding in the 2–3 plane is formally written as
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$$
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\Phi_ {\mathrm{C}} = \mathrm{P} _ {\mathrm{C}} (\mathbf {f}) - \mu \mathrm{F} _ {\mathrm{N} _ {\mathrm{C}}} \leq 0,
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$$
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where the potential $\mathrm { P } _ { \mathrm { C } } ( \mathbf { f } )$ represents the magnitude of the frictional tangential tractions in the connector in a direction tangent to the local 2–3 plane on which contact occurs, $\mathrm { F _ { N _ { C } } }$ is the frictionproducing normal force on the same plane, and $\mu$ is the friction coefficient. Frictional stick occurs if $\Phi _ { \mathrm { { C } } } < 0 ;$ and sliding occurs if $\Phi _ { \mathrm { { C } } } = 0$ , in which case the friction force (CSFC) is $\mu \mathrm { F _ { N _ { C } } }$ .
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The normal force $\mathrm { F _ { N _ { C } } }$ is the sum of a magnitude measure of force-producing connector forces, $\mathrm { F } _ { \mathrm { C } } = g ( \mathbf { f } )$ , and a self-equilibrated internal contact force, $\mathrm { F _ { C } ^ { \mathrm { i n t } } }$ :
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