Summary

SLIDE-PLANE
Basic, assembled, or complex:	Basic
Kinematic constraints:	$x = x_0$
Constraint force output:	$f_1$
Available components:	$u_2, u_3$
Kinetic force output:	$f_2, f_3$
Orientation at a:	Required
Orientation at b:	Ignored
Connector stops:	$l_2^{min} \leq y \leq l_2^{max}$ , $l_3^{min} \leq z \leq l_3^{max}$
Constitutive reference lengths:	$l_2^{ref}, l_3^{ref}$
Predefined friction parameters:	Optional: $F_{C}^{int}$
Contact force for predefined friction:	$F_{C}$

# SLIPRING Connection type SLIPRING provides a connection between two nodes that models material flow and stretching between two points of a belt system. It can be used to model seat belts (see “Seat belt analysis of a simplified crash dummy,” Section 3.3.1 of the Abaqus Example Problems Guide), pulley systems, and taut cable systems. The angle between two adjacent belt segments is used only for friction calculations. By default, the angle, , is computed automatically from the nodal coordinates as an angle between and . Alternatively, you can specify the angle between two adjacent belt segments (in radians) as part of the connector section definition. You can use this option to specify wrapping angles larger than . This connection type activates the material flow degree of freedom (10) at both nodes of the 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 other SLIPRING connectors that are part of the belt system, attaching it to a RETRACTOR (FLOW-CONVERTER) connector, or applying a boundary condition. SLIPRING connections cannot be used in two-dimensional and axisymmetric analyses in Abaqus/Explicit. 

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a Ψa radius ignored b Ψb α c

Figure 31.1.5–29 Connection type SLIPRING. # Description The SLIPRING connection does not constrain any component of relative motion. Hence, there is no restriction on the position of the connector nodes. The distance between nodes is $$ d _ {a b} = \left\| \mathbf {x} _ {b} - \mathbf {x} _ {a} \right\|. $$ The belt material can flow and stretch between nodes a and b. Flow can occur with no stretching (such as in a rigid belt), stretching can occur with no flow (such as when the flow is constrained at both nodes of the connector), or both flow and stretching can occur simultaneously (such as in compliant belts). By convention, the material flow at node a is positive if it enters segment and is positive at node b if it exits the segment. A reference length can be defined in incremental fashion as $$ l _ {n e w} ^ {r e f} = l _ {o l d} ^ {r e f} + \Delta \Psi_ {a} - \Delta \Psi_ {b}, $$ where $l _ { n e w } ^ { r e f }$ is the reference length at the end of the current increment, $l _ { o l d } ^ { r e f }$ is the reference length at the beginning of the current increment, $\Delta \Psi _ { a }$ is the incremental flow at node ${ \pmb a } ,$ and $\Delta \Psi _ { b }$ is the incremental flow at node b. The stretch in the belt can then be defined as $$ d = \frac {d _ {a b}}{l _ {n e w} ^ {r e f}}, $$ and the “strain” in the belt can be computed as $$ u _ {1} = u _ {1} ^ {m a t} = d - 1. $$ At the beginning of the analysis, the reference length at $t = 0$ is $$ l ^ {r e f} | _ {t = 0} = \frac {d _ {a b} | _ {t = 0}}{d _ {p}}, $$ where $d _ { p }$ is the initial stretch of the belt. By default, the initial stretch is $d _ { p } = 1 . 0$ meaning that there are no initial strains in the belt. You can specify initial strains in the belt, $u _ { 1 } | _ { t = 0 }$ , by specifying a connector constitutive reference. The initial stretch is then computed using $$ d _ {p} = u _ {1} | _ {t = 0} + 1. $$ The second available component of relative motion is simply the material flow past node b, $$ u _ {2} = u _ {2} ^ {m a t} = \Psi_ {b}. $$ The third component of relative motion is the material flow into node a and is used only for output: $$ u _ {3} = u _ {3} ^ {m a t} = \Psi_ {a}. $$ The kinetic force is $$ \mathbf {f} _ {s l i p r i n g} = f _ {1} l _ {n e w} ^ {r e f} \mathbf {q}, \quad \mathrm{where} \quad \mathbf {q} = \frac {1}{\left\| \mathbf {x} _ {b} - \mathbf {x} _ {a} \right\|} (\mathbf {x} _ {b} - \mathbf {x} _ {a}). $$ # Limitations At most two SLIPRING connectors can share a common node. The following limitations apply with respect to the kinetic behavior that can be defined in the SLIPRING connection type: • Only predefined friction can be defined in the second component of relative motion as outlined below. • In Abaqus/Explicit plasticity, damage and lock connector behavior cannot be specified. • The connectivities of the two adjacent SLIPRING connector elements sharing a common node b (Figure 31.1.5–29) should be in the typical order $\scriptstyle a - b$ and $_ { b - c . }$ In addition, any two adjacent SLIPRING connector elements must refer to the same connector behavior except for the friction data. # Friction Predefined Coulomb-like friction in the SLIPRING connection relates the tension in the belt segment (kinetic force $f _ { 1 }$ in component 1) to the tension in the adjacent belt segment . In the simpler case of frictionless sliding, the two tensions are equal (apart from inertial effects due to the motion of the belt in dynamic analyses). If frictional effects are included as material flows past node $b ,$ the two tensions differ by the total friction force (CSF2) over the contact arch between the belt and the ring (angle ). The Coulomb-like frictional effect is a well-known analytical result. In the case when frictional sliding occurs in the direction illustrated in Figure 31.1.5–29, the tensions in the two segments, $f _ { a b } = f _ { 1 }$ and $f _ { b c }$ , are related as follows: $$ f _ {a b} = f _ {b c} e ^ {- \mu \alpha}, $$ where $\mu$ is the friction coefficient. The friction force is simply the difference $$ C S F 2 = f _ {b c} - f _ {a b}. $$ More formally, the frictional relationship is modeled by considering the potential function $$ \Phi = f _ {a b} - f _ {b c} e ^ {- \mu \alpha}. $$ Frictional stick occurs if $\Phi < 0 ;$ and sliding occurs if $\Phi = 0$ , in which case the tension force $f _ { a b } =$ $f _ { b c } e ^ { - \mu \alpha }$ . Friction forces do not develop if the kinetic force $f _ { 1 }$ is compressive. When sliding occurs in the opposite direction, the sign of the exponent in the potential equation changes. The friction force is reported as $f _ { 2 }$ in this connection type. The friction-generating “contact force” is reported as $\mathrm { C N F } 2 { = } f _ { 1 }$ . In Abaqus/Explicit, by default, the distance between the two nodes of the SLIPRING is not allowed to become less then one hundredth of the original distance between the nodes, which prevents the SLIPRING from collapsing to zero length during the analysis. The two nodes of the SLIPRING can move apart after coming to the minimum distance configuration during the analysis. In addition, the belt can continue to slip over the nodes while they are stopped at the minimum distance configuration. This default value of the minimum distance can be overridden by specifying a lower limit of the connector stop in component 1 of the SLIPRING. # Output Some of the connector output variables have a somewhat different meaning for this connection type than usual, as follows: • CP1 is the current distance between the nodes; • CP2 is the material flow at node $\begin{array} { r } { \pmb { b } ; } \end{array}$ • CP3 is the material flow at node ${ \pmb a } _ { \ u { \gamma } }$ and • CU1 is the strain (dimensionless) in the segment . Summary

SLIPRING
Basic, assembled, or complex:	Complex
Kinematic constraints:	None
Constraint force output:	None
Available components:	$u_{1}, u_{2}, u_{3}$
Kinetic force output:	$f_{1}, f_{2}$
Orientation at a:	Ignored
Orientation at b:	Ignored
Connector stops:	None
Constitutive reference lengths:	$d_{p} - 1$
Predefined friction parameters:	None
Contact force for predefined friction:	$f_{1}$

# SLOT Connection type SLOT provides a connection where node b stays on the line defined by the orientation of node a and the initial position of node b. The line of action of the slot is the ${ \bf e } _ { 1 } ^ { a }$ -direction. In three-dimensional analysis node b cannot move in the direction normal to the slot; i.e., the ${ \bf e } _ { 3 } ^ { a }$ direction. If node b is free to move in the normal direction, connection type SLIDE-PLANE should be used. 

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e² a e¹ y₀ b u₁

Figure 31.1.5–30 Connection type SLOT. # Description The line of the slot is defined by the first local direction at node ${ \pmb a } , { \bf e } _ { 1 } ^ { a }$ , and the initial position of node b. The SLOT connection constrains the position of node $\pmb { b } , \mathbf { x } _ { b }$ , to remain on the line of the slot. Therefore, the relative position of node b is fixed in the directions perpendicular to the slot: $$ y = \mathbf {e} _ {2} ^ {a} \cdot (\mathbf {x} _ {b} - \mathbf {x} _ {a}) = y _ {0}, $$ where $y _ { 0 }$ is the initial distance from node a to the slot in the local 2-direction. In three dimensions $$ z = \mathbf {e} _ {3} ^ {a} \cdot (\mathbf {x} _ {b} - \mathbf {x} _ {a}) = z _ {0}, $$ where $z _ { 0 }$ is the initial distance from node a to the slot in the local 3-direction. The constraint force in the slot is $$ \bar {\mathbf {f}} = f _ {2} \mathbf {e} _ {2} ^ {a} + f _ {3} \mathbf {e} _ {3} ^ {a}, $$ where $f _ { 3 } = 0$ in two-dimensional analysis. Node b can move along the line of the slot. The relative position in the slot is the distance between node b and node a along the ${ \bf e } _ { 1 } ^ { a }$ -direction and is defined as $$ x = \mathbf {e} _ {1} ^ {a} \cdot (\mathbf {x} _ {b} - \mathbf {x} _ {a}). $$ The available component of relative motion is the displacement $u _ { 1 }$ , which measures the change of the relative position in length along the slot and is defined as $$ u _ {1} = x - x _ {0}, $$ where $x _ { 0 }$ is the initial distance between node b and node a along the slot. The connector constitutive displacement is $$ u _ {1} ^ {m a t} = x - l _ {1} ^ {r e f}. $$ The kinetic force in the slot is $$ \mathbf {f} _ {s l o t} = f _ {1} \mathbf {e} _ {1} ^ {a}. $$ # Friction 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 $$ \Phi = \mathrm{P} (\mathbf {f}) - \mu \mathrm{F} _ {\mathrm{N}} \leq 0, $$ where the potential $\mathrm { P } ( \mathbf { f } )$ represents the magnitude of the frictional tangential tractions in the connector in a direction tangent to the slot axis along which contact occurs, $\mathrm { F _ { N } }$ is the friction-producing normal (contact) force in the direction normal to the slot, and $\mu$ is the friction coefficient. Frictional stick occurs if $\Phi < 0 ;$ ; and sliding occurs if $\Phi = 0$ , in which case the friction force is $\mu \mathrm { F _ { N } }$ . The normal force $\mathrm { F _ { N } }$ is the sum of a magnitude measure of the friction-producing connector force, $\mathrm { F } _ { \mathrm { C } } = g ( \mathbf { f } )$ , and a self-equilibrated internal contact force, $\mathrm { F _ { C } ^ { i n t } }$ : $$ \mathrm {F_ {N}} = | \mathrm {F_ {C}} + \mathrm {F_ {C} ^ {int}} | = | g (\mathbf {f}) + \mathrm {F_ {C} ^ {int}} |. $$ The force magnitude $\mathrm { F _ { C } }$ is computed using $$ \mathrm {F_ {C}} = \sqrt {f _ {2} ^ {2} + f _ {3} ^ {2}}. $$ The magnitude of the frictional tangential tractions $\mathrm { P } ( \mathbf { f } ) = | f _ { 1 } |$ . 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. Summary

SLOT
Basic, assembled, or complex:	Basic
Kinematic constraints:	$y = y_0, z = z_0$
Constraint force output:	$f_2, f_3$
Available components:	$u_1$
Kinetic force output:	$f_1$
Orientation at $a$ :	Required
Orientation at $b$ :	Ignored
Connector stops:	$l_{1}^{min} \leq l \leq l_{1}^{max}$
Constitutive reference lengths:	$l_{1}^{ref}$
Predefined friction parameters:	Optional: $F_{C}^{int}$
Contact force for predefined friction:	$F_{C}$

# TRANSLATOR Connection type TRANSLATOR provides a slot constraint between two nodes and aligns their local directions. 

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e₂ᵃ a e₁ᵃ e₁ᵇ b e₃ᵃ e₃ᵇ e₂ᵇ u₁

Figure 31.1.5–31 Connection type TRANSLATOR. # Description 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 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 TRANSLATOR constraint. Thus, in most cases the connector output associated with a TRANSLATOR 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. # Friction Predefined Coulomb-like friction in the TRANSLATOR connection relates the kinematic constraint forces and moments in the connector to the friction force (CSF1) in the translation along the slot. The frictional effect is formally written as $$ \Phi = \mathrm{P} (\mathbf {f}) - \mu \mathrm{F} _ {\mathrm{N}} \leq 0, $$ where the potential $\mathrm { P } ( \mathbf { f } )$ represents the magnitude of the frictional tangential traction in the connector in the local 1-direction, $\mathrm { F _ { N } }$ is the friction-producing normal (contact) force in the direction normal to the slot, and $\mu$ is the friction coefficient. Frictional stick occurs if $\Phi < 0 ;$ ; and sliding occurs if $\Phi = 0$ , in which case the friction force is $\mu \mathrm { F _ { N } }$ . The normal force $\mathrm { F _ { N } }$ is the sum of a magnitude measure of contact friction-producing connector forces, $\mathrm { F } _ { \mathrm { C } } = g ( \mathbf { f } )$ , and a self-equilibrated internal contact force, $\mathrm { F _ { C } ^ { i n t } }$ : $$ \mathrm {F_ {N}} = | \mathrm {F_ {C}} + \mathrm {F_ {C} ^ {int}} | = | g (\mathbf {f}) + \mathrm {F_ {C} ^ {int}} |. $$ The contact force magnitude $\mathrm { F _ { C } }$ is defined by summing the following three contributions: • a force contribution from torque, $F _ { t o r q }$ , obtained by scaling the torque constraint moment about the 1-direction, $M _ { t o r q }$ , by a length factor, as follows: $$ M _ {t o r q} = | m _ {1} |, $$ $$ F _ {t o r q} = \frac {M _ {t o r q}}{R _ {r}}, $$ where $R _ { r }$ represents the effective radius of the shaft cross-section in the local 2–3 plane (if $R _ { r }$ is $0 . 0 , M _ { t o r q }$ is ignored); • a radial force contribution, $F _ { r }$ (the magnitude of the constraint forces enforcing the SLOT constraint): $$ F _ {r} = \sqrt {f _ {2} ^ {2} + f _ {3} ^ {2}}; $$ and • a force contribution from “bending,” $F _ { b e n d }$ , obtained by scaling the bending constraint moment, $M _ { b e n d }$ , by a length factor, as follows: $$ M _ {b e n d} = \sqrt {m _ {2} ^ {2} + m _ {3} ^ {2}}, $$ $$ F _ {b e n d} = 2 \frac {M _ {b e n d}}{L}, $$ where L represents a characteristic overlapping length in the slot direction. If L is $0 . 0 , M _ { b e n d }$ is ignored. Thus, $$ \mathrm {F_ {C}} = g (\mathbf {f}) = F _ {t o r q} + F _ {r} + F _ {b e n d} = | \frac {m _ {1}}{R _ {r}} | + \sqrt {f _ {2} ^ {2} + f _ {3} ^ {2}} + \sqrt {(\beta m _ {2}) ^ {2} + (\beta m _ {3}) ^ {2}}, $$ where $\begin{array} { r } { \beta = \frac { 2 } { L } } \end{array}$ The magnitude of the frictional tangential tractions, is $\left| f _ { 1 } \right|$ .