/INTER/TYPE23

Block Format Keyword Defines a contact interface for airbag fabrics, modeling contact between a main surface and a secondary surface which are supposed to belong to an airbag.

This is a soft penalty contact which can deal with penetrations and intersections often coming in the folded airbag mesh. This interface can be used for self-impacting.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/INTER/TYPE23/inter_ID/unit_ID
inter_title
surf_IDs surf_IDm Istf Igap Ibag Idel
Fscalegap Gapmax Fpenmax
Stmin Stmax
Stfac Fric Gapmin Tstart Tstop
IBC Inacti VISs Bumult
Ifric Ifiltr Xfreq
Read this input only if Ifric > 0
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
C1 C2 C3 C4 C5
Read this input only if Ifric > 1
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
C6
Blank
Blank
Blank

Definition

Field Contents SI Unit Example
inter_ID Interface identifier.

(Integer, maximum 10 digits)

unit_ID Unit Identifier.

(Integer, maximum 10 digits)

inter_title Interface title.

(Character, maximum 100 characters)

surf_IDs Secondary surface identifier.

(Integer)

surf_IDm Main surface identifier.

(Integer)

Istf Stiffness definition flag.
= 0
Stfac is a stiffness scale factor and the stiffness is computed according to the secondary side characteristics.
= 1
Stfac is a stiffness value.

(Integer)

Igap Gap/element option flag. 3
= 0
Gap is constant and equal to the minimum gap.
= 1
Variable gap varies according to the characteristics of the impacted main surface and the impacting secondary node.

(Integer)

Ibag Airbag vent holes closure flag in case of contact.
= 0 (Default)
No closure.
= 1
Closure.

(Integer)

Idel Node deletion flag.
= 0 (Default)
No deletion.
= 1
Non-connected nodes are removed from the secondary side of the interface.

(Integer)

Fscalegap Gap scale factor.

Default = 1.0 (Real)

Gapmax Maximum gap.
= 0
No maximum value for the gap.

(Real)

[ m ]
Fpenmax Maximum fraction of initial penetration. 4

(Real)

Stmin Minimum stiffness.

(Real)

[ N m ]
Stmax Maximum stiffness.

Default = 1030 (Real)

[ N m ]
Stfac Interface stiffness (if Istf = 1).

Default set to 0.0 (Real)

[ N m ]
Stiffness scale factor for the interface (if Istf = 0).

Default = 1.0 (Real)

Fric Coulomb friction.

(Real)

Gapmin Minimum gap for impact activation.

(Real)

[ m ]
Tstart Start time.

(Real)

[ s ]
Tstop Time for temporary deactivation.

(Real)

[ s ]
IBC Deactivation flag of boundary conditions at impact.

(Boolean)

Inacti Stiffness deactivation flag of stiffness in case of initial penetrations. 4
= 0
No action.
= 1
Deactivation of stiffness on nodes.
= 5
Gap is variable with time and initial gap is computed as:
g a p 0 = G a p P 0 with P 0 the initial penetration.
= 6
Gap is variable with time but initial penetration is computed as (the node is slightly depenetrated):
g a p 0 = G a p P 0 5 % ( G a p P 0 ) .

(Integer)

VISs Critical damping coefficient on interface stiffness.

Default set to 1.0 (Real)

Bumult Sorting factor. 5 6

Default set to 0.20 (Real)

Ifric Friction formulation flag. 8 9
= 0 (Default)
Static Coulomb friction law.
= 1
Generalized viscous friction law.
= 2
(Modified) Darmstad friction law.
= 3
Renard friction law.
= 4
Exponential decay friction law.

(Integer)

Ifiltr Friction filtering flag. 10
= 0 (Default)
No filter is used.
= 1
Simple numerical filter.
= 2
Standard -3dB filter with filtering period.
= 3
Standard -3dB filter with cutting frequency.

(Integer)

Xfreq Filtering coefficient.

A value should be between 0 and 1.

(Real)

C1 - C6 (Optional) Friction law coefficient.

(Real)

See Table 1

Flags for Deactivation of Boundary Conditions: IBC

(1)-1 (1)-2 (1)-3 (1)-4 (1)-5 (1)-6 (1)-7 (1)-8 (1)-9 (1)-10
IBCX IBCY IBCZ

Definition

Field Contents SI Unit Example
IBCX Deactivation flag of X boundary condition at impact.
= 0
Free DOF
= 1
Fixed DOF

(Boolean)

IBCY Deactivation flag of Y boundary condition at impact.
= 0
Free DOF
= 1
Fixed DOF

(Boolean)

IBCZ Deactivation flag of Z boundary condition at impact.
= 0
Free DOF
= 1
Fixed DOF

(Boolean)

Comments

  1. For contact stiffness:

    K = Stfac K s if Istf = 0.

    While,

    K s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbWaaS baaSqaaiaadohaaeqaaaaa@3852@ is an equivalent nodal stiffness of the secondary component computed as:

    K s = Stfac 0.5 E t when node is connected to a shell element.

    Where,
    E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@3757@
    Young's modulus
    B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@3757@
    Bulk modulus of the secondary component
    t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@3757@
    Shell thickness
    V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@3757@
    Solid element volume
  2. If Gapmin is not specified or set to zero, a default value is computed as the minimum of t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@3757@ (average thickness of the secondary shell elements).
  3. If Igap = 1, variable gap is computed as:
    max [ G a p min , min ( F s c a l e g a p g s , G a p max ) ]

    While,

    g s MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGZbaabeaaaaa@3803@ : secondary node gap:

    g s = t 2 with t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@3757@ is the largest thickness of the shell elements connected to the secondary node.

    If the secondary node is connected to multiple shells, the largest computed secondary gap is used.

    The variable gap is always at least equal to Gapmin.

  4. Inacti = 6 is recommended, in order to avoid numerical (high frequency) effects into the interface before inflation.
    Figure 1.


    If Inacti = 5 or 6 and if Fpenmax is not equal to zero, nodes stiffness is deactivated if:

    Penetration Fpenmax Gap

  5. The sorting factor, Bumult is used to speed up the sorting algorithm.
  6. The sorting factor Bumult is machine dependent.
  7. One node can belong to the two surfaces at the same time.
  8. For friction formulation
    • If the friction flag Ifric = 0 (default), the old static friction formulation is used:

      F t μ F n with μ = Fric ( μ is Coulomb friction coefficient).

    • For flag Ifric > 0, new friction models are introduced. In this case, the friction coefficient is set by a function μ = μ ( ρ , V )
      Where,
      p
      Pressure of the normal force on the main segment
      V
      Tangential velocity of the secondary node relative to the main segment
  9. Currently, the coefficients C1 through C6 are used to define a variable friction coefficient μ for new friction formulations.
    The following formulations are available:
    • Ifric = 1 (generalized viscous friction law):
      μ = Fric + C 1 p + C 2 V + C 3 p V + C 4 p 2 + C 5 V 2
    • Ifric = 2 (Modified Darmstad law):
      μ=Fric+ C 1 e ( C 2 V ) p 2 + C 3 e ( C 4 V ) p+ C 5 e ( C 6 V ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBcq GH9aqpcaWGgbGaamOCaiaadMgacaWGJbGaey4kaSIaam4qamaaBaaa leaacaaIXaaabeaakiabgwSixlaadwgadaahaaWcbeqaamaabmaaba Gaam4qamaaBaaameaacaaIYaaabeaaliaadAfaaiaawIcacaGLPaaa aaGccqGHflY1caWGWbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaam 4qamaaBaaaleaacaaIZaaabeaakiabgwSixlaadwgadaahaaWcbeqa amaabmaabaGaam4qamaaBaaameaacaaI0aaabeaaliaadAfaaiaawI cacaGLPaaaaaGccqGHflY1caWGWbGaey4kaSIaam4qamaaBaaaleaa caaI1aaabeaakiabgwSixlaadwgadaahaaWcbeqaamaabmaabaGaam 4qamaaBaaameaacaaI2aaabeaaliaadAfaaiaawIcacaGLPaaaaaaa aa@6298@
    • Ifric = 3 (Renard law):

      μ = C 1 + ( C 3 C 1 ) V C 5 ( 2 V C 5 ) if V [ 0 , C 5 ]

      μ = C 3 ( ( C 3 C 4 ) ( V C 5 C 6 C 5 ) 2 ( 3 2 V C 5 C 6 C 5 ) ) if V [ C 5 , C 6 ]

      μ = C 2 1 1 C 2 C 4 + ( V C 6 ) 2 if V C 6

      Where,
      C 1 = μ s
      C 2 = μ d
      C 3 = μ max
      C 4 = μ min
      C 5 = V cr 1
      C 6 = V c r 2

      First critical velocity V c r 1 = C 5 must be different to 0 ( C 5 0 ).

      First critical velocity V c r 1 = C 5 must be lower than the second critical velocity V c r 2 = C 6 ( C 5 < C 6 ).

      The static friction coefficient C 1 and the dynamic friction coefficient C 2 , must be lower than the maximum friction C 3 ( C 1 C 3 and C 2 C 3 ).

      The minimum friction coefficient C 4

    • Ifric = 4 (Exponential decay friction law)

      The frictional coefficient is assumed to be dependent on the relative velocity V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvaaaa@36D1@ of the surfaces in contact according to:

      μ = C 1 + F r i c C 1 e C 2 V MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBcq GH9aqpcaWGdbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSYaaeWaaeaa caWGgbGaamOCaiaadMgacaWGJbGaeyOeI0Iaam4qamaaBaaaleaaca aIXaaabeaaaOGaayjkaiaawMcaaiabgwSixlaadwgadaahaaWcbeqa amaabmaabaGaeyOeI0Iaam4qamaaBaaameaacaaIYaaabeaalmaaem aabaGaamOvaaGaay5bSlaawIa7aaGaayjkaiaawMcaaaaaaaa@4F0A@

    Table 1. Units for Friction Formulations
    Ifric Fric C1 C2 C3 C4 C5 C6
    1 [ 1 P a ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaaGymaaqaaiaaccfacaGGHbaaaaGaay5waiaaw2faaaaa @3AD5@ [ s m ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaae4Caaqaaiaab2gaaaaacaGLBbGaayzxaaaaaa@3A46@ [ s Pa m ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaae4CaaqaaiaabcfacaqGHbGaeyyXICTaaeyBaaaaaiaa wUfacaGLDbaaaaa@3E47@ [ 1 Pa 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaaGymaaqaaiaabcfacaqGHbWaaWbaaSqabeaacaaIYaaa aaaaaOGaay5waiaaw2faaaaa@3BC6@ [ s 2 m 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaae4CamaaCaaaleqabaGaaGOmaaaaaOqaaiaab2gadaah aaWcbeqaaiaaikdaaaaaaaGccaGLBbGaayzxaaaaaa@3C2C@
    2 [ 1 Pa 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaaGymaaqaaiaabcfacaqGHbWaaWbaaSqabeaacaaIYaaa aaaaaOGaay5waiaaw2faaaaa@3BC6@ [ s m ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaae4Caaqaaiaab2gaaaaacaGLBbGaayzxaaaaaa@3A46@ [ 1 P a ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaaGymaaqaaiaaccfacaGGHbaaaaGaay5waiaaw2faaaaa @3AD5@ [ s m ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaae4Caaqaaiaab2gaaaaacaGLBbGaayzxaaaaaa@3A46@ [ s m ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaae4Caaqaaiaab2gaaaaacaGLBbGaayzxaaaaaa@3A46@
    3 [ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@ [ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@
    4 [ s m ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaam aalaaabaGaae4Caaqaaiaab2gaaaaacaGLBbGaayzxaaaaaa@3A46@
  10. Friction filtering

    If Ifiltr0, the tangential forces are smoothed using a filter:

    F T f = α F T ( t ) + 1 α F T f ( t d t ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOramaaBa aaleaacaWGubGaamOzaaqabaGccqGH9aqpcqaHXoqycaWHgbWaaSba aSqaaiaadsfaaeqaaOGaaiikaiaadshacaGGPaGaey4kaSYaaeWaae aacaaIXaGaeyOeI0IaeqySdegacaGLOaGaayzkaaGaaCOramaaBaaa leaacaWGubGaamOzaaqabaGccaGGOaGaamiDaiabgkHiTiaadsgaca WG0bGaaiykaaaa@4D2D@

    Where,
    F T f MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOramaaBa aaleaacaWGubGaamOzaaqabaaaaa@38B2@
    Filtered tangential force
    F T ( t ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOramaaBa aaleaacaWGubaabeaakiaacIcacaWG0bGaaiykaaaa@3A23@
    Calculated tangential force at time t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36EC@ before filtering
    F T f ( t d t ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOramaaBa aaleaacaWGubGaamOzaaqabaGccaGGOaGaamiDaiabgkHiTiaadsga caWG0bGaaiykaaaa@3DDD@
    Filtered tangential force at the previous time step
    t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36EC@
    Current simulation time
    d t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaads haaaa@37D5@
    Current simulation time step
    α MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3793@
    Filtering coefficient
    Where, α MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3793@ coefficient is calculated from, if:
    • Ifiltr =1 α = X f r e q , simple numerical filter with a value between 0 and 1.
    • Ifiltr =2 α = 2 π X f r e q , standard -3dB filter, with the number of time steps to filter defined as X f r e q = d t T , and T = filtering period
    • Ifiltr =3 α = 2 π X f r e q d t standard -3dB filter, with Xfreq = cutting frequency
  11. The type of friction penalty formulation is based on the incremental stiffness formulation:

    The friction forces are:

    F t n e w = min ( μ F n , F a d h )

    While an adhesion force is computed as:

    F a d h = F t o l d + Δ F t with Δ F t = K V t d t

    Where, V t is the tangential velocity of the secondary node relative to the main segment.