Force: Joint Friction

Format

<Force_JointFriction
     id                        = "integer"   
   [ label                     = "string" ]     
     joint_id                  = "integer"
     mu_static                 = "real"
     mu_dynamic                = "real"   
   [ max_stiction_deform       = "real" ]
   [ bristle_stiffness         = "real" ]
   [ damping_effects           = "real" ]   
   [ viscous_effects           = "real" ]
   [ stiction_trans_vel        = "real"]
   [ inactive                  = "None | Static" ]
   [ effect                    = "All|Stiction |Sliding" ]
   [ inputs                    = "4-bit integer" ]
   [ torque_preload            = "real" ]
   [ friction_arm              = "real" ]
   [ bending_reaction_arm      = "real" ]
   [ pin_radius                = "real" ]
/>

<Force_JointFriction
     id                        = "integer"   
   [ label                     = "string" ]     
     joint_id                  = "integer"
     mu_static                 = "real"
     mu_dynamic                = "real"    
   [ max_stiction_deform       = "real" ]   
   [ bristle_stiffness         = "real" ]
   [ damping_effects           = "real" ]
   [ viscous_effects           = "real" ]
   [ stiction_trans_vel        = "real"]
   [ inactive                  = "None |Static" ]
   [ effect                    = "All |Stiction|Sliding" ]
   [ inputs                    = "4-bit integer" ]    
   [ torque_preload            = "real" ]
   [ ball_radius               = "real" ]
/>

<Force_JointFriction
     id                        = "integer"   
   [ label                     = "string" ]     
     joint_id                  = "integer"
     mu_static                 = "real"
     mu_dynamic                = "real"    
   [ max_stiction_deform       = "real" ]
   [ bristle_stiffness         = "real" ]   
   [ damping_effects           = "real" ]
   [ viscous_effects           = "real" ]
   [ stiction_trans_vel        = "real" ]
   [ inactive                  = "None|Static"]
   [ effect                    = "All|Stiction|Sliding" ]
   [ inputs                    = "4-bit integer" ]     
   [ force_preload             = "real" ]
   [ reaction_arm              = "real" ]
   [ initial_overlap           = "real" ]
   [ overlap_delta             = "Constant|Increase|Decrease" ]
/>

<Force_JointFriction
     id                        = "integer"   
   [ label                     = "string" ]     
     joint_id                  = "integer"
     mu_static                 = "real"
     mu_dynamic                = "real"    
   [ max_stiction_deform       = "real" ]
   [ bristle_stiffness         = "real" ]
   [ damping_effects           = "real" ]
   [ viscous_effects           = "real" ]
   [ stiction_trans_vel        = "real"]
   [ inactive                  = "None|Static"]
   [ effect                    = "All|Stiction|Sliding" ]
   [ inputs                    = "4-bit integer" ]
   [ force_preload             = "real" ]
   [ torque_preload            = "real" ]
   [ pin_radius                = "real" ]
   [ initial_overlap           = "real" ]
   [ overlap_delta             = "Constant|Increase|Decrease" ]
/>

<Force_JointFriction
     id                        = "integer"   
   [ label                     = "string" ]     
     joint_id                  = "integer"
     mu_static                 = "real"
     mu_dynamic                = "real"    
   [ max_stiction_deform       = "real" ]
   [ bristle_stiffness         = "real"]   
   [ damping_effects           = "real"]   
   [ viscous_effects           = "real"]   
   [ stiction_trans_vel        = "real"]   
   [ inactive                  = "None|Static"]
   [ effect                    = "All|Stiction|Sliding"]
   [ inputs                    = "4-bit integer" ]
   [ torque_preload            = "real" ]
   [ friction_arm              = "real" ]
   [ bending_reaction_arm      = "real" ]
   [ pin_radius                = "real" ]
   [ yoke_choice               = "i_yoke|j_yoke"]
/>

Attributes

id

Element identification number, (integer > 0). This number is unique among all Force_JointFriction elements.

label

A string containing the element description.

joint_id

Specifies the joint ID of the joint where the friction force is applied.

mu_static

Specifies the static friction coefficient (μ_s) in the joint.

mu_static > 0.

Default: NONE

mu_dynamic

Specifies the dynamic friction coefficient (μ_d) in the joint.

mu_dynamic > 0.

Default: NONE

max_stiction_deform

Specifies the maximum deformation that can occur in a joint for static friction.

If specified to be non-zero, a finite static friction force is applied even if the relative velocity is zero.

Default = 0.01 length units

stiction_trans_vel

Real value of the absolute velocity below, which the friction transitions from dynamic friction to static friction.

Default = 0.1 length/time.

inactive

Specify "STATIC" to disable joint friction during static analysis.

Default: NONE

effect

Specifies the frictional effect. By default, both static and dynamic friction is considered. Stiction uses only static friction, sliding uses only dynamic friction.

Default: ALL

inputs

A 4-bit integer value specifies the input forces to the friction model. By default, all the preload and joint reaction forces are included. Valid values are 0 or 1 for each bit.

Default: 1111

There are four choices:

• 1st bit: Preload
• 2nd bit: Reaction_Force
• 3rd bit: Bending_Moment
• 4th bit: Torsional_Moment

Setting a 0 or 1 for each bit of the inputs value controls individual inputs.

Thus:

• 1111 - All four inputs are included
• 0000 - None of the four inputs are included
• 1000 - Only include Preload
• 1101 - Include Preload, Reaction_Force and Torsional_Moment, but not Bending_Moment
• 1001 - Include Preload and Torsional_Moment, but not Reaction_Force and Bending_Moment

Options available per joint type are as defined below:

• Translational - Preload, Reaction_Force, Bending_Moment, Torsional_Moment
• Cylindrical, Revolute, Universal, Hooke - Preload, Reaction_Force, Bending_Moment
• Spherical: Preload, Reaction_Force

force_preload

Specifies the preload friction force for translational and cylindrical joints.

Default : 0.0 force units

torque_preload

Specifies the preload friction torque for revolute, cylindrical, universal and spherical joints.

Default: 0.0 force*length units

friction_arm

Specifies the moment arm used to compute axial friction torque in revolute, universal joints.

Default: 1.0 length units

reaction_arm

Specifies the moment arm of the reaction torque about the translation joint axial axes.

Default: 1.0 length units

bending_reaction_arm

Specifies the moment arm to compute the bending moment in revolute, hooke and universal joints.

Default: 1.0 length units

pin_radius

Specifies the radius of the pin for revolute, cylindrical, hooke and universal joints.

Default: 1.0 length units

ball_radius

Specifies the radius of the spherical joint.

Default: 1.0 length units

initial_overlap

Specifies the initial overlap of the sliding parts in translation and cylindrical joints.

Default: 1000.0 length units

overlap_delta

Specifies friction characteristics in the sliding joint.

INCREASE

Overlap increases as the I marker translates in the positive direction of the z-axis of the J marker.

DECREASE

Overlap decreases in the positive direction of the J marker.

Default: CONSTANT

yoke_choice

Specifies the yoke choice for hooke and universal joints.

bristle_stiffness

Specifies the bristle stiffness in the LuGre model, σ₀. See Comments 1-4 for more information.

Default: 100

damping_effects

Defines the damping coefficient for the pre-displacement (or stiction) regime. Its main role is to damp out bristle vibrations in the pre-displacement regime. Represented as σ₁ in this documents. See Comments 1-4 for more information.

Default: 0.316

viscous_effects

Specifies the coefficient for the viscous damping force that occurs when relative sliding actually begins. Represented as σ₂ in this document. See Comments 1-4 for more information. In order to guarantee passivity of the LuGre model (for example, it always dissipates energy) the following condition must be met:

$${\sigma }_1<\frac{{\sigma }_2{\mu }_d}{{\mu }_s-{\mu }_d}$$

Default: 0.0004

Examples

<Force_JointFriction
    id                     = "1"

    joint_id               = "301001"

    mu_static              = "0.5"

    mu_dynamic             = "0.3"

    inactive               = "NONE"

    stiction_trans_vel     = "10."

    effect                 = "All"

    torque_preload         = "0."

    friction_arm           = "7."

    pin_radius             = "5."

    bending_reaction_arm   = "5."

    inputs                 = "1111"   
 />

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  <Param_Transient
     integrator_type     = "DSTIFF"
     integr_tol          = "0.0001"
     h_max               = "0.1"
     h0_max              = "0.005"
     h_min               = "1.0000E-06"
     max_order           = "6"
     vel_tol_factor      = "1000."
     rel_abs_tol_ratio   = "0.01"
     central_difference  = "FALSE"
     dae_constr_tol      = "0.000001"
     dae_corrector_maxit = "10"
     dae_corrector_minit = "0"
     dae_index           = "3"
     dae_vel_ctrl        = "TRUE"
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     hht_alpha           = "-0.33"
     newmark_beta        = "0.36"
     newmark_gamma       = "0.7"
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     force_unit          = "NEWTON"
     mass_unit           = "KILOGRAM"
     length_unit         = "MILLIMETER"
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  <Reference_Marker
     id                  = "30101010"
     label               = "Global Frame"
     body_id             = "30101"
     body_type           = "RigidBody"
  />
  <Reference_Marker
     id                  = "30101020"
     label               = "Joint 0-Marker J"
     body_id             = "30101"
     body_type           = "RigidBody"
     a00                 = "-1."
     a10                 = "0."
     a20                 = "0."
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     id                  = "30102020"
     label               = "Joint 0-Marker I"
     body_id             = "30102"
     body_type           = "RigidBody"
     a00                 = "-1."
     a10                 = "0."
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     a22                 = "0."
  />
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     id                  = "30102030"
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     body_id             = "30102"
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     body_id             = "30102"
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     a20                 = "0."
     a02                 = "1."
     a12                 = "0."
     a22                 = "0."
  />
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     id                  = "90000001"
     body_id             = "30102"
     body_type           = "RigidBody"
     pos_x               = "-100."
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     a00                 = "0."
     a10                 = "1."
     a20                 = "0."
     a02                 = "1."
     a12                 = "0."
     a22                 = "0."
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  <Reference_Marker
     id                  = "90000002"
     body_id             = "30102"
     body_type           = "RigidBody"
     a00                 = "0."
     a10                 = "-1."
     a20                 = "0."
     a02                 = "-1."
     a12                 = "0."
     a22                 = "0."
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  <Reference_Marker
     id                  = "90000012"
     body_id             = "30102"
     body_type           = "RigidBody"
     pos_x               = "-100."
     pos_y               = "-9.86361"
     pos_z               = "-9.86361"
  />

<Body_Rigid
     id                  = "30101"
     label               = "Ground"
     IsGround            = "TRUE"
  />
  <Body_Rigid
     id                  = "30102"
     label               = "bob"
     cg_id               = "30102030"
     im_id               = "30102030"
     lprf_id             = "73780000"
     mass                = "1."
     inertia_xx          = "1000."
     inertia_yy          = "1000."
     inertia_zz          = "1000."
  />
  <Constraint_Joint
     id                  = "301001"
     label               = "Joint 0"
     type                = "REVOLUTE"
     i_marker_id         = "30102020"
     j_marker_id         = "30101020"
  />
  <Force_Gravity
     grav_x              = "0."
     grav_y              = "0."
     grav_z              = "-9810."
  />
 <Force_JointFriction
     id                  = "1"
     joint_id            = "301001"
     mu_static           = "0.5"
     mu_dynamic          = "0.3"
     inactive            = "NONE"
     stiction_trans_vel  = "10."
     effect              = "All"
     torque_preload      = "0."
     friction_arm        = "7."
     pin_radius          = "5."
     bending_reaction_arm= "5."
     inputs              = "1111"
  />
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Comments

1. MotionSolve uses the LuGre (Lundt-Grenoble) model for friction. This model may be summarized as follows:
   A bristle model is used to idealize friction. Friction is modeled as the average deflection force of elastic springs between two mating surfaces. When a tangential motion is applied, the bristles will deflect like springs. If the deflection is sufficiently large, the bristles start to slip. The slip velocity between the mating surfaces, v_slip, determines the average bristle deflection for a steady state motion. It is lower at low velocities, which implies that the steady state deflection decreases with increasing velocity. Figure 2 depicts the bristle model.

   
   
   

2. The LuGre model is capable of representing several different effects:
   • The effect of the mating surfaces being pushed apart by lubricant.
   • The Stribeck effect (at very low speed). When partial fluid lubrication exists, contact between the surfaces decreases and thus friction decreases exponentially from stiction.
   • Rate dependent friction phenomena, such as varying break-away force and frictional lag Static friction between two surfaces.
3. The LuGre model, for friction in one dimension, is expressed as follows:

   $F=LuGre({\mu }_s,{\mu }_d,\sigma ,N,v,z)$ $F=N({\sigma }_{0}z+{\sigma }_1\frac{dz}{dt})+{\sigma }_{2}v$ $\sigma =\left[{\sigma }_0{\sigma }_1{\sigma }_2\right]$ $g(v)={\mu }_d+\left({\mu }_s-u_d\right)e^{-{(\raisebox{1ex}{$v$}\!\left/ \!\raisebox{-1ex}{$vs$}\right.)}^2}$ $\frac{dz}{dt}=v-{\sigma }_0\frac{\left|v\right|z}{g(v)}$

   States z = bristle deflection Inputs v = slip velocity vs = static > dynamic friction transition slip velocity Parameters s0 = bristle stiffness s1 = damping coefficient s2 = viscosity coefficient vs = stiction transition velocity Outputs F = friction force

4. The LuGre function may be explained as follows: the function g(v) defines the shape of the friction function.

   
   
   

   As shown in Figure 3, g(v) can model Coulomb friction 2(a), Stiction 2(b), and the Stribeck effect (2c). From its definition: ${\mu }_d\le g(v)\le {\mu }_s$ .

   At the microscopic level, two surfaces make contact at various asperities. These asperities are represented with bristles, and the bristles deflect like a spring when there is a relative velocity between the two surfaces. The deflection of the springs gives rise to the friction force. If the deflection is sufficiently large, then the bristles will slip in a highly random manner because of the irregular surfaces. Although the deflection of the bristles is random, the LuGre model only considers the average deflection. The average deflection of the bristles, z, is modeled by a first-order differential equation that relates bristle deflection to slip velocity, coefficients of friction, and normal force.

   • The parameter σ₀ represents the bristle stiffness, and it models the stiffness resisting micro-deformation in the friction element.
   • The parameter σ₁ represents the damping in the pre-displacement (or stiction) regime. When the friction state is far away from this regime, its influence is negligible since $\dot{z}$ tends to zero on a faster time-scale than the slip velocity,v, as the system leaves the pre-displacement zone where the velocity v, is close to zero.
   • The parameter σ₂ represents viscous damping. This is primarily responsible for the increase in friction force with the increase in the slip velocity,v. The viscous effect dominates the damping when there is a slip velocity between the two surfaces.
5. The LuGre model can capture hysteresis as shown in Figure 4.

   
   
   

6. The LuGre model also captures frequency dependent hysteresis as shown in Figure 5.

   
   
   

7. The LuGre model can also be used to describe friction in N -Dimension as follows: Let the 3D space of relevance.
   • Slip velocity $v_r={\left[v_x{v}_y{v}_z\right]}^T$ , $v_m=\Vert v_r\Vert$
   • Bristle states $Z_r={\left[Z_x{Z}_y{Z}_z\right]}^T$ ,
   Now define the following terms:

   • $g(v_m)={\mu }_d+\left({\mu }_s-u_d\right)e^{-{(\raisebox{1ex}{$v$}\!\left/ \!\raisebox{-1ex}{$vs$}\right.)}^2}$
   The 3D LuGre friction model is defined by the equations below:

   • $\frac{d{z}_i}{dt}=v_i-{\sigma }_0\frac{\left|v_m\right|z_i}{g(v_m)}$
   • $F_{x,y,z}=LuGre3d({\mu }_s,{\mu }_d,\sigma ,N,v,z)$ , $F_i=N({\sigma }_0{z}_i+{\sigma }_1\frac{d{z}_i}{dt})+{\sigma }_2{v}_i$
   • i=x,y,z
8. Friction Force Coupling.

   The coupling effect leads to the familiar friction ellipse that defines the limiting forces when sliding is seen in x, y, z directions, as shown in Figure 6.

   
   
   

9. Friction in a Revolute Joint.

   Figure 7 below describes the geometry of a Revolute joint. A circular pin, shown in grey, holds two bodies (labeled 1 and 2) so that they are able to rotate about the z-axis, which is normal to the plane of the picture. The inset shows a y-z section of the joint cutting through the pin.

   
   
   

   The following geometrical properties are important for friction calculations:

   • The radius of the pin, denoted as r_p.
   • The bending reaction arm, r_b. The width of the inside part, Part-1, is 2*r_b.
   • The friction arm, r_n. This is the effective radius at which the axial force acts.
   In a multibody simulation, a joint can be loaded in several ways, thus generating reactions forces and torques. Each of these contributes to the interface dynamics by means of friction forces. For a revolute joint, there are four contributing effects:

   • Friction on the cylindrical surface of the pin, caused by the normal force in the xy plane, Nxy.
   • Friction on the end-caps of the pin, caused by the axial force, Nz.
   • Friction caused by the force couple equivalent to the moment arm in the xy plane, Nrxy.
   • Friction caused by the assembly preload in the joint.
   The calculation of each of these effects is explained next.

   Friction on the cylindrical surface of the pin, caused by the normal force in the xy plane, N_xy

   Figure 8(a) describes the contribution coming from the loading in the x-y plane. The contact surface is at the pin and hence the slip velocity depends on the pin radius and joint relative angular velocity ω_r.

   The slip velocity in the x-y plane is = v_slip1=ω_rr_p.

   The force normal to the pin surface, $N_{xy}=\sqrt{\left(N_x^2+N_v^2\right)}$ .

   The friction force is: $F_{xy1}=LuGre({\mu }_s,{\mu }_d,\sigma ,N_{xy},v_{slip1},z)$ .

   The torque generated by this friction force will be: T_z1=F_x1r_p.

   
   
   

   Friction on the end-caps of the pin, caused by the axial force, N_z

   From Figure 8(b) we see that friction due to axial loading leads to an annular disc-like contact surface with an effective radius of r_n. The slip velocity in the axial direction is therefore: v_slip1=ω_rr_n. The force normal to the pin surface is N_z.

   The friction force magnitude to axial loading is $F_{xy2}=LuGre({\mu }_s,{\mu }_d,\sigma ,N_z,v_{slip2},z)$ .

   The friction torque due to axial loading is T_z2=F_xy2r_n.

   Friction caused by the force couple equivalent to the moment arm in the xy plane, N_rxy

   The reaction moments can introduce additional frictional torque. This is shown in Figure 9.

   
   
   

   In this case, we need to first identify the individual reactions obtained from the reaction torques. Based on Figure 7, it is seen that since the moment arm of the reaction moment is r_b, the equivalent force in the couple would be:

   $N_{rxy}=\frac{\sqrt{\left(N_{rx}^2+N_{ry}^2\right)}}{r_b};N_{rxy}$ is the reaction force acting on the contact surface.

   The normal force acts at a radius r_p. Hence, the friction force can be expressed as:

   $F_{rxy}=LuGre({\mu }_s,{\mu }_d,\sigma ,N_{rxy},{\omega }_p{r}_p,z)$ .

   The frictional torque caused by this friction force is T_z3=F_rxyr_p.

   Friction caused by the assembly preload in the joint.

   The torque preload is divided by μ_s to calculate an equivalent force normal to the cylindrical surface of the pin. Friction forces on the surface are calculated next, using the LuGre function. Finally, frictional forces on the cylindrical surface are calculated by computing the moment of the frictional forces.

   $N{}_{preload}=F_{preload}/\left({\mu }_s{r}_p\right)$

   $F_{preload}=LuGre({\mu }_s,{\mu }_d,\sigma ,N_{preload},{\omega }_p{r}_p,z)$

   T_z4=F_preloadr_p

   The friction force due to preload is T_z4.

   The total frictional torque in the revolute joint is T_Jfric=T_z1+T_z2+T_z3+T_z4.

10. Friction in a Translational Joint.

    Figure 10 describes the geometry of a translational joint. A slider with a rectangular slot, shown in brick red, slides on a rectangular body, shown in green. The z-axis of marker, J, on the green body, defines the axis of translation. The inset shows an x-y section of the joint cutting through the slider.

    
    
    

    The following geometrical properties are important for friction calculations:

    • The reaction arm, denoted as r_n.
    • The size of the slider, along the axis of translation, is denoted as x_s0.
    The translational joint can be loaded in several ways, each of which generates a friction force. The total friction force is a sum of each of these effects. For a translational joint, there are four contributing effects:

    • Friction in the xy plane, caused by the rubbing of the block on the slider, Figure 11(a).
    • Friction in the xy plane, caused by the torsion of the block on the slider, Figure 11(b).
    • Friction in the yz plane, caused by the bending of the slider, Figure 11(c).
    • Friction caused by the assembly preload in the joint.

    
    
    

    Friction in the xy plane, caused by the rubbing of the block on the slider

    The normal force acting in the xy plane is shown in Figure 10(a). N_x and N_y are the constraint forces preventing relative motion between block and slider in the x- and y-directions respectively.

    The normal force thus is: $N=\sqrt{\left(N_x^2+N_y^2\right)}$ .

    The friction force magnitude to axial loading is: $=F_{z1}=LuGre({\mu }_s,{\mu }_d,\sigma ,N,v_z,z)$ .

    Friction in the xy plane, caused by the torsion of the block on the slider.

    Referring to Figure 10(b), the torsional moment between the block and the slider causes contact at two of the corners of the block. Assuming that the distance between the contact points is the constant r_n, the torsional moment N_rz can be replaced by a couple of equal and opposite forces N_xy acting at each of the contact points.

    $N_{xy}=\frac{N_{rz}}{2_{rn}}$

    The friction force due to torsion is: $=F_{z2}=LuGre({\mu }_s,{\mu }_d,\sigma ,N_{xv},v_z,z)$ .

    Friction in the xz and yz planes, caused by the bending of the slider.

    Referring to Figure 10(c), the bending moment tends to bend the block relative to the slider. Contact occurs at two points separated by a distance that is a function of initial_overlap and overlap_delta. Let the distance between the contacts be denoted as x_s. Replacing the bending moment by a couple with a moment arm x_s gives:

    $$N_{xy3}=\frac{\sqrt{\left(N_{rx}^2+N_{ry}^2\right)}}{x_s}$$

    The friction force due to bending is: $=F_{z3}=LuGre({\mu }_s,{\mu }_d,\sigma ,N_{xy3},v_z,z)$ .

    Friction caused by the assembly preload in the joint.

    The force preload is divided by μ_s to calculate an equivalent force normal to the contact surface if the joint. Friction forces on the surface are calculated next, using the LuGre function. Thus:

    $$N_{preload}=\raisebox{1ex}{$T_{preload}$}\!\left/ \!\raisebox{-1ex}{${\mu }_s$}\right.$$

    The friction force due to preload is: $=F_{z4}=LuGre({\mu }_s,{\mu }_d,\sigma ,N_{preload},v_z,z)$ .

    The total frictional force in the translational joint is: F_Jfric=F_z1+F_z2+F_z3+F_z4.

11. Friction in a Cylindrical Joint.

    Figure 12 describes the geometry of a cylindrical joint. A slider with a circular slot, shown in brick red, slides on a circular shaft, shown in green. The z-axis of Marker J on the green shaft defines the axis of translation and rotation. The inset shows an x-y section of the joint cutting through the slider.

    The following geometrical properties are important for friction calculations:

    • The pin radius, denoted as r_p.
    • The bending moment arm, along the axis of translation, is denoted as, x₀.

    
    
    

    The cylindrical joint can be loaded in several ways, each of which generates a friction force. The total friction force is a sum of each of these effects. For a cylindrical joint, there are three contributing effects:

    • Friction Force/Torque, caused by the radial load, Figure 12(a).
    • Friction Force/Torque, caused by the bending of the slider, Figure 12(b).
    • Friction Force/Torque, caused by the assembly preload in the joint.

    
    
    

    Friction Force/Torque, caused by the radial load

    The normal force acting in the xy plane is shown in Figure 13(a). N_x and N_y are the constraint forces preventing relative motion between block and slider in the x- and y-directions respectively

    The normal force thus is $N=\sqrt{\left(N_x^2+N_y^2\right)}$ and the slip velocity at the contact point is $v={\left[\omega r_{p}0v_z\right]}^T$ .

    Define the bristle deflections $Z={\left[Z_{rot}0Z_{trans}\right]}^T$ .

    The friction force due to radial loading is $F_{zy1,0z1}=LuGre3d({\mu }_s,{\mu }_d,\sigma ,N,v_z,z)$ .

    The friction torque due to radial loading is $=T_{z1}=r_p\cdot F_{xy1}$ .

    Friction Force/Torque caused by the bending of the slider.

    Referring to Figure 11(b), the bending moment tends to bend the block relative to the slider. Contact occurs at two points separated by a distance that is a function of initial_overlap and overlap_delta. Let the distance between the contacts be denoted as x_s. Replacing the bending moment by a couple with a moment arm x_s gives:

    $$N_{xy2}=\frac{\sqrt{\left(N_{rx}^2+N_{ry}^2\right)}}{x_s}$$

    The friction force due to bending is $F_{xy2,0z2}=LuGre3d({\mu }_s,{\mu }_d,\sigma ,N,v,z)$ .

    The friction torque due to bending is $=T_{z2}=r_p\cdot F_{xy2}$ .

    Friction Force/Torque caused by the assembly preload in the joint.

    The force preload is divided by μ_s to calculate an equivalent force normal to the contact surface if the joint. Friction forces on the surface are calculated next, using the LuGre function. Thus:

    $$N_{preload}=\raisebox{1ex}{$F_{preload}$}\!\left/ \!\raisebox{-1ex}{${\mu }_s$}\right.$$

    The friction force due to preload is $F_{xy3,0z3}=LuGre3d({\mu }_s,{\mu }_d,\sigma ,N_{preload},v,z)$ .

    The friction torque due to preload is $=T_{z3}=r_p\cdot F_{xy3}$ .

    The total circumferential friction force in the cylindrical joint =F_xy1+F_xy2+F_xy3=0.

    The total axial frictional force in the cylindrical joint F_Jfric=F_z1+F_z2+F_z3.

    The total frictional torque in the cylindrical joint =T_Jfric=T_z1+T_z2+T_z3 .

12. Friction in a Universal Joint.

    A universal joint can be conceptually thought of as being composed of two revolute joints. The axis of one revolute joint is along the I-Yoke, and the axis of the second revolute joint is along the J-Yoke. Friction on each of the yokes is modeled as for a revolute joint. For more information on modeling friction in the revolute joint, see Comment 8.

13. Friction in a Spherical Joint.
    A spherical joint allows rotations about all the three axes and arrests all translational degrees of freedom. Hence, dynamics around a spherical joint can be cast in terms of relative motion on the spherical surface of the ball. A typical loading scenario for a spherical joint is described in Figure 13. In this case, referring to Figure 14below, the following geometrical properties are important for friction calculations:

    • The ball radius, denoted as r_b.

      The spherical joint can be loaded because of the constraint force between the ball and the socket and the joint assembly preload. The total friction force is a sum of each of these effects. For the sake of simplicity, we define the normal reaction as the sum of the joint reaction force and the preload force. This is represented as $\overrightarrow{N}$ in the figure below.

      Let $N=\Vert \overrightarrow{N}\Vert$

      The angular velocity of the ball with respect to the socket be ${\omega }_r=\left[{\omega }_x{\omega }_y{\omega }_z\right]$ .

      
      
      

      The components of the slip velocity can be computed in the global coordinate system as follows:

      Slip velocity $v_r={\left[r_b{\omega }_x{r}_b{\omega }_y{r}_b{\omega }_z\right]}^T$ , $v_m=\Vert v_r\Vert$ Bristle states $Z={\left[Z_x{Z}_y{Z}_z\right]}^T$ .

      The frictional force in the spherical joint, following the conventions described in Comment 6, is .

      And the frictional torque is therefore .

14. Friction in a Hook Joint.

    A hook joint can be conceptually thought of as being composed of two revolute joints. The axis of one revolute joint is along the I-Yoke, and the axis of the second revolute joint is along the J-Yoke. Friction on each of the yokes is modeled as for a revolute joint. For more information on modeling friction in the revolute joint, see Comment 8.

15. References:
    • C. Canudas de Wit, H. Olsson, K. J. Aström, and P. Lischinsky,A New Model for Control of Systems with Friction. IEEE Transactions on Automatic Control, March 1995, vol.40, No.3, pp.419-425.
    • Velenis E., Tsiotras P., Canudas-de-Wit C.(2005), "Extension of the LuGre Dynamic Tire Friction Model to 2D Motion", Vehicle System Dynamics, 2005.
    • Bliman, P. A. Sorine, M., "Friction modeling by hysteresis operators. Application to Dahl, sticktion and Stribeck effects", Pitman Research Notes in Mathematics Series, 1993, Issue 286, pages 10
    • Olsson, H. Aström, K. J. Canudas De Wit, C. Gaefvert, M. Lischinsky, P., "Friction Models and Friction Compensation", European Journal of Control, 1998, Vol. 4; Number 3, pages 176-195
    • J. C. Trinkle, J.-S. Pang, S. Sudarsky, G. Lo, "On Dynamic Multi-Rigid-Body Contact Problems with Coulomb Friction", ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Volume 77, Issue 4, pp. 267-279, 1997