In this model, adhesion friction and viscous friction are considered, as these two terms. In this model, adhesion friction and viscous friction are considered, since these two terms represent the most used friction model. This simple model is not always sufficient for describing and compensating for friction. The hysteresis effect is considered a characteristic of the pre-sliding regime. The Stribeck velocity limit represents the start-up phase of the mechanism. The terms that make up this new set are:

The static friction denoted by par $F_s$, the viscous friction denoted by $F_v$, and the Stribeck effect defined by the terms $C_1$ and $C_2$ with ${\stackrel{\dot{}}{X}}_s$ the Stribeck velocity limit. The expression of the friction force $F_{fro}$ is given by:

$F_{fro}=F_s+F_v+C_1+C_2$. (1)

hence:

$F_{fro}={\mu }_s{F}_N+{\mu }_v\stackrel{\dot{}}{X}+C_1+C_2$. (2)

With $C_1$ and $C_2$

$C_1=\{\begin{array}{ll}{C}_1{\text{e}}^{-{\left(\frac{\stackrel{\dot{}}{X}}{{\stackrel{\dot{}}{X}}_s}\right)}^2}\hfill & \text{si}{\stackrel{\dot{}}{X}}_t>0\hfill \\ 0\hfill & \text{si}{\stackrel{\dot{}}{X}}_t\le 0\hfill \end{array}$ (3)

$C_2=\{\begin{array}{ll}0\hfill & \text{si}{\stackrel{\dot{}}{X}}_t\ge 0\hfill \\ -C_2{\text{e}}^{-{\left(\frac{\stackrel{\dot{}}{X}}{{\stackrel{\dot{}}{X}}_s}\right)}^2}\hfill & \text{si}{\stackrel{\dot{}}{X}}_t<0\hfill \end{array}$ (4)

With X being the virtual displacement defined as

Figure 2 shows the variation of the friction force as a function of velocity. A sharp drop in friction is observed near zero velocity, representing the Stribeck effect, followed by stabilization and a linear growth at higher velocities, which reflects the combination of Coulomb friction with a viscous contribution.

The simulation was performed using a 4th-order Runge-Kutta time integration method, which provides a good balance between accuracy and numerical efficiency. A time step of Δt = 10⁻³ s was adopted to properly capture the rapid transitions around zero velocity while ensuring numerical stability. The temporal discretization was therefore chosen fine enough to reproduce the critical transient phenomena. In addition, a relative tolerance of 10⁻⁶ was imposed in the solver to control numerical errors.

Convergence was verified by comparing results obtained with different time steps and tolerance levels. Since the differences between the curves were negligible, it was concluded that the solution is robust and independent of the numerical parameters.

In this model, Coulomb friction and static friction are considered, since these two terms represent the most commonly used friction model. This simple model is not always sufficient for describing and compensating for friction. The stick-slip effect is accounted for by the static friction model characteristic. The start-up phase of the mechanism is represented by the Stribeck velocity limit ( Figure 3 ). The hysteresis effect is expressed by introducing velocity into the terms $C_1$ and $C_2$.

$F_{fro}=F_C+F_S+C_1+C_2$. (5)

$F_{fro}=\{\begin{array}{ll}{F}_{ex}\hfill & \text{if}\stackrel{\dot{}}{X}=0\text{and}\left|F_{ex}\right|<F_s\hfill \\ {\mu }_c\text{sign}\left(\stackrel{\dot{}}{X}\right)+C_1+C_2\hfill & \text{if}\stackrel{\dot{}}{X}\ne 0\text{and}\left|F_{ex}\right|\ge F_s\hfill \end{array}$. (6)

And

$C_1=\{\begin{array}{ll}{C}_1{\text{e}}^{-{\left(\frac{\stackrel{\dot{}}{X}}{{\stackrel{\dot{}}{X}}_s}\right)}^2}\hfill & \text{if}{\stackrel{\dot{}}{X}}_t>0\hfill \\ 0\hfill & \text{if}{\stackrel{\dot{}}{X}}_t\le 0\hfill \end{array}$

$C_2=\{\begin{array}{ll}0\hfill & \text{if}{\stackrel{\dot{}}{X}}_t\ge 0\hfill \\ -C_2{\text{e}}^{-{\left(\frac{\stackrel{\dot{}}{X}}{{\stackrel{\dot{}}{X}}_s}\right)}^2}\hfill & \text{if}{\stackrel{\dot{}}{X}}_t<0\hfill \end{array}$

The figure illustrates the evolution of the friction force as a function of velocity. A rapid decrease of the force is observed near zero velocity, representing the Stribeck effect, followed by stabilisation toward a constant value in the dynamic regime (Coulomb friction). The simulation was carried out using a 4th-order Runge–Kutta integration method with a time step Δt = 10⁻³ s, ensuring both numerical stability and accuracy. The temporal discretisation was chosen sufficiently fine to accurately capture the critical zone around zero velocity. A relative tolerance of 10⁻⁶ was imposed in the solver to control numerical error. Convergence was verified by comparing results obtained with different time steps and tolerances: as the variations between curves were negligible, the robustness and independence of the solution with respect to the numerical parameters were confirmed.

We then consider the three most well-known and commonly used friction models: Coulomb friction, viscous friction, and static friction. These simple models are not always sufficient to describe phenomena related to friction. In the terms $C_1$ and $C_2$, the Stribeck effect appears through the introduction of velocity, as the hysteresis effect does not occur during the velocity decay phase ( Figure 4 ). The expression for the friction force is given by:

$F_{fro}=\{\begin{array}{ll}\text{sign}\left(F_{ex}\right)\min \left(\left|F_{ex}\right|,F_s\right)\hfill & \text{if}\stackrel{\dot{}}{X}=0\hfill \\ {\mu }_c\text{sign}\left(\stackrel{\dot{}}{X}\right)+{\mu }_v\stackrel{\dot{}}{X}+C_1+C_2\hfill & \text{if}\stackrel{\dot{}}{X}\ne 0\hfill \end{array}$. (7)

And $C_1=\{\begin{array}{ll}{C}_1{\text{e}}^{-{\left(\frac{\stackrel{\dot{}}{X}}{{\stackrel{\dot{}}{X}}_s}\right)}^2}\hfill & \text{if}{\stackrel{\dot{}}{X}}_t>0\hfill \\ 0\hfill & \text{if}{\stackrel{\dot{}}{X}}_t\le 0\hfill \end{array}$

$C_2=\{\begin{array}{ll}0\hfill & \text{if}{\stackrel{\dot{}}{X}}_t\ge 0\hfill \\ -C_2{\text{e}}^{-{\left(\frac{\stackrel{\dot{}}{X}}{{\stackrel{\dot{}}{X}}_s}\right)}^2}\hfill & \text{if}{\stackrel{\dot{}}{X}}_t<0\hfill \end{array}$

The red curve shows the friction law as a function of velocity:

Such a law F(v) is typically used in nonlinear dynamics with dry friction.

The time step must be small enough to resolve the rapid change around zero velocity.

In this model, we consider the P. Dahl friction model along with the viscous model. Since the P. Dahl model does not account for the viscous behavior of friction, the viscous friction model is added to capture this important [4] physical phenomenon when the contact is lubricated. This model is therefore expressed as:

$F_{fro}={\mu }_v\stackrel{\dot{}}{X}+F_c\left(1-\exp \left(\frac{-\sigma \left|X\right|}{F_c}\right)\right)\text{sign}\left(\stackrel{\dot{}}{X}\right)$ (8)

$\text{sign}\left(\stackrel{\dot{}}{X}\right)=\{\begin{array}{ll}1\hfill & \text{pour}\stackrel{\dot{}}{X}>0\hfill \\ 0\hfill & \text{pour}\stackrel{\dot{}}{X}=0\hfill \\ -1\hfill & \text{pour}\stackrel{\dot{}}{X}<0\hfill \end{array}$ (9)

The equations are an exception to the prescribed specifications of this template. Thus, at a velocity different from zero, the model would be linearly dependent on the relative velocity. For this combination, friction is a function of two variables: velocity and displacement, both of which are time dependent. Therefore, for the simulation of this model, the displacement is defined as a function of time, and then the duration over which the movement between the two surfaces occurs is specified. For this work, we chose the velocity $\stackrel{\dot{}}{X}\left(t\right)=t\left(t+1\right)$ over the time interval from zero to two seconds: t = [0, 3] seconds, giving a velocity $\stackrel{\dot{}}{X}=26\text{m}/\text{s}$. Since $\stackrel{\dot{}}{X}=\frac{\text{d}X}{\text{d}t}$, the [5] [6] , displacement is: $X\left(t\right)=\frac{1}{3}{t}^3+\frac{1}{2}{t}^2$

The red curve shows how friction evolves with displacement (See Figure 5 ):

Implicit time integration (Backward Euler, Newmark, BDF).

Newton-Raphson iterations at each step to solve the nonlinear equilibrium equations.

In this case, we consider the static friction model together with the Dahl model. Since the P. Dahl model does not account for the static friction phenomenon, this combined model consists of the Dahl model and the static [7] [8] friction model. The expression for this model is as follows:

$F_{fro}={\mu }_s{F}_N+F_{fc}\left(1-\exp \left(\frac{-\sigma \left|X\right|}{F_{fc}}\right)\right)\text{sign}\left(\stackrel{\dot{}}{X}\right)$ (10)

The red curve represents a nonlinear friction law as a function of displacement (See Figure 6 ):

This reflects a nonlinear friction law with a threshold, characteristic of the stick–slip transition (adhesion → sliding).

Because the problem is highly nonlinear with a discontinuity, an implicit incremental integration scheme is most suitable.

Typically:

Explicit schemes would require extremely small step sizes and would be unstable near the discontinuity.

The time step/displacement increment must be very small near Shift = 0, where the force changes abruptly.

An adaptive stepping strategy is generally applied:

In this final proposed model, we consider the combination of the static and viscous models, associated with the Dahl model, to address the two phenomena not accounted for by the Dahl model: the viscous and static behavior of friction. However, the Stribeck effect remains unsatisfied [9] - [11] .

$F_{fro}={\mu }_s{F}_N+{\mu }_v\stackrel{\dot{}}{X}+F_{fc}\left(1-\exp \left(\frac{-\sigma \left|X\right|}{F_{fc}}\right)\right)\text{sign}\left(\stackrel{\dot{}}{X}\right)$ (11)

The red curve shows the nonlinear friction law as a function of displacement (See Figure 7 ):

This behavior is typical of a regularized dry-friction model, with a clear stick–slip transition at the origin.

The blue point indicates the initial or equilibrium state.

Since this is a highly nonlinear problem with discontinuities, an implicit incremental integration scheme is generally required:

Explicit methods would be unstable here or require extremely small step sizes.

He time step/displacement increment must be very small near the discontinuity to capture the sudden transition in force.

Typically, an adaptive stepping strategy is used:

If increments are too large, the solution will jump across discontinuity and lose accuracy.