Dahl and LuGre Dynamic Friction Simulator

Model and simulation inputs

All quantities use SI units. Select Calculate after changing inputs.

Simulation summary

A model response under the entered constant parameters.

Peak absolute friction—

Friction time history

Force–displacement hysteresis

Equations used

Positive displacement, velocity, spring force, and external force share one axis. The signed friction force F_f is subtracted in the motion equation. SI units are used internally.

LuGre: g(v) = F_c + (F_s − F_c) exp[−(|v|/v_s)^p]

LuGre: ż = v − σ₀|v|z/g(v)

LuGre: F_f = σ₀z + σ₁ż + σ₂v

Dahl: ż = v − σ₀|v|z/F_c; F_f = σ₀z + σ₂v

ẋ = v; m v̇ = k(v_dt − x) + A sin(2πft) − F_f

Symbols: z is average bristle deflection; σ₀, σ₁, and σ₂ are bristle stiffness, microdamping, and viscous coefficients; F_c and F_s are Coulomb and static friction; v_s and p shape the Stribeck curve; m is mass; k is spring stiffness; v_d is support speed; and A and f define the sinusoidal external force. Classical fixed-step RK4 integrates x, v, and z.

Assumptions and limits

The model is lumped, deterministic, one-dimensional, isothermal, and uses constant calibrated parameters. It omits changing normal load, wear, temperature, lubricant supply, impacts, backlash, and contact geometry. Near-stick fraction depends on the selected numerical threshold. Use no more than 30 s and verify sensitivity to a smaller time step. Do not use uncalibrated output for safety, life, control-stability, or component qualification decisions.

References

Canudas de Wit et al. (1995), doi:10.1109/9.376053.
Dahl (1968), A Solid Friction Model, TOR-0158(3107-18)-1.

How to use

Select LuGre for Stribeck decay and bristle damping, or Dahl for the simpler presliding-memory relation.
Enter Coulomb and static friction in N. Static friction must be at least Coulomb friction; LuGre uses both, while Dahl uses the Coulomb level.
Enter bristle stiffness in N/m, damping and viscous coefficients in N·s/m, and LuGre Stribeck velocity and exponent.
Enter slider mass, drive-spring stiffness, and constant support speed. Set sinusoidal force amplitude to zero when no external forcing is required.
Set the initial displacement, velocity, and bristle deflection, then choose duration and RK4 time step in seconds.
Select Calculate to update the summary and both plots. Correct any red input warning before interpreting the model.
Read peak friction, final state, near-stick sample fraction, velocity reversals, and integration details. Plot axes show explicit SI units.
Use Reset example to restore the supplied LuGre demonstration inputs and recalculate.
Repeat with half the time step; materially changing outputs indicate insufficient integration resolution or a stiff parameter set.
Do not use this simulator where parameters are uncalibrated, normal load or temperature changes strongly, impacts/backlash dominate, or a safety-critical prediction is required.

Interpretation

A force–displacement loop indicates frictional memory. Resolved velocity reversals and a high near-stick fraction can indicate oscillatory stick–slip, but the chosen threshold and time step affect those metrics. Compare model variants only with identical physical inputs and calibrated coefficients.