17_0410 A Viscoelastic Tyre Friction Model Based on a Partial Differential-Algebraic Inclusion

Presented at EUROMECH Colloquium 578: Rolling Contact Mechanics for Multibody System Dynamics, April 2017 (Madeira, Portugal).
http://dx.doi.org/10.13140/RG.2.2.18618.88006

Also, a more sophisticated mathematical derivation is available here:
- Ryo Kikuuwe: "A Brush-Type Tire Model with Nonsmooth Representation," Mathematical Problems in Engineering, Vol.2019, Article 9747605, December 2019
https://www.hindawi.com/journals/mpe/2019/9747605/

Transcript

1. 1 A Viscoelastic Tyre Friction Model A Viscoelastic Tyre Friction Model Based on Based on a Partial Differential-Algebraic Inclusion a Partial Differential-Algebraic Inclusion Department of Mechanical Engineering Department of Mechanical Engineering Kyushu University, Japan Kyushu University, Japan Ryo Kikuuwe & Kahhaw Hoo Ryo Kikuuwe & Kahhaw Hoo http://rk.mech.kyushu-u.ac.jp/ http://rk.mech.kyushu-u.ac.jp/ http://www.youtube.com/kikuuwe/ http://www.youtube.com/kikuuwe/
2. 2 Tyre-Road Interaction Tyre-Road Interaction  Usually represented by 3 components of force and torque  Generated by complicated factors such as:  viscoelastic deformation  friction  Physics model is needed lateral force Fy longitudinal force Fx self-aligning torque Mz distributed along the contact patch
3. 3  Pacejka's Magic Formula  Empirically known to be good if the parameters (B, C, etc) are appropriately chosen.  Drawbacks:  No basis on physics.  Valid only in steady state.  Indeterminate at zero velocity.  Cannot deal with dry steering Conventional Approach Conventional Approach [Bakker, Pacejka & Linder, 1989] slip ratio · slip angle ® slip angle ® Y X Fx , Fy , Mz Fy Fx Mz
4. 4 What Are Happening There? What Are Happening There? ! V F R »  Each point is pulled by Coulomb-like friction force and displaced viscoelastically.  Partially sticking and partially sliding  Max friction force depends on the normal pressure at each point. [Bottom View] moving direction sticking slipping slip angle e(», t) » contact patch [Bottom View] [Side View] pressure distribution
5. 5  A Differential-Algebraic Model  Needs implicit integration  More faithful to the force balance  Reproduces stiction Friction + Viscoelasticity Friction + Viscoelasticity  LuGre Model  just an ODE. So, it’s convenient  No stiction [Kikuuwe et al., 2006] [Canudas de Wit et al., 1995] v K B e f velocity v Differential-Algebraic Inclusion
6. 6 Extension to Tyre Friction Extension to Tyre Friction e.g., [Velenis et al., 2005]  Point Contact  Tyre (Surface Contact) New PDAI Model New PDAI Model  ODE (LuGre)  DAI (Kikuuwe et al.)  PDE
7. 7 New Tyre Model New Tyre Model ground speed of the small part displacement rate of the small part  Friction force (distributed)  Viscoelastic force (distributed)  Total forces » Partial Differential-Algebraic Inclusion
8. 8 Implicit Euler Discretization Implicit Euler Discretization where along time t: along space »: Algebraic Inclusion Partial Differential- Algebraic Inclusion Discretization
9. 9 Algebraic Inclusion: Solvable Algebraic Inclusion: Solvable where  This solution realizes the stiction, thanks to the set-valuedness of ¡. Algebraic Inclusion Its Solution
10. 10 New Algorithm New Algorithm Implicit Euler discretization in time and space Algorithm Algorithm Algebraic Inclusion Algebraic Inclusion Partial Differential Algebraic Inclusion analytical solution Input: tyre velocities Output:{Fx ,Fy ,Mz } (The Model)
11. 11 Parameter Fitting Parameter Fitting  Needs to fit with Magic Formula in the steady state.  Parameter sets are provided by Bakker et al. [1987], under 4 vertical loads (2 to 8 kN), at the speeds:  60 km/h, for Fx  70 km/h, for Fy and Mz ®-Fy ∙-Fx ®-Mz
12. 12 Parameter Fitting (Cont’d) Parameter Fitting (Cont’d) {Fx ,Fy } shapes with high {∙,®} Stribeck curves in {x,y} directions Slopes of {Fx ,Fy ,Mz } curves at small {∙,®,®} Peak value and its location of Mz curve Stiffness K in {x,y} directions & Contact patch length L Normal force distribution ®-Fy ∙-Fx ®-Mz
13. 13 Results of Parameter Fitting Results of Parameter Fitting  Largely OK.  But still needs some improvements especially in Mz Gray: Magic Formula Red : New Model ®-Fy ∙-Fx ®-Mz
14. 14 Concluding Remarks Concluding Remarks  We have proposed:  PDAI representation of rolling friction of tyres  Its spatio-temporal integration algorithm  Parameter fitting scheme referring to Magic Formula.  Physically plausible  advantageous over Magic Formula  Proper in zero-velocity (stiction) treatment  advantageous over Magic Formula & LuGre model  Open problems are:  Improvement of parameter fitting scheme  Validation of transient response