lugre_tire_model
TRANSCRIPT
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ARM Internal PresentationNovember, 2006
Dynamic Tire-Road Friction Model for the Dynamic Tire-Road Friction Model for the Simulation of Vehicle Handling Simulation of Vehicle Handling
PerformancePerformance
Ragnar LedesmaRagnar LedesmaCVS Advanced Engineering
ArvinMeritor, Inc.
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ARM Internal PresentationNovember, 2006
Problem Statement
• Pacejka tire model requires extensive tire testing for every tire being considered
• Pacejka coefficients do not have physical significance – they are essentially curve fitting parameters
• Pacejka coefficients are not readily available – tire manufacturers sometimes provide raw data showing lateral forces and aligning moments versus slip angles
• We need a tire model based on first principles, i.e., model the mechanics of friction between the tire and the road
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ARM Internal PresentationNovember, 2006
Objectives
• Develop an analytical tire model for vehicle handling simulation, subject to the following requirements:• Minimum number of states (internal variables) for fast run-times
• Minimum number of model parameters
• Model parameters should have physical meaning
• Accurate in both transient and steady-state conditions
• Capable for combined braking and turning maneuvers
• Captures hysteresis effects and tire relaxation lags
• Tire-road friction forces can be scaled with normal force
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ARM Internal PresentationNovember, 2006
LuGre Tire-Road Friction Model
• Dynamic friction model for determining the tire-road friction forces along the fore-aft and lateral directions
• Model inputs include: • Tire normal force
• Translational velocity of wheel center
• Angular velocity of wheel
• Tire slip angle
• Model outputs include:• Tire longitudinal force
• Tire lateral force
• Tire aligning moment
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ARM Internal PresentationNovember, 2006
LuGre Tire Model Parameters
• Tire-road friction parameters• Static friction coefficient
∀ µsx – along x-axis (longitudinal direction)
∀ µsy – along y-axis (lateral direction)
• Kinetic friction coefficient
∀ µkx – along x-axis (longitudinal direction)
∀ µky – along y-axis (lateral direction)
• Stribeck velocity, vs
• Stribeck exponent, γ
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ARM Internal PresentationNovember, 2006
LuGre Tire Model Parameters
• Brush model parameters (model for carcass and thread deformation) • Stiffness w.r.t. elastic deformation of bristles
∀ σ0x – along x-axis (longitudinal direction)
∀ σ0y – along y-axis (lateral direction)
• Viscous damping rate w.r.t. elastic deformation of bristles
∀ σ1x – along x-axis (longitudinal direction)
∀ σ1y – along y-axis (lateral direction)
• Equivalent viscous damping rate representing the Coulomb friction due to sliding between the tire and the road
∀ σ2x – along x-axis (longitudinal direction)
∀ σ2y – along y-axis (lateral direction)
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ARM Internal PresentationNovember, 2006
LuGre Tire Model Parameters
• Tire geometry parameters • Static loaded radius, R
• Contact patch length, L – function of static loaded radius and tire radial deformation
• Tire-road contact pressure distribution, fn(ξ)
• Uniform
• Trapezoidal
• Parabolic
• Trigonometric
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ARM Internal PresentationNovember, 2006
LuGre Tire Model
• LuGre tire model can be expressed in 3 forms:• Distributed model (infinite number of states)
• Discrete model (large, finite number of states)
• Average lumped model (minimum number of states)
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ARM Internal PresentationNovember, 2006
Distributed LuGre Tire Model
• Partial differential equations for bristle deformation
• Expressions for the instantaneous friction coefficient
• Expressions for the longitudinal and lateral friction forces
• Expression for the tire aligning moment
ξξωξ
µσλξ
∂∂
−−=∂
∂ ),(),(
)(),(2
0 tzRtz
vv
t
tz ii
ki
irri
iyxi ,=
riii
iiii vt
tztzt *
),(*),(*),( 210 σξσξσξµ +
∂∂
+= yxi ,=
ξξξµ dfttF n
L
ii *)(*),()(0∫= yxi ,=
ξξξξµ dLfttM n
L
yz *)2/(*)(*),()(0
−= ∫
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ARM Internal PresentationNovember, 2006
Discrete LuGre Tire Model
• Partial differential equations for bristle deformation
• Expressions for the instantaneous friction coefficient
• Expressions for the longitudinal and lateral friction forces
• Expression for the tire aligning moment
)(1)(
),1(,,20
, ijijijki
irriij zz
L
NRz
vvz −−−−−= ω
µσλ
,,...,2 Nj = ,0,1 =iz yxi ,=
riiijiijiij vzz *** 2,1,0, σσσµ ++= ,,...,2 Nj = yxi ,=
∑= −
=N
jjniji N
LfF
2,, )
1(*µ yxi ,=
∑= −
−−=N
jjnijz N
LNLjLfM
2,, )
1(*))1/(*2/(**µ
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ARM Internal PresentationNovember, 2006
Average Lumped LuGre Tire Model
• Partial differential equations for bristle deformation
• Expressions for the instantaneous friction coefficient
• Expressions for the longitudinal and lateral friction forces
• Expression for the tire aligning moment
iiiki
irrii ztRz
vvz *)(*
)(2
0 κωµ
σλ −−=
LzRztRzv
vLF
Gz ytit
ky
yrry
nt /**)(*
)(2
0 ωυωµ
σλ+−−=
yxi ,=
riiiiiii vzz *** 210 σσσµ ++=
)***()( 210 riiiiiini vzzFtF σσσ ++= yxi ,=
)}2
1(*)
2
1(*)
2
1(*{)( 210 LF
GvzzzzLFtM
nryytyytyynz −+−+−= σσσ
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ARM Internal PresentationNovember, 2006
Example: Lu-Gre Tire Model Parameters
• Tire-road dynamic friction model parametersLongitudinal direction (x-axis):
σx0 555 (1/m) – stiffness w.r.t. bristle elastic deformation σx1 0.033 (sec/m) – damping w.r.t. bristle deformation rate σx2 0.001 (sec/m) – equiv. viscous damping w.r.t. sliding velocity µsx 1.35 – static friction coefficient µkx 0.75 – dynamic friction coefficient
Lateral direction (y-axis): σy0 470 (1/m) – stiffness w.r.t. bristle elastic deformation σy1 0.033 (sec/m) – damping w.r.t. bristle deformation rate σy2 0.001 (sec/m) – equiv. viscous damping w.r.t. sliding velocity µsy 1.40 – static friction coefficient µky 0.75 – dynamic friction coefficient
Other parameters vs 3.96 (m/sec) – Stribeck velocity γ 1.0 – Stribeck exponent L 0.15 (m) – contact patch length R 0.50 (m) – static loaded radius
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ARM Internal PresentationNovember, 2006
Example: Longitudinal Forces
• Fx versus slip ratio curves (constant slip angle)
Fn=2000 N
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ARM Internal PresentationNovember, 2006
Example: Friction Ellipse (combined slip)
• Fx versus Fy curves (constant slip angle)
Fn=2000 N
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ARM Internal PresentationNovember, 2006
Example: Lateral Forces
• Fy versus slip angle (constant slip ratio)
Fn=22,240 N
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ARM Internal PresentationNovember, 2006
Example: Aligning Moment
• Mz versus slip angle (constant slip ratio)
Fn=22,240 N
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ARM Internal PresentationNovember, 2006
Next Steps
• Implement the tire model in TruckSim
• Compare results with built-in TruckSim tire model
• Check if tire relaxation effects are captured
• Implement tire model in ADAMS• Average lumped model
• Discrete LuGre tire model