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MF-Tyre/MF-Swift Copyright TNO, 2013 MF-Tyre/MF-Swift Copyright TNO, 2013 MF-Tyre/MF-Swift Dr. Antoine Schmeitz 2 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Introduction TNO’s tyre modelling toolchain tyre (virtual) testing parameter fitting + tyre model signal tyre MBS database MF-Tyre processing TYDEX files property solver file MF-Swift MF-Tool Measurement Identification Simulation Copyright TNO, 2013 1 MF-Tyre/MF-Swift 3 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Introduction What is MF-Tyre/MF-Swift? MF-Tyre/MF-Swift is an all-encompassing tyre model for use in vehicle dynamics simulations This means: emphasis on an accurate representation of the generated (spindle) forces tyre model is relatively fast can handle continuously varying inputs model is robust for extreme inputs model the tyre as simple as possible, but not simpler for the intended vehicle dynamics applications 4 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Introduction Model usage and intended range of application All kind of vehicle handling simulations: e.g. ISO tests like steady-state cornering, lane changes, J-turn, braking, etc. Sine with Dwell, mu split, low mu, rollover, fishhook, etc. Vehicle behaviour on uneven roads: ride comfort analyses durability load calculations (fatigue spectra and load cases) Simulations with control systems, e.g. ABS, ESP, etc. Analysis of drive line vibrations Analysis of (aircraft) shimmy vibrations; typically about 10-25 Hz Used for passenger car, truck, motorcycle and aircraft tyres Copyright TNO, 2013 2 MF-Tyre/MF-Swift 5 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Modelling aspects and contents (1) 1. Basic tyre properties and constitutive relations for tyre radii, contact patch size, stiffness, rolling resistance nonlinear, effects of loads, velocity, inflation pressure 2. Inclusion of measured tyre steady-state slip characteristics described by load and inflation pressure dependent Magic Formula responses to sideslip, longitudinal slip, camber and turn slip 3. Tyre transients / relaxation length properties carcass compliance proper contact transient properties due to finite contact length 4. Inclusion of belt dynamics primary natural frequencies (rigid ring modes) and gyroscopic effects 5. Tyre rolling over uneven road surface tyre deformation on obstacles, arbitrary uneven roads 6 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Modelling aspects and contents (2) 6. Model usage and availability selection of complexity level road definition 7. Model parameterisation measurement requirements MF-Tool 8. Concluding remarks Copyright TNO, 2013 3 MF-Tyre/MF-Swift 7 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift 1. Basic tyre properties and constitutive relations 8 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Some definitions For a freely normal to the road rolling wheel: • free tyre radius RΩ V R = x • loaded radius R RΩ e Ω l Ω γ • effective rolling radius R ρ = − e RΩ Rl • tyre deflection ρ Vx F Rl • forward velocity Vx z Re • wheel spin velocity Ω Mz ρ C • longitudinal slip velocity Vsx M Fx x S Fy • lateral slip velocity Vsy My Vsx Vsy • longitudinal force Fx • lateral force Fy • vertical force Fz • overturning moment Mx • self aligning moment Mz γ • rolling resistance moment My • inclination angle Copyright TNO, 2013 4 MF-Tyre/MF-Swift 9 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Constitutive relations Tyre radii, stiffness, contact patch dimensions, rolling resistance, etc. are non- constant for different operating conditions (load, velocity, etc.) In MF-Swift nonlinear empirical relations are used to describe these basic tyre properties coefficient from curve fitting Examples: vertical force = (ρ Ω ) Fz Fz f ,,,,FxF yp i 2 2 R q F q F F = 1+ q 0 Ω − Fcx x − Fcy y ⋅ z v2 V0 Fz0 Fz0 ρ ρ 2 q + q ()1+ p dp F Fz 1 Fz 2 Fz 1 i z0 R0 R0 10 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Constitutive relations effective rolling radius coefficient from curve fitting F F F = − z0 z + z Re RΩ Dreff arctan Breff Freff cz Fz0 Fz0 2 ΩR = + 0 RΩ R0 qre 0 qv1 V 0 Fz Copyright TNO, 2013 5 MF-Tyre/MF-Swift 11 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift 2. Inclusion of measured tyre steady-state slip characteristics ( magic formula ) 12 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Magic Formula Descriptions are available for the three steady-state modes of slip γ Equations depend on vertical load Fz, inclination angle and inflation κ γ ϕ pressure pi ; [ Fx , Fy , Mx , My , Mz ] = MF ( Fz , , α, , , pi, V) + combinations side slip V V sy turn slip ψ& α V ϕ −= α = sy + camber t arctan V Vx V R Mz x spin ψ Fy & Ω V Fy V − x Mz κ −= Vsx −= Vx Vr Fy F F t Vx Vx x y 0 Ω− Vsx M −= Vx Re z Mz Vx Fx 0 0.1α 0.2 0.3 0 0.1 0.2 0.3 aϕ = a/R longitudinal slip 0 0.1κ 0.2 0.3 Copyright TNO, 2013 6 MF-Tyre/MF-Swift 13 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Magic Formula κ α α Notion: the base tyre characteristics Fx=f( ), Fy=f( ) and Mz=f( ) have a sinusoidal shape, with a “stretched” horizontal axis for large values of slip This consideration is the basis for a tyre model known as “Magic Formula ” Some notes: First versions developed by Egbert Bakker (Volvo) and prof. Pacejka (TU Delft, TNO) MF-Tyre software developed and distributed by TNO since 1996 Probably the most popular tyre model for vehicle handling simulations (worldwide!) 14 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Base Magic Formula f = D sin[ C arctan { Bx – E( Bx – arctan( Bx) )} ] F(x) = f( x) + S V X : input, e.g. α or κ D : peak factor x = X + SH C : shape factor F : output, e.g. Fy or Fx f F B : stiffness factor E : curvature factor SH SH,V : hor./vert. shift D f∞ x Note: S • C determines the limit value when x →∞ V arctan( BCD) X • BCD determines the slope near the origin • B, E & C determine the location of the peak Copyright TNO, 2013 7 MF-Tyre/MF-Swift 15 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Stretching the sine… parameters in this example: B=8, C=1.5, D=5500, E=-2 16 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Example: pure longitudinal slip characteristics where: • coefficients p are determined in curve fitting process • scaling coefficients λ: • equal 1 during fitting process • may be used to adjust tyre characteristics note: • vertical load increment: • K = C : longitudinal slip stiffness F − F x fκ df = z z0 z F • Bx is calculated from Cx, Dx and Kx z0 • F is nominal load • equations simplified for educational reasons… z0 Copyright TNO, 2013 8 MF-Tyre/MF-Swift 17 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Example: pure longitudinal slip characteristics result after fitting coefficients: ⇒ data reduction: 495 measurement points ⇒ 11 coefficients interpolation/extrapolation (load, braking/driving): 18 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Magic Formula Similar equations exist for Fx, Fy, Mx, My and Mz Combined slip: reduction of Fx and Fy by applying a weighting function G to the ‘pure’ characteristics = ⋅ Example: Fx Gxα Fx0 In total 144 parameters that are curve fitted using an automated process; typical fit errors of a few percent vertical force Fz κ longitudinal slip Fx longitudinal force side slip angle α F lateral force Magic y inclination angle γ M overturning moment Formula x turn slip ϕ rolling resistance moment t My forward velocity self aligning moment Vx Mz inflation pressure pi parameters: general (4), Fx (29), Fy (50), Mx (15), My (8), Mz (38) Copyright TNO, 2013 9 MF-Tyre/MF-Swift 19 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift 3. Tyre transients / relaxation length properties 20 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Tyre transients Measure step response to side slip angle Procedure: 1. Apply steering angle of 1 deg. 2. Load the tyre 3. Start rolling the track at low velocity (e.g. 0.05 m/s) Copyright TNO, 2013 10 MF-Tyre/MF-Swift 21 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Tyre transients • Tyre cannot instantaneously react to changes in slip • Travelled distance required to build up forces ϕ γ . • Relaxation effects exist for all modes of slip -ψ wheel plane This is due to: belt 1. carcass compliance 2. finite contact length _ _ Vc In MF-Swift both are considered: α V* 1. carcass stiffness → springs α* C 2. contact patch slip model ϕ* (Magic Formula (MF) slip model and contact patch transients) 22 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Simplified example carcass stiffness • Assume linear tyre model for small slip angles: v F = Cα α −= y u • Create dynamic tyre model by accounting for tyre sidewall stiffness: . F = Cα′ = kε Cα′ = kε y . v + ε • Dynamic side slip angle: α′ −= wheel plane u C 1 ′ ′ v α& +α −= = α k u u contact patch • Introducing the relaxation length σ (=C/k) and V≈u: σ . α′ +α′ =α F = Cα′ V y Copyright TNO, 2013 11 MF-Tyre/MF-Swift 23 Copyright TNO, 2013 Dr. Antoine Schmeitz MF-Tyre/MF-Swift Simplified example carcass stiffness • First order dynamics between lateral force and side slip angle input, transfer function: C σ H α s)( = time constant: Fy , σ s +1 V V • Relaxation length σ does not depend on forward velocity V: • response time reduces when increasing V; • travelled distance required to build up the lateral force remains the same.
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