Energy balance and tyre motions during shimmy

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Franklin Stafford
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1 Energy balance and tyre motions during shimmy Shenhai Ran I.J.M. Besselink H. Nijmeijer Dynamics & Control Eindhoven University of Technology April 2, 4th International Tyre Colloquium - Guildford - UK

2 Introduction Motivation of my PhD research: Tyre models for shimmy analysis linear and non-linear Fy &M z with relaxation behaviour April 2, 215 1/18

3 Introduction Motivation of my PhD research: Tyre models for shimmy analysis linear and non-linear Fy &M z with relaxation behaviour enhanced relaxation behaviour with contact patch dynamics including turn slip and its interaction with side slip belt dynamics with a rigid ring approach fundamentals of shimmy April 2, 215 1/18

4 Introduction Motivation of my PhD research: Tyre models for shimmy analysis linear and non-linear Fy &M z with relaxation behaviour enhanced relaxation behaviour with contact patch dynamics including turn slip and its interaction with side slip belt dynamics with a rigid ring approach fundamentals of shimmy Question to myself: How to compare and evaluate tyre models? steady state characteristics, step response & frequency response For shimmy? April 2, 215 1/18

5 Introduction Motivation of my PhD research: Tyre models for shimmy analysis linear and non-linear Fy &M z with relaxation behaviour enhanced relaxation behaviour with contact patch dynamics including turn slip and its interaction with side slip belt dynamics with a rigid ring approach fundamentals of shimmy Question to myself: How to compare and evaluate tyre models? steady state characteristics, step response & frequency response For shimmy? from an energy point of view April 2, 215 1/18

6 Contents Motivation Von Schlippe Tyre Model The Energy Flow Method Energy transfer through tyre Shimmy energy Stability analysis Stability with only yaw degree of freedom Stability with lateral flexibility Conclusions & Outlook April 2, 215 2/18

7 Von Schlippe tyre model Governing equations: σ V ẏ1 + y 1 = y c + (σ + a)ψ y 2 (t) = y 1 (t 2a V ) No sliding between contact line and ground April 2, 215 3/18

8 Von Schlippe tyre model No sliding between contact line and ground Governing equations: σ V ẏ1 + y 1 = y c + (σ + a)ψ y 2 (t) = y 1 (t 2a V ) ( ) ( v1 + v 2 y1 + y 2 F y = c v = c v 2 2 ) M z = c β ( v1 v 2 2a = c β ( y1 y 2 2a y c ) ) ψ April 2, 215 3/18

9 Energy transfer through tyre April 2, 215 4/18

10 Energy transfer through tyre Energy and work at wheel center: F a cos ψẏ c t + F d ẋ c t + M a ψ t = U + E k April 2, 215 4/18

11 Energy transfer through tyre Energy and work at wheel center: F a cos ψẏ c t + F d ẋ c t + M a ψ t = U + E k Force and moment equilibrium: } F a = F y cos ψ M a = M z F d ẋ c t = F y cos ψẏ c t + M z ψ t + U + E k April 2, 215 4/18

12 The shimmy energy Suppose the wheel exhibits periodic motions, U and E k vanish: T F d ẋ c dt = T (F y cos ψẏ c + M z ψ)dt April 2, 215 5/18

13 The shimmy energy Suppose the wheel exhibits periodic motions, U and E k vanish: T F d ẋ c dt = Shimmy energy is defined as follows: W T T (F y cos ψẏ c + M z ψ)dt (F y ẏ c + M z ψ)dt W > energy flows into lateral-yaw motion from forward motion April 2, 215 5/18

14 Sinusoidal motion at wheel center The wheel moves with prescribed sinusoidal motions: } y c (t) = Aη sin(ωt + ξ) ψ(t) = A sin(ωt) To calculate W, transfer functions F y &M z w.r.t lateral and yaw motions used: [ ] [ ] [ ] FY (s) H11 (s) H = 12 (s) YC (s) M Z (s) H 21 (s) H 22 (s) Ψ(s) and H mn = H mn (jω), θ mn = H mn (jω); m, n = 1, 2 April 2, 215 6/18

15 Transfer Functions ( 2as ) 2σ s V + 1 e V H 11 (s) = H Fy,y c (s) = c v 2(σ s V + 1) ( (σ + a)(1 + e 2as ) V ) H 12 (s) = H Fy,ψ(s) = c v 2(σ s V + 1) ( 1 e 2as ) V H 21 (s) = H Mz,y c (s) = c β 2a(σ s V + 1) ( 2as ) 2aσ s V (σ a) + (σ + a)e V H 22 (s) = H Mz,ψ(s) = c β 2a(σ s V + 1) April 2, 215 7/18

16 Transfer Functions ( 2as ) 2σ s V + 1 e V H 11 (s) = H Fy,y c (s) = c v 2(σ s V + 1) ( (σ + a)(1 + e 2as ) V ) H 12 (s) = H Fy,ψ(s) = c v 2(σ s V + 1) H 21 (s) = H Mz,y c (s) = c β ( 1 e 2as V 2a(σ s V + 1) ( 2as 2aσ s V (σ a) + (σ + a)e V H 22 (s) = H Mz,ψ(s) = c β 2a(σ s V + 1) s V jω V ) path wavelength λ 2πω V ) April 2, 215 7/18

17 Energy components where W = W 11 + W 12 + W 21 + W 22 W 11 = W 12 = W 21 = W 22 = T T T T ( ) AηH 11 sin(ωt + ξ + θ 11 ) Aωη cos(ωt + ξ) dt = πη 2 A 2 H 11 sin θ 11 ( ) AH 12 sin(ωt + θ 12 ) Aωη cos(ωt + ξ) dt = πηa 2 H 12 sin(ξ θ 12 ) ( ) AηH 21 sin(ωt + ξ + θ 21 ) Aω cos(ωt) dt = πηa 2 H 21 sin(ξ + θ 21 ) ( ) AH 22 sin(ωt + θ 22 ) Aω cos(ωt) dt = πa 2 H 22 sin θ 22 April 2, 215 8/18

18 Pure lateral and yaw motion of wheel center shimmy energy [J].5 pure yaw W 22 pure lateral W a/λ [ ] Shimmy can only occur at large λ for pure yaw motion; not possible if only lateral motion exists! April 2, 215 9/18

19 Zero shimmy energy boundary of combined input In the case of combined motion, solving W = leads to: η = H 12 sin(ξ θ 12 ) H 21 sin(ξ + θ 21 ) ± 2H 11 sin θ 11 where = ( ) H 12 sin(ξ θ 12 ) H 21 sin(ξ + θ 21 ) 2 4H 11 H 22 sin θ 11 sin θ 22 It is a circle in a polar plot, where the distance to the origin is η and the angle to the positive x-axis represents ξ. April 2, 215 1/18

20 Zero shimmy energy boundary of combined input W > η ξ W < λ =4a λ =2a λ =1a 3 33 circles in the polar plot inside the circles W > position depends on λ if the tyre parameters are known pure yaw case: origin pure lateral case: infinitely outside April 2, /18

21 Stability of a trailing wheel suspension representative model for shimmy analysis yaw or yaw-lateral degrees of freedom linearised system stability determined by the eigenvalues, O April 2, /18

22 Stability with only yaw degree of freedom (y c = eψ) yaw stiffness k ψ = 2 KNm/rad 14 velocity V [km/h] B 2 B1 V [km/h] λ/a η ξ B π π π B π mechanical trail e [m] gray: unstable April 2, /18

23 Stability with only yaw degree of freedom (y c = eψ) yaw stiffness k ψ = 2 KNm/rad 14 velocity V [km/h] B 2 B1 V [km/h] λ/a η ξ B π π π B π mechanical trail e [m] gray: unstable April 2, /18

24 Stability with only yaw degree of freedom (y c = eψ) yaw stiffness k ψ = 2 KNm/rad velocity V [km/h] B 2 B1 W> 12 km/h η 2 km/h 7 km/h ξ 4 B mechanical trail e [m] B 2 gray: unstable April 2, /18

25 Stability with lateral flexibility 6 velocity = 2 km/h 6 velocity = 7 km/h 6 velocity = 12 km/h yaw stiffness [knm/rad] B 4 B mechanical trail e [m] mechanical trail e [m] Boundary k ψ [knm/rad] λ/a η ξ π B π B π π π mechanical trail e [m] April 2, /18

26 Stability with lateral flexibility 6 velocity = 2 km/h 6 velocity = 7 km/h 6 velocity = 12 km/h yaw stiffness [knm/rad] B 4 B mechanical trail e [m] mechanical trail e [m] Boundary k ψ [knm/rad] λ/a η ξ π B π B π π π mechanical trail e [m] April 2, /18

27 Stability with lateral flexibility yaw stiffness [knm/rad] velocity = 7 km/h η ξ B 3 18 B 4 1kNm/rad 2kNm/rad kNm/rad mechanical trail e [m] April 2, /18

28 Motion of contact line at B 1 &B 3 σ V ẏ1 + y 1 = y c + (σ + a)ψ the contact line remain straight equivalent pure yaw oscillation around an imaginary steering axis no energy transfers from the forward motion April 2, /18

29 Conclusions & Outlook A framework to evaluate tyre models for shimmy application, based on the energy flow method: April 2, /18

30 Conclusions & Outlook A framework to evaluate tyre models for shimmy application, based on the energy flow method: Energy flow method Transfer functions linear tyre models Stability April 2, /18

31 Conclusions & Outlook A framework to evaluate tyre models for shimmy application, based on the energy flow method: Energy flow method Transfer functions linear tyre models Stability Generally applicable to non-linear tyre model with numerical instead of analytical solutions. Dedicated set-up for experimental validation. April 2, /18

32 Thanks! Questions? April 2, /18

P321(b), Assignement 1

P321(b), Assignement 1 P31(b), Assignement 1 1 Exercise 3.1 (Fetter and Walecka) a) The problem is that of a point mass rotating along a circle of radius a, rotating with a constant angular velocity Ω. Generally, 3 coordinates