energy balance and tyre motions during shimmy - … balance... · energy balance and tyre motions...

Energy balance and tyre motionsduring shimmy

Shenhai RanI.J.M. BesselinkH. Nijmeijer

Dynamics & ControlEindhoven University of Technology

April 20, 2015 @ 4th International Tyre Colloquium - Guildford - UK

April 20, 2015 1/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK

Introduction

Motivation of my PhD research: Tyre models for shimmy analysisI linear and non-linear Fy&Mz with relaxation behaviour

I enhanced relaxation behaviour with contact patch dynamicsI including turn slip and its interaction with side slipI belt dynamics with a rigid ring approachI fundamentals of shimmy

Question to myself: How to compare and evaluate tyre models?I steady state characteristics, step response & frequency responseI For shimmy? from an energy point of view


Introduction

Motivation of my PhD research: Tyre models for shimmy analysisI linear and non-linear Fy&Mz with relaxation behaviourI enhanced relaxation behaviour with contact patch dynamicsI including turn slip and its interaction with side slipI belt dynamics with a rigid ring approachI fundamentals of shimmy



Introduction


Question to myself: How to compare and evaluate tyre models?I steady state characteristics, step response & frequency responseI For shimmy?

from an energy point of view


Introduction




Contents

Motivation

Von Schlippe Tyre Model

The Energy Flow MethodEnergy transfer through tyreShimmy energy

Stability analysisStability with only yaw degree of freedomStability with lateral flexibility

Conclusions & Outlook


Von Schlippe tyre model

�

��

��

�

��

No sliding between contact line and ground

Governing equations:

σ

Vy1 + y1 = yc + (σ+ a)ψ

y2(t) = y1(t−2a

V)

Fy = cv

(v1 + v22

)= cv

(y1 + y22

− yc

)Mz = cβ

(v1 − v22a

)= cβ

(y1 − y22a

−ψ

)


Energy transfer through tyre

�

�

��

��

��

�

�

��

� ��

Energy and work at wheel center:

Fa cosψyc∆t+ Fdxc∆t+Maψ∆t = ∆U+ ∆Ek

Force and moment equilibrium:

Fa = −Fy cosψ

Ma = −Mz

}

Fdxc∆t = Fy cosψyc∆t+Mzψ∆t+ ∆U+ ∆Ek


The shimmy energy

Suppose the wheel exhibits periodic motions, ∆U and ∆Ek vanish:∫T0

Fdxcdt =∫T0

(Fy cosψyc +Mzψ)dt

Shimmy energy is defined as follows:

W ,∫T0

(Fyyc +Mzψ)dt

W > 0⇒ energy flows into lateral-yaw motion from forward motion


Sinusoidal motion at wheel center

The wheel moves with prescribed sinusoidal motions:

yc(t) = Aη sin(ωt+ ξ)

ψ(t) = A sin(ωt)

}

To calculateW, transfer functions Fy&Mz w.r.t lateral and yaw motions used:[FY(s)MZ(s)

]=

[H11(s) H12(s)H21(s) H22(s)

] [YC(s)Ψ(s)

]and

Hmn = |Hmn(jω)|, θmn = ∠Hmn(jω); m,n = 1, 2


Transfer Functions

H11(s) = HFy,yc(s) = cv

(2σ sV+ 1− e−

2asV

2(σ sV+ 1)

)

H12(s) = HFy,ψ(s) = cv

((σ+ a)(1+ e−

2asV )

2(σ sV+ 1)

)

H21(s) = HMz,yc(s) = −cβ

(1− e−

2asV

2a(σ sV+ 1)

)

H22(s) = HMz,ψ(s) = −cβ

(2aσ s

V− (σ− a) + (σ+ a)e−

2asV

2a(σ sV+ 1)

)

s

V⇒ jω

V⇒ path wavelength λ ,

2πω

V


Transfer Functions

H11(s) = HFy,yc(s) = cv

(2σ sV+ 1− e−

2asV

2(σ sV+ 1)

)

H12(s) = HFy,ψ(s) = cv

((σ+ a)(1+ e−

2asV )

2(σ sV+ 1)

)

H21(s) = HMz,yc(s) = −cβ

(1− e−

2asV

2a(σ sV+ 1)

)

H22(s) = HMz,ψ(s) = −cβ

(2aσ s

V− (σ− a) + (σ+ a)e−

2asV

2a(σ sV+ 1)

)s

V⇒ jω

V⇒ path wavelength λ ,

2πω

V


Energy components

W =W11 +W12 +W21 +W22

where

W11 =

∫T0

(AηH11 sin(ωt+ ξ+ θ11) ·Aωη cos(ωt+ ξ)

)dt = πη2A2H11 sin θ11

W12 =

∫T0

(AH12 sin(ωt+ θ12) ·Aωη cos(ωt+ ξ)

)dt = −πηA2H12 sin(ξ− θ12)

W21 =

∫T0

(AηH21 sin(ωt+ ξ+ θ21) ·Aω cos(ωt)

)dt = πηA2H21 sin(ξ+ θ21)

W22 =

∫T0

(AH22 sin(ωt+ θ22) ·Aω cos(ωt)

)dt = πA2H22 sin θ22


Pure lateral and yaw motion of wheel center

10−2

10−1

100

−0.5

0

0.5

a/λ [−]

shim

my

ener

gy [J

]

pure yaw W22

pure lateral W11

Shimmy can only occur at large λ for pure yaw motion; not possible if only lateral motion exists!


Zero shimmy energy boundary of combined input

In the case of combined motion, solvingW = 0 leads to:

η =H12 sin(ξ− θ12) −H21 sin(ξ+ θ21)±

√∆

2H11 sin θ11

where∆ =

(H12 sin(ξ− θ12) −H21 sin(ξ+ θ21)

)2 − 4H11H22 sin θ11 sin θ22

It is a circle in a polar plot, where the distance to the origin is η and the angle to the positive x-axisrepresents ξ.


Zero shimmy energy boundary of combined input

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

η

ξ

W > 0

W < 0

λ = 40a

λ = 20a

λ = 10a

I circles in the polar plotI inside the circlesW > 0

I position depends on λ if the tyre parametersare known

I pure yaw case: originI pure lateral case: infinitely outside


Stability of a trailing wheel suspension

��, ��

�

�

O

I representative model for shimmy analysisI yaw or yaw-lateral degrees of freedomI linearised systemI stability determined by the eigenvalues


Stability with only yaw degree of freedom(yc = −eψ)

mechanical trail e [m]

velo

city

V [k

m/h

]

yaw stiffness kψ = 20 KNm/rad

−0.4 −0.2 0 0.2 0.4 0.6

20

40

60

80

100

120

140

B2 B

1

gray: unstable

V [km/h] λ/a η ξ

B1

20 11.4266 0.5278 π70 39.9930 0.5278 π

120 68.5594 0.5278 π

B2

20 8.5903 0.2158 π70 22.7869 0.0049 0

120 39.2878 0.0218 0


Stability with only yaw degree of freedom(yc = −eψ)


velo

city

V [k

m/h

]

yaw stiffness kψ = 20 KNm/rad

−0.4 −0.2 0 0.2 0.4 0.6

20

40

60

80

100

120

140

B2 B

1

gray: unstable

0.2

0.4

0.6

0.8

1

210

60

240

90

270

120

300

150

330

180 0

20 km/h70 km/h120 km/h

η

ξW>0

B1

B2


Stability with lateral flexibility


yaw

stif

fnes

s [k

Nm

/rad

]

velocity = 20 km/h

−0.1 0 0.1 0.2 0.30

10

20

30

40

50

60


velocity = 120 km/h

−0.1 0 0.1 0.2 0.30

10

20

30

40

50

60


velocity = 70 km/h

−0.1 0 0.1 0.2 0.30

10

20

30

40

50

60

B4

B3

Boundary kψ [kNm/rad] λ/a η ξ

B310 25.7983 0.5278 π20 27.2170 0.5278 π

B4

10 32.1553 0.1978 1.4448π20 22.7186 0.1420 0.5301π30 18.5927 0.0401 0.4961π


Stability with lateral flexibility


yaw

stif

fnes

s [k

Nm

/rad

]

velocity = 70 km/h

−0.1 0 0.1 0.2 0.30

10

20

30

40

50

60

0.2

0.4

0.6

30

210

60

240

90

270

120

300

150

330

180 0

10kNm/rad20kNm/rad30kNm/rad

ξ

η

B3

B4


Motion of contact line at B1&B3

� � �

�

��

��

σ

Vy1 + y1 = yc + (σ+ a)ψ

I the contact line remain straightI equivalent pure yaw oscillation

around an imaginary steering axisI no energy transfers from the

forward motion


Conclusions & Outlook

A framework to evaluate tyre models for shimmy application, based on the energy flow method:

Energy flowmethod

Transferfunctions

Stabilitylinear tyre models

Generally applicable to non-linear tyre model with numerical instead of analytical solutions.Dedicated set-up for experimental validation.


Thanks!

Questions?

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