Energy balance and tyre motions during shimmy - … BALANCE... · Energy balance and tyre motions...
Transcript of 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
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 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
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
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 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
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
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 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
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
April 20, 2015 2/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
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
April 20, 2015 3/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
Von Schlippe tyre model
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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
)
April 20, 2015 3/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
Von Schlippe tyre model
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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
)
April 20, 2015 4/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
Energy transfer through tyre
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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
April 20, 2015 4/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
Energy transfer through tyre
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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
April 20, 2015 4/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
Energy transfer through tyre
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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
April 20, 2015 5/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
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
April 20, 2015 5/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
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
April 20, 2015 6/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
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
April 20, 2015 7/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
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
April 20, 2015 7/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
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
April 20, 2015 8/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
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
April 20, 2015 9/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
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!
April 20, 2015 10/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
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.
April 20, 2015 11/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
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
April 20, 2015 12/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
Stability of a trailing wheel suspension
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I representative model for shimmy analysisI yaw or yaw-lateral degrees of freedomI linearised systemI stability determined by the eigenvalues
April 20, 2015 13/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
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
April 20, 2015 13/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
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
April 20, 2015 13/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
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
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
April 20, 2015 14/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
Stability with lateral flexibility
mechanical trail e [m]
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
mechanical trail e [m]
velocity = 120 km/h
−0.1 0 0.1 0.2 0.30
10
20
30
40
50
60
mechanical trail e [m]
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
April 20, 2015 14/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
Stability with lateral flexibility
mechanical trail e [m]
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
mechanical trail e [m]
velocity = 120 km/h
−0.1 0 0.1 0.2 0.30
10
20
30
40
50
60
mechanical trail e [m]
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
April 20, 2015 15/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
Stability with lateral flexibility
mechanical trail e [m]
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
April 20, 2015 16/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
Motion of contact line at B1&B3
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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
April 20, 2015 17/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
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.
April 20, 2015 17/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
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.
April 20, 2015 17/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
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.
April 20, 2015 18/18/w Section Dynamics & Control 4th International Tyre Colloquium - Guildford - UK
Thanks!
Questions?