Nonlinear effects on the self-excited chatter oscillations in motorcycle dynamics, including tyre relaxation

Abstract

Nonlinear effects on the stability of the motorcycle rear ‘chatter’ phenomenon are investigated by means of a minimal two degrees of freedom model, including tyre relaxation since this aspect has not been investigated in-depth in previous studies. Motorcycle ‘chatter’ manifests itself as a self-excited oscillation, which arises during braking in the frequency range between 17 and 22 Hz, affecting safety and performance. The study of the linearised system gives indications on the initiation of self-excited vibrations and thus helps to prevent them using proper design techniques. Post-bifurcation behaviour is analysed focusing on limit cycles and bifurcation diagrams, studied by means of specific applications of the harmonic balance method and Floquet theory. This allows the detection of the validity of the linear results, and to identify the meaningful parameters in limit cycle generation, their amplitude, and their stability.

Appendices

Appendix A: Nomenclature

α, α0	Swingarm rotation angle, and its stationary value
α s0	Preload angle of the equivalent rotational spring of the suspension, acting on the joint S
θ, θ0	Wheel rotation angle, and its stationary value
ω 0	Angular speed of rear sprocket (wheel), when in stationary conditions
ω p	Angular speed of front sprocket (pinion), always in stationary conditions
Ω	Angular frequency
H(Ω)	Frequency response function
φ	Phase delay
β p	Inclination angle of the line connecting the centre of pinion to the swingarm pivot
ψ	Chain angle between the taut (lower) branch and the swingarm direction
l sa	Swingarm length
z sa	Height of swingarm pivot with respect to the ground (positive)
l p	Offset distance of centre of pinion from the swingarm pivot
r p	Radius of front sprocket (pinion)
r c	Radius of rear sprocket (wheel)
d	Centre distance between the sprockets
l tc	Tangential distance between the sprockets
l c	Total dynamic length of the chain
l fc	Total free length of the chain
R r	Outer radius of the wheel
V x0	Constant travelling speed of the translating frame
V x	Actual longitudinal travelling speed of wheel centre
Vsx, Vsxr	Longitudinal slip velocity, and its value with relaxation
J α	Equivalent moment of inertia of swingarm and wheel about the axis of pivot S
J θ	Wheel moment of inertia (rim, tyre, and rear sprocket) about the axis of joint A
c s	Equivalent rotational damping coefficients of the suspension, acting on the joint S
k s	Equivalent rotational stiffness of the suspension, acting on the joint S
k z	Tyre vertical stiffness
k c	Chain stiffness
F c	Chain elastic force
M cα	Moment due to the chain elastic force, about the swingarm pivot
M cθ	Moment due to the chain elastic force, about the wheel centre
Fz, Fz0	Normal ground force, and its stationary value
Fx, Fxr, Fx0	Longitudinal ground force, its value with relaxation and its stationary value
κ, κr, κ0	Longitudinal slip coefficient, its value with relaxation and its stationary value
λx, λx0	Relaxation length of the tyre, and its stationary value
K, K0	Tyre ground longitudinal stiffness, and its stationary value
u	Displacement of tyre ground contact point, in the Maxwell model
η F	Floquet multiplier
t	Time
T	Period of oscillation
Bx, Cx, Dx, Ex	MF semiempirical coefficients, depending on the working conditions of the tyre
F z0P	MF Tyre nominal load
pkx1	MF nominal tyre slip stiffness
pkx2	MF Variation of slip stiffness with load
pkx3	MF Exponent of slip stiffness with load
pcx1	MF shape factor
pdx1	MF nominal tyre friction
pdx2	MF Variation of friction with load
pex1	MF Nominal tyre force curvature
pex2	MF Variation of curvature with load
pex3	MF Quadratic variation of curvature with load
pcfx1	MF linear coefficient for tyre ground longitudinal stiffness K
pcfx2	MF quadratic coefficient for tyre ground longitudinal stiffness K
Cκ, Cκ0	Component with respect to slip κ of the tyre force Fx gradient, and its stationary value
CFz, CFz0	Component with respect to Fz of the tyre force Fx gradient, and its stationary value
C K	Derivative of K with respect to Fz
C Vsx	Partial derivative of Fx with respect to slip velocity Vsx
I	Identity matrix
C *	Equivalent damping matrix
K *	Equivalent stiffness matrix

Appendix B: Adopted values for the parameters

Model parameter	Value	Model parameter	Value
J α	8.56 kg m2	l sa	0.65 m
J θ	0.95 kg m2	r c	0.10 m
c s	420 Nms rad–1	r p	0.04 m
k s	8100 Nm rad–1	l p	0.10 m
k c	1.15 × 106 Nm–1	β p	0.40 rad
k z	1.70 × 105 Nm–1	R r	0.32 m

MF parameter	Value	MF parameter	Value
F z0P	1475 N	pdx1	1.28
pkx1	25.4	pdx2	–7.82 × 10–3
pkx2	1.10	pex1	0.47
pkx3	0.20	pex2	9.39 × 10–5
pcx1	1.77	pex3	6.62 × 10–2

Relax. parameter	Value	Relax. parameter	Value
λ x0	0.0387347 m	pcfx1	0.152
K 0	1.48 × 105 Nm–1	pcfx2	0.378

Input parameter	Value	Computed parameter	Value
F z0	− 400 N	F x0	–269 N
κ 0	− 0.035	ω 0	78.4 rad s–1
V x0	26 ms–1	ω p	196 rad s–1
α 0	0.20 rad	θ 0	7.43 × 10–3 rad
C κ0	6248.7 N	z sa	0.447 m
C Fz0	0.70918	α s0	0.243 rad

Appendix C: Nonlinear equations of motion

Nonlinear equations of motion expanded in Taylor series up to the third order, case without relaxation:

$$ \begin{aligned} \left\{ \begin{gathered} \ddot{\tilde{\alpha }} + f_{1} (\tilde{\alpha },\dot{\tilde{\alpha }},\tilde{\theta },\dot{\tilde{\theta }}) = 0 \hfill \\ \ddot{\tilde{\theta }} + f_{2} (\tilde{\alpha },\dot{\tilde{\alpha }},\tilde{\theta },\dot{\tilde{\theta }}) = 0 \hfill \\ \end{gathered} \right. \\ f_{j} (\tilde{\alpha },\dot{\tilde{\alpha }},\tilde{\theta },\dot{\tilde{\theta }}) & = a_{j - 1}^{(1)} \tilde{\alpha } + a_{j - 2}^{(1)} \dot{\tilde{\alpha }} + a_{j - 3}^{(1)} \tilde{\theta } + a_{j - 4}^{(1)} \dot{\tilde{\theta }} \\ & \quad + a_{j - 1}^{(2)} \tilde{\alpha }^{2} + a_{j - 2}^{(2)} \dot{\tilde{\alpha }}^{2} + a_{j - 3}^{(2)} \dot{\tilde{\theta }}^{2} \\ & \quad + a_{j - 4}^{(2)} \tilde{\alpha }\dot{\tilde{\alpha }} + a_{j - 5}^{(2)} \tilde{\alpha }\tilde{\theta } + a_{j - 6}^{(2)} \tilde{\alpha }\dot{\tilde{\theta }} + a_{j - 7}^{(2)} \dot{\tilde{\alpha }}\dot{\tilde{\theta }} \\ & \quad + a_{j - 1}^{(3)} \tilde{\alpha }^{3} + a_{j - 2}^{(3)} \dot{\tilde{\alpha }}^{3} + a_{j - 3}^{(3)} \dot{\tilde{\theta }}^{3} \\ & \quad + a_{j - 4}^{(3)} \tilde{\alpha }^{2} \dot{\tilde{\alpha }} + a_{j - 5}^{(3)} \tilde{\alpha }^{2} \tilde{\theta } + a_{j - 6}^{(3)} \tilde{\alpha }^{2} \dot{\tilde{\theta }} + a_{j - 7}^{(3)} \tilde{\alpha }\dot{\tilde{\alpha }}^{2} \\ & \quad + a_{j - 8}^{(3)} \dot{\tilde{\alpha }}^{2} \dot{\tilde{\theta }} + a_{j - 9}^{(3)} \tilde{\alpha }\dot{\tilde{\theta }}^{2} + a_{j - 10}^{(3)} \dot{\tilde{\alpha }}\dot{\tilde{\theta }}^{2} + a_{j - 11}^{(3)} \tilde{\alpha }\dot{\tilde{\alpha }}\dot{\tilde{\theta }} \\ \end{aligned} $$

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First order coefficients of f₁ and f₂:

Term	f1	Value	f2	Value	Unit
$\tilde{\alpha }$	a(1)1–1	10,815.1	a(1)2–1	–17,110.4	s–2
$\dot{\tilde{\alpha }}$	a(1)1–2	49.5172	a(1)2–2	–10.0140	s–1
$\tilde{\theta }$	a(1)1–3	931.071	a(1)2–3	12,105.3	s–2
$\dot{\tilde{\theta }}$	a(1)1–4	–1.16021	a(1)2–4	25.7151	s–1

Second order coefficients of f₁ and f₂:

Term	f1	Value	f2	Value	Unit
$\tilde{\alpha }^{2}$	a(2)1–1	18,291.2	a(2)2–1	–313,868	s–2
$\dot{\tilde{\alpha }}^{2}$	a(2)1–2	–2.01941 × 10–2	a(2)2–2	0.447585	–
$\dot{\tilde{\theta }}^{2}$	a(2)1–3	–0.118365	a(2)2–3	2.62346	–
$\tilde{\alpha }\dot{\tilde{\alpha }}$	a(2)1–4	130.343	a(2)2–4	–2819.47	s–1
$\tilde{\alpha }\tilde{\theta }$	a(2)1–5	–1159.07	a(2)2–5	0	s–2
$\tilde{\alpha }\dot{\tilde{\theta }}$	a(2)1–6	–328.986	a(2)2–6	7113.25	s–1
$\dot{\tilde{\alpha }}\dot{\tilde{\theta }}$	a(2)1–7	9.79505 × 10–2	a(2)2–7	–2.17099	–

Third order coefficients of f₁ and f₂:

Term	f1	Value	f2	Value	Unit
$\tilde{\alpha }^{3}$	a(3)1–1	116,980	a(3)2–1	− 217,387	s–2
$\dot{\tilde{\alpha }}^{3}$	a(3)1–2	2.92983 × 10–5	a(3)2–2	6.49371 × 10–4	s
$\dot{\tilde{\theta }}^{3}$	a(3)1–3	2.71189 × 10–3	a(3)2–3	− 6.01067 × 10–2	s
$\tilde{\alpha }^{2} \dot{\tilde{\alpha }}$	a(3)1–4	2131.64	a(3)2–4	− 27,550.4	s–1
$\tilde{\alpha }^{2} \tilde{\theta }$	a(3)1–5	–41.7690	a(3)2–5	0	s–2
$\tilde{\alpha }^{2} \dot{\tilde{\theta }}$	a(3)1–6	–3851.49	a(3)2–6	35,668.8	s–1
$\tilde{\alpha }\dot{\tilde{\alpha }}^{2}$	a(3)1–7	–6.48427	a(3)2–7	140.613	–
$\dot{\tilde{\alpha }}^{2} \dot{\tilde{\theta }}$	a(3)1–8	2.89409 × 10–4	a(3)2–8	− 6.41451 × 10–3	s
$\tilde{\alpha }\dot{\tilde{\theta }}^{2}$	a(3)1–9	–37.2483	a(3)2–9	807.374	–
$\dot{\tilde{\alpha }}\dot{\tilde{\theta }}^{2}$	a(3)1–10	–1.9924 × 10–3	a(3)2–10	4.41608 × 10–2	s
$\tilde{\alpha }\dot{\tilde{\alpha }}\dot{\tilde{\theta }}$	a(3)1–11	31.1279	a(3)2–11	− 674.859	–

Nonlinear equations of motion expanded in Taylor series up to the third order, case with relaxation (semi-nonlinear model with λ_x0 = 0.0387347 m, as reported in Appendix B):

$$ \begin{aligned} \left\{ \begin{gathered} \ddot{\tilde{\alpha }} + g_{1} (\tilde{\alpha },\dot{\tilde{\alpha }},\tilde{\theta },\dot{\tilde{\theta }},\tilde{\kappa }_{r} ) = 0 \hfill \\ \ddot{\tilde{\theta }} + g_{2} (\tilde{\alpha },\dot{\tilde{\alpha }},\tilde{\theta },\dot{\tilde{\theta }},\tilde{\kappa }_{r} ) = 0 \hfill \\ \dot{\tilde{\kappa }}_{r} + g_{3} (\tilde{\alpha },\dot{\tilde{\alpha }},\tilde{\theta },\dot{\tilde{\theta }},\tilde{\kappa }_{r} ) = 0,\quad g_{3} = V_{x0} \lambda_{x0}^{ - 1} [\tilde{\kappa }_{r} - \tilde{\kappa }(\tilde{\alpha },\dot{\tilde{\alpha }},\dot{\tilde{\theta }})] \hfill \\ \end{gathered} \right. \\ g_{j} (\tilde{\alpha },\dot{\tilde{\alpha }},\tilde{\theta },\dot{\tilde{\theta }},\tilde{\kappa }_{r} ) & = b_{j - 1}^{(1)} \tilde{\alpha } + b_{j - 2}^{(1)} \dot{\tilde{\alpha }} + b_{j - 3}^{(1)} \tilde{\theta } + b_{j - 4}^{(1)} \dot{\tilde{\theta }} + b_{j - 5}^{(1)} \tilde{\kappa }_{r} \\ & \quad + b_{j - 1}^{(2)} \tilde{\alpha }^{2} + b_{j - 2}^{(2)} \dot{\tilde{\alpha }}^{2} + b_{j - 3}^{(2)} \tilde{\kappa }_{r}^{2} \\ & \quad + b_{j - 4}^{(2)} \tilde{\alpha }\dot{\tilde{\alpha }} + b_{j - 5}^{(2)} \tilde{\alpha }\tilde{\theta } + b_{j - 6}^{(2)} \dot{\tilde{\alpha }}\dot{\tilde{\theta }} + b_{j - 7}^{(2)} \tilde{\alpha }\tilde{\kappa }_{r} \\ & \quad + b_{j - 1}^{(3)} \tilde{\alpha }^{3} + b_{j - 2}^{(3)} \dot{\tilde{\alpha }}^{3} + b_{j - 3}^{(3)} \tilde{\kappa }_{r}^{3} \\ & \quad + b_{j - 4}^{(3)} \tilde{\alpha }^{2} \dot{\tilde{\alpha }} + b_{j - 5}^{(3)} \tilde{\alpha }^{2} \dot{\tilde{\theta }} + b_{j - 6}^{(3)} \tilde{\alpha }\dot{\tilde{\alpha }}^{2} + b_{j - 7}^{(3)} \dot{\tilde{\alpha }}^{2} \dot{\tilde{\theta }} \\ & \quad + b_{j - 8}^{(3)} \tilde{\alpha }\tilde{\kappa }_{r}^{2} + b_{j - 9}^{(3)} \tilde{\alpha }^{2} \tilde{\kappa }_{r} + b_{j - 10}^{(3)} \tilde{\alpha }\dot{\tilde{\alpha }}\dot{\tilde{\theta }} \\ \end{aligned} $$

(49)

First order coefficients of g₁, g₂ and g₃:

Term	g1	Value	g2	Value	Unit	g3	Value	Unit
$\tilde{\alpha }$	b(1)1–1	10,815.1	b(1)2–1	− 17,110.4	s–2	b(1)3–1	0	s–1
$\dot{\tilde{\alpha }}$	b(1)1–2	49.5172	b(1)2–2	0	s–1	b(1)3–2	3.21715	–
$\tilde{\theta }$	b(1)1–3	931.071	b(1)2–3	12,105.3	s–2	b(1)3–3	0	s–1
$\dot{\tilde{\theta }}$	b(1)1–4	0	b(1)2–4	0	s–1	b(1)3–4	− 8.26132	–
$\tilde{\kappa }_{r}$	b(1)1–5	− 94.2671	b(1)2–5	2089.35	s–2	b(1)3–5	671.232	s–1

Second order coefficients of g₁, g₂ and g₃:

Term	g1	Value	g2	Value	Unit	g3	Value	Unit
$\tilde{\alpha }^{2}$	b(2)1–1	18,291.2	b(2)2–1	–313,868	s–2	b(2)3–1	0	s–1
$\dot{\tilde{\alpha }}^{2}$	b(2)1–2	0	b(2)2–2	0	–	b(2)3–2	–1.59787 × 10–2	s
$\tilde{\kappa }_{r}^{2}$	b(2)1–3	–781.394	b(2)2–3	17,318.9	s–2	b(2)3–3	0	s–1
$\tilde{\alpha }\dot{\tilde{\alpha }}$	b(2)1–4	0	b(2)2–4	0	s–1	b(2)3–4	15.8707	–
$\tilde{\alpha }\tilde{\theta }$	b(2)1–5	–1159.07	b(2)2–5	0	s–2	b(2)3–5	0	s–1
$\dot{\tilde{\alpha }}\dot{\tilde{\theta }}$	b(2)1–6	0	b(2)2–6	0	–	b(2)3–6	4.10318 × 10–2	s
$\tilde{\alpha }\tilde{\kappa }_{r}$	b(2)1–7	–26,730.1	b(2)2–7	577,952	s–2	b(2)3–7	0	s–1

Third order coefficients of g₁, g₂ and g₃:

Term	g1	Value	g2	Value	Unit	g3	Value	Unit
$\tilde{\alpha }^{3}$	b(3)1–1	116,980	b(3)2–1	–217,387	s–2	b(3)3–1	0	s–1
$\dot{\tilde{\alpha }}^{3}$	b(3)1–2	0	b(3)2–2	0	s	b(3)3–2	7.93620 × 10–5	s2
$\tilde{\kappa }_{r}^{3}$	b(3)1–3	1454.60	b(3)2–3	–32,239.9	s–2	b(3)3–3	0	s–1
$\tilde{\alpha }^{2} \dot{\tilde{\alpha }}$	b(3)1–4	0	b(3)2–4	0	s–1	b(3)3–4	− 1.60857	–
$\tilde{\alpha }^{2} \tilde{\theta }$	b(3)1–5	− 41.7690	b(3)2–5	0	s–2	b(3)3–5	0	s–1
$\tilde{\alpha }\dot{\tilde{\alpha }}^{2}$	b(3)1–6	0	b(3)2–6	0	–	b(3)3–6	− 0.157651	s
$\dot{\tilde{\alpha }}^{2} \dot{\tilde{\theta }}$	b(3)1–7	0	b(3)2–7	0	s	b(3)3–7	− 2.03794 × 10–4	s2
$\tilde{\alpha }\tilde{\kappa }_{r}^{2}$	b(3)1–8	− 245,897	b(3)2–8	5.32993 × 106	s–2	b(3)3–8	0	s–1
$\tilde{\alpha }^{2} \tilde{\kappa }_{r}$	b(3)1–9	− 312,933	b(3)2–9	2.89809 × 106	s–2	b(3)3–9	0	s–1
$\tilde{\alpha }\dot{\tilde{\alpha }}\dot{\tilde{\theta }}$	b(3)1–10	0	b(3)2–10	0	–	b(3)3–10	0.202416	s

Appendix D: Linearized model with relaxation

The equations of motion Eq. (6), coupled with the relaxation model Eq. (15) are analytically linearized as follows.

Linearized moments caused by the chain force in Eq. (5):

$$ \begin{aligned} \tilde{M}_{c\alpha } & = M_{c\alpha } - M_{c\alpha 0} \cong \left. {\frac{{\partial M_{c\alpha } }}{\partial \alpha }} \right|_{\,0} \tilde{\alpha } + \left. {\frac{{\partial M_{c\alpha } }}{\partial \theta }} \right|_{\,0} \tilde{\theta } \\ \tilde{M}_{c\theta } & = M_{c\theta } - M_{c\theta 0} \cong \left. {\frac{{\partial M_{c\theta } }}{\partial \alpha }} \right|_{\,0} \tilde{\alpha } + \left. {\frac{{\partial M_{c\theta } }}{\partial \theta }} \right|_{\,0} \tilde{\theta } \\ \end{aligned} $$

(50)

with coefficients:

$$ \begin{aligned} \left. {\frac{{\partial M_{c\alpha } }}{\partial \alpha }} \right|_{\,0} & \cong - k_{c} l_{sa}^{2} \sin^{2} \psi (\alpha_{0} ),\quad \left. {\frac{{\partial M_{c\alpha } }}{\partial \theta }} \right|_{\,0} = - k_{c} l_{sa} r_{c} \sin \psi (\alpha_{0} ), \\ \left. {\frac{{\partial M_{c\theta } }}{\partial \alpha }} \right|_{\,0} & = - k_{c} l_{sa} r_{c} \sin \psi (\alpha_{0} ),\quad \left. {\frac{{\partial M_{c\theta } }}{\partial \theta }} \right|_{\,0} = - k_{c} r_{c}^{2} \\ \end{aligned} $$

(51)

Linearized non-stationary components of the ground forces:

$$ \begin{aligned} \tilde{F}_{z} & = F_{z} - F_{z0} = - (k_{z} l_{sa} \cos \alpha_{0} )\tilde{\alpha }, \\ \tilde{F}_{x} & = F_{x} - F_{x0} = C_{\kappa 0} \tilde{\kappa }_{r} + C_{{F_{z0} }} \tilde{F}_{z} = C_{\kappa 0} \tilde{\kappa }_{r} - C_{{F_{z0} }} (k_{z} l_{sa} \cos \alpha_{0} )\tilde{\alpha } \\ \end{aligned} $$

(52)

Linearized equations of motion Eq. (6), considering the relaxation model described by Eqs. (15) and (23):

$$ \left\{ \begin{aligned}& J_{\alpha } {\ddot{\tilde{\alpha }}} + c_{s} {\dot{\tilde{\alpha }}} + c_{1\alpha } \tilde{\alpha } + c_{1\theta } \tilde{\theta } + c_{{1\kappa_{r} }} \tilde{\kappa }_{r} = 0 \hfill \\ &J_{\theta } {\ddot{\tilde{\theta }}} + c_{2\alpha } \tilde{\alpha } + c_{2\theta } \tilde{\theta } + c_{{2\kappa_{r} }} \tilde{\kappa }_{r} = 0 \end{aligned} \right. $$

(53)

with coefficients:

$$ \begin{aligned} c_{1\alpha } & = k_{s} + k_{c} l_{sa}^{2} \sin^{2} \psi (\alpha_{0} ) \\ & + k_{z} \\ & l_{sa}^{2} \cos \alpha_{0} (\cos \alpha_{0} + C_{{F_{z0} }} \sin \alpha_{0} ) - l_{sa} (F_{x0} \cos \alpha_{0} - F_{z0} \sin \alpha_{0} ) \\ c_{1\theta } & = k_{c} l_{sa} r_{c} \sin \psi (\alpha_{0} ) \\ c_{{1\kappa_{r} }} &= - C_{\kappa 0} l_{sa} \sin \alpha_{0} \\ c_{2\alpha } & = k_{c} l_{sa} r_{c} \sin \psi (\alpha_{0} )\\ & - C_{{F_{z0} }} k_{z} l_{sa} \cos \alpha_{0} (z_{sa} - l_{sa} \sin \alpha_{0} ) - F_{x0} l_{sa} \cos \alpha_{0} \\ c_{2\theta } & = k_{c} r_{c}^{2} \\ c_{{2\kappa_{r} }} & = C_{\kappa 0} (z_{sa} - l_{sa} \sin \alpha_{0} ) \\ \end{aligned} $$

(54)

Linearized relaxation equation Eq. (15), obtained after linearization of the slip coefficient κ in Eq. (1):

$$ \frac{{\lambda_{x0} }}{{V_{x0} }}\dot{\tilde{\kappa }}_{r} + \tilde{\kappa }_{r} = \tilde{\kappa } = \kappa - \kappa_{0} = - c_{{3\dot{\alpha }}} \dot{\tilde{\alpha }} + c_{{3\dot{\theta }}} \dot{\tilde{\theta }} $$

(55)

with coefficients:

$$ \begin{aligned} &c_{{3\dot{\alpha }}} = \frac{{R_{r} \omega_{0} l_{sa} \sin \alpha_{0} }}{{V_{x0}^{2} }} \\ & c_{{3\dot{\theta }}} = \frac{{R_{r} }}{{V_{x0} }} \end{aligned} $$

(56)

The linearized equations of motion with relaxation can the be expressed in state-space form, as in Sect. 2.2, Eqs. (24) and (25).