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Research Papers

# Influence of Road Camber on Motorcycle Stability

[+] Author and Article Information
Simos Evangelou

Department of Electrical and Electronic Engineering, and Department of Mechanical Engineering, Imperial College London, London SW7 2AZ, UKs.evangelou@imperial.ac.uk

David J. Limebeer1

Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, U.Kd.limebeer@imperial.ac.uk

Maria Tomas Rodriguez

Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, U.Kmaria.tomas@imperial.ac.uk

http://www.imperial.ac.uk/controlandpower/motorcycles/

1

Corresponding author.

J. Appl. Mech 75(6), 061020 (Aug 21, 2008) (12 pages) doi:10.1115/1.2937140 History: Received July 18, 2007; Revised January 07, 2008; Published August 21, 2008

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## Figures

Figure 1

Scaled diagrammatic side-view of the motorcycle model in its nominal configuration. The seven constituent bodies are shown as (dark) circles, with their radii proportional to their mass. All the points critical to building the model are individually marked. For example, the largest mass is the rear frame with its mass center located at p8.

Figure 16

Real parts of the wobble- and weave-mode eigenvalues as a function of speed for seven different road camber angles and a motorcycle roll angle of ϕ=30deg. The real part of the wobble-mode eigenvalue is shown in the (a) figure, while the weave-mode behavior is shown in the (b) plot. The road camber curves are annotated as −15deg, ∗; −10deg, △; −5deg, ●; 0deg, ×; 5deg, ○; 10deg, +; 15deg, ◇.

Figure 2

The road surface used for cambered road stability studies is a right circular cone (as illustrated, the cone is inverted for positive camber angles in the range 90deg>θ⩾0). The central axis of the cone is aligned with the inertial axis nz, with its vertex at the origin n0 of the inertial reference system. For camber angles in the range 90deg>θ⩾0, the motorcycle is assumed to ride on the interior surface of the cone along a circular trajectory. Only positive yaw rate operating conditions are considered (clockwise when seen from above), which means that for positive roll angles the machine leans toward the central axis of the cone. The nominal rear-wheel ground-contact point is P. The actual rear-wheel ground-contact point is assumed to move over the tangent plane T(P); the normal to the plane T(P) is the vector ∇SP. A second tangent plane is used to describe locally the road surface under the front wheel. The vector rp⊥ is the projection of rp onto the ground plane, while vp⊥ is the velocity of P projected onto the ground plane.

Figure 3

Contact between an inclined road surface and a motorcycle tire in a single-wheel model. The total mass of the machine and rider is M, the total weight is therefore Mg, and the centripetal force is Mv2∕r. The tire crown radius is ρ, and the distance between the motorcycle’s mass center C and the center of the tire crown is lo. The road camber angle is θ, while the motorcycle roll angle is ϕ; the motorcycle comes out of the page for positive yaw rates (ψ̇>0). The tire’s normal load and side force are given by Fz and Fy, respectively, and are applied at the contact point O. The force Fz is normal to the road surface, while Fy is tangent to it.

Figure 4

Roll-angle feedback loop used in the simulation model. The steering torque Ts is generated from the difference between the roll angle ϕ and the adaptive roll angle reference ϕref(1+ks⟨rp⊥,vp⊥⟩); rp⊥ and vp⊥ are defined in Fig. 2. If the motorcycle is moving toward the inertial axis nz, the adaptive gain term ks⟨rp⊥,vp⊥⟩ adjusts the roll-angle reference so as to steer the machine away from it. Conversely, if the motorcycle is moving away from nz, the adaptive gain term ks⟨rp⊥,vp⊥⟩ steers the machine toward it. The adaptive roll-angle term thus has the effect of centering the machine trajectory on nz and becomes noncontributory once the machine’s trajectory has been centered; in this event, ⟨rp⊥,vp⊥⟩=0.

Figure 5

Adaptive roll-reference controller centering the motorcycle trajectory on nz. (a) The machine trajectory begins at the initial point (180, 150) in the ground plane, which is outside the origin-centered equilibrium circle for the motorcycle cornering at 10m∕s with a roll angle of 15deg; under these conditions, the equilibrium radius of curvature is 45m. In this simulation, the adaptive roll-angle reference gain is ks=5.0×10−4 (see Fig. 4). (b) Motorcycle running on the surface of a cone with camber angle θ=5deg. In this simulation, the machine accelerates from 5m∕sto75m∕s at 0.005m∕s2, with the motorcycle roll angle maintained at ϕ=0deg; this trajectory begins at (0,0) in the ground plane.

Figure 6

Approximate and exact side force and normal loads as functions of camber angle; the machine roll angle is ϕ=10deg. (a) Normalized side force; the dashed curve comes from Eq. 12, while the solid curve is computed using the high-fidelity model at a forward speed of 10m∕s. (b) Normalized normal load; the dashed curve comes from Eq. 13, while the solid curve is computed using the high-fidelity model.

Figure 7

Roll-angle stability limits. (a) Condition 15 shows that one static roll stability limit is reached when the machine mass center lies directly above the ground contact. It follows from Eq. 11 that under these conditions the machine’s path curvature approaches zero (r→∞) for all operating points along this stability boundary. (b) The condition locosϕ+ρcosθ>0 shows that a static roll stability limit is approached when the centripetal force passes through the ground-contact point. This operating condition can only be approached asymptotically as v2∕r→∞; note that the gravitational force is ignored due to its negligible influence in this case.

Figure 8

Static stability limits. In order for a stable trim-state to exist, the motorcycle must operate within the cross-hatched region illustrated. This region is defined by the following: a◯ is the friction limit given by ϕ−θ>−arcsin(μlimitρ∕(lo(1+μlimit2)1∕2))−arctan(μlimit), in which lo=0.4316m, ρ=0.0775m and μlimit=1.6 are used for illustration; b◯ is the friction limit given by ϕ−θ<arcsin(μlimitρ∕(lo(1+μlimit2)1∕2))+arctan(μlimit). The limits a◯ and b◯ taken together come from Eq. 14. The boundary c◯ is the vertical roll stability limit given by inequality losinϕ+ρsinθ⩾0, (see Fig. 7); d◯ horizontal roll stability limit given by inequality locosϕ+ρcosθ>0 (see Fig. 7). Under wall of death conditions, the road camber angle is given by θ=90deg, and stable roll angles exist between ϕ=23.5deg and roll angles approaching 90deg.

Figure 9

Stable operation under wall of death type conditions. (a) shows the equilibrium roll angle as a function of v2∕(rg), and was generated using Eq. 10 for three different road camber angles, and with lo=0.4316m and ρ=0.0775m. The dotted curve corresponds to a camber angle of θ=70deg, the dot-dash curve to θ=90deg, and the dashed curve to θ=110deg. For θ>90deg, the vehicle traverses the interior of a noninverted cone; the lowest viable value of v2∕(rg) comes from the friction limit condition ϕ−θ>−arcsin(μlimitρ∕(lo(1+μlimit2)1∕2))−arctan(μlimit) (boundary a◯ in Fig. 8), while the limiting value associated with v2∕(rg)→∞ and the corresponding highest roll angles come from the horizontal roll stability limit (boundary d◯). (b) shows the motorcycle in roll equilibrium against the wall of death; the resultant of the gravitational and centripetal forces act through the tire contact point O.

Figure 10

Root loci for four road camber angles showing the wobble- and weave-mode eigenvalues as a function of speed. The motorcycle roll angle is ϕ=0deg, and the speed is varied from 5m∕sto75m∕s. The highest speed is marked with a ⋆ and the lowest speed with a ◻. The road cambers are annotated as 0deg, ×; 5deg, ○; 10deg, +; 15deg, ◇.

Figure 11

Real parts of the wobble- and weave-mode eigenvalues as a function of speed for four different road camber angles, and a motorcycle roll angle of ϕ=0deg. The real part of the wobble-mode eigenvalue is shown in (a), while the real part of the weave-mode eigenvalue is shown in (b). The road camber curves are annotated as 0deg, ×; 5deg, ○; 10deg, +; 15deg, ◇.

Figure 12

Root loci for seven road camber angles, with a motorcycle roll angle of ϕ=5deg (a), and ϕ=15deg (b). The speed is varied from 5m∕sto75m∕s; the highest speed is marked with a ⋆ and the lowest speed with a ◻. The road cambers are annotated as −15deg, ∗; −10deg, △; −5deg, ●; 0deg, ×; 5deg, ○; 10deg, +; 15deg, ◇.

Figure 13

Real part of the wobble-mode eigenvalue as a function of the road camber angle for four different machine roll angles. The motorcycle’s forward speed is constant at (a) 10m∕s and (b) 75m∕s. The motorcycle roll-angle curves are annotated as 0deg, ×; 5deg, ○; 10deg, +; 15deg, ◇.

Figure 14

Real part of the weave-mode eigenvalue as a function of the road camber angle for four different machine roll angles. The motorcycle’s forward speed is constant at (a) 10m∕s and (b) 75m∕s. The motorcycle roll-angle curves are annotated as 0deg, ×; 5deg, ○; 10deg, +; 15deg, ◇.

Figure 15

Root-loci for seven road camber angles, with a motorcycle roll angle of ϕ=30deg; the speed is varied from 5m∕sto75m∕s. The highest speed is marked with a ⋆ and the lowest speed with a ◻. The road cambers are annotated as −15deg, ∗; −10deg, △; −5deg, ●; 0deg, ×; 5deg, ○; 10deg, +; 15deg, ◇.

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