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Research Papers

Burst Oscillations in the Accelerating Bicycle

[+] Author and Article Information
David J. N. Limebeer

Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJdavid.limebeer@eng.ox.ac.uk

Amrit Sharma

Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJamrit.sharma82@gmail.com

J. Appl. Mech 77(6), 061012 (Sep 01, 2010) (20 pages) doi:10.1115/1.4000909 History: Received January 24, 2009; Revised December 10, 2009; Published September 01, 2010; Online September 01, 2010

The purpose of this paper is to study the dynamics of the accelerating bicycle. It is shown that time-scale separation can be used to study the oscillatory characteristics of the accelerating machine using time-invariant models. These models are used to explain practically observed wobble-mode bursting oscillations that are associated most frequently with down-hill riding. If the vehicle is cornering under constant acceleration, at a fixed roll angle, it is shown that for low values of acceleration (and braking), it follows closely a logarithmic spiral shaped trajectory. The studies presented are facilitated by a novel adaptive control scheme that centers the machine’s trajectory on any arbitrary point in the ground plane. The influences of cambered road surfaces are also investigated. The bicycle model employed is an extension of that originally developed by Whipple, and comprises two road wheels and two laterally-symmetric frame assemblies that are free to rotate relative to each other along an inclined steering axis. For the most part, the front frame is treated as being flexible and the bicycle is fitted with force generating road tires, rather than classical nonholonomic rolling constraints. This research provides the ground work required for generating more complex dynamic models for high-performance motorcycle studies.

Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Steering oscillations in a braking bicycle model. The top plot shows the steering angle behavior as the bicycle decelerates from 15 m/s to 3 m/s at −1 ms−2. The bottom plot shows a time-frequency plot of the steering angle signal. The color bar on the right gives the spectral amplitudes in dB.

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Figure 2

The simple bicycle model with its degrees of freedom. The model comprises of two frames hinged together along an inclined steering axis. The rider is rigidly attached to the rear frame, and each wheel is assumed to be in point contact with the road.

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Figure 3

Eigenvalues of the simple straight-running bicycle model as a function of speed. The real parts are shown as (blue) dots and the imaginary parts as (red) crosses. This plot is the same as that illustrated in Fig. 3 of Ref. 3.

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Figure 4

Eigenvalues plots for the simple straight-running bicycle under braking and acceleration: (a) the case of straight-running acceleration at 2.0 m/s2. The real parts of the frozen-time eigenvalues for the time-varying model given in Eq. 3 are shown as the (black) dots, while the imaginary parts are given by the (red) crosses. The real parts of the eigenvalues for the time-invariant model given in Eq. 13 by the (blue) stars, while the imaginary parts are given by the (green) circles; (b) the case of straight-running braking at −2.0 m/s2

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Figure 5

Kinematics of a bicycle tire during acceleration and cornering: (a) key velocity and force vectors in the x-y plane; (b) side view (x-z plane) of a spinning wheel and the associated drive force −Fx

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Figure 6

Flexible front frame assembly and inclined steering axis. The frame flex angle is ϕ, the steer angle is δ, the wheel spin angle is ξ, and the steer axis tilt angle is λ. The origin of a main-frame-fixed axis system Oxyz is at O with the y-axis pointing out of the page. The mass of the moving wheel assembly is mf; its mass center is located at G with coordinates (xf,0,zf). The road contact point is at P and the wheel spin axis is at C. The road speed is assumed to be v.

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Figure 8

Rolling contact between a bicycle’s (front) tire and a cambered road surface with inclination angle θ: (a) physical situation in which the (front) wheel’s camber angle is φf, and the superscripts f and r denote the front and rear, respectively. The tire makes contact with the road at P, with the center of the tire’s crown labelled Q; (b) corresponding “thin-wheel” representation with a flexible tire carcass. The (front) carcass force is Frf, and the tire’s normal load and side forces are given by Fzf and Fyf, respectively. These forces are applied at the contact point P′ with Fzf normal to the road surface and Fyf tangent to it. The tire carcass has stiffness Kf and damping Df. In the SAE sign convention, the bicycle comes out of the page for positive yaw rates (ψ̇>0).

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Figure 10

Eigenvalue loci for the straight-running accelerating bicycle. The (blue) crossed curve corresponds to the constant speed case, the (red) starred locus illustrates an acceleration of 1 m/s2, and the (green) circled plot corresponds to a deceleration of −1 m/s2. The speed of the bicycle varies from 0 ms−1 (squares) to 25 ms−1 (diamonds). Note that the frame structural damping has been reduce to 17.6 Nmsr−1 from the value given in Table 1.

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Figure 13

Trajectory of the bicycle rear-wheel ground contact point. The machine is operating at a roll angle of 5 deg and a constant speed of 5 ms−1; the adaptive steering controller gain is set to ks=0.001. The plot shows how the adaptive steering control loop centers the bicycle’s trajectory on the inertial z-axis.

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Figure 14

Relationship between the intrinsic and polar coordinates. Illustrated is a curve C and a general point P on that curve. The origin of the polar coordinate system is O, with r and ϑ as the polar coordinates of P. The instantaneous center of rotation is R, the instantaneous radius of curvature is ρ, with κ as the angle between ρ and the horizontal x-axis. The angles ζ and η define the orientation of a tangent to C at P with respect to r and the x-axis, respectively.

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Figure 17

Road surface used for camber-road stability studies. The central axis of a right circular-cone in aligned with the inertial z-axis with its vertex at the origin n0 of the inertial reference system. For camber angles in the range 90 deg>θ≥0 deg, the constant-speed bicycle 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 bicycle leans towards 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 T(P) is the vector ∇SP. A second tangent plane (not shown) is used to locally describe 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 on to the ground plane; these vectors are used in the adaptive roll angle control loop described in Sec. 5.

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Figure 18

Side force and normal loads as functions of camber angle; the bicycle roll angle is 10 deg: (a) the dashed (red) curve comes from the approximate Eq. 25, while the solid (blue) curve is computed using the high-fidelity bicycle model; (b) the dashed (red) curve comes from the approximate Eq. 26, while the solid (blue) curve is computed using the high-fidelity bicycle model

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Figure 19

Rear-wheel ground-contact point P in a plane tangent to a conical road surface. The tangent plane is defined by the normal vector ∇SP (see Eq. 40). The vector vlong is in the bicycle’s direction of travel and vlat is orthogonal to it; both vectors lie in the tangent plane. The contact point velocity vP is shown with its components in cylindrical coordinates; the er and eϑ components are in the ground plane, with the ez component perpendicular to it. The velocity vector has longitudinal and lateral components vlong and vlat, and make angles γ and ξ with the ground plane, respectively.

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Figure 20

Trajectory of the accelerating bicycle on a conical road surface. The road camber angle is 15 deg and the bicycle is accelerating from 5 ms−1 to 25 ms−1 at 0.1 ms−2. The bicycle’s rear-wheel ground-contact point trajectory is the solid (red) curve, while the 3D logarithmic spiral generated by Eqs. 35,44 is the dashed (blue) curve.

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Figure 21

The frozen-time and constant-speed eigenvalue loci for the accelerating and decelerating bicycles. The frozen-time loci are shown as (blue) crosses, while the constant-speed loci are shown as (red) stars. The roll angle is 10 deg, the road camber angle is 15 deg and the speed of the bicycle varies from 5 ms−1 (squares) to 25 ms−1 (diamonds): (a) deceleration at −0.5 ms−2; (b) acceleration of 0.5 ms−2.

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Figure 22

Eigenvalue loci for the accelerating bicycle with a fixed roll angle of 20 deg. The (blue) crossed curve corresponds to the constant speed case, the (red) starred locus illustrates an acceleration of 1 m/s2, and the (green) circled plot corresponds to a deceleration of −1 m/s2. The speed of the bicycle varies from 5 ms−1 (squares) to 25 ms−1 (diamonds). Note that the frame structural damping has been reduce to 17.6 Nmsr−1 from the value given in Fig. 2.

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Figure 23

Free body diagram of an accelerating road wheel. The force F accelerates the wheel’s mass, while the couple F′ results in the angular acceleration Ω̇.

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Figure 7

Root loci for the front frame assembly system. Only the roots associated with the shimmy mode with positive imaginary part are shown. The speed is swept from 0 m/s to 20 m/s, with a square marking the low-speed end of the loci. The (blue) x-curve corresponds to zero acceleration, the (red)  ∗-curve corresponds to an acceleration of 2 ms−2 and the (green) o-curve corresponds to an acceleration of −2 ms−2.

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Figure 9

Tire-force system for the inverted-pendulum model. The rider-machine mass center is located at C, the whole-machine weight is Mg, the centripetal force is Mv2/ρ, in which ρ is the turn radius of curvature, the bicycle’s roll angle is φ, the road camber angle is θ, and the tire makes contact with the road at P′. The distance CP′ between the mass center and the ground contact point is lo. The combined tire normal load and side forces are given by Fz and Fy, respectively, and are applied at P′. The force Fz is normal to the road surface, while Fy is tangent to it.

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Figure 11

Eigenvalues of the straight-running bicycle under braking and acceleration, and ascending and descending a hill. The (red) starred locus illustrates an acceleration of 1 m/s2, and the (green) circled plot corresponds to a deceleration of −1 m/s2. The triangles correspond to the machine ascending a hill, with inclination angle 5.851 deg, while the crosses illustrate the machine descending a hill with inclination angle −5.851 deg. The speed of the bicycle varies from 0 ms−1 (squares) to 25 ms−1 (diamonds).

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Figure 12

Roll-angle feedback loop with an adaptive roll angle reference. 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⟩), where rp and vp are the desired position and velocity vectors of the bicycle, respectively. If the bicycle is moving toward the desired center of rotation, the adaptive gain term ks⟨rp,vp⟩ adjusts the roll-angle reference so as to steer the machine away from it. Conversely, if the bicycle is moving away from the desired center of rotation, the adaptive gain term ks⟨rp,vp⟩ steers the machine towards it. The adaptive roll-angle term becomes noncontributory once the machine’s trajectory has been centered; in this event ⟨rp,vp⟩=0.

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Figure 15

Trajectory of the accelerating cornering bicycle; the initial velocity is 5 ms−1 and the final velocity is 25 ms−1: (a) the trajectory of the rear-wheel ground-contact point for the bicycle accelerating at 0.1 ms−2, with a fixed roll angle of 10 deg is shown as the solid (red) curve. The corresponding logarithmic spiral generated by (35) is shown as the dashed (blue) curve. The (black) dot-dashed curve is generated by the bicycle model with both its wheel spin inertias increased by a factor of ten; the bicycle is again accelerating at 0.1 ms−2, with a fixed roll angle of 10 deg; (b) the (blue) dashed line corresponds to a fixed lateral acceleration of 1.73 ms−2, the solid (red) curve shows the lateral acceleration of the bicycle accelerating at 0.1 ms−2, and the dot-dash line illustrates the lateral acceleration of the bicycle with both its wheel spin inertias increased by a factor of ten.

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Figure 16

Eigenvalue loci for the accelerating bicycle with a roll angle of 20 deg and a road camber angle of 15 deg. The (blue) crossed curve corresponds to the constant speed case, the (red) starred locus illustrates an acceleration of 0.5 m/s2, and the (green) circled plot corresponds to a deceleration of −0.5 ms−2. The speed of the bicycle varies from 5 ms−1 (squares) to 25 ms−1 (diamonds).

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