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Research Papers

Suppression of Burst Oscillations in Racing Motorcycles

[+] Author and Article Information
Simos A. Evangelou

Department of Electrical and Electronic, and Mechanical Engineering,
Imperial College,
London, SW7 2AZ, UKe-mail: s.evangelou@imperial.ac.uk

David J. N. Limebeer

Professor of Control Engineering,
Department of Engineering Science,
University of Oxford,
Parks Road, Oxford, OX1 3PJ, UK
e-mail: david.limebeer@eng.ox.ac.uk

Maria Tomas-Rodriguez

Mathematics and Engineering Department,
City University,
London, EC1V 0HB, UK
e-mail:Maria.Tomas-Rodriguez.1@city.ac.uk

Contributed by the Applied Mechanics Division of ASME for publication in the Journalof Applied Mechanics. Manuscript received July 3, 2010; final manuscript received November 26, 2011; accepted manuscript posted April 2, 2012; published online October 29, 2012. Assoc. Editor: Wei-Chau Xie. To view figures in color, please refer to the online version of this article.

J. Appl. Mech 80(1), 011003 (Oct 29, 2012) (14 pages) Paper No: JAM-10-1224; doi: 10.1115/1.4006491 History: Received July 03, 2010; Revised November 26, 2011; Accepted April 02, 2012

Burst oscillations occurring at high speed, and under firm acceleration, can be suppressed with a mechanical steering compensator. Burst instabilities in the subject racing motorcycle are the result of interactions between the wobble and weave modes under firm-acceleration at high speed. Under accelerating conditions, the wobble-mode frequency (of the subject motorcycle) decreases, while the weave mode frequency increases so that destabilizing interactions can occur. The design analysis is based on a time-separation principle, which assumes that bursting occurs on time scales over which speed variations can be neglected. Even under braking and acceleration conditions linear time-invariant models corresponding to constant-speed operation can be utilized in the design process. The influences of braking and acceleration are modeled using d’Alembert-type inertial forces that are applied at the mass centers of each of the model’s constituent bodies. The resulting steering compensator is a simple mechanical network that comprises a conventional steering damper in series with a linear spring. In control theoretic terms, this network is a mechanical lag compensator. A robust control framework was used to optimize the compensator design because it is necessary to address the inevitable uncertainties in the motorcycle model, as well as the nonlinearities that influence the machine’s local behavior as the vehicle ranges over its operating envelope.

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References

Anderson, B. D. O., 1985, “Adaptive Systems, Lack of Persistency of Excitation and Bursting Phenomena,” Automatica, 21, pp. 247–258. [CrossRef]
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Evangelou, S., Limebeer, D. J. N., Sharp, R. S., and Smith, M. C., 2007, “Mechanical Steering Compensators for High-Performance Motorcycles,” ASME J. Appl. Mech., 74(5), pp. 332–346. [CrossRef]
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Figures

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Fig. 4

Kinematic freedoms of the subject motorcycle including the suspension system, the frame flexibility and the chain drive

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Fig. 3

Side-view of the motorcycle model in its nominal configuration. The six constituent bodies are shown as (red) circles with their radii proportional to the cube root of their masses; the massless drive sprockets are shown as green circles. The points p1 through p16 are used primarily to define the kinematic constraints used in the model; we do not propose going into these details here.

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Fig. 2

Short-period Fourier transform of the damper stroke rate signal illustrated in Fig. 1. A weave-like burst oscillation occurs at approximately 8 s and persists for approximately 4 s; the signal amplitude is given in decibels.

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Fig. 1

Measured test data for the subject racing motorcycle. The green trace shows the speed in m/s, the red trace shows the acceleration ( × 10 m/s2), the black trace is the motorcycle’s roll angle (degs) and the blue trace represents the steering damper velocity in mm/s. Oscillatory steering bursting is clearly visible at 8 s and corresponds to a speed of approximately 70 m/s under approximately 5 m/s2 of acceleration. The frequency of the bursting is 28 rad/s and corresponds to a weave-type oscillation.

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Fig. 5

Front and rear suspension damper characteristics showing damper force against damper velocity. Negative velocities correspond to the compression of the damper unit.

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Fig. 6

Front and rear spring force as a function of the suspension units’ displacement. The dotted lines represent measured data, while the solid lines are given by the empirical relationship given in Eq. (2). Positive displacements correspond to suspension unit compression.

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Fig. 7

The tire road-contact geometry. The computation of the tire forces and moments depends on the kinematics of the wheels’ spinal center point Psc, the tires’ crown center point Pcc, and the ground contact point Pcp. The wheel radius vector Rrw points from Psc to Pcc with the wheel radius given by |Rrw|. The tire carcass height vector Rth points from Pcc to Pcp, while the ground contact vector Rcp points from the wheel spinal center Psc to the ground contact point Pcp; Rcp=Rrw+Rth. The ground contact point can be regarded as a moving point in the rear wheel, or as a moving but ‘unspun’ point in the rear swing arm body.

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Fig. 8

Generalized regulator popularized by the robust control community [27]. In the context of this study, the exogenous road forcing disturbances are represented by the signal d(s), the steering angle is δ(s), while the steering-compensator-induced torque is given by t(s) in which ‘s’ is the Laplace transform variable. The generalized plant P(s) is a linearized motorcycle model, while k(s) is the steering compensator.

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Fig. 9

Constant-speed straight-running root-loci for the subject motorcycle showing the influence of the steering damper—speed is the swept parameter. The characteristics of the open-loop machine are shown in the + (red) plot, while those corresponding to the subject motorcycle fitted with a steering damper are given by the × (blue) plot. The speed is varied from 10 m/s to 95 m/s in steps of 2 m/s; the top speed is marked with a star, while the lowest speed is marked with a square symbol. The subject motorcycle, when fitted with a steering damper alone, has adequate constant-speed damping in both the wobble and weave modes.

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Fig. 15

Straight-running weave-mode eigenvector loci for the subject motorcycle (with a steering damper fitted) with acceleration-related inertial force the varied parameter for speeds of 70 m/s (blue) × and 80 m/s (green) ○. The acceleration-related inertial forces are varied between −4 m/s2 and 4 m/s2. The 13 eigenvector components corresponding to the generalized coordinates are shown; the eigenvectors are normalized so that the steer angle component is +1. The largest acceleration-related inertial force (4 m/s2) is marked with a ⋆, while braking at −4 m/s2 is marked with a □. As with the classical weave mode, the five dominant components are: the machine’s lateral translation, and the yaw, roll, frame twist, and steer angles. The acceleration-related inertial forces reduce the amplitude of the lateral translation, and the yaw, roll and twist angle components. Under acceleration the weave mode of the subject motorcycle becomes ‘wobble like.’

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Fig. 12

Root-loci for the subject motorcycle (with a steering damper fitted) under constant-speed conditions. Speed is the swept parameter for roll angles of 0 deg (blue) × , 15 deg (green) ○, 30 deg (red) + and 45 deg (black) ⋄. The speed is varied from 9 m/s to 95 m/s in steps of 2 m/s. The highest speed is marked with a ⋆ and the lowest speed with a □.

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Fig. 13

Straight-running root-loci showing the wobble- and weave-mode eigenvalues of the subject motorcycle with the speed varied between 9 m/s and 95 m/s. A steering damper is fitted. The curves show the effect of acceleration-related inertial forces as follows: −4 m/s2 (pink) ▽, −2 m/s2 (black) ⋄, 0 m/s2 (blue) × , 2 m/s2 (green) ○ and 4 m/s2 (red +). The highest speed is marked with a ⋆ and the lowest speed with a □.

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Fig. 14

Root-loci showing the effect of acceleration-related d’Alembert forces at five different speeds. The acceleration is swept over the inteval ±5 m/s2; the highest acceleration is marked with a ⋆ and the lowest acceleration with a □. The subject motorcycle is fitted with a steering damper. The purple curve, annotated by ▽s, corresponds to a speed of 50 m/s. The black curve, annotated with ⋄s, corresponds to a speed of 60 m/s. The blue curve, annotated by × s, corresponds to a speed of 70 m/s. The green curve, annotated with ○s, corresponds to a speed of 80 m/s. The red curve, annotated with + s, corresponds to a speed of 90 m/s. The light blue curve, annotated by △s, is a reference plot corresponding to the zero-acceleration over the 10 m/s to 90 m/s speed range; the highest speed is marked with a ⋆ and the lowest speed with a □.

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Fig. 10

Simulated weave-mode burst oscillations in the subject motorcycle under straight-running conditions. In this simulation the speed is varied according to v = v0 + A sin(0.2t) m/s, while a small steering torque of the form tp = ε cos(2π4.5t) N-m is applied simultaneously. The speed is shown as the solid (green) line, 10 × the acceleration is shown as the dot-dashed (red) line, while 10 × the steering velocity is shown as the solid (blue) bursting characteristic (in rad/s). The parameter values used in the simulation are v0 = 70 m/s, A = 25 m/s, and ε = 5.0 × 10−5 N-m. The discontinuities in the acceleration signal are caused by chain snap that occurs as chain drive switches between driving and braking.

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Fig. 11

Short-period Fourier transform of the steering velocity signal given in Fig. 10. Data between 10 s and 20 s was extracted for analysis; bursting is apparent between 14 s and 18 s (in Fig. 10). The bursting frequency increases from approximately 33 rad/s to approximately 38 rad/s in sympathy with the speed increase on this time interval.

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Fig. 18

Steering compensation networks. The conventional scheme is the pure damper shown in diagram (a), while the spring-damper lag compensator, which comprises a series-connected spring and damper combination is shown in diagram (b). The optimal damper value was found to be c1 = 8.06 Nms/rad, while the optimized spring-damper values are k = 921.6 Nm/rad and c2 = 7.30 Nms/rad.

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Fig. 19

Region of stability. Closed-loop eigenvalues are excluded from the cross hatched region.

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Fig. 16

Root-loci of the subject motorcycle with a steering damper fitted. Speed is the varied parameter for roll angles of 0 deg (blue) × , 15 deg (green) ○, 30 deg (red) + and 45 deg (black) ⋄. The speed is varied from 9 m/s to 95 m/s in steps of 2 m/s. The highest speed is marked with a ⋆ and the lowest speed with a □. In (a) the braking-related acceleration is −4 m/s2, while in (b), the acceleration is 4 m/s2.

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Fig. 17

Nyquist diagrams of the steering compensation loop for the straight-running subject motorcycle at 80 m/s. Four acceleration-related inertial forces are illustrated: −2 m/s2 (black) dotted line, 0 m/s2 (blue) solid line, 2 m/s2 (green) dashed line and 4 m/s2 (red) dashed-dotted line. Note that the machine is unstable at 80 m/s and 4 m/s2 of acceleration for any value of steering damper.

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Fig. 24

Straight-running root-loci with speed the varied paremeter for acceleration-related inertial force of −4 m/s2 (pink ▽), −2 m/s2 (black ⋄), 0 m/s2 (blue × ), 2 m/s2 (green ○) and 4 m/s2 (red + ). The compensator is fitted. The speed is varied from 9 m/s to 95 m/s in steps of 2 m/s. The highest speed is marked with a ⋆ and the lowest speed with a □.

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Fig. 20

Nyquist diagram for the cases when the machine is fitted with a steering damper (blue solid line), and the machine fitted with the steering compensator (green dashed line). The machine’s speed is 80 m/s, the roll angle is 0 deg and the acceleration-related inertial force is 4 m/s2. This diagram illustrates the improvement of the stability margins brought about by a steering compensator at high speed and high acceleration.

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Fig. 21

Nyquist diagrams for the case of the subject motorcycle fitted with an optimized steering compensator. The acceleration-related inertial forces are: −2 m/s2 (black) dotted line, 0 m/s2 (blue) solid line, 2 m/s2 (green) dashed line and 4 m/s2 (red) dashed-dotted line. The machine speed is 80 m/s and its roll angle is 0 deg.

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Fig. 22

Influence of a steering compensator at low speed and moderate braking. The Nyquist diagram for the case when the machine is fitted with the steering damper (blue) solid line and the compensator (green) dashed line. The machine’s speed is 30 m/s, the roll angle is 0 deg and the acceleration-related inertial force is −4 m/s2. This figure illustrates the detrimental influence of the steering compensator at low speeds and firm braking.

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Fig. 23

Root-loci showing the wobble- and weave-mode eigenvalues as a function of the acceleration-related inertial force, which is varied from −5 m/s2 to 5 m/s2. The machine’s roll angle is 0 deg and the speed is 80 m/s. The blue × plot corresponds to the case when the steering damper (8.0566 Nms) is fitted and green ○ is the case when compensator is fitted (damper 7.2956 Nms; spring 921.58 Nm/rad). The −5 m/s2 case is shown with □ and the 5 m/s2 case is shown with ⋆.

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Fig. 25.

Straight-running root-loci comparing the behavior of the subject motorcycle fitted with a damper and a steering compensator. The speed is varied from 9 m/s to 95 m/s in steps of 2 m/s; the highest speed is marked with a ⋆ and the lowest speed with a □. The response corresponding to the damper-equipped machine is shown as the (blue) × plot, while the compensator-equipped vehicle is shown as the (green) ○ plot. (a) corresponds to −4 m/s2, while the (b) plot corresponds to 4 m/s2.

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