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TECHNICAL PAPERS

Mechanical Steering Compensators for High-Performance Motorcycles

[+] Author and Article Information
Simos Evangelou

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

David J. N. Limebeer1

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

Robin S. Sharp

Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, UKrobin.sharp@imperial.ac.uk

Malcolm C. Smith

Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UKmcs@eng.cam.ac.uk

The immittance function of a network refers to either its impedance or admittance function.

1

Author to whom correspondence should be addressed.

J. Appl. Mech 74(2), 332-346 (Mar 01, 2006) (15 pages) doi:10.1115/1.2198547 History: Received November 04, 2004; Revised March 01, 2006

This paper introduces the idea of using mechanical steering compensators to improve the dynamic behavior of high-performance motorcycles. These compensators are seen as possible replacements for a conventional steering damper and comprise networks of springs, dampers, and a less familiar component called the inerter. The inerter was recently introduced to allow the synthesis of arbitrary passive mechanical impedances, and finds a potential application in the present work. The design and synthesis of these compensation systems make use of the analogy between passive electrical and mechanical networks. This analogy is reviewed alongside the links between passivity, positive reality, and network synthesis. Compensator design methods that are based on classical Bode-Nyquist frequency-response ideas are presented. Initial designs are subsequently optimized using a sequential quadratic programing algorithm. This optimization process ensures improved performance over the machine’s entire operating regime. The investigation is developed from an analysis of specific mechanical networks to the class of all biquadratic positive real functions. This aspect of the research is directed to answering the question: “What is the best possible system performance achievable using any simple passive mechanical network compensator?” The study makes use of computer simulations, which exploit a state-of-the-art motorcycle model whose parameter set is based on a Suzuki GSX-R1000 sports machine. The results show that, compared to a conventional steering damper, it is possible to obtain significant improvements in the dynamic properties of the primary oscillatory modes, known as “wobble” and “weave.”

Copyright © 2007 by American Society of Mechanical Engineers
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References

Figures

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Figure 9

Straight-running root-loci with speed the varied parameter. The speed is increased from 5m∕s(◻) to 75m∕s(⋆). The × locus represents the nominal machine without a steering damper, but fitted instead with a torsional spring of 100Nm∕rad, ∘ represents a spring of 200Nm∕rad, and + corresponds to a spring of 400Nm∕rad.

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Figure 8

Straight-running root-loci with speed the varied parameter. The speed is increased from 5m∕s(◻) to 75m∕s(⋆). The × locus represents the nominal machine damping value, ∘ refers to a steering damping decrease of 3Nms∕rad, and + to a steering damping reduction of 6Nms∕rad.

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Figure 18

Normalized frequency-response characteristics of the network SC-2 for three values of damping ratio

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Figure 19

(a) Stability constraint region and (b) damping ratio optimization region

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Figure 21

Cost function Jt for 7–75m∕s and 0–45deg lean with the motorcycle fitted with the optimized SC-1 compensated system

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Figure 22

Cost function Jt for 7–75m∕s and 0–45deg lean with the motorcycle fitted with the optimized SC-2 compensated system

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Figure 24

The vertical axis shows changes in the damping ratio of the wobble mode for single-parameter variations in each of the time-domain optimized networks: nominal steering damper SC-1 to SC-4. The trim condition is 45deg and 11m∕s; the parameter values are adjusted by 0.01%. The order of the parameters in each group is the same as that given in Table 3.

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Figure 1

Schematic of an inerter embodiment

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Figure 2

Circuit symbols and electromechanical correspondences with defining equations and admittances Y(s)

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Figure 3

Scaled diagrammatic motorcycle and rider in side view. The motorcycle-and-rider model shows the machine layout with each of the masses depicted as a proportionately scaled shaded circle.

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Figure 4

Feedback arrangement in which P(s) is the linearized motorcycle model and K(s) is the steering compensator. The signal d(s) represents vertical road displacement disturbances, Ts(s) is the steering torque and δ(s) is the steering angle.

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Figure 5

Root-locus plot for the straight-running motorcycle with speed the varied parameter. No steering damper is fitted. The speed is increased from 5m∕s(◻) to 85m∕s(⋆).

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Figure 6

Root-locus plots for: Straight running (×), 15deg(∘), 30deg(+), and 45deg(◇) of roll angle with speed the varied parameter. The nominal steering damper is fitted. The speed is increased from 7m∕s(◻) to 75m∕s(⋆).

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Figure 7

Straight-running Nyquist diagrams for the open-loop motorcycle model for four different values of forward speed. On the 75m∕s locus, the frequency at A is 47.6rad∕s, at B it is 33.8rad∕s, and at C it reduces to 28.4rad∕s.

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Figure 10

Straight-running root-loci with speed the varied parameter; the speed is increased from 5m∕s(◻) to 75m∕s(⋆). The × locus represents the nominal machine without a steering damper, ∘ represents the effect of an inertance of 0.1kgm2, and + the influence of an inertance of 0.2kgm2.

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Figure 11

Simple steering compensation networks

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Figure 12

Normalized frequency responses for SC-3 with unity gain and resonant frequency ωn=1 for three values of damping ratio ζ

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Figure 13

Nyquist diagram for the straight-running open-loop motorcycle at a forward speed of 75m∕s. The solid line represents the nominal machine with a unity-gain steering damper and the dashed line the compensated system using the SC-3 network given in Fig. 1 (ωn=33.8, ζ=0.5, c=1).

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Figure 14

Root-locus plots for: Straight running (×), 15deg(∘), 30deg(+), and 45deg(◇) with speed the varied parameter from 7m∕s(◻) to 75m∕s(⋆). The machine is fitted with the SC-3 network with the parameter values given in Table 1.

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Figure 15

Frequency responses for SC-4 with unity gain, resonant frequency ωn=1, denominator damping ratio ζ2=0.75 for three values of numerator damping ratio ζ1

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Figure 16

Nyquist diagram for the straight-running open-loop motorcycle at a forward speed of 75m∕s. The solid line represents the nominal machine with a unity-gain steering damper and the dashed line the compensated system using the SC-4 network given in Fig. 1. The network parameters are c1=1, ζ1 =0.1,ζ2 =0.4, and ωn=25.

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Figure 17

Root-locus plots for: Straight running (×), 15deg(∘), 30deg(+), and 45deg(◇) with speed the varied parameter for the compensated machine fitted with the network SC-4 with the parameters of Table 2. The speed is increased from 7m∕s(◻) to 75m∕s(⋆).

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Figure 20

Root-locus plots for: Straight running (×), 15deg(∘), 30deg(+), and 45deg(◇) of roll angle with speed the varied parameter. The network SC-2 is fitted and was optimized to minimize the performance index 4. The speed is increased from 7m∕s(◻) to 75m∕s(⋆).

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Figure 23

Nyquist diagrams for the open-loop motorcycle with a forward speed of 75m∕s and a roll angle of 15deg. The solid line represents the machine fitted with the nominal steering damper, and the dashed line represents the machine fitted with the time-domain optimized SC-2 network.

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Figure 25

Root-locus plots for: Straight running (×), 15deg(∘), 30deg(+), and 45deg(◇) of roll angle, with the speed varied from 7m∕s(◻) to 75m∕s(⋆). The network SC-2 is fitted with its parameters set to minimize Jf in 5.

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Figure 26

Road forcing gain for 7–75m∕s and 3–45deg roll angle for the frequency-domain optimized SC-2 compensated system

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Figure 27

Bode magnitude plot for road displacement forcing (0dB=1rad∕m). The machine is operating at a forward speed of 15m∕s and a roll angle of 45deg. The solid line represents the nominal machine, and the dashed line the machine fitted with the frequency-domain optimized SC-2 network.

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Figure 28

Bode magnitude plot for road displacement forcing (0dB=1rad∕m). The machine is operating at a forward speed of 75m∕s and a roll angle of 15deg. The solid line represents the nominal machine, and the dashed line the machine fitted with the frequency-domain optimized SC-2 network.

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Figure 29

Compensator parameter sensitivities for the wobble mode at 45deg and 11m∕s. The vertical axis shows the change in the damping ratio for a 0.01% change in each parameter. The frequency-domain optimized networks are used; the first network is the nominal steering damper. The parameters in each of the four network groups (SC-1 to SC-4) adopt the ordering in Table 4.

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Figure 30

Wobble-mode damping ratio sensitivities at 45deg and 11m∕s. The vertical axis shows the change in the damping ratio for a 0.01% change in each of the parameters shown in the tables below the subfigures. The bars are shown in groups of five; each group represents the steering-compensated vehicle with the frequency-domain optimized networks in the order: nominal steering damper, SC-2, SC-3, SC-4, and SC-5.

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