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Experimental and model-based analysis of motorcycle wobbleand weaveCitation for published version (APA):Jansen, T. J., & Nijmeijer, H. (2014). Experimental and model-based analysis of motorcycle wobble and weave.(D&C; Vol. 2014.003). Eindhoven University of Technology.

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Where innovation starts

Department of Mechanical EngineeringDen Dolech 2, 5612 AZ EindhovenP.O. Box 513, 5600 MB EindhovenThe Netherlands

AuthorTim JANSEN [email protected]

DateJune 6, 2014

VersionFinal version

SupervisorProf. Dr. H. NijmeijerEindhoven Universityof Technology

CoachesProf. V. CossalterM. MassaroUniversity of Padova

Experimental and model-basedanalysis of motorcycle wobble andweave

Report number: DC 2014.003

T.J. Jansen BSc

Abstract

This report presents analysis of motorcycle wobble and weave in different ways. The first methodis the identification of wobble and weave via Stochastic Subspace Identification (SSI) and thesecond makes use of the software FastBike. Three SSI variants have been outlined and imple-mented in Matlab namely, Covariance (COV), Unweighted Principal Component (UPC) and theConanical Variate Analysis (CVA). Next, pole selection using stabilisation diagrams is discussedand the different variants are compared. It is concluded that the covariance driven variant showsthe most promising results. Finally, SSI-COV is used to identify weave and wobble of experimen-tal data for different motorcycles. The second method presented in this report uses two modelsfor comparing wobble and weave. The model by R.S. Sharp has been implemented in Fast-Bike and is compared to the original model. It has been concluded that the motorcycle stabilitymodel by Sharp is a good baseline because it corresponds to the results provided by the modelimplemented in FastBike.

ii


Table of contents

TitleExperimental and model-basedanalysis of motorcycle wobble andweave

1 Introduction 1

2 Stochastic Subspace Identification variants 3

2.1 Covariance driven method (SSI-COV) . . . . . . . . . . . . . . 3

2.2 Data driven method (SSI-DATA) . . . . . . . . . . . . . . . . . . 4

2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Identification of weave and wobble modes 7

3.1 Construction of a stabilisation diagram . . . . . . . . . . . . . . 7

3.2 Comparison between methods . . . . . . . . . . . . . . . . . . 8

3.3 Reducing calculation time . . . . . . . . . . . . . . . . . . . . . 10

3.3.1 Computational gain . . . . . . . . . . . . . . . . . . . . 11

3.4 Hands-off compared to Hands-on the handlebar . . . . . . . . 11

3.5 Effect of shorter data series . . . . . . . . . . . . . . . . . . . . 11

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Experiments 15

4.1 Suzuki SV650S . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.1.1 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Kawasaki KLR 650 Tengai . . . . . . . . . . . . . . . . . . . . . 19

4.2.1 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Sharp model in FastBike 23

5.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Conclusions and Recommendations 33

6.1 Stochastic Subspace Identification . . . . . . . . . . . . . . . . 33

6.2 Sharp 1994 model and FastBike comparison . . . . . . . . . . 33

6.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . 34


Table of contents

TitleExperimental and model-basedanalysis of motorcycle wobble andweave

7 Appendices 35

A Matlab file ssicov.m . . . . . . . . . . . . . . . . . . . . . . . . 35

B Matlab file ssicva.m . . . . . . . . . . . . . . . . . . . . . . . . 37

C Matlab file ssiupc.m . . . . . . . . . . . . . . . . . . . . . . . . 38

D Matlab file process_sys.m . . . . . . . . . . . . . . . . . . . . . 39

List of Figures

3.1 Results with COV method with hands-off data at 80 mph . . . . . . . . . . . . . . . 8

3.2 Results with CVA method with hands-off data at 80 mph . . . . . . . . . . . . . . . 9

3.3 Results with UPC method with hands-off data at 80 mph . . . . . . . . . . . . . . . 9

3.4 Results for the weave mode including shortcut . . . . . . . . . . . . . . . . . . . . 10

3.5 Comparison between hands-off and hands-on situation . . . . . . . . . . . . . . . 12

3.6 Stabilisation diagrams with COV-method and shortcut for data series of shorter time 12

4.1 Testbikes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Speed profile of the straight driving tests for different speeds . . . . . . . . . . . . 16

4.3 Details of the coast-down experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.4 Stabilisation diagram of the straight driving test at different speeds . . . . . . . . . 17

4.5 Stabilisation diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.6 Stabilisation diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.7 Wobble mode frequency and damping ratio for coast-down experiment . . . . . . . 19

4.8 Gathered data with KLR 650 testing . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.9 Results of the KLR 650 test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.10 Stabilisation diagrams of the Kawasaki Tengai at different speeds . . . . . . . . . . 21

5.1 Geometry of the LISP programming model . . . . . . . . . . . . . . . . . . . . . . 25

5.2 Geometry of the FastBike model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.3 Front and rear tyre data and fitted curves. . . . . . . . . . . . . . . . . . . . . . . . 28



5.6 Root-loci for weave and wobble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.7 Normal tyre force for front and rear tyre for increasing speed . . . . . . . . . . . . 30

5.8 Root-loci for weave and wobble including aerodynamics . . . . . . . . . . . . . . . 31

5.9 Root-loci for weave and wobble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.10 Root-loci for weave and wobble with unlocked suspension . . . . . . . . . . . . . . 32

vi

List of Tables

2.1 The In- and outputs of the ssicov.m Matlab function . . . . . . . . . . . . . . . . . . 4

3.1 Inputs of ’process_sys.m’ Matlab function . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Different methods and corresponding computation times . . . . . . . . . . . . . . . 11

3.3 Computer specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.1 Suzuki SV650S specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Kawasaki KLR 650 Tengai specifications . . . . . . . . . . . . . . . . . . . . . . . . 20

5.1 Values for masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.2 Values for masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.3 Values for moments of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.4 Values for compliances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.5 Values for damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.6 Other values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

vii

List of symbols, abbreviations and variables

Symbols

A State transition matrixC Output matrixCi Covariance matrix0i Extended observability matrix1i Reversed extended stochastic controllability matrix† Moore-Penrose pseudo-inverseGz Rotational speed around 2D datalogger z-axisζ Damping ratioN Number of data entries in signalH Output block Hankel matrixi Data output signal indexS0 Initial amplitudeS(t) Sine wave approximationt TimeT Identity matrixφ Phase angley Data output signalYp Part of data output signal in which p refers to "Past"Y f Part of data output signal in which f refers to "Future"ωn Natural frequency

Abbreviations

COV Covariance driven methodCVA Conanical variate analysisDATA Data driven methodSSI Stochastic Subspace IdentificationUPC Unweighted principal component

Matlab variables

c f First column of "Future"-Hankel matrixcp First column of "Past"-Hankel matrixdT Sample timef actor Natural frequency is considered stable when present in at least

’1/factor’ of identifications

viii

idx Index value of identified modesnmax maximum order size of approximationnmin minimum order size of approximationr f Last row of "Future"-Hankel matrixr p Last row of "Past"-Hankel matrixsys State space system output of Matlab function (SSI output)weave_max Maximum estimated weave frequencyweave_min Minimum estimated weave frequencywobble_max Maximum estimated wobble frequencywobble_min Minimum estimated wobble frequencyW n_tol Tolerance for half the natural frequency window sizezeta_tol Tolerance for half the damping ratio window size

FastBike variables

M f Mass of front frameM f Mass of rear frameMb Mass of rear wheel assemblyMp Mass of rider upper bodyI f x Front frame inertia about x-axisI f z Front frame inertia about z-axisI f xz Front frame inertia productIr x Rear frame inertia about x-axisIr z Rear frame inertia about z-axisIr xz Rear frame inertia productIpx Rider upper body inertia about x-axisIpz Rider upper body inertia about z-axisIpxz Rider upper body inertia productirwx Rear wheel inertia about x-axisirwy Rear wheel spin inertiai fwy Front wheel spin inertiair y Effective engine flywheel inertiaSv Steering head lateral complianceSγ Steering head torsional twist complianceSζ Rider upper body restraint complianceSλ Rear wheel assembly twist complianceDv Steering head lateral dampingDγ Steering head torsional twist dampingDζ Rider upper body restraint dampingDλ Rear wheel assembly twist dampingksteer Steering stiffnessdsteer Steer damperDc Aerodynamic drag coefficientLc Aerodynamic lift coefficientC f v f Front tyre cornering stiffnessC f vr Rear tyre cornering stiffnessCmv f Front tyre aligning moment stiffnessCmvr Rear tyre aligning moment stiffnessC f 1 Front tyre camber stiffnessCr1 Rear tyre camber stiffnessC f 2 Front tyre twisting stiffnessCr2 Rear tyre twisting stiffness

ix

Cy f Front tyre lateral stiffnessCyr Rear tyre lateral stiffnessσ f Front tyre relaxation lengthσr Rear tyre relaxation lengthO2_O3x Distance in x-direction between the origin of the handlebar reference

frame on the handlebar axis and the origin of the rear assembly ref-erence frame

O2_O3z Distance in z-direction between the origin of the handlebar referenceframe on the handlebar axis and the origin of the rear assembly ref-erence frame

O3_O6x Distance in x-direction between the origin of the handlebar referenceframe on the handlebar axis and the origin of the wheel center refer-ence frame

O3_O6z Distance in z-direction between the origin of the handlebar referenceframe on the handlebar axis and the origin of the wheel center refer-ence frame

O5_C Swingarm length

x

1 Introduction

In motorcycle stability, two oscillatory motions are most important namely wobble and weave[3, 4]. Wobble is an oscillation of the front-steering assembly of the motorcycle with a frequencybetween 6 and 10 Hz. It can be compared with shimmy of a tyre and it typically happens withhands-off the handlebar. Wobble has a higher chance of occurring at lower speeds instead ofweave, which usually happens at high speeds. The weave mode oscillates in the frequency rangeof 2 to 4 Hz and it not only consists of steering assembly motion but roll, yaw and lateral motionof the rear frame as well. It can occur hands-off as well as hands-on the handlebar and becomesless damped at higher speeds.

The goal of the assignment described in this report is to develop an algorithm that can be used toidentify the wobble and weave modes from experimental data. In order to accomplish this goal,Stochastic Subspace Identification (SSI)[2, 10] has been applied to raw steering angle data. SSIcan be divided into three categories: covariance (COV), unweighted principal component (UPC)and the canonical variate analysis (CVA). The three variants have been applied to hands-offand hands-on motorcycle steering angle data, acquired under steady-state conditions, and acomparison has been made between methods. Also, different datasets have been used, namelyfour different speeds for the hands-off as well as the hands-on situation.

First, the different SSI methods have been implemented in Matlab, compared and the influenceof computational shortcuts has been examined. After a method has been determined to generatethe best results, different data sets are compared. The experimental data used has been providedby Harley Davidson and consists of hands on and hands off the handlebar at different speeds.Influences on the results, by decreasing the length of the steering angle dataset, have also beenobserved. Finally, new experimental data of less quality has been generated and wobble andweave identifications have been carried out.

In the last part of this report, two motorcycle models for wobble and weave analysis have beencompared. The first model is the baseline model by R.S. Sharp [13] in 1994 and the second,a software utility called FastBike[7, 5]. Properties of the 1994 model have been implementedin FastBike, calculations have been made and the results have been compared with the originalmodel.

1

2 Stochastic Subspace Identification variants

In this chapter, an outline is presented of the mentioned SSI variants. Section 2.1 shows theCovariance method and in section 2.2, the SSI-data methods CVA and UPC can be found insection 2.2.

2.1 Covariance driven method (SSI-COV)

The covariance method is based on singular value decomposition (SVD) in particular. For astochastic system, the discrete-time state-space model is given as

xk+1 = Axk + wkyk = Cxk + vk

(2.1)

in which A ε <n×n and C ε <l×n are system matrices, xk ε <n is the discrete state vector, yk ε <

l

is the measured discrete output and wk ε <n , vk ε <

l are process and measurement white noise,respectively. The dimension l stands for the number of output signals.In order to identify the system matrix A, the first step is to assemble an output block Hankel matrixfrom the output signal y

y = [ y0 y1 · · · yN ] (2.2)

where the subscript N corresponds to the number of data points of the measured steer angle.The following partitioned output block Hankel matrix

H =(

Yp

Y f

)=

y0 y1 · · · yN−2i...

......

...

yi−2 yi−1 · · · yN−i−2yi−1 yi · · · yN−i−1yi yi+1 · · · yN−i

yi+1 yi+2 · · · yN−i+1...

......

...

y2i−1 y2i · · · yN−1

(2.3)

can be defined using (2.2). Herein, the subscript i is the model order which is defined by the user.The subscript p in Yp refers to "past" and f relates to "future", since all values are smaller thani in the upper half and greater or equal to i in the bottom half of the Hankel matrix. The Matlabfunction ssicov.m is shown in appendix A and the in- and outputs can be found in table 2.1. First,the indices of the Hankel matrix are defined which are used to calculate the upper part of the totalHankel matrix, Yp. The inputs of the Hankel command cp and rp are the first column and last rowof the Hankel matrix, respectively.

3

Table 2.1: The In- and outputs of the ssicov.m Matlab function

Inputs y measured output signaldT sample timenmin minimum order size of approximationnmax maximum order size of approximation

Outputs sys state space system

Y f can be calculated in the same manner but cp and rp are different from cf and rf. See [2, 10]for more information. It is shown in [11] that the impulse response sequences of the time invariantdeterministic system,

xi+1 = Axi + Guiyi = Cxi +30ui

(2.4)

are equivalent to the output covariance sequences of the discrete-time state-space model in(2.1). In (2.4), A and C are identical to the stochastic system in (2.1) and G is the next-stateoutput covariance matrix. 30 is the output covariance matrix at lag zero. The equivalence canalso be expressed in terms of the system matrices A, G and C,

3i = yi = C Ai−1G (2.5)

The next step is to compute the covariance matrix Ci and rewrite using (2.5)

Ci = limN→∞

1N(Y f Y T

p ) = 0i ·1ci =

C

C A...

C Ai−1

· (Ai−1G · · · A2G AG G)

(2.6)

in which 0i is the extended observability matrix and 1ci is the reversed extended stochastic con-

trollability matrix. Perfoming a Singular Value Decomposition (SVD) on Ci gives

Ci = (U1U2)

(S1 00 0

)(V T

1V T

2

)= U1S1V T

1 (2.7)

By factoring, 0i and 1ci can be calculated,

0i = U1S1/21 T (2.8)

1ci = T−1S1/2

1 V T1 (2.9)

where T is a nonsingular similarity transformation matrix which can be defined to be an identitymatrix. Combining the result with equation 2.6, the system matrix A follows as shown below,

CC A...

C Ai−2

· A =

C AC A2

...

C Ai−1

⇒ A =

C

C A...

C Ai−2

†

C AC A2

...

C Ai−1

(2.10)

where † denotes the Moore-Penrose pseudo-inverse. The matrix C can be calculated using 0ias shown in appendix A.

2.2 Data driven method (SSI-DATA)

The major difference between SSI-DATA and SSI-COV is the fact that the covariances are notexplicitly calculated in the data driven method and this provides a numerical advantage. SSI-DATA uses a special form of the Kalman filter [11] to identify the system matrices of the unknown

4

stochastic system using Kalman state estimates calculated directly from the measured outputdata. This is opposite to the typical application of the Kalman filter where the filter is used toestimate the states using known system matrices and noise covariances [10]. More informationand derivations can be found in [2, 11]

The first step is, again, to construct the output block Hankel matrix from the output system yas shown in 2.1. The Hankel matrix is similar to the one in SSI-COV but it is additionally par-titioned and LQ-decomposition is applied. LQ-decomposition is the numerical matrix version ofthe Gramm-Schmidt orthogonalization procedure [9], as shown in the following

H =(

YpY f

)= L QT

=

L11 0 0L21 L22 0L31 L32 L33

QT1

QT2

QT3

(2.11)

Matlab does not include a function for LQ-decomposition but QR-decomposition is built-in. Asshown by [1], LQ-decomposition of H is similar to QR-decomposition of HT . In Matlab, the ’triu’-command is used to extract the upper triangular part of the matrix like equation 2.11. The Matlabfile ’ssicva.m’ can be found in appendix B. The matrix sizes of the different parts (e.g. L21 andL31) are defined similar to [2].

A = (L0) Q ↔ AT= QT

(LT

0

)(2.12)

The next step is to define the weighting matrix and this is where the UPC method and CVA methodare different. In the UPC method, the weighting matrix is selected as the idendity matrix. TheCVA algorithm uses a second LQ-decomposition on the L factors that correspond to Y f in orderto find the weighting matrix, see the weighting matrix part in appendix B. Once the weightingmatrix is known, SVD can be carried out, as shown below.

W(

L21L31

)= (U1U2)

(S1 00 0

)(V T

1V T

2

)= U1S1V T

1 (2.13)

The system matrices A and C can be found using the following, it is beyond the scope of thisreport to show detailed derivation of how this is accomplished [2].(

AC

)=

( (W−1U1S1/2

1

)†L31

L21

)(L21L31

)†

W−1U1S1/21 (2.14)

2.3 Summary

In this chapter, two different methods for SSI have been provided. The first method is the covari-ance driven method which is particularly based on singular value decomposistion. The secondmethod is the data driven method and it can be divided into two subcategories. Both methodsdiffer from the SSI-COV because the covariances are not explicitly calculated. The diffenrencebetween both SSI-DATA method lies in the determination of the weighting matrix.

5

3 Identification of weave and wobble modes

In this chapter, stabilisation diagrams and the construction will be explained. Also, the differentSSI methods have been compared. Computational shortcuts have been implemented and theyhave been compared to the results of the original methods as well. The advantages of theshortcut, time wise, have been calculated. Sections 3.4 and 3.5 discuss the differences betweenthe hands-off and hands-on cases and the effect of shorter data series on the results. The inputdata used is the relative steering angle, between the front assembly and the rear assembly. Thetests to record the data have been carried out at 4 different speeds and hands-off and hands-onthe handlebar in straight running. Due to confidentiality, the speeds cannot be revealed.

3.1 Construction of a stabilisation diagram

In modal analysis, engineers often use stabilisation diagrams. These diagrams are constructedfrom identified poles from the same data set with different model order sizes. In general, stabil-isation diagrams will show vertical lines of poles which are classified as true modes because ithas been identified for each model order. To construct a stabilisation diagram, the state spacesystem output from the SSI has to be processed and the Matlab function ’process_sys.m’ hasbeen used. The inputs for this function are shown the table below. The output of the functionare four structure arrays: ’weave’, ’wobble’, ’other ’ and ’all ’. Each structure array contains thefollowing data: ’Wn’ with the natural frecuency, an array ’zeta’ with the damping ratio and ’i ’ withthe associated order sizes.

Table 3.1: Inputs of ’process_sys.m’ Matlab function

Input Descriptionsys State space system (SSI output)Wn_tol Tolerance for half the natural frequency window size (see [10])zeta_tol Tolerance for half the damping ratio window sizefactor Natural frequency is considered stable when present in at least ’1/factor’ of

identificationsweave_min Minimum estimated weave frequencyweave_max Maximum estimated weave frequencywobble_min Minimum estimated wobble frequencywobble_max Maximum estimated wobble frequency

The stabilisation diagram will show all identified poles, which means that the model order has notbeen taken into account. This is where, in [2, 10], the term "stable poles" is introduced. This termcould be considered misleading because it does not apply to the physical stability of the pole butit indicates that the natural frequency and damping ratio of the pole are within the user specifiedtolerances as stated in table 3.1. Whether if an identified pole can be classified as a stable poledepends on the fact if the pole exists for all order sizes. The Matlab function in which the stablepoles have been determined can be found in appendix D and a description can be found below.

7

The ’process_sys.m’-file uses the ’damp’-command in Matlab to calculate the natural frequencies,damping ratios and poles. The next step is to identify the weave and wobble mode. This hasbeen done by sorting the array with the natural frequencies and determining differences with the’diff ’-command. When the difference is greater than two times the natural frequency tolerance(’Wn_tol ’), it is considered to be the next mode. The index of these steps in modes is stored inthe ’idx ’-value. The different natural frequencies are stored in ’modes’ when they lie within thetolerances.

The next step is to associate the damping ratio and order size to the identified natural frequen-cies and aggregated for weave, wobble and other. Therefore, these are default values for theminimum and maximum estimated frequencies for the inputs in table 3.1. As mentioned before,typical weave frequencies lie within a frequency range of 2-4 Hz and characteristic frequenciesfor wobble are 6-10 Hz. The final step is to remove incorrect poles, called "spurious poles"[2], bychecking the damping ratio. If the damping ratio of a, seemingly, stable pole does not correlateto the damping ratio of the identified poles at different order sizes, it is removed. In other words,if a pole that has been classified as a weave pole does not correspond to the average dampingratio for weave poles, it is not stored in the weave structure array. The same approach has beenapplied to the wobble mode.

3.2 Comparison between methods

The results for the COV method, on the hands-off data at 80 mph, can be found in figure 3.1. Infigure 3.1a, the stabilisation diagram is shown. There are three clear modes identified as shownby vertical lines of blue stars. The vertical lines of red circles are considered "spurious poles",due to deviating damping ratios the have not been identified as "stable poles". The first one isthe weave mode around 3 Hz, the second is the wobble mode around 8.5 Hz and the third modeat 17 Hz corresponds to the wheel speed which is represented by the dashed line. In order tocalculate the wheel speed, a wheel rolling radius of 0.33 m has been assumed. In figure 3.1b,a close up of the weave mode is shown as well as the corresponding damping ratio. It becomesclear that the best approximation has been found between order sizes 15 and 25 because thedeviation in the damping ratio is very small.

0 5 10 15 20 250

5

10

15

20

25

30

Pole natural frequency [Hz]

Mod

el o

rder

(i)

(a) Stabilisation diagram with SSI-COV method

2.8 3 3.2 3.4 3.60

5

10

15

20

25

30

Ord

er s

ize

Frequency [Hz]0.1 0.2 0.3

0

5

10

15

20

25

30

Damping ratio

(b) Weave mode frequency and damping ratio

Figure 3.1: Results with COV method with hands-off data at 80 mph

Figure 3.2 shows the results for the CVA method with the same data as above. The weave

8

mode has been identified from model order 8 up to 30 but the best approximation is, again, foundbetween 15 and 25 Hz. This can be made clear from the damping ratio which is fairly constantbetween these order sizes. The identification of the wobble mode is not as successful as with theCOV method, as shown in the stabilisation diagram. The wheel speed has not been identified bythe algorithm due to the results interfering with each other. The interference can be seen in figure3.2a, where at some order sizes, two circles are shown at 17 Hz and therefore the mode has notbeen identified by the algorithm.

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(a) Stabilisation diagram with SSI-CVA method

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Figure 3.2: Results with CVA method with hands-off data at 80 mph

The results for the UPC method are shown in figure 3.3 and high similarity can be seen comparedto the CVA method. This is caused by the the fact that both methods are based on the samecalculation.

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(a) Stabilisation diagram with SSI-UPC method

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Figure 3.3: Results with UPC method with hands-off data at 80 mph

9

3.3 Reducing calculation time

For both methods, SSI-COV and SSI-DATA, the possibility exists to implement a computationalshortcut which provides a significant reduction in calculation time. This shortcut is based oncomputation of only the highest order of the approximation, which is the user defined "nmax" andscaling down the lower order sizes. In case of the SSI-COV method, the SVD is only calculated forthe highest order and the output is scaled down to the lower desired order sizes. For SSI-DATA,the LQ-decomposition and SVD are carried out at only the highest order size and truncated forthe desired lower order sizes. Figure 3.4 shows a comparison between methods with the shortcutimplemented. It becomes clear that the SSI-COV method produces the best results because thedeviation in frequency and damping ratio is smaller compared to the CVA and UPC method. Itcan be seen that the shortcut provides better results than SSI-COV without the shortcut. This canbe clarified by the fact that only a single SVD has been carried out at the highest order size. Theidentification at smaller order sizes has been derived from the highest order size and therefore,the results will be closer to each other. In the original algorithm, the SVD will be carried out atevery order size between "nmin" and "nmax".

2.8 3 3.2 3.4 3.6 3.80

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(a) The Covariance driven method

2.8 3 3.2 3.4 3.6 3.80

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(b) The Canonical Variate Analysis method

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(c) The Unweighted Principal Component method

Figure 3.4: Results for the weave mode including shortcut

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3.3.1 Computational gain

A comparison between methods, with and without shortcut, is shown in table 3.2. Five runshave been performed for each method and the average is shown and it can be made clear thatthe shortcut is very beneficial compared to the original method. Table 3.3 gives the computerspecifications.

Table 3.2: Different methods and corresponding computation times

Method Computational time [s]COV 5.27COV shortcut 1.59CVA 7.29CVA shortcut 1.79UPC 7.25UPC shortcut 1.70

Table 3.3: Computer specifications

Operating system Windows 8 Pro Version 6.2 32 bitsProcessor Intel Core 2 Duo CPU @ 2.20GHzMemory 2048MBMatlab version 8.1.0.604 (R2013a)

3.4 Hands-off compared to Hands-on the handlebar

The SSI-COV method including shortcut has been used for different speeds and for the hands-offand hands-on situation. The result is shown in figure 3.5. On the horizontal axis, the speedsare shown, starting at "Speed 1" increasing in steps of 10 mph (16 km/h) until "Speed 4". Forboth situations, the natural frequency increases and the damping decreases for increasing speedwhich corresponds to weave mode theory. As can be seen below, the damping ratio for the hands-on case is lower than the hands-off situation. The hands-on situation could be compared to amotorcycle equipped with a steering damper which is known to decrease weave mode stability[3]. This accounts for the weave mode only, as can be seen by the frequency which is around 3Hz. For the wobble mode, the hands-on case has a higher damping ratio than the hands-off casedue to the fact that the wobble mode is a pure rotation of the steering assembly.

3.5 Effect of shorter data series

In [2], it is mentioned that a 5-10 minutes of data is necessary to get a good identification. Theoriginal steering angle data set at speed 2 is equal to twelve minutes and in figure 3.6, six minutesand three minutes are shown. It can be pointed out that less natural frequencies are beingidentified with the shorter data series. The stabilisation diagrams show less clearly identifiedmodes than the original with twelve minutes of data.

11

1 2 3 41

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ping

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io [%

]

Hands−Off FrequencyHands−On FrequencyHands−Off DampingHands−On Damping

Figure 3.5: Comparison between hands-off and hands-on situation

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(a) Six minutes of data

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(b) Three minutes of data

Figure 3.6: Stabilisation diagrams with COV-method and shortcut for data series of shorter time

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3.6 Summary

In this chapter, the different SSI methods as provided in chapter 3, have been compared. Thecovariance driven method shows the most promising results, with and without implemented short-cut. Also, different experiments have been compared, namely the hands-off and hands-on situa-tion. It can be concluded that the hands-on situation is less stable for the weave mode. In the lastsection of this chapter, the effect of shorter input data has been examined. It has become clearthat the identification is less accurate as mentioned in [2].

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4 Experiments

In order to test the SSI algorithm for inputs with quality that varies, a new data set had to beacquired. This has been done by attaching a 2D datalogger with accelerometers to a SuzukiSV650S and a Kawasaki Tengai 650. The results of the tests are discussed in this chapter. Bothmotorcycles are shown in figure 4.1.

(a) Suzuki SV650S test motorcycle (b) Kawasaki Tengai 650 test motorcycle

Figure 4.1: Testbikes

4.1 Suzuki SV650S

The Suzuki SV650S test motorcycle has been produced in 2003, it is a sports-tour motorcyclefor a reasonable budget. The fact that the purchase price is not very high has an effect on thesuspension of the bike which is very soft in its original settings. On this particular motorcycle,the front suspension has been replaced with Showa suspension. In test trim, the motorcycle hasbeen fitted with luggage in order to decrease the weave and wobble stability. This has been doneto increase the chance of capturing the wobble or weave mode due to the increased possibility ofoccurrence. In table 4.1, specifications of the Suzuki motorcycle can be found.

Table 4.1: Suzuki SV650S specifications

Engine power 54.7 kWEngine torque 64 NmWheelbase 1440 mmWeight 198 kgFront tyre Bridgestone S20 120/60-ZR17Rear tyre Bridgestone T30 160/60-ZR17

15

4.1.1 Testing

Different experiments have been carried out with this motorcycle. The first test is a straightdriving test at constant speed with hands-on the handlebar. The data has been recorded at threedifferent velocities namely, 90 km/h, 70 km/h and 50 km/h. The second experiment is a coast-down in which a wobble input has been applied to the handlebar with hands-off the handlebar atan initial speed of 45 km/h.

The speed profile straight driving tests are shown in figure 4.2. As can be seen, the first oneconsists of 7 minutes of data and an average speed of 88 km/h. and 9 minutes of data with andaverage speed of 71 km/h, respectively. The 50 km/h test has been performed on straight citystreets and can also be found in figure 4.2. It consists of 6.5 minutes of data and the averagespeed is equal to 49 km/h.

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(c) 50 km/h test

Figure 4.2: Speed profile of the straight driving tests for different speeds

The speed profile of the coast down has been plotted in figure 4.3 along with a part of the rota-tional speed about the steering axis (Gz) with a wobble input.

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Gz

[deg

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(b) Rotational speed about the steering axis

Figure 4.3: Details of the coast-down experiment

4.1.2 Analysis

The quality of the data captured by the data logger on the Suzuki is not as high as the dataprovided by Harley Davidson. This can be concluded by looking at the stabilisation diagrams infigure 4.4. The algorithm used is the COV method because this method produced the best resultswith the Harley Davidson data. Figure 4.4a shows the stabilisation diagram of the 90 km/h test,it can be seen that the identification is not succesful because there are very little "stable poles"visible. The identified points around 9.5 Hz correspond to the wobble mode but has not beenidentified for order sizes below 15. In figure 4.4b, the 70 km/h test is shown and one very clearmode can be seen, around 11.5 Hz. It does not seem to be a pure wobble or weave mode thoughand it does not correspond to the wheel speed as well. A possible explanation could be the factthat the wobble frequency can change on acceleration and deceleration [6]. Since the speedis not exactly constant, a higher frequency wobble could have been captured on decelerating.Further experiments can point out the cause of the increased wobble frequency.

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Figure 4.4: Stabilisation diagram of the straight driving test at different speeds

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In figure 4.5, the stabilisation diagrams of the 50 km/h and coast-down test are shown. The dataof the 50 km/h test is very noisy due to a different test location from the 90 km/h and 70 km/htest and therefore figure 4.5a does not show any results. The coast down test on the other handshows a very promising mode around 9.5 Hz. Since a small wobble input has been applied to thehandlebar, the wobble frequency is clearly captured by the algorithm and therefore this is visiblein the stabilisation diagram. It can be concluded that the wobble mode of the Suzuki SV650motorcycle at the configuration in which it has been tested, is present at 9.5 Hz. The 90 km/h testshows signs of the wobble being present and the coast-down test confirms this once more.

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(a) Straight driving test at 50 km/h

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Figure 4.5: Stabilisation diagrams

A remark has to be made about the coast-down identification. The captured data consists of anexternal input because the wobble had to be excited by the testrider, as shown in figure 4.3b. TheSSI method has been designed for input data without any excitation by the rider and therefore ithas to be checked if the SSI algorithm can be used for identification in the coast-down experiment.This has been done by fitting the steering angle by the function

S(t) = S0e(−ωnζ t)· sin

[(

√1− ζ 2)t − φ

](4.1)

In figure 4.6, one excitation of the handlebar has been plotted along with the calculation of thedamping ratio and two fitted curves. The data contains some noise and therefore, the fit doesnot correspond perfectly. However, the damping ratio and the natural frequency can be extracted.The natural frequency of the data is equal to that of the fitted curve and has a value of 9.18Hz with a damping ratio of 0.0764. The natural frequency of the SSI is equal to 9.62 Hz with adamping ratio of 0.0763, as shown in figure 4.7. The difference in natural frequencies is causedby the length of the data series, which is very short for one excitation compared to the total 120seconds. Therefore, it can be confirmed that the SSI algorithm can be used to identify the wobblemode with sufficient accuracy in the coast down test, in which an input has been applied to thehandlebar.

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−60

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DataPeaksDamping curve

(a) The coast-down data along with the determina-tion of the damping ratio

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DataFitted curveFit with SSI−result

(b) Plot of the original data, fitted curve and result ofthe SSI

Figure 4.6: Stabilisation diagrams

9.2 9.4 9.6 9.8 100

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Figure 4.7: Wobble mode frequency and damping ratio for coast-down experiment

4.2 Kawasaki KLR 650 Tengai

The Kawasaki KLR 650 Tengai is a motorcycle in the adventure touring segment. It has beenproduced in 1989 and is known for being light on the rear wheel. It is often compared to Honda’s

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much more famous Transalp though the handling feels much lighter. The specifications for theTengai can be found in table 4.2.

Table 4.2: Kawasaki KLR 650 Tengai specifications

Engine power 35.8 kWEngine torque 55 NmWheelbase 1496 mmWeight 182 kgFront tyre Heidenau K60 scout M+S 90/90-ZR21Rear tyre Heidenau K60 scout M+S 130/80-ZR17

4.2.1 Testing

This motorcycle has been tested with the focus on identifying the weave mode because thismotorcycle is experiencing weave. A data logger with accelerometers has been attached to therear frame of the motorcycle and around eight minutes of data has been gathered at 130 km/h asshown in figure 4.8 in which Gz is the rotational speed around the vertical axis of the rear frame.

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(b) Part of the captured data of Gz

Figure 4.8: Gathered data with KLR 650 testing

4.2.2 Analysis

The stabilisation diagram of the eight minutes test at 130 km/h is shown in figure 4.9 along withthe weave frequency and damping ratio. The identification of the weave motion is very clear andit can be seen that the damping ratio for the weave of the Tengai is very low. Two other tests havebeen carried out with the Kawasaki Tengai and the results are shown in figure 4.10. They consistof only one minute of data but due to the low damped weave on this particular motorcycle, theresults are still good. The weave mode has been identified as good as the long dataset at 130km/h at both speeds. The only difference is the identification of the wheel speed. In the 110 km/htest, there is a stable pole at the dashed line but in the 130 km/h test there are no poles at all atthe wheel speed, represented by the vertical dashed line. An explanation for this difference is thefairly constant speed of the first test compared to a more noisy speed profile of the 130 km/h test.

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(a) Stabilisation diagram of the KLR 650 at 130 km/h

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(b) Close up of the weave mode and weave damping

Figure 4.9: Results of the KLR 650 test

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Figure 4.10: Stabilisation diagrams of the Kawasaki Tengai at different speeds

4.3 Summary

In this chapter, the results of experiments with two motorcycles in in different segments havebeen shown. The Suzuki shows a very clear wobble mode on a coast-down test and a very clearweave mode has been identified on the Kawasaki. It can be concluded that the SSI algorithmalso computes good identifications on other experimental data than the data provided by HarleyDavidson. Therefore, it could be used in future experiments as well.

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5 Sharp model in FastBike

One of the first studies on lateral motions of motorcycles has been executed by R.S. Sharp in1971 [12]. It was followed by his paper in 1994 [13] and this model has been programmed in theAutosim package by S. Evangelou and D. Limebeer [8], referred to as the LISP model. Furthermodeling on motorcycle weave and wobble has been done with FastBike [7], non-linear multibodysoftware specialized on motorcycle dynamics developed at DIMEG. In Fastbike, three differentkind of simulations are possible: steady state simulations, stability and frequency domain simu-lations and time domain simulations. The model includes both rigid and flexible bodies, differentkind of suspension linkages, a special model for the road-tyre interaction and a transmissionmodel which includes the compliance as well. Therefore, all kinds of motorcycles can be mod-eled and the original model by R.S. Sharp of 1994 has been implemented in FastBike and acomparison has been made.

5.1 Model description

The "Sharp 1994" model consists of the following bodies:

• Handlebars, front forks and front wheel.

• Rear frame containing the engine, the rider’s legs and lower body.

• The rear wheel assembly

• The rider’s upper body

The rear frame is the central body to which the other bodies are connected. The arms of therider are represented by the damping and stiffness of the revolute joint between the rear frameand the front assembly. The rear wheel is connected with a hinge and the upper body of the riderwith a longitudinal hinge at saddle height. The muscular activity of the rider in remaining uprightis represented by a spring-damper system. The degrees of freedom allowed in the model are:

• Forward and lateral motion of the reference point O as shown in figure 5.1.

• Yaw of the rear frame.

• Roll of the rear frame.

• Lateral displacement of the steering axis relative to the rear frame.

• Twist displacement of the steering axis relative to the rear frame.

• Steering displacement of the front frame relative to the rear frame.

• Twist displacement of the rear wheel assembly relative to the rear frame

• Roll displacement of the rider’s upper body relative to the rear frame.

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The tyre side force, aligning moment and overturning moment are included proportional to theside-slip angle. These are dependent on the tyre load which is affected by aerodynamic effects.The tyre forces are lagged in order to add the relaxation length into the model. The camber angleis implemented as well, of which the response is not lagged because it is not dependent of tyredistortions.

In the FastBike model, the following bodies are considered:

• The chassis

• The front frame including the handlebar and the upper part of the front suspension.

• The front unsprung mass which is the lower part of the front suspension.

• The front wheel.

• The rear unsprung mass, i.e. the swing arm.

• The rear wheel.

• The lower rider, from feet to hip.

• The upper rider, from hip to head.

The degrees of freedom in de model are as follows, starting with the gross motion of the motor-cycle:

• The position and orientation of the chassis (6 DOF).

• The steering rotation.

• The front suspension travel.

• The rear suspension travel.

• The front wheel rotation.

• The rear wheel rotation.

The structural compliance requires additional degrees of freedom related to the stiffness:

• Chassis deflections (3 DOF).

• The rider lateral displacement.

• Yaw and roll angles (3 DOF).

• Deflection of the front and rear rim (4 DOF).

• Deflection of the transmission (2 DOF).

• The rear suspension linkage deflections (2 DOF).

• The steering head torsion deflection.

• The front suspension linkage deflection.

The degrees of freedom as stated above can be included or locked, which causes the amountof degrees of freedom to vary between 11 and 29 whereas the number of degrees of freedom inthe LISP model is 9. Regarding tyre modelling, the relaxation length is not directly implementedin FastBike but the tyres are modelled with a tuned lateral stiffness in order to account for therelaxation length.

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5.2 Implementation

The first part of the implementation is to convert the geometrical parameters from the LISP modelto the FastBike model, as shown in figure 5.1 and 5.2. This has to be done because both modelsuse different multi-body packages and therefore the relations between components change.

Figure 5.1: Geometry of the LISP programming model

O2_O3z

O3_O6x

O3_O6zO2_O3x

O5_C

MbMf

Mr

Mp

O3

O2O5

Figure 5.2: Geometry of the FastBike model

The first assumption made is that the rear swingarm angle, α0r , is equal to zero which corre-sponds to an ε1 of 90 degrees. The symbols used here correspond to the ones used in Fastbikeand by Sharp. The next step is to calculate the general parameters such as the swingarm length,O5_C, and the length of the front fork, O3_O6z. The dimensions in figure 5.1 are know andtherefore have been used after the following equations have been derived. The values can befound in table 5.1.

O3_O6x = R f sin(ε)− t (5.1)

O3_O6z = −t sin(ε)− s + R f cos(ε)2

cos(ε)(5.2)

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O2_O3x = −−s + Rr

tan(ε)(5.3)

O2_O3z = −s + Rr (5.4)

O5_C =l sin(ε) cos(ε)+ t sin(ε)+ b sin(ε) cos(ε)− s + Rr cos(ε)2

cos(ε) sin ε(5.5)

Table 5.1: Values for masses

Parameter Value [m]O3_O6x 0.0582O3_O6z 0.5163O2_O3x 0.8839O2_O3z -0.4490

O5_C 0.2654

Now that the main geometry is known, the center of mass of the different bodies has to betransformed to the FastBike environment. This means that they have to be expressed in the localframes as shown in figure 5.2. The distances off all masses are known with respect to the pointO in figure 5.1. The front frame assembly mass, M f , has to be expressed in local frame 3. Therear frame, Mr , and the upper rider mass, Mp, in frame 2 and the rear wheel assembly mass, Mb,in frame 5.

The next step is to enter the values of the variables such as the mass, inertia, stiffness anddamping. On the inertia of the rear frame, a remark has to be made considering the LISP programpaper [8] because it has been switched with the rider upper body inertia. Another difference isthe sign of the cross moment of inertia, Ir xz , which has to be changed as stated in the Errata ofthe LISP paper [8]. Also, in FastBike, instead of using the stiffness, the compliance has to becalculated which is one divided by the stiffness. Some assumptions have been made, namelythe steering head is considered to be rigid because the influence on weave and wobble can beneglected which will be shown later. For simplicity, at first aerodynamics have not been taken intoaccount as well as the suspension, which has been locked. The implemented parameters alongwith the corresponding values can be found in the tables below and [8].

Table 5.2: Values for masses

Parameter Symbol Value [kg]Mass of front frame M f 40.59Mass of rear frame M f 170.3Mass of rear wheel assembly Mb 25.0Mass of rider upper body Mp 50.0

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Table 5.3: Values for moments of inertia

Parameter Symbol Value [kg·m2]Front frame inertia about x-axis I f x 3.97Front frame inertia about z-axis I f z hands-off: 0.71

hands-on: 0.91Front frame inertia product I f xz 0.0Rear frame inertia about x-axis Ir x 7.43Rear frame inertia about z-axis Ir z 11.63Rear frame inertia product Ir xz -7.4Rider upper body inertia about x-axis Ipx 1.96Rider upper body inertia about z-axis Ipz 0.55Rider upper body inertia product Ipxz -0.26Rear wheel inertia about x-axis irwx 0.4Rear wheel spin inertia irwy 0.65Front wheel spin inertia i fwy 0.58Effective engine flywheel inertia ir y 0.41

Table 5.4: Values for compliances

Parameter Symbol Value [rad/Nm]Steering head lateral compliance Sv 9.615·10−7

Steering head torsional twist compliance Sγ 1.634·10−5

Rider upper body restraint compliance Sζ 1.0·10−4

Rear wheel assembly twist compliance Sλ 2.174·10−5

Table 5.5: Values for damping

Parameter Symbol Value [Nm·s/rad]Steering head lateral damping Dv 456Steering head torsional twist damping Dγ 44.1Rider upper body restraint damping Dζ 156Rear wheel assembly twist damping Dλ 17.7

Table 5.6: Other values

Parameter Symbol ValueSteering stiffness ksteer hands-off: 0.0 Nm/rad

hands-on: 50.0 Nm/radSteer damper dsteer hands-off: 1.0 Nm·s/rad

hands-on: 6.0 Nm·s/radAerodynamic drag coefficient Dc 0.377 m2

Aerodynamic lift coefficient Lc 0.05 m2

The tyre properties are the next values to be implemented. The tyre data to fit can be found inthe LISP programming paper [8], the equations below and are shown in figures 5.3a-5.5b. Ascan be seen, the quality of the fit varies between properties. This is caused by the fact whetherthe fit has to be used in FastBike or Matlab. The way the fit is implemented in FastBike limits thequality of some fits, for example the camber stiffness which is approximated by a linear equation.

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As shown in the equations below, the tyre properties are dependent on the normal tyre forces,Z f and Zr .

C f v f = −300+ 28.577Z f − 0.0144Z2f + 1.431 · 10−6 Z3

f + 3.347 · 10−10 Z4f (5.6)

C f vr = −92.9+ 23.129Zr −4.6631000

Z2r − 6.457 · 10−7 Z3

r + 1.887 · 10−10 Z4r (5.7)

Cmv f = −0.281+ 0.2442Z f + 8.575 · 10−5 Z2f (5.8)

Cmvr = 9+ 0.3573Zr + 3.378 · 10−5 Z2r (5.9)

C f 1 = −13.25+ 1.302Z f − 1.39 · 10−4 Z2f (5.10)

Cr1 = 27.38+ 0.9727Zr − 4 · 10−6 Z2r (5.11)

C f 2 = 2.788+ 0.0165Z f + 3.9 · 10−6 Z2f (5.12)

Cr2 = 2.056+ 0.01282Zr + 4.928 ·−6 Z2r (5.13)

σ f = 0.1012+ 1.297 · 10−4 Z f − 3.267 · 10−8 Z2f (5.14)

σr = 0.03594+ 1.941 · 10−4 Zr − 5.667 · 10−8 Z2r + 5.728 · 10−12 Z3

r (5.15)

Cy f =C f v f

σ f(5.16)

Cyr =C f vr

σr(5.17)

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

30

Tyre load [kN]

Cor

nerin

g st

iffne

ss [k

N/r

ad]

Rear tyre dataRear tyre fitFront tyre dataFront tyre fit

(a) Cornering stiffness, C f v .

0 0.5 1 1.5 2 2.5 3 3.5 4−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

Tyre load [kN]

Alig

ning

mom

ent c

oeffi

cien

ts [k

Nm

/rad

]


(b) Aligning moment stiffness, Cmv .

Figure 5.3: Front and rear tyre data and fitted curves.

28

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

Tyre load [kN]

Cam

ber

stiff

ness

[kN

/rad

]


(a) Camber stiffness, C1.

0 0.5 1 1.5 2 2.5 3 3.5 40

20

40

60

80

100

120

Tyre force [kN]

Tw

istin

g st

iffne

ss [N

m/r

ad]


(b) Twisting stiffness, C2.


0 0.5 1 1.5 2 2.5 3 3.5 40

0.05

0.1

0.15

0.2

0.25

0.3

Tyre load [kN]

Rel

axat

ion

leng

th [m

]


(a) Relaxation length, σ .

0 0.5 1 1.5 2 2.5 3 3.5 40

20

40

60

80

100

120

Tyre load [kN]

Late

ral s

tifne

ss [N

/m r

ad]

Rear lateral stiffness dataRear lateral stiffness fitFront lateral stiffness dataFront lateral stiffness fit

(b) Lateral stiffness, Cy .


5.3 Results

After all parameters have been implemented, simulations can be made and the results are com-pared to the original Sharp model. The root-loci for hands-off and hands-on can be found infigure 5.6 and they show a lot of similarity with the original model. The root-loci are based onlinearized models. In Fastbike, first the equilibrium trim at a certain speed is computed by solvingthe equations in steady state condition and the state space matrices are found after linearisationof the system about this configuration [7]. In [8], two models are used, one non-linear and a linearmodel which is used to create the root-locus. The model is linearized using an equilibrium statein which the motorcycle is moving with constant forward speed. In the figures below, the modelhas been simplified by neglecting aerodynamics, suspension and steering head stiffness. Thenext step is to implement these elements as well.

29

−40 −35 −30 −25 −20 −15 −10 −5 0 50

2

4

6

8

10

12

Real parts [rad/s]

Fre

quen

cy [H

z]

Weave

Wobble

Fastbike 2013Sharp 1994

(a) Hands-off the handlebar

−40 −35 −30 −25 −20 −15 −10 −5 0 50

2

4

6

8

10

12

Real parts [rad/s]

Fre

quen

cy [H

z]

Weave

Wobble


(b) Hands-on the handlebar

Figure 5.6: Root-loci for weave and wobble

First, the aerodynamics have been added, the parameters can be found in table 5.6. In figure5.7, the normal tyre force as a function of speed is shown for the front and rear tyre. Withoutaerodynamics, the normal tyre forces do not change over speed. With aerodynamics though,the normal tyre forces become dependent on forward velocity. Due to the aerodynamic lift of themotorcycle, the weight distribution shifts to the rear of the bike and therefore the normal forceschange.

Due to this change in normal force, the root-loci change as well as shown in figure 5.8. Thebiggest difference is the wobble frequency at higher speeds which is lower than without aerody-namics. This can be explained by looking back at figure 5.7 because the normal forces changeat higher speeds as well. Comparing [13] and FastBike, it is noticeable that the results show ahigher similarity.

0 10 20 30 40 50 60

800

1000

1200

1400

1600

1800

2000

Speed [m/s]

Nor

mal

forc

e [N

]

Front tyreRear tyre

(a) Without aerodynamics

0 10 20 30 40 50 60

800

1000

1200

1400

1600

1800

2000

Speed [m/s]

Nor

mal

forc

e [N

]

Front tyreRear tyre

(b) With aerodynamics

Figure 5.7: Normal tyre force for front and rear tyre for increasing speed

30

−40 −35 −30 −25 −20 −15 −10 −5 0 50

2

4

6

8

10

12

Real parts [rad/s]

Fre

quen

cy [H

z]

Weave

Wobble



−40 −35 −30 −25 −20 −15 −10 −5 0 50

2

4

6

8

10

12

Real parts [rad/s]

Fre

quen

cy [H

z]

Weave

Wobble



Figure 5.8: Root-loci for weave and wobble including aerodynamics

Figure 5.9 shows the results of the original by R.S. Sharp [13] simulation along with the results ofa modified simulation with a rigid steering head. As can be seen, the original simulation consistsof an extra mode with a frequency around 48 Hz which is the lateral motion of the steering head.As shown, the lateral flexibility of the steering head does not have an influence on the wobble andweave of the motorcycle and therefore can be neglected.

Another assumption has been the locked suspension because there are no values known forsuspension parameters of the motorcycle. Comparing figures 5.8 and 5.10, it becomes clear thatthe suspension does not result in a big difference. This means that the locked suspension is avalid assumption and more probable since no suspension stiffness and damping is known.

−40 −35 −30 −25 −20 −15 −10 −5 0 50

10

20

30

40

50

60

Fre

quen

cy [H

z]

Real parts [rad/s]

Steering head lateral motion

(a) Original Sharp model with steering head flexibilityin lateral direction

−40 −35 −30 −25 −20 −15 −10 −5 0 50

10

20

30

40

50

60

Fre

quen

cy [H

z]

Real parts [rad/s]

(b) Modified Sharp model with rigid steering head inlateral direction

Figure 5.9: Root-loci for weave and wobble

31

−40 −35 −30 −25 −20 −15 −10 −5 0 50

2

4

6

8

10

12

Real parts [rad/s]

Fre

quen

cy [H

z]

Weave

Wobble



−40 −35 −30 −25 −20 −15 −10 −5 0 50

2

4

6

8

10

12

Real parts [rad/s]

Fre

quen

cy [H

z]

Weave

Wobble



Figure 5.10: Root-loci for weave and wobble with unlocked suspension

5.4 Summary

In this chapter, the model in [13] has been validated using [8] and Fastbike. It can be concludedthat the "Sharp 1994" model shows similar results as an advanced multibody model developed inrecent years.

Looking back at figure 3.5, it can be seen that the same stability properties can be seen asfigure 5.10. The hands-off damping for weave is lower than the hands-on case and vice-versa forwobble which is less damped in the hands-off case. Also, the wobble frequency fluctuates withforward velocity as shown in figure 4.4.

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6 Conclusions and Recommendations

This report has approached motorcycle stability, in particular wobble and weave, from differentangles. In the first part, an algorithm has been developed in order to identify lateral motionsfrom experimental data. In the second part, a comparison has been made between the Sharpmotorcycle stability model from 1994 [13] and Fastbike.

6.1 Stochastic Subspace Identification

Three different algorithms for stochastic subspace identification have been implemented in Mat-lab, tested and compared. The method that produces the best results on the Harley Davidsondata is SSI-COV based on SVD with a computational shortcut. The best weave mode identifi-cation has been calculated as shown in chapter four. However, the quality of the identificationdepends on the captured data from the experiment. It became clear that the data provided byHarley Davidson was of higher quality than the data gathered with the experiments.

For the same Harley Davidson data J.C. Brendelson and A.K. Dhingra [2] found slightly differentresults because their findings show that the CVA method is superior. Although the differencesare small, one would expect the same results. The SSI implementation is the same but theconstruction of the stabilisation diagrams, in particular the boundaries and selection for "stablepoles", is different and this explains the distinction in the results.

For a good identification, a minimum of ten minutes of data is necessary. Otherwise, modes arenot recognized or even clearly calculated by the SSI method. An exception can be made when asmall input is provided and a typical motion is forced to happen as shown in section 5.1.2.

6.2 Sharp 1994 model and FastBike comparison

After implementing the Sharp model in FastBike by using the parameters given by [8] and as-sumptions being validated, a comparison has been made. The results show that the 1994 Sharpanalysis has been a very good baseline because the more developed FastBike software showssimilar results.

A remark has to be made concerning tyre properties, which are not clear from the Sharp modeland do not correspond to the LISP programming paper. The equations for the tyre propertiesin the Sharp paper do not correspond to the figures which has been stated in the LISP paperas well. The small difference that has been found is likely to be caused by the tyre modeling,also because small changes in tyre cornering stiffness provide a significant change in motorcycleweave stability.

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6.3 Recommendations

In order to make a better comparison between the identifications of the Harley Davidson data inchapter 3.2 and the experiments in chapter 4, capturing the data should be carried out in a moresimilar manner. In other words, the data gathering experiment should take place on a closedtrack which contains a straight in which several minutes of straight driving at a constant speedcould be captured because the experiment has to be carried out under steady-state conditions.Also, the rider on both motorcycles has to be the same instead of 2 different riders which havebeen used in order to acquire the data in chapter 4.

In chapter 5, the way in which the hands-off and hands-on cases have been programmed inFastbike are different due to difficulties in the implementation. In order to implement the hands-off case, additional lines of code had to be added because in standard trim, the rider is modeledwith hands-on the handlebars. After implementation it is made possible to remove the couplingbetween the handlebar and the rider. With the additional code, the hands-on situation shouldalso be possible to simulate but the results did not correspond to the previous calculations whichcould be considered to be correct. Therefore, in case of the hands-on simulations, the additionallines have not been implemented and the previous version has been used. It could be consideredadequate to implement the additional code and find proper results in the hands-on case as well.

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7 Appendices

A Matlab file ssicov.m

The following code shows the Matlab implementation of the covariance driven SSI method.

function [sys] = ssicov(y , dT, nmin, nmax)%This function approximates measuring data with a (partial) state

space system as output. The computational method used is Covariancedriven SSI.

%% Define orderN = size(y,1);l = size(y,2);for i = nmin:nmax%% Compute Hankel parts%% Ypcp = 0:i-1;rp = i-1:N-i-1;Yp_idx = hankel(cp,rp)+1;Yp = zeros(size(Yp_idx));for k = 1:size(Yp_idx,1)

for m = 1:size(Yp_idx,2)Yp(k,m) = y(Yp_idx(k,m));

endend%% Yfcf = i:2*i-1;rf = 2*i-1:N-1;Yf_idx = hankel(cf,rf)+1;Yf = zeros(size(Yf_idx));for k = 1:size(Yf_idx,1)

for m = 1:size(Yf_idx,2)Yf(k,m) = y(Yf_idx(k,m));

endend%% Compute Covariance matrixCi = (1/N)*(Yf*Yp');%% Singular value decomposition[U1, S1, V1] = svd(Ci);%% FactoringT = eye(size(U1));Gammai = U1*sqrt(S1)*T;Deltaci = inv(T)*sqrt(S1)*V1';%% Output

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n = size(y,2);A = pinv(Gammai(1:l*(i-1),:))*Gammai(l+1:l*i,:);B = [];C = Gammai(1:l,:);D = [];sys(i).ss = ss(A,B,C,D,dT);endend

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B Matlab file ssicva.m

The Matlab code below shows the function for the Data driven SSI method, the Canonical VariateAnalysis in particular.

function [sys] = ssicva(y , dT, nmin, nmax)%This function approximates measuring data with a (partial) state

space system as output. The computational method used is CanonicalVariate Analysis (CVA)

%% Define orderl = size(y,2);for i = nmin:nmax%% Compute Hankel matrixc = 0:2*i-1;r = 2*i-1:length(y)-1;H_ind = hankel(c,r)+1;Ha = eye(size(H_ind));for k = 1:size(H_ind,1)

for m = 1:size(H_ind,2)Ha(k,m) = y(H_ind(k,m));

endend%% LQ decompositionL1 = triu(qr(Ha'))';m = l*i + l + i-1;n = l*i + l + l*(i-1);L1 = L1(1:m,1:n);%% Split up matrix L1L21 = L1(i+l,1:i);L22 = L1(i+l,i+l);L31 = L1(i+2*l:2*i,1:i);L32 = L1(i+2*l:2*i,i+l);L33 = L1(i+2*l:2*i,i+2*l:2*i);%% Find weighting matrixL_part = [L21 L22 zeros(l,i-l); L31 L32 L33];L2 = triu(qr(L_part'))';L2 = L2(1:i,1:i);W = inv(L2);%% Singular value decomposition[U1, S1, V1] = svd(L2\[L21;L31]);%% Calcutate the system matricesinvWU1 = W\U1;invWU1 = invWU1(1:end-1,:);AC = [pinv(invWU1*sqrt(S1))*L31; L21]*pinv([L21;L31])*(W\U1*sqrt(S1));%% Outputn = size(y,2);A = AC(1:end-n,:);B = [];C = AC(end-n+1,:);D = [];sys(i).ss = ss(A,B,C,D,dT);endend

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C Matlab file ssiupc.m

In this appendix, the implementation of the Matlab function for the Unweighted Principal Compo-nent method can be found.

function [sys] = ssiupc(y , dT, nmin, nmax)%This function approximates measuring data with a (partial) state

space system as output. The computational method used is UnweightedPrincipal Component (UPC)

%% Define orderl = size(y,2);for i = nmin:nmax%% Compute Hankel matrixc = 0:2*i-1;r = 2*i-1:length(y)-1;H_ind = hankel(c,r)+1;Ha = eye(size(H_ind));for k = 1:size(H_ind,1)

for m = 1:size(H_ind,2)Ha(k,m) = y(H_ind(k,m));

endend%% LQ decompositionL1 = triu(qr(Ha'))';m = l*i + l + i-1;n = l*i + l + l*(i-1);L1 = L1(1:m,1:n);%% Split up matrix L1L21 = L1(i+l,1:i);L22 = L1(i+l,i+l);L31 = L1(i+2*l:2*i,1:i);L32 = L1(i+2*l:2*i,i+l);L33 = L1(i+2*l:2*i,i+2*l:2*i);%% Find weighting matrixW = eye(size([L21;L31]));%% Singular value decomposition[U1, S1, V1] = svd(W*[L21;L31]);%% Calcutate the system matricesinvWU1 = W\U1;invWU1 = invWU1(1:end-1,:);AC = [pinv(invWU1*sqrt(S1))*L31; L21]*pinv([L21;L31])*(W\U1*sqrt(S1));%% Outputn = size(y,2);A = AC(1:end-n,:);B = [];C = AC(end-n+1,:);D = [];sys(i).ss = ss(A,B,C,D,dT);endend

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D Matlab file process_sys.m

This appendix shows the Matlab file process_sys which is used to identify modes.

function [weave, wobble, other, all] = process_sys(sys, Wn_tol,zeta_tol,...factor, weave_min, weave_max, wobble_min, wobble_max)

%%% % Calculate natural frequency and damping ratio for different values

of% % the order size for construction of a stabilisation diagram.%%for i = 1:length(sys)

[Wn_t, zeta_t, P_t] = damp(sys(i).ss); %Use damp commandfor j = 1:length(Wn_t)

Wn(j,i) = Wn_t(j)/2/pi; % from rad/s to Hzzeta(j,i) = zeta_t(j);P(j,i) = P_t(j);

endendWn(Wn==0) = nan;%%mode = Wn;mode(mode == 0) = nan;r_damp = zeta;r_damp(r_damp == 0) = nan;%%mode(isnan(mode)) = []; % Remove nan valuesmode = sort(mode, 'ascend'); % Sort frequenciesmode = mode(1:2:length(mode)); % Remove double inputsdmode = diff(mode); % Determine changes in natural frequenciesidx = find(dmode>2*Wn_tol); % Get index of changesidx = [0 idx idx(end)+1];for i = 1:length(idx)-1

modes_t = mode(idx(i)+1:idx(i+1)); % Collect 1 frequencyWn_mean = mean(modes_t);if length(modes_t) > length(sys)/factor

for j = 1:length(modes_t)if modes_t(j)>Wn_mean-Wn_tol&&modes_t(j)<Wn_mean+Wn_tol% Natural frequency is considered an actual natural frequency% when present in at least '1/factor' of identifications% and be within 'Wn_tol' of the mean valuemodes(j,i) = modes_t(j);end

endend

endmodes(modes==0)=[];%%for i = 1:length(modes)

[row, col] = find(Wn == modes(i));order(i) = col(1); % Get associated order sizefor j = 1:length(row)

damps_t(j) = zeta(row(j),col(j)); % Get associated damping ratio

39

endif size(damps_t)>1

if damps_t(1) == damps_t(2)zetas(i) = damps_t(1);

elsezetas(i) = [];modes(i) = [];

endelse zetas(i) = damps_t(1);end

end%%% % Pre-allocation for speed % %Wn_weave_t = zeros(size(modes));zeta_weave_t = zeros(size(modes));i_weave_t = zeros(size(modes));Wn_wobble_t = zeros(size(modes));zeta_wobble_t = zeros(size(modes));i_wobble_t= zeros(size(modes));Wn_other = zeros(size(modes));i_other = zeros(size(modes));

for i = 1:length(modes)if modes(i)>weave_min&&modes(i)<weave_max % Aggregate weave

Wn_weave_t(i) = modes(i);zeta_weave_t(i) = zetas(i);i_weave_t(i) = order(i);

elseif modes(i)>wobble_min&&modes(i)<wobble_max % Aggregate wobbleWn_wobble_t(i) = modes(i);zeta_wobble_t(i) = zetas(i);i_wobble_t(i) = order(i);

elseWn_other(i) = modes(i); % Aggregate otheri_other(i) = order(i);

endend

% % Remove empty values % %Wn_weave_t(Wn_weave_t==0) = [];zeta_weave_t(zeta_weave_t==0) = [];i_weave_t(i_weave_t==0) = [];Wn_wobble_t(Wn_wobble_t==0) = [];zeta_wobble_t(zeta_wobble_t==0) = [];i_wobble_t(i_wobble_t==0) = [];Wn_other(Wn_other==0) = [];i_other(i_other==0) = [];

%% Check if Wn and zeta correspond for weaveweave_check = mean(zeta_weave_t);for i = 1:length(zeta_weave_t)

if zeta_weave_t(i)>weave_check-zeta_tol&&zeta_weave_t(i)<weave_check+zeta_tolWn_weave(i) = Wn_weave_t(i);zeta_weave(i) = zeta_weave_t(i);

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i_weave(i) = i_weave_t(i);end

endif exist('Wn_weave')==0

%case wobble not identifiedWn_weave (1)=0;zeta_weave(1)=0;i_weave (1)=0;

endWn_weave(Wn_weave==0) = [];zeta_weave(zeta_weave==0) = [];i_weave(i_weave==0) = [];

wobble_check = mean(zeta_wobble_t);for i = 1:length(zeta_wobble_t)

if zeta_wobble_t(i)>wobble_check-zeta_tol&&zeta_wobble_t(i)<wobble_check+zeta_tolWn_wobble(i) = Wn_wobble_t(i);zeta_wobble(i) = zeta_wobble_t(i);i_wobble(i) = i_wobble_t(i);

endendif exist('Wn_wobble')==0

%case wobble not identifiedWn_wobble (1)=0;zeta_wobble(1)=0;i_wobble (1)=0;

endWn_wobble(Wn_wobble==0) = [];zeta_wobble(zeta_wobble==0) = [];i_wobble(i_wobble==0) = [];

%% Save outputweave.Wn = Wn_weave;weave.zeta = zeta_weave;weave.i = i_weave;wobble.Wn = Wn_wobble;wobble.zeta = zeta_wobble;wobble.i = i_wobble;other.Wn = Wn_other;other.i = i_other;all.Wn = Wn;all.zeta = zeta;end

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Bibliography

[1] S. Blackfort. Lq factorization. 1999.

[2] J.C. Brendelson and A.K. Dhingra. Stochastic subspace identification applied to the weavemode of motorcycles. ASME Journal of Dynamic Systems, Measurement and Control,135:1–9, 2013.

[3] V. Cossalter. Motorcycle Dynamics, 2nd ed. 2006.

[4] V. Cossalter, A. Doria, M. Formentini, and M. Peretto. Experimental and numerical anal-ysis of the influence of tyres’ properties on the straight running stability of a sport-touringmotorcycle. Vehicle Systems Dynamics, 50(3):357–375, 2012.

[5] V. Cossalter, G. Dalla Torre, R. Lot, and M. Massaro. An advanced multibod model forthe analysis of motorcycle dynamics. 3rd ICMEM International Conference on MechanicalEngineering and Mechanics, 2009.

[6] S. Evangelou D.J.N. Limebeer, R.S. Sharp. The stability of motorcycles under accelerationand braking. Proceedings of the Institution of Mechanical Engineers, Part C, 215:1095–1109, 2001.

[7] Dynamotion. Fastbike. 2009.

[8] S. Evangelou and D.J.N. Limebeer. Lisp programming of the "sharp 1994" motorcycle model.

[9] B. De Moor K. De Cock. Subspace identification methods. UNESCO Encyclopedia of LifeSupport Systems, 1:933–979, 2003.

[10] M. Massaro, R. Lot, V. Cossalter, J. Brendelson, and J. Sadauckas. Numerical and experi-mental investigation of passive rider effects on motorcycle weave. Vehicle System Dynamics:International Journal of Vehicle Mechanics and Mobility, pages 215–217, 2012.

[11] P. Van Overschee and B. De Moor. Subspace Identification For Linear Systems: Theory Im-plementation and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands,1996.

[12] R.S. Sharp. The stability and control of motorcycles. Journal of Mechanical EngineeringScience, 13(5):316 – 329, 1971.

[13] R.S. Sharp. Vibrational modes of motorcycles and their design parameter sensitivities. Ve-hicle NVH and Refinement, pages 107 – 121, 1994.

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