Frequency analysis is of uttermost importance when dealing with controllability of a dynamic arrangement in changing working domains. This becomes relevant due to the non-linearity in damping, stiffness and inertia properties of mechanisms with broad move and under heavy external conditions, such as a motorbike running on off-road terrains, braking and transmitting traction to uneven road profiles. A relatively unproblematic but complete planar model of the vehicle on uneven terrain has been here adopted, which takes into account the whole dynamics and not-linear rigid body kinematics of the motorcycle. This complete planar model is the computational engine for running the eigenvalue analysis of the time-varying eigenproblem. Tracking the eigensolution variability along the working space becomes a relevant insight on the behaviour of the motorbike under the unlike conditions. Theoretical formulation and simulations are discussed in particular.

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Frequency assay of a motorbike under motion

conditions

A. Zanarini, E. Brugnoni

University of Bologna, Section of Mechanical Engineering,

Viale Risorgimento ii, I-40136, Bologna, Italian republic

e-postal service: a.zanarini@unibo.it

Abstract

Frequency analysis is of farthest importance when dealing with controllability of a dynamic arrangement in

changing working domains. This becomes relevant due to the not-linearity in damping, stiffness and inertia

backdrop of mechanisms with broad movement and under heavy external weather condition, such as a motorcycle

running on off-route terrains, braking and transmitting traction to uneven road profiles.

A relatively simple but complete planar model of the vehicle on uneven terrain has been here adopted,

which takes into account the whole dynamics and non-linear rigid body kinematics of the motorbike. This

consummate planar model is the computational engine for running the eigenvalue assay of the time-fivearying

eigenproblem. Tracking the eigensolution variability along the working space becomes a relevant insight on

the behaviour of the motorbike under the different conditions. Theoretical conception and simulations are

discussed in particular.

1 Introduction

It is common practice nowadays to approach complex dynamic problems like vehicle design with multibody

modelling [1, ii], retaining dynamic backdrop and generalised forces in 3D-space, albeit bringing in a high

complexity in the models and high computational time & costs. On the opposite side, when simplicity and

low price simulations are asked, extremely simplified models [3] are ofttimes used for the first assessment of

the dynamic behaviour of a generalised vehicle. The latter simple models may retain none of the kinematic,

inertial, stiffness and assimilation non-linearities of spatial mechanisms, and in which the lumped properties

are sometime not equally elementary every bit it can seem to be evaluated without solving the complete set of differential

motility equations. Indeed, similar quarter car models or similar, they oversimplify the pause kinematics

(only vertical move and pitch effectually the centroid are considered, without a precise definition of the pitch

axis due to the suspension constraints) up to the farthermost consequence that it becomes not straightforward to

introduce real role properties into the simple linear motion equation and that the models do non reflect the

variability of the auto behaviour along with the kinematic position. A better approach to codify the

linear motility model should suggest to lump the spatial kineto-static backdrop of all the parts that brand the

assembly into quantities that are evaluated past means of a linearisation of the move equations around the

working point of involvement, keeping in mind the limited validity of the lumped backdrop, which should be

re-calculated every time the system moves away from the neighbourhoods of the linearisation point due to a

modify in the kinematic configuration.

It is thus axiomatic that oversimplified models can not take into account the kinematic based 5ariability of the

system dynamics, losing the hazard to highlight the contributions that a spatial mechanism, such is a existent

2291

Figure 1: Degrees-of-freedom in the model

pause, can offering in tuning the vehicle dynamics and NVH characteristics; furthermore, this might also

induce the incorrect conclusion that all pause architectures are equivalent.

A relatively unproblematic but complete planar model of the vehicle on uneven terrain, approached by means of

Lagrangian non-linear move equations, has been here adopted; the modelled motorcycle has its rigid frame,

suspensions mechanisms, shock absorber packs and deformable tires. The shock absorber backdrop are

besides non-linear, with attending paid to the extension and rebound conditions. The kinematics of the attaching

points of the spring-damper assembly on the rear bike suspension gives a non linear behaviour to the elastic

and assimilation forces. Further, traction, braking and aerodynamic effects can be taken into account in the

model.

By means of 4 degree-of-freedoms (dofs) the non-linear dynamic response, e.m. to the travelling on united nations-

fifty-fifty terrain together with braking/traction force or vertical dispatch in the plane of the motorcycle due to

cornering, can exist integrated. The transient solution of motility equations provides all the motion conditions

that are relevant to due eastvaluate a time-varying transfer function matrix and to codify a time-5arying eigen-

trouble. At each time instant at which the movement equations are integrated a linearisation of the Lagrange

equations effectually the working point can be performed, obtaining the linearised mass, damping and stiffness

matrices, from which to calculate the transfer role matrix and formulate the eigenproblem. The instant

eigensolution can be evaluated with the properly updated backdrop, with a moderate computational toll, and

without the limits of a fourth dimension-frequency analysis based merely on the fourth dimension histories, since at that place are no constraints

on the frequency resolution due to transforming operations. The variability of the transfer role matrix

and of the eigenproblem solution tin can exist evaluated along the fourth dimension axis or transient dynamics. The eigenvalues

might testify stable damped oscillations, but as well the conditions for rising instabilities, which might non evolve

into cluttered motion just because of the express time-validity for those conditions, merely examination the potentiality to

relevant undamped oscillations; they might as well underline changes into the restraining conditions and related

unstable not-oscillatory motions. The eigenvectors might likewise bear witness the alter in motion distribution due

to the external running conditions. From the drawn analysis, the effort of the pilot or of an aiding automated

command system in reaching targets about comfort, traction and braking issues might also be assessed like in

[four].

ii Planar model of a motorbike with interruption kinematics

The motorbike is modelled by ways of 5 rigid bodies: the suspended mass (frame, engine, upper sliders

of the front end fork and airplane pilot), the rear wheel, the front wheel, the swing-arm of the rear pause, the lower

stanchions of the front fork. These bodies are connected by revolute joints or by a translational joint (forepart

fork). The tires are both considered as elastic bodies with their stiffness and damping properties. The shock

absorbers in the front fork and in the rear swing arm have also elastic and damping characteristics.

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Figure 2: Positions of remarkable points of the vehicle on the global reference frame

If the motility in the symmetry plane of the motorbike is sought, disregarding the horizontal move and the

bike rotations, the kinematics of the vehicle can exist analysed past means of just four dofs as in Fig.one: the vertical

displacement z of the suspended mass, the pitch angle θof the suspended mass, the vertical displacement zf

of the front end wheel and the vertical displacement zr of the rear wheel. The location of the contact patches of

the tires are considered by the displacements yr and yf , equally office of the basis shaping.

2.1 Displacements & kinetic energy in the motorbike assembly

The displacements of the points of interest are described in the global coordinate organisation, whose origin is

located at the contact patch between the rear tyre and the flat footing, equally in Fig.two. The location (xsa ,z

sa) T of

the pivot linking the swing-arm depends on the vertical movement and on the pitch of the suspended mass:

x sa

zsa = x Yard

z + cos θ−sin θ

sin θ cos θ . 10 sa0 − x G0

zsa0 − z0 (ane)

where (xG ,z )T is the instantaneous position of the suspended mass centroid, (xsa0 ,z

sa0) T is the constant

location of the swing-arm pin in the motorcycle frame reference when at time t=0 the latter is parallel

to the global reference, (xYard0 ,z

0) T is the initial absolute position of the suspended mass centroid, (10 sa 0−

xK0 ,z

sa0−z 0) T is the relative location of the swing-arm pivot in the motorcycle frame local reference. If the

angle θsa of the swing-arm (of lenght lsa ) to the horizontal axis is defined as θsa = arcsin(( zsa − zr ) /lsa ) −

θsa0 , being θ sa0 = arccos((x sa0 −xr ) /50 sa ), it also follows that xsa = 50sa cos( θsa +θsa0 ).

The location (xGsa ,z

Gsa) T of the swing-arm centroid is defined by:

x Gsa

zGsa = 10 r

zr + cos θ sa −sin θsa

sin θsa cos θsa . x Gsa0 − x r0

zGsa0 − zr0 (2)

The location (xGs ,z

Gs) T of the centroid of the lower stanchions in the front end fork can be defined as:

ten Gs

zGs = x f

zf + cos θ s −sin θs

sin θsouth cos θs . x Gs0 − x f0

zGs0 − zf0 (3)

where the location (tensf ,z

sf ) T of a betoken on the upper role of the forepart fork can be defined equally x sf

zsf =

x One thousand

z + cos θ−sin θ

sin θ cos θ . x sf 0

zsf 0 ,θs = θs0 + θand xf = xsf +( zsf − zf )tanorthθs .

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constant value

(xG0 ,z

0)(0. 696,0. 741)[ chiliad]

(xsa0 ,z

sa0)(0. 541,0. 471)[ m]

(tenGsa0 ,z

Gsa0)(0. 350,0. 450)[ g]

lsa 0.565[one thousand]

(xr0 ,z

r0)(0, 0. 307)[ m]

(xf0 ,z

f0)(one. 548,0. 294)[ m]

(xGs0 ,z

Gs0)(ane. 400,0. 800)[ m]

(xsf 0 ,z

sf 0)(1. 251,0. 900)[ one thousand]

θs0 26[◦]

Table 1: Kinematic constants

constant valueast

mt 250[ kg ]

grandsa viii[kg]

ms iv[kg]

mrt 12[ kg]

chiliadft ten[ kg]

It 40[ kgmtwo ]

Isa 0.3[ kgm2 ]

Is 0.1[ kgmii ]

Table two: Inertial backdrop

constant valueast

ksf 13000[ N/m]

chiliadsr 85000[ N/yard]

1000pf 172916[ N/m]

thoupr 180664[ N/m]

csf k[ Ns/m ]

csr 4000[ Ns/k ]

cpf 150[ Ns/m ]

cpr 200[ Ns/m ]

Table three: Stiffness & damping

The total kinetic energy Ekin of the system will thus exist a non-linear function of the (z,θ,z r,z

f) T dofs,

through the explicit constraints above written and disregarding the forward motion of the motorbikeastward & the

rotation of the wheels, as:

Ekin =1

2(mt ˙ ztwo + It ˙

θ2 +m sa ˙ z2

sa +I sa ˙

θii

sa +g s ˙ z2

Gs +I s ˙

θ2

s+m rt ˙ zii

r+m ft ˙ z2

f)(4)

In Eqn.four mt &It are the mass and the moment of inertia (around the centroid) of the suspended part in the

motorcycle with its pilot; msa &Isa are the mass and the moment of inertia of the swing-arm; ms &I south

are the mass and the moment of inertia of the lower stanchions of the front fork; mrt is the rear wheel&tire

mass; mft is the front bicycle&tire mass. The values of the constants in the implemented model can exist found

in Tab.1, while the inertial properties of the sub-parts are outlined in Tab.2.

2.2 Pitch axis location

The pitch axis for the suspended mass can exist easily identified if the positions of the wheels' pivots are fixed

in a kinematic compatible configuration[5, half dozen, 7, 8]; in this mode the machinery has one dof only, and the

pitch axis location (xcir ,z

cir)is found at the intersection of the line that passes through the centres of the

rotational joints on the swing-arm ((xr ,z

r)&(10 sa,z

sa)) with the line that passes through the centre of the

forepart wheel pin and that is orthogonal to the prismatic joint given by the stanchions of the front fork.

If the actual wheelbase p is evaluated as p= lsa cos θsa +( xsf − xsa )+ zsf tan θsouth , introducing the distances

fiftyr and lf of the pitch axis from the rear and front cycle pivots, information technology follows the system of closure equations that

describes the position of the pitch axis (also known as centre of instantaneous rotation):

fiftyr cos θ sa =p +fiftyf cos θs (5)

zr +lr sin θ sa =zf +lf sin θdue south (6)

which gives lr =ptan θdue south +zr −z f

cos θsa tan θs − sin θsa and finally (x cir ,z

cir) T =( tenr + 50r cos θsa ,z

r+l rsin θ sa) T .

2.3 Suspensions and tires with stiffness and damping

The elastic and absorption properties of the vehicle are lumped in the shock absorbers inside the front &

rear interruption and in the compressible tires. Generally linear springs are mounted, whereas a non-linear

effect on the swing-arm forces is obtained with the design of the kinematics of the extremes of the shock

absorber packs; in this model also a linear behaviour of the tire stiffness is modelled; in Tab.iii thousandsf ,thousandsr are

the stiffness constant in the front and rear break, thousandpf and kpr in the front and rear tires; csf ,csr are the

damping coefficient in the front end and rear suspension, cpf and cpr in the front and rear tires. The model takes

into account an increase of stiffness in the shock absorbers when the relative displacement is locked at the

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opposite sides of the range; further, the damping coefficients can be different in compression and extension,

or modelled by means of feature curves, east.g. by bi-linear curves by each branch. By introducing the

relative lengths fiftysf and lsr at the extremes of the front and rear shock absorbers every bit in Fig.ii, expressed by lsf =

(zsf − zf )ii +( tensf − xf )2 and lsr = ( zshb − zsht )two +( xshb − xsht )2 , where ten shb

zshb = x r

zr +

cos θ sa −sin θsa

sin θsa cos θsa . x shb0 − ten r

zshb0 − zr0 and ten sht

zsht = 10 G

z + cos θ−sin θ

sin θ cos θ . x sht0 − x G0

zsht0 − zG0 ,

the elastic energy potential can be formulated as follows:

Eelas =1

ii(ksf l2

sf +one thousand sr l ii

sr +one thousand pf (z f −y f ) 2 +k pr (z r −y r ) ii )(7)

The Rayleigh function of the free energy dissipated in the shock absorbers and in the tires tin be written as

follows:

Due westdiss =1

two(csf ˙

fifty2

sf +c sr ˙

lii

sr +c pf (˙ zf −˙ yf )ii + cpr (˙ zr − ˙ yr )2 )(8)

The motion due to the compliance of the rubberband elements causes also a change in the gravitational potential,

which can be written as follows:

Uyard =thout gz +m sa gz Gsa +one thousands gz Gs +m rt gzr +m ft gzf (9)

The total potential energy Uof the system is thus the sum of the elastic energy of Eqn.7 and of the gravita-

tional potential of Eqn.9:

U= Due eastelas +U1000 (10)

2.4 Ground shaping

With the hypothesis of a constant forward velocity of the fiveehicle, the relative move confronting bumps or on an

uneven terrain tin be easily modelled, taking into account the spatial filibuster between the front and rear tires

due to the wheel base p. Thus, by imposing the velocity, the role that shapes the footing gives the instant

values of the yr and yf quantities.

ii.v External forces

The modelling can have into business relationship different external forces on the vehicle: chain pulling forces (both in

traction and in braking weather), horizontal inertial forces, aerodynamic forces and braking torques on

both wheels. In order to include these forces in the model, the virtual work principle was adopted. Although

the model does not take the horizontal translational dof, the application point of these forces accept a complete

description of its motion in the local coordinate organisation used; there follows that all the external forces do

virtual work and contribute to the local dynamics of the organization.

2.5.one Traction

The propulsive torque Mm of the engine is transmitted through a primary gear ratio τgrand and a final ratio τf to

the pinion: MP =Mchiliad τthousand τf . Then the chain transmission has a farther ratio, which depends on the geometry

(radius rp and rsouthward ) of the pinion and sprocket: Gs = Mp rsouth /rp . Information technology follows that the chain tension T, interim on

both the motorbike chassis and on the rear bicycle pivot with opposite signs, has module T= Mp /rp and its

direction is that of the actual tense branch of the concatenation, hither simplified past the direction that passes through

the points (xp ,z

p+r p) T and (x r,z

r+r s) T . Finally, disregarding the friction in the pivots and the rolling

resistance in the basis contact, the horizontal (friction) component of the constrain forcefulness of the ground on

the rear tire becomes St =Ms /rer , where rer is the external radius of the rear tire.

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2.v.two Braking

During braking, with radial callipers, there is a pure braking torque applied to the wheels. Disregarding the

friction in the pivots and the rolling resistance of the ground on the tires, the braking torques are counterbalanced

by the horizontal component of the constraint reactions given past the ground to each bike Southwardbf =Mbf /ref

and Due southbr = 1000br /rer , where ref is the external radius of the front tire, and by the contrary torques of reaction

Thousandbfr =−Thousandbf and Chiliadbrr =−Mbr , given by the front fork and the swing-arm.

2.5.3 Aerodynamic force

The aerodynamic force Fa , applied in the eye of pressure (10cp ,z

cp) T on the suspended mass, can be

considered in its drag and lift components. The drag component Fadvertisement =one /2ρcdAv2 is horizontal and contrary

to the forward direction of the motorbike, the lift component Fal =1 /2ρclAv2 is vertical and upwardly, where

ρis the air density, cd is the drag coefficient, Ais the area of the frontal section of the vehicle, v is the frontward

velocity of the motorbike and c50 is the aerodynamic lift coefficient.

2.5.iv Horizontal inertial forces

With the assumption of no rolling resistance of the ground on the tires and of no dissipation on the pivots,

the horizontal inertial force Fi , practical to the centroid of the suspended mass, has been false as being

opposite to the sum of vectors Due southt ,Southbf and Southbr ;Fi has thus positive sign in braking and negative sign in

traction.

2.five.v Virtual work of external forces

The full virtual work δWe of the external forces on the organisation is the sum of each component virtual due westork:

δWe = δW tc +δW b+δW a+δW if (xi)

where δWtc =−T .δXpc +T.δ10sc is the virtual piece of work of the chain pulling strength, δ Xpc is the virtual move

of the bespeak on the tense co-operative tangent to the pinion, δXsc is the virtual movement of the betoken on the tense

co-operative tangent to the sprocket; δWb =−Mbfr .δθ −Kbrr .δθsa is the virtual work of the braking torques;

δWa =Fa .δXcp is the virtual work of the aerodynamic force; δWif =− ( Southt + Sbf +Sbr ) δxg is the virtual

piece of work of the horizontal inertial forces here modelled.

iii Equations of motility

With the kinetic energy of Eqn.4, the conservative potential of Eqn.ten, the Rayleigh dissipated energy of

Eqn.eight and with the full virtual work of external forces of Eqn.eleven, the i-th equation of motility has the

post-obit Lagrange formulation, where qT =( z,θ,z f,z

r) T are the generalised lagrangian coordinates:

d

dt ∂E kinorth

∂˙ qi − ∂East kin

∂qi

+∂U

∂qi

+∂West diss

∂˙ qi

=∂δW east

∂qi

(12)

With the explicit constraint formulation adopted, basis shaping and time dependent external forces, an

ordinary differential equation (ODE) fix can be integrated to obtain the time-domain motion solution of the

modelled vehicle.

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four Linearisation of the model & eigenproblem formulation

The transformation of time-domain data into frequency domain poses many restrictions in terms of frequency

resolution; the transient response of a vehicle can have a limited lapse with very fast changing on relative

positions, thus requiring too small-scale integrator steps to have just a rough frequency resolution, and high com-

putation costs. In order to obtain a detailed frequency analysis of the behaviour of the vehicle in changing

weather condition the approach here proposed is to formulate a linearised fourth dimension-varying dynamic problem, whose

solution is followed along the time axis in terms of transfer function matrix and eigenproblem. Instead of

pursuing the linearisation around kinematic compatible positions but, as was done in [9] where a position-

frequency domain mapping was proposed, in this paper the linearisation was done around points of interest

in the time histories of the generalised coordinates qthat solve Eqn.12, with the evaluated (q , ˙

q,¨

q).

4.1 Linearisation of displacements and college order contributions

Around each position of interest, the formulation of the displacements is linearised by ways of a first lodge

Taylor serial expansion, while in free energy related functions but 2nd order terms are kept, disregarding the

products of different variables. In that location follows a linearised gear up of ODEs for gratuitous oscillations, which can be

easily represented in matrix formulation past 4x4 mass M, damping C and stiffness K constant matrices:

M¨

q+ C˙

q+ Kq = 0 (13)

By ways of a transformation of Eqn.13 into the Laplace domain, the dynamic stiffness Z(p ) tin exist obtained

as:

Z(p )=p2 M+p C+ Chiliad(xiv)

from which there follows the transfer office matrix H(p )=[ Z(p )]−1 ; the transfer function matrix is

changing with the linearisation point [10] and turns into the frequency response function matrix with the

Laplace variable p= iω , with ievery bit imaginary unit of measurement.

4.2 Country space solution of the eigenproblem

Being in the case of full general damping due to the high damping in the stupor absorbers, i.e. non-proportional

damping and C =αM +β K , where α and β are constants, the eigenproblem results in a complex-valued

representation, thus approached by ways of a state space formulation of the motion equations:

CM

M0

 ˙

q

¨

q + K0

0− G q

˙

q = 0(15)

where CM

M0

can exist defined as modal grandA 8x8 matrix, while K0

0− 1000 as modal 1000B 8x8 matrix.

The eigensolution is based on the generalised eigenproblem formulation:

chiliad−1

Am B Φ=λΦ(16)

where Φ is the state space 8x1 eigenvector (ψT ,λψ

T) Tcontaining the spatial domain 4x1 eigenvector ψand

the eigenvalue λ , both every bit complex-valued quantities. The eigensolution occurs [11] in circuitous-cohabit

pairs when the constraints are properly defined and mA &mB matrices are symmetrical. In the disproportion

of the modal thousandA &thousandB matrices, left and right eigenvectors tin can differ [12], merely remain orthogonal.

Stable oscillations occur with complex-valued eigenvalues having a negative real part, while unstable oscilla-

tion are present if the real part turns positive. The generalised modal parameters of the oscillatory behaviour

of the system tin be obtained from the circuitous-valued eigenvalues of the type λj =−ζj ωn j ∓iωnorth j i−ζ 2

j:

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Effigy 3: Time history evaluation of the motorcycle motion in overtaking a bump at constant speed

the undamped natural frequency of the j-th mode is ωnorth j = one thousandj /mj = Re 2

λj +Im ii

λj =|λ j |, where

Reλ j =−ζj ωnorthward j =−cj / (2mj ),Im λj=ω due northj 1−ζ ii

j=ω d j,one thousand j ,c j ,k j ,ζ j =c j /(2 thou j 1000 j )and ω d jare

respectively the modal mass, modal damping coefficient, modal stiffness coefficient, modal damping ratio

and damped frequency of the j-th mode. But there might as well be a linearisation point around which the

imaginary parts of the complex-conjugate eigenvalues completely vanish, beingness the eigenvalues simply real-

valued like λj =−ζj ωn j ∓ωn j ζ 2

j−one, highlighting a change in the behaviour of the system; in the instance

eigenvalue is negative, there will be no oscillation and the system shows a modal over-damping; when the

eigenvalue is positive, instead, an unstable non-oscillatory motion is present around that linearisation bespeak.

At each position of interest the eigensolution circuitous-valued pairs (λ, Φ, Φ∗ ) (symmetrical matrices) or

(λ, Φl ,Φr )(asymmetrical matrices) can be evaluated with no restriction in frequency resolution, giving a

first order approximation of the instant dynamic behaviour of the system around that position.

5 Test case and results

The frequency assay arroyo here presented was tested on the model of a motorbike with reliable values

taken from a mass product motorcycle. The analysed manoeuvre is the overtaking of a bump with 0.3m

acme and 2m length, at a constant horizontal speed of 10m/south, selected for its completeness of behaviours.

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Figure 4: FRFs maps of the time-varying state-space eigenproblem: a) H11 ,b)H12 ,c)H13 ,d)Hxiv ,eastward)H22 ,

f) H23 ,1000)H24 ,h)H33 ,i)H34 ,l)H44 .

five.1 Time-domain motions

As can be seen in Fig.3, the motorcycle is in steady-land earlier jumping on the bump, it compresses first

the front fork & tire and and so it transfers load to the rear interruption & tire, with the front cycle (in light-green)

released from the footing and the rear shock absorber compressed to its maximum (thus in blood-red). There is so

a flying interval (both tires are green), during which the suspensions extend to their limits (becoming red)

and then they oscillate effectually an intermediate position while the body of the motorcycle is turning forwards.

At the instant when the forepart wheel lands, in that location is the compression of the front fork & tire, later followed by

the bouncing of the rear suspension & tire, while the body returns in its steady-land configuration.

Information technology is clear that whatsoever fourth dimension i wheel is discrete from the ground the dynamic properties are changing dramat-

ically, equally volition be follomidweek by the frequency analysis. It is likewise understandable how the flying status is

dominated by the gravity, which induces an unstable non-oscillatory motility to the system, with increasing

aamplitude up to the landing instant.

5.ii Frequency Response Function variability

In Fig.iv the FRFs components of the H matrix are depicted in the time (steps) - frequency domain, with

logarithmic amplitudes. It can exist noticed that the cross terms manifest an higher variability of the FRFs,

which can be linked to the nonlinear kinematics of the suspension mechanisms. However, a dominant change

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  



20









20

  



 

 

1



 

      





  

Figure v: Eigenvalues of the time-varying state-space eigenproblem on the complex plane: a) 1st &iind ,b)

3rd &ivthursday ,c)5th &6th and d) 7thursday &8th

in the dynamic behaviour is due to the lifting of the wheels from the footing, with a dramatic change in the

dynamic properties of the system, in terms of low frequency contribution for the driving point FRFs, merely

likewise in terms of influences at higher frequencies for the cross FRFs. Indeed it must be remembered that the

suspensions can still oscillate even when the motorcycle is in the flying motion, the rotation of the body can

have only some little restrain from the interruption inertial, damping and rubberband forces, while the vertical

motion of the body has even weaker restrains.

5.3 Eigenproblem variability

The variability of the eigenproblem is here analysed in terms of changes in the eigenvalues, natural frequen-

cies, modal damping ratios and eigenvectors at the time step of linearisation.

5.iii.1 Eigenvalues, natural frequencies & modal damping

The eigenvalues are plotted in pairs on the completen plane, as in Fig.5, with the first eigenvalue of the pair

being represented in a blue circle, while the second in a cerise square. The first iv eigenvalues have a shut

natural frequency of support when both wheels are in contact with the ground, as tin can be seen in Fig.6a-b,

which corresponds to that of the suspension dynamics towards the suspended motorbike body, existence the rear

suspension a fiddling more than harder (ωnorthward 12 =18 .5Hz) than the front 1 ( ωn 34 =16 .eightHz) in the model; the modal

damping ratio of the front end intermission is higher than in the rear, with ζ34 ∼

=0. 45 against ζ 12 ∼

=0. 26 on the

rear; thus the damped frequencies are a bit more than detached (ωd 12 =17 .9Hz vs ωd 34 =15 .0Hz). This to

say that in the automated solution of the eigenproblem there might exist some switching in the ranking of the

eigenvalues, sorted in descending magnitude, reflected in Fig.5a-b, where the eigenvalue distribution does not

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Figure 6: Modal natural frequencies [Hz] of the time-varying state-space eigenproblem along the time steps:

a) 1st &2nd ,b)3rd &4th ,c)5th &6thursday and d) 7thursday &eightth

appear symmetric in regards to the existent axis just can be reconstructed by means of following the symmetric

patters in both pictures. Merely the fourth eigenvalue gets close to the origin, underlining that the first iv

eigenvalues represent stable damped behaviours of the system. It must be pointed out that, when both wheels

are in contact with the footing, the modal stiffness is college due to the tire contributions, thus reducing the

modal damping ratio ζ12 &ζ34 of the modes linked to the predominant motion of the suspensions; on the

other side, when the wheels do not touch the ground, the shock absorbers practice an over-damped action on the

suspensions, as can be followed in Fig.7 where the effective value of ζ12 &ζ34 can be too higher than unity.

The fifth and sixth eigenvalues behave fairly symmetrically on the complex plane of Fig.5c, never being

source of unstable oscillations in these simulations. These eigenvalues are mostly linked to the pitching

modes, with a natural frequency ωnorth 56 ∼

=three.5Hz when the vehicle has both wheels on the footing. It is

interesting to observe that the variability of ωn 56 follows the leap of the motorcycle, from raising first the

front wheel from the ground, to the landing with the front tire and the sudden rebound afterwards of the

rear suspension: at that place results a smoother transition, every bit can be seen in Fig.6c. Fifty-fifty during the flight over

the crash-land ωnorthward 56 does not drop completely to nada, equally said before about the kineto-elastic contributions of the

suspensions. The modal damping ratio ζ56 of the pitching in Fig.7c is always greater than goose egg and for a

bigger part of the time-history is bound nether the critical damping, meaning that well damped oscillations

can be present; also, at the middle of the flight, over-damping is acting on the pitching related modes, when

the break oscillation are extinguished and less effective restrains are on the rotation of the motorbike

body, in pair with the drib of ωnorth 56 .

The seventh and eight eigenvalues are mostly related to the vertical move of the vehicle, with quite a low

natural frequency ωdue north 78 ∼

=i.68Hz in steady-state conditions, as in Fig.6d. The frequency analysis depicted

on the complex-airplane of Fig.5d clearly tells that a limited time was spent on the two wheels with a damped

oscillatory behaviour; most of the evaluations have a null imaginary function, and the eight eigenvalue shows

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Effigy 7: Modal damping factors of the time-varying state-space eigenproblem along the fourth dimension steps: a) 1st

&2nd ,b)3rd &fourth ,c)5th &half-dozenth and d) seventh &viiith

bones unstable not-oscillatory behaviour: it is the flight of the motorbike over the bump and the niggling rebound

when landed, which has an increasing modal damping ratio ζ78 in Fig.7d before reaching the top of the flying

trajectory, but a negative modal damping ζ78 afterwards the climax of the flight, when gravity expands the vertical

downward motion.

5.3.2 Eigenvectors

In Fig.eight-9 are presented, by means of superimposed images, the four pairs of circuitous-valued eigenvectors

of the linearised country-infinite model in different fourth dimension instants (and location of the motorcycle with regards to

the crash-land); the first film of each sequence is the 1st &2nd eigenvector, and so follow the threerd &4th in the

second paradigm, the fivethursday &half-dozenthursday in the third and the 7thursday &8thursday in the terminal. Being complex-valued, the eigenvectors

conduct too a distributed potential filibuster within the shape, when represented equally fourth dimension-domain animation, which

tin be a further check on the consistency of the assay; exist aware that the magnitude of the displacements

in the picture is augmented to permit the reader place the motorbike modal behaviour, thus in that location also follows

the pinch or the lift of the tire from the ground, with coded colours (blood-red for maximal pinch,

green when free from contacts).

In detail, the eigenvectors in the first group of Fig.viii are pictured in the steady state condition before the

bump, when both the wheels are in contact with the ground, and the dynamic behaviour tin exist decomposed

in four damped oscillations. The 2d group of Fig.8 is related to the rising of the forepart wheel on the bump,

while in the third group the forepart wheel as definitely left the ground, showing the unstable macro movement

of the body, which is a composition of displacement and pitch. In Fig.9 the behaviour in flight and landing

is shown; during flight information technology is remarked by the eigenvectors how the pitching and autumn are weakly restrained and

unstable motions, while the suspensions still accept a relative motion regarding the frame.

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Figure 8: The 4 pairs of eigenmodes of the time-varying state-infinite eigenproblem forth the fourth dimension steps:

a) steady-state before the bump, b) front wheel on the crash-land, c) front end bicycle not in contact

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Figure 9: The iv pairs of eigenmodes of the time-varying land-infinite eigenproblem along the fourth dimension steps:

d) flying weather condition, e) front bike landing

5.4 Computational cost

A limited computational burden was indeed asked by the exam instance evaluation on a dual Intel Xeon Paxville

DP (dual core 2.8GHz CPUs) workstation: 3.33GB of retention allocated and 55 minutes were taken for

the complete calculations on the fourth dimension domain (for an integration lasting 10s) and 200 H(iω ) matrices &

eigenproblems solved.

vi Conclusions

The test case here presented has proved the effectiveness of the frequency assay approach on a fourth dimension-

varying mechanism as circuitous as an off-road motorbike in the action of jumping over a bump. The analysis

approach behaves consistently with the asked targets of getting detailed data in the frequency domain.

The formulation of the linearised equations of motion around a point of interest, obtained from a detailed

nonlinear dynamic assay, indeed gives the run a risk to explore the frequency domain with the resolution non

available with the time-history transformation simply. Farther efforts will be dedicated to test the approach

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too on other manoeuvres and different shapes of the ground. This planar vehicle model gives promising

hints for developing this thought for the implementation on larger models and on spatial mechanisms. Too, the

computational burden required is limited in comparison to inquire the integrator of the movement equations to give

all the data for obtaining such frequency meaningful content.

Acknowledgements

The first author wishes to thank Mr. Brugnoni for his diligent and smart piece of work during the preparation of

his BA dissertation [13], which constitutes in this research newspaper the calculation engine for the time-domain

solution of the motorbike model, and the starting betoken for the further dynamic analysis in the frequency

domain carried on by the first author.

References

[1] R. Southward. Sharp, South. Evangelou, D. J. N. Limebeer, Advances in the Modelling of Motorcycle Dynamics,

Multibody Arrangement Dynamics, Fiveol. 12, Kluwer Bookish Publishers (2004), pp. 251-283.

[2] V. Cossalter, R. Lot, M. Massaro, An advanced multibody code for handling and stability analysis of

motorcycles, Meccanica, Fiveol. 46, Springer (2011), pp. 943-958.

[3] 50. WU, W.-J. Zhang, Hierarchical modeling of semi-active control of a full motorbike break with

six degrees of fredooms, Int. J. Automotiveastward Engineering science, Vol. xi, No.1, KSAE (2010), pp. 27-32.

[4] M. Corno, S. M. Savaresi, G. J. Balas, On linear-parameter-varying (LPV) sideslip-controller design for

two-wheeled vehicles, Int. J. Robust Nonlinear Control, Fiveol. 19, John Wiley & Sons (2009), pp.1313-

1336.

[5] J. R. Ellis, Vehicle Dynamics, London Business Books Ltd., London (1969).

[half-dozen] T. D. Gillespie, Fundamentals of 5ehicle Dynamics, SAE, Westwardarrandale, PA-Usa, (1992).

[7] W. F. Milliken, D. L. Milliken, Race car vehicle dynamics, SAE, Westarrandale, PA-The states, (1995).

[8] A. Zanarini, Analisi cinetostatica grafica per meccanismi piani , Societ´

a Editrice Esculapio, Bologna

(2012).

[9] A. Zanarini, Kinematics-based variability of the dynamic behaviour in spatial mechanisms , in P. Sas,

editor, Proceedings of the ISMA2008 Conference - Multy-body dynamics and control, Leuven, Belgium,

2008 September fifteen-17, Leuven (2008), pp. 2175-2190.

[10] A. Carrella, D. J. Ewins, Identifying and quantifying structural nonlinearities in engineering science applica-

tions from measured frequency response functions, Mechanical Systems and Point Processing, 5ol.

25, Elsevier (2011), pp. 1011-1027.

[11] F. Tisseur, K. Meerbergen, The Quadric Eigenvalue Problem, SIAM Review, 5ol. 43, No. 2, Lodge

for Industrial and Applied Mathematics (2001), pp. 235-286.

[12] Fifty. Starek, D. J. Inman, Symmetric inverse eigenvalue vibration problem and its application, Mechanical

Systems and Bespeak Processing, Vol. 15, Elsevier (2001), pp. 11-29.

[xiii] E. Brugnoni, Analisi del comportamento dinamico di un motoveicolo, A. Zanarini supervisor, BA final

dissertation in Fundamentals of Machine Mechanics, Degree in Free energy Engineering, University of

Bologna - Italia, (2011).

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... Information technology tin can be conspicuously seen, also in the higher up conception that reduces the multibody inertia contributions to a one degree of freedom system, how the turbine has a variable reduced inertia, therefore showing different dynamic backdrop along the whole revolution. The problem might also be approached by multibody modelling every bit done by the author in other studies [14][15][16][17][eighteen]. From the expression of the aerodynamic torque contributions, the total piece of work of the aerodynamic forces follows equally: ...

  • Alessandro Zanarini Alessandro Zanarini

Not simply the well known and spread horizontal centrality machines are present in the current of air turbine scenario, but other architectures have found their niche. Amongst them, the Darrieus solution is known for its relatively compact design and pocket-size size, just, fifty-fifty more than, for its vertical axis, every bit well as for the vertical blades distributed on the side of a rotating cylinder, even when twisted with special 3D helical shapes. Reverse to the horizontal centrality solution mostly as 3-bladed version, the Darrieus design tin non take reward of the axis orientation control of the blade plane, considering information technology accepts air current from whatsoever management. But in club to maximise the efficiency in harvesting energy from the wind, the Darrieus turbine searches for the all-time attack bending betwixt each blade and the wind flow at a specific speed, optimised forth the complete rotation of the bract around the primal axis. Together with a simple self-command towards the wind flow direction, a solution to improve the energy conversion into torque comes from the blueprint of a controlled guidance of the attack bending of the vertical blades towards the flow. In this work the mutual use of an old mechanism, the 4-bar linkage, is analysed to generate the passive control of each of the iii blades in a uncomplicated Darrieus turbine pattern to improve the efficiency. This activity was made to understand, with relatively simple models, the advantages and drawbacks of such machinery architecture for the optimal guidance of each bract, as a preliminary and feasibility methodological study for a potential research path in wind energy harvesting alternative solutions. The work therefore deals with the selection of a common wing profile and a 3-bladed machine scheme, together with the adoption of iv-bar linkage for each passively controlled blade. The aerodynamic force distribution and torque simulation follow, as functions of the athwart position, at ii different wind speeds in a complete turbine rotation. The assumptions, the formulation and results for the simple Darrieus machine with an set on bending guidance mechanism are discussed in particular.

... Automobiles are called since they are widely used and evidence that the model works with any conventional spring-damper system. The value is also not different off-road motorbike damping systems, which is some other instance of equipment that is widely used [24]. The characteristics of the sample damper are shown in Table one and the details of the simulation parameters are explained in the simulation results department. ...

... The to a higher place mentioned variability of the torsional stiffness of the contacts has furnishings on the dynamic signature of the planetary gear single phase. As in [15][16][17], a kinematics based variability of the eigenproblem tin can exist addressed. A lumped model of the planetary gear (one planet only be considered) can be set upwardly following the scheme in Effigy xi, and the motility equations for small-scale torsional oscillation around the kinematic position of interest can be drawn by means of Lagrangian approach, one planet only existence considered. ...

  • Alessandro Zanarini Alessandro Zanarini

Gear mesh flexibility is a varying and periodic quantity, which depends on several geometric, motion and material parameters; in NVH and structural analyses of systems with transmission problems this stiffness variability tin play an important function on the design targets, such as sound quality and fatigue life interpretation. Many times the tooth mesh flexibility in gears is modelled by ways of lumped parameters formulas obtained from literature or by means of complex finite element models of the contact. In this piece of work the roots of lumped parameters formulas were investigated, equally a preliminary and feasibility study for a potential research path in gear condition monitoring, noise source & structural dynamics excitation modelling and manual errors. Starting from the kinematics of mating rigid teeth in planetary gears with a single stage, the time-varying flexibility contribution of each deflected pair is added with simple assumptions, resulting in the parametric modelling of the torsional stiffness as a kinematics based periodic quantity. Parameters were extracted from this approach, to explore a viable and computationally cost effective geometry based method to model the torsional vibrations and manual error in planetary gears. Proceedings DOI: 10.6092/unibo/amsacta/5963

... Therefore physics based, but more efficient, approaches are sought after for general flexible mechanisms. While the authors have already worked on kinematics dependant models of mechanisms [three][four][5][half-dozen][seven][viii], where the elasticity was lumped mainly in betwixt the rigid parts, the approach here used is focused on modelling the elasticity of each moving parts, giving a relevant contribution on the whole system behaviour in the time domain. In [9,x] a linearly independent set of shape functions was introduced to continuously model the flexible components of local reference displacements equally a superposition of shape functions; a novel formulation of the flexible properties of axle-like components then followed, with a reduced size motion equation prepare and no restriction in the geometrical model. ...

Flexible multibody dynamics is ofttimes approached in instance of minor flexible deformations past means of the finite-element modelling method in component mode synthesis techniques to reduce the size of the computations. In this work, by means of proper shape functions bases, the elastic behaviour of slender beams is approximated in its local axial and transversal displacement components. Attention is paid to the inertial contributions to enhance the driving role. An ideal quick return machinery is hither modelled with the meshless arroyo. The results of different modelling choices are collected, to testify how this shape functions based formulation tin handle the added flexibility by means of a modest fix of degrees of freedom and motion equations. Some aspects are detailed and discussed. Proceedings DOI: 10.6092/unibo/amsacta/5963

... Furthermore, the proper selection of the effective dynamic modeshapes to exist included in the reduced set requires a highly skilled attention, becoming a kind of art of trained personnel. While the authors have already worked on kinematics dependant models of mechanisms [3][4][5][6][7][viii], where the elasticity was lumped mainly in between the rigid parts, the approach hither presented is focused on modelling the inner elasticity of each moving parts, giving a relevant contribution on the whole system behaviour in the time domain. ...

Flexible multibody dynamics is often approached in case of pocket-sized flexible defor- mations by means of the finite-chemical element modelling method or with a component modal base in component mode synthesis techniques. In these approaches the size of the motion equations can be high or general description can be poor. In this paper, instead, a linearly independent set up of shape functions is introduced to continu- ously model the flexible components of local reference displacements as a superposition of shape functions; a novel formulation of the flexible properties of axle-like components and then follows, with a reduced size motion equation prepare and no restriction in the spatial sampling. All of the modelling aspects, such as kinematics, inertia and stiffness distribution and generalised forces, are extensively discussed. Test cases prove the effectiveness of the proposed approach.

  • Alessandro Zanarini Alessandro Zanarini

The intrinsic variability of dynamic backdrop in spatial systems is faced in this paper past means of a kinematics-based modal approach. An effective analytical-numerical procedure is here presented to mathematically ascertain a local linear model of a spatial mechanism, composed of rigid bodies connected past standard kinematic and lumped elastic constraints; the linearization is effective simply in the closeness of the kinematically admissible position of interest and thus yields to a varying generalised eigenproblem, function of the kinematical configuration of the whole organization through the working domain. An extended assay is reported to obtain the motion equations of an unconstrained rigid body, thus linear equations of constrained movement are obtained from linearized displacements and constraint equations. The eigensystem is evaluated in every specific kinematical configuration of interest, mapping the variability of the dynamic performances across the working domain; the kinematical configuration is provided at any footstep of a userdefined motion law past standard non-linear kinematics. Real industrial applications, consisting in ii 6 degree-of-freedom robots, are tested by means of the outlined methodology. Results are reported and discussed in detail.

  • Robin Due south. Precipitous
  • Simos Evangelou Simos Evangelou
  • D.J.N. Limebeer

Starting from an existing avant-garde motorbike dynamics model, which allows simulation of reasonably general motions and stability, modal and response computations for small perturbations from any trim condition, improvements are described. These concern (a) tyre/road contact geometry, (b) tyre shear force and moment descriptions, as functions of load, slip and camber, (c) tyre relaxation backdrop, (d) a new analytic treatment of the monoshock rear suspension mechanism with sample results, (due east) parameter values describing a contemporary high performance auto and passenger, (f) steady-state equilibrium and power checking and (thou) steering control. In detail, the Magic Formula motorcycle tyre model is utilised and complete sets of parameter values for gimmicky tyres are derived past identification methods. The new model is used for steady turning, stability, pattern parameter sensitivity and response to road forcing calculations. The results show the predictions of the model to be in full general agreement with observations of motorcycle behaviour from the field and they advise that frame flexibility remains an of import design and analysis area, despite improvements in frame designs over recent years. Motorcycle rider parameters have meaning influences on the behaviour, with results consistent with a commonly held view, that lightweight riders are more likely to endure oscillation problems than heavyweight ones.

This work illustrates the modelling aspects of an advanced motorbike multibody model which captures the almost important features of the vehicle, including 5 unlike suspensions schemes (telescopic fork, telelever, duolever, swingarm and paralever) all provided with their own flexibilities. Several lumped stiffness are used to model the structural compliance of the frame as well as the flexibility of the manual arrangement. A passive rider model which simulates the interaction of the passenger with the handlebar and the chassis and a 3D route-tire model which takes into business relationship both the carcass geometry and compliance are besides included. The presented mathematical model performs non-linear time domain simulations, not-linear steady country analysis, linear stability and frequency domain analyses. Some simulation examples are presented: a steady country cornering manoeuvre where the decomposition of steering torque in its main components is carried out, a stability assay where the effects of rider'southward impedance and structural flexibilities on weave and wobble vibration modes is highlighted, a stability analysis of a braking motorcycle which experiences a chatter instability and a lane change manoeuvre where the main handling indexes are computed.

We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, circuitous, Hermitian, skew-Hermitian) and the spectral properties of the trouble. We allocate numerical methods and catalogue available software.

This paper describes the application of linear-parameter-varying (LPV) control design techniques to the problem of slip control for two-wheeled vehicles. A nonlinear multi-body motorcycle simulator is employed to derive a control-oriented dynamic model. Information technology is shown that, in order to devise a robust controller with practiced operation, it is necessary to accept into account the dependence of the model on the velocity and on the bicycle skid. This dependence is modeled via an LPV system constructed from Jacobian linearizations at unlike velocities and sideslip values. The control trouble is formulated every bit a model-matching control problem within the LPV framework; a specific modification of the LPV command synthesis algorithm is proposed to alleviate controller interpolation problems. Linear and nonlinear simulations indicate that the synthesized controller achieves the required robustness and functioning. Copyright © 2008 John Wiley & Sons, Ltd.

  • Alex Carrella Alex Carrella
  • D. J. Ewins

Engineering structures seldom behave linearly and, as a result, linearity checks are common do in the testing of critical structures exposed to dynamic loading to ascertain the boundary of validity of the linear authorities. However, in large scale industrial applications, there is no general methodology for dynamicists to extract nonlinear parameters from measured vibration information so that these can be then included in the associated numerical models. In this paper, a simple method based on the information contained in the frequency response function (FRF) properties of a structure is studied. This technique falls within the category of unmarried-degree-of-liberty (SDOF) modal analysis methods. The principle upon which information technology is based is effectively a linearisation whereby information technology is causeless that at given amplitude of displacement response the organisation responds at the same frequency every bit the excitation and that stiffness and damping are constants. In and then doing, past extracting this information at different amplitudes of vibration response, it is possible to estimate the aamplitude-dependent 'natural' frequency and modal loss factor. Considering of its mathematical simplicity and practical implementation during standard vibration testing, this method is particularly suitable for practical applications. In this paper, the method is illustrated and new analyses are carried out to validate its performance on numerical simulations before applying it to data measured on a complex aerospace test construction also as a total-scale helicopter.

  • Thomas D. Gillespie

This volume attempts to find a heart basis by balancing applied science principles and equations of use to every automotive engineer with practical explanations of the mechanics involved, so that those without a formal applied science degree can still comprehend and use most of the principles discussed. Either as an introductory text or a applied professional person overview, this book is an ideal reference.

This paper summarises the authors' previous effort on inverse eigenvalue problem for linear vibrating systems described by a vector differential equation with abiding coefficient matrices and non-proportional damping. The inverse problem of interest here is that of determining real symmetric coefficient matrices assumed to represent mass normalised velocity and position coefficient matrices, given a set of specified complex eigenvalues and eigenvectors. There are given two solutions of a symmetric changed eigenvalue problem presented past Starek and Inman [one, ii].The theory of inverse eigenvalue trouble is practical to the model updating problem. The goal of this paper is to recognise that the model updating problem is a subset of the inverse eigenvalue problem. The approach proposed here is to utilize the results of changed eigenvalue trouble to develop methods for model updating.Comments are made on how their process may be used to solve the damage detection problem.

  • 50. Wu
  • W. -J. Zhang

Hierarchical command is a new control framework in the vehicle vibration control field. In this newspaper, a hierarchical modeling method is presented to form a different motorcycle model, compared to the traditional model with six degrees of freedoms (DOF), so as to construct hierarchical modeling control. The whole control framework is composed of a central command, two local controls and two uncontrollable parts. The front and rear wheel systems of a motorbike are all dealt with by using ii contained local 2-DOF systems. The commuter and engine act every bit uncontrollable passive parts. The primal control is composed of an algorithm made upwardly of some dynamic equations that harmonize local relations. The vertical and pitch accelerations of the intermission middle are treated equally central control objects. With the help of Linear Quadratic Gaussian (LQG) algorithms adopted by two local controls, respectively, and Matlab software, some results of the simulation prove that hierarchical modeling command requires less CPU time, reduces respond fourth dimension and improves ride quality. Fundamental WordsHierarchical modeling method-Motorcycle suspension-Semi-active control-Six degree of freedoms-Simulation