Heave and Pitch Dynamics

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Vehicle System DynamicsPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713659010

Heave-Pitch-Roll Dynamics of Flexible Vehicle in Variable Velocity RunD. Yadav; S. Kamle; S. Talukdar

Online publication date: 09 August 2010

To cite this Article Yadav, D. , Kamle, S. and Talukdar, S.(2000) 'Heave-Pitch-Roll Dynamics of Flexible Vehicle in VariableVelocity Run', Vehicle System Dynamics, 33: 1, 1 28

To link to this Article: DOI: 10.1076/0042-3114(200001)33:1;1-5;FT001URL: http://dx.doi.org/10.1076/0042-3114(200001)33:1;1-5;FT001

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.Vehicle System Dynamics, 33 2000 , pp. 1 28 0042-3114r00r3301-028$15.00q Swets & Zeitlinger

Heave-Pitch-Roll Dynamics of Flexible Vehicle in

Variable Velocity Run

D. YADAV 1, S. KAMLE 2 and S. TALUKDAR 3

SUMMARY

Analysis for response statistics evaluation of a flexible vehicle travelling with variable velocity overnonhomogeneously profiled flexible track is presented with a heave-pitch-roll model. The vehicle bodyis idealised as a flexible member with variable cross-section, inertia, damping and stiffness distribu-tions. The vehicle may also have variable section slender elastic attachments. Coupled dynamics withrigid body heave-pitch-roll modes and elastic bending-torsion modes of the vehicle body along withcoupled bending-torsion modes of the attachments are considered. Equivalent linear suspension systemcharacteristics are employed for developing the analysis. Numerical results are presented for an aircraftwith tricycle landing gear arrangements and comparison is made with other models.

1. INTRODUCTION

Unevenness in the track induces vibration in the moving vehicle and the flexible

track system. With increasing operation speeds and lighter constructions, predic-

tion of the vehicle response has become significant for the assessment of its ride

quality, design and control. The knowledge of track dynamics is important for

design, life and maintenance of the track system.

The vehicle-track system has three main components - vehicle super-structure,

suspension system and the track-foundation system. The super-structure, in gen-

eral, is flexible and may have appendages in specific cases. Its dynamics has

heave, pitch and roll as rigid body modes with superimposed bending and torsion

as flexible modes. The suspension system generally has some shock absorber with

spring and damper arrangement mounted over pneumatic wheels. The track is laid

over a subgrade supported by a foundation. The track foundation system under-

goes deflection due to the moving vehicle loading. The track surface has undula-

tions that are random in nature.

The sophistication of the system model to be adopted depends on the degree of

accuracy desired. The analytical study was initiated with simple approximate rigid

1Professor; Department of Aerospace Engineering, I.I.T. Kanpur, India - 208016.

2Associate Professor; Department of Aerospace Engineering, I.I.T. Kanpur, India - 208016.

3Graduate Student; Department of Aerospace Engineering, I.I.T. Kanpur, India - 208016.

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D. YADAV ET AL.2

lumped mass models. Both constant and variable velocity runs have been consid-w xered with random track inputs 16 . Rigid multi-axled multi-wheeled vehicle

w xmodels with heave-pitch-roll modes have been analysed 7 10 . Structural flexibil-

ity of ground vehicle models have been taken into account with inclusion ofw xbending modes in heave-pitch models 1114 . The effect of track flexibility has

come under consideration with the study of flexible beams under moving loadsw x1526 . The interaction of flexible vehicle body with flexible track has been

w xstudied with various degrees of freedom for nonstationary inputs 2730 . The

vehicle models employed have ranged from single lumped mass to continuous

nonuniform elements and tracks models have ranged from rigid to flexible

members with stiffness, damping and inertial properties.w xThe present paper, extending the earlier work presented in Ref. 27 , introduces

improvement in the vehicle model by considering heave-pitch-roll dynamics of the

flexible vehicle with slender attachments. The track and foundation have been

assumed flexible with track unevenness a nonhomogeneous random process .specified by the generalised power spectral density PSD function. The present

analysis and modelling approach is applicable to most ground vehiclertrack

systems. The authors consider an addition to analysis techniques in this area. The

approach has been illustrated by obtaining the second order response statistics for

a small aircraft in ground runs. Comparison has been drawn with some published

work.

2. SYSTEM MODEL

The system consists of a vehicle travelling over an uneven and flexible track

supported by elastic subgrade and foundation. The vehicle super-structure is

treated as a flexible member. It is allowed to have nonuniform distribution of

mass, stiffness and damping. It has all the possible rigid body modes in heave,

pitch and roll along with coupled flexible bending and torsion modes. Two slender

attachments modelled as cantilever beams with variable mass, stiffness and

damping distributions are also considered. These also have point mass loadings at

.arbitrary locations to simulate stores loads carried by the attachments. A similarapproach is usable when the attachments are different in number. The multi-axle

multi-wheel effect is incorporated with a tricycle wheel arrangement, as in nose

wheel type aircraft landing gears.

The shock absorber and the pneumatic wheel in vehicle suspension have

nonlinear behaviour in stiffness and damping. As nonlinear suspension model is

not directly amenable to closed form solution, a linearised model based on

equivalent energy per unit cycle has been adopted for this analytical study. The

linear model is capable of producing realistic and meaningful results for judging

the system performance. The masses of the undercarriages and wheels are lumped .at their center of gravity c.g. locations to incorporate their inertial effects during

dynamics.




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HEAVE-PITCH-ROLL DYNAMICS 3

The track-foundation system has a prepared track slab laid over a subgrade

supported by a foundation. The limitations in the construction process and variable

settlements of the subgrade and the foundation result in unevenness in the track

surface. The surface undulations are best modelled as a nonhomogeneous random

process of the space coordinates. The track-foundation system is subjected to

deflection under loading. The track length is much larger than the width and the

track slab has been idealised as a beam member supported on a foundation with

distributed stiffness and damping.

Figure 1 shows the system model for an aircraft with nose wheel type of

landing gears in ground runs. A similar model may be constructed for other

multi-wheeled multi-axled vehicles also. A rectangular coordinate system with

origin at the vehicle c.g. is employed for developing the system equations. The

deflections considered are w - deflection of the vehicle body due to heave, pitchband bending modes in the vertical plane, u - rotational deflection of the vehiclebbody about the longitudinal axis due to roll and torque, w and w - verticalL Rdeflections of the left and right wing attachments in bending and heave modes, uL

Fig. 1. System Model.




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D. YADAV ET AL.4

and u - torsional rotation for the left and right wing attachments, z - the wheel R i jc.g. displacements with the first suffix 1 for the front axle and 2 for the rear axle

and second suffix 1 for the center, 2 for the left and 3 for the right wheel. y

indicates the vertical deflection of the track. The distances are measured as r

along the vehicle axis and g transverse to it, s along the wing attachment axisbfrom its root and e transverse to it, and x along the longitudinal direction of thew

track.

A vehicle forward motion pattern may be described in terms of the location .x t of its center of gravity indicating the distance travelled at the time instant t.c

Any general forward motion, like constant velocity, accelerating, decelerating, etc.

may be modelled by expressing the c.g. location by a polynomial series

mkx t s a t . 1 . .c k

ks0

It can be seen that m s 1 gives a constant velocity forward motion and m s 2

gives a constant accelerationrdeceleration motion pattern. The size of m and the

value of the coefficients a may be selected to fit any desired type of forwardkmotion pattern.

.The nonhomogeneous random unevenness h x of the track can be expressed inw xthe Stiltjes integral form 31

`

h x s h x q exp jV x dS V , 2 . . . . .Hmy`

.where h x is a deterministic function describing the mean level of the track andm .S V is related to the zero mean random part of the unevenness with

U E dS V s 0, and E dS V dS V s F V ,V dV dV , . . . .1 2 h h 1 2 1 2R R

where asterisk denotes the complex conjugate and F is the generalised PSD ofh hR Rthe random track. As mentioned before, the expression admits nonhomogeneous

track profiles. The mean track profile depends on the ground characteristics, track

preparation technique and the settlement of the subgrade and the foundation. Itmay take various shapes and for the present study is expressed by a polynomial

series

ll

ih x s h x . 3 . .m iis0

The size of the series and the value of the coefficients can be matched to represent

a desired mean shape.

In a tricycle type heave-pitch-roll model, vehicle wheels do not follow the same

axial line. This leads to unsymmetrical roughness inputs to the wheels. Mean trackw xheights experienced by a wheel can be determined 27 as a time function using

. .Eqs. 1 and 3 .




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The track and foundation deformations under the transverse loading induced by

the moving wheels are small because of its large stiffness. The main interest in

considering the track and foundation to be flexible in the present analysis is to

capture the interaction effects on the vehicle response. A simple track model is

expected to be adequate for this purpose. The track model is a continuous member

and is assumed to have uniform distributions of mass, stiffness and damping

neglecting any variations in these quantities. It is further assumed that the track

deforms in a vertical direction under moving wheel loads and any torsion on the

track due to all the wheels not being in the same line has been neglected.

3. SYSTEM EQUATIONS

The vehicle body is a beam type of member with distributed damping. Its equation

.of motion for transverse vibration w r,t in the vertical plane can be described bybw x32

E2

E2 w E2 w Ew Eb b b

E I r q m r q c r sf r,t q M r,t , . . . . .b b b b b b2 2 2 5 Et ErEr Er Et4 .

in which E I is the flexural stiffness, m is the mass and c is the viscousb b b bdamping per unit length. The force f and moment M due to the reactions fromb bthe shock absorbers and the flexible appendages are point loads acting at theattachments locations. These may be expressed as distributed loads with delta

functions

f r,t s y C w r,t yz q K w r,t yz d ry 1 4 4 . . . . b s11 b 11 s11 b 11 13

y C w r,t yz q K w r,t yz d ry 1 4 4 . . . s2 i 0 2 i s2 i 0 2 i 2is2

2E E w

Ly E I s .w L w L L 2 5Es EsL L s s0L2

E E wRq E I s P d ry 1 , 5 . . .w R w R R 02 5Es EsR R s s0R

and

Eu EuL RM r,t s y G J s q G J s d ry 1 , . . . .b w L w L L w R w R R 0

5 5Es EsL Rs s0 s s0L R

6 .




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D. YADAV ET AL.6

where C and K are the suspension damping and stiffness. The two numerals ins sthe suffixes are used to denote axle location and wheel location on the axle as

elaborated in the model description. E I and G J are the flexural and torsionalw w w wstiffness of the flexible attachments undergoing bending transverse displacement w

and twist u relative to vehicle body. Subscript L and R is used for left and right

attachments. zs denote the vertical displacement of the unsprung masses. 1 is0the distance of the elastic axis root of the flexible attachment from the aircraft c.g.

and 1 and 1 are the locations of the front and the rear axles. The quantity w is1 2 0the total vertical deflection of the vehicle body from its longitudinal axis and is

given by

w r,t s w r,t q g r u r,t , . . . .0 b b b

.where g r is the distance of the point on the vehicle body at station r, measuredb .transverse to the longitudinal axis. The torsional deformation u r,t of theb

vehicle body is governed by the differential equation

E Eu E2u Eub b bG J r yI r y q r s yG r,t , 7 . . . . .b b b r b b2 5

Er Er EtEt

where G I , I and q are torsional stiffness, mass moment of inertia andb b b r brotational viscous damping per unit length of the beam. The reaction torque Gbper unit length on the vehicle body from the landing gears and the attachments is

given by

3

G r,t s C w r,t yz q K w r,t y z e r d ry 1 4 4 . . . . . b s2 i 0 2 i s2 i 0 2 i i 2is2

2 2E w E wL R

q E I s q E I s d ry 1 , . . .w L w L L w R w R R 02 2 5 5Es EsL Rs s0 s s0L R8 .

where e and e are the distance of the wheels on the rear axle from the2 3longitudinal axis of the vehicle body.

. .It may be noted that Eqs. 4 and 7 also include the rigid body heave andpitch modes for the vehicle body.

The equations of motion for the front and rear wheels are

M z q C z y h x yy x ,t q K z y h x yy x ,t 4 . . . . 4 u11 11 u11 11 11 1 1 u11 11 11 1 1

q C z y w l ,t q K z y w l ,t s 0, 9 4 4 . . . s11 11 b 1 s11 11 b 1




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and

M z q C z y h x yy x ,t q K z y h x yy x ,t 4 . . . . 4 u2 i 2 i u2 i 2 i 2 i 2 2 u2 i 2 i 2 i 2 2

q C z y w l ,t y w s ,t q K z y w l ,t . . 4 .s2 i 2 i 0 2 i g i s 2 i 2 i 0 2

yw s ,t s 0, i s 2,3 10 .4 .i g i

while w s w , w s w , s s s , and s s s .2 L 3 R g 2 g L g 3 g RThe coupled bending-torsional oscillation of the flexible attachment carrying

w xarbitrary masses at arbitrary locations can be written as 15

E2

E2 w E2 w E2u Ew

E I s q m s y m s e s q c s . . . . .w w w w w w2 2 2 2 5 EtEs Es Et EtEu

y e s c s sf s,t , 11 . . . .w w wEt

and

E Eu E2u E2 wG J s yI s q m s e s y q s . . . . .w w wa w w2 2 5

Es Es Et Et

E

uE

w2qe s c s q c s e s s e s f s,t y G s,t , 12 . . . . . . . .4w w w w w w wEt Et

where e is the offset of the shear center from the c.g. of the cross-section, c andw wq are the distributed viscous damping for translational and rotational motion, and

m and I is the mass and mass moment of inertia of cross-section about thew waelastic axis.

The reaction force f and torque G per unit length of the span are given byw w

E2

Xf s,t s ym s w l ,t y c s w l ,t q m s e s w l ,t . . . . . . . . w w 0 0 w 0 w w 0 02Et

pE

Xq e s c s w l ,t y M w l ,t q w s,t 4 . . . . . w w 0 0 k 0 0

Etks1

2E

X y M e s w l ,t q u s,t d s y s y C w l ,t . . . . .k w 0 0 k s2 i 0 02 5Etqw s,t yz q K w l ,t q w s,t yz d s y s4 4 . . . . 2 i s 2 i 0 0 2 i g i

q V s,t , 13 . .ac




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D. YADAV ET AL.8

and

E2

XG s,t s y I s w l ,t q m s e s w l ,t y q s . . . . . . .w wa 0 0 w w o 0 w2

Et

EX2qe s c s w l ,t q c s e s w l ,t . . . . . .4 w w 0 0 w w 0 0

Et

2p pE

X y I w l ,t q u s,t y M e s w l ,t . . . . k 0 0 k w 0 02 5Etks1 ks1

qw s,t d s y s qM s,t , 144 . . . . k ac

where M is the kth concentrated mass and I is its polar moment of inertia aboutk k . .the elastic axis. Eqs. 11 to 14 may be made particular to the left or rightattachment by using subscript L or R and i as 2 for the left and 3 for the right

attachment.

V and M are the aerodynamic forces and moments. These can be expressedac acas a function of vehicle forward velocity, exposed surface characteristics and

w xaerodynamic coefficients 27 .

As mentioned earlier, the pavement is idealised as of a slender member on

elastic supports. The equation of motion for the transverse oscillation can bew xwritten as 9

E4y E2y E y

E I q m q c q k y sf x ,t , 15 . . p p p f f p4 2EtEx Et

where E I , m , c and k are the track flexural rigidity, mass and distributed p p p f f damping and spring constant of the subgrade foundation per unit length. These are

assumed to be uniform along the track length. The impressed force per unit length

is given by

f x ,t s y C z y h x yy x ,t q K z y h x . . . . 4

p u11 11 11 u11 11 11

3

yy x ,t d x yx y C z y h x yy x ,t4 . . . . 4 1 u2 i 2 i 2 iis1

qK z y h x yy x ,t d x yx 16 4 . . . .u2 i 2 i 2 i 2

4. SOLUTION TECHNIQUE

The solution of the linear systems represented by the set of partial and ordinarydifferential equations can be obtained with known initial and boundary conditions.

In the method followed, first the partial differential equations are discretized in




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terms of their time dependent normal coordinates. Thereafter, complete system

equations are decoupled using modal approach and analytical expressions for the

response and its statistics are developed.

4.1. Discretization of the Equation of Motion

. . . . .The governing system equations of motion are Eqs. 4 , 7 , 9 12 and 15 . . .Out of these Eqs. 9 and 10 are ordinary differential equations for lumped

masses while the rest are partial differential equations for continuous elements. A

set of ordinary differential equations can be obtained for the modal amplitudes byw xusing the modal decomposition technique 33 . Considering the vehicle body first,

let its deformation be expressed as

` `

w r,t s W r h t , and u r,t s T r h t , 17 . . . . . . . b b i b i b i t iis0 is0

. .where h t and h t are the generalised coordinate associated with the ithb i t ibending and the ith torsional modes.

. . . .Substituting Eq. 17 in Eqs. 4 and 7 , multiplying these by W r andb k .T r respectively, integrating both over the length domain of the vehicle bodyb k

and using the orthogonality conditions of the normal modes yields the following

w xtwo discretized equations for the vehicle body 33

`

2h t q D h t q v h t s Q t , 18 . . . . . b i i k b i b i b i b iks1

`

X 2h t q D h t q v h t s Q t , i s 1,2, . . . 19 . . . . . t i i k t i t i t i t iks1

where v and v are the natural frequencies; Q and Q are the generalisedb i t i b i t iforces; D and D

Xare modal damping coefficients in the bending and torsionali k i k

modes of the vehicle body. The expressions for these generalised quantities arew xgiven in Appendix 1 33 .

The vehicle body performs rigid body heave-pitch-roll motion and the first two

bending modes represent heave and pitch motion while the first mode in torsion

represents rigid roll. These rigid body modes correspond to zero natural frequen-

cies. Other finite nonzero frequencies correspond to elastic modes. For the vehicle

body modelled as a variable section beam the bending and torsional modes and the

natural frequencies can be found by solving the free vibration equations. Thevariations in cross-sectional properties need to be fitted to a power series with

w xfinite number of terms as in Ref. 27 .




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D. YADAV ET AL.10

Following the above mentioned sequence of steps the equations for the other . .elastic elements have also been discretized. Eqs. 11 and 12 for the attachments

take the following form for the left member

`

L L L 2 L Lh t q D h t q v h t s Q t , i s 1,2, . . . 20 . . . . . w i i k w k L i w i w iks1

in which h are the attachment generalised coordinates, v are the naturalw i ifrequencies corresponding to the ith normal mode in coupled bending-torsion. As

mentioned earlier the extension L is for the left attachment. The relations for the

right attachment is obtained by replacing L by R. The expressions for the

generalised quantities for the attachment are placed in Appendix 2. .Eq. 15 for the track may be discretized in terms of the normal coordinates as

h t qBh t q v2 y j 2 yBj yjv 2 j qB h t s Q t , 21 . . . . . . 5 . p i p i p i i i p i i p i p i

where B s c rm . The parameter j is related to the foundation damping and v f p pis the damped natural frequency of the track beam. The parameters v , j and thep

.normal mode function c x can be found by solving the free vibration equation ofthe track beam with appropriate boundary conditions. The generalised mass Mp iand the generalised force Q associated with the ith mode are given in Appendixp i3.

4.2. System Response

A continuous system has infinite modes. However, the lower modes are known to

have dominant contribution as compared to the higher modes. Let the first nbmodes for the vehicle body in bending and n in torsion, first n coupledt wbending-torsion modes for each attachments and first n modes for the track bepconsidered for analysis. This leads to the system degrees of freedom n as

n s n q n q 3 q 2 n q n .b t w p

The discretized equations for the vehicle body, unsprung masses, left and right . . . . .attachments and track - Eqs. 18 , 19 , 9 , 10 and 20 for left and right

.attachment and 21 can then be combined and presented as a matrix equation

Mq t q Cq t qK q t sF t , 22 . . . . .

.where q nx1 is the response vector of the system generalised coordinates, F . .nx1 is the generalised excitation vector and M, C and K nxn are system mass,damping and stiffness matrices respectively.




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.With decoupling of the system Eq. 22 , into 2n first order equations the systemw xgeneralised response can be expressed as 31

2 n 2 n n n`

q t s X u exp ya t q u u m H v,t dS F w , . . . . . Hm o i m i i m i i r r k i k 1 1y`

is1 is1 rs1 ks1

m s 1 , 2 , . . . n; m s m q n , 23 .1

where X are the constants of integration to be determined from the initialo i .conditions. a are the eigenvalues of the matrix A 2 n = 2 ni

y1 y1 M C M K A syI 0

4and u are the eigenvectors corresponding to the eigenvalue a , U is the modaln i i 4matrix formed by the eigenvectors as its column, u represents the column ofn i

y1 y1 .U and m are the elements of M . H v,t is the transient frequency responsei jw xfunction given by 31

1H v,t s exp jvt y exp ya ty t . 24 . . . . .i i 0

a qjv

4.3. Response Statistics

One can now consider the expectation of the response. The mean of the displace-w xment may be expressed as 27

2 n 2 n n n

m t s X u exp ya t q u u m I t , m s 1 , 2 , . . . ,n . . . q o i m i i m i i r r k i k m 1 1is1 is1 rs1 ks1

25 .

where

2 3

I t s A TT t qB TT t q C TT t . 26 . . . . . . i k k p s 1 p s k p s 2 p s k 3ps1ss1

Coefficients A , B and C and components TT , TT and TT for different valuesk k k 1 2 3of p and s are given in Appendix 4 and Appendix 5 respectively.

w xThe expression for the response covariance is 27

2 n n 2 n n n n

K t ,t s u u m u u m . q q 1 2 i ll llr r s k p p g g wi k n nrs1ps1ss1gs1ws1lls1

= I t ,t y I t I t , i ,ks 1 , 2 , . . . ,n 27 . . . . 4s , g 1 2 lls 1 p g 2




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D. YADAV ET AL.12

.where i s i q n, k s kq n, I and I are defined in Eq. 26 and integral In n lls p g s, gis given by

` `U

I t ,t s H v ,t H v ,t F v ,v dv dv . 28 . . . . .H Hs , g 1 2 ll 1 1 p 2 2 F F 1 2 1 2s gy` y`

The above integral can be evaluated with the knowledge of the input PSD. The

required input PSD are placed in the Appendix 6 using the track PSD of the formw x9

F V , V sAXexp ybV2 d V y V , 29 . . . .h h 1 2 1 1 2R R

where AX

and b are roughness constant and correlation index for a particular class

of track. The integral I is evaluated using Cauchys residue theorem. The basics, gterms needed for the covariance evaluation are

allX X X<


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large number of dynamic modes have to be analysed. The full presentation of the

results, however, would require a very large space and would become impractical.

Hence, the results have been presented here for a few selected initial modes only,

which are known to be more significant compared to the higher modes. Results

have been obtained for constant velocity taxi, accelerating take-off and decelerat-

ing landing runs. The first two - taxi and take-off - are used to present the model

response behaviour while the landing run is used for drawing a comparison of the

present model with other models. The touch down impact at landing is expected to

show clearly the differences in the performance of the various models.

5.1. Taxi Run

In this run the aircraft is assumed to move at constant velocity. Results have been

obtained with three forward velocities - 40 Kmrh, 60 Kmrh and 80 Kmrh. Only

the mean and variance of the displacement response are presented here for the

initial 20 seconds period in Figures 2 and 3 for the first three flexural and two

Fig. 2. Mean Displacement Response - Taxi Run. Key :- Vehicle speed: - - - - 40 Kmrh; ... 60 Kmrh;80 Kmrh.




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D. YADAV ET AL.14

Fig. 3. Displacement Response Variance - Taxi Run. Key as in Figure 2.

torsional modes of the airframe, three coupled bending-torsional modes of the left

wing, one flexural mode of the track and heave modes of the three unsprung

masses. Out of the airframe modes the first two flexural modes represent fuselage

rigid body heave and pitch while the first one in torsion is for the rigid rolling ofthe airplane. The right wing modes are similar to the left wing for the present case

and have not been presented to save space.

The mean track level selected has an increasing slope with superimposed

sinusoidal profile. All the mean displacements of the aircraft and the pavement

show influence of the track mean profile to various degrees. Out of the two

angular rigid body modes, rolling motion is more affected by the change in vehicle

forward velocity. The elastic fuselage modes in bending and torsion reveal the

presence of high frequency components in mean response for the initial phase

which gradually diminishes with time, leading to a steady state pattern. In mostcases, a single dominant frequency can be observed in the steady state part which

varies with the change in vehicle forward velocity. High frequency oscillations in




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the transient stage are less sensitive to the change in vehicle velocity. This

indicates that they are caused by the system structural modes rather than external

excitation.

The response of the unsprung masses shows a similarity in pattern though the

two main wheels mean response characteristics are not found to be identical. This

may be due to coupled rolling of the vehicle.

The wing mean displacements exhibit clear steady state pattern in the taxi run.

The track normal coordinate displacement mean under the aircraft c.g. also

reflects the pattern of the track mean profile.

The displacement variances show that in the early stages, response is oscillatory

with a high peak at the onset of the vehicle motion. Subsequently the response

subsides to steady asymptotic values dependent on the vehicle forward velocity.

5.2. Take-off Run

Take-off run is an accelerating condition of the vehicle. The aircraft starts from

rest. It is assumed to run with uniform acceleration until the lift-off velocity is

achieved and the plane takes off. Three different forward accelerations 1.6, 1.8 and

2.0 mrs2 with a take-off velocity 216 Kmrh have been considered. The time

history plots are presented from start of the motion to lift-off.

Figure 4 presents mean and variance for the displacement during take-off for

some selected responses - three bending and two torsional modes of the fuselage,

nose wheel, left main wheel, first wing bending mode and first pavement bendingmode. At the initiation of the motion the mean response shows low transience.

Mean response magnitudes are seen to build up gradually with the increase in the

vehicle forward speed and increase in track mean level with rising gradient.

However, as the aircraft proceeds to take-off, the mean displacement response

shows a decrease in amplitude. This indicates the reduction in the strength of

ground input as the wheels start loosing contact with the ground. Rigid body

motion in roll seems to be more pronounced with increasing velocity. The track

normal coordinates mean response shows an initial low value which subsequently

rises as the vehicle vibration increases with increase in speed imposing moredynamic load. The high frequency initial transience is unaffected by the changes in

forward velocity and are inherent to the structures, while the low frequency

transients reflect the track mean profile input.

The variances for the vehicle response show very low values in the initial stage.

As the vehicle accelerates from rest, the track input keeps increasing in strength

with the increase in the forward velocity. The variance characteristics, in general,

seems to follow this pattern. Variances are seen to increase faster with forward

acceleration of the vehicle but at take-off instances, differences in magnitudes are

not large in all cases.The system displacement variances exhibit some high frequency variation in

their response characteristics that are quickly damped out. The vehicle pitch




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D. YADAV ET AL.16

. .Fig. 4. Displacement Response characteristics - Takeoff Run. a Mean, b Variance. Key :- Vehicle2 2 2forward acceleration: ---- 1.6 mrs ; . . . 1.8 mrs ; 2.0 mrs .

response, however, has more persistent oscillations compared to the other re-

sponses.

5.3. Comparison of Model Behaviours in Landing Run

The accuracy of the response analysis depends on the model employed and hencethe selection of the model is very important. A comparative study of the

displacement and acceleration response of different vehicle models is presented in




18/29


this section to gain some insight into their relative behaviour. The landing run with

touch-down impact, the most severe of the ground runs, has been selected for this

study. Four different vehicle models have been considered. These are: .a Model 1- Two point heave-pitch model with aircraft fuselage as a rigid

w xmember 27 . .b Model 2- Two point input heave-pitch model as in Model 1 with bending

flexibility included for the fuselage. .c Model 3- Model 2 along with roll effect. Gives a three point heave-pitch-roll

model with fuselage bending and rigid roll. .d Model 4- Improves Model 3 by including fuselage torsional flexibility to

have a three point heave-pitch-roll model including fuselage

bending and torsional flexibility. This is the model analysed in

the present study.

The four models describe a ground vehicle with increasing degree of accuracy.

Model 1 and 2 are two point input models and ignore the roll effect. Model 3 and

4 are three point input models and incorporate the roll effects.

Displacement and acceleration mean and variance of heave, pitch, roll and one

bending mode of the fuselage, nose and one main wheel, first coupled bending -

torsion modes of one wing and track first normal coordinate have been presented

for comparison. The landing touch-down condition is assumed to have vertical

sink velocity 1.2 mrs and forward glide velocity 216 kmrhr. The run is assumed

to be with uniform slowing rate 1.5 mrs2. Plots are generated from the touch-down

instant to stoppage of the vehicle forward motion.

Figure 5 presents the mean and the variance of the displacement response. Thebehaviour of the three point input models are different from the two point input

models for the fuselage response. The roll model indicates smaller values of initial

peak displacements compared to the no roll models. The two point input models,

in general, possess more higher frequency components than the three point input

models. The variance response of the fuselage shows greater differences between

the two sets of models compared to the mean response. The above observations

are also true for the nose wheel and the wing response. The mean response of the

main wheel for the two types of models, however, are not very different.

The track response also indicates the presence of strong initial impact attouch-down of the aircraft for the first two models. However, the frequency

content for the pavement mean response due the two sets of models are close.

Mean and variance of the rigid body rolling of the vehicle in Model 3 and 4

have similar behaviour at touchdown impact with little difference in the amplitude.

Flexible wings normal coordinate mean and response variance in the two point

input models show strong sensitivity to the landing impact. In the latter phase of

landing run, displacement response characteristics in all the models are close. This

is expected as the impact energy is quickly dissipated and the vehicle forward

motion also slow down.The acceleration response mean and variance are placed in Figure 6. The

comparative observations between the two types of models for the displacement




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D. YADAV ET AL.18

. .Fig. 5. Comparison of Displacement response Characteristics in landing run. a Mean, b Variance.Key :- Model 1 - - - -; Model 2 . . . ; Model 3 ; Model 4 . . . .

characteristics are generally valid for the acceleration response also. The accelera-tion decays faster than the displacement. The damping dissipation of the roll

models seems to be stronger than the no roll models.




20/29


. .Fig. 6. Comparison of Acceleration response Characteristics in landing run. a Mean, b Variance.Key as in Figure 5.

The comparison shows that inclusion of roll mode is important for study of thevehicle behaviour. In general, magnitudes of initial impact response mean and

variance of heave-pitch vehicle models are higher than the heave-pitch-roll




21/29

D. YADAV ET AL.20

models. Inclusion of the elastic torsional mode of the vehicle shows minor

influence on the response behaviour as compared to the elastic bending mode.

6. CONCLUSIONS

An analytical method has been worked out to evaluate the response statistics of

flexible vehicle - flexible track models with multiaxled multiwheeled configura-

tions. Any deterministic variation of vehicle forward motion and non homogeneity

of ground unevenness can be handled. Some conclusions arrived at in the study are .1 Interactions of heave, pitch and roll modes are significant for response

prediction in case of unsymmetrical distribution of roughness across the track. .2 Bending flexibility is more significant compared to torsional flexibility of

the long and slender vehicle for accurate description of the response behaviour. .3 Vehicle and pavement response is highly dependent on the vehicle speed

and track roughness input. .4 Periodicity present in the track mean shape is reflected in the mean response

of the vehicle and the pavement. Removal of any periodic component in the track

mean is an important part of maintenance as at some particular vehicle speed the

input may have frequencies close to some system frequency and cause unaccept-

ably large response.

It can be summarised that a working technique has been presented for the first

time to analytically obtain the response characteristics of a multi-wheeled and

multi-axled vehicle moving over a flexible nonhomogeneous track. This has great

significance for design of light bodied vehicles, like aircrafts, as displacements andthe associated stresses and strains can be provided accurately in the probabilistic

environment.

REFERENCES

1. Silby, N.S., An analytical study of effects of some airplane and landing gear factors of theresponse to runway roughness with application to supersonic transport, NASA TND - 1492,1962.

2. Tung, C.C., Penzien, J., Horenjeff, R., The effect of runway unevenness on the dynamic response

of the supersonic aircraft, NASA CR - 119, 1964.3. Virchis, V.J., Robson, J.D. Response of an accelerating vehicle to random road undulations,

Journal of Sound and Vibrations, 18, 1971, pp. 423427.4. Sobczyk, K. and Macvean, D.S., Nonstationary random vibration of system travelling with

variable velocity, Stochastic Problem on Dynamics, University of Southampton, Ed. B.L.Clarkson, 1976, pp. 412432.

5. Yadav, D., Nigam, N.C., Ground induced nonstationary response of vehicles, Journal of Soundand Vibrations, 61, 1978, pp. 117126.

.6. Harrison, R.F., Hammond, J.K., Evolutionary frequencyrtime spectral analysis of the responseof vehicles moving on rough ground using covariance equivalent modelling, Journal of Soundand Vibrations, 107, 1986, pp. 2938.

7. Kirk, C.L., The random heave - pitch response of aircraft to runway roughness The Aeronauti-

cal Journal, RAS 75, 1971, pp. 476483.8. Pintado, P. and Benitez, F.G., Optimization for vehicle suspensions I Time domain Vehicle

System Dynamics, 19, 1990, pp. 273288.9. Yadav, D. and Upadhyay, H.C., Dynamics of vehicles in variable velocity runs over nonhomoge-




22/29


neous flexible track and foundation with two point input models Journal of Sound andVibrations, 156, 1992, pp. 247268.

10. Yadav, D. and Upadhyay, H.C., Heave - pitch - roll dynamics of a vehicle with a variablevelocity over nonhomogeneously profiled track, Journal of Sound and Vibrations, 164, 1993 pp.337348.

11. Wilson, J.F. and Biggers, S.B., Dynamic interactions between long, high speed trains of

aircushion vehicles and their guideways, Journal of Dynamic System, Measurement and Control,ASME, 93, 1971, pp. 1624.12. Khulief, Y.A. and Sun, S.P., Finite element modelling and semiactive control of vibrations in

road vehicles, Journal of Dynamic System, Measurement and Control, ASME, 111, 1989. pp.521527.

13. Hac, A., Youn, I. and Chen, H.H., Control of suspensions for vehicles with flexible bodies-Part-I:Active suspensions Journal of Dynamic System, Measurement and Control, ASME, 118, 1996,pp. 508517.

14. Cole, D.J. and Cebon, D., Validation of an articulated vehicle simulation, Vehicle SystemDynamics, 21, 1992, pp. 197223.

15. Criner, H.E., McCann, G.D., Rails on elastic foundation under the influence of high- speedtravelling loads, Journal of Applied Mechanics, 20, 1953, pp. 1322.

16. Kenney, J.T., Steady-state vibrations of a beam on elastic foundation for a moving load,Journal of Applied mechanics, 21, 1954, pp. 359364.17. Saito, H., Murakami, T., Vibration of an infinite beam on an elastic foundation with considera-

tion of mass of a foundation, Bulletin of Japan Society of Mechanical Engineers, 12, 1969, pp.200205.

18. Rades, M., Steady state response of a finite beam on a Posternak-type foundation InternationalJournal of Solids and Structures, 6, 1970, pp. 739756.

19. Kerr, A.D., The continuously supported rail subjected to an axial force and a moving load,International Journal of Mechanical Science, 14, 1972, pp. 7178.

20. Rades, M., Dynamic analysis of an inertial foundation model International Journal of Solidsand Structures, 8, 1972, pp. 13531372.

21. Fryba, L., Vibration of solids and structures under moving loads, Noordhoff InternationalPublishing, Groningen, 1972.

22. Fryba, L., Non-stationary response of a beam to a moving random load, Journal of Sound andVibrations, 46, 1976, pp. 323338.

23. Suzuki, S.J., Dynamic behaviour of a finite beam subjected to travelling loads with acceleration,Journal of Sound and Vibrations, 55, 1977, pp. 6570.

24. Hino, J., Yoshimura, T., Ananthnarayan, N., Vibration analysis of non-linear beams subjected tomoving loads using the finite element method, Journal of Sound and Vibrations, 100, 1985, pp.477491.

25. Iyengar, R.N., Pranesh, M.R., Dynamic response of a beam on a foundation of finite depth, .Indian Geotechnical Journal, 15 2 , 1985, pp. 5363.

26. Yoshimura, T., Hino, J., Kamata, T., Ananthnarayana, N. Random vibration of a non-linear beamsubjected to a moving load: a finite element analysis, Journal of Sound and Vibrations, 122,1988, pp. 317329.

27. Talukdar, S., Kamle, S. and Yadav, D., Track induced heave - pitch dynamics of vehicles withvariable section flexible attachements, Vehicle System Dynamics, 28, 1998, pp. 126.

28. Kortum, W., Wormley, D.N., Dynamic interaction between travelling vehicles and guidewaysystem, Vehicle System Dynamics, 10, 1981, pp. 285317.

29. Diana, G., Cheli, F., Dynamic interaction of railway systems with large bridges, Vehicle SystemDynamics, 18, 1989, pp. 71106.

30. Diana, G., Cheli, F., Bruni, S., Collina, A., Interaction between railroad super-structure andrailway vehicles, The Dynamics of Vehicles on Roads and Tracks, Supplement to Vehicle System

Dynamics, 23, 1994, pp. 7586.31. Nigam, N.C., Introduction to Random Vibration, MIT Press Cambridge 1983.32. Fung, Y.C., An Introduction to the theory of aeroelasticity, 1955, New York, Dover Publica-

tion.

33. Bishop, R.E.D., Price, W.G., Coupled bending and twisting of a timoshenko beam, Journal ofSound and Vibration, 5o, 1877, pp. 469477.

34. Talukdar, S., Dynamic response of flexible aircraft in ground runs over nonhomogeneouslyprofiled flexible runway Ph.D. Thesis, Department of Aerospace Engineering, I.I.T. Kanpur,1997.




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D. YADAV ET AL.22

LIST OF SYMBOLS

A State matrix

AX

Roughness constant

A Aspect ratioTa Coefficients of describing polynomialk

c System damping matrix

C , C Lift and aerodynamic moment coefficients L m acC Viscous damping

c Viscous damping per unit length

c Mean chord lengthhD Modal damping coefficient

E Youngs modulus of elasticity

e Shear center offset

F Generalised force vector

f Distributed applied loading

G Shear modulus

g Distance measured perpendicular to vehicle longitudinal axis

H Frequency response function

h Track elevation

h Mean track unevennessmh Random track roughnessR

.I Polar moment of inertia MI of the concentrated mass about the

shear center; Wing MI; Track cross section MII Aircraft pitch MI about c.g.sI Aircraft roll MI about c.g.rI Wing mass polar MI about elastic axiswaJ Torsional constant

j Imaginary unit

K Covariance, Spring constant

K System stiffness matrix

k Distributed spring stiffness, counting index

L Lengthl ,l ,l Wing elastic axis and axle distance from the vehicle c.g0 1 2M System mass matrix

M Distributed bending moment

m Mass per unit length, counting index

n Dimensions of the system matrices, number of normal modes

Q Generalised force

q Response vector

S Surface area

s Space coordinateT Torsional mode shape

t Time instant




24/29


U Modal matrix

u Eigen vector

V Vehicle forward velocity

V Lift force per unit length on the wingacW Bending displacement function

w Transverse displacement

X Integration constantox Distance along the track

x Location of vehicle c.g along the trackcy Track transverse displacement

z Unsprung mass vertical displacement

a Eigen values

d Dirac delta function; Kronecker delta

G Distributed torque

h Generalised coordinates

u Angle of twist

q Distributed rotational damping

m, s Mean and standard deviation

r Mass density

F Power spectral density

f Torsional displacement functions

c Track displacement function in the ith mode

V Spatial frequency of the track input

v Temporal frequency of the track input; natural frequency

Subscripts

L Left

R Right

1 Front axle, Center wheel

2 Rear axle, Left wheel

3 Right wheel

f Foundation

i ith mode

p Track

s Suspension

u Tyre

b Vehicle body bending

t Vehicle body torsion

w Wing

k kth concentrated mass

Superscript

g Generalised quantityL Left wing

R Right wing




25/29

D. YADAV ET AL.24

APPENDIX

1. Generalised Quantities for Vehicle Body in ith mode:

Mass Mg s m r W2 r dr A1 . . .Hb i b b iLb

Moment of Inertia Ig s I r T2 r dr A2 . . .Hb i b i b iLb

1Bending Damping D s c r W r W r dr A3 . . . .Hi k b b i bk gM Lb i b

1Torsional Damping D9 s q r T r T r dr A4 . . . .Hi k b bi bk gI Lb i b

1X

Force Q s f r,t W r qM r,t W r dr A5 4 . . . . .Hb i b b i b b iM gbi Lb

1Torque Q s G r,t T r dr A6 . . .Ht i b b ig

I Lb i b

2. Generalised Quantities for the Attachments in ith Mode:

Mass Mg s m s W2 s qI s f2 s y 2 m s e s f s W s ds . . . . . . . . 4Hi w i w i w w i iLW

A7 .

gDamping D s 1rM c s W s W s q q s . . . . .Hi k i w i k Lw

qe2 s c s f s f s y e s c s f s W s . . . . . . . .4w w i k w w i k

qW s f s ds A84 . . .i K

gForce Q s 1r M f s,t W s y e s f s q G s,t f s ds 4 . . . . . . .

Hw i i w i w i w i

Lw

A9 .




26/29


3. Generalised Quantities for the Track beam in ith Mode:

Mass M s m c2 x dx A10 . .H p i p iLp

Force Q s 1r M f x,t c x dx A11 . . . .H p i p i p iLp

4. Coefficients for the mean forcing function

A s 0 for ks1 , 2 , . . . n q n ,n q n q 4 , . . . , n q n q 3 q 2 n and all p , sk p s b t b t b t w

s K for ksn q n q 1, and ps ss1; ksn q n q 2, and p s su p s b t b t

s 2; ksn q n q 3, and ps2, ss3b t

s c x K rM for ks n q n q 4 q 2 n , . . . ,n; .ky j p u p s k y j b t w

and p s s s 1,2; p s 2, s s 3; j s n q n q 3 q 2 n A12 .b t w

.B - Expression as in Eq. A12 with K replaced by C.kps

c s 0 for ks1 , 2 , . . . ,n q n q 3, n q n q 4 q 2 n , . . . ,nk b t b t w

1r2g L L L 2 2s L r M W s y e s f s 1 y s r L ds . . . 4 4 . H0 kyj ky j L w L L k y j L L w LLw

g L Lq M r M c s f s ds , for ksn q n q 4 , . . . , n q n . . .H0 ky j h L k y j L L b t b t Lw

q 3 q n ; jsn q n q 3w b t

1r2g R R R 2 2s L r M W s y e s f s 1 y s r L ds . . . 4 4 . H0 kyj ky j R w R R k y j R R w RLwL

g R 2 Rq M r M c s f s ds for ksn q n q 4 , . . . , n q n . . .H0 ky j h R k y j R R b t b t LwR

q 3 q 2 n ; jsn q n q 3 q n A13 .w b t w

with

L s r C S r2pL , M s r C r20 a L W W 0 a m ac




27/29

D. YADAV ET AL.26

where r is air density, C and C are aerodynamic coefficients for lift anda L m acmoment, S and L are plan form areas and semi span of wing.w w

5. Components of integral I for the mean responsei k

ll ll kk

TT s h q h c J q h c J , . . . . . p s p s p s p s p s1 p s 0 i i ,0 1 i i , r 2is1 is1 rs1

ll ll kky1

TT s h c J q h c rq 1 J , . . . . . p s p s p s p s2 p s i i ,1 1 i i , rq1 2is1 is1 rs1

.2 my1X X

TT s c J q c J , A14 .3 0 1 r 2rs1

..with J s 1 y exp ya t ra , kks m.i1 i i

rr rykrt y1 r! t r! . rJ s q y y1 exp ya t , . .2 ikq1 rq1a ry k ! a a .i i iks1

and

r

ic s a , c s 1ra r ki y rq k a c ; rG 1 . . . . . . p s p s p si ,0 p 0 i , r p0 p k i , rykks1

a s a q 1 , a s a s a y 1 ; a s a s a s a , ks 1 , 2 , . . . ,m10 0 1 20 30 0 2 1 k 2 k 3 k k

r2X X X X X X

c s a ; c s 1rra 3ky r a c , rG 1; a s kq 1 a . . . .0 0 r 0 k ryk k kq1ks1

6. Input PSD in terms of track characteristics

f v ,v s 0 for s, gs1 , 2 , . . . n q n ; n q n q 4 , . . . , n q n q 3 q 2 n .F F 1 2 b t b t b t ws g

X X X X X X 4s C C v v qj v C K y v K Cu r k u r k 1 2 1 u r k u r k 2 u rk u r k

X XqK K F v , v .XXu r k u r k h h 1 2kr k r

for s s g with s s n q n q 1 , . . . , n q n q 3;b t b t

s s n q n q 1, rs rX

s ks kX

s 1;b t

s s n q n q 2, rs rXs ks k

Xs 2;b t




28/29


and s s n q n q 3, rs rXs 2, ks k

Xs 3; for s / g ,b t

s s n q n q 1, gs s q 1, rsks1, rXsk

Xs2;b t

s s n q n q 1, gs s q 2, rsks1, rXs2, k

Xs3;b t

s s n q n q 2, g s s q 2, rs ks 2, rXs 2, k

Xs 3b t

s 1rM c x F v , v q c x F v ,v . . . . . p , gyk gyk 1 F F 1 2 gyk 2 F F 1 2s j s jq 1

qF v , v . 4F F 1 2s jq2

for s s n q n q 1 , . . . , n q n q 3; gsn q n q 4 q 2 n , . . . ,n;b t b t b t w

ks n q n q 3 q 2 n ; jsn q n q 1b t w b t

s 1rM M c x c x F v ,v q c x c x . . . . . .p , syk p , gyk syk 1 gyk 1 f f 1 2 syk 1 gyk 2j j

qc x c x F v ,v q F v , v . . . .4 4sy k 2 gyk 1 F F 1 2 F F 1 2j jq1 j jq2

qF x c x F v ,v . . .sy k 2 gyk 2 F F 1 2jq 1 jq1

q2F v ,v q F v , v . . 4F F 1 2 F F 1 2jq 1 jq2 jq2 jq2

for s, g s n q n q 4 q 2 n , . . . ,n A15 .b t w

w x7. System Data used for obtaining numerical results 34

I. Aircraft

1. Fuselage5 .Mass M : 1.46 = 10 Kgb

6 2

.Pitch Moment of inertia I : 7.178 = 10 Kgms 5 2 .Roll moment of inertia I : 6.325 = 10 KgmsLength : 44.35 m

10 2 ..Flexural rigidity E I 0 : 9.135 = 10 Nmb b9 2 ..Torsional rigidity G J 0 : 11.6 = 10 Nmb b

-1 . ..Viscous damping c 0 rm 0 : 0.298 s .b b2. Landing gears

Nose gear Main gear .Distance from arc c.g. 1 ,1 16.68 1.32 m1 2

Main gear tread: 6.3 m 3 .Mass M ,M sM 0.248 1.1105 = 10 Kgu11 u22 u236 .Stiffness K , K s K 2.36 14.5 = 10 Nrms11 s22 s23




29/29

D. YADAV ET AL.28

. 4damping C ,C s C 8.24 50.1 = 10 Nsrms11 s22 s 236 .Tyre stiffnesses K , K s K 1.34 8.5 = 10 Nrmu11 u22 u23

6 .Tyre damping C ,C , C 0.98 7.86 = 10 Nsrmu11 u22 u233. Wing

.Semi span L : 10.0 mw .Aspect ratio A : 7.08T ..Mean chord c 0 : 3.0 mh

6 2 ..Flexural rigidity E I 0 : 7.174 = 10 Nmw w6 2 ..Torsional ridigity G J 0 : 4.662 = 10 Nmw w

2 ..Mass m. i. about elastic axis I 0 : 8.643 kgm rmwa ..Massrlength m 0 : 42.73 kgrmw

..Shear center offset e 0 : 0.25 mw-1 . ..Linear damping c 0 rm 0 : 0.16 sw w

-1 . ..Torsional damping G J 0 rI 0 : 0.008 sw w wa

.Aerodynamic coefficients lift C : 0.8,L .moment C : 0.1mac

3 .Air density r : 1.12 kgrma

II. Runway

1. Properties .Length L : 1400.0 mp

.Massrlength m : 3620 kgrm07 2 .Flexural rigidity E I : 1.38 = 10 Nmp p

-1 .Foundation damping c rm : 0.04 sf 08 2 .Foundation stiffness k : 1.705 = 10 Nrmf

X -5 .Roughness constant A : 0.5025 = 10 .Correlation index b : 0.1012

2. Mean profile

The mean track is assumed to be a sinusoidal profile over a uniform slope. This is .represented with the following parameters in Eq. 3

h s h s h s . . . s 0,0 2 43 .h s 0.001 q 2pA rW , h s y2pA r 3!W ,1 0 1 3 0 1

5

.h s 2pA r 5!W etc. where w s 100 pm5 0 1 1A s 0.075, 0.04, 0.06 m for nose, left main and right main wheel paths.0



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Heave and Pitch Dynamics