MEHANIKA VO NJE - Odsek za puteve, eleznice i aerodrome · Poduºna dinamika ²inskih vozila...

Poduºna dinamika ²inskih vozilaKretanje voza u krivini

Interakcija ²inskih vozila i kolosekaVinklerova podloga

Naponski talasi u tlu

MEHANIKA VO�NJE

Odsek za puteve, ºeleznice i aerodrome

Prof dr Stanko Br£i¢Doc dr Stanko �ori¢Doc dr Anina Glumac

Gra�evinski fakultetUniverzitet u Beogradu

�k. god. 2018/19

S.�ori¢ Mehanika voºnje




Sadrºaj

1 Poduºna dinamika ²inskih vozilaVoz kao jedna materijalna ta£kaVoz kao sistem materijalnih ta£aka

2 Kretanje voza u kriviniUravnoteºuju¢a brzina

3 Interakcija ²inskih vozila i kolosekaNajjednostavniji model: pokretne sileModeli polovine vagonaSloºeniji ra£unski modeli

4 Vinklerova podlogaRazli£iti modeli kolosekaDiferencijalna jedna£ina ²tapa na Vinklerovoj podloziVozovi velikih brzina: kriti£na brzina

5 Naponski talasi u tluVrste naponskih talasaBrzine prostiranja talasaPropagacija naponskih talasa kroz tlo





Voz kao jedna materijalna ta£kaVoz kao sistem materijalnih ta£aka

Sadrºaj











Voz posmatran kao materijalna ta£ka






Poduºna dinamika ²inskih vozila

Voz posmatran kao materijalna ta£ka: pretpostavke

Zanemaruju se veze izme�u lokomotive i vagona

Voz se posmatra kao jedna materijalna ta£ka

Materijalna ta£ka je koncentrisana u sredi²tu mase voza

Masa mat. ta£ke je jednaka ukupnoj masi vozaPrilikom pravolinijskog kretanja voza na voz deluju sile:

- Vu£na sila Ze

- Sila otpora kretanju W- Sila ko£enja Bk

Gravitacione sile deluju (samo) prilikom kretanja voza ponagibu







Osnovni otpori kretanju voza

Otpori trenja u osovinama i vezama

Otpori kotrljanja to£kova po ²inama

Otpori usled trenja klizanja to£kova po ²inama

Aerodinami£ki otpori

Otpori usled elasti£nosti (ugiba) koloseka






Relativno u£e²¢e u otporima kretanju






Voz posmatran kao materijalna ta£ka

Diferencijalna jedna£ina kretanja

Diferencijalna jedna£ina kretanja:

m~a = ~FR odn. m~r = ~Ze + ~W

Prikazano u skalarnom obliku (x je pravac kretanja):

mx = Ze −W

Po£etni uslovi kretanja (obi£no su homogeni):

t = 0 : x(0) = x0 x(0) = v0






Sadrºaj












Voz kao sistem materijalnih ta£aka

Voz se posmatra kao skup materijalnih ta£aka

Svaki vagon i jedna ili vi²e lokomotiva se posmatraju kao pojedna materijalna ta£ka

Posmatra se samo translatorno kretanje vagona i lokomotive(zato su mat. ta£ke dovoljne)

Vagoni i lokomotiva, odn. mat. ta£ke su me�usobno povezani

Veze izme�u vagona i lokomotive su opruge i viskozniprigu²iva£i

Opruge i viskozni prigu²iva£i mogu da budu linearni ilinelinearni






Model voza sa lokomotivom i dva vagona








Model voza sa tri mase (sa lokomotivom i dva vagona) jereprezent proizvoljne kompozicijeU kompoziciji voza postoji

- vozilo na £elu kompozicije (masa m1)- vozilo unutar kompozicije (masa m2)- vozilo na kraju kompozicije (masa m3)

Lokomotiva moºe da bude na bilo kojoj poziciji (u ra£unskommodelu)

U realnosti je lokomotiva na po£etku i/ili na kraju kompozicije








Sile koje deluju na vagone i lokomotivu:Sila otpora kretanju Fri

- sila otpora kotrljanju ("rolling resistance")- sila aerodinami£kog otpora ("aerodynamic resistance")

U sile otpora kretanju spada i sila (pneumatskog) ko£enjavagona (ne i lokomotive!)

Komponenta sile gravitacije Fgi (ako je trasa u nagibu:usponu ili padu)

Vu£na sila ili sila (dinami£kog) ko£enja lokomotive Ft/db






Komponente sile gravitacije: kretanje u nagibu






Sistem sa dve materijalne ta£ke

12/06/2005Prof dr Stanko Brcic 2

Systems with Two DOF –Undamped vibration

k1m1 F1(t)

x1

m2F2(t)

k2 k3x2








Systems with Two DOF

m

a1 = x1”

F1(t)F1em

a2 = x2”

F2(t)

x1 x2

F2e F2e F3e

• F1e = k1 x1• F2e = k2 (x2-x1)• F3e = k3 x2

mi ai = Fri(i = 1,2)









• Differential equations of motion:

• or, 23122222

11122111)(

)(xkxxkFxm

xkxxkFxm−−−=−−+=

22231222

12212111)(

)(Fxkkxkxm

Fxkxkkxm=++−=−++









• Matrix form of differential equations:

• where

[ ]{ } [ ]{ } { })(tFxKxM =+

[ ] [ ]

{ } { }⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

=

⎥⎦

⎤⎢⎣

⎡+−

−+=⎥

⎦

⎤⎢⎣

⎡=

)()(

)()()(

)(

00

2

1

2

1

322

221

2

1

tFtF

tFtxtx

tx

kkkkkk

Km

mM









• Matrices [M] and [K] are constant, symmetric and positive definite matrices:

[ ] [ ] [ ] [ ]

22132222111

2112222111

12

0

)2,1,(

kkkkkkkkkmmmmmm

jikKmM ijij

−==+=+=====

===








Sile koje deluju na vagone i lokomotivu:Sila otpora kretanju Fr,i


U sile otpora kretanju spada i sila (pneumatskog) ko£enjavagona (ne i lokomotive!)

Komponenta sile gravitacije Fg,i (ako je trasa u nagibu:usponu ili padu)

Vu£na sila ili sila (dinami£kog) ko£enja lokomotive Ft/db








Generalisane koordinate: translacija vozila x1(t), x2(t) i x3(t)

Diferencijalne jedna£ine kretanja - razdvajanje vozila

Prvo vozilo (lokomotiva)

m1a1 + c1(v1 − v2) + k1(x1 − x2) = Ft/db − Fr1 ± Fg1

Srednje vozilo (vagon)

m2a2 + c1(v2 − v1) + c2(v2 − v3)+ k1(x2 − x1) + k2(x2 − x3) = −Fr2 ± Fg2

Poslednje vozilo (vagon)

m3a3 + c2(v3 − v2) + k2(x3 − x2) = −Fr3 ± Fg3S.�ori¢ Mehanika voºnje







Brzine i ubrzanja vozila su ai = xi vi = xi

Matri£ni oblik diferencijalnih jedna£ina kretanja

[M ]{x}+ [C]{x}+ [K]{x} = {F }t/db − {F }r ± {F }g

gde su, redom,

Matrica mase i vektor ubrzanja:

[M ] =

m1

m2

m3

{x} = {a} =

x1x2x3








Matrica prigu²enja i vektor brzine:

[C] =

c1 −c1 0−c1 (c1 + c2) −c20 −c2 c3

{x} = {v} =

x1x2x3

Matrica krutosti i vektor pomeranja:

[K] =

k1 −k1 0−k1 (k1 + k2) −k20 −k2 k3

{x} =

x1x2x3








Vektori optere¢enja: vu£na sila ili sila ko£enja, kao i sila otporakretanju (otpor kotrljanja i aerodinami£ki otpor)

{F }t/db =

Ft/db,1

00

{F }r =

Fr,1Fr,2Fr,3

Komponenta gravitacione sile (±mg sinα): ako se voz kre¢epo usponu, znak "-" ili po padu, znak "+"

{F }g =

Fg,1Fg,2Fg,3







Model sa proizvoljnim brojem lokomotiva i vagona

Posmatra se kompozicija sa proizvoljnim brojem lokomotiva ivagona

Lokomotiva moºe da se na�e na bilo kojoj poziciji ukompoziciji (napred, nazad ili u sredini, iako je to nerealno)

Generalisane koordinate: translacija svakog vozilaxi(t), (i = 1, 2, . . . , n)

Diferencijalne jedna£ine kretanja se izvode razdvajanjem vozila(posmatraju se kao slobodne mat. ta£ke)

Sile veze izme�u vozila su sile u oprugama i prigu²iva£ima(linearno proporcionalne sa pomeranjima i brzinama vozila)








Prvo vozilo (i=1)

m1a1 + c1(v1 − v2) + k1(x1 − x2) = Ft/db,1 − Fr,1 ± Fg,1

Srednja vozila (broj i: i=2,3,. . . , n-1)

miai + ci−1(vi − vi−1) + ci(vi − vi+1)

+ ki−1(xi − xi−1) + ki(xi − xi+1) = Ft/db,i − Fr,i ± Fg,i

Poslednje vozilo (i=n)

mnan+cn−1(vn−vn−1)+kn−1(xn−xn−1) = Ft/db,n−Fr,n±Fg,n








Matri£ni oblik diferencijalnih jedna£ina kretanja

[M ]{x}+ [C]{x}+ [K]{x} = {F (t)}t/db − {F }r ± {F }g(1)

Matrica mase i vektor ubrzanja (analogno i vektori brzine ipomeranja):

[M ] =

m1

. . .mi

. . .mn

{x} =

x1...xi...xn








Matrica prigu²enja (tri-dijagonalna struktura) [C] =

c1 −c1−c1 (c1 + c2) −c2

. . .−ci−1 (ci + ci+1) −ci+1

−cn−2 (cn−1 + cn) −cn−cn−1 cn








Matrica krutosti (tri-dijagonalna struktura) [K] =

k1 −k1−k1 (k1 + k2) −k2

. . .−ki−1 (ki + ki+1) −ki+1

−kn−2 (kn−1 + kn) −kn−kn−1 kn








Vektori optere¢enja: vu£na sila ili sila ko£enja, kao i sila otporakretanju (otpor kotrljanja i aerodinami£ki otpor)

{F }t/db =

Ft/db,1...

Ft/db,i...

Ft/db,n

{F }r =

Fr,1...Fr,i...

Fr,3

U vektoru {F (t)}t/db su elementi 6= 0 samo na mestima gde senalazi lokomotiva u kompoziciji








Optere¢enje usled gravitacione sile (komponenta ±mg sinα)postoji samo ako se voz kre¢e po usponu, znak "-" ili po padu,znak "+"

{F }g =

Fg,1...Fg,i...

Fg,n








Diferencijalne jedna£ine kretanja (1) su sistem obi£nih linearnihdiferencijalnih jedna£ina sa konstantnim koe�cijentima

Matrica mase je dijagonalna matrica

Matrica prigu²enja i matrica krutosti su simetri£ne ipozitivno-de�nitne matrice tro-dijagonalne strukture

Problem je u odre�ivanju odgovaraju¢ih numeri£kih podatakaza konstante ci i kiProblem je i u odgovaraju¢em de�nisanju vektora optere¢enja(kako vu£ne sile, ili sile ko£enja, tako i u sili otpora kretanju)







Nelinearni model sa proizvoljnim brojem lokomotiva i vagona

Imaju¢i u vidu stvarne veze izme�u vagona, odn. izme�uvagona i lokomotive, linearne sile veze oblika F = kx, kao iF = cv nisu dovoljno dobra aproksimacija

Formuli²u se nelinearne relacije za sile veze izme�u vozila

U nelinearnom modeliranju kretanja voza, sile veze izme�uvozila (vagona i lokomotive) prikazuju se nelinearnimfunkcijama zavisnim od brzina i ubrzanja vozila F = F (v, x)

Dobija se sistem nelinearnih diferencijalnih jedna£ina kretanja








Sile veze izme�u pojedinih vozila kompozicije:- za prvo vozilo (i = 1)

Fwc,1 = Fwc,1(v1, v2, x1, x2)

- za vozilo unutar kopozicije (i=2,3,. . . ,n-1)

Fwc,i = Fwc,i(vi−1, vi, vi+1, xi−1, xi, xi+1)

- za poslednje vozilo (i = n)

Fwc,n = Fwc,n(vn−1, vn, xn−1, xn)

Linijska struktura kompozicije voza generi²e trakastu(trodijagonalnu) strukturu u silama veze








Prvo vozilo (i=1)

m1 a1 + Fwc,1 = Ft/db,1 − Fr,1 ± Fg,1

Srednja vozila (broj i: i=2,3,. . . , n-1)

mi ai + Fwc,i = Fr/tb,i − Fr,i ± Fg,i

Poslednje vozilo (i=n)

mnan + Fwc,n = Ft/db,n − Fr,n ± Fg,n








Jedna£ine kretanja se napi²u za sve slobodne ta£ke i prikaºu seu matri£nom obliku

Matri£ni oblik nelinearnih diferencijalnih jedna£ina kretanja

[M ]{x}+ {F }wx(x,x) = {F (t)}t/db − {F }r ± {F }g

U linearnom pristupu vektor nelinearnih sila veze izme�uvagona {F }wx(x,x) se dobija u vidu linearnog zbira

{F }wx(x,x) = [C]{x}+ [K]{x}






Veze izme�u vozila (vagona i lokomotive)







Sila otpora kretanju FrSila otpora kretanju Fr,i ("propulsion resistance")


Sila otpora kretanju se obi£no prikazuje u obliku zavisnosti odbrzine kretanja v:

Fr = A+B v + C v2

gde su A,B, C empirijski coe�cijenti






Sila otpora kretanju Fr







Vu£na sila i sila ko£enja lokomotive Ft/db

Vu£na sila i sila ko£enja lokomotive Ft/db ("traction anddynamic braking") se prikazuju kao ista sila u ra£unskommodelu, ali se razlikuju po smeru

Vu£na sila i sila ko£enja lokomotive se komplikovano de�n²u ura£unskim modelima

Dizel elektri£ne lokomotive imaju obi£no 8 nivoa pode²avanjavu£ne sile ("osam brzina", "throttle notch levels")

Pri manjim brzinama vu£na sila je proporcionalna sa nivoom(stepenom) N i skoro je nezavisna od brzine

Pri ve¢im brzinama vu£na sila opada sa porastom brzine






Vu£na sila kod dizel elektri£ne lokomotive






Sila ko£enja kod dizel elektri£ne lokomotive






Sila ko£enja kod elektri£ne lokomotive






Sudar vagona i mogu¢e "penjanje" vagona







Udobnost voºnje u poduºnom pravcu

Udobnost voºnje se obi£no izraºava preko ubrzanja uvertikalnom i bo£nom pravcu

Zbog male athezije to£ka i ²ine, u poduºnom pravcu sumogu¢a relativno mala ubrzanja

Sama lokomotiva moºe da ostvari ubrzanje od oko ≈ 0.3 g(odn. red veli£ine 0.1÷ 1.0m/s2), uz pogon na sve to£kove

U slu£aju ko£enja, usporenja su ne²to manja - reda veli£ine0.1÷ 0.6m/s2

Prihvatljive amplitude poduºnog oscilovanja oko 75 mm






Udobnost voºnje - granice ubrzanja






Udobnost voºnje - granice amplituda oscilovanja





Uravnoteºuju¢a brzina

Sadrºaj











Kretanje voza u krivini

Ravnoteºa sila u krivini

Ako se voz kre¢e po krivini sa konstantnim radijusom R i sakonstantnom brzinom v, normalno ubrzanje centra masevagona je

an =v2

R

Normalno ubrzanje je usmereno ka centru krivine

U skladu sa D'Alambert-ovim principom, normalnom ubrzanjuodgovara centrifugalna sila

Fin =mv2

R








Centrifugalna sila teºi da prevrne vozilo na spolja²nju stranu(od centra krivine)

Kolosek mora da se izvede u krivini sa nadvi²enjem spolja²nje²ine h

Ugao popre£nog nagiba koloseka je φkolAko je razmak ²ina G, onda je sinφkol =

hG

Mogu¢a je ravnoteºa izme�u komponente gravitacione sileteºine i komponente centrifugalne (inercijalne) sile

Brzina kretanja vozila pri takvoj ravnoteºi je uravnoteºuju¢abrzina






Ravnoteºa sila u krivini - nadvi²enje spolja²nje ²ine








Ravnoteºa komponente centrifugalne sile i komponentesopstvene teºine

mv2

Rcosφ = mg sinφ (2)

U principu, ugao φ je jednak zbiru ugla popre£nog nagibakoloseka φkol i ugla kotrljanja ("roll angle") vagona, kojipostoji zbog elasti£nih veza izme�u vagona i postolja

Ako se pretpostavi da je φ ≈ φkol, pri £emu je ugao popre£nognagiba relativno mali, onda je

cosφ ≈ 1 sinφ =h

G(3)








Sa h je ozna£eno nadvi²enje ²ine, a G je razmak izme�u ²ina

Unose¢i (3) u jedna£inu ravnoteºe (2) dobija seuravnoteºuju¢a brzina vozila:

v =

√g hR

G(4)

Ako se vozilo kre¢e sa manjom brziom od brzine (4), ondavozilo ima vi²ak nadvi²enja

Ako se voz kre¢e sa brzinom koja je ve¢a od brzine (4), ondapostoji nedostatak nadvi²enja








Nedostatak nadvi²enja spolja²nje ²ine izaziva ve¢e bo£ne silena spolja²njoj ²ini

Takve bo£ne sile mogu da izazovu neºeljeno kretanje to£ka pospolja²njoj ²ini

Mogu¢a posledica je penjanje to£ka na spolja²nju ²inu iispadanje vagona iz koloseka

Pri kretanju u krivini, javljaju se i poduºne sile na kontaktu sa²inama

Spolja²nji to£ak se kre¢e po ve¢em radijusu - prelazi duºi put








Osovina se obr¢e sa konstantnom ugaonom brzinom, pa zbograzlike u pre�enom putu jedan ili oba to£ka na osovini mora daprokliza

Klizanje to£kova se smanjuje ukoliko se promene radijusikotrljanja to£kova - zbog konusnog oblika to£kova upopre£nom preseku

Vagon se u krivini pomera ka spolja²njoj ²ini, £ime se pove¢avaradijus kotrljanja spolja²njeg to£ka

Time se pove¢ava i njegova brzina u poduºnom pravcu uodnosu na unutra²nji to£ak

Na taj na£in se smanjuje proklizavanje i habanje to£kova, odn.dobija se bolje pona²anje u krivini





Najjednostavniji model: pokretne sileModeli polovine vagonaSloºeniji ra£unski modeli

Sadrºaj











Vertikalna dinamika ²inskih vozila

Kretanje ²inskih vozila u vertikalnoj ravni

Zanemaruju se bo£ne sile

Zanemaruje se oscilovanje vagona oko poduºne ose X

Zanemaruje se oscilovanje oko popre£ne ose Y

Vagoni se posmatraju kao kruta tela

Uticaj voza na kolosek: pokretne vertikalne sile






Kretanje voza - sistem pokretnih sila

Najjednostavnija interakcija ²inskih vozila i koloseka: sistempokretnih sila






Nominalne osovinske sile






Sadrºaj











Svaki vagon sa po dva stepena slobode

Realan model: sanduk vagona je kruto telo viskoelasti£no vezano zaobrtna postolja sa to£kovima






Model sa izdvojenom "polovinom" vagona

Svaka "polovina" vagona se posmatra kao jedna mat. ta£kaviskoelasti£no vezana za postolje sa to£kom






Model sa izdvojenom "polovinom" vagona






Prikaz unutra²njih sila veze

December 2, 2005 9:20 Vehicle–Bridge Interaction Dynamics bk04-001

Vehicle–Bridge Interaction Element Considering Pitching Effect 203

y

z

x

element i

element j

fc2fc3 fc1fc4 WW

xcixcj

ui

u j

rirj

xci

xcj

WW

2W

yv

θv

Fig. 7.3. Free body diagrams for components of the VBI system.

used to denote quantities associated with the front and rear wheels

(or wheel assemblies) of the vehicle under consideration. As can be

seen from Fig. 7.2, both elements i and j of the bridge, on which the

vehicle is acting will be affected by the pitching motion of the car

body via the two suspension units. The reverse is also true.

Figure 7.3 shows the free body diagrams for the components of the

VBI system that are of interest, in which 2W denotes the weight of

the car body (i.e., W = 0.5 Mvg, with Mv indicating the total mass

of the car body, and g the acceleration of gravity), xc the contact

position of each set of wheels (or wheel assemblies) on the beam

element, and u the vertical displacement of the beam element. As

can be seen, the (rigid) car body is acted upon by the contact forces

fc1, fc2, fc3, fc4, and the front and rear wheels by the contact forces






Sadrºaj











Sloºeniji model vagona i koloseka

Modern Railway Track 6 DYNAMIC TRACK DESIGN

107

6 DYNAMIC TRACK DESIGN

6.1 Introduction

When dealing with track mechanics most of the problems are related in one way or another to dynam-ics. The dynamic interaction between vehicle and track can be described reasonably well in the verti-cal direction using mathematical models. Figure 6.1 gives an example of such a model made up of adiscrete mass-spring system for the vehicle, a discretely supported beam to describe the track, and aHertzian spring acting in the wheel/rail contact area.

Dynamic behaviour occurs in a fairly wideband ranging from very low frequencies ofthe order of 0.5-1 Hz for lateral and verticalcar body accelerations to 2000 Hz as a con-sequence of geometrical irregularities in railsand wheel treads. The suspension systembetween wheelset and bogie is the firstspring/damper combination to reduce vibra-tions originating from the wheel/rail interac-tion and is therefore called primarysuspension. The reduction of the vibrationsof lower frequency is dealt with in the secondstage between bogie and car body and iscalled secondary suspension. This terminol-ogy can be applied to the track part of the model in the same way. The railpad and railclip representthe primary suspension of the track and the ballast layer or comparable medium represent the sec-ondary suspension of the track.

Actual dynamic calculation is, however, extremely complex and is by no means generally accessible.Most analyses are limited to quasi-static considerations. Real dynamic problems are for the most partapproached in a very pragmatic way by carrying out measurements.

In this chapter attention is given to the basic ingredients of the dynamic behaviour of railway track.Section 6.2 deals with some fundamental aspects. The 1-mass spring system, presented in Section6.2.2, can be regarded as the most elementary system with the aid of which a number of practicalproblems can be considered. Extensions can be made in two directions: the construction can beenhanced to a multi degree of freedom system, and the load can be made more complex in terms ofimpact loads, and loads with a random character.

In Section 6.3 the track is modelled with relatively simple beam models consisting of the beam on anelastic foundation, a double beam, and a discretely supported track structure. The transfer functionbetween track load and track displacement is discussed. Also the effect of a moving load running onthe track is considered, as the track is considered to be infinitely stiff.

Track and rolling stock should in fact not be considered separately, but as one consistent system. Forthis reason the interaction between vehicle and track is introduced here without going into all thedetails required for a full treatment of this complex matter. After the introduction of the Hertzian spring,the physics of which were discussed earlier in Chapter 2, the transfer function between wheel and railis derived in Section 6.4. This relationship plays an important role when interpreting track recordingcar data.

In Section 6.5 a concept is developed from which the relevant vehicle reactions can be calculated inreal time using transfer functions based on track geometry measured independently of speed. Atransfer function represents the contribution made by a geometry component to a vehicle reaction inthe frequency domain. Geometry components include cant, level, alignment, and track gauge, andvehicle reactions include Q forces, Y forces, and horizontal and vertical vehicle body accelerations.

Figure 6.1: Dynamic model of vehicle-track interaction

x

yr

Rail padSleeperBallast

Car body

Bogie

WheelsetHertzian spring

Primary suspension

Secondary suspension






Sloºeniji model vagona i koloseka






3D model vagona i koloseka






3D model vagona i koloseka

December 2, 2005 9:20 Vehicle–Bridge Interaction Dynamics bk04-001

412 Vehicle–Bridge Interaction Dynamics

each is composed of one car body, two bogies and four wheelsets, as

shown in Fig. 11.2. The total number of degrees of freedom (DOFs)

implied by each vehicle is 27. Such a model enables us to simulate

the vertical, lateral, rolling, yawing and pitching motions of the car

body, as well as the vertical and lateral contact forces between the

rails and wheels.

C Body

Bogie

Wheelset

1V3V

5V7V Sleeper

Rail

Ballast

Bridge or Soil Roadbed( )

(a)

1H

2H

3H

4H

5H

6H

7H

8H ATrack

BTrack

Sleeper

1

26

7

8 4

35

Rails

1st wheelset

2nd wheelset

3rd wheelset4th wheelset

(b)

Fig. 11.2. Train car, track and bridge models: (a) side view, (b) top view, and(c) rear view.





Razli£iti modeli kolosekaDiferencijalna jedna£ina ²tapa na Vinklerovoj podloziVozovi velikih brzina: kriti£na brzina

Analiza koloseka usled kretanja voza

Modern Railway Track 13 NUMERICAL OPTIMIZATION OF RAILWAY TRACK

417

Numerical modelsHere the response quantities relating to cost efficiency, acoustic properties, and maintenance effortare considered of importance to the optimum performance of ERS. To estimate the performance of anERS design, static and dynamic models have been developed. The static response quantities suchas stresses and displacements of an embedded rail structure under various loading conditions havebeen obtained using a general purpose finite element package ANSYS.The 2-D and 3-D FE models of ERS are shown in Figure 13.5. Before these models were included inthe optimization process, they were verified by comparing the results of laboratory tests and finite ele-ment calculations [175].

It should be noted that the elastic strip under the rail, which is a common part of existing designs, isnow replaced by elastic compound. In [175] it is demonstrated that the same behaviour of a structurecan be achieved by only using a compound with adjusted E modulus and Poisson ratio.

Three loading cases have been considered to obtain the static response quantities of a structure forassessment of ERS design (Figure 13.8). The dynamic responses of ERS have been obtained usinga finite element program RAIL that is described in Section 6.9. The numerical mode of ERS, builtusing RAIL, is shown in Figure 13.6. Here, the application of RAIL focuses on two aspects, namelyacoustic noise produced by a track and wheel-rail wear.

In order to ensure that the static and dynamic models describe the behaviour of the same ERS, theyhave been coupled to each other by adjusting geometrical properties such as cross-sectional momentof inertia, etc. of the rails in the dynamic analysis based on the parameters of the static model. Also,the static and dynamic vertical stiffness of ERS has been correlated. To determine the static stiffnessof a track the vertical load has been applied at the top of the rail head as shown in Figure 13.8a.The static ( ) and dynamic ( ) vertical stiffness are then calculated as and

( is the vertical displacement of the rail corresponding to this loading case).

Figure 13.5: 3-D (a) and 2-D (b) finite element models of ERS with SA42 rail

Figure 13.6: RAIL model of ERS (moving train loading case)

Elastic bed

Rail

Slab

Elastic compound

Kstat Kdyn Kstat Fy uy 1,⁄=Kdyn 2Kstat= uy 1,






Sadrºaj












Analiza koloseka usled kretanja voza

Najprostija analiza koloseka:�ine se posmatraju kao kontinualni nosa£ na ∞ krutihoslonaca (pragovi, ` ≈ 0.60m)�ine se posmatraju kao kontinualan nosa£ na ∞ elasti£nihoslonaca (pragovi, ` ≈ 0.60m)�ine se posmatraju kao beskona£no dug ²tap na kontinualnimelasti£nim osloncimaAnaliza ²tapa na Winkler-ovoj podlozi (linearna zavisnost sila -ugib)






�tap na kontinualnoj Winklerovoj podlozi






Winklerovoa podloga






�tap na diskretnoj Winklerovoj podlozi






Sloºenije modeliranje koloseka






Modeliranje koloseka primenom metode kona£nihelemenata






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Diferencijalna jedna£ina ravnoteºe (Winkler, 1867)

EJd4w

dx4+ kw(x) = q(x) (5)

- EJ . . . krutost ²ine na savijanje

- w = w(x) . . . ugib ²ine

- k krutost podloge (kN/m2, odn. kN/m po m duºine ²ine)

- q(x) . . . raspodeljeno optere¢enje na ²inu








Uvode¢i oznaku

β =

(k

4EJ

) 14

Op²ti integral dif. jed. (5), odn. re²enje homogene jedna£ine,dato je u obliku

wh(x) = eβx(C1 sinβx+ C2 cosβx)

+ e−βx(C3 sinβx+ C4 cosβx)(6)

gde su C1 do C4 nepoznate integracione konstante








Integracione konstante C1 do C4 se odre�uju iz odgovaraju¢ihgrani£nih uslova

Re²enje nehomogene dif. jedna£ine (5), odn. partikularniintegral, dat je sa wp(x)

Kona£no re²enje dif. jedna£ine je dato sa zbirom

w(x) = wh(x) + wp(x) (7)

Iz re²enja (7) za ugibe se zatim odre�uju momenti savijanja itransverzalne sile






Sadrºaj











High speed trains: v>250 km/h


121

For tracks of good quality the critical speed lies far beyond the operating speed, but with poor soilconditions or other mass/spring configurations the critical speed can be so low that special measuresare required. In case the train speed approaches the wave propagation speed, the soil may experi-ence a liquefaction type of phenomenon as seen in Figure 6.19. An actual measurement in track onsoft soil is shown in Figure 6.20.

For the undamped case (left column ofFigure 6.17) a simple formula exists [98] for thedynamic amplification:

(6.56)

6.3.4 Discrete support

The model in Figure 6.10(c), in which the rail is supported in a discrete manner, gives the bestapproximation. Such an approach also lends itself to the application of standard element programsprograms which will be discussed later in Section 6.9. These element method programs give greatflexibility as regards load forms and support conditions.

6.4 Vertical wheel response

6.4.1 Hertzian contact spring

During vehicle/track interaction the forces are transmitted by means of the wheel/rail contact area. Onaccount of the geometry of the contact area between the round wheel and the rail, the relationshipbetween force and compression, represented by the Hertzian contact spring, is not linear as hasalready been discussed in Section 2.7. The relationship between force F and indentation y of the con-tact surface can be written as:

(6.57)

in which cH [Nm-3/2] is a constant depending on the radii and the material properties.

Figure 6.20: Actual measurement on soft soil

-14

-13

-12

-11

-10

-9

-8

-7

-6

-5

120 150 180 210 240

Running speed [km/h]

Verti

cal d

ispl

acem

ent [

mm

]

Critical train speed

225

High speed trainIC train

-14

-13

-12

-11

-10

-9

-8

-7

-6

-5

120 150 180 210 240


Verti

cal d

ispl

acem

ent [

mm

]


225

High speed trainIC trainHigh speed trainIC train

Figure 6.19: Wave propagation at high speed

c GT =

ρc G

T =ρ

wdynwstat------------ 1

1 vvcr-------

2–

---------------------------------=

F cH y3 2⁄=






High speed trains: kriti£na brzina


121

For tracks of good quality the critical speed lies far beyond the operating speed, but with poor soilconditions or other mass/spring configurations the critical speed can be so low that special measuresare required. In case the train speed approaches the wave propagation speed, the soil may experi-ence a liquefaction type of phenomenon as seen in Figure 6.19. An actual measurement in track onsoft soil is shown in Figure 6.20.

For the undamped case (left column ofFigure 6.17) a simple formula exists [98] for thedynamic amplification:

(6.56)

6.3.4 Discrete support

The model in Figure 6.10(c), in which the rail is supported in a discrete manner, gives the bestapproximation. Such an approach also lends itself to the application of standard element programsprograms which will be discussed later in Section 6.9. These element method programs give greatflexibility as regards load forms and support conditions.

6.4 Vertical wheel response

6.4.1 Hertzian contact spring

During vehicle/track interaction the forces are transmitted by means of the wheel/rail contact area. Onaccount of the geometry of the contact area between the round wheel and the rail, the relationshipbetween force and compression, represented by the Hertzian contact spring, is not linear as hasalready been discussed in Section 2.7. The relationship between force F and indentation y of the con-tact surface can be written as:

(6.57)

in which cH [Nm-3/2] is a constant depending on the radii and the material properties.

Figure 6.20: Actual measurement on soft soil

-14

-13

-12

-11

-10

-9

-8

-7

-6

-5

120 150 180 210 240


Verti

cal d

ispl

acem

ent [

mm

]


225

High speed trainIC train

-14

-13

-12

-11

-10

-9

-8

-7

-6

-5

120 150 180 210 240


Verti

cal d

ispl

acem

ent [

mm

]


225

High speed trainIC trainHigh speed trainIC train

Figure 6.19: Wave propagation at high speed

c GT =

ρc G

T =ρ

wdynwstat------------ 1

1 vvcr-------

2–

---------------------------------=

F cH y3 2⁄=







High speed trains: kriti£na brzina

Kriti£na brzina voza: ako je bliska sa brzinom prostiranjaRaylegih-evih (R) talasa u tlu

Mogu da nastanu veoma izraºene vibracije koloseka i okolnogtla

Fenomen je sli£an probijanju zvu£nog zida supersoni£nogaviona

Uo£eno je na pojedinim deonicama sa lo²im (neodgovaraju¢im)tlom - slojevi glinovitog tla

Fenomen je uo£en u �vedskoj (Swedish West Coast Line)





Vrste naponskih talasaBrzine prostiranja talasaPropagacija naponskih talasa kroz tlo

Sadrºaj












Zapreminski i povr²inski talasi

Naponski talasi u tlu su oscilatorna kretanja £estica tla

Nastaju usled zemljotresa, eksplozija, udara u tlo (pobijanje²ipova, industrijski £eki¢i i razne ma²ine), kretanja vozila,posebno ²inskih, . . .Dve osnovne vrste naponskih talasa

Zapreminski talasi- Primarni (kompresioni, longitudinalni) ili P talasi- Sekundarni (smi£u¢i, transverzalni) ili S talasi

Povr²inski talasi- Rayleigh-evi ili R talasi (ili P-SV talasi)Love-ovi ili L talasi (ili SH talasi)






Zapreminski P i S talasi






Povr²inski R i L talasi






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Brzina prostiranja talasa kroz tlo

Brzina prostiranja P i S talasa

vP =

√λ+ 2µ

ρ=

√M

ρvS =

√µ

ρ=

√G

ρ(8)

gde su λ i µ Lameove konstante, a ρ gustina sredine kroz kojuse prostire talas

E i G su modul elasti£nosti i modul smicanja tla,

dok je M modulus dat sa

M =1− ν

(1 + ν)(1− 2ν)E






Sadrºaj












Brzina prostiranja talasa kroz tlo

Brzina prostiranja R talasa je oko 0.95 vS

Prema tome, vP > vS > vR

Zapreminski talasi (iz ta£kastog izvora) se prostiru pravolinijskiradijalno sa sfernim talasnim frontom

Povr²inski R talasi se prostiru pravolinijski radijalno sacilindri£nim talasnim frontom

Oko 67% energije se prenosi sa povr²inskim R talasima, oko26% preko zapreminskih S talasa i oko 7% energije se prenosiputem P talasa

Povr²inski R talasi su najzna£ajniji naponski poreme¢aj napovr²ini tla







Propagacija naponskih talasa kroz tlo

Zakon propagacije (atenuacije) naponskih talasa kroz tlo je

v = v1

(R1

R

)ne−α(R−R1) (9)

- v i v1 amplitude brzina na rastojanju R i R1 (izvor talasa)

- n koe�cijent geometrijske atenuacije (obi£no n = 0.5)

- α koe�cijent materijalne atenuacije dat sa α = 2πξλ

- ξ koe�cijent relativnog prigu²enja tla

- λ talasna tuºina talasa, data sa λ = cf gde su c brzina

prostiranja talasa, a f dominantna frekvencija







Propagacija naponskih talasa kroz tlo

Iz zakona (9) se vidi da se amplitude oscilovanja talasasmanjuju sa duºinom propagacije

Frekventni sastav talasa se ne menja sa duºinom prostiranja

Oscilovanje tla se prenosi (horizontalno) do temelja obliºnjihzgrada

Oscilovanje temelja se prenosi vertikalno kroz zidove i stubove,a zatim horizontalno na svaku tavanicu

Oscilovanje tavanica moºe da smeta ljudima






Gradili²te u okolini Moskve






Kretanje te²kog vozila preko prepreke






Merenje vibracija kod prepreke i na gradili²tu






�elezni£ka stanica Prokop u Beogradu






Vertikalne vibracije temelja stubova za te²ki voz






Ra£unski model lamele plo£a na koti 105






Ra£unska simulacija ubrzanja na plo£i na koti 105






Pri£vrsni sistem "Vanguard"


MEHANIKA VO NJE - Odsek za puteve, eleznice i aerodrome · Poduºna dinamika ²inskih vozila...

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