Development of a track dynamics model using Mindlin plate theory and its application to coupled vehicle-floating slab track systems

Mechanical Systems and Signal Processing 140 (2020) 106641

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Development of a track dynamics model using Mindlin plate theory and its application to coupled vehicle-floating slab track systems Jun Luo, Shengyang Zhu ⇑, Wanming Zhai Train and Track Research Institute, State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, PR China

a r t i c l e

i n f o

Article history: Received 18 March 2019 Received in revised form 22 November 2019 Accepted 10 January 2020

Keywords: Floating slab track Vehicle-track coupled dynamics Mindlin plate theory Timoshenko beam functions Rayleigh-ritz method Size effect

a b s t r a c t Unlike track slabs in high speed railways, floating slabs in metro lines are usually relatively thicker and their thickness can often reach 0.5 m or more. When the thickness of the track slab is relatively large compared to the length and width, the influence of shear effect and rotatory inertia will become significant and it is more reliable to regard the track slab as an elastic thick plate than adopt the classical thin plate model in the simulation. This paper presents a three-dimensional dynamic model for coupled vehicle-floating slab track (CVFST) systems on the basis of Mindlin plate theory for the first time. The floating slab is described as an elastic rectangle Mindlin plate with free boundary conditions resting on steel springs, the mode shapes of the Mindlin plate are approximated by series of products of Timoshenko beam functions, and the corresponding vibration equations are solved by Rayleigh-Ritz method in time domain. The vehicle subsystem and track subsystem are coupled via wheel-rail nonlinear interactions. The effectiveness of the proposed model is validated by comparing with the measured data from an impact test and numerical results in the previous literature. Influence of size effect on the slab natural frequencies and dynamic responses of the CVFST system are investigated, and discrepancies of the calculation results between the developed model and the traditional thin plate model are simultaneously discussed. Some practical conclusions are drawn and the developed model may serve as a potent tool for more accurate assessment of train-induced vibrations in metro lines. Ó 2020 Elsevier Ltd. All rights reserved.

1. Introduction As one of the important means of urban transportation, the subway not only facilitates people’s travel, but also alleviates the serious pressure of urban environmental pollution. It has broad application prospects and has been developed rapidly in China’s major cities in recent years. However, on the other hand, train-induced vibrations will propagate through foundation, tunnel and other media in the form of elastic waves, and can be eventually transmitted to the ground and surrounding buildings, which may influence the living environment of neighboring residents and even threaten normal operation of some sophisticated instruments. Train-induced vibrations in metro lines is an important environmental issue and has attracted wide attention of scholars [1–4]. Floating slabs have been gradually popularized and applied in the construction of subways

⇑ Corresponding author. E-mail address: [email protected] (S. Zhu). https://doi.org/10.1016/j.ymssp.2020.106641 0888-3270/Ó 2020 Elsevier Ltd. All rights reserved.

2

J. Luo et al. / Mechanical Systems and Signal Processing 140 (2020) 106641

owing to its good performance of vibration and noise reduction. In order to expand the effective frequency band of vibration isolation, reducing the natural frequencies of the floating slab by increasing the thickness is a commonly used approach in engineering practice. Hence, for the floating slab in metro lines, it’s usually relatively thicker with a thickness of 0.5 m or more and can be regarded as an elastic rectangular plate in the dynamic analysis. The dynamic analysis of rectangular plates as basic structures is of great significance in the engineering applications such as bridge decks, floor slabs and track slabs. The plate structures are normally divided into two categories, yielding thin plates, described by the Kirchhoff plate theory, and thick plates, mainly described by the Mindlin plate theory [5,6]. It is well known that the Kirchhoff theory has been successfully applied to the vibration analysis of a great number of thin plate structures in engineering such as the track slabs in high speed railways. Their thickness is much smaller compared to the width and length, and are usually regarded as elastic rectangular thin plates in three-dimensional model based on finite element method [7,8] or modal superposition method [9,10]. The classical thin plate theory is based on the assumption that the straight lines originally normal to the middle surface remain unchanged during deformation, which indicates that it does not take into account either the effect of transverse shear deformation or rotatory inertia, and hence it becomes inaccurate for thicker plates. Mindlin [5,6] proposed an improved plate theory by considering both effects and introducing a shear factor to take account of the fact that the shear strain distribution along the plate thickness is not uniform. Comprehensive research works have been devoted to the free vibration analysis of thick plates with different boundary conditions. Srinivas et al. [11] developed a three-dimensional linear, small deformation theory of elasticity solution by the direct method for the free vibration of simply-supported thick rectangular plates. Liew et al. presented sets of accurate vibration frequencies for thick rectangular plates subjected to 21 boundary conditions involving all possible combinations of clamped, simply supported and free edges [12], oblique internal line supports [13] and internal ring supports [14] using pb-2 Rayleigh-Ritz method. Gorman et al. [15] obtained accurate solutions for the natural frequencies and mode shapes of the completely free Mindlin plate based on the superposition method. Dawe et al. [16] applied the Rayleigh-Ritz method to the prediction of the natural frequencies of flexural vibrations of square plates with genernal boundary conditions. The spatial variations of the thick plate deflection and the two rotations are assumed to be series of products of Timoshenko beam functions, which were also employed by the studies of Zhou et al. [17,18] and Cheung et al. [19]. Additionally, many scholars have also contributed to the forced vibration of thick plates subjected to external forces. Shen et al. [20] dealt with the dynamic response of Reissner-Mindlin plates exposed to thermomechanical loading and resting on a Pasternak-type elastic foundation by using both the modal superposition approach and state variable approach. Park et al. [21] applied the Rayleigh-Ritz method to predict the vibration response of the Mindlin plate subjected to distributed random forces with imposed spectral characteristics. Dyniewicz et al. [22] presented a Mindlin plate subjected to a pair of inertial loads traveling at a constant high speed in opposite directions along straight and curved trajectories. Amiri et al. [23] and Gbadeyan et al. [24] investigated dynamic responses of an undamped Mindlin plate under a moving mass. For more complex situations, Zaman et al. [25] and Van et al. [26] established mathematical models to study Mindlin plates on viscoelastic foundations subjected to suspended lumped masses. In view of aforementioned literature review, vibration problems of Mindlin plates have been investigated from various aspects. However, to the best of the authors’ knowledge, few attempts have been focused on dynamic responses of thick plates induced by vehicle-track interactions, which consist of a large number of degrees of freedom (DOFs) and is very likely to be encountered in floating slab track system in metro lines. In previous studies, the floating slab is often regarded as an Euler beam [27–29], a thin plate [30], or a solid model based on commercial finite element software [3,31]. The beam model is easy to be established, but it differs from the actual structure and is only suitable for the analysis in a 2-D space. Its accuracy and application scope can be further improved by using plate or solid models. The finite element software has superiority in model establishment, however, its disadvantages in low computational efficiency and high computer memory consumption are a non-trivial problem. The thin plate model is applicable for spatial dynamic analysis with satisfactory computational efficiency, but the effectiveness is still required to be verified with the increase of plate thickness. The aim of this paper is to develop a three-dimensional dynamics model for coupled vehicle-floating slab track (CVFST) systems on the basis of Mindlin plate theory, to correctly and efficiently predict the vibrations induced by metro vehicles. The paper is organized as follows: In Section 2, we deduce the Timoshenko beam functions with free boundary conditions, which will be applied as the mode shapes of the Mindlin plate. Then the elaborate procedures to solve the free and forced vibrations of the Mindlin plate using Rayleigh-Ritz method are presented in Section 3, and the CVFST dynamics model is set up in Section 4. In Section 5, the developed model is validated by an impact test and published literature, and size effect of the floating slab are fully investigated according to both thin plate and Mindlin plate models. Finally, some conclusions are drawn in Section 6. 2. Timoshenko beam functions with free-free boundary conditions Selection of appropriate trail functions is always the first and core step to solve free or forced vibration of plates by Rayleigh-Ritz method. In this paper, Timoshenko beam functions with free-free edges are adopted as the trail functions to solve the Mindlin plate vibrations considering the boundary constraints of the floating slab. When the effect of rotatory inertia and shear deformation of a beam is taken into account, a pair of coupled differential equations for the deflection w and the bending slope w are given by Timoshenko [32] as

J. Luo et al. / Mechanical Systems and Signal Processing 140 (2020) 106641

EI


 @2w @w @2w þ G j A  w  Iq 2 ¼ 0 r 2 @x @x @t

! @2w @ 2 w @w ¼0 qA 2  Gjr A  @x2 @x @t

3

ð1aÞ

ð1bÞ

in which, E is the modulus of elasticity, G is the shear modulus, I is the area moment of inertia of cross section, q is the density, A is the cross-section area, and jr is the shear factor. After some algebraic manipulations (seen in Appendix A), the bending slope w and the deflection w can be eliminated from Eq. (1a) and (1b), respectively.

EI


 @4w @2w EIq @ 4 w Iq2 @ 4 w þ q A  I q þ þ ¼0 2 2 4 2 @x Gjr @x @t Gjr @t 4 @t

ð2aÞ

EI


 4 @4w @2w EIq @ w Iq2 @ 4 w þ q A  I q þ þ ¼0 2 2 @x4 Gjr @x2 @t Gjr @t4 @t

ð2bÞ

Let

w ¼ Xeixt w ¼ Weixt n ¼ x=L

ð3Þ

X ¼ A1 cosh ban þ A2 sinh ban þ A3 cos bbn þ A4 sin bbn

ð4Þ

W ¼ A01 sinh ban þ A02 cosh ban þ A03 sin bbn þ A04 cos bbn

ð5Þ

pffiffiffiffiffiffiffi where X is the mode function of w; W is the mode function of w; i is 1; x is the angular eigen-frequency of the beam; L is the length of the beam; and n is the non-dimensional coordinate. Substituting the Eq. (3) into Eq. (2), the general solutions of the mode functions X(n) and W(n) are given by [33]

where




a b

r2 ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 u 4 ¼ pffiffiffi tðr2 þ s2 Þ þ ðr2  s2 Þ þ 2 2 b I 2

AL

s2 ¼

EI

2

2

Gjr AL

b ¼

qAL4 EI

x2

ð6Þ

Only half of the constants in Eqs. (4) and (5) are independent, and the relations can be established by plugging Eqs. (3)–(5) into Eq. (1). Thus, the mode function W(n) can be further expressed as

W ¼ k1 A1 sinh ban þ k1 A2 cosh ban  k2 A3 sin bbn þ k2 A4 cos bbn baA1 baA2 i sinhban þ h i coshban ¼ h 2 2 2 2 2 2 L 1  b s ða þ r Þ L 1  b s ða2 þ r 2 Þ bbA3 bbA4  h i sinbbn þ h i cosbbn 2  2  L 1 þ b s2 b 2  r 2 L 1 þ b s2 b2  r2

ð7Þ

By applying the boundary conditions of a free-free Timoshenko beam, one can arrive at W0 ð0Þ ¼ 0 W0 ð1Þ ¼ 0

X 0 ð0Þ X 0 ð1Þ  Wð0Þ ¼ 0  Wð1Þ ¼ 0 L L

ð8Þ

Nonzero solutions of the integration coefficients A1–A4 can be obtained by evaluating the fourth-order coefficient determinant of Eq. (8), and thus the frequency equation takes the following form

h 2  i b 2  2  2coshbacosbb þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b r 2 r 2  s2 þ 3r 2  s2 sinhbasinbb ¼ 0 2 2 2 1b r s

ð9Þ

The roots of the above transcendental equation indicate the values of a series of frequency parameters b1,b2,. . .bk,. . ., and the corresponding mode functions can be determined accordingly.

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J. Luo et al. / Mechanical Systems and Signal Processing 140 (2020) 106641

1 X ¼ coshban þ kdsinhban þ cosbbn þ dsinbbn f

W ¼ k1 sinhban þ k1 kdcoshban 

k2 sinbbn þ k2 dcosbbn f

ð10Þ

ð11Þ

in which

k¼

a b

f¼

a2 þ r 2 a2 þ s2

d¼

coshba  cosbb ksinhba  fsinbb

ð12Þ

It should be noted that the free-free beam modes given by Eqs. (10) and (11) do not constitute the complete set of mode functions required for the analysis of plates having a pair of opposite edges free. The above modes supply the third and higher trail functions and can be supplemented by the first two order rigid modes

8 1 > < X1 ¼ p ffiffiffi X 2 ¼ 3ð1  2nÞ > : X ¼ coshb an þ kdsinhb an þ 1 cosb bn þ dsinb bn k P 3 k k2 k2 k2 k2 f

ð13Þ

8 > < Wx1 ¼ 0 Wx2 ¼ 1 > : W ¼ k sinhb an þ k kdcoshb an  k2 sinb bn þ k dcosb bn k P 3 1 1 2 xk k2 k2 k2 k2 f

ð14Þ

Moreover, it can be seen in Eq. (6) that the s2 and r2 represent the participation of shear effect and rotatory inertia in Timoshenko beam, respectively. By setting s2 and r2 to zero, namely, the influence of shear effect and rotatory inertia are eliminated, the derived Timoshenko beam function in Eq. (13) can be readily reduced to the Euler-Bernoulli beam functions, which are suitable for the solution of thin plates as introduced in Ref. [9]. Hence, the trail functions adopted here have great advantages since they provide a unified approximate approach to solve the vibrations of both thin plate and Mindlin plate. 3. Vibration equations for Mindlin plate using the Rayleigh-Ritz method Different from the classical thin plate, the inclusion of shear effects in Mindlin plate theory means that the cross-sectional rotations bx, by can no longer be expressed solely in terms of the deflection w of the mid-plane, thus, three independent variables namely w, bx and by are introduced to represent the deformations of the plate. The three differential equations that govern the vibration of the Mindlin plate can be expressed as

Gjr hs

@ 2 w @ 2 w @bx @by þ 2 þ þ @x2 @y @x @y

!

 qs hs

@2w þ qðx; y; t Þ ¼ 0 @t2

ð15aÞ

" !# !
 2 2 D @ 2 bx @ by @ 2 bx @ by @w @2b  qs J 2x ¼ 0  G þ 1 þ ð1  mÞ þ ð m Þ þ j h b þ r s x 2 2 2 2 @x @x @y @x @x@y @t

ð15bÞ

! " !#
 @ 2 by @ 2 by @ 2 by @ 2 bx @ 2 by D @w  qs J 2 ¼ 0 ð1  mÞ þ þ þ ð1 þ mÞ  Gjr hs by þ 2 2 2 2 @y @x @y @y @x@y @t

ð15cÞ

with 3

D¼

Ehs 1 3 h J¼ 12 s 12ð1  m2 Þ

where hs, qs, D and m represent the thickness, density, flexural rigidity and Poisson’s ratio of the plate, respectively; q is the external uniform load acting on the plate; and other notations can be referred to Section 2. For free vibration of the plate (q = 0), the three independent variables, namely w, bx and by, take the following forms by introducing the separation variable method

wðx; y; t Þ ¼ W ðx; yÞeixt

ð16aÞ

bx ðx; y; t Þ ¼ Ux ðx; yÞeixt

ð16bÞ

by ðx; y; t Þ ¼ Uy ðx; yÞeixt

ð16cÞ

where W(x,y), Ux(x,y) and Uy(x,y) are the plate mode functions, and x is the angular eigen-frequency of the plate. The displacement field and strain field of a Mindlin plate are given by

J. Luo et al. / Mechanical Systems and Signal Processing 140 (2020) 106641

v ¼ zby ðx; y; tÞ;

u ¼ zbx ðx; y; t Þ;

w ¼ wðx; y; tÞ

n

eyy eyz exz exy g ¼ z @b@xx z @b@yy

f exx

@w @y

5

ð17Þ @w @x

þ by

þ bx

z



@bx @y

þ

@by @x

o

ð18Þ

where u and v denote longitudinal and lateral displacements of the plate, respectively; and eii and eij denote in-plane and shear strain components, respectively. The normal and shear stresses can be obtained from the following constitutive relations:

2

3

2

1 m rxx 6r 7 6m 1 6 yy 7 6 6 7 6 6 ryz 7 ¼ E 6 6 7 1  m2 6 6 7 6 4 rxz 5 4 rxy

1m 2 1m 2

9 38 exx > > > > > 7> >e > > > 7> yy > = 7< 7 eyz 7> > 7> > exz > > > 5> > > > > ; 1m :

ð19Þ

exy

2

The plate strain energy U and kinetic energy T can be expressed based on Eqs. (17)–(19)

U¼

1 2

ZZZ





rxx exx þ ryy eyy þ rxy exy þ jr rxz exz þ jr ryz eyz dxdydz

V


2 # ZZ "
2
@by 2 D @bx @bx @by 1  m @bx @by Gjr hs dxdy þ ¼ þ þ 2m þ þ 2 @x @y @x @y 2 @y @x 2 S 2
2 # ZZ "
@w @w þ bx þ þ by dxdy  @x @y

ð20aÞ

S

qs hs

T¼

ZZ "


2 S

2
2
2 #  # ZZ "
2
ZZ
2 @by 2 @u @v @w qs hs @w qs J @bx dxdy ¼ dxdy þ þ dxdy þ þ @t @t @t 2 @t 2 @t @t S

ð20bÞ

S

Based on Eqs. (16) and (20), the energy functional P for a Mindlin plate can be written in terms of the maximum strain energy Umax and the maximum kinetic energy Tmax as

P ¼ T max  U max U max ¼ þ

T max ¼

D 2

RR @Ux 2 S

Gjr hs 2

qs hs 2

ð21aÞ

 2

@ Uy 1m @ Ux dxdy þ þ @x 2 @y @x

2 2  RR @W dxdy þ Ux þ @W þ Uy @x @y þ



@ Uy @y

2

Ux þ 2m @@x

@ Uy @y

ð21bÞ

S

ZZ W 2 dxdy þ

x2

qs J 2

x2

S

ZZ 



U2x þ U2y dxdy

ð21cÞ

S

Moreover, the mode functions of the Mindlin plate corresponding to the mnth natural frequency xmn can be further equivalent to the product of the Timoshenko beam functions along the length and width of the plate, respectively.

W mn ðx; yÞ ¼ Amn X m ðxÞY n ðyÞ

ð22aÞ

Uxmn ðx; yÞ ¼ Bmn Wxm ðxÞY n ðyÞ

ð22bÞ

Uymn ðx; yÞ ¼ C mn X m ðxÞWyn ðyÞ

ð22cÞ

where the Amn, Bmn, Cmn are the modal coefficients representing the arbitrary amplitudes of deflection and rotation; the four unidirectional functions Xm(x), Yn(y), Wxm(x), and Wyn(y) are those Timoshenko beam functions derived in Section 2, which are appropriate to the vibration of thick plates with free boundary conditions. Introduce the Hamilton principle [34]

Z

t2

d t1

Z ðT  U Þdt þ d

t2

W nc dt ¼ 0

ð23Þ

t1

where d denotes the variation symbol; T and U denote the kinetic and potential energy for a dynamic system, respectively; Wnc denotes the virtual work done by the non-conservative forces; and t1–t2 is an arbitrary integration time period.

6

J. Luo et al. / Mechanical Systems and Signal Processing 140 (2020) 106641

The plate free vibration satisfies the Hamilton principle of conservative systems. Substitution of Eqs. (21) and (22) into Eq. (23) yields homogeneous linear equations with respect to the modal coefficients that, after some algebraic manipulation, can be arranged in the form

2 6 6 4

Gjr hs ðB3m B1n þ B1m B3n Þ  qs hs x2mn B1m B1n

Gjr hs B5m B1n

Gjr hs B5m B1n

mÞ DB4m B1n þ Gjr hs B2m B1n þ Dð1 B2m B3n  qs J x2mn B2m B1n 2

Gjr hs B1m B5n

mÞ DmB6m B6n þ Dð1 B5m B5n 2

9 38 > < Amn > = 7 5 Bmn ¼ 0 > > : ; mÞ C mn DB1m B4n þ Gjr hs B1m B2n þ Dð1 B3m B2n  qs J x2mn B1m B2n 2 Gjr hs B1m B5n mÞ DmB6m B6n þ Dð1 B5m B5n 2

ð24Þ

where B1m, B1n, B2m, B2n, B3m, B3n, B4m, B4n, B5m, B5n, B6m and B6n are the integral constants with respect to the Timoshenko beam functions, which take the form

B1m ¼

R Ls 0

R Ws

X 2m ðxÞdx

B1n ¼

R Ws 0

R Ls

Y 2n ðyÞdy

B2m ¼

R Ls 0

W2xm ðxÞdx

R Ws

B3m ¼ 0 X 02 B3n ¼ 0 Y 02 m ðxÞdx n ðyÞdy R R Ls Ws 02 02 B4m ¼ 0 Wxm ðxÞdx B4n ¼ 0 Wyn ðyÞdy B5m ¼ 0 Wxm ðxÞX 0 m ðxÞdx R Ws R RW L B5n ¼ 0 Wyn ðyÞY 0 n ðyÞdy B6m ¼ 0 s W0 xm ðxÞX m ðxÞdx B6n ¼ 0 s W0 yn ðyÞY n ðyÞdy B2n ¼

0

R Ls

W2yn ðyÞdy

ð25Þ

In order to obtain the nonzero solutions of the modal coefficients, the coefficient determinant of Eq. (24) must be zero

  Dmnð11Þ    Dmnð21Þ   Dmnð31Þ

Dmnð12Þ Dmnð22Þ Dmnð32Þ

 Dmnð13Þ   Dmnð23Þ  ¼ 0  Dmnð33Þ 

ð26Þ

Eq. (26) is an univariate cubic equation related to the x2, for any set of (m,n), three sequent frequencies can be solved xmn(k) (k = 1,2,3, m = 1,2,3,. . ., n = 1,2,3,. . .), namely, low, medium and high ones in which the low one represents the flexural frequency and the other two represent thickness-shear frequencies. Once a series of frequencies are obtained, the proportional relationship between the modal coefficients can be determined by substituting them back to Eq. (24), expressed as

Bmn

  Dmnð21Þ  D mnð31Þ ¼   Dmnð22Þ  D mnð32Þ

 Dmnð23Þ   Dmnð33Þ  A Dmnð23Þ  mn  Dmnð33Þ 

C mn

  Dmnð21Þ  D mnð31Þ ¼   Dmnð23Þ  D mnð33Þ

 Dmnð23Þ   Dmnð33Þ  A Dmnð22Þ  mn  Dmnð32Þ 

ð27Þ

Meanwhile, the mode functions in Eq. (22) can also be determined. Now, to consider the plate forced vibration (q – 0), the solutions of the dynamic vertical and angular displacements are expanded separately by Rayleigh-Ritz method [20], given as

wðx; y; t Þ ¼

Ny X Nx X 3 X

kÞ kÞ W ðmn ðx; yÞT ðmn ðt Þ ¼

m¼1 n¼1 k¼1

bx ðx; y; t Þ ¼

Ny X Nx X 3 X

Ny X Nx X 3 X

kÞ kÞ Aðmn X m ðxÞY n ðyÞT ðmn ðt Þ

ð28aÞ

kÞ kÞ Bðmn Wxm ðxÞY n ðyÞT ðmn ðt Þ

ð28bÞ

kÞ kÞ C ðmn X m ðxÞWyn ðyÞT ðmn ðt Þ

ð28cÞ

m¼1 n¼1 k¼1

kÞ kÞ Uðxmn ðx; yÞT ðmn ðt Þ ¼

m¼1 n¼1 k¼1

by ðx; y; t Þ ¼

Ny X Nx X 3 X

Ny X Nx X 3 X m¼1 n¼1 k¼1

kÞ kÞ Uðymn ðx; yÞT ðmn ðt Þ ¼

m¼1 n¼1 k¼1

Ny X Nx X 3 X m¼1 n¼1 k¼1

where Tmn(t) is time dependent modal coordinate; Nx or Ny is the truncated mode numbers of Xm(x) and Wxm(x) or Yn(y) and

Wyn(y), respectively. Note that the following mode equations can be established by substituting Eq. (16) into Eq. (15)

Gjr hs

@ 2 W @ 2 W @ Ux @ Uy þ þ þ @x2 @y2 @x @y

!

¼ qs hs x2 W

" !# !
 D @ 2 Ux @ 2 Uy @ 2 Ux @ 2 Uy @W  G þ 1 þ ¼ qs J x2 Ux ð1  mÞ þ ð m Þ þ j h U þ r s x 2 @x @x2 @y2 @x2 @x@y

ð29aÞ

ð29bÞ

J. Luo et al. / Mechanical Systems and Signal Processing 140 (2020) 106641

" !# !
 D @ 2 Uy @ 2 Uy @ 2 Uy @ 2 Ux @W  Gjr hs Uy þ þ 1 þ ¼ qs J x2 Uy ð1  m Þ þ ð m Þ þ 2 2 2 2 @y @x @y @y @x@y

7

ð29cÞ

Incorporation of Eqs. (28) and (29) into Eq. (15), one is led to Ny X Nx X 3 h X

i ðkÞ kÞ kÞ2 ðkÞ T€ mn ðt Þ þ xðmn T mn ðt Þ qs hs W ðmn  qðx; y; tÞ ¼ 0

ð30aÞ

i ðkÞ kÞ kÞ2 ðkÞ T€ mn ðt Þ þ xðmn T mn ðt Þ qs J Uðxmn ¼0

ð30bÞ

i ðkÞ kÞ kÞ2 ðkÞ T€ mn ðt Þ þ xðmn T mn ðt Þ qs J Uðymn ¼0

ð30cÞ

m¼1 n¼1 k¼1 Ny X Nx X 3 h X m¼1 n¼1 k¼1 Ny X Nx X 3 h X m¼1 n¼1 k¼1

By utilizing orthogonality condition of the mode shapes of the Mindlin plates

ZZ h



i



ðgÞ gÞ gÞ kÞ kÞ kÞ qs hs W ðmn W kl þ qs J Uðxmn Uðxkl þ Uðymn Uðykl dxdy

S

¼ 0 m–k or n–l or k–g –0 m ¼ k and n ¼ l and k ¼ g

ð31Þ

the three modal coordinates can be decoupled into the following three independent equations ultimately.

RR

ðkÞ

kÞ2 ðkÞ T€ mn ðtÞ þ xðmn T mn ðt Þ ¼ RR h

kÞ qðx; y; t ÞW ðmn dxdy  i k ¼ 1; 2; 3 kÞ2 kÞ2 kÞ2 dxdy qs hs W ðmn þ qs J Uðxmn þ Uðymn S

ð32Þ

S

where k = 1,2,3 denote the vertical motion, torsional motion in x and y directions, respectively. 4. Coupled dynamics model for vehicle-steel spring floating slab track Based on the vehicle-track coupled dynamics theory proposed by Zhai et al. [9], the deduced vibration equations of the Mindlin plate are further applied to dynamics modelling of the floating slab and are implemented into the CVFST system. The modelling strategies will be introduced in Sections 4.1–4.4. Fig. 1 schematically shows the developed spatial model of the CVFST system, which is solved by means of a new fast explicit integration method [35] for its stability, accuracy and efficiency. 4.1. Vibration equations of the vehicle According to the multi-body system dynamics, the vehicle subsystem is built by considering seven rigid parts involving a car body, two bogies, and four wheelsets with the primary and the secondary suspensions. Each component is respectively assigned with 5 DOFs involving the vertical displacement Z, the lateral displacement Y, the roll angle U, the yaw angle W and the pitch angle b. Therefore, the vehicle subsystem has a total of 35 DOFs, as listed in Table 1. The longitudinal motion is supposed to be known and characterized by the vehicle constant velocity v. Without loss of generality, the equations of the vehicle subsystem can be depicted in the form of second-order differential equations in the time domain

Mv Av þ Cv Vv þ Kv Xv ¼ Fv

ð33Þ

where Mv, Cv and Kv are the mass matrix, the damping matrix and the stiffness matrix of the vehicle subsystem, respectively; Av, Vv and Xv denote the displacement, velocity and acceleration vectors of the vehicle system, respectively; and Fv denotes the force vector representing the gravity of each component and the nonlinear wheel-rail contact forces acting on the wheelsets. 4.2. Vibration equations of the rail The rail is regarded as a simply supported Euler-Bernoulli beam subjected to the dynamic wheel-rail interaction forces and fastener forces. The fourth-order partial equations of the rail can be converted into the second order ordinary equations by adopting a modal superposition approach. The vertical, lateral and torsional vibration equations of the rail are described in the generalized coordinate as follows [9]:

€Vk ðt Þ þ q

sffiffiffiffiffiffiffiffiffiffi"
4
 X
# Np 4 X EIY kp 2 kpxsi kpxwj þ k ¼ 1  NV  qVk ðtÞ ¼ PrVi ðt Þsin Pj ðt Þsin mr Lr mr Lr Lr Lr i¼1 j¼1

ð34aÞ

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Fig. 1. Three dimensional dynamics model of the vehicle-floating slab track system.

Table 1 Degrees of freedom of the vehicle model. Vehicle component

Car body Bogie frame (i = 1,2) Wheelset (i = 1,2,3,4)

Type of motion Vertical

Lateral

Roll

Yaw

Pitch

Zc Zti Zwi

Yc Yti Ywi

Uc Uti Uwi

Wc Wti Wwi

bc bti bwi

sffiffiffiffiffiffiffiffiffiffi"
4
 X
# Np 4 X EIZ kp 2 kpxsi kpxwj €Lk ðt Þ þ  q þ k ¼ 1  NL qLk ðtÞ ¼ PrLi ðt Þsin Q j ðt Þsin mr Lr mr Lr Lr Lr i¼1 j¼1

ð34bÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffi"
2
 X
# Np 4 X GIt kp 2 kpxsi kpxwj þ k ¼ 1  NT qTk ðt Þ ¼ M ðtÞsin M wj ðtÞsin  qTk ðt Þ þ qr I0 Lr qr I0 Lr i¼1 si Lr Lr j¼1

ð34cÞ

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where qVk(t), qLk(t) and qTk(t) are the generalized coordinates describing the vertical, lateral and torsional motions of the rail, respectively; EIY, EIZ and GIt are the vertical bending, lateral bending and torsional stiffness of the rail, respectively; mr, Lr, I0 and qr are the mass per unit length, the calculation length, the torsional inertia and the density of the rail, respectively; xsi are the coordinates of the rail-supporting points; xwj are the coordinates of the contact points of the wheel and rail; Np is the total number of rail fasteners on the slab; NV, NL and NT are the truncated number of the rail mode functions; Pj(t) and Qj(t) are the vertical and lateral wheel-rail forces, respectively; Msi and Mwj are the equivalent moments acting on the rail; and PrVi(t) and PrLi(t) are the vertical and lateral rail-supporting forces, respectively. 4.3. Vibration equations of the floating slab Fig. 2 shows a floating slab subjected to the vertical forces from fasteners and steel springs. In this paper, only the vertical motion of the slab with corresponding flexural frequencies is considered. By taking the classical modal damping into account, the forced vibration equations of the slab in the generalized coordinate can be recast as a series of second-order ordinary differential equations based on the Eqs. (13), (14), (22) and (32).

"

T€ mn ðtÞ þ 2fmn xmn T_ mn ðtÞ þ x2mn T mn ðt Þ ¼

Np P

i¼1

PrVi ðtÞX m ðxsi ÞY n ðysi Þ 

Nb P







F sVj ðt ÞX m xbj Y n ybj

# 

j¼1

  qs hs B1m B1n þ qs J B2mn B2m B1n þ C 2mn B1m B2n

ð35Þ

where fmn denotes the damping ratio of the floating slab; xsi and ysi are the x-coordinates and y-coordinates of the railsupporting points, respectively; xbj and ybj are the x-coordinates and y-coordinates of the steel springs, respectively; Nb is the total number of steel springs beneath the slab; and PrVi(t) and FsVj(t) are the fastener forces and steel spring forces, respectively, which can be expressed as

PrVi ðtÞ ¼ kp

NV X

Z k ðxsi ÞqVk ðt Þ 

þ cp

Z k ðxsi Þq_ Vk ðtÞ 

k¼1

F sVj ðtÞ ¼ kss

! X m ðxsi ÞY n ðysi ÞT mn ðtÞ

m¼1 n¼1

k¼1 NV X

Ny Nx X X

Ny Nx X X

Ny Nx X X

! _ X m ðxsi ÞY n ðysi ÞT mn ðt Þ

ð36Þ

m¼1 n¼1 Ny Nx X X         X m xbj Y n ybj T mn ðt Þ þ css X m xbj Y n ybj T_ mn ðt Þ

m¼1 n¼1

ð37Þ

m¼1 n¼1

where kp and cp are the stiffness and damping of the rail pad, respectively; kss and css are the stiffness and damping of the steel spring, respectively. Regarding the lateral motion of the floating slab, rigid behavior [9] is assumed due to its large bending stiffness in the lateral direction. In order to compare the response differences according to the two plate theories in follow-up numerical discussion, vibration equations of the floating slab based on the Kirchhoff thin plate theory are also deduced in Appendix B. 4.4. Wheel-rail nonlinear interaction model In the dynamics model, the wheel-rail interactions are mainly characterized by the wheel-rail normal forces based on the nonlinear Hertz contact theory [36], and the tangential wheel-rail creep forces, which are first formulated by Kalker linear

Fig. 2. The floating slab subjected to the vertical forces from fasteners and steel springs.

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rolling contact theory [37], and then modified by the Shen-Hedrick-Elkins non-linear model [38]. For more details about the calculation of wheel-rail interaction forces, monograph [39] can be consulted for readers. 5. Numerical results and discussion 5.1. Model validation In order to validate the reliability of the proposed Mindlin plate dynamics model, vibration tests of the floating slab system were conducted using the M + P International VibPilot as the data acquisition and analysis system, as shown in Fig. 3. Three acceleration sensors were attached to the point A1, A2 and A3 respectively on the floating slab as shown in Fig. 4.

Fig. 3. Vibration test of the floating slab with data acquisition system.

9060 630

650

650

650

650

650

650

650

650

650

650

650

650

630

A2 650 Fig. 4. Geometry of the tested steel spring floating slab and distribution of the accelerometers.

Fig. 5. Time histories of the impact force.

2000

B A3

3150

Steel springs

A1

305

495

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During the test, an impact force was exerted on the predefined point B on the floating slab by using a hammer, and the time histories of the force can be seen in Fig. 5. The corresponding induced accelerations of the floating slab were collected by the system with accelerometers. Fig. 6 shows the comparisons of time histories and frequency contents of the slab acceleration between the measured data and simulation results according to Mindlin plate and thin plate theories. It can be seen in Fig. 6(a) that the measured time histories of the slab acceleration mainly fluctuate in the range of 1.05 m/s2 and 1.35 m/s2, while the simulated amplitudes mainly oscillate in the range of 0.92 m/s2 and 1.13 m/s2. In all, the simulated results based on the two plate models are basically consistent with the tested results. Analogous conclusions can also be drawn from Fig. 6(c). Regarding the frequency domain of the thin plate model as portrayed in Fig. 6(b) and (d), it coincides well with the measured data in the frequency range below 100 Hz, while better agreements can be clearly identified both in the positions of the peak values and their amplitudes of the slab acceleration for the Mindlin plate model in the frequency range around 0–250 Hz, which covers the major concerned band for railway-induced vibration in urban areas. Some limited differences can also be observed in the medium and high frequency range over 250 Hz. This is probably because several cylindrical cavities are excavated from the floating slab to install the steel springs as portrayed in Fig. 4, these cavities will inevitably affect the slab vibration responses to some extent. Furthermore, the present solutions for natural frequencies of the Mindlin plate are also verified by the available numerical results from Liew et al. [12], in which thickness ratio g1 = hs/Ws and aspect ratio g2 = Ls/Ws are the two indexes defined to depict size effect on the non-dimensional frequency parameters X

xW 2 X¼ 2s p

rffiffiffiffiffiffiffiffiffi qs hs D

ð38Þ

where Ls and Ws are the length and width of the plate, respectively. The first ten frequency parameters of the Mindlin plate with free boundary conditions (FFFF) are presented in Fig. 7, in which g1 = 0.001, 0.1, and 0.2, and g2 = 1.5, 2.0, and 2.5, respectively. It can be observed that the results in this paper are

Fig. 6. Comparison of the slab accelerations between measured data and numerical results: (a) time histories of point A3; (b) frequency contents of point A3; (c) time histories of point A1/A2; (d) frequency contents of point A1/A2.

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Fig. 7. Comparison of frequency parameters of FFFF Mindlin plate: (a) g1 = 0.001; (b) g1 = 0.1;(c) g1 = 0.2.

in good agreement with those obtained from Liew et al. both in the quantities and variation trends. In general, the comparison results involving the free and forced vibrations have properly demonstrated the reliability of the developed model, which will be adopted to predict the dynamic responses of the CVFST system in following study. 5.2. Dynamic analysis of CVFST system In this section, dynamic responses of the CVFST system based on Mindlin plate and thin plate theories are simultaneously investigated under the excitation of track random irregularities. The geometric dimension of the floating slab in the simulation is shown schematically in Fig. 4. The irregularities are assumed to move backward at the speed of 80 km/h to simulate the vehicle traveling along the track at a constant velocity. The American track spectrum class 6 [40] is adopted in the metro lines with the wavelengths between 1 m and 30 m, additionally Eq. (39) proposed by the Chinese Railway Science Academy is used for the wavelength of track elevation irregularity in the range of 0.01–1 m [41].

Sv ðf Þ ¼ 0:036 f

3:15

ð39Þ

where f (1/m) denotes spatial frequency of track irregularities. It can be found that the CVFST system based the two plate theories have negligible influence on the dynamic responses above the floating slab such as rail supporting forces, wheel-rail interaction forces, or vehicle responses. Here, the comparisons of wheel-rail vertical force and rail supporting forces are shown in Figs. 8 and 9 as examples, which demonstrate that no apparent discrepancies can be observed both in time histories and frequency contents. Hence, we select two representative dynamic indexes to evaluate the response differences, namely, the slab acceleration and the steel spring supporting force. For the steel spring force shown in Fig. 10, apparent enlargement can be found in time domain. The maximum value, and mean value according to thin plate model are found to be 18.97 kN and 17.32 kN, respectively, which increase by 7.78% and 6.78% compared to the proposed model. As for the frequency components, Mindlin plate model predicts slightly smaller amplitude than the thin plate model except some differences can be observed around 10 Hz. As portrayed in Fig. 11(a), the maximum value and mean value of slab acceleration using the Mindlin plate theory are found to be 5.29 m/s2 and 4.69  104 m/s2, respectively, which are about 1.11 and 1.09 times as large as those using the thin plate theory. The peak value position in the frequency range over 200 Hz is considerably altered as shown in Fig. 11(b). As the root mean square

Fig. 8. Comparison of wheel-rail vertical force: (a) time histories; (b) frequency contents.

J. Luo et al. / Mechanical Systems and Signal Processing 140 (2020) 106641

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Fig. 9. Comparison of rail supporting force: (a) time histories; (b) frequency contents.

Fig. 10. Comparison of steel spring force: (a) time histories; (b) frequency contents.

Fig. 11. Comparison of slab acceleration: (a) time histories; (b) frequency contents.

(RMS) acceleration is able to objectively reflect structure vibration energies during a period of time [3], and is widely used in the analysis of metro vibration problems Hence, the comparison of the RMS acceleration level in the one-third octave band is also displayed Fig. 12. It can be seen that the two plate models mainly lead to differences in the frequency range below 20 Hz and over 200 Hz, and the RMS acceleration level reaches the maximum value of 116.8 dB at around 63 Hz. Due to the fact that the steel spring floating slabs with various dimensions are widely applied to the metro lines worldwide, in order to give a comprehensive insight into the dynamic response differences of floating slabs with various dimensions according to the two plate theories, size effect on the slab vibration will be investigated in the following study. 5.3. Size effect on natural frequencies of floating slab Natural frequency is one of the most important indicators to evaluate vibration characteristics of structures. A frequency ratio kf is defined to describe the proximity of the natural frequencies calculated by the two plate theories. Smaller values imply larger differences or more significant effects of shear deformation and rotatory inertia of the plate in other words.

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J. Luo et al. / Mechanical Systems and Signal Processing 140 (2020) 106641

Fig. 12. Comparison of RMS acceleration level of floating slab in one-third octave band.

kf ¼

f MindN f ThinN

ð40Þ

where fThinN and fMindN denote the slab natural frequencies calculated from thin plate theory and Mindlin plate theory, respectively; N denotes the N-th order frequency of the slab. By substituting Nx = 10 and Ny = 10 into Eq. (26) and Eq. (B8), respectively, the obtained first fifty natural frequencies based on the two plate theories and corresponding frequency ratios are illustrated in Figs. 13–15. As depicted in Fig. 13, the thickness of the slab varies from 0.2 m to 0.6 m with an increment of 0.2 m, while the width and length are kept unchanged with the values of 3.15 m and 9.06 m, respectively. The frequency ratio decreases from 0.998 to 0.985 for N = 1, but decreases from 0.882 to 0.550 for N = 50, which seems to indicate the thick plate effect becomes more significant with higher frequency orders and larger slab thickness. Fig. 14 shows three different slab sizes with the same size ratios, i.e.,

Fig. 13. Comparison of plate natural frequency using the two plate theories with various plate thickness: (a) hs = 0.2 m, Ws = 3.15 m, Ls = 9.06 m; (b) hs = 0.4 m, Ws = 3.15 m, Ls = 9.06 m; (c) hs = 0.6 m, Ws = 3.15 m, Ls = 9.06 m.

Fig. 14. Comparison of plate natural frequency using the two plate theories with the same size ratios: (a) hs = 0.2 m, Ws = 2.5 m, Ls = 5 m; (b) hs = 0.4 m, Ws = 5 m, Ls = 10 m; (c) hs = 0.6 m, Ws = 7.5 m, Ls = 15 m.

J. Luo et al. / Mechanical Systems and Signal Processing 140 (2020) 106641

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Fig. 15. Comparison of plate natural frequency using the two plate theories with the same thickness and various size ratios. (a) hs = 0.3 m, Ws = 6 m, Ls = 12 m; (b) hs = 0.3 m, Ws = 3 m, Ls = 12 m; (c) hs = 0.3 m, Ws = 2 m, Ls = 12 m.

g1 = 0.08 and g2 = 2.0. It can be seen that the slab frequencies are considerably altered with the change of slab sizes. However, the differences between the frequency ratios in the three cases are inconspicuous although the plate thickness varies from 0.2 m to 0.6 m. Fig. 15 shows three different size ratios with the same plate thickness of 0.3 m, i.e., g1 = 0.05, g2 = 2.0; g1 = 0.10, g2 = 4.0; and g1 = 0.15, g2 = 6.0, respectively. Obvious discrepancies of the frequency ratio curves in Fig. 15(a)– (c) can be observed although the plate thickness is kept unchanged. With the frequency order ranging from 1 to 50, the frequency ratio decreases from 0.998 to 0.904, 0.783, and 0.652, respectively. Conclusions can be reached that the definition of thin plate and thick plate is dependent on the relative dimension rather than the absolute dimension, and the frequency differences are closely related to thickness ratio g1 and aspect ratio g2. Then, the two size ratios involving thickness ratio g1 and aspect ratio g2 are introduced to investigate size effect on the slab natural frequencies. For the sake of generality, relatively wider size ratios are taken into account. The thickness ratio

Fig. 16. Comparison of frequency ratio with various size ratios: (a) g2 = 2.0; (b) g2 = 4.0; (c) g2 = 6.0; (d) g2 = 8.0.

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varies from 0.05 to 0.25 with an increment of 0.05, and the aspect ratio varies from 2.0 to 8.0 with an increment of 2.0. As shown in Fig. 16, the frequency ratio kf is smaller than 1.0 for all discussed cases, and the value is approximate to 1.0 for the fundamental frequency (N = 1) and decreases considerably as the frequency order increases. For the same aspect ratio, the frequency ratio decreases with the increase of thickness ratio, and the reduction becomes more notable with larger frequency order. For examples as seen in Fig. 16(b), with thickness ratio varying from 0.05 to 0.25, the frequency ratio decreases from 0.911 to 0.837 for N = 10, and decreases from 0.926 to 0.453 for N = 50. On the other hand, for the same thickness ratio, the frequency ratio undergoes an increase as aspect ratio increases, however, the trend becomes

Table 2 Three-dimensional size and corresponding size ratios of the floating slab in the simulation. Plate number

hs/m

Ws/m

Ls/m

g1

g2

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15

0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50

3.15 3.15 3.15 3.15 3.15 3.15 3.15 3.15 3.15 3.15 3.15 3.15 3.15 3.15 3.15

9.06 9.06 9.06 9.06 9.06 9.06 9.06 6.46 7.76 10.36 11.66 12.96 14.26 19.46 24.66

0.063 0.079 0.095 0.111 0.127 0.143 0.159 0.159 0.159 0.159 0.159 0.159 0.159 0.159 0.159

2.876 2.876 2.876 2.876 2.876 2.876 2.876 2.050 2.463 3.289 3.702 4.114 4.527 6.178 7.829

Fig. 17. Influence of thickness ratio on RMS acceleration level of slab using two plate models: (a) P1; (b) P3; (c) P5; (d) P7.

J. Luo et al. / Mechanical Systems and Signal Processing 140 (2020) 106641

17

inapparent when its value is larger than 6.0. For examples when g1 = 0.15, with g2 varying from 2.0 to 6.0, the frequency ratio increases from 0.780 to 0.911 for N = 20, and it increases from 0.911 to 0.924 for N = 20 with g2 varying from 6.0 to 8.0, as seen in Fig. 16(a)–(d). From the above analysis, it can be concluded that slab natural frequencies will decrease when shear deformation and rotatory inertia are considered based on Mindlin plate theory, and high order frequencies tend to undergo more reduction than the low order ones. The frequency discrepancies obtained from the two plate theories will increase with the augment of thickness ratio and decrease with the augment of aspect ratio. 5.4. Size effect on dynamic responses of CVFST system Size effect of the floating slab on dynamic responses of the CVFST system under a moving vehicle will be further studied in this section. Based on the aforementioned analysis, floating slabs with 15 different size ratios are considered as listed in Table 2. For the floating slab numbered from P1 to P7, its thickness varies from 0.20 m to 0.50 m with the same aspect ratio of 2.876, and for the floating slab numbered from P7 to P15, its length varies from 6.46 m to 24.66 m with the same thickness ratio of 0.159. The comparisons of RMS acceleration level of floating slabs with various thickness ratios and aspect ratios are shown in Figs. 17 and 18. It can be seen in Fig. 17(a) that the frequency contents of both models almost coincide with each other in all the frequency ranges, which can be attributed to the fact that the slab natural frequencies are approaching when the thickness ratio is small. With the thickness ratio ranging from 0.063 to 0.095, some acceptable differences can be observed as displayed in Fig. 17(b). As the thickness ratio continues to increase in Fig. 17(c) and (d), obvious differences are found in the frequency range below 20 Hz and above 200 Hz. For examples as seen in Fig. 17(d), the RMS acceleration levels calculated by the Mindlin plate model at 10 Hz and 250 Hz are found to be 104.6 dB and 91.4 dB, respectively, which increase 7.2 dB and decrease 10.1 dB, respectively, compared to those in the thin plate model. Regarding the floating slab with the same thickness ratio of 0.159 as portrayed in Fig. 18, significant discrepancies of the RMS acceleration level can be found in both low and high frequency ranges when the aspect ratio is 2.050. With the aspect ratio varying from 2.050 to 7.829, the RMS acceleration level attained by the thin plate model becomes gradually well-

Fig. 18. Influence of aspect ratio on RMS acceleration level of slab using two plate models: (a) P8; (b) P10; (c) P12; (d) P15.

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Fig. 19. Influence of size ratios on maximum value of steel spring force using two plate models: (a) thickness ratio; (b) aspect ratio.

coincident with that from the Mindlin plate model in the low and medium frequency bands when the aspect ratio is larger than 4.114, and some differences can still be observed in the frequency range over 200 Hz. Such phenomenon is mainly owing to the fact that increasing aspect ratio will narrow the differences of the slab natural frequencies calculated by the two plate models but the degree of influence is limited as mentioned in the last section. In this scenario, it is deduced that the floating slab can be modeled as an elastic thin plate for simplicity if the concerned frequency is lower than 200 Hz. Then, the influence of size ratios on maximum value of steel spring force is discussed. As shown in Fig. 19(a), when the aspect ratio is 2.876, the maximum value of steel spring force decreases from 32.6 kN to 21.94 kN for the thin plate model, and from 32.46 kN to 20.95 kN for the Mindlin plate model with the thickness ratio ranging from 0.063 to 0.159. The thin plate model always leads to larger values, and the difference between the two plate models is negligible when g1 = 0.063, and becomes gradually pronounced as the thickness ratio increases. In terms of the influence of aspect ratio as seen in Fig. 19(b), when thickness ratio is kept unchanged with the value of 0.159, the difference of the steel spring force between the two plate models tends to decrease when the aspect ratio is larger than 4.114, and the results nearly coincide with each other as the ratio approaches 7.829.

6. Conclusions The innovation of the present study is laid on providing methodologies to the establishment of a CVFST dynamics model by implementing the Mindlin plate theory into the conventional vehicle-track coupled dynamics. Experimental tests and exsisting numerical results are employed to validate the forced and free vibrations of the developed model, and the comparisons indicate that they match well with each other. Size effect on the slab natural frequencies and dynamic responses obtained from Kirchhoff thin plate and Mindlin plate models has been fully investigated. According to the theoretical analysis, certain key points can be concluded as follows. 1. The floating slab natural frequencies will decrease when shear deformation and rotatory inertia are considered, and high order frequencies tend to undergo more reduction than the low order ones. 2. The frequency discrepancies calculated by the thin plate and Mindlin plate models are rather sensitive to the size ratios and become conspicuous with larger thickness ratio and smaller aspect ratio, which indicates that the boundary between thin plate and thick plate is dependent on the relative dimension rather than the absolute dimension. 3. The participation of shear effect and rotatory inertia of the slab can hardly alter the dynamic responses above the floating slab, but will inevitably affect the slab flexural vibrations and steel spring supporting forces to a certain extent, owing to the variation of slab natural frequencies and mode shapes compared to the thin plate model. 4. Rational modelling of the track structures is of great significance to accurately evaluate train-induced vibrations. When the thickness ratio of the floating slab is smaller than 0.10, it is feasible to apply the thin plate model to simulate its dynamic behavior with acceptable accuracy. For the slab with thickness ratio larger than 0.10, the two plate models will lead to obvious differences in steel spring force and RMS acceleration level of the slab when the aspect ratio is smaller than 4.0, and the Mindlin plate model is suggested in this scenario. Moreover, the differences will be narrowed when the aspect ratio is larger than 4.0, and the thin plate model will still be a good choice if concerned frequency of the slab is lower than 200 Hz.

CRediT authorship contribution statement Jun Luo: Formal analysis, Data curation, Investigation, Methodology, Software, Validation, Visualization, Writing - original draft, Writing - review & editing. Shengyang Zhu: Conceptualization, Funding acquisition, Writing - review & editing,

J. Luo et al. / Mechanical Systems and Signal Processing 140 (2020) 106641

19

Supervision, Resources, Project administration. Wanming Zhai: Conceptualization, Funding acquisition, Supervision, Resources, Project administration. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 11790283, No. 51978587, No. 51708457, No. 51778194), the Fund from State Key Laboratory of Traction Power (2019TPL-T16), the Young Elite Scientists Sponsorship Program by CAST (2018QNRC001), which is gratefully acknowledged by the authors. Appendix A The appendix presents a procedure to decouple the deflection w and the bending slope w in differential equations of Timoshenko beam, and Eq. (1) is given here again for convenience:

EI


 @2w @w @2w  w  Iq 2 ¼ 0 þ G j A r 2 @x @x @t

ðA1Þ

! @2w @ 2 w @w ¼0 qA 2  Gjr A  @x2 @x @t

ðA2Þ

The following equation can be obtained based on Eqs. (A1) and (A2).

! @w 1 @2w @2w ¼ Iq 2  EI 2 þ w @x Gjr A @x @t

ðA3Þ

@w @ 2 w q @2w ¼ 2  @x @x Gjr @t 2

ðA4Þ

Moreover, the third-order partial derivative with respect to x for w and w and can be expressed as

! @3w 1 @4w @4w @2w ¼ Iq 2 2  EI 4 þ 2 3 @x Gjr A @x @x @x @t

ðA5Þ

@3w @4w q @4w ¼ 4  3 @x @x Gjr @x2 @t2

ðA6Þ

The first-order partial derivative with respect to x for Eq. (A1) and the second-order partial derivative with respect to t for Eq. (A2) are given by

EI

! @3w @ 2 w @w @3w  Iq þ G j A  ¼0 r 3 2 @x @x @x @x@t2

@3w @4w q @4w ¼ 2 2 2 Gjr @t4 @x@t @x @t

ðA7Þ

ðA8Þ

Substitution of Eqs. (A4), (A6) and (A8) into Eq. (A7) yields

EI


 @4w @2w EIq @ 4 w Iq2 @ 4 w þ q A  I q þ þ ¼0 2 2 4 2 @x Gjr @x @t Gjr @t 4 @t

ðA9Þ

The second-order partial derivative with respect to t for Eq. (A1) and the first-order partial derivative with respect to x for Eq. (A2) are given by

@3w 1 @4w @4w ¼ Iq 4  EI 2 2 2 G j A @x@t @t @x @t r

!

þ

@2w @t2

ðA10Þ

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J. Luo et al. / Mechanical Systems and Signal Processing 140 (2020) 106641

qA

! @3w @3w @2w ¼0  G j A  r @x3 @x2 @x@t2

ðA11Þ

Substitution of Eqs. (A5) and (A10) into Eq. (A11) leads to

EI


 4 @4w @2w EIq @ w Iq2 @ 4 w þ q A  I q þ þ ¼0 2 2 4 2 @x Gjr @x @t Gjr @t 4 @t

ðA12Þ

Thus, the deflection w and the bending slope w in differential equations of Timoshenko beam can be decoupled into two independent equations as shown in Eqs. (A9) and (A12). Appendix B The differential equation that governs the vibration of a rectangular thin plate can be expressed as

! @4w @4w @4w @2w D þ 2 þ q h  qðx; y; t Þ ¼ 0 þ s s @x4 @x2 @y2 @y4 @t2

ðB1Þ

For free vibration of the plate (q = 0), the energy functional P can be written in terms of the maximum strain energy Umax and the maximum kinetic energy Tmax as

P ¼ T max  U max U max

D ¼ 2

ZZ S

T max ¼

qs hs 2

ðB2Þ

2

2 2 4 @ W þ@ W @x2 @y2

!2

0

@2W @2W @2W  2ð1  mÞ@ 2   @x @y2 @x@y

!2 1 3 A5dxdy

ðB3Þ

ZZ W 2 dxdy

x2

ðB4Þ

S

Moreover, the mode functions of the thin plate corresponding to the mnth natural frequency xmn can be further equivalent to the product of the Euler-Bernoulli beam functions along the length and width of the plate, respectively.

W mn ðx; yÞ ¼ Amn X m ðxÞY n ðyÞ

ðB5Þ

By introducing the Hamilton principle, and substituting Eqs. (B3)–(B5) into Eq. (23), homogeneous linear equations with respect to the modal coefficient Amn can be arranged in the form



qs hs 2

D 2

x2mn B1m B1n  ½B2m B1n þ B1m B2n þ 2ðmB3m B3n þ ð1  mÞB4m B4n Þ Amn ¼ 0

ðB6Þ

where B1m, B1n, B2m, B2n, B3m, B3n, B4m, B4n are the integral constants with respect to the Euler-Bernoulli beam functions, which take the form

RW RL B1n ¼ 0 s Y 2n ðyÞdy B2m ¼ 0 s X 002 m ðxÞdx R W s 002 R Ls 00 R W s 00 B2n ¼ 0 Y n ðyÞdy B3m ¼ 0 X m ðxÞX m ðxÞdx B3n ¼ 0 Y n ðyÞY n ðyÞdy RL RW B4m ¼ 0 s X 02 B4n ¼ 0 s Y 02 m ðxÞdx n ðyÞdy

B1m ¼

R Ls 0

X 2m ðxÞdx

ðB7Þ

Based on Eq. (B6), the natural frequency of the thin plate is given by

xmn

sffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D B2m B1n þ B1m B2n þ 2ðmB3m B3n þ ð1  mÞB4m B4n Þ ¼ B1m B1n q s hs

ðB8Þ

Now consider the forced vibration of the thin plate (q – 0). The solutions of the dynamic vertical displacements can be expanded separately by Ritz method, given as

wðx; y; t Þ ¼

Ny Nx X X

W mn ðx; yÞT mn ðtÞ ¼

m¼1 n¼1

Ny Nx X X

X m ðxÞY n ðyÞT mn ðt Þ

ðB9Þ

m¼1 n¼1

Note that the following mode equations can be established by substituting Eq. (16a) into Eq. (B1)

@4W @4W @4W D þ 2 þ @x4 @x2 @y2 @y4

!

 qs hs x2 W ¼ 0

ðB10Þ

J. Luo et al. / Mechanical Systems and Signal Processing 140 (2020) 106641

21

Substitution of Eqs. (B9) and (B10) into Eq. (B1) yields Ny h Nx X X

i T€ mn ðt Þ þ x2mn T mn ðt Þ qs hs W mn  qðx; y; tÞ ¼ 0

ðB11Þ

m¼1 n¼1

By utilizing orthogonality condition of the mode shapes of the thin plates,



ZZ

qs hs W mn W kl dxdy S

¼0

m–k or n–l

–0

m ¼ k and n ¼ l

ðB12Þ

the second-order ordinary differential equations of the slab vertical vibration in terms of the generalized coordinate can be obtained as follows:

RR

T€ mn ðtÞ þ x

2 mn T mn ðt Þ

¼

S

qðx; y; tÞW mn dxdy RR qs hs W 2mn dxdy

ðB13Þ

S

Furthermore, by taking the classical modal damping into account, the forced vibration equations of the floating slab according to the thin plate theory can be recast as

" T€ mn ðtÞ þ 2fmn xmn T_ mn ðtÞ þ x2mn T mn ðt Þ ¼

Np P

i¼1

PrVi ðtÞX m ðxsi ÞY n ðysi Þ 

Nb P

#     F sVj ðt ÞX m xbj Y n ybj

j¼1

qs hs B1m B1n

ðB14Þ

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