Dynamic response of sleepers in a track with uneven rail irregularities using a 3D vehicle–track model with sleeper beams

Abstract

Employing a 3D model of the vehicle–track interaction, dynamic responses of sleepers in railway tracks are determined due to the effect of uneven irregularities (cross-levels) in left/right rails. An integrated mathematical model of the vehicle–track system is used to calculate the dynamic behavior of sleepers in the railway track. The general methodology of the dynamic analysis is presented by extracting the mathematical equations and by introducing the coupling process in the wheel–rail interface. Besides the vertical (bounce) and pitch motions, roll degrees of freedom of the components are taken into account. Except rails and sleepers, other components of the vehicle–track system are simulated with the rigid-body elements, linked to each other within the resilient elements (springs and dashpots). The rails and sleepers are modeled with the beam elements, making the model more detailed in the track subsystem. A numerical integration algorithm based on Newmark’s method is developed, capable of solving the dynamic interaction of the entire system in time domain. Applying various irregularity profiles from the field measurement data as the source of excitation, the dynamic response of sleepers in track structure is treated in the longitudinal and traverse directions. Dynamic displacements under sleepers are compared with the analytical solution of the beam on elastic foundation (static response) to highlight the dynamic amplifications. The results of analysis show that the profile unevenness in left/right rails plays an important role in dynamic stability of the sleepers in railway track, particularly when the severity of the profile unevenness was increased.

Appendices

Appendix 1: Equations of the vehicle [31, 37]

Vertical motion of the car-body:

$$\begin{aligned} M_{Y}^{c} \ddot{Y}^{c} + 2c_{{2Yj}} \dot{Y}^{c} - 2c_{{2Yj}} \dot{Y}_{j}^{b} + 2k_{{2Yj}} Y^{c} - 2k_{{2Yj}} Y_{j}^{b} = 0,\quad j = f,r \end{aligned}$$

(14)

Pitching motion of the car-body:

$$\begin{aligned} J_{Z}^{c} \ddot{R}_{Z}^{c} + 2s_{j}^{2} c_{{2Yj}} \dot{R}_{Z}^{c} + 2s_{j}^{2} k_{{2Yj}} R_{Z}^{c} = 0,\quad j = f,r. \end{aligned}$$

(15)

Rolling motion of the car-body:

$$\begin{aligned}&J_{X}^{c} \ddot{R}_{X}^{c} + 2\left( b_{{2j}}^{2} c_{{2Yj}} + h_{{1j}}^{2} c_{{2Zj}} \right) \dot{R}_{X}^{c} + \left( 2h_{{1j}} h_{{2j}} c_{{2Zj}} - 2b_{{2j}}^{2} c_{{2Yj}} \right) \dot{R}_{{Xj}}^{b}\nonumber \\&\qquad + 2\left( b_{{2j}}^{2} k_{{2Yj}} + h_{{1j}}^{2} k_{{2Zj}} \right) R_{X}^{c} + \left( 2h_{{1j}} h_{{2j}} k_{{2Zj}} - 2b_{{2j}}^{2} k_{{2Yj}} \right) R_{{Xj}}^{b} = 0,\quad j = f,r \end{aligned}$$

(16)

Vertical motion of the jth bogie:

$$\begin{aligned}&M_{{Yj}}^{b} \ddot{Y}_{j}^{b} + \left( 2c_{{2Yj}} + 4c_{{1Yj}} \right) \dot{Y}_{j}^{b} - 2c_{{2Yj}} \dot{Y}^{c} + ( - 1)^{j} 2s_{j} c_{{2Yj}} \dot{R}_{Z}^{c} \nonumber \\&\qquad - 2c_{{1Yj}} \dot{Y}_{{jk}}^{w} + \left( 2k_{{2Yj}} + 4k_{{1Yj}} \right) Y_{j}^{b} - 2k_{{2Yj}} Y^{c} + ( - 1)^{j} 2s_{j} k_{{2Yj}} R_{Z}^{c} = 0 \nonumber \\&\qquad j = f,r,\quad k = 1,2 \end{aligned}$$

(17)

Pitching motion of the jth bogie:

$$\begin{aligned} J_{{Zj}}^{b} \ddot{R}_{{Zj}}^{b} + 4t_{j}^{2} c_{{1Yj}} \dot{R}_{{Zj}}^{b} + ( - 1)^{k} 2t_{j} c_{{1Yj}} \dot{Y}_{{jk}}^{w} + 4t_{j}^{2} k_{{1Yj}} R_{{Zj}}^{b} + ( - 1)^{k} 2t_{j} k_{{1Yj}} Y_{{jk}}^{w} = 0. \end{aligned}$$

(18)

Rolling motion of the jth bogie:

$$\begin{aligned}&J_{{Xj}}^{b} \ddot{R}_{{Xj}}^{b} + \left( 2b_{{2j}}^{2} c_{{2Yj}} + 2h_{{2j}}^{2} c_{{2Zj}} + 4h_{{3j}}^{2} c_{{1Zj}} + 4b_{{1j}}^{2} c_{{1Yj}} \right) \dot{R}_{{Xj}}^{b} + 2\left( h_{{1j}} h_{{2j}} c_{{2Zj}} - b_{{2j}}^{2} c_{{2Yj}} \right) \dot{R}_{X}^{c} \nonumber \\&\qquad - 2b_{{1j}}^{2} c_{{1Yj}} \dot{R}_{{Xjk}}^{w} + \left( 2b_{{2j}}^{2} k_{{2Yj}} + 2h_{{2j}}^{2} k_{{2Zj}} + 4h_{{3j}}^{2} k_{{1Zj}} + 4b_{{1j}}^{2} k_{{1Yj}}\right) R_{{Xj}}^{b} \nonumber \\&\qquad -2\left( h_{{1j}} h_{{2j}} k_{{2Zj}} - b_{{2j}}^{2} k_{{2Yj}} \right) R_{X}^{c} - 2b_{{1j}}^{2} k_{{1Yj}} R_{{Xjk}}^{w} = 0,\quad j = f,r,\;\; k = 1,2 \end{aligned}$$

(19)

Vertical motion of the kth wheelset:

$$\begin{aligned}&M_{{Y,jk}}^{w} \ddot{Y}_{{jk}}^{w} + 2c_{{1Y}} \dot{Y}_{{jk}}^{w} - 2c_{{1Y}} \dot{Y}_{j}^{b} + ( - 1)^{k} 2c_{{1Y}} t_{{jk}} \dot{R}_{{Zj}}^{b} + 2k_{{1Y}} Y_{{jk}}^{w} - 2k_{{1Y}} Y_{j}^{b} \nonumber \\&\qquad + ( - 1)^{k} 2k_{{1Y}} t_{{jk}} R_{{Zj}}^{b} = F_{{Y,jk}}^{{w - r}} . \end{aligned}$$

(20)

Rolling motion of the kth wheelset:

$$\begin{aligned} J_{X}^{w} \ddot{R}_{{Xjk}}^{w} + 2c_{{1Y}} b_{1}^{2} \dot{R}_{{Xjk}}^{w} - 2c_{{1Y}} b_{1}^{2} \dot{R}_{{Xj}}^{b} + 2k_{{1Y}} b_{1}^{2} R_{{Xjk}}^{w} - 2k_{{1Y}} b_{1}^{2} R_{{Xj}}^{b} = M_{{X,jk}}^{{w - r}} \end{aligned}$$

(21)

Matrix-form equation of the entire vehicle:

$$\begin{aligned}{}[M^{v} ]\{ \ddot{U}^{v} \} + [C^{v}]\{\dot{U}^{v} \} + [K^{v} ]\{ U^{v} \} = \{ F^{v} \} \end{aligned}$$

(22)

Displacement vector of the car-body:

$$\begin{aligned} \{ U^{v} \} = \left\langle \begin{array}{lllllll} {u_{{f1}}^{w} } &{} \;\; {u_{{f2}}^{w} } &{} \;\; {u_{f}^{b} } &{}\;\; {u^{c} } &{}\;\; {u_{r}^{b} } &{}\;\; {u_{{r1}}^{w} } &{}\;\; {u_{{r2}}^{w} } \\ \end{array}\right\rangle ^{\mathrm{T}}_{(17,1)} . \end{aligned}$$

(23)

Mass matrix of the vehicle:

$$\begin{aligned}{}[M^{v} ] = \hbox {diag}\left[ \begin{array}{ccccccc} {M_{{f1}}^{w} } &{}\;\; {M_{{f2}}^{w} } &{}\;\; {M_{f}^{b} } &{}\;\; {M^{c} } &{}\;\; {M_{r}^{b} } &{}\;\; {M_{{r1}}^{w} } &{}\;\; {M_{{r2}}^{w} } \\ \end{array} \right] \end{aligned}$$

(24)

Stiffness matrix of the vehicle:

$$\begin{aligned}{}[K^v ]=\left[ {\begin{array}{ccccccc} {K_{f1}^w }&{}\;\; 0&{}\;\; {K_{f1}^{wb} }&{}\;\; 0&{}\;\; 0&{}\;\; 0&{}\;\; 0 \\ &{}\;\; {K_{f2}^w }&{}\;\; {K_{f2}^{wb} }&{}\;\; 0&{}\;\; 0&{}\;\; 0&{}\;\; 0 \\ &{} &{}\;\; {K_f^b }&{}\;\; {K_f^{bc} }&{}\;\; 0&{}\;\; 0&{}\;\; 0 \\ &{} &{} &{}\;\; {K^c }&{}\;\; {K_r^{bc} }&{}\;\; 0&{}\;\; 0 \\ &{} &{} &{} &{}\;\; {K_r^b }&{}\;\; {K_{r1}^{wb} }&{}\;\; {K_{r2}^{wb} } \\ &{} &{} &{} &{} &{}\;\; {K_{r1}^w }&{}\;\; 0 \\ {\hbox {sym.}}&{} &{} &{} &{} &{} &{}\;\; {K_{r2}^w } \\ \end{array} } \right] \end{aligned}$$

(25)

External dynamic force vector of the vehicle:

$$\begin{aligned} \left\{ F^v \right\} = \left\langle {\begin{array}{lllllll} {F_{f1}^w }&{}\;\; {F_{f2}^w }&{}\;\; {F_f^b }&{}\;\; {F^c }&{}\;\; {F_r^b }&{}\;\; {F_{r1}^w }&{}\;\; {F_{r2}^w } \\ \end{array} }\right\rangle _{(17,1)}^{\mathrm{T}} \end{aligned}$$

(26)

Effective force vectors in the integration problem of this study:

$$\begin{aligned} \left\{ F^{c} \right\}= & {} \{ 0\} \quad \hbox {and} \quad \left\{ F_{j}^{b} \right\} = \{0\}\nonumber \\ \left\{ F_{{jk}}^{w} \right\}= & {} \left\{ F_{{jk}}^{{w - r}} \right\} = \left\langle {M_{X}^{{w - r}} }\;\; {F_{Y}^{w - r}} \right\rangle _{{jk}}^{\mathrm{T}} \end{aligned}$$

(27)

Appendix 2: Equations of the track [31, 47]

Vertical/bending motion of the rail beam:

$$\begin{aligned} m^{r} \frac{{\partial ^{2} y^{r} (x,t)}}{{\partial t^{2} }} = \sum \limits _{N_{\mathrm{w}} }^{{j = 1}} F_{\mathrm{wr}zj} (t)\delta (x - x_{\mathrm{w}_{j} } )\rho ^{r} I_{y} ^{r} \frac{{\partial ^{2} R_{z}^{r} (x,t)}}{{\partial t^{2} }} - EI_{y} \frac{{\partial ^{4} R_{z}^{r} (x,t)}}{{\partial x^{4} }} = 0 \end{aligned}$$

(28)

Torsional motion (rolling) of the rail beam:

$$\begin{aligned} \rho ^{r}I_0^{r}\frac{\partial ^{2}R_x^r (x,t)}{\partial t^{2}}=-\sum \limits _{N_\mathrm{s} }^{i=1} M_{\mathrm{s}i} (t)\delta \times (x-x_{\mathrm{s}i} )+\sum \limits _{N_\mathrm{w} }^{j=1} M_{\mathrm{G}j} (t)\delta (x-x_{\mathrm{w}_j } ) \end{aligned}$$

(29)

Rail beam shape functions for deflection, bending and torsion (20 mode functions were superposed):

$$\begin{aligned} y^{r}(x,t)= & {} \sum \limits _{20}^{k=1} Y_k^{r}(x)q_{yk}^{r}(t)\qquad Y_k^{r}(x)=\sqrt{\frac{2}{m^{r}l_e^r }}\sin \left( \frac{k\pi }{l_e^r }x\right) \nonumber \\ r_z^r (x,t)= & {} \sum \limits _{20}^{k=1} R_{zk}^{r}(x)w_{zk}^{r}(t)\qquad R_{zk}^{r}(x)=\sqrt{\frac{2}{\rho I_z^{r}l_e^r }}\cos \left( \frac{k\pi }{l_e^r }x\right) \nonumber \\ r_x^r (x,t)= & {} \sum \limits _{20}^{k=1} R_{xk}^{r}(x)q_{\mathrm{x}k}^{r}(t)\qquad R_{xk}^{r}(x)=\sqrt{\frac{2}{\rho I_0^{r}l_e^r }}\sin \left( \frac{k\pi }{l_e^r }x\right) \end{aligned}$$

(30)

Vertical/bending motion of the sleeper beam (z is the major axis of the sleeper):

$$\begin{aligned} m^{s}\frac{\partial ^{2}y^{s}(z,t)}{\partial t^{2}}= \sum \limits _{N_\mathrm{r} (2)}^{j=1} F_{\mathrm{rs}zj} (t)\delta (z-z_{\mathrm{w}_j } )\rho ^{s}I_z^{s}\frac{\partial ^{2}R_x^s (z,t)}{\partial t^{2}}-E^{s}I_z^{s}\frac{\partial ^{4}R_x^s (z,t)}{\partial z^{4}}=0 \end{aligned}$$

(31)

Sleeper beam shape functions for deflection, bending and torsion (20 mode functions were superposed):

$$\begin{aligned} y^{s}(z,t)= & {} \sum \limits _{20}^{k=1} Y_k^{s}(z)q_{yk}^{s}(t)\qquad Y_k^{s}(z)=\sqrt{\frac{2}{m^{s}l_e^s }}\sin \left( \frac{k\pi }{l_e^s }z\right) \nonumber \\ r_x^s (z,t)= & {} \sum \limits _{20}^{k=1} R_{xk}^{s}(x)q_{\mathrm{x}k}^{s}(t)\qquad R_{xk}^{s}(z)=\sqrt{\frac{2}{\rho ^{s}I_0^{s}l_e^s }}\sin \left( \frac{k\pi }{l_e^s }z\right) \end{aligned}$$

(32)

Dynamic equilibrium of the ith ballast block in vertical motion:

$$\begin{aligned}&M_{{Yij}}^{{bl}} \ddot{Y}_{{ij}}^{{bl}} + (c_{{bYij}} + 2c_{{fYij}} )\dot{Y}_{{ij}}^{{bl}} - c_{{bYij}} \dot{Y}_{{ij}}^{r} + (k_{{bYij}} + 2k_{{fYij}} )Y_{{ij}}^{{bl}} \nonumber \\&\qquad - k_{{bYij}} Y_{{ij}}^{r} = 0,\quad i = 1,2,\ldots ,N_{s} ,\;\; j = 1,2,\ldots ,5 \end{aligned}$$

(33)

Mass matrix of the rail and sleeper elements:

$$\begin{aligned} \left[ M^{r/s} \right] =\frac{m.l_e^{r/s}}{420}\left[ { \begin{array}{llll} {156}&{}\quad {22l_e^{r/s} }&{}\quad {54}&{}\quad {-13l_e^{r/s} } \\ {22l_e^{r/s} }&{}\quad {4l_e^{r/s2}}&{}\quad {13l_e^{r/s} }&{}\quad {-3l_e^{r/s2}} \\ {54}&{}\quad {13l_e^{r/s} }&{}\quad {2l_e^{r/s2}}&{}\quad {-22l_e^{r/s} } \\ {-13l_e^{r/s} }&{}\quad {-3l_e^{r/s2}}&{}\quad {-22l_e^{r/s} }&{}\quad {4l_e^{r/s2}} \\ \end{array} } \right] \end{aligned}$$

(34)

Stiffness matrix of the rail and sleeper elements:

$$\begin{aligned} \left[ K^{r/s} \right] =\frac{2E^{r/s}I^{r/s}}{l_e^{r/s3}}\left[ {{\begin{array}{cccc} 6&{} {-6}&{} {3l_e^{r/s} }&{} {3l_e^{r/s} } \\ {-6}&{} 6&{} {-3l_e^{r/s} }&{} {-3l_e^{r/s} } \\ {3l_e^{r/s} }&{} {-3l_e^{r/s} }&{} {2l_e^{r/s2}}&{} {l_e^{r/s2}} \\ {3l_e^{r/s} }&{} {-3l_e^{r/s} }&{} {l_e^{r/s2}}&{} {2l_e^{r/s2}} \\ \end{array} }} \right] \end{aligned}$$

(35)

Damping matrix of the jth rail and sleeper beams, Rayleigh damping

$$\begin{aligned} C_j^{r/s} =\alpha M_j^{r/s} +\beta K_j^{r/s} \end{aligned}$$

(36)

Mass matrices of the entire ballast elements (The entire ballast mass divided to five separate blocks) (Table 4):

$$\begin{aligned} \left[ M_j^{Bl} \right] =\left[ {{\begin{array}{cccc} {M_1^{bl} }&{} 0&{} \cdots &{} 0 \\ 0&{} {M_2^{bl} }&{} \cdots &{} 0 \\ \cdot &{} \cdot &{} \cdot &{} \cdot \\ \cdot &{} \cdot &{} \cdot &{} \cdot \\ \cdot &{} \cdot &{} \cdot &{} \cdot \\ 0&{} 0&{} \cdots &{} {M_{N_s }^{bl} } \\ \end{array} }} \right] , \;\; j=1,2,\ldots ,5 \end{aligned}$$

(37)

Manipulating configurations of the track matrices:

$$\begin{aligned} \left[ M^{T}\right] =\left[ {{\begin{array}{lll} {[M^{r}]}&{} \quad {[M^{r}/M^{s}]}&{} \quad 0 \\ {[M^{r}/M^{s}]}&{} \quad {[M^{s}]}&{} \quad {[M^{s}/M^{bl}]} \\ 0&{} \quad {[M^{s}/M^{bl}]}&{} \quad {[M^{bl}]} \\ \end{array} }} \right] \end{aligned}$$

(38)

Equation of motion of the entire track:

$$\begin{aligned} \left[ M^{t} \right] \{ \ddot{U}^{t} \} + [C^{t} ]\{ \dot{U}^{t} \} + [K^{t} ]\{ U^{t} \} = \{ F^{t} \} \end{aligned}$$

(39)

Table 4 Notations