1 Introduction
1.1 Research background
1.2 State-of-the-art review
1.3 Goal of this work
2 Modeling of the train–track–substructure dynamic interaction
2.1 Train model
2.2 Track model
2.3 Bridge model
2.4 Tunnel model
2.5 General methods for the coupling of sub-systems
2.5.1 Wheel–rail coupling matrices
2.5.2 Interaction matrices between the track and substructures
3 Method for achieving large-scale train–track–substructure dynamic interaction
3.1 Park integration method
3.2 Mapping relation of the degrees of freedom with respect to various coordinate systems
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When \(x_{4} \le L_{x,1}\) is satisfied,\({{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{\text{l}} - X_{{\text{l}}} - Y_{{\text{l}}} - Z_{{\text{l}}} }} = {{\varvec{\varPhi}}}_{\text{c,1}}^{\text{l}} :{{\varvec{\varPhi}}}_{\text{c,2}}^{{\text{l}}} ,\, {\varvec{\varPhi}}_{\text{c,1}}^{\text{l}} { = }1\), \({{\varvec{\varPhi}}}_{{\text{c,2}}}^{{\text{l}}} { = }\left( {t_{2} - n_{0} + 1} \right)N_{{\text{c}}}\),\({{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{m}}} - X_{{\text{m}}} - Y_{{\text{m}}} - Z_{{\text{m}}} }} = {{\varvec{\varPhi}}}_{{\text{c,1}}}^{{\text{m}}} :{{\varvec{\varPhi}}}_{{\text{c,2}}}^{{\text{m}}}\), \({{\varvec{\varPhi}}}_{{{\text{c}} ,1}}^{{\text{m}}} { = }\left( {n_{0} - t_{1} } \right)N_{{\text{c}}} + 1\), \({{\varvec{\varPhi}}}_{{\text{c,2}}}^{{\text{m}}} { = }\left( {t_{2} - t_{1} + 1} \right)N_{{\text{c}}}\),with \(t_{1} { = }\left[ {x_{4} /\left( {L_{{\text{t}}} + l_{{\text{t}}}^{^{\prime}} } \right)} \right] + 1\), \(t_{2} { = }\left[ {x_{1} /\left( {L_{{\text{t}}} + l_{{\text{t}}}^{^{\prime}} } \right)} \right] + 1\),where the symbol “:” denotes an operator of the left number to right number with increment of 1; \(t_{1}\) and \(t_{2}\) denote the track slab number with respect to the positions of \(x_{4}\) and \(x_{1}\), respectively; \(N_{{\text{c}}}\) denotes the total number of DOFs for a baseplate; \(n_{0}\) denotes the initial baseplate against the start position of the substructure.
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When \(L_{x,1} < x_{4} < L_{x,2}\) is satisfied,\({{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{l}}} - X_{{\text{l}}} - Y_{{\text{l}}} - Z_{{\text{l}}} }} = {{\varvec{\varPhi}}}_{{\text{c,1}}}^{{\text{l}}} :{{\varvec{\varPhi}}}_{{\text{c,2}}}^{{\text{l}}}, \, {{\varvec{\varPhi}}}_{{\text{c,1}}}^{{\text{l}}} { = }\left( {t_{1} - n_{0} } \right)N_{{\text{c}}} + 1\), \({{\varvec{\varPhi}}}_{{\text{c,2}}}^{{\text{l}}} { = }\left( {t_{2} - n_{0} + 1} \right)N_{{\text{c}}},\, {{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{m}}} - X_{{\text{m}}} - Y_{{\text{m}}} - Z_{{\text{m}}} }} = {{\varvec{\varPhi}}}_{{\text{c,1}}}^{{\text{m}}} :{{\varvec{\varPhi}}}_{{\text{c,2}}}^{{\text{m}}}, \,{{\varvec{\varPhi}}}_{{\text{c,1}}}^{{\text{m}}} { = }1,\)$${{\varvec{\varPhi}}}_{{{\text{c}} ,2}}^{{\text{m}}} { = }\left( {t_{2} - t_{1} + 1} \right)N_{{\text{c}}}.$$For conditions \(x_{1} \ge L_{x,2}\):
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When \(x_{4} < L_{x,1}\) is satisfied,\({{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{l}}} - X_{{\text{l}}} - Y_{{\text{l}}} - Z_{{\text{l}}} }} = {{\varvec{\varPhi}}}_{{\text{c,1}}}^{{\text{l}}} :{{\varvec{\varPhi}}}_{{\text{c,2}}}^{{\text{l}}}\), \({{\varvec{\varPhi}}}_{{\text{c,1}}}^{{\text{l}}} { = }1\), \({{\varvec{\varPhi}}}_{{\text{c,2}}}^{{\text{l}}} { = }\left( {n_{1} - n_{0} + 1} \right)N_{{\text{c}}}\),\({{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{m}}} - X_{{\text{m}}} - Y_{{\text{m}}} - Z_{{\text{m}}} }} = {{\varvec{\varPhi}}}_{{\text{c,1}}}^{{\text{m}}} :{{\varvec{\varPhi}}}_{{\text{c,2}}}^{{\text{m}}}\), \({{\varvec{\varPhi}}}_{{\text{c,1}}}^{{\text{m}}} { = }\left( {n_{0} - t_{1} } \right)N_{{\text{c}}} + 1\),where \(n_{1}\) denotes the end baseplate number against the end position of the substructure.$${{\varvec{\varPhi}}}_{{\text{c,2}}}^{{\text{m}}} { = }\left( {n_{1} - t_{1} + 1} \right)N_{{\text{c}}},$$
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When \(L_{x,1} \le x_{4} < L_{x,2}\) is satisfied,\({{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{l}}} - X_{{\text{l}}} - Y_{{\text{l}}} - Z_{{\text{l}}} }} = {{\varvec{\varPhi}}}_{{\text{c,1}}}^{{\text{l}}} :{{\varvec{\varPhi}}}_{{\text{c,2}}}^{{\text{l}}}\), \({{\varvec{\varPhi}}}_{{\text{c,1}}}^{{\text{l}}} { = }\left( {t_{1} - n_{0} } \right)N_{{\text{c}}} + 1\), \(\quad{{\varvec{\varPhi}}}_{{\text{c,2}}}^{{\text{l}}} { = }\left( {n_{1} - n_{0} + 1} \right)N_{{\text{c}}}\),\({{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{m}}} - X_{{\text{m}}} - Y_{{\text{m}}} - Z_{{\text{m}}} }} = {{\varvec{\varPhi}}}_{{\text{c,1}}}^{{\text{m}}} :{{\varvec{\varPhi}}}_{{\text{c,2}}}^{{\text{m}}}\), \({{\varvec{\varPhi}}}_{{\text{c,1}}}^{{\text{m}}} { = }1\),$${{\varvec{\varPhi}}}_{{\text{c,2}}}^{{\text{m}}} { = }\left( {n_{1} - t_{1} + 1} \right)N_{{\text{c}}}.$$
3.3 Iterative procedures for this large-scale dynamic system
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Step 1 Set \({\varvec{X}}_{n} \left( {{{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{0} - X_{0} - Y_{0} - Z_{0} }} } \right)\) as convergence index.If \({\varvec{X}}_{n} \left( {\varvec{\varPhi}_{{\text{c}}}^{{O_{0} - X_{0} - Y_{0} - Z_{0} }} } \right) \le \varepsilon\), where \(\varepsilon { = 10}^{{{ - }8}}\), go to step 5; or go to step 2.
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The train–track system DOFs in the \(O_{0} - X_{0} - Y_{0} - Z_{0}\)and \(O_{{\text{m}}} - X_{{\text{m}}} - Y_{{\text{m}}} - Z_{{\text{m}}}\) coordinate systems are, respectively, represented as \({{\varvec{\varPhi}}}_{\text{TT}}\) and \({{\varvec{\varPhi}}}_{{{\text{TT}}}}^{^{\prime}}\). The force vector for the train–track system includes the wheel–rail force vector \({\varvec{F}}_{0}\) and the boundary force exerted by the substructure, that is,with$${\varvec{F}}_{{{\text{TT}}}} \left( {n{}_{{\text{d}}} + {{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{m}}} - X_{{\text{m}}} - Y_{{\text{m}}} - Z_{{\text{m}}} }} } \right) = {\varvec{F}}_{0} \left( {n{}_{{\text{d}}} + {{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{m}}} - X_{{\text{m}}} - Y_{{\text{m}}} - Z_{{\text{m}}} }} } \right) - {\varvec{F}}_{{\text{m}}} - {\varvec{F}}_{{\text{c}}} - {\varvec{F}}_{{\text{k}}}$$(16)$${\varvec{F}}_{{\text{m}}}\, { = }\,{\varvec{M}}_{{{\text{cS}}}} \left( {{{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{l}}} - X_{{\text{l}}} - Y_{{\text{l}}} - Z_{{\text{l}}} }} ,n_{{\text{d}}}^{^{\prime}} + 1:n_{{\text{d}}}^{^{\prime}} + \varvec{N}_{{{\text{gp}}}} } \right){\ddot{\varvec{X}}} \left( {{{\varvec{\varPhi}}}_{{\text{r}}} + {{\varvec{\varPhi}}}_{{\text{t}}} + {{\varvec{\varPhi}}}_{{\text{c}}} + 1:{{\varvec{\varPhi}}}_{{\text{r}}} + {{\varvec{\varPhi}}}_{{\text{t}}} + {{\varvec{\varPhi}}}_{{\text{c}}} + \varvec{N}_{{{\text{gp}}}} } \right),$$$${\varvec{F}}_{{\text{c}}} = {\varvec{C}}_{{{\text{cS}}}} \left( {{{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{l}}} - X_{{\text{l}}} - Y_{{\text{l}}} - Z_{{\text{l}}} }} ,n_{{\text{d}}}^{^{\prime}} + 1:n_{{\text{d}}}^{^{\prime}} + \varvec{N}_{{{\text{gp}}}} } \right){\dot{\varvec{X}}}\left( {{{\varvec{\varPhi}}}_{{\text{r}}} + {{\varvec{\varPhi}}}_{{\text{t}}} + {{\varvec{\varPhi}}}_{{\text{c}}} + 1:{{\varvec{\varPhi}}}_{{\text{r}}} + {{\varvec{\varPhi}}}_{{\text{t}}} + {{\varvec{\varPhi}}}_{{\text{c}}} + \varvec{N}_{{{\text{gp}}}} } \right),$$$${\varvec{F}}_{{\text{k}}} { = }{\varvec{K}}_{{{\text{cS}}}} \left( {{{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{l}}} - X_{{\text{l}}} - Y_{{\text{l}}} - Z_{{\text{l}}} }} ,n_{{\text{d}}}^{^{\prime}} + 1:n_{{\text{d}}}^{^{\prime}} + \varvec{N}_{{{\text{gp}}}} } \right){\varvec{X}}\left( {{{\varvec{\varPhi}}}_{{\text{r}}} + {{\varvec{\varPhi}}}_{{\text{t}}} + {{\varvec{\varPhi}}}_{{\text{c}}} + 1:{{\varvec{\varPhi}}}_{{\text{r}}} + {{\varvec{\varPhi}}}_{{\text{t}}} + {{\varvec{\varPhi}}}_{{\text{c}}} + \varvec{N}_{{{\text{gp}}}} } \right),$$where \(n{}_{{\text{d}}}\) and \(n_{{\text{d}}}^{^{\prime}}\) are, respectively, the total number of DOFs of the rail and track slab in the \(O_{{\text{m}}} - X_{{\text{m}}} - Y_{{\text{m}}} - Z_{{\text{m}}}\) and \(O_{{\text{l}}} - X_{{\text{l}}} - Y_{{\text{l}}} - Z_{{\text{l}}}\) coordinate systems; \({{\varvec{\varPhi}}}_{{\text{r}}}\), \({{\varvec{\varPhi}}}_{{\text{t}}}\) and \({{\varvec{\varPhi}}}_{{\text{c}}}\) denote the total number of DOFs of the rail, track slab and support layer in \(O_{0} - X_{0} - Y_{0} - Z_{0}\) coordinate system, respectively; \(N_{{{\text{gp}}}}\) is the total number of the substructural DOFs; \({\varvec{M}}_{{{\text{cS}}}}\), \({\varvec{C}}_{{{\text{cS}}}}\) and \({\varvec{K}}_{{{\text{cS}}}}\) denote the mass, damping and stiffness matrices of the supporting layer–substructure coupling system, respectively.
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Then TTS system responses with train–track system solution update can be obtained by following Eq. (14) as$$\left( {{\ddot{\varvec{x}}} _{n + 1} ,{\dot{\varvec{x}}}_{n + 1} ,{\varvec{x}}_{n + 1} } \right){ = }P\left( {{\varvec{M}}_{{{\text{TT}}}} ,{\varvec{C}}_{{{\text{TT}}}} ,{\varvec{K}}_{{{\text{TT}}}} ,{\varvec{F}}_{{{\text{TT}}}} ,{\varvec{x}}_{n\sim n - 2} ,{\dot{\varvec{x}}}_{n\sim n - 2} ,{{\varvec{\varPhi}}}_{{{\text{TT}}}} ,{{\varvec{\varPhi}}}_{{{\text{TT}}}}^{^{\prime}} } \right) .$$(17)
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Step 3 Calculate the substructural system response.The force vector of the substructural system excited by the supporting layer iswith$${\varvec{F}}_{\text S} = - {\varvec{F}}_{{\text{m}}}^{^{\prime}} - {\varvec{F}}_{{\text{m}}}^{^{\prime}} - {\varvec{F}}_{{\text{m}}}^{^{\prime}}$$(18)$${\varvec{F}}_{{\text{m}}}^{^{\prime}} { = }{\varvec{M}}_{{{\text{cS}}}} \left( {n_{{\text{d}}}^{^{\prime}} + 1:n_{{\text{d}}}^{^{\prime}} + N_{{{\text{gp}}}} ,{{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{l}}} - X_{{\text{l}}} - Y_{{\text{l}}} - Z_{{\text{l}}} }} } \right){\ddot{\varvec{X}}} \left( {{{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{0} - X_{0} - Y_{0} - Z_{0} }} {|}_{{{{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{m}}} - X_{{\text{m}}} - Y_{{\text{m}}} - Z_{{\text{m}}} }} }} } \right),$$$${\varvec{F}}_{{\text{c}}}^{^{\prime}} { = }{\varvec{C}}_{{{\text{cS}}}} \left( {n_{{\text{d}}}^{^{\prime}} + 1:n_{{\text{d}}}^{^{\prime}} + N_{{{\text{gp}}}} ,{{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{l}}} - X_{{\text{l}}} - Y_{{\text{l}}} - Z_{{\text{l}}} }} } \right){\dot{\varvec{X}}}\left( {{{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{0} - X_{0} - Y_{0} - Z_{0} }} {|}_{{{{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{m}}} - X_{{\text{m}}} - Y_{{\text{m}}} - Z_{{\text{m}}} }} }} } \right),$$$${\varvec{F}}_{{\text{k}}}^{^{\prime}} { = }{\varvec{K}}_{{{\text{cS}}}} \left( {n_{{\text{d}}}^{^{\prime}} + 1:n_{{\text{d}}}^{^{\prime}} + N_{{{\text{gp}}}} ,{{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{l}}} - X_{{\text{l}}} - Y_{{\text{l}}} - Z_{{\text{l}}} }} } \right){\varvec{X}}\left( {{{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{0} - X_{0} - Y_{0} - Z_{0} }} {|}_{{{{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{m}}} - X_{{\text{m}}} - Y_{{\text{m}}} - Z_{{\text{m}}} }} }} } \right),$$where \({{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{0} - X_{0} - Y_{0} - Z_{0} }} {|}_{{{{\varvec{\varPhi}}}_{{\text{c}}}^{{O_{{\text{m}}} - X_{{\text{m}}} - Y_{{\text{m}}} - Z_{{\text{m}}} }} }}\) denotes the DoF vector of supporting layer chosen from coordinate of \(O_{0} - X_{0} - Y_{0} - Z_{0}\).
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Consequently, the TTS system responses by updating the substructural system solution can be obtained byin which \({{\varvec{\varPhi}}}_{\text{S}} { = }{{\varvec{\varPhi}}}_{{\text{r}}} + {{\varvec{\varPhi}}}_{{\text{t}}} + {{\varvec{\varPhi}}}_{{\text{c}}} + 1:{{\varvec{\varPhi}}}_{{\text{r}}} + {{\varvec{\varPhi}}}_{{\text{t}}} + {{\varvec{\varPhi}}}_{{\text{c}}} + N_{{{\text{gp}}}}\), and \({\varvec{\varPhi}}_{\text{S}}^{\prime} { = }n_{\text{d}}^{\prime} + 1:n_{\text{d}}^{\prime} + N_{\text{gp}}\).$$\left( {{\ddot{\varvec{x}}} _{n + 1} ,{\dot{\varvec{x}}}_{n + 1} ,{\varvec{x}}_{n + 1} } \right){ = }P\left( {{\varvec{M}}_{{{\text{cS}}}} ,{\varvec{C}}_{{{\text{cS}}}} ,{\varvec{K}}_{{{\text{cS}}}} ,{\varvec{F}}_{\text{S}} ,{\varvec{x}}_{n\sim n - 2} ,{\dot{\varvec{x}}}_{n\sim n - 2} ,{{\varvec{\varPhi}}}_{\text{S}} ,{{\varvec{\varPhi}}}_{\text{S}}^{^{\prime}} } \right),$$(19)
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Step 4 Calculate the maximum absolute value
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Step 5 Jump out of the iterative loop and update the displacement and velocity response vector in the previous three steps, preparing for the next Park integration, namely$$\left\{ \begin{gathered} {\varvec{x}}_{n - 2} { = }{\varvec{x}}_{n - 1} \hfill \\{\varvec{x}}_{n - 1} { = }{\varvec{x}}_{n} \hfill \\{\varvec{x}}_{n} { = }{\varvec{x}}_{{n{ + }1}} \hfill \\ \end{gathered} \right.,\;\rm {and}\;\left\{ \begin{gathered} {\dot{\varvec{x}}}_{n - 2} { = }{\dot{\varvec{x}}}_{n - 1} \hfill \\{\dot{\varvec{x}}}_{n - 1} { = }{\dot{\varvec{x}}}_{n}\hfill \\{\dot{\varvec{x}}}_{n} { = }{\dot{\varvec{x}}}_{{n{ + }1}} \hfill \\ \end{gathered} \right..$$(20)
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Step 6 Perform non-iterative computation illustrated in [56], or go to step 1 to conduct the next iteration solution.
4 Numerical examples
4.1 Validation of the proposed model
4.2 Clarification of the influence of substructures on train–track responses
4.3 Influence of track irregularities on TTS dynamic performance
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China high-speed spectrum [57]:where S denotes the power spectral density; \(f\) denotes the spatial frequency; \(A\) and \(k\) denote the coefficients as shown in Tables 1 and 2.$$S(f) = Af^{ - k},$$(12)Table 1The coefficients for the fitting formula of the track irregularity spectrumItemThe 1st sectionThe 2nd sectionThe 3rd sectionThe 4th sectionA (× 10–5)kA (× 10–3)kA (× 10–4)kA(× 10–4)kGauge5497.80.82825.07011.90371.87784.5948––Cross-level361.481.727843.6851.046145.8672.0939––Alignment395.131.867011.0471.53547.56332.8171––Vertical profile1.05443.38913.55881.9271197.841.36433.94883.4516Table 2The spatial frequency and corresponding wavelength of sectional points for the spectrumItemThe 1st–2nd sectionThe 2nd–3rd sectionThe 3rd–4th sectionFrequency (m−1)Wavelength (m)Frequency (m−1)concluavelength (m)Frequency (m−1)Wavelength (m)Gauge0.10909.20.29383.4––Cross-level0.025838.80.11638.6––Alignment0.045022.20.12348.1––Vertical profile0.018753.50.047421.10.15336.5
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German high-speed low-disturbance spectrum [2]:where \(S_{{\text{v}}}\), \(S_{{\text{a}}}\) and \(S_{{\text{x}}}\) denote the power spectral density of vertical profile irregularity, alignment irregularity and cross-level irregularity, respectively; \(\varOmega\) denotes the spatial wavenumber, in rad/m; truncated wavenumbers \(\varOmega_{{\text{c}}} = 0.8246{\text{ rad/m}}\) and \(\varOmega_{{\text{r}}} = 0.0206{\text{ rad/m}}\); and for low-disturbance spectrum, coefficients \(A_{{\text{v}}} = 4.032 \times 10^{ - 7} {\text{ m}} \cdot {\text{rad}}\), \(A_{{\text{a}}} = 2.119 \times 10^{ - 7} {\text{ m}} \cdot {\text{rad}}\), and \(b = 0.75{\text{ m}}\).$$\left\{ {\begin{array}{*{20}l} {S_{\text{v}} (\varvec {\varOmega} ) = \frac{{A_{\text{v}} \varvec {\varOmega} _{\text{c}}^{2} }}{{(\varvec {\varOmega} ^{2} + \varvec {\varOmega} _{\text{r}}^{2} )(\varvec {\varOmega} ^{2} + \varvec {\varOmega} _{\text{c}}^{2} )}}} \hfill & {({\text{vertical}}\;{\text{profile}})} \hfill \\ {S_{\text{a}} (\varvec {\varOmega} ) = \frac{{A_{\text{a}} \varvec {\varOmega} _{\text{c}}^{2} }}{{(\varvec {\varOmega} ^{2} + \varvec {\varOmega} _{\text{r}}^{2} )(\varvec {\varOmega} ^{2} + \varvec {\varOmega} _{\text{c}}^{2} )}}} \hfill & {{\text{(alignment)}}} \hfill \\ {S_{\text{x}} (\varvec {\varOmega} ) = \frac{{A_{\text{v}} \varvec {\varOmega} _{\text{c}}^{2} \varvec {\varOmega} ^{2} }}{{b^{2} (\varvec {\varOmega} ^{2} + \varvec {\varOmega} _{\text{r}}^{2} )(\varvec {\varOmega} _{\text{c}}^{2} + \varvec {\varOmega} _{\text{c}}^{2} )(\varvec {\varOmega} _{\text{c}}^{2} + \varvec {\varOmega} _{\text{s}}^{2} )}}} \hfill & {({\text{cross-level}})} \hfill \\ \end{array} } \right. ,$$(13)
Track Component | China high-speed spectrum | German low-disturbance spectrum | ||
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Lateral (m/s2) | Vertical (m/s2) | Lateral (m/s2) | Vertical (m/s2) | |
Track slab | 1.01 | 4.89 | 0.66 | 7.24 |
Tunnel | 1.30 | 3.73 | 0.60 | 6.20 |
Bridge | 0.39 | 0.62 | 0.59 | 0.81 |