Random Vibration Analysis of a Vehicle–Bridge Interaction System Subjected to Traveling Seismic Ground Motions Using Pseudo-excitation Method

Abstract

For long, multi-span bridges, traveling seismic waves arrive at different bridge support points at different times. To study this difference, random dynamic vibration analysis of a vehicle–bridge interaction system under traveling seismic ground motions was performed in the present paper. A vehicle model with 27 degrees of freedom is used, while three-dimensional Euler beams are used to model the track and the bridge. The equation of motion of the vehicle–bridge interaction system was established through the wheel-rail relationship. The expression of the standard deviation of the system vibrations and the running safety factor is derived by the pseudo-excitation method. The proposed method is validated by comparing random bridge vibrations using the Monte-Carlo method. As a case study, a Chinese-made electric multiple unit train running on a ten-span simply supported bridge is analyzed under track irregularities and seismic ground motions with consideration of the effects of different train speeds, different seismic intensities, and different seismic wave propagation velocities. The results show that wave propagation velocities significantly affect the random vibration performances and the running safety of the vehicle-bridge interaction system. Therefore, it is important to include wave propagation velocities when calculating the random seismic vibrations of a vehicle-bridge interaction system.

Appendix: Components in Eq. (32)

$${\mathbf{X}}_{vb} = \left[ {\begin{array}{*{20}c} {{\mathbf{u}}_{v} } & {{\mathbf{q}}_{r} } & {{\mathbf{q}}_{b} } \\ \end{array} } \right]^{{\text{T}}} ;\quad {\dot{\mathbf{X}}}_{vb} = \left[ {\begin{array}{*{20}c} {{\dot{\mathbf{u}}}_{v} } & {{\dot{\mathbf{q}}}_{r} } & {{\dot{\mathbf{q}}}_{b} } \\ \end{array} } \right]^{{\text{T}}} ;\quad {\mathbf{\ddot{X}}}_{vb} = \left[ {\begin{array}{*{20}c} {{\mathbf{\ddot{u}}}_{v} } & {{\mathbf{\ddot{q}}}_{r} } & {{\mathbf{\ddot{q}}}_{b} } \\ \end{array} } \right]^{{\text{T}}} ;$$

(a.1)

$${\mathbf{M}}_{svb} = {\mathbf{M}}_{vbs}^{{\text{T}}} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{M}}_{sb} {{\varvec{\Phi}}}_{b} } \\ \end{array} } \right];\quad$$

(a.2)

$${\mathbf{C}}_{svb} = {\mathbf{C}}_{vbs}^{{\text{T}}} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{C}}_{sb} {{\varvec{\Phi}}}_{b} } \\ \end{array} } \right];$$

(a.3)

$${\mathbf{K}}_{svb} = {\mathbf{K}}_{vbs}^{{\text{T}}} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{K}}_{sb} {{\varvec{\Phi}}}_{b} } \\ \end{array} } \right];$$

(a.4)

$${\mathbf{M^{\prime}}}_{vb} = diag\left[ {\begin{array}{*{20}c} {{\mathbf{M}}_{vv} } & {{{\varvec{\Phi}}}_{r}^{{\text{T}}} {\mathbf{M}}_{rr} {{\varvec{\Phi}}}_{r} + {{\varvec{\Phi}}}_{r}^{{\text{T}}} {\mathbf{N}}^{{\text{T}}} {{\varvec{\Gamma}}}^{ - 1} {\mathbf{M}}_{ww} {\mathbf{\Gamma N\Phi }}_{r} } & {{{\varvec{\Phi}}}_{b}^{{\text{T}}} {\mathbf{M}}_{bb} {{\varvec{\Phi}}}_{b} } \\ \end{array} } \right];\quad$$

(a.5)

$$\begin{gathered} {\mathbf{C^{\prime}}}_{vb} = \left[ {\begin{array}{*{20}c} {{\mathbf{C}}_{vv} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{{\varvec{\Phi}}}_{r}^{{\text{T}}} {\mathbf{C}}_{rr} {{\varvec{\Phi}}}_{r} } & {{{\varvec{\Phi}}}_{r}^{{\text{T}}} {\mathbf{C}}_{rb} {{\varvec{\Phi}}}_{b} } \\ {\mathbf{0}} & {{{\varvec{\Phi}}}_{b}^{{\text{T}}} {\mathbf{C}}_{br} {{\varvec{\Phi}}}_{r} } & {{{\varvec{\Phi}}}_{b}^{{\text{T}}} {\mathbf{C}}_{bb} {{\varvec{\Phi}}}_{b} } \\ \end{array} } \right] \hfill \\ + \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {{\mathbf{C}}_{vw} {\mathbf{\Gamma N\Phi }}_{r} } & {\mathbf{0}} \\ {{{\varvec{\Phi}}}_{r}^{{\text{T}}} {\mathbf{N}}^{{\text{T}}} {{\varvec{\Gamma}}}^{ - 1} {\mathbf{C}}_{wv} } & {{{\varvec{\Phi}}}_{r}^{{\text{T}}} {\mathbf{N}}^{{\text{T}}} {{\varvec{\Gamma}}}^{ - 1} \left( {2V_{t} {\mathbf{M}}_{ww} {\mathbf{\Gamma N}}_{,x} + {\mathbf{C}}_{ww} {\mathbf{\Gamma N}}} \right){{\varvec{\Phi}}}_{r} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right];\quad \hfill \\ \end{gathered}$$

(a.6)

$$\begin{gathered} {\mathbf{K^{\prime}}}_{vb} = \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{vv} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{{\varvec{\Phi}}}_{r}^{{\text{T}}} {\mathbf{K}}_{rr} {{\varvec{\Phi}}}_{r} } & {{{\varvec{\Phi}}}_{r}^{{\text{T}}} {\mathbf{K}}_{rb} {{\varvec{\Phi}}}_{b} } \\ {\mathbf{0}} & {{{\varvec{\Phi}}}_{b}^{{\text{T}}} {\mathbf{K}}_{br} {{\varvec{\Phi}}}_{r} } & {{{\varvec{\Phi}}}_{b}^{{\text{T}}} {\mathbf{K}}_{bb} {{\varvec{\Phi}}}_{b} } \\ \end{array} } \right] \hfill \\ + \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {\left( {V_{t} {\mathbf{C}}_{vw} {\mathbf{\Gamma N}}_{,x} + {\mathbf{K}}_{vw} {\mathbf{\Gamma N}}} \right){{\varvec{\Phi}}}_{r} } & {\mathbf{0}} \\ {{{\varvec{\Phi}}}_{r}^{{\text{T}}} {\mathbf{N}}^{{\text{T}}} {{\varvec{\Gamma}}}^{ - 1} {\mathbf{K}}_{wv} } & {{{\varvec{\Phi}}}_{r}^{{\text{T}}} {\mathbf{N}}^{{\text{T}}} {{\varvec{\Gamma}}}^{ - 1} \left( {V_{t}^{2} {\mathbf{M}}_{ww} {\mathbf{\Gamma N}}_{,xx} + V_{t} {\mathbf{C}}_{ww} {\mathbf{\Gamma N}}_{,x} + {\mathbf{K}}_{ww} {\mathbf{\Gamma N}}} \right){{\varvec{\Phi}}}_{r} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right]; \hfill \\ \end{gathered}$$

(a.7)

$$\begin{gathered} {\mathbf{T}} = \left[ {\begin{array}{*{20}c} {\mathbf{I}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{{\varvec{\Phi}}}_{r}^{{\text{T}}} {\mathbf{N}}^{{\text{T}}} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right];\quad \hfill \\ {\mathbf{R}} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {{\mathbf{C}}_{vw} } & {{\mathbf{K}}_{vw} } \\ {{{\varvec{\Gamma}}}^{ - 1} {\mathbf{M}}_{ww} } & {{{\varvec{\Gamma}}}^{ - 1} {\mathbf{C}}_{ww} } & {{{\varvec{\Gamma}}}^{ - 1} {\mathbf{K}}_{ww} } \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right]; \hfill \\ \end{gathered}$$

(a.8)

$$\begin{gathered} {\mathbf{F}}_{sw} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {{{\varvec{\Gamma}}}^{ - 1} {\mathbf{F}}_{w} } & {\mathbf{0}} \\ \end{array} } \right]^{{\text{T}}} ;\quad \hfill \\ {\mathbf{F}}_{r} = \left[ {\begin{array}{*{20}c} {{\mathbf{\ddot{r}}}} & {{\dot{\mathbf{r}}}} & {\mathbf{r}} \\ \end{array} } \right]^{{\text{T}}} \hfill \\ \end{gathered}$$

(a.9)