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Research on mechanism and control methods of carbody chattering of an electric multiple-unit train

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Abstract

Carbody chattering is an abnormal vibration that severely deteriorates the ride quality of a railway vehicle. However, systematic studies on the mechanisms and control methods of carbody chattering are inadequate. Hence, in-situ tests, wheel and rail profile tests, modal parameter tests, and root locus analyses were conducted for an electric multiple-unit train to study the carbody chattering mechanism. Results show significant concave wear on wheel treads that have not yet met their wheel-turning mileages. When the vehicle moves from a carbody non-chattering to a chattering section, the wheel–rail contact positions are scattered and jumping is observed; then, the wheel–rail contact conicity increases rapidly, causing the modal damping ratio of the bogie hunting motion to reduce to 0, the bogie to change from stable to critical-unstable state, and bogie hunting motion frequency to increase close to the modal frequency of the carbody diamond-shaped deformation, thereby triggering synchronous movement. This amplifies the modal vibration, causing carbody chattering. Therefore, three control methods are proposed for carbody chattering—turning worn wheels; grinding rail profiles in the carbody chattering section; and synchronous optimisation of the primary longitudinal and lateral positioning stiffness, node stiffness, and damping coefficient of the yaw damper—according to the multi-objective synchronisation optimisation method to improve operational stability and ride quality. Test results show that all three methods effectively control carbody chattering; compared to the original vehicle, the amplitude of carbody chattering acceleration at 10 Hz can be reduced by 90%, 40% and 60% for the three methods.

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Acknowledgements

The authors acknowledge the financial support provided by National Natural Science Foundation of China (Grant No. 51805373).

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Authors and Affiliations

  1. Institute of Rail Transit, Tongji University, No. 4800, Cao An Road, Shanghai, 201804, People’s Republic of China

    Dao Gong, Guangyu Liu & Jinsong Zhou

Authors
  1. Dao Gong
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  2. Guangyu Liu
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  3. Jinsong Zhou
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Correspondence to Dao Gong.

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Appendices

Appendix A

Table 8 Dynamic parameters of the carbody
Full size table

Appendix B

$$\begin{aligned} \mathbf{M} = &\mathrm{diag}\left [ m_{t} \quad m_{t} \quad I_{tx} \quad I_{ty} \quad I_{tz} \quad m_{w} \quad I_{wz} \quad m_{w} \quad I_{wz} \quad m_{t} \quad m_{t} \quad I_{tx} \quad I_{ty} \quad I_{tz} \quad m_{w} \right. \\ &\left. \qquad I_{wz} \quad m_{w}\quad I_{wz} \quad m_{c} \quad m_{c} \quad I_{cx} \quad I_{cy} \quad I_{cz} \right ] \end{aligned}$$
$$ \left ( \mathbf{C} + \mathbf{C}_{\mathbf{W} - \mathbf{R}}/v \right )_{1\sim 3} \!=\! \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} 2c_{ys} & 0 & 2c_{ys}h_{tsc} \\ 0 & 4c_{zp} + 2c_{zs} & 0 \\ 2c_{ys}h_{tsc} & 0 & 4c_{zp}d_{pc}^{2} + 2c_{zs}d_{sc}^{2} + 2c_{ys}h_{tsc}^{2} \\ 0 & 0 & - 2c_{zs}d_{sc}l_{tsv} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ - 2c_{ys} & 0 & - 2c_{ys}h_{tsc} \\ 0 & - 2c_{zs} & 0 \\ - 2c_{ys}h_{csc} & 0 & - 2c_{zs}d_{sc}^{2} - 2c_{ys}h_{tsc}h_{csc} \\ 0 & c_{zs}\left ( l_{scv1} + l_{scv2} \right ) & c_{zs}d_{sc}\left ( l_{scv1} - l_{scv2} \right ) \\ - c_{ys}\left ( l_{sc1} + l_{sc2} \right ) & 0 & - c_{ys}h_{tsc}\left ( l_{sc1} + l_{sc2} \right ) \end{array}\displaystyle \right ], $$
$$ \left ( \mathbf{C} + \mathbf{C}_{\mathbf{W} - \mathbf{R}}/v \right )_{4\sim 6} = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ - 2c_{zs}l_{tsv}d_{sc} & 0 & 0 \\ 4c_{zp}l_{c}^{2} + 2c_{zs}l_{tsv}^{2} & 0 & 0 \\ 0 & c_{fs} + 2c_{ys}l_{tsc}^{2} & 0 \\ 0 & 0 & 2f_{22}/v \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 2c_{zs}l_{tsv}d_{sc} & 0 & 0 \\ c_{zs}l_{tsv}\left ( l_{scv2} - l_{scv1} \right ) & 0 & 0 \\ 0 & c_{ys}l_{tsc}\left ( l_{sc2} - l_{sc1} \right ) - c_{fs} & 0 \end{array}\displaystyle \right ] $$
$$ \left ( \mathbf{C} + \mathbf{C}_{\mathbf{W} - \mathbf{R}}/v \right )_{7\sim 9} = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 2b^{2}f_{11}/v & 0 & 0 \\ 0 & 2f_{22}/v & 0 \\ 0 & 0 & 2b^{2}f_{11}/v \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\displaystyle \right ], $$
$$ \left ( \mathbf{C} \!+\! \mathbf{C}_{\mathbf{W} - \mathbf{R}}/v \right )_{10\sim 12} \!=\! \left [\! \textstyle\begin{array}{c@{\quad }c@{\quad }c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 2c_{ys} & 0 & 2c_{ys}h_{tsc} \\ 0 & 4c_{zp} + 2c_{zs} & 0 \\ 2c_{ys}h_{tsc} & 0 & 4c_{zp}d_{pc}^{2} + 2c_{zs}d_{sc}^{2} + 2c_{ys}h_{tsc}^{2} \\ 0 & 0 & - 2c_{zs}d_{sc}l_{tsv} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ - 2c_{ys} & 0 & - 2c_{ys}h_{tsc} \\ 0 & - 2c_{zs} & 0 \\ - 2c_{ys}h_{\csc } & 0 & - 2c_{zs}d_{sc}^{2} - 2c_{ys}h_{tsc}h_{csc} \\ 0 & - c_{zs}\left ( l_{scv1} + l_{scv2} \right ) & c_{zs}d_{sc}\left ( l_{scv1} - l_{scv2} \right ) \\ c_{ys}\left ( l_{sc1} + l_{sc2} \right ) & 0 & c_{ys}h_{tsc}\left ( l_{sc1} + l_{sc2} \right ) \end{array}\displaystyle \! \right ] $$
$$ \left ( \mathbf{C} + \mathbf{C}_{\mathbf{W} - \mathbf{R}}/v \right )_{13\sim 15} = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ - 2c_{zs}l_{tsv}d_{sc} & 0 & 0 \\ 4c_{zp}l_{c}^{2} + 2c_{zs}l_{tsv}^{2} & 0 & 0 \\ 0 & c_{fs} + 2c_{ys}l_{tsc}^{2} & 0 \\ 0 & 0 & 2f_{22}/v \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 2c_{zs}l_{tsv}d_{sc} & 0 & 0 \\ c_{zs}l_{tsv}\left ( l_{scv2} - l_{scv1} \right ) & 0 & 0 \\ 0 & c_{ys}l_{tsc}\left ( l_{sc2} - l_{sc1} \right ) - c_{fs} & 0 \end{array}\displaystyle \right ] $$
$$ \left ( \mathbf{C} + \mathbf{C}_{\mathbf{W} - \mathbf{R}}/v \right )_{16\sim 18} = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 2b^{2}f_{11}/v & 0 & 0 \\ 0 & 2f_{22}/v & 0 \\ 0 & 0 & 2b^{2}f_{11}/v \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\displaystyle \right ] $$
$$ \left ( \mathbf{C} + \mathbf{C}_{\mathbf{W} - \mathbf{R}}/v \right )_{19\sim 21} = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} - 2c_{ys} & 0 & - 2c_{ys}h_{csc} \\ 0 & - 2c_{zs} & 0 \\ - 2c_{ys}h_{tsc} & 0 & - 2c_{zs}d_{sc}^{2} - 2c_{ys}h_{\csc } h_{tsc} \\ 0 & 0 & 2c_{zs}d_{sc}l_{tsv} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ - 2c_{ys} & 0 & - 2c_{ys}h_{csc} \\ 0 & - 2c_{zs} & 0 \\ - 2c_{ys}h_{tsc} & 0 & - 2c_{zs}d_{sc}^{2} - 2c_{ys}h_{\csc } h_{tsc} \\ 0 & 0 & 2c_{zs}d_{sc}l_{tsv} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 4c_{ys} & 0 & 4c_{ys}h_{csc} \\ 0 & 4c_{zs} & 0 \\ 4c_{ys}h_{csc} & 0 & 4c_{zs}d_{sc}^{2} + 4c_{ys}h_{csc}^{2} \\ 0 & 0 & 2c_{zs}d_{sc}\left ( l_{scv2} - l_{scv1} \right ) \\ 0 & 0 & 0 \end{array}\displaystyle \right ], $$
$$ \left ( \mathbf{C} + \mathbf{C}_{\mathbf{W} - \mathbf{R}}/v \right )_{22\sim 23} = \left [ \textstyle\begin{array}{c@{\quad }c} 0 & - c_{ys}\left ( l_{sc1} + l_{sc2} \right ) \\ c_{zs}\left ( l_{scv1} + l_{scv2} \right ) & 0 \\ c_{zs}d_{sc}\left ( l_{scv1} - l_{scv2} \right ) & - c_{ys}h_{tsc}\left ( l_{sc1} + l_{sc2} \right ) \\ c_{zs}l_{tsv}\left ( l_{scv2} - l_{scv1} \right ) & 0 \\ 0 & c_{ys}l_{tsc}\left ( l_{sc2} - l_{sc1} \right ) - c_{fs} \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & c_{ys}\left ( l_{sc1} + l_{sc2} \right ) \\ - c_{zs}\left ( l_{scv1} + l_{scv2} \right ) & 0 \\ c_{zs}d_{sc}\left ( l_{scv1} - l_{scv2} \right ) & c_{ys}h_{tsc}\left ( l_{sc1} + l_{sc2} \right ) \\ c_{zs}l_{tsv}\left ( l_{scv2} - l_{scv1} \right ) & 0 \\ 0 & c_{ys}l_{tsc}\left ( l_{sc2} - l_{sc1} \right ) - c_{fs} \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 2c_{zs}d_{sc}\left ( l_{scv2} - l_{scv1} \right ) & 0 \\ 2c_{zs}\left ( l_{scv1}^{2} + l_{scv2}^{2} \right ) & 0 \\ 0 & 2c_{fs} + 2c_{ys}\left ( l_{sc1}^{2} + l_{sc2}^{2} \right ) \end{array}\displaystyle \right ] $$
$$\begin{aligned} &\left ( \mathbf{K} + \mathbf{K}_{\mathbf{W} - \mathbf{R}} \right )_{1\sim 3}\\ &\quad = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} k_{ty} + 2k_{yp} + k_{ys} & 0 & 2k_{yp}h_{tp} - k_{ty}h_{ttk} - k_{ys}h_{tsk} \\ 0 & 4k_{zp} + 2k_{zs} & 0 \\ 2k_{yp}h_{tp} - k_{ty}h_{ttk} - k_{ys}h_{tsk} & 0 & k_{tb} + 4k_{zp}d_{p}^{2} + 2k_{zs}d_{s}^{2} + 2k_{yp}h_{tp}^{2} + 2k_{ys}h_{tsk}^{2} \\ 0 & 0 & - 2k_{zs}d_{s}l_{tsk} \\ 0 & 0 & 0 \\ - k_{yp} & 0 & - k_{yp}h_{tp} \\ 0 & 0 & 0 \\ - k_{yp} & 0 & - k_{yp}h_{tp} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ - k_{ty} - k_{ys} & 0 & k_{ty}h_{ttk} + k_{ys}h_{tsk} \\ 0 & - 2k_{zs} & 0 \\ - k_{ty}h_{ctr} - k_{ys}h_{csk} & 0 & k_{tb} - 2k_{zs}d_{s}^{2} + k_{ty}h_{ttk}h_{ctr} + k_{ys}h_{tsk}h_{csk} \\ 0 & k_{zs}\left ( l_{skv1} + l_{skv2} \right ) & k_{zs}d_{s}\left ( l_{sck1} - l_{sck2} \right ) \\ - \left ( k_{ty} + k_{ys} \right )l_{ss} & 0 & \left ( k_{ty}h_{ttk} + k_{ys}h_{tsk} \right )l_{ss} \end{array}\displaystyle \right ] \end{aligned}$$
$$ \left ( \mathbf{K} + \mathbf{K}_{\mathbf{W} - \mathbf{R}} \right )_{4\sim 6} = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} 0 & 0 & - k_{yp} \\ 0 & 0 & 0 \\ - 2k_{zs}l_{tsk}d_{s} & 0 & - k_{yp}h_{tp} \\ 4k_{zp}a^{2} + 2k_{zs}l_{tsk}^{2} & 0 & 0 \\ 0 & k_{fs} + 2k_{yp}a^{2} + 4k_{xp}d_{p}^{2} & - k_{yp}a \\ 0 & - k_{yp}a & k_{yp} \\ 0 & - 2k_{xp}d_{p}^{2} & 2bf_{11}s/r_{0} \\ 0 & k_{yp}a & 0 \\ 0 & - 2k_{xp}d_{p}^{2} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 2k_{zs}l_{tsk}d_{s} & 0 & 0 \\ k_{zs}l_{tsk}\left ( l_{skv2} - l_{skv1} \right ) & 0 & 0 \\ 0 & - k_{fs} & 0 \end{array}\displaystyle \right ], $$
$$ \left ( \mathbf{K} + \mathbf{K}_{\mathbf{W} - \mathbf{R}} \right )_{7\sim 9} = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} 0 & - k_{yp} & 0 \\ 0 & 0 & 0 \\ 0 & - k_{yp}h_{tp} & 0 \\ 0 & 0 & 0 \\ - 2k_{xp}d_{p}^{2} & k_{yp}a & - 2k_{xp}d_{p}^{2} \\ - 2f_{22} & 0 & 0 \\ 2k_{xp}d_{p}^{2} & 0 & 0 \\ 0 & k_{yp} & - 2f_{22} \\ 0 & 2bf_{11}s/r_{0} & 2k_{xp}d_{p}^{2} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\displaystyle \right ] $$
$$\begin{aligned} &\left ( \mathbf{K} + \mathbf{K}_{\mathbf{W} - \mathbf{R}} \right )_{10\sim 12} \\ &\quad = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ k_{ty} + 2k_{yp} + k_{ys} & 0 & 2k_{yp}h_{tp} - k_{ty}h_{ttk} - k_{ys}h_{tsk} \\ 0 & 4k_{zp} + 2k_{zs} & 0 \\ 2k_{yp}h_{tp} - k_{ty}h_{ttk} - k_{ys}h_{tsk} & 0 & k_{tb} + 4k_{zp}d_{p}^{2} + 2k_{zs}d_{s}^{2} + 2k_{yp}h_{tp}^{2} + 2k_{ys}h_{tsk}^{2} \\ 0 & 0 & - 2k_{zs}d_{s}l_{tsk} \\ 0 & 0 & 0 \\ - k_{yp} & 0 & - k_{yp}h_{tp} \\ 0 & 0 & 0 \\ - k_{yp} & 0 & - k_{yp}h_{tp} \\ 0 & 0 & 0 \\ - k_{ty} - k_{ys} & 0 & k_{ty}h_{ttk} + k_{ys}h_{tsk} \\ 0 & - 2k_{zs} & 0 \\ - k_{ty}h_{ctr} - k_{ys}h_{csk} & 0 & k_{tb} - 2k_{zs}d_{s}^{2} + k_{ty}h_{ttk}h_{ctr} + k_{ys}h_{tsk}h_{csk} \\ 0 & - k_{zs}\left ( l_{skv1} + l_{skv2} \right ) & k_{zs}d_{s}\left ( l_{sck1} - l_{sck2} \right ) \\ \left ( k_{ty} + k_{ys} \right )l_{ss} & 0 & - \left ( k_{ty}h_{ttk} + k_{ys}h_{tsk} \right )l_{ss} \end{array}\displaystyle \right ] \end{aligned}$$
$$ \left ( \mathbf{K} + \mathbf{K}_{\mathbf{W} - \mathbf{R}} \right )_{13\sim 15} = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & - k_{yp} \\ 0 & 0 & 0 \\ - 2k_{zs}l_{tsk}d_{s} & 0 & - k_{yp}h_{tp} \\ 4k_{zp}a^{2} + 2k_{zs}l_{tsk}^{2} & 0 & 0 \\ 0 & k_{fs} + 2k_{yp}a^{2} + 4k_{xp}d_{p}^{2} & - k_{yp}a \\ 0 & - k_{yp}a & k_{yp} \\ 0 & - 2k_{xp}d_{p}^{2} & 2bf_{11}s/r_{0} \\ 0 & k_{yp}a & 0 \\ 0 & - 2k_{xp}d_{p}^{2} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 2k_{zs}l_{tsk}d_{s} & 0 & 0 \\ k_{zs}l_{tsk}\left ( l_{skv2} - l_{skv1} \right ) & 0 & 0 \\ 0 & - k_{fs} & 0 \end{array}\displaystyle \right ], $$
$$ \left ( \mathbf{K} + \mathbf{K}_{\mathbf{W} - \mathbf{R}} \right )_{16\sim 18} = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & - k_{yp} & 0 \\ 0 & 0 & 0 \\ 0 & - k_{yp}h_{tp} & 0 \\ 0 & 0 & 0 \\ - 2k_{xp}d_{p}^{2} & k_{yp}a & - 2k_{xp}d_{p}^{2} \\ - 2f_{22} & 0 & 0 \\ 2k_{xp}d_{p}^{2} & 0 & 0 \\ 0 & k_{yp} & - 2f_{22} \\ 0 & 2bf_{11}s/r_{0} & 2k_{xp}d_{p}^{2} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\displaystyle \right ] $$
$$\begin{aligned} &\left ( \mathbf{K} + \mathbf{K}_{\mathbf{W} - \mathbf{R}} \right )_{19\sim 21} \\ &\quad = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} - k_{ty} - k_{ys} & 0 & - k_{ty}h_{ctr} - k_{ys}h_{csk} \\ 0 & - 2k_{zs} & 0 \\ k_{ty}h_{ttk} + k_{ys}h_{tsk} & 0 & k_{tb} - 2k_{zs}d_{s}^{2} + k_{yp}h_{ctr}h_{ttk} + k_{ys}h_{csk}h_{tsk} \\ 0 & 0 & 2k_{zs}d_{s}l_{tsk} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ - k_{ty} - k_{ys} & 0 & - k_{ty}h_{ctr} - k_{ys}h_{csk} \\ 0 & - 2k_{zs} & 0 \\ k_{ty}h_{ttk} + k_{ys}h_{tsk} & 0 & k_{tb} - 2k_{zs}d_{s}^{2} + k_{yp}h_{ctr}h_{ttk} + k_{ys}h_{csk}h_{tsk} \\ 0 & 0 & 2k_{zs}d_{s}l_{tsk} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 2k_{ty} + 2k_{ys} & 0 & 2k_{ty}h_{ctr} + 2k_{ys}h_{csk} \\ 0 & 4k_{zs} & 0 \\ 2k_{ty}h_{ctr} + 2k_{ys}h_{csk} & 0 & 2k + 4k_{zs}d_{s}^{2} + 2k_{yp}h_{ctr}h_{ttk} + 2k_{ys}h_{csk}h_{tsk} \\ 0 & 0 & 2k_{zs}d_{s}\left ( l_{sck2} - l_{sck1} \right ) \\ 0 & 0 & 0 \end{array}\displaystyle \right ], \end{aligned}$$
$$ \left ( \mathbf{K} + \mathbf{K}_{\mathbf{W} - \mathbf{R}} \right )_{22\sim 23} = \left [ \textstyle\begin{array}{c@{\quad }c} 0 & - \left ( k_{ty} + k_{ys} \right )l_{ss} \\ k_{zs}\left ( l_{sck1} + l_{sck2} \right ) & 0 \\ k_{zs}d_{s}\left ( l_{skv1} - l_{skv2} \right ) & \left ( k_{ty}h_{ttk} + k_{ys}h_{tsk} \right )l_{ss} \\ k_{zs}l_{tsk}\left ( l_{skv2} - l_{skv1} \right ) & 0 \\ 0 & - k_{fs} \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & \left ( k_{ty} + k_{ys} \right )l_{ss} \\ - k_{zs}\left ( l_{skv1} + l_{skv2} \right ) & 0 \\ k_{zs}d_{s}\left ( l_{skv1} - l_{skv2} \right ) & - \left ( k_{ty}h_{ttk} + k_{ys}h_{tsk} \right )l_{ss} \\ k_{zs}l_{tsk}\left ( l_{skv2} - l_{skv1} \right ) & 0 \\ 0 & - k_{fs} \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 2k_{zs}d_{s}\left ( l_{skv2} - l_{skv1} \right ) & 0 \\ 2k_{zs}\left ( l_{skv1}^{2} + l_{skv2}^{2} \right ) & 0 \\ 0 & 2k_{fs} + 2\left ( k_{ty} + k_{ys} \right )l_{ss}^{2} \end{array}\displaystyle \right ] $$

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Gong, D., Liu, G. & Zhou, J. Research on mechanism and control methods of carbody chattering of an electric multiple-unit train. Multibody Syst Dyn 53, 135–172 (2021). https://doi.org/10.1007/s11044-021-09779-9

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