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Bifurcation analysis of high-speed railway wheel-set

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Abstract

In this paper, the lateral mathematical model of a railway wheel-set with two degrees of freedom has been built. To study the effect of yaw damper on the stability of high-speed wheel-set, the constrain between the wheel-set and the bogie is assumed as rigid. In this lateral model, only the nonlinear wheel/rail contact relationship has been taken into consideration in the lateral direction, due to both the nonlinear relationship between lateral displacement and contact angle of the wheel and the rail, and the nonlinear relationship between lateral displacement and the equivalent radius of the wheel at the contact point. And the nonlinear parameters are attained by using polynomial interpolation. By using Center Manifold Theorem, the method of Normal Form and Poincaré method, the model is reduced to a planar dynamical system, and the symbolic expression of the first-order fine focus is given, which can be used to determine which kind of bifurcation will occur at the critical speed. The simulation is established by using the parameters of the wheel-set of Chinese high-speed railway vehicle CRH3, and the result is consistent with the determination of the first-order fine focus. At last, the influence of different parameters on the stability of CRH3 wheel-set is investigated.

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Authors and Affiliations

  1. State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, 610031, Sichuan, People’s Republic of China

    Tingting Zhang & Huanyun Dai

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  1. Tingting Zhang
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  2. Huanyun Dai
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Correspondence to Tingting Zhang.

Additional information

Supported by China Railway Corporation (Grant No. 2014J004-A).

Appendix

Appendix

(1) Expression of \(E_i (i=1,\ldots ,6)\) in Sect. 3.1

$$\begin{aligned} E_1= & {} \displaystyle \frac{3}{8[(\omega ^2-q)^2+p^2\omega ^2]}[p\overline{\mu }_c^2\\&-(p^2-q+\omega ^2)\overline{\mu }_c+p\omega ^2-p],\\ E_5= & {} \displaystyle \frac{3}{8\overline{p}_2^3[(\omega ^2-q)^2+p^2\omega ^2]}\{[\overline{\mu }_c^2\omega ^2(1-\omega ^2)\\&+\,(1-\omega ^2)^3][-p\overline{\mu }_c^2 +(p-q+\omega ^2)\overline{\mu }_c\\&-\,p\omega ^2+p]+[\overline{\mu }_c^3\omega ^2+\overline{\mu }_c(1-\omega ^2)^2][(q-\omega ^2)\overline{\mu }_c^2\\&-\,pq\overline{\mu }_c+(p^2-q+1)\omega ^2+q^2-q]\},\\ E_6= & {} \displaystyle \frac{3p_2p}{8[(\omega ^2-q)^2+p^2\omega ^2]}. \end{aligned}$$

(2) Expression of \(F_i (i=1,\ldots ,6)\) in Sect. 3.2

$$\begin{aligned} E_2= & {} \displaystyle \frac{1}{8\overline{p}_2^2[(\omega ^2-q)^2+p^2\omega ^2]}\{[\overline{\mu }_c^2\omega ^2\\&+3(1-\omega ^2)^2][-p\overline{\mu }_c^2 +(p-q+\omega ^2)\overline{\mu }_c\\&-p\omega ^2+p]+2\overline{\mu }_c(1-\omega ^2)[(q-\omega ^2)\overline{\mu }_c^2\\&-pq\overline{\mu }_c+(p^2-q+1)\omega ^2+q^2-q]\},\\ E_3= & {} \displaystyle \frac{1}{8\overline{p}_2[(\omega ^2-q)^2+p^2\omega ^2]}\{3(1-\omega ^2)\\&\times [-p\overline{\mu }_c^2+(p-q+\omega ^2)\overline{\mu }_c-p\omega ^2+p]\\&+\overline{\mu }_c[(q-\omega ^2)\overline{\mu }_c^2-pq\overline{\mu }_c\\&+(p^2-q+1)\omega ^2+q^2-q]\},\\ E_4= & {} \displaystyle \frac{1}{8\overline{p}_2^2[(\omega ^2-q)^2+p^2\omega ^2]}\{2\overline{\mu }_c\omega ^2(1-\omega ^2) \\&\times [-p\overline{\mu }_c^2+(p-q+\omega ^2)\overline{\mu }_c-p\omega ^2+p]\\&+[3\overline{\mu }_c^2\omega ^2+(1-\omega ^2)^2][(q-\omega ^2)\overline{\mu }_c^2-pq\overline{\mu }_c\\&+(p^2-q+1)\omega ^2+q^2-q]\}, \end{aligned}$$
$$\begin{aligned} F_1= & {} \displaystyle \frac{3}{8[\alpha ^2+(\omega _1-\omega _2)^2][\alpha ^2+(\omega _1+\omega _2)^2]} \\&\times (-3\alpha ^2\overline{\mu }_c-2\alpha \overline{\mu }_c^2 -2\alpha \omega _1^2-\overline{\mu }_c\omega _1^2+\overline{\mu }_c\omega _2^2+2\alpha ),\\ F_2= & {} \displaystyle \frac{1}{8\overline{p}_2^2[\alpha ^2+(\omega _1-\omega _2)^2][\alpha ^2+(\omega _1+\omega _2)^2]}\\&\times \{[3(1-\omega _1^2)^2+\omega _1^2\overline{\mu }_c^2](3\alpha ^2\overline{\mu }_c +2\alpha \overline{\mu }_c^2+2\alpha \omega _1^2\\&+\overline{\mu }_c\omega _1^2- \overline{\mu }_c\omega _2^2-2\alpha )+2\overline{\mu }_c(1-\omega _1^2) (\alpha ^4\\&+2\alpha ^3\overline{\mu }_c+\alpha ^2\overline{\mu }_c^2+3\alpha ^2\omega _1^2 +2\alpha ^2\omega _2^2\\&+2\alpha \overline{\mu }_c\omega _2^2-\overline{\mu }_c^2\omega _1^2 +\overline{\mu }_c^2\omega _2^2-\omega _1^2\omega _2^2\\&+\omega _2^4-\alpha ^2+\omega _1^2-\omega _2^2)\},\\ F_3= & {} \displaystyle \frac{1}{8\overline{p}_2[\alpha ^2+(\omega _1-\omega _2)^2][\alpha ^2+(\omega _1+\omega _2)^2]}\\&\times [3(1-\omega _1^2)(3\alpha ^2\overline{\mu }_c+2\alpha \overline{\mu }_c^2 +2\alpha \omega _1^2+\overline{\mu }_c\omega _1^2\\&-\overline{\mu }_c\omega _2^2-2\alpha )+\overline{\mu }_c (\alpha ^4+2\alpha ^3\overline{\mu }_c+\alpha ^2\overline{\mu }_c^2\\&+3\alpha ^2\omega _1^2 +2\alpha ^2\omega _2^2+2\alpha \overline{\mu }_c\omega _2^2-\overline{\mu }_c^2\omega _1^2\\&+\overline{\mu }_c^2\omega _2^2 -\omega _1^2\omega _2^2+\omega _2^4-\alpha ^2+\omega _1^2-\omega _2^2)],\\ F_4= & {} \displaystyle \frac{1}{8\overline{p}_2^2[\alpha ^2+(\omega _1-\omega _2)^2][\alpha ^2+(\omega _1+\omega _2)^2]}\\&\times \{2\omega _1\overline{\mu }_c(1-\omega _1^2)(3\alpha ^2\overline{\mu }_c+2\alpha \overline{\mu }_c^2 +2\alpha \omega _1^2\\&+\overline{\mu }_c\omega _1^2-\overline{\mu }_c\omega _2^2-2\alpha ) +[3\omega _1\overline{\mu }_c^2+(1-\omega _1^2)^2](\alpha ^4\\&+2\alpha ^3\overline{\mu }_c+\alpha ^2\overline{\mu }_c^2+3\alpha ^2\omega _1^2\\&+2\alpha ^2\omega _2^2+2\alpha \overline{\mu }_c\omega _2^2-\overline{\mu }_c^2\omega _1^2 +\overline{\mu }_c^2\omega _2^2-\omega _1^2\omega _2^2\\&+\omega _2^4-\alpha ^2+\omega _1^2-\omega _2^2)\},\\ F_5= & {} \displaystyle \frac{3}{8\overline{p}_2^3[\alpha ^2+(\omega _1-\omega _2)^2][\alpha ^2+(\omega _1+\omega _2)^2]}\\&\times \{[(1-\omega _1^2)^3+\omega _1^2\overline{\mu }_c^2(1-\omega _1^2)](3\alpha ^2\overline{\mu }_c+2\alpha \overline{\mu }_c^2\\&+2\alpha \omega _1^2+\overline{\mu }_c\omega _1^2-\overline{\mu }_c\omega _2^2-2\alpha )+[\overline{\mu }_c(1-\omega _1^2)^2\\&+\omega _1^2\overline{\mu }_c^3](\alpha ^4+2\alpha ^3\overline{\mu }_c+\alpha ^2\overline{\mu }_c^2+3\alpha ^2\omega _1^2\\&+2\alpha ^2\omega _2^2+2\alpha \overline{\mu }_c\omega _2^2-\overline{\mu }_c^2\omega _1^2 +\overline{\mu }_c^2\omega _2^2\\&-\omega _1^2\omega _2^2+\omega _2^4-\alpha ^2+\omega _1^2-\omega _2^2)\},\\ F_6= & {} -\displaystyle \frac{3\alpha \overline{p}_2}{4[\alpha ^2+(\omega _1-\omega _2)^2][\alpha ^2+(\omega _1+\omega _2)^2]}. \end{aligned}$$

(3) Parameters of CRH3 wheel-set in Sect. 4

Parameter

Comments

Value

M

Wheel-set mass

5279 kg

I

Yaw moment of wheel-set

5460 N m

\(r_0\)

Wheel radius without lateral displacement

460 mm

a

Half of the rolling cycle gauge

0.7465 m

b

Half of the air spring arm

0.95 m

Parameter

Comments

Value

\(K_{2x}\)

Secondary longitudinal stiffness

4.383\(\mathrm{MN}\,\mathrm{m}^{-1}\)

\(K_{2y}\)

Secondary lateral stiffness

0.133 MN \(\mathrm{m}^{-1}\)

\(C_{2x}\)

Yaw damper

0.33 MN/(m/s)

\(f_{11}\)

Longitudinal creep coefficient

3 MN

\(f_{22}\)

Longitudinal creep coefficient

3 MN

k

Coefficient relative to cosine value of wheel–rail contact angle

1.994

g

Gravitational acceleration

9.81 \(\mathrm{m/s}^{2}\)

\(e_0\)

Coefficient relative to tangent value of wheel–rail contact angle

22.77

\(e_2\)

Coefficient relative to tangent value of wheel–rail contact angle

294502

\(\xi _1\)

Coefficient relative to the equivalent radius of left/right wheel

0.20

\(\xi _2\)

Coefficient relative to the equivalent radius of left/right wheel

11.77

\(\xi _3\)

Coefficient relative to the equivalent radius of left/right wheel

−6140.54

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Zhang, T., Dai, H. Bifurcation analysis of high-speed railway wheel-set. Nonlinear Dyn 83, 1511–1528 (2016). https://doi.org/10.1007/s11071-015-2425-2

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