Integration of uniform design and quantum-behaved particle swarm optimization to the robust design for a railway vehicle suspension system under different wheel conicities and wheel rolling radii

Abstract

This paper proposes a systematic method, integrating the uniform design (UD) of experiments and quantum-behaved particle swarm optimization (QPSO), to solve the problem of a robust design for a railway vehicle suspension system. Based on the new nonlinear creep model derived from combining Hertz contact theory, Kalker’s linear theory and a heuristic nonlinear creep model, the modeling and dynamic analysis of a 24 degree-of-freedom railway vehicle system were investigated. The Lyapunov indirect method was used to examine the effects of suspension parameters, wheel conicities and wheel rolling radii on critical hunting speeds. Generally, the critical hunting speeds of a vehicle system resulting from worn wheels with different wheel rolling radii are lower than those of a vehicle system having original wheels without different wheel rolling radii. Because of worn wheels, the critical hunting speed of a running railway vehicle substantially declines over the long term. For safety reasons, it is necessary to design the suspension system parameters to increase the robustness of the system and decrease the sensitive of wheel noises. By applying UD and QPSO, the nominal-the-best signal-to-noise ratio of the system was increased from −48.17 to −34.05 dB. The rate of improvement was 29.31%. This study has demonstrated that the integration of UD and QPSO can successfully reveal the optimal solution of suspension parameters for solving the robust design problem of a railway vehicle suspension system.

Appendices

Appendix A: System parameters (Refs. [10, 28, 29])

Parameters	Value
Wheelset mass	$m_w =1117.9$ kg
Bogie frame mass	$m_t =350.26$ kg
Car body mass	$m_c =8041.3$ kg
Roll moment of the inertia of the wheelset	$I_{wx} =608.1 \,\,\hbox {kg}\cdot \hbox {m}^{2}$
Spin moment of the inertia of the wheelset	$I_{wy} =72 \,\,\hbox {kg}\cdot \hbox {m}^{2}$
Yaw moment of the inertia of the wheelset	$I_{wz} =608.1 \,\,\hbox {kg}\cdot \hbox {m}^{2}$
Roll moment of the inertia of the bogie frame	$I_{tx} =300 \,\,\hbox {kg}\cdot \hbox {m}^{2}$
Yaw moment of the inertia of the bogie frame	$I_{tz} =105.2 \,\,\hbox {kg}\cdot \hbox {m}^{2}$
Roll moment of the inertia of the car body	$I_{cx} =14270 \,\,\hbox {kg}\cdot \hbox {m}^{2}$
Yaw moment of the inertia of the car body	$I_{cz} =123760.5 \,\,\hbox {kg}\cdot \hbox {m}^{2}$

Wheel radius	$r_0 =0.43$ m
Half of track gauge	$a=0.7175$ m
Wheel conicity	$\lambda =0.05$
Half of primary longitudinal spring arm	$b_1 =1.0$ m
Half of primary longitudinal damping arm	$b_1 =1.0$ m
Half of primary vertical spring arm	$b_1 =1.0$ m
Half of primary vertical damping arm	$b_1 =1.0$ m
Half of secondary longitudinal spring arm	$b_2 =1.18$ m
Half of secondary longitudinal damping arm	$b_3 =1.4$ m
Half of secondary vertical spring arm	$b_2 =1.18$ m
Half of secondary vertical damping arm	$b_3 =1.4$ m
Half of primary lateral spring arm	$L_1 =1.5$ m
Half of primary lateral damping arm	$L_2 =1.5$ m
Longitudinal distance from wheelset center of gravity to car body	$L_c =4.2$ m
Height of external weight above center of gravity of wheelset	$h=1.4$ m
Vertical distance from wheelset center of gravity to car body	$h_C =1.67$ m
Vertical distance from wheelset center of gravity to secondary suspension	$h_T =0.47$ m
Longitudinal stiffness of primary suspension	$K_{px} = 9\times 10^{5}$ N/m
Lateral stiffness of primary suspension	$K_{py} =3.9\times 10^{5}$ N/m
Vertical stiffness of primary suspension	$K_{pz} =6\times 10^{5}$ N/m
Vertical damping of primary suspension	$C_{pz} =4\times 10^{4} \,\,\hbox {N}\cdot \hbox {s}$/m
Longitudinal stiffness of secondary suspension	$K_{sx} =3.5\times 10^{4}$ N/m
Lateral stiffness of secondary suspension	$K_{sy} =3.5\times 10^{4}$ N/m
Vertical stiffness of secondary suspension	$K_{sz} =3.5\times 10^{5}$ N/m
Longitudinal damping of secondary suspension	$C_{sx} =3.2\times 10^{4} \,\,\hbox {N}\cdot \hbox {s}$/m
Lateral damping of secondary suspension	$C_{sy} =1\times 10^{4} \,\,\hbox {N}\cdot \hbox {s}$/m
Vertical damping of secondary suspension	$C_{sz} =4\times 10^{4} \,\,\hbox {N}\cdot \hbox {s}$/m
Radius of curved tracks	$R=1500$ m
Superelevation angle of curved track	$\phi _{se} =0.0524$ rad
External load	$W_\mathrm{ext} =5.6\times 10^{4}$ N
Coefficient of friction	$f=0.2$
Normal force acting on wheelset in equilibrium state	$N={W_\mathrm{ext} }/2$ N
Young’s modulus of wheels	$E_w =2.058\times 10^{11}$ Pa

Young’s modulus of rails	$E_R =2.058\times 10^{11}$ Pa
Poisson’s ratio of wheels	$\sigma _R =0.3$
Poisson’s ratio of rails	$\sigma _w =0.3$
Principal rolling radius of the wheel	$R_1 =r_0 =0.43$ m
Principal transverse radius of curvature of the wheel profile at the point of contact	$R^{\prime }_1 =\infty$
Principal rolling radius of the rail at the point of contact	$R_2 =\infty$
Principal transverse radius of curvature of the rail profile at the point of contact	$R^{\prime }_2 =0.6$ m

Appendix B

Normal forces:

Assuming the equilibrium speed motion of wheelsets on the circular curved tracks (Fig. 2), the vertical components of normal force for each wheelset are calculated and expressed as

$$\begin{aligned} N_{Lzij}^*=N_{Rzij}^*=\frac{1}{2}\left( {W_\mathrm{ext} +m_w g+\frac{V^{2}W_\mathrm{ext} }{gR}\phi _{se} } \right) . \end{aligned}$$

(B1)

Speed-dependent and linear creep forces and creep moments:

Considering the various wheel conicities $\lambda _{ij}$ and wheel rolling radii $r_{ij}$ at wheel/rail contact surfaces after wheel turning, the linear creep forces and moments directly calculated via Kalker’s linear theory are calculated and expressed as

$$\begin{aligned} F_{Lxij}^*= & {} -\frac{f_{33,ij} }{V}\left[ {V\left( 1+\frac{a}{R}-\frac{r_{Lij} }{r_{ij} }\right) -a{\dot{\psi }}_{wij} } \right] , \end{aligned}$$

(B2a)

$$\begin{aligned} F_{Lyij}^*= & {} -\frac{f_{11,ij} }{V}({\dot{y}}_{wij} +r_{Lij} \dot{\phi }_{wij}-V\psi _{wij})\nonumber \\&-\frac{f_{12,ij} }{V}\left( \dot{\psi }_{wij} -\frac{V}{R}-\frac{V}{r_{ij} }\delta _{Lij}\right) , \end{aligned}$$

(B2b)

$$\begin{aligned} M_{Lzij}^*= & {} \frac{f_{12,ij} }{V}\left[ {{\dot{y}}_{wij} -V\psi _{wij} +r_{Lij} {\dot{\phi }}_{wij} } \right] \nonumber \\&-\frac{f_{22,ij} }{V}\left[ {{\dot{\psi }}_{wij} -\frac{V}{R}-\frac{V}{r_{ij} }\delta _{Lij} } \right] , \end{aligned}$$

(B2c)

$$\begin{aligned} F_{Rxij}^*= & {} -\frac{f_{33,ij} }{V}\left[ {V\left( 1-\frac{a}{R}-\frac{r_{Rij} }{r_{ij} }\right) +a{\dot{\psi }}_{wij} } \right] , \end{aligned}$$

(B3a)

$$\begin{aligned} F_{Ryij}^*= & {} -\frac{f_{11,ij} }{V}({\dot{y}}_{wij} +r_{Rij} \dot{\phi }_{wij} -V\psi _{wij} )\nonumber \\&-\frac{f_{12,ij} }{V}\left( \dot{\psi }_{wij} -\frac{V}{R}+\frac{V}{r_{ij} }\delta _{Rij}\right) , \end{aligned}$$

(B3b)

$$\begin{aligned} M_{Rzij}^*= & {} \frac{f_{12,ij} }{V}\left[ {{\dot{y}}_{wij} -V\psi _{wij} +r_{Rij} {\dot{\phi }}_{wij} } \right] \nonumber \\&-\frac{f_{22,ij} }{V}\left[ {{\dot{\psi }}_{wij} -\frac{V}{R}+\frac{V}{r_{ij} }\delta _{Rij} } \right] , \end{aligned}$$

(B3c)

where $r_{Lij} =r_{ij} +\lambda _{ij} y_{wij}$, $r_{Rij} =r_{ij} -\lambda _{ij} y_{wij}$, $\delta _{Lij} =\delta _{Rij} =\lambda _{ij}$, $\delta _{Lij} =\delta _{Rij} =\lambda _{ij}$.

Suspension forces and moments:

Wheelsets

$$\begin{aligned} F_{syij}= & {} -2K_{py} y_{wij} -(-1)^{j}2K_{py} L_1 \psi _{ti} +2K_{py} y_{ti} \nonumber \\&-2C_{py} {\dot{y}}_{wij} -(-1)^{j}2C_{py} L_2 {\dot{\psi }}_{ti} +2C_{py} {\dot{y}}_{ti} \nonumber \\&+2K_{py} h_T \phi _{ti} +2C_{py} h_T {\dot{\phi }}_{ti}, \end{aligned}$$

(B4)

$$\begin{aligned} M_{sxij}= & {} 2K_{pz} b_1^2 \phi _{ti} +2C_{pz} b_1^2 {\dot{\phi }}_{ti}\nonumber \\&-2b_1^2 K_{pz} \phi _{wij} -2b_1^2 C_{pz} {\dot{\phi }}_{wij} , \end{aligned}$$

(B5)

$$\begin{aligned} M_{szij}= & {} 2K_{px} b_1^2 \psi _{ti} +2C_{px} b_1^2 {\dot{\psi }}_{ti}\nonumber \\&-2K_{px} b_1^2 \psi _{wij} -2C_{px} b_1^2 {\dot{\psi }}_{wij} , \end{aligned}$$

(B6)

$$\begin{aligned} M_{exij}= & {} hW_\mathrm{ext} \left( {\frac{V^{2}}{gR}-\phi _{se} -\phi _{wij} } \right) . \end{aligned}$$

(B7)

Bogie frame

$$\begin{aligned} F_{syti}= & {} 2K_{py} y_{wi1} +2C_{py} {\dot{y}}_{wi1} +2K_{py} y_{wi2} +2C_{py} {\dot{y}}_{wi2} \nonumber \\&+(-4K_{py} -2K_{sy} )y_{ti} +(-4C_{py} -2C_{sy} ){\dot{y}}_{ti} \nonumber \\&+2\left( {-1} \right) ^{i}K_{sy} L_c \psi _c +2\left( {-1} \right) ^{i}C_{sy} L_c {\dot{\psi }}_c \nonumber \\&+2K_{sy} y_c +2C_{sy} \dot{y}_c \nonumber \\&+2K_{sy} \left( {h_c -h_T } \right) \phi _c +2C_{sy} \left( {h_c -h_T } \right) {\dot{\phi }}_c \nonumber \\&-4K_{py} h_T \phi _{ti} -4C_{py} h_T {\dot{\phi }}_{ti}, \end{aligned}$$

(B8)

$$\begin{aligned} F_{szti}= & {} 2K_{sz} z_c +2C_{sz} {\dot{z}}_c -2\left( {K_{sz} +2K_{pz} } \right) z_{ti} \nonumber \\&-2\left( {C_{sz} +2C_{pz} } \right) {\dot{z}}_{ti} \nonumber \\&+2K_{pz} z_{wi1} +2C_{pz} {\dot{z}}_{wi1} +2K_{pz} z_{wi2} +2C_{pz} {\dot{z}}_{wi2},\nonumber \\ \end{aligned}$$

(B9)

$$\begin{aligned} M_{sxti}= & {} 2K_{sz} b_2^2 \phi _c +2C_{sz} b_3^2 {\dot{\phi }}_c -2K_{sz} b_2^2 \phi _{ti} -2C_{sz} b_3^2 {\dot{\phi }}_{ti}\nonumber \\&+2K_{py} h_T y_{wi1} +2C_{py} h_T {\dot{y}}_{wi1} \nonumber \\&+2K_{py} h_T y_{wi2} +2C_{py} h_T {\dot{y}}_{wi2} \nonumber \\&-4K_{py} h_T y_{ti} -4C_{py} h_T {\dot{y}}_{ti} -4K_{py} h_T^2 \phi _{ti} \nonumber \\&-4C_{py} h_T^2 {\dot{\phi }}_{ti} -4K_{pz} b_1^2 \phi _{ti} -4C_{pz} b_1^2 {\dot{\phi }}_{ti} \nonumber \\&+2K_{pz} b_1^2 \phi _{wi1} +2C_{pz} b_1^2 {\dot{\phi }}_{wi1} \nonumber \\&+2K_{pz} b_1^2 \phi _{wi2} +2C_{pz} b_1^2 {\dot{\phi }}_{wi2}, \end{aligned}$$

(B10)

$$\begin{aligned} M_{szti}= & {} \left( -4K_{py} L_1^2 -4K_{px} b_1^2 -2K_{sx} b_2^2\right) \psi _{ti}\nonumber \\&+\left( -4C_{py} L_2^2 -4C_{px} b_1^2 -2C_{sx} b_3^2\right) {\dot{\psi }}_{ti} \nonumber \\&+2K_{py} L_1 y_{wi1} +2C_{py} L_2 {\dot{y}}_{wi1} \nonumber \\&+2K_{px} b_1^2 \psi _{wi1} +2C_{px} b_1^2 {\dot{\psi }}_{wi1} \nonumber \\&-2K_{py} L_1 y_{wi2} -2C_{py} L_2 {\dot{y}}_{wi2} \nonumber \\&+2K_{px} b_1^2 \psi _{wi2} +2C_{px} b_1^2 {\dot{\psi }}_{wi2} \nonumber \\&+2K_{sx} b_2^2 \psi _c +2C_{sx} b_3^2 {\dot{\psi }}_c. \end{aligned}$$

(B11)

Car body

$$\begin{aligned} F_{syc}= & {} -2K_{sy} (2y_c -y_{t1} -y_{t2} )-4K_{sy} \left( {h_c -h_T } \right) \phi _c \nonumber \\&-2C_{sy} (2{\dot{y}}_c -{\dot{y}}_{t1} -{\dot{y}}_{t2} )-4C_{sy} \left( {h_c -h_T } \right) {\dot{\phi }}_c ,\nonumber \\ \end{aligned}$$

(B12)

$$\begin{aligned} F_{szc}= & {} -4K_{sz} z_c -4C_{sz} {\dot{z}}_c +2K_{sz} z_{t1} \nonumber \\&+2C_{sz} {\dot{z}}_{t1} +2K_{sz} z_{t2} +2C_{sz} {\dot{z}}_{t2} , \end{aligned}$$

(B13)

$$\begin{aligned} M_{sxc}= & {} 2K_{sz} b_2^2 \phi _{t1} +2C_{sz} b_3^2 {\dot{\phi }}_{t1}\nonumber \\&+2K_{sz} b_2^2 \phi _{t2} +2C_{sz} b_3^2 {\dot{\phi }}_{t2} \nonumber \\&-4K_{sz} b_2^2 \phi _c -4C_{sz} b_3^2 {\dot{\phi }}_c \nonumber \\&-4K_{sy} \left( {h_c -h_T } \right) y_c -4C_{sy} \left( {h_c -h_T } \right) {\dot{y}}_c \nonumber \\&+2K_{sy} \left( {h_c -h_T } \right) y_{t1} \nonumber \\&+2C_{sy} \left( {h_c -h_T } \right) {\dot{y}}_{t1} +2K_{sy} \left( {h_c -h_T } \right) y_{t2} \nonumber \\&+2C_{sy} \left( {h_c -h_T } \right) {\dot{y}}_{t2} \nonumber \\&-4K_{sy} \left( {h_c -h_T } \right) ^{2}\phi _c -4C_{sy} \left( {h_c -h_T } \right) ^{2}{\dot{\phi }}_c , \end{aligned}$$

(B14)

$$\begin{aligned} M_{szc}= & {} -4K_{sy} \psi _c L_c^2 -4C_{sy} {\dot{\psi }}_c L_c^2\nonumber \\&-2K_{sx} b_2^2 (2\psi _c -\psi _{t1} -\psi _{t2} ) \nonumber \\&-2C_{sx} b_3^2 (2{\dot{\psi }}_c -{\dot{\psi }}_{t1} -\dot{\psi }_{t2} )\nonumber \\&-2K_{sy} L_c (-y_{t1} +y_{t2} )-2C_{sy} L_c (-{\dot{y}}_{t1} +{\dot{y}}_{t2} ).\nonumber \\ \end{aligned}$$

(B15)