Effect of the nonlinear displacement-dependent characteristics of a hydraulic damper on high-speed rail pantograph dynamics

Abstract

A new simplified parametric model, which is more suitable for pantograph–catenary dynamics simulation, is proposed to describe the nonlinear displacement-dependent damping characteristics of a pantograph hydraulic damper and validated by the experimental results in this study. Then, a full mathematical model of the pantograph–catenary system, which incorporates the new damper model, is established to simulate the effect of the damping characteristics on the pantograph dynamics. The simulation results show that large $F_{\mathrm{const}}$ (saturation damping force of the damper during compression) and $C_{\mathrm{0}}$ (initial damping coefficient of the damper during extension) in the pantograph damper model can improve both the raising performance and contact quality of the pantograph, whereas a large $C_{\mathrm{0}}$ has no obvious effect on the lowering time of the pantograph; the nonlinear displacement-dependent damping characteristics described by the second item in the new damper model have dominating effects on the total lowering time, maximum acceleration and maximum impact acceleration of the pantograph. Thus, within the constraint of total lowering time, increasing the nonlinear displacement-dependent damping coefficient of the damper will improve the lowering performance of the pantograph and reduce excessive impact between the pantograph and its base frame. In addition, damping performance of the new damper model would vary with the vehicle speeds, when operating beyond the nominal-speed range of the vehicle, the damping performance would deteriorate obviously. The proposed concise pantograph hydraulic damper model appears to be more adaptive to working conditions of the pantograph, and more complete and accurate than the previous single-parameter linear model, so it is more useful in the context of pantograph–catenary dynamics simulation and further parameter optimizations. The obtained simulation results are also valuable and instructive for further optimal specification of railway pantograph hydraulic dampers.

Appendix

Parameters and values in the pantograph–catenary dynamics modelling and simulation

Notation (Unit)	Description	Value	Remarks
$m_{1}$ (kg)	Coupling rod mass	3.90
$J_{1}\hbox { (kg~m}^{2})$	Coupling rod moment of inertia	1.88
$l_{1}$ (m)	Coupling rod length	1.20
$l_{\mathrm{m1}}$ (m)	Length from the centre of gravity of the coupling rod to joint A	6.03E−001
$\theta _{2} ({^{\circ }})$	Angle from the coupling rod to level	Variable
$l_{2}$ (m)	Length of connecting rod BC	3.40E−001
$\theta _{1} ({^{\circ }})$	Angle from the connecting rod BC to level	Variable
$m_{3}$ (kg)	Lower arm mass	2.10E+001
$J_{3}$ $\hbox {(kg m}^{2})$	Lower arm moment of inertia	1.75E+001
$l_{3}$ (m)	Lower arm length	1.58
$l_{\mathrm{m3}}$ (m)	Length from the centre of gravity of the lower arm to joint D	7.93E-001
$\alpha ({^{\circ }})$	Rising angle of the lower arm (pantograph)	Variable
$h_{0}$ (m)	Vertical distance of joints A and D	1.30E−001
$l_{0}$ (m)	Horizontal distance of joints A and D	7.20E−001
$m_{4}$ (kg)	Upper arm mass	1.60E+001
$J_{4}\hbox { (kg~m}^{2})$	Upper arm moment of inertia	2.02E+001
$l_{4}$ (m)	Upper arm length	1.95
$l_{\mathrm{m4}}$ (m)	Length from the centre of gravity of the upper arm to joint C	9.14E−001
$\beta ({^{\circ }})$	Angle from the connecting rod BC to upper arm	1.15E+001
$y_{\mathrm{e}}$ (m)	Height of joint E	Variable
$m_{\mathrm{h}}$ (kg)	Pan-head mass	5.00
$k_{\mathrm{h}}$ (N/m)	Equivalent stiffness of the pan-head suspension	7.60E+003
$c_{\mathrm{h}}$ (N/m)	Equivalent damping coefficient of the pan-head suspension	5.00E+001
$l_{\mathrm{h}}$ (m)	Height between the collector and joint E	1.00E−001
$y_{\mathrm{h}}$ (m)	Height of the pantograph collector	Variable
$y_{\mathrm{c}}$ (m)	Height of the catenary	1.70
$L_{\mathrm{c}}$ (m)	Span length of the catenary	6.30E+001
$L_{\mathrm{d}}$ (m)	Dropper interval	9.00
$k_{\mathrm{c}}$ (N/m)	Catenary stiffness	Variable
$k_{\mathrm{0}}$ (N/m)	Static stiffness of the catenary	3.6845E+003
$a_{1}$	Coefficient	4.665E−001
$a_{2}$	Coefficient	8.32E−002
$a_{3}$	Coefficient	2.603E−001
$a_{4}$	Coefficient	− 2.801E−001
$a_{5}$	Coefficient	− 3.364E−001
$F_{c}$ (N)	pantograph–catenary contact force	Variable
v (km/h)	Vehicle speed	2.00E+002
L	Lagrangian function	Function
T (J)	Kinetic energy of the framework	Variable
U (J)	Potential energy of the framework	Variable
$G_{\mathrm{F}}$ (N m)	Generalized force	Variable
$J_{\mathrm{f}}\hbox { (kg~m}^{2})$	Equivalent moment of inertia of the framework	Variable
$U_{\mathrm{f}}\hbox {(N~s}^{2}/\hbox {m})$	Coefficient of ${\dot{y}_{\mathrm{e}}}^{2}$ in dynamic model of the framework	Variable
$F_{\mathrm{f}}$ (N m)	Equivalent generalized force of the framework	Variable
$M_{\upalpha }$ (N m)	Uplift moment	Variable
$F_{\mathrm{u}}$ (N)	Static uplift force	7.00E+001
$C_{\mathrm{f}}$ (N s/m)	Equivalent damping coefficient of the framework	Variable
$F_{\mathrm{d}}$ (N)	Damping force of the hydraulic damper	Variable
$g \hbox {(m/s}^{2})$	Acceleration of gravity	9.80
l (m)	Length of the connection rod DP	1.80E−001
$x_{\mathrm{d}}$ (m)	Horizontal distance between point P and joint N	3.50E−001
$y_{\mathrm{d}}$ (m)	Vertical distance between joints D and N	0.00	Joints D and N are on the same level
s (m)	Instantaneous length of the hydraulic damper	Variable
$\gamma ({^{\circ }})$	Angle from the connection rod DP to lower arm	5.56E+001
$s_{0}$ (m)	Hydraulic damper length when the pantograph is completely raised	3.65E-001
x(t) (m)	Instantaneous displacement of the hydraulic damper	Variable
$k_{1}-k_{13}$	Coefficients	Variable	The unit depends on concrete meaning of the coefficient
t (s)	Time	Variable
$A_{\mathrm{c}} (\hbox {m}^{2})$	Pressure action area of the piston during the extension stroke of the damper	Variable
$A_{\mathrm{f}} (\hbox {m}^{2})$	Cross-section area of the orifices in the rod for fluid outflow	Variable
$A_{1}-A_{n} (\hbox {m}^{2})$	Changeable cross-section area of the orifices in the rod for fluid inflow	Variable	$n=1, 2,{\ldots } 6$ in this work
$A_{\mathrm{x}} (\hbox {m}^{2})$	Pressure action area of the piston during the compression stroke of the damper	Variable
$C_{\mathrm{com}}$ (N s/m)	Damping coefficient of the damper during compression	Variable
$C_{\mathrm{d1}}$	Discharge coefficient of the orifice	7.20E−001
$C_{\mathrm{d2}}$	Discharge coefficient of the shim-stack valve	6.10E−001
$C_{\mathrm{e}}$	Equivalent-pressure correction factor	3.15E−001	FEA identified
$C_{\mathrm{ext}}$ (N s/m)	Damping coefficient of the damper during extension	Variable
$C_{\mathrm{w}} (\hbox {m}^{6}/\hbox {N})$	Deflection coefficient of the shim	Variable
$C_{0}$ (N s/m)	Initial damping coefficient of the damper during extension	Variable
D (m)	Piston diameter	3.60E−002
E (Pa)	Elastic modulus of the shim	2.00E+011
$F_{\mathrm{const}}$ (N)	Damping force of the damper during compression	Variable
P (Pa)	Instantaneous working pressure of the damper	Variable
$P_{\mathrm{i}}$ (Pa)	Instantaneous pressure in the hollow passage of the rod	Variable
$Q_{\mathrm{work}} (\hbox {m}^{3}/\hbox {s})$	Instantaneous working flow of the damper	Variable
d (m)	Rod diameter	1.58E−002
$d_{0}$ (m)	Diameter of the orifice in the inner tube	6.00E−004
$d_{1}$ (m)	Diameter of the orifice in the rod for fluid inflow	1.10E−003
$d_{2}$ (m)	Diameter of the orifice in the rod for fluid outflow	1.20E−003
$d_{3}$ (m)	Diameter of the orifice in the rod for fluid outflow	1.10E−003
$d_{4}$ (m)	Diameter of the orifice in the rod for fluid outflow	1.10E−003
$h_{1}-h_{\mathrm{n}}$ (m)	Thickness of the shims in a shim-stack	5.00E−004
$r_{\mathrm{s}}$ (m)	Outer radius of the shim	8.00E−003
$s_{\mathrm{a}}$ (m)	Displacement amplitude of the damper	5.00E−002
$s_{1}$ (m)	Distance from the first orifice in the rod to point (0, $s_{\mathrm{a}}/2)$	2.10E−002
$\Delta {s} _{1}$ (m)	Orifice interval	1.40E−003
$\Delta {s} _{2}$ (m)	Orifice interval	3.20E−003
$\rho$ $(\hbox {kg/m}^{3})$	Oil density	8.75+002