Hunting stability analysis of partially filled tank wagon on curved track using coupled CFD-MBD method

Abstract

In this study, we develop an innovative numerical method for the investigation of the stability of a partially filled tank wagon moving on a curved track. The calculations are carried out in two subsystems including a dynamic system and fluid sloshing. We analyze the wagon dynamic system the multibody dynamic (MBD) model with 21 degrees of freedom (21-DOFs), which takes into account the lateral, vertical, roll, pitch, and yaw motions. The heuristic creep theory is used for the wheel–rail contact model. We adopt the fourth-order Runge–Kutta method for solving of this model. The transient fluid slosh is simulated by the computational fluid dynamic (CFD) model. The volume of fluid (VOF) technique is used for tracking the free surface of the fluid. This model is validated experimentally using the sloshing test setup. Then the simultaneous interaction between the dynamic system and the transient fluid slosh is analyzed by coupling the CFD model with the MBD model. By the parametric study on the filled-volume and wagon velocity, the critical hunting speed is derived by the Lyapunov indirect method. The results show that a higher filled volume decreases the critical hunting speed. Also, at the instability condition, an increasing trend for the phase trajectory of the wagon components is evident.

Appendix: Nomenclature

$I_{cx}$ :

$X$-axis inertia moments for car body

$I_{cy}$ :

$Y$-axis inertia moments for car body

$I_{cz}$ :

$Z$-axis inertia moments for car body

$L_{t}$ :

Length of cylindrical tank

$m_{c}$ :

Mass of car body

$R_{t}$ :

Radius of cylindrical tank

$I_{bx}$ :

$X$-axis inertia moments for bogie

$I_{bz}$ :

$Z$-axis inertia moments for bogie

$m_{b}$ :

Mass of bogie

$a$ :

Half of track gauge

$f_{11}$ :

Lateral creep coefficient

$f_{12}$ :

Lateral/spin creep coefficient

$f_{22}$ :

Spin creep coefficient

$f_{33}$ :

Longitudinal creep coefficient

$I_{wx}$ :

$X$-axis inertia moment for wheel-set

$I_{wz}$ :

$Z$-axis inertia moment for wheel-set

$m_{w}$ :

Mass of wheel-set

$r_{0}$ :

Wheel radius

$R$ :

Radius of curved track

$w_{ext}$ :

External load (weight of the elements of mechanical and electrical system)

$\lambda$ :

Wheel conicity

$\phi_{se}$ :

Superelevation angle of curved track

$b_{1}$ :

Half distance of primary longitudinal suspension

$b_{2}$ :

Half distance of secondary longitudinal suspension

$c_{px}$ :

Longitudinal primary suspension damp

$c_{py}$ :

Lateral primary suspension damping

$c_{pz}$ :

Vertical primary suspension damping

$c_{sx}$ :

Longitudinal secondary suspension damping

$c_{sy}$ :

Lateral secondary suspension damping

$c_{sz}$ :

Vertical secondary suspension damping

$h_{c}$ :

Vertical distance from wheel-set center of gravity to car body

$h_{t}$ :

Vertical distance from wheel-set center of gravity to secondary suspension

$k_{px}$ :

Longitudinal primary suspension stiffness

$k_{py}$ :

Lateral primary suspension stiffness

$k_{pz}$ :

Vertical primary suspension stiffness

$k_{sx}$ :

Longitudinal secondary suspension stiffness

$k_{sy}$ :

Lateral secondary suspension stiffness

$k_{sz}$ :

Vertical secondary suspension stiffness

$l_{1}$ :

Half distance of primary lateral suspension

$l_{c}$ :

Longitudinal distance from wheel-set center of gravity to car body

$A$ :

Stability matrix

$A_{r}$ :

Rotation matrix for coordinate system

$g$ :

Gravity acceleration

$V$ :

Fluid flow velocity

$v$ :

Wagon velocity

$v^{*}$ :

Hunting speed

$x_{0}$ :

Equilibrium point

$x(t)$ :

Vector of state variables

$X\_Acc$ :

Accelerometer data (longitudinal)

$y_{b}$ :

Lateral coordinate of bogies

$y_{c}$ :

Lateral coordinate of car body

$y_{w}$ :

Lateral coordinate of wheel-sets

$Y\_Acc$ :

Accelerometer data (lateral)

$z_{b}$ :

Vertical coordinate of bogies (front and rear)

$z_{c}$ :

Vertical coordinate of car body

$Z\_Acc$ :

Accelerometer data (vertical)

$\theta_{c}$ :

$Y$-axis rotational coordinate (pitch angle) of car body

$\phi_{b}$ :

$X$-axis rotational coordinate (roll angle) of bogies

$\phi_{c}$ :

$X$-axis rotational coordinate (roll angle) of car body

$\psi_{c}$ :

$Z$-axis rotational coordinate (yaw angle) of car body

$\psi_{b}$ :

$Z$-axis rotational coordinate (yaw angle) of bogies

$\psi_{w}$ :

$Z$-axis rotational coordinate (yaw angle) of wheel-sets

$.^{(\text{Dot})}$ :

First-order derivative

${..}^{(\text{Double dot})}$ :

Second-order derivative

$A_{i}$ :

Area vector of the $i$th wall cell

$CG_{y}$ :

Fluid’s center of gravity coordinate (lateral)

$CG_{z}$ :

Fluid’s center of gravity coordinate (vertical)

$f$ :

Volume fraction of the liquid phase for a cell

$f_{1}$ :

Volume fraction for air

$F(t)$ :

Fluid slosh force

$F_{f}$ :

External force for CFD model

$F_{i}$ :

Force vector of ith wall cell

$F_{fy}$ :

Lateral sloshing force

$F_{fz}$ :

Vertical sloshing force

$I_{fx}$ :

$X$-axis inertia moment for fluid

$I_{fy}$ :

$Y$-axis inertia moment for fluid

$I_{fz}$ :

$Z$-axis inertia moment for fluid

$M(t)$ :

Fluid slosh moment

$m_{f}$ :

Mass of fluid in car body

$M_{fx}$ :

Fluid slosh moment about $X$ axis

$M_{fy}$ :

Fluid slosh moment about $Y$ axis

$M_{fz}$ :

Fluid slosh moment about $Z$ axis

$p$ :

Pressure

$P_{i}$ :

Pressure of the ith wall cell

$Q$ :

Wetted face on tank wall

$Q_{m}$ :

Mesh quality

$r$ :

Position vector of a fluid particle

$\bar{r}_{i}$ :

Position vector of wall cell from tank coordinate

$t$ :

Time

$t_{i}$ :

Initial time

$t_{i + 1}$ :

Time in the next step

$U$ :

Translational velocity vector

$\nabla$ :

Gradient operator

$\Delta t$ :

Time step

$\rho$ :

Mass density of fluid

$\rho_{1}$ :

Density of air

$\rho_{2}$ :

Density of water

$\mu$ :

Viscosity

$\mu_{1}$ :

Dynamic viscosity for air

$\mu_{2}$ :

Dynamic viscosity for water

$\varOmega$ :

Angular velocity vector