# Dynamical Effects of the Increase of the Axle Load on European Freight Railway Vehicles
## 2 Vehicle Model
### Primary Suspension  
The primary suspension, shown in Figure 2 for a single axle-box, consists of four springs in concentric pairs. The internal spring constitutes the \"full load\" spring and acts only after a certain deflection of the external spring thanks to the clearance between the internal spring and the upper support. This clearance is calculated to ensure that the spring does not act when the wagon is in a tare condition, but only after a certain minimum load, thus creating a suspension stage with non-linear vertical stiffness. A third value of the stiffness is achieved when the external spring reaches the solid height condition, as there is no clearance left between the coils; this condition does not occur in the normal operation of the bogie and it has been simulated adopting a very high stiffness value (\(10^{7}\) N/m). The external spring of each group of springs arranged on the inner side of the bogie is not directly supported by the structure of the bogie frame, but by a plate support which is movable and connected to the bogie frame by means of two chain links (Lenoir-Link). The link constitutes a substantially rigid kinematic constraint, whose constraint reaction force, which keeps the spring support in equilibrium, acts along the line joining the two pins of the link. Therefore, the vertical force applied by the spring is split along two components of which the first (vertical) is directly applied to the bogie frame, while the second one (horizontal) is discharged on a piston that is in contact with the side of the axle-box. The purpose of this mechanism is to generate a friction force, dependent on the vehicle load (as it depends on the deflection of the spring), which damps the vertical motions of the bogie frame. The friction force develops on two surfaces: the first one is constituted by the contact zone between the piston and the axle-box, while the second one is the opposite surface of the axle-box which is pushed in contact with the bogie frame. All surfaces are coated with wear-resistant material (Si-Mg steel plates). The mechanism has been simulated by neglecting the inertia of the piston and of the cup spring support, by means of a series of forces applied between the axle-box and the bogie frame defined by explicit functions. The force applied to the piston F\({}_{\mathrm{XP}}\) depends on the vertical force applied to the external spring \(F_{Z2}\) and the inclination of the link, and can be calculated according to Equation (1). \[F_{\mathrm{XP}}=F_{Z2}\cdot\left(\frac{x+x_{0}}{h_{0}}\right) \tag{1}\]  
where \(x\) is the longitudinal displacement of the piston (assumed equal to the displacement of the axle-box), \(x_{0}\) is the nominal distance of the link pins in the longitudinal direction (27 mm), and h\({}_{0}\) is the nominal distance in the vertical direction (68 mm). Figure 2: Primary suspension and Lenoir-Link device. The force \(F_{Z2}\) is then calculated according to Equation (2). \[F_{Z2}=ke\cdot(z^{+}+z_{0}) \tag{2}\]  
where \(ke\) is the stiffness of the external spring; \(z^{+}\) is the vertical displacement of the bogie due to the acting load; and \(z_{0}\) is the deformation of the spring in the initial conditions (at preload). The reaction force on the second friction surface of the axle-box is calculated as:  
\[F_{XR}=-fimpact(x)-D_{i}\dot{x} \tag{3}\]  
The function _fimpact_ (\(x\)) represents an impact with a metallic surface; it has been defined according to Equation (4) and is plotted in Figure 3 on the right. \[fimpact(x)= \begin{cases}0&if\ x>0\\ 5\cdot 10^{8}\cdot x^{3}+3\cdot 10^{8}\cdot x^{2}-2\cdot 10^{7}\cdot x&if\ x\leq 0 \end{cases}  \tag{4}\]  
\(D_{i}\) represents the damping, mainly due to the sliding of the piston in its seat, and is assumed as viscous and constant and equal to 10 Ns/m. Both friction forces are applied between axle-box and bogie frame on the axis of action of the piston at a longitudinal coordinate corresponding to the considered friction surface on the axle-box. These forces are defined by Equations (5)-(8).