# Evaluation of Dynamics of a Freight Wagon Model with Viscous Damping
## 2 Wagon Model  
The impact of the nature of suspension damping on vehicle dynamics was investigated using a model of a two-axle wagon with geometric and mass-inertia parameters typical of a two-axle coal wagon. The rationale for choosing a two-axle wagon for the study was to reduce potential factors that could affect the running dynamics of the wagon, such as body-bogie yaw torque, or the coupling of bogie and body vibrations. The multibody wagon model, being implemented in the VI-Rail MD 2010 13.0 software (based on the Adams MD 2010 environment [14; 15]), is relatively simple and does not take into account the suspension kinematics, with leaf springs and links (or double links) forming the technical pendulum [16], as found in the standard designs of two-axle freight wagons. Thus, there are no constraints in the suspension system due to the specifics of the suspension design; only spring and damping elements. Two wagon loading conditions were assumed for the tests: empty with a static axle load of 56 kN and laden with a static axle load of 219 kN.  
The physical model of the wagon is depicted in Figure 6 and its main parameters are shown in Table 1--values referring to the laden wagon are placed in parenthesis.  
Figure 5: TF 25 bogie [4].  
Figure 6: Physical model of the wagon: (**a**) Simulation model in VI-Rail MD 2010 13.0 software; (**b**) Diagram of the wagon model.  
The Differential Algebraic Equations (DAEs) describing the dynamics of the individual mass elements of the model are generated automatically in the Adams environment (Hexagon, formerly MSC Software) based on the Euler-Lagrange formalism. The general form of the equations is given by:  
\[\frac{d}{dt}\left[\left(\frac{\partial L}{\partial\dot{\mathbf{q}}}\right)^{ \mathrm{T}}\right]-\left(\frac{\partial L}{\partial\dot{\mathbf{q}}}\right)^{ \mathrm{T}}+\mathbf{\Phi_{q}^{\mathrm{T}}\lambda}=\mathbf{Q}, \tag{1}\]  
where \(\mathbf{q}\)--column matrix of \(n\) generalized coordinates of the rigid bodies; \(L\)--Lagrangian (the difference in kinetic and potential energies: \(L=T-V\)); \(\mathbf{\Phi_{q}}\)--Jacobian matrix of constraints; \(\lambda\)--vector of Lagrange multipliers; \(\mathbf{Q}\)--vector of external forces.  
In order to solve (integrate) DAE in Adams, the default GSTIFF solver with I3 formulation was employed, and its error control is on the displacement states. GSTIFF is the integrator of variable-order, variable-step, multi-step, and its maximum integration order is six. However, the results of simulations are generated with a constant time step. The potential forces due to gravity are included in the vector of external forces \(\mathbf{Q}\) in Adams solver. The wheel-rail creep forces are computed using the built-in procedure derived from the [PERSON]'s FASTSIM algorithm. This wheel-rail contact model represents a common way to calculate wheel-rail forces by means of a simulation approach [17; 18; 19; 20; 21]. At each simulation step, contact kinematics, required for creep force calculation, are computed using actual wheel and rail profiles. The implemented procedure allows for the analysis of the multipoint wheel-rail contact.