# Modeling and Characterization of Complex Dynamical Properties of Railway Ballast
## 2 Modeling of Complex Dynamical Properties of the Railway Track and Ballast  
Understanding the interaction between vehicles and the track and the track system stiffness is vital to the safety and comfort of trains [46; 47; 48; 49; 2; 4; 46; 2]. The stiffness of the ballasted railway track is primarily provided by resilient components such as the rail pads and ballast. The stiffness of track components changes as their condition deteriorates due to ballast damage and contamination. On the other hand, deviations in track component stiffness can also accelerate the track degradation process and lead to new track defects. Hence, track stiffness should be appropriately monitored during the whole life cycle of railway tracks. In addition, most of the dynamic train-track interaction models require the stiffness values of ballast as inputs. Therefore, accurate modeling and precise identification of the stiffness of track components from field measurement are very much desired. Most vehicle dynamics commercial programs have limited track components, such as ballasts. Assumptions must be made to simulate the vehicle and track performances when using these programs. The stiffness of vertical tracks or track components (\(k\)) can be defined in the simplest form as the ratio between load (\(F\)) and deflection (\(z\)) as a function of time (\(t\)) [2; 7; 24; 25]:  
\[k(t)=\frac{F(t)}{z(t)} \tag{1}\]  
Commonly, the stiffness of the different components of the track structure is nonlinear. Moreover, at significant load intensities, increases in nonlinear stiffness are a consequence of ballast compaction. As the load value increases, the ballast deformations increase these contact surfaces, thus increasing the ballast stiffness. In addition to the load value, ballast stiffness depends on excitation frequency (\(f\)). The time/space-varying ballast stiffness is one of the basic causes of differential track settlement, which has a primary influence on track geometry deterioration. The basic cause of the emergence of spatially varying track stiffness is the change of variable ballast. Traditionally, track behavior analysis under vertical load is based on the beam on an elastic foundation model and Winkler's hypothesis. This approach assumes that the behavior of the superstructure and substructure elements is linear and that there is proportionality between the load and the deflection, with the track and component stiffness having constant values. The improved approach to the stiffness definition includes both inelastic and nonlinear behavior of the superstructure and substructure elements, the difference between the stiffness under static and under dynamic load, and the variations of the stiffness along the track. A typical ballasted railway track system can be modeled as a layered structure that comprises rails, fasteners/pads/sleepers, and ballast, as well as foundations. _Track stiffness_ can be defined as the combined effect of different track components. Figure 1 shows the schematic model of the Quart vehicle-track system. It has various system components in a vertical direction, in which subsystems describing the Quart vehicle and the track are spatially coupled by the wheel-rail interface. In the vehicle sub-model, two double-axle bogies support both ends of the body, and the bogies are connected to the wheels at one end and the body at the other end through the suspension. The track sub-model in Figure 1 represents the typical ballasted track structure, consisting of rails, pads/sleepers, ballast, and the subgrade/foundation. The rail can be regarded as a continuous Bernoulli-Euler beam, which is discretely supported at the rail-sleeper junctions by three layers of springs and dampers, representing the elasticity and damping of the pad/sleeper, the ballast, and the subgrade/foundation.