# Qualitative Prediction Model for Dynamic Behavior of Ballasted Tracks
## 2 Qualitative Analysis
### Application of Qualitative Analysis to Track Dynamics
The structural dynamic equations are represented as:  
\[M\ddot{u}+C\dot{u}+Ku=F \tag{9}\]  
Figure 1: Ballasted track model. where \(M\), \(C\), and \(K\) are the mass, damping, and stiffness matrices of the track, respectively, and \(F\) is the applied load vector [9]. In the frequency domain, this can be represented as [9]:  
\[k(t)=\frac{Q(t)}{w(t)}\;or\;k(f)=\frac{Q(f)}{w(f)} \tag{2}\]0  
Assuming that \(F\) is a unit load vector acting at an excitation position on the rail, the track receptance is the solution of \(U\) at the loading position; the dynamic track stiffness was then calculated as the inverse of the track receptance [9]. The rail is defined by Young's modulus \(E\) (kN/cm\({}^{2}\)), shear modulus G (kN/cm\({}^{2}\)), density \(\rho\) (kN/cm\({}^{3}\)), cross-sectional area \(A\) (cm\({}^{2}\)), second moment of area of section \(I\) (cm\({}^{4}\)), and shear coefficient \(\kappa\); the bending stiffness is represented by \(B=EI\) and the shear stiffness by \(K=GA\kappa\)[10]. The sleeper was calculated by mass \(ms\), whereas the rail pad and the ballast were modeled by their respective stiffness and loss factor \(k_{pad}\) and \(\eta_{pad}\), and \(k_{bal}\) and \(\eta_{bal}\), respectively [10]:  
\[k(t)=\frac{Q(t)}{w(t)}\;or\;k(f)=\frac{Q(f)}{w(f)} \tag{2}\]1  
The wheel and point forces are assumed to be exerted on each support of the rail. The dynamic stiffness (impedance) \(S(\omega)\) is related to the force \(\hat{F}_{n}\) at each support point and corresponding displacement \(\hat{u}_{n}\) as [10]:  
\[k(t)=\frac{Q(t)}{w(t)}\;or\;k(f)=\frac{Q(f)}{w(f)} \tag{2}\]2  
\[k(t)=\frac{Q(t)}{w(t)}\;or\;k(f)=\frac{Q(f)}{w(f)} \tag{2}\]3  
The dynamic response of the in-service ballasted track was relatively random or higher than the results of FE analysis, and was, therefore, distributed more roughly and over a wider range than its initial design value (i.e., the value considered during track construction) [1, 2, 3, 18, 22, 43]. Therefore, the dynamic response of in-service ballasted tracks does not depend on a single track structure parameter. The constitutive track model was then applied to define the continuous equivalent stiffness properties in the longitudinal direction based on an anisotropic formulation, since the sleepers are usually held in contact with the ballast by a linking element. The connection between the rail and the sleepers was assumed to be a single vector response on the rail and to only correspond to the vertical displacement; thus, the motion of the rail in the transformed domain is expressed as [1, 2, 9, 44]:  
\[\left(\underbrace{\left[ \begin{array}{cc}EIk_{1}^{4}&-k_{p}^{*}\\ -k_{p}^{*}&k_{p}^{*}\end{array} \right]}_{[K]_{rail}}-\omega\underbrace{\left[ \begin{array}{cc}m_{r}&0\\ 0&0\end{array} \right]}_{[M]_{rail}}\right)\underbrace{\left( \begin{array}{ cc}u_{rail}\\ u_{sleeper}\end{array} \right)}_{u}=\underbrace{\left[P\right]}_{f} \tag{14}\]  
where \(Elk_{1}\) is the bending stiffness of the rail (kN/cm\({}^{2}\)), \(k_{p}\) is the complex stiffness of the rail pad (kN), \(m_{r}\) is the rail's mass per unit length (kN/cm), and the vectors \(u\) and \(f\) represent the displacement and load on the rail (m), respectively [35]. The superscript * in \(k_{p}\) represents the complex stiffness), which accounts for the damping properties of the rail pad; \(k_{p}^{*}=k_{p}+i\alpha c_{p}\), where \(c_{p}\) represents the viscous damping factor. The stiffness and mass matrices of the rail were then assembled to the dynamic system functions [1, 2, 41, 44, 45]. The qualitative analysis was defined by the display method of the analytical matrices over a discrete surface area (space solution) [46, 47, 48]. As analyzing the dynamic response of the ballasted track requires the relationship between the track component and force parameters and the track behavior, a numerical model based on conventional track dynamics theory was compared with the FE analysis and the field measurements.