# Qualitative Prediction Model for Dynamic Behavior of Ballasted Tracks
## 2 Qualitative Analysis
### Application of Qualitative Analysis to Track Dynamics  
Track dynamics theory (i.e., beam on continuous elastic foundation as defined by [PERSON]) and qualitative analysis were then applied to simulate track dynamics. The qualitative analysis mainly focuses on track dynamics theory describing the track stiffness, i.e., related to the rail pads, ballast and subgrade stiffness, and track impact factor, whose response depends on the track stress, displacement, and resonance in the vertical plane. The model is then used to calculate the beam bending stiffness \(EI\) (N\(\cdot\)m\({}^{2}\)) and the foundation stiffness (i.e., the ballast bed modulus) \(k\) (N/m\({}^{2}\), i.e., N/m per unit meter rail) [1; 2]. The rail deflection \(w(x)\) (where \(x\) is the length coordinate) was then obtained from the following differential Equation (1):  
\[EI\frac{d^{4}w(x)}{dx^{4}}+kw(x)=q(x) \tag{1}\]  
where \(q(x)\) is the distributed load on the rail (N/m). The discrete rail support model, a continuous model, reproduces the pinned\(-\)pinned frequency on distributed layers [1; 2; 39]. In their investigation of track support behavior, [PERSON] and [PERSON] [39] concluded that poorly damped sleeper resonances lead to higher sleeper stress [1; 2; 40]. The rails, sleepers, and rail pads (and ballasts) were modeled as a beam (via Euler-Bernoulli or Rayleigh-Timoshenko beam theory), a rigid mass, and a spring damper, respectively [1; 2; 41]. Track support stiffness \(k\) (kN/mm) (i.e., the precise dynamic track stiffness) can be defined as the ratio between the vertical force \(Q\) (kN) on the rail and the vertical track displacement \(w\) (mm) as [9]:  
\[k(t)=\frac{Q(t)}{w(t)}\;or\;k(f)=\frac{Q(f)}{w(f)} \tag{2}\]  
Therefore, the dynamic track stiffness is dependent on the applied load, or a function of the excitation frequency \(f\) in the time \(t\) domain. Conventional track calculations were based on a static approach developed by [PERSON] [9]. The ballasted track model, as represented by [PERSON] et al. [4], is shown in Figure 1, where a spring damper element represents a track component [4], the sleeper spacing is represented by \(a\), and the total of series support stiffness of the rail pad, ballast, and subgrade is \(k_{\rm S}\). The governing set of differential equations accounting for these boundary conditions defines the rail displacement \(w\) (mm), sectional moment of the rail \(M\) (cm\({}^{4}\)), and pressure load on sleeper \(F\) (kN) as [9]:  
\[w(x)=\frac{QL^{3}}{8 EI}e^{-(x/L)}\biggl{(}\cos\frac{x}{L}+\sin\frac{x}{L}\biggr{)} \tag{3}\]  
\[M(x)=\frac{QL}{4}e^{-(x/L)}\biggl{(}\cos\frac{x}{L}-\sin\frac{x}{L}\biggr{)} \tag{4}\]  
\[F(x)=\frac{Qa}{2L}e^{-(x/L)}\biggl{(}\cos\frac{x}{L}+\sin\frac{x}{L}\biggr{)} \tag{5}\]  
where \(L\) is the characteristic length of the track (mm), as [3; 9]:  
\[L=\sqrt[4]{\frac{4 EI}{k_{c}}}=\sqrt[4]{\frac{4 Ela}{k_{s}}} \tag{6}\]  
and \(k_{s}\) is determined from the pad stiffness \(k_{pad}\) and ballast/subgrade stiffness \(k_{bs}\) (kN/mm), as [3; 9]:  
\[\frac{1}{k_{s}}=\frac{1}{k_{pad}}+\frac{1}{k_{bs}} \tag{7}\]  
Using Equation (2), the static track stiffness can thus be calculated as [3; 9]:  
\[k=\frac{8 EI}{L^{3}}=2\sqrt[4]{\frac{4}{8}}Elk_{c}^{3} \tag{8}\]  
Thus, the track support stiffness is a function of the structural properties of the rail, rail pad, sleeper, and ballast and subgrade. Recently, track models presented for dynamic vehicle-track interaction have been used to investigate the dynamic track stiffness, in which a finite element (FE) model is conducted by the half of track model, the rail is modeled by Rayleigh-Timoshenko beam theory, and the sleepers are considered as rigid masses [9; 10]. Here, the FE model of ballasted tracks consisted of one rail of finite length, discretely supported via rail pads by sleepers on the ballast/subgrade [1; 2; 42]. The rail pad and ballast/subgrade were also modeled as a series of elastic springs and viscous dampers [9; 10].