*2.3.2.1 Dynamic vertical force (P1 and P2)*

Due to [23] and [22], the P1 and P2 force is applied in the TDM. Eqs. (12)-(16) depict the composition and calculation scheme of the vertical impact forces P1 and P2.

$$\text{Iteration}: \mathbf{P\_{1,S}} = \mathbf{P\_0} + \mathbf{S} \times \mathbf{2a} \times \sqrt{\frac{\mathbf{K\_H} \times \mathbf{m\_e}}{\mathbf{1} + \mathbf{m\_{v\_{m\_u}}}}} \tag{12}$$

*Traffic Load and Its Impact on Track Maintenance DOI: http://dx.doi.org/10.5772/intechopen.110800*

$$\mathbf{m\_e} \approx \mathbf{0}.4 \times \left(\mathbf{m\_r} + \frac{\mathbf{m\_s}}{\mathbf{l}}\right) \tag{13}$$

$$\mathbf{K}\_{\rm H} = \frac{\mathbf{P}\_{\rm 1,est} - \mathbf{P}\_0}{\mathbf{G} \times \left(\mathbf{P}\_{\rm 1,est}^{2/3} - \mathbf{P}\_0^{2/3}\right)} \tag{14}$$

$$\mathbf{G} = \frac{\mathbf{3.86}}{\mathbf{R}\_{\text{Wheel}} \, ^{0.115}} \times \mathbf{10}^{-8} \tag{15}$$

$$\mathbf{P\_{2,S}} = \mathbf{P\_0} + \mathbf{S} \times 2\mathbf{a} \times \sqrt{\frac{\mathbf{m\_u}}{\mathbf{m\_u} + \mathbf{m\_t}}} \times \left(\mathbf{1} - \frac{\mathbf{c\_t} \times \boldsymbol{\pi}}{\mathbf{4} \times \sqrt{\mathbf{K\_t} \times (\mathbf{m\_u} + \mathbf{m\_t})}}\right) \times \sqrt{\mathbf{K\_t} \times \mathbf{m\_u}} \tag{16}$$

P1,S,P2,S—vehicle dynamic wheel forces (N); P0—vehicle static wheel force (N); S —relevant speed (limited by vehicle or track alignment) (m/s); 2α—total joint angle (rad); mu—unsprung mass per vehicle wheel (kg); mt—effective vertical track mass per vehicle wheel (kg); ct—effective track damping per vehicle wheel (Ns/m); Kt—effective vertical track stiffness per vehicle wheel (N/m); me—effective track mass per vehicle wheel (kg); mr—rail mass per unit length (kg/m); ms—mass of half a sleeper (kg); l—sleeper spacing (m); P1,est—estimated vertical dynamic wheel force (N); G—Hertzian flexibility constant (for worn tyre profiles) (m/N2/3); RWheel—wheel radius (m); KH—linearized Hertzian contact stiffness per vehicle wheel (N/m).

The total vertical force that arises due to the interaction between rail and wheel comprises not only the static gravitational loading (P0) but also the dynamic forces activated by speed (S), unsprung mass (mu) and the rail's alignment (2α). Both calculation schemes of the vertical wheel forces P1 (Eq. (12)) and P2 (Eq. (16)) are based on this approach. While P2 does also depend on track parameters (track stiffness Kt, track damping ct and track mass mt per wheel), P1 is additionally conditioned by the effective track mass (me) and the linearized Hertzian contact stiffness (KH) per wheel. The Hertzian contact stiffness KH is subject to an iterative calculation between Eqs. (12) and (14) and therefore depending on the P1 force. P1 can be summarized to be the high-frequency portion of the impact force that is primarily responsible for surface wear and local stress peaks of the rail material. P2 is the lowfrequency and long-waved force component that mainly stresses the sleepers and ballast bed [22, 28].

In the TDM the included constants mt, Kt, ct and 2α are applied according to [23]. Further, rail mass (mr) belongs to UIC60 rail, the mass of half a sleeper (ms) to concrete sleepers and the sleeper spacing (l) to the standard spacing in Austria. The included constants in the TDM have the following values:


As the approach for P1 and P2 depends on the unsprung mass (mu), the model differs between the wheels of powered and unpowered axles. Wheels of powered and unpowered axles therefore show varying values in both, P1 and P2. Section 2.4 gives an example calculation of the P1 and P2 force for a universal locomotive at a speed of 90 kmph.
