# Evaluation of Dynamics of a Freight Wagon Model with Viscous Damping
## 3 Vibration Damping Models
### Dry Friction--The Kolsch Model  
The dry friction model of suspension damping elements, implemented in VI-Rail software, is based on the model of structural members exhibiting static hysteresis, developed by [PERSON]. Hysteresis is a known phenomenon occurring during a spring and damper operation, and it is usually included in models [24; 26]. This model can also be used to reproduce the behavior of friction elements such as wedges, leaf, springs, and rubber components. The original form of the mathematical model includes ordinary differential equations--the constitutive equations of \(i=1\ldots n\), connected in parallel, \(n\)-forces \(K_{i}\), essential for energy dissipation [27]. The \(K_{i}\) forces are state variables, called 'hidden variables', and cannot be directly measured in the process of model identification in contrast to the total force \(F\) to which \(K_{i}\) contributes. Since the forces \(K_{i}\) are originally associated with the structural damping of structural members exhibiting hysteresis, in this study, in order to model the dry friction damping element of suspension, only one \(K\) term is adopted in the [PERSON] model. In this regard, it is no longer intended to reflect internal damping but the damping force generated in the interface of contacting suspension elements' surfaces, and thus, it is henceforth denoted as friction force \(F_{f}\). Adapted to the rail vehicle suspension, the [PERSON]'s model consists of a series connection of a spring (with stiffness \(k_{1}\)) and a friction component (with the hold force \(H\)). These are parallel-connected by another spring with stiffness \(k_{0}\), representing the main spring suspension element (Figure 9). As long as the force in the spring \(k_{1}\) does not reach the hold force \(H\), the system behaves like a spring with an equivalent stiffness \(k=k_{0}+k_{1}\). When the force in the spring \(k_{1}\) exceeds the hold force \(H\), the friction element will start to slip, generating a friction force \(F_{f}\). The hold force \(H\) is defined as (8):  
\[H=\mu\Big{(}F_{pre}+k_{0}x+F_{f}\Big{)}, \tag{8}\]  
Figure 8: Characteristics of the Maxwell model with constant damping \(c\) and for different stiffness \(k\) values, an excitation frequency \(f=1\) Hz, and an amplitude \(A=0.005\) m.  
Figure 9: The Kolsch dry friction model. where \(\mu\)--coefficient of friction, \(F_{prc}\)--preload friction force, \(F_{f}\)--friction force in the friction element. The friction force in the Kolsch model is described by expression (9):  
\[\dot{F}_{f}=k_{1}\dot{x}\bigg{\{}A(x)-\Big{[}\beta+\alpha\text{sgn}\Big{(}F_{f} \dot{x}\Big{)}\Big{]}\bigg{|}\frac{F_{f}}{H(x)}\bigg{|}^{m}\bigg{\}}, \tag{9}\]  
where \(m\)--an exponent of the transition; \(\alpha\), \(\beta\)--constants; \(A(x)\)--a parameter depending on spring deflection. The total force \(F\) in the suspension with the friction element is the sum of spring \(k_{0}\) force and force in the friction element (10):  
\[F=k_{0}x+F_{f}, \tag{10}\]  
Due to the lack of the Kolsch model parameter values obtained experimentally, the parameters implicitly implemented in the simulation software were used. These parameters are listed in Table 2.  
The friction and total force characteristics of the Kolsch model, with the parameters presented in Table 2, are shown in Figure 10. The friction force characteristic is not ideally rectangular, but it is slightly sloping, reproducing elastic micro-deformation by spring \(k_{1}\). If the stiffness of this spring is reduced, the skewness of the hysteresis loop increases. When the stiffness \(k_{1}\) is infinite, then the loop will have a rectangular shape. The characteristics of the total suspension force exhibit a noticeable force-displacement relationship, increasing monotonically (spring \(k_{0}\)), while forming a hysteresis loop as a result of spring \(k_{1}\) and the friction force element.