## 1 Introduction  
In railway applications, if the track is ballasted, the role to support track systems is widely played by a ballast layer, so-called ballast bed. It is composed of crushed and graded rock particles providing high friction to absorb the energy produced by dynamic forces owing to train-track interactions, and adequate strength to support track system. In ideal case, it is assumed that the ballast layer is laid uniformly along the track, which secures constant support forces for the track via sleepers, and a natural drainage. In reality, tracks are exposed to environmental effects leading to progressive deterioration in the ballast bed. In this sense, a well-known process is'mud pumping' that is associated with ballast fouling due to foreign particles, filling the voids inside the ballast bed and inhibiting the drainage. The process changes the mechanical properties of the ballast bed and induces more volatile dynamic forces that incur track settlements, which accumulate mud pumping. This cascade of events repeat perpetually and eventually, terminates in a failure of the ballast bed that cannot provide any support for the sleepers/bearers. In that case, sleepers/bearers lose the contact with the ballast bed, which amplifies the wheel-rail contact forces and increases risks for safety. This phenomenon is also known as 'hanging sleepers'. Numerous studies have been conducted to understand the effect of unsupported sleepers on dynamic behavior of track components, and train-track interaction. In a pioneering study, it has found that a significant difference between dynamic response of a supported track and a track with unsupported sleepers occurs at a frequency of 200 Hz at which sleepers are likely to have a considerable movement. The situation could be more critical at pinned-pinned frequency owing to greater dynamic strains for the track with unsupported sleepers [1]. Most importantly, the ballast is a primary source of damping, in the absence of which vibration of unsupported sleepers are almost undamped, and consequently, strains in components of the unsupported sleepers are expected to be higher and result in lower component life [1]. In fact, the phenomenon is more sophisticated as sleepers could be partially supported in various forms. The type and length of void under partially supported sleepers influence the natural frequencies and occasionally, cause a \"mode swapping\" [2]. Only effective option to investigate various forms of void under unsupported sleepers seems to be the numerical modeling, as well as fully unsupported sleepers. Other options such as in-situ and laboratory measurements present a great challenge in terms of a methodology and a suitable experimental setup due to the nature of ballast bed deterioration where the forming process does not follow the same pattern in each case. It is believed that a well-calibrated numerical model with in-situ or laboratory measurements could represent the real track with a good precision. The calibration is strongly related to the level of numerical modeling. For instance, it has found that a quasi-static analysis underestimates the contact forces in comparison to a dynamic analysis for unsupported sleepers [1]. Likewise, a model of a simple mass on a Hertzian spring could largely neglects the dynamic effects of train-track interaction [3]. Apart from the level of a numerical model, it should be noted that the calibration is also sensitive to environmental conditions during measurements. In an experiment with a 1/5 scale experimental setup, it was observed that the outcome of the experiments were diverging at higher frequencies [4]. Similar problem was observed in [1] when they made measurements on frozen ballast bed. Admittedly, higher level of numerical modeling complicates the solution process. Nevertheless, it gives an advantage that more advance numerical models enable researchers to investigate the unsupported sleepers for various scenarios.