# Influence of Non-Uniform Rail Loads on the Rotation of Railway Sleepers
## 1 Introduction  
The issue of railway mechanics is the subject of many studies, both regarding modelling and model analysis and experimental works in laboratory and operational conditions [1]. There are also theoretical and experimental studies being conducted that are related to the identification of railway track parameters that allow for improving the applied calculation models [2]. In the first models related to railways, analyses focused on a beam on an elastic foundation with static and dynamic loads (see, e.g., [3]). Apart from that, simplified models with focused parameters were analysed. Examples of such analyses are also the works conducted in the second half of the 20 th century, e.g., [4, 5]. These models, despite many simplifications, allowed for obtaining interesting results, introducing a substitute unevenness at the point of contact of the wheel and the rail, taking into account both the geometrical imperfections of the rolling surface of the rail head and the mechanical imperfections resulting from the uneven support of sleepers in the ballast. These studies allowed to expand the knowledge base in the scope of interaction between the vehicle and the track, also taking into consideration the stochastic nature of substitute unevenness and the relationship between interactions and train speeds.  
Most of the works related to railway dynamics deal with the development of the concept of a beam on an elastic foundation. A broad review of these works was presented, e.g., [5, 6]. Generalisations were applied in relation to introducing an axial force in rails [7, 8]; a mobile load; and, first, one oscillating force, e.g., [9, 10], and then multiple forces, e.g., [11].  
Other generalisations concern the analysis of two layers (rail as a beam and sleeper layer)--see, e.g., [11, 12]--as well as a multi-layer structure, e.g., [13, 14, 15]. Also, the problem of a critical load speed [4, 9, 11, 16, 17, 18, 19] and the stochastic properties of a rail foundation, e.g., [20], were analysed. Apart from linear models, cases of the unevenness of the rail foundation were also analysed, e.g., [1, 10]. In the featured works, the rail was modelled as the Euler-Bernoulli beam; the analysis of the Timoshenko beam was presented, e.g., in [6, 20]. Various methods were employed to analytically solve the problem of a variable mobile force. The basic method is to use the Fourier transform, e.g., [9, 16], also with the application of wavelet approximation [13, 19, 20]. The described solutions and load functions in the form of Fourier series in a definite but sufficiently long range were appliedsuccessfully in [5] and specifically in [11], where the good consistency of the calculation results with the results of experimental tests was revealed. Examples of numerical analyses of the problem of a dynamic response and their experimental verifications were presented, e.g., in works [14; 15; 21].  
This paper deals with a significant generalisation of the analysis presented in paper [11]. This generalisation consists of introducing sleeper rotations as stiffblocks in a multi-layer structure. Sleeper rotation is the result of uneven loads on rails, taking place specifically in curves.