## 1 Introduction  
Bolted joints are widely used in various technological contexts. To ensure their high durability, among other things, it is necessary to prevent their loosening. Loosening within these joints changes the load distribution not only in the joint itself, but also in the objects in which these fasteners are used, potentially leading to damage [1, 2, 3]. Bolted joints should also be able to carry the highest possible loads. A key design and assembly factor for meeting these requirements is ensuring that the initial load is sufficiently high [1]. In addition, the coefficient of friction within the parts to be joined plays an important role in ensuring the stability of these joints. These parameters are interrelated, as an increase in the coefficient of friction reduces the initial load [4]. However, while a decrease in the coefficient of friction increases the tendency to loosen, there is a specific range for this parameter in which its decrease does not increase the risk of joint loosening [5]. The coefficient of friction is affected by the type of friction present. This, in turn, depends primarily on the condition of the surface, the presence of lubricants, the type and quantity of the joints, and the possible use of coatings [5, 6, 7, 8, 9, 10]. At the same time, it should be noted that during operations, especially with successive tightening and loosening cycles of bolt connection elements, the clamping force may decrease [11, 12]. The effects of lubricants on the stability of the friction coefficient value during multiple tightening and loosening cycles have been observed. Consistently, the coefficient of friction remains stable or decreases in the presence of lubricants. However, in their absence, an increase in the coefficient of friction can be observed with successive tightening and loosening cycles of bolted joints [13]. One study [14] has presented a method for evaluating the course of loosening bolted connections. It is primarily applicable at an early stage. This post-initial state of bolted joint loosening is characterized by small relative movements and a relatively small loading degree. As it advances, the relative displacements in the bolted joints become larger and larger, relatively large loading degrees occur, and bolt tension begins to decrease rapidly [12; 15; 16]. Analyses of the course of bolted joint self-loosening indicate that this phenomenon is most common in transverse loading [15; 17]. In addition, oscillations within the bolted joint can intensify its wear, taking the form of fitting, which makes self-loosening even more important for the correct operation of the joint [18; 19; 20]. For these reasons, analyzing the distribution of loads acting within a bolted joint is crucial for its safe and reliable operation [21; 22]. This problem is of significant industrial importance as, among other things, about 25% of service problems in the automotive industry are associated with the loosening of fasteners. For example, loose fasteners have been found in more than 10% of newly manufactured cars [23]. Another fundamental reason for the loosening of bolted joints is vibrations [2; 17; 24; 25; 26; 27]. Low thread angles increase the risk of loosening of those joints as well [28]. This is a significant problem, given that there is no fully effective method to protect against the loosening of bolted joints [24]. While self-loosening increases in bolted joints as rotational vibration amplitude rises, a high initial preload impedes the self-loosening of bolted joints [25]. Vibrations are fundamental, being one of the most common causes of failure in dynamically loaded bolted joints [29], and a finite element method study of bolted joints influenced by vibration is presented in [30]. Bolted joints have many damage mechanisms, as analyzed in [31], which characterizes the most important problems relating to the use of these joints.