## 1 Introduction  
High-speed railways in India have achieved higher speeds, which not only help in the saving of energy but also the protection of the environment. As one of the longest railway networks in the world, vehicle safety and passenger comfort are key components of the service. The high speed of the railway has also aided in the achievement of extensive and profound development in India. As the speed of the network is increasing day by day, the need for safety measures and reliability is also increasing. The operating efficiency of rail vehicle multibody models is primarily influenced by the rail-wheel contact model and the suspension element model [1]. The suspension system comprises a rubber, coil, air spring and hydraulic damper. The coil spring and hydraulic damper are commonly used as vibration control elements in the rail vehicle. Moreover, these are essential parts for ensuring safety and improving ride quality. However, significant numbers of coil spring breakages have recently been found in the primary suspension during rail vehicle operations [1]. When fatigue calculations are carried out, the maximum stress is calculated according to the static displacements or loads in the axial and transverse directions. However, the internal resonances within coil springs that can be excited at higher frequencies are not taken into account [2]. Neglecting the high-frequency excitation that occurs in working conditions can lead to improper design [3, 4]. The dynamic stiffness of resilient elements can be measured using direct or indirect measurement methods. Enhancing the material and processing procedures of coil springs is necessary to increase their service life, but it is also essential to study and gather information about the causes behind the occurrences of the high stresses and the resonances inside the springs [5; 6; 7; 8; 9; 10]. Researchers have used wave motion and spring dynamic response to study the dynamics of vehicular components. To assess the axial and transverse dynamic stiffness of a suspension spring, the dynamic stiffness matrix method has been used, and the suspension springs showed an obvious dynamic stiffening [11]. A few researchers have used these models to perform experiments to understand the stiffening of springs during the vibration transmission of high-speed rails. Numerical and experimental studies have been conducted to gain a deeper insight into how short-pitch rail corrugation could drastically decrease the wear and tear of the springs due to the vibrations induced by rail corrugation [5]. Several studies in the railway sector have focused on the fatigue analysis of rubber springs rather than metallic ones [12; 13; 14; 15]. In experiments carried out by scientists, it was found that the very first circle or loop of the spring experiences the most contact stresses within the primary suspension systems of high-speed trains [16; 17; 18; 19; 20; 21]. It has been observed that not many studies have been carried out to identify the specific causes of the fatigue failure of coil springs while combining the vehicle-track coupled dynamic system, coil spring models and other suspension components [22]. Thus, it is necessary to acknowledge the dependence of the damage of springs on the impact of the dynamic properties of the spring itself, as the primary suspension is between the frame of the bogie and the wheelest, leading to the close contact of the wheels and the rail components [23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34]. As far as the spring coil is concerned, the stiffness varies with the exciting frequency because its dynamic properties are dictated by the mass distribution of the spring itself. The irregularities of a random track are spread over the whole frequency spectrum; therefore, coil spring structural modes may be excited at a particular frequency from the track input and may largely alter the coil spring stiffness and result in the decay of the vehicle performance and reduce the working life of the spring. [PERSON] et al.