###### Abstract
These types of springs are among the oldest elastic elements [17]. They consist of several elements that interact, making their structure complex compared to the other elastic elements [18]. There are several types of leaf springs used in utility vehicles. Single-leaf or mono-leaf springs are commonly found in light vehicles such as passenger cars, vans, and light trucks. Multi-leaf springs, made of several graduated leaves, are used across various vehicles including trucks, trailers, and heavy-duty vehicles. In contrast to conventional leaf springs that maintain a uniform thickness throughout, tapered leaf springs are thicker in the center and gradually their thickness decreases towards the ends. They are commonly used in various vehicles, including trucks, trailers, and passenger cars [19]. In modeling software, a coil spring is simpler than a leaf spring, which consists of direct input of characteristics such as rigidity, preload, and their coordinates into the model. However, modeling a leaf spring takes into account all elements that it contains, such as the number of leaves with their coordinates, the number of elements for each blade, characteristics of shackles, clips, and bushings, the spring's position relative to the axis, etc., which makes its modeling more complex. Therefore, to ensure that the leaf spring model has the same characteristics as the real spring installed in the vehicle, it should be validated before being used in the full vehicle model. It is challenging to design a leaf spring with all the desired characteristics such as vertical stiffness, torsional stiffness, etc. Several researchers have developed different methods to model leaf springs and find their characteristics. The Fancher method [20], also known as the empirical method, uses equations or functions capable of representing experimentally obtained force-displacement characteristics. The mathematical model applies these equations to describe both the spring force and the Coulomb damping force, employing linear approximations based on experimental results of force-displacement tests conducted on the studied leaf springs. The beam theory studies the leaf spring as a beam to determine its characteristics, and several beam theories have been used for this purpose. [PERSON]'s theory [21] considers the leaf spring as a cantilever beam and assesses its properties. In [22], a fusion of [PERSON]'s and [PERSON]'s theories are employed to enhance the visualization of the physical dynamic behavior of leaf springs. Another theory applies the second theorem of [PERSON] [23], which consists of determining the beam's deformation energy when subjected to external force. The displacement of the beam is obtained as the derivative of the deformation energy expressed in terms of force and the expression of beam stiffness is also derived. This theory is used in [24] to replace the multi-leaf spring with five leaves with a tapered leaf consisting of three leaves, ensuring that the new leaf spring maintains the same stiffness as the original leaf spring. The accuracy of this method is verified experimentally and by simulation in ADAMS. The finite element method (FEM) is widely used to solve complex physical problems for which obtaining analytical solutions is difficult [25], if not impossible. This method is used to analyze the leaf spring's characteristics. In [26], a comparative study of leaf springs examined steel and composite materials. The leaf spring was modeled using CATIA, and static analysis was performed using ANSYS Workbench to compare it with the developed analytical model for the steel material. Then, a static analysis of a leaf spring with the same design but using composite materials was carried out using the same software. It was observed that the composite material performed better than steel. The discrete method involves dividing the leaf spring into rigid elements, which are then linked together using components such as torsion springs and dampers [27].