## 1 Introduction
Typical applications of the damper in building structures to resist earthquakes and as a shock absorber in a vehicle are discussed. [PERSON] et al. [3] proposed an analytical technique for the frequency analysis and design of nonlinear dampers to further improve vehicle dynamic performance in automotive applications. Using the Energy Balance Method (EBM), a methodology for determining the equivalent linear damping coefficient of any nonlinear damper was developed. Using the quarter car model, the design principles of position-dependent, velocity-dependent, and position-velocity-dependent dampers were proposed. A number of attempts have been made to develop nonlinear dampers in which the gap for viscous fluid flow varies with the stroke of the piston. In this way, a nonlinear damping characteristic is obtained. [PERSON] et al. [4] developed a nonlinear variable damping device called the variable damping viscous dashpot (VDVD) for use as a bridge vibration absorber. This device is insensitive to frequency detuning effects. The basic idea is that two conical surfaces with a common axis--inner and outer--form the gap for fluid flow. When the surfaces are axially shifted, the gap for fluid flow changes. [PERSON] and [PERSON] [5] proposed a nonlinear damper based on a cylindrical piston moving in a conical cylinder. The results of experimental research on a nonlinear damper for a twin-tube motorcycle shock absorber presented in [6] identify phenomena in the damper caused by external force, frictional forces in the main cylinder and compensation cylinder, and the compliance of the seal of the rebound chamber. In [7], a study of large-scale nonlinear viscous dampers for application in earthquake-resistant building structures is presented. The concept of a cylindrical damper with a stroke-dependent annual gap between the piston and the cylinder for fluid flow is presented by [PERSON] [8]. Hysteresis behavior under different excitation conditions and significantly higher energy dissipation compared to linear fluid dampers were proved. Another intensively researched way to achieve the nonlinearity effect in damping is the development of control strategies for active/actively controlled systems. [PERSON] et al. [9] present an active damping method to minimize velocity overshoot and achieve an optimal damping ratio at shock loading by introducing a control valve into a hydraulic system with a symmetrical cylinder. Using a multi-objective optimization method, the control parameters (gain and time constant) were determined, which allowed the improved damping ratio and reduced dynamic impact. [PERSON] and [PERSON] [10] developed an Adjustable Bypass Fluid Damper (ABFD) with a pair of external fluid-flow tubes and flow-control valves to control the fluid flow. This allowed the damping function to be adjusted according to the displacement of the structure. The other way to achieve damping nonlinearity is by using semi-active dampers based on the application of controllable parameter fluids, such as magnetorheological (MR) fluids. By applying a magnetic field, the viscosity of these fluids is changed, enabling the diversity of their application, e.g., for dampers, batteries, valves and brakes. [PERSON] et al. [11] present a thorough review of MR fluids, detailing their composition, preparation and operation principles, and multiple application cases, including MR dampers with controllable damping characteristics. Another review of MR technology for dampers was carried out by [PERSON] [12], in which a cylinder-shaped MR damper with variable damping is discussed in detail. [PERSON] et al. [13] developed a new phenomenological model to describe the nonlinear behavior of magnetorheological (MR) dampers with fluid deficiency. The damper analyzed is a multi-channel bypass damper with an MR valve that controls the damping modes of the damper by controlling the bypass flow. The typical application of MR dampers is in semi-active vehicle suspension systems.