## 1 Introduction  
For many decades, squeeze film dampers (SFDs) have been successfully employed in high-speed turbomachinery to reduce excessive vibrations caused by unbalance. In SFDs, a thin lubricant film is squeezed between two non-rotating surfaces, creating pressure and a resulting reaction force that dampens the vibrations of the rotor. The SFD is equipped with an elastic suspension concept through the squirrel cage springs. To maintain the position of the rotor journal in the centre of the damper clearance during rest, a suitable preload is applied through a retaining spring. As a result of the unbalance force generated by the rotor during operation, the precession motion can exhibit a circular centred orbit. The arrangement of the squirrel cage springs, the operating conditions of SFD, and its dynamic behaviour are investigated with various computational methods based on the theory of lubrication and are also obtained by experimental works, as described in [1, 2]. SFDs are the topic of numerous research works due to their nonlinear behaviour, namely, jump phenomena, bistable operation, and nonsynchronous motion, which have been investigated both theoretically and experimentally in [3; 4; 5; 6; 7; 8; 9; 10; 11; 12]. The nonsynchronous motion can accelerate rotor failure due to alternating bending stresses, while the bistable operation can increase the transmitted forces to the foundation and the radiated noise. In dampers with narrow clearance, abrupt changes in operational speeds or technological forces can induce conditions for which the damper was not designed. As a consequence, the vibration response may exhibit a large orbit and lead to contact between the journal and the housing. The researchers [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20] have paid significant attention to the observation of various types of nonsynchronous responses, such as subsynchronous, super-subsynchronous, quasiperiodic, and chaotic, through both experimental and computational methods. The nonlinear effects of SFD were studied by [PERSON] et al. [16] that developed the mathematical model of a rigid rotor with consideration of the fluid inertia influence. The bifurcation behaviour of the rotor was analytically solved with the averaging technique and C-L method only near the resonance. The three different regimes of the frequency response curve were found. Similarly, the work [21] deals with the flexible rotor discretised with finite elements. The steady-state and transient vibration response was numerically computed by the time integration. The results, however, do not demonstrate the nonlinear phenomena in the rotor response. Recent work [22] investigates a circular centred orbit with a small amplitude of the finite length SFD. The pressure distribution is numerically computed, and the achieved results are compared with the analytical predictions. A closed-form expression of the oil forces using the momentum approximation method and the perturbation method is employed in the work [23], by the same authors. Moreover, [PERSON] et al. in paper [13] consider harmonic base excitation in the computational model and showed that an increase in frequency can change the synchronous response to an irregular one. As a result of technological constraints or manufacturing imprecision, the rotor disc may be mounted asymmetrically on the shaft, or varying support stiffness may be employed at each support. This leads to the gyroscopic effect, which causes the splitting of rotor mode shapes as the rotational speed increases. The bifurcation analysis using singularity methods is conducted in the study [4], which focuses on the behaviour of a rigid rotor with an asymmetrical spring stiffness support of SFD. The results directly provide the parameters to avoid harmful phenomena caused by the nonlinear coupling between the translational and precessional motion of the asymmetrical rotor.