## 1 Introduction  
Atomic force microscopy (AFM) is widely used to image the surface morphology of polymers, ceramics, glass, and biological cells [1; 2; 3]. It has three common operation modes: contact, non-contact, and tapping [4; 5; 6]. The core component of AFM is a microcantilever beam [7]. The tapping mode is preferred for experiments involving biological samples in liquids. The TM-AFM is the most widely used atomic force microscope [8; 9; 10].  
The accuracy of TM-AFM is limited to the angstrom level due to various factors [11; 12]. Mathematical modeling can account for these factors as dynamic parameters in the system. The proper matching of dynamic parameters is important to avoid complex dynamics and long steady-state times.  
In the past 30 years, researchers have conducted extensive research on the dynamic characteristics of the TM-AFM microcantilever beam system from various perspectives. Based on the JKR (Johnson-Kendall-Roberts) contact model, [PERSON] et al. [13] proposed an interaction model that considers the adhesion effect between the tip and the tested sample. [PERSON] et al. [14] analyzed a special mathematical map that exhibits square root singularities, which directly leads to an infinite series period-doubling bifurcation in the system. [PERSON] et al. [15; 16] discussed the three-dimensional characteristic modes of both tip and tuning fork tips in TM-AFM microcantilever beams, and summarized their interaction with samples as well as their nonlinear dynamics when moving in liquid environments. [PERSON] et al. [17] measured the vibration of a TM-AFM microcantilever beam in air and water using a scanning laser Doppler vibrometer (Polytec MSA-400 Micro System Analyzer from Polytec Gmbh, Waldbronn, Germany.)  
Gmbh, Waldbronn, Germany. [PERSON] et al. [18; 19; 20], based on the JKR contact model, provided energy loading and unloading curves under tip-sample interaction conditions to study energy dissipation caused by adhesion, plastic deformation, liquid bridge formation, and air-damping effects. However, the effect of changes in the damping coefficient on the steady-state behavior of the TM-AFM cantilever has not been analyzed from a multiple stability perspective.  
The purpose of this paper is to investigate the impact of the damping coefficient on the steady-state behavior of a TM-AFM microcantilever beam through extensive numerical simulations. We also aim to uncover the underlying mechanism by utilizing various stability-related theories and methods. Our objective is to identify the optimal range for the damping coefficient, which is a crucial dynamic parameter during TM-AFM motion. This will serve as a valuable reference for improving imaging quality, measurement accuracy, operability, and troubleshooting potential faults associated with the TM-AFM system.  
This paper is organized as follows: Section 2 introduces the mechanical model and motion of the TM-AFM microcantilever beam system. Section 3 analyzes the effect of linear damping coefficients on the motion stability of the system. Section 4 analyzes the effect of the pressure film damping coefficient on the stability of the system and compares it with the effect of the linear damping coefficient. Section 5 analyzes the effect of the coupled action of the dual damping parameters on the stability of the system and determines the boundary conditions for the transition of the system's motion from an unstable to a stable state. Finally, Section 6 discusses the results and summarizes the whole paper.