## 6 Discussion and Conclusions  
Combined with the physical significance of system (2), the first-order resonant angular frequency of a microcantilever beam is usually (1000-20,000) Hz, and the corresponding vibration period is (0.0003-0.0060) s, which depends on whether or not there is a coating or using the coating material that is not used. According to the analysis in Section 3, we know that when \(p=0.006\), the motion state of system (2) can be stabilized in (0.15-3) s (\(M=500\)) if \(r=0.006\); if \(r=0.001\), it takes (1.5-30) s (\(M=5000\)) to be stabilized. When measuring the surface shape of an object under test, the TM-AFM needs to measure thousands of feature points. However, if measuring each feature point requires several seconds or even tens of seconds of stabilization time to ensure accurate and reliable measurement data, such inefficiency may not meet the requirements of practical working conditions. Finally, we can conclude the following by combining the coupling nature of the double damping parameters under the actual working conditions and the found boundary conditions that to ensure the accuracy and reliability of the single measurement data in 500 iteration cycles, the value of the dimensionless linear damping coefficient \(r\) must be greater than 0.01014. The paper studies the effect of the damping coefficient on the motion characteristics of the TM-AFM microcantilever beam system using multiple stability theory. By keeping the number of iteration periods constant and varying the dimensionless linear damping coefficient and the dimensionless piezoelectric membrane damping coefficient, we found that increasing the dimensionless linear damping coefficient or increasing the dimensionless piezoelectric membrane damping coefficient can shorten the time required for the system to transition from an unsteady to a stable state under the given parameter conditions. In addition, we obtained the boundary conditions for the transition of the system from the unstable state to the stable state by using the bifurcation diagram of the dual damping parameter coupling. Finally, after considering the actual operating conditions of the TM-AFM, we conclude that the dimensionless damping coefficient of the operating medium of the TM-AFM microcantilever beam must be greater than 0.01014 to ensure the accuracy and reliability of the measured data in 500 iteration cycles. The research findings presented in this paper provide valuable references for media selection in TM-AFM measurements, enhancing imaging quality, measurement accuracy, and troubleshooting. Based on Equation (2), it is evident that there are numerous dynamic parameters present in system (2). The multiple stability theory discussed in this paper can be effectively used to analyze the impact of these dynamic parameters on the distinctive properties of  
Figure 11: Two-parameter bifurcation diagrams of system (2) in the \(r\times p\) parameter plane: (**a**) \(r\in(0,0.02)\), \(p\in(0,0.002)\); (**b**) \(r\in(0,0.006)\), \(p\in(0,0.0001)\). system motion. In the future, we intend to conduct additional research to explore these topics in greater detail. Conceptualization, P.S. and Y.C. ; methodology, P.S. ; software, P.S. ; validation, Y.C. and X.L. ; formal analysis, P.S. ; investigation, P.S. ; resources, J.C. and X.L. ; data curation, P.S. ; writing--original draft preparation, P.S. ; writing--review and editing, P.S. and K.C. ; visualization, P.S. ; supervision, Y.C. ; project administration, X.L. ; funding acquisition, X.L., P.S. and J.C. All authors have read and agreed to the published version of the manuscript. This research was funded by the Gansu Youth Science and Technology Fund (23)JRRA1710), the Scientific Research Project of Gansu Provincial Institute of Metrology (GY2023003), in part by the Natural Science Foundation of Tibet Autonomous Region (XZ202301 YD0004C), in part by the Fundamental Research Funds for National Institute of Metrology program of China (AKYZD1804-1). Not applicable.